FLAC and Numerical Modeling Geomecahnic_2003

FLAC AND NUMERICAL MODELING IN GEOMECHANICS

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Copyright © 2003 Swets & Zeitlinger B.V., Lisse, The Netherlands

PROCEEDINGS OF THE THIRD INTERNATIONAL FLAC SYMPOSIUM,21–24 OCTOBER 2003, SUDBURY, ONTARIO, CANADA

FLAC and Numerical Modeling in Geomechanics

Edited by

Richard Brummer & Patrick AndrieuxItasca Consulting Canada Inc., Sudbury, Ontario, Canada

Christine Detournay & Roger HartItasca Consulting Group Inc., Minneapolis, Minnesota, USA

A.A. BALKEMA PUBLISHERS LISSE / ABINGDON / EXTON (PA) / TOKYO

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Published by: A.A. Balkema, a member of Swets & Zeitlinger Publisherswww.balkema.nl and www.szp.swets.nl

ISBN 90 5809 581 9

Printed in the Netherlands

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FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitlinger, Lisse, ISBN 90 5809 581 9

V

Table of contents

Preface IX

Organisation XI

Constitutive modelsCompensation grouting analysis with FLAC3D 3X. Borrás, B. Celada, P. Varona & M. Senís

An automated procedure for 3-dimensional mesh generation 9A.K. Chugh & T.D. Stark

A new constitutive model based on the Hoek-Brown criterion 17P. Cundall, C. Carranza-Torres & R. Hart

A study of compaction band formation with the Double-Yield model 27C. Detournay, P. Cundall & J. Parra

A new viscoplastic model for rocks: application to the Mine-by-Test of AECL-URL 35F. Laigle

Prediction of deformations induced by tunneling using a time-dependent model 45A. Purwodihardjo & B. Cambou

Modeling of anhydrite swelling with FLAC 55J.M. Rodríguez-Ortiz, P. Varona & P. Velasco

Scenario testing of fluid-flow and deformation during mineralization: from simple tocomplex geometries 63P.M. Schaubs, A. Ord & G.H. German

Constitutive models for rock mass: numerical implementation, verification and validation 71M. Souley, K. Su, M. Ghoreychi & G. Armand

Slope stabilityA parametric study of slope stability under circular failure condition by a numerical method 83M. Aksoy & G. Once

Numerical modeling of seepage-induced liquefaction and slope failure 91S.A. Bastani & B.L. Kutter

Complex geology slope stability analysis by shear strength reduction 99M. Cala & J. Flisiak

Analysis of hydraulic fracture risk in a zoned dam with FLAC3D 103C. Peybernes

Mesh geometry effects on slope stability calculation by FLAC strength reduction method – linear and non-linear failure criteria 109R. Shukha & R. Baker

3D slope stability analysis at Boinás East gold mine 117A. Varela Suárez & L.I. Alonso González

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Underground cavity designThe effect of tunnel inclination and “k” ratio on the behavior of surrounding rock mass 127M. Iphar, M. Aksoy, M. Yavuz & G. Once

Numerical analysis of the volume loss influence on building during tunnel excavation 135O. Jenck & D. Dias

Application of FLAC3D on HLW underground repository concept development 145S. Kwon, J.H. Park, J.W. Choi & W.J. Cho

Numerical simulation of radial bolting: Application to the Tartaiguille railway tunnel 153F. Laigle & A. Saïtta

Recent experiences of the prediction of tunneling induced ground movements 161C. Pound & J.P. Beveridge

Numerical modeling of remedial measures in a failed tunnel 169Y. Sun & P.J.N. Pells

Mining applicationsSill pillar design at the Niobec mine using FLAC3D 181P. Frenette & R. Corthésy

Stability analyses of undermined sill mats for base metal mining 189R.K. Brummer, P.P. Andrieux & C.P. O’Connor

FLAC numerical simulations of tunneling through paste backfill at Brunswick Mine 197P. Andrieux, R. Brummer, A. Mortazavi, B. Simser & P. George

FLAC3D numerical simulations of ore pillars at Laronde Mine 205R.K. Brummer, C.P. O’Connor, J. Bastien, L. Bourguignon & A. Cossette

Modeling arching effects in narrow backfilled stopes with FLAC 211L. Li, M. Aubertin, R. Simon, B. Bussière & T. Belem

FLAC3D numerical simulations of deep mining at Laronde Mine 221C.P. O’Connor, R.K. Brummer, P.P. Andrieux, R. Emond & B. McLaughlin

Three-dimensional strain softening modeling of deep longwall coal mine layouts 233S. Badr, U. Ozbay, S. Kieffer & M. Salamon

FISH functions for FLAC3D analyses of irregular narrow vein mining 241H. Zhu & P.P. Andrieux

Soil structure interactionA calibrated FLAC model for geosynthetic reinforced soil modular block walls at end of construction 251K. Hatami, R.J. Bathurst & T. Allen

Three-dimensional modeling of an excavation adjacent to a major structure 261J.P. Hsi & M.A. Coulthard

Pile installation using FLAC 273A. Klar & I. Einav

Axial tension development in the liner of a proposed Cedar Hills regional municipal solid waste landfill expansion 279F. Ma

VI

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The usability analyses of HDPE leachate collection pipes in a solid waste landfill 287F. Ma

FLAC numerical simulations of the behavior of a spray-on liner for rock support 295C.P. O’Connor, R.K. Brummer, G. Swan & G. Doyle

A numerical study of the influence of piles in the passive zone of embedded retaining walls 301T.Y. Yap & C. Pound

Dynamic and thermal analysisA practice orientated modified linear elastic constitutive model for fire loads and its application in tunnel construction 313E. Abazovic & A. Amon

Seismic liquefaction: centrifuge and numerical modeling 321P.M. Byrne, S.S. Park & M. Beaty

Modeling the dynamic response of cantilever earth-retaining walls using FLAC 333R.A. Green & R.M. Ebeling

VII

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IX

Preface

The first two International FLAC Symposia were held in Minneapolis (USA) in September 1999, and in Lyon(France) in October 2001. In 2003, the third International Symposium on FLAC and Numerical Modeling inGeomechanics returned to North America and was held in Sudbury (Ontario, Canada) from October 21 toOctober 24, 2003, with two days of short courses before the symposium.

Technical contributions to the conference were received from a wide range of different disciplines, repre-senting virtually the entire globe. A volunteer Technical Committee reviewed the papers, and where necessaryclarifications were suggested to the authors prior to finalization of their manuscripts.

The contributions in this volume cover seven main topics:

• Constitutive Models• Slope Stability• Underground Cavity Design• Mining Applications• Soil Structure Interaction• Dynamic Analyses• Thermal Analyses

The FLAC conferences provide all FLAC and FLAC3D users with an opportunity to meet and learn from eachother and from the people who develop the code. Conversely, they also allow Itasca staff members to learn fromthe practical experiences of code users “out there in the real world”. These interactions improve our collectiveknowledge and allow us to improve the performance of these numerical models in simulating the behavior ofgeomaterials. These proceedings contain a comprehensive collection of FLAC & FLAC3D applications – casestudies as well as research presentations. We believe that this publication will help users by documenting a valu-able resource for the solution of geomechanical problems.

The compilation presented here would not have been possible without the efforts of our authors and our TechnicalCommittee, and we thank them. We particularly thank and recognize the efforts of Michele Nelson, who served asan extremely capable and efficient Technical Editor.

Richard Brummer Roger HartPatrick Andrieux Christine DetournayItasca Consulting Canada Inc. Itasca Consulting Group Inc.

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XI

Organisation

Conference Technical Committee:

The following individuals provided technical input to the conference, and scientific overview and reviews of the abstracts and papers.

Patrick Andrieux, Itasca Consulting Canada Inc.Daniel Billaux, Itasca Consultants SARichard Brummer, Itasca Consulting Canada Inc.Peter Cundall, Itasca Consulting Group Inc.Christine Detournay, Itasca Consulting Group Inc.Samantha Espley, INCO LimitedRoger Hart, Itasca Consulting Group Inc.Ugur Ozbay, Colorado School of MinesChris Pound, Mott MacDonald LimitedGraham Swan, Falconbridge Limited

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Constitutive models

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3

Compensation grouting analysis with FLAC3D

X. BorrásGestió D’Infrastructures SA (GISA), Barcelona, Spain

B. CeladaGeocontrol SA, Madrid, Spain

P. Varona & M. SenísItasca Consultores SL, Asturias, Spain

ABSTRACT: The Barcelona Metro Line 3 extension was excavated crossing 6.5 meters below a main watersupply pipe. Compensation grouting was used to minimize the deformations in the gallery. A FLAC3D modelwas developed to investigate the efficiency of this process. The model was first calibrated to reproduce theextensometer measurements and was later re-run without the compensation grouting in order to assess theeffectiveness of such treatment.

1 INTRODUCTION

The Barcelona Metro Line 3 extension was excavatedcrossing 6.5 meters below a main water supply pipe(Borrás et al. 2001). This is one of the two pipes ofAigües Ter-Llobregat (ATLL) which supply water tothe city of Barcelona.

Due to the importance of this water pipe, duringthe construction of the tunnel, compensation groutingwas used to minimize the deformations induced bythe excavation process in the existing gallery thatcontains the water pipe.

A FLAC3D model simulating the whole excavationsequence and the compensation grouting process hasbeen developed in order to evaluate the effects of theconstruction and the treatment in the pipe, calibratingthe model with the actual instrumentation resultsobtained during the excavation. The instrumentationinstalled consisted of 17 rod extensometers located0.5 m below the bottom of the ATLL gallery.

2 FLAC3D MODEL

2.1 Geometry of the model

Figure 1 presents the problem geometry with thedimensions of both tunnels and their location. TheFLAC3D model (Figs. 2 & 3) reproduces this geometry.In plan view the pipe gallery forms a 35° angle withthe axis of the tunnel (Fig. 4).

2.2 Excavation sequence

The numerical model considers the sequential exca-vation of the metro tunnel:

– Excavation of the heading in steps of 1 m length

20 m

4.7 mIn

t eri

0.5 m

6.5 m

20 m

4.7 mInjection depth

RodExtensometerlocation

0.5 m

6.5 m

Figure 1. Problem description.

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– Installation of the support: shotcrete with a thicknessof 30 cm and TH-29 steel arches.

– Installation of a 15 cm thick shotcrete lining andprovisional invert, 10 m behind the excavation face.

Figure 5 shows a detail of the excavation sequencefollowed in the model.

2.3 Material properties

The geological profile assumed is shown in Figure 6.The Mohr-Coulomb constitutive model has beenassigned to all the soils. The properties assumed areshown in Table 1.

Both the shotcrete and the lining have been mod-eled using regular elements with an elastic constitu-tive model. The aging of the shotcrete has beensimulated by the Young Modulus evolution law shownin Figure 7 (based on Estefanía 2000).

4

Figure 2. FLAC3D model. General view.

Figure 3. FLAC3D model. Tunnels geometry.

Figure 4. FLAC3D model. Plan view.

Figure 5. Excavation sequence.

Figure 6. Geological profile.

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2.4 Simulation of the compensation grouting

Compensation grouting injections consist of a mix-ture of cement and bentonite. To simulate these injec-tions a methodology which reproduces the injectionprocess carried out during the real construction hasbeen developed. Figure 8 shows a plan view of thetreatment area with the location of the drills andsleeves used.

This real geometry has been reproduced in theFLAC3D model. Figure 9 shows the location of all the drills considered and the sleeves used in one of the injection cycles.

The methodology used for the simulation of theinjection process is based in the bulb expansionmodel proposed by Buchet et al. (1999). According tothem the injection effect can be modeled by increas-ing the volume of the elements in which the injectionis made. This volume increment is carried out applyingsome “fictitious” hydrostatic stresses in the element,which makes it expand.

These stresses are applied instantaneously, initial-izing an hydrostatic stress increment of the element aspulses and then reaching a mechanical equilibrium.This process is repeated until the volumetric straininduced in the element is the one corresponding to afraction of the volume injected. The volumetric strainincrement due to the injection is defined by:

(1)

where Vi is the injected volume, V0 is the initial vol-ume of the element and � is the efficiency of theinjection. The process followed during the injectionmodeling is shown in Figure 10.

Figure 11 shows, as an example of the process, theincrements applied to the vertical stress and their later

5

Table 1. Geotechnical properties assigned to the soils.

E � c �(MPa) v (°) (t/m2) (t/m3)

Quaternary 50 0.33 25 1.5 2.1Natural fills 30 0.35 32 1.0 2.0Weathered Granite (V) 75 0.30 37 1.5 2.2Weathered Granite (IV) 100 0.30 37 2.8 2.3Weathered Granite (II/III) 300 0.25 37 7.5 2.6

y = 6644.4Ln(x) + 11076R2 = 0.9262

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

3.0E+04

3.5E+04

4.0E+04

0 5 10 15 20 25 30 35

Distancia al frente (m)

E (M

Pa)

Figure 7. Hardening law applied to the concrete.

01

02

03

04 05 06 07 08 09 10 11 12 13 14 15 16 17 1819

20

21

Figure 8. Drills and sleeves location.

Figure 9. Location of drills and sleeves in the model.

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relaxation until the equilibrium is reached. Figure 12presents the evolution of the volumetric strain, show-ing the successive increments produced until thestrain corresponding to the injected volume is reached.

Following the real injection scheme, the drills andsleeves that are injected every cycle are reproduced inthe FLAC3D model, finding the closest element to theposition of the sleeve and proceeding in the waydescribed above. As an example of the modeling,Figure 13 shows the increment of the vertical dis-placements (at extensometer depth) produced duringone of the injection cycles (the location of the sleevesinjected in the cycle is shown too). Figure 14 shows,for the same cycle, the heave produced at the groundsurface.

These two figures show how the heave is less pro-nounced but affects a larger area as the distance fromthe sleeves increases.

6

T

V0

�Vi�

INI Hydrostatic stress

Calculation of ��v produced

noT

End of injection

yes

��v � ��v

��v

Figure 10. Modeling the injection process.

FLAC3D 2.00

Itasca Consulting Group, Inc.Minneapolis, MN USA

Step 927118:50:04 Mon Nov 13 2000

History

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0N�Pasos �10^3

1.376

1.378

1.380

1.382

1.384

1.386

1.388

1.390

1.392

1.394

Tension Vertical �10^5

Rev 5 tension_media (FISH function) Line style

1.375e+005 <-> 1.396e+005

Vs. Step

1.000e+001 <-> 9.270e+003

Figure 11. Vertical stresses during the injection process.

FLAC 3D 2.00


Step 927118:48:40 Mon Nov 13 2000

History

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Deformacion volumetrica �10^-2

1Volumetric Strain Inc. Zone 113 Line style

1.347e-003 <-> 5.443e-002

Vs. Step

1.000e+001 <-> 9.270e+003

N�Pasos �10^3

Figure 12. Evolution of the volumetric strain.

FLAC 3D 2.00


Step 54106 Model Projection09:45:05 Tue Nov 14 2000

Center:X: 0.000e+000Y: 4.000e+001Z: 4.885e+000

Rotation:X : 90.000Y : 0.000Z : 0.000

Dist: 2.964e+002 Size: 4.830e+001

Plane Origin:X: 0.000e+000Y: 0.000e+000Z: 1.077e+001

Plane Orientation:Dip: 0.000DD: 0.000

Contour of Z-Displacement

Plane: on0.0000e+000 to 5.0000e-004 5.0000e-004 to 1.0000e-0031.0000e-003 to 1.5000e-003 1.5000e-003 to 2.0000e-003 2.0000e-003 to 2.5000e-003 2.5000e-003 to 3.0000e-0033.0000e-003 to 3.5000e-003 3.5000e-003 to 4.0000e-0034.0000e-003 to 4.5000e-0034.5000e-003 to 5.0000e-003 5.0000e-003 to 5.5000e-003 5.5000e-003 to 5.6775e-003

Interval = 5.0e-004

Taladros

Figure 13. Vertical displacement increments in one of theinjection cycles, at extensometer depth.

FLAC 3D 2.00


Step 54106 Model Projection09:46:43 Tue Nov 14 2000

Center:X: 0.000e+000Y: 4.000e+001Z: 4.885e+000

Rotation:X: 90.000Y: 0.000Z: 0.000

Dist: 2.964e+002 Size: 4.830e+001

Plane Origin:X: 0.000e+000Y: 0.000e+000 Z: 2.470e+001

Plane Orientation:Dip: 0.000 DD: 0.000

Contour of Z-DisplacementPlane: on

0.0000e+000 to 2.5000e-005 2.5000e-005 to 5.0000e-005 5.0000e-005 to 7.5000e-005 7.5000e-005 to 1.0000e-004 1.0000e-004 to 1.2500e-004 1.2500e-004 to 1.5000e-004 1.5000e-004 to 1.7500e-004 1.7500e-004 to 2.0000e-004 2.0000e-004 to 2.2500e-004 2.2500e-004 to 2.5000e-004 2.5000e-004 to 2.7500e-004 2.7500e-004 to 3.0000e-004 3.0000e-004 to 3.2500e-004 3.2500e-004 to 3.5000e-004 3.5000e-004 to 3.5961e-004

Figure 14. Vertical displacement increments in one of theinjection cycles, at ground surface.

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3 RESULTS

3.1 Comparison with instrumentation

Figure 15 shows the location of the 17 rod exten-someters used to monitor the compensation groutingprocess. The model was first calibrated varying thegrout efficiency �, in order to match the actual mea-surements with the calculated values, achieving agood fit. A second run was made without the com-pensation grouting in order to calculate what defor-mations would have been induced without anytreatment.

Figure 16 shows the evolution of extensometer E4located outside the treated area. A vertical displace-ment of 10 mm was measured and without grouting,12 mm are predicted.

Figure 17 shows the evolution of extensometer E6located in a relatively stiff material within the treatedarea. Here 8 mm settlement was recorded comparedto 18 mm predicted without compensation grouting.In a softer material, extensometer E10 (Fig. 18), thedifference between measured (8 mm) and predictedwithout treatment (22 mm) is even larger.

3.2 Calculation of the efficiency

An average grouting efficiency can be defined as the ratio of difference between the volume of the set-tlement trough without (Vs

NI) and with compensationVs

WI to the injected volume:

(2)

In the present analysis the efficiency obtained has been:

(3)

This empirical parameter is crucial for predictive stud-ies, and a sufficient database for a given soil type isnecessary before such analysis should be attempted.

3.3 Calculation of volume loss

The volume loss can be defined as the ratio of the vol-ume of the settlement trough to the excavated volume.

Figure 19 shows the volume loss calculated alongthe tunnel axis for both hypotheses (with and withoutgrouting). In both cases the volume loss depends on

7

FLAC3D 2.00


Step 17228 Model Projection10:34:27 Wed Nov15 2000

Center:X: 0.000e+000Y: 4.000e+001Z: 4.885e+000

Rotation:X: 90.000Y:0.000Z: 0.000

Dist: 2.964e+002

X: 0.000e+000Y: 0.000e+000Z: 1.077e+001

Plane Orientation:Dip: 0.000DD: 0.000

GeologíaPlane: on

1.111000e+0071.880000e+0072.884500e+007

Taladros

Extensometros

E1

E1B

E2

E2B

E3

E3B

E4

E5

E6

E7

E8

E9

E10

E11

E12

E13

E14

Tunel

Galeria ATLL

Size: 4.830e+001

Plane Origin:

Figure 15. Location of the rod extensometers.

Extensometer E4

-0.022

-0.020

-0.018

-0.016

-0.014

-0.012

-0.010

-0.006

-0.008

-0.004

-0.002

0.000

0.002

10/9

11/9

12/9

13/9

14/9

15/9

16/9

17/9

18/9

19/9

20/9

21/9

22/9

23/9

24/9

25/9

26/9

27/9

28/9

29/9

30/9

1/10

2/10

3/10

4/10

5/10

6/10

7/10

8/10

9/10

10/1

011

/10

12/1

013

/10

14/1

015

/10

16/1

017

/10

18/1

019

/10

20/1

021

/10

22/1

023

/10

Date

Ver

tica

l d

isp

lace

men

t (m

)

Measured FLAC FLAC No Injections

Figure 16. Results obtained for extensometer E4.

Extensometer E6

-0.022

-0.020

-0.018

-0.016

-0.014

-0.012

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

10/9

11/9

12/9

13/9

14/9

15/9

16/9

17/9

18/9

19/9

20/9

21/9

22/9

23/9

24/9

25/9

26/9

27/9

28/9

29/9

30/9

1/10

2/10

3/10

4/10

5/10

6/10

7/10

8/10

9/10

10/1

011

/10

12/1

013

/10

14/1

015

/10

16/1

017

/10

18/1

019

/10

20/1

021

/10

22/1

023

/10Date

Ver

tica

l dis

pla

cem

ent

(m)



Extensometer E10

- 0.022

- 0.020

- 0.018

-0.016

- 0.014

- 0.012

- 0.010

- 0.008

- 0.006

- 0.004

- 0.002

0.000

0.002

10/9

11/9

12/9

13/9

14/9

15/9

16/9

17/9

18/9

19/9

20/9

21/9

22/9

23/9

24/9

25/9

26/9

27/9

28/9

29/9

30/9

1/10

2/10

3/10

4/10

5/10

6/10

7/10

8/10

9/10

10/1

011

/10

12/1

013

/10

14/1

015

/10

16/1

017

/10

18/1

019

/10

20/1

021

/10

22/1

023

/10Date

Ver

tica

l d

isp

lace

men

t (m

)



09069-01.qxd 08/11/2003 20:13 PM Page 7


the characteristics of the soil above the tunnel. So in thearea in which the tunnel is excavated in natural fill thevolume loss is about 2.4%; and as the tunnel runstowards stiffer soils (weathered granite) this relationdecreases to 0.8%. These values agree with the onesdescribed in the literature. For example, Oteo (2000)reports values of 1–2% for stiff clays and 1–5% forgranular soils above the water table.

The maximum effect of the compensation groutingis a reduction of the volume loss of 0.4% from 1.2%to 0.8% at chainage 1708.

3.4 Effects on the gallery

Figure 20 shows the vertical displacements producedon the bottom of the ATLL gallery along its axis, with

maximum values of 19 mm in the hypothesis withouttreatment and 15 mm in the one with compensationgrouting. In the same figure the corresponding hori-zontal strains have been represented too. These horizontal strains have been calculated as:

(4)

where L is the initial distance between two pointsalong the gallery, and L is the distance once the dis-placement has occurred. The strains show low valuesin both cases, although the induced tensile strains arehigher in the hypothesis with injections.

4 CONCLUSIONS

The main conclusions that can be obtained from thisanalysis are:

– The expected grout efficiency (30%) was muchhigher than the actual efficiency (10%).

– The expected volume loss (0.07–0.2%) was muchlower than the actual volume loss (1–2%).

– The expected volume to inject (13.7 m3) was muchlower than the actual volume injected (68 m3). Stillonly partial compensation was achieved.

– According to the comparison between the modelwith compensation grouting and the model with-out the treatment area has been insufficient.

– Numerical models should play an important role in the design of compensation grouting providingaccurate estimates of the ground loss and of therequired treatment area.

REFERENCES

Borrás, X., Pérez, A., Magro, J.A., Celada, B. & Varona, P.2001. Construcción del tramo Montbau-Canyelles de laLínea 3 del Metro de Barcelona. In Ingeopres N° 92,Abril 2001, Madrid: 54–64.

Buchet, G. & Van Cotthem, A. 1999. 3D “Steady State”numerical modeling of tunneling and compensationgrouting. In Detournay & Hart (eds), FLAC andNumerical Modeling in Geomechanics; Proc. intern.symp., Minneapolis, MN, 1–3 September 1999: 255–261.Rotterdam: Balkema.

Estefania, S. 2000. Utilización de Métodos Numéricos en elProceso Constructivo. Proc. III Curso sobre Ingenieríade Túneles. Madrid, 12–14 June 2000.

Oteo, C. 2000. Subsidencia producida por los túneles. InJimeno (ed), Manual de túneles y obras subterráneas.U.D. Proyectos, E.T.S.I. Minas, U.P.M., Madrid.

8

NaturalFills

Quaternary

A. GraniteV

A. GraniteI V

A. Granite III

-2.5

-2.3

-2.0

-1.8

-1.5

-1.3

-1.0

-0.8

-0.5

-0.3

0.0

1668 1673 1678 1683 1693 1698 1703 1708 1713 1718 1723 1728 1733 1738 1743 17481688Chainage

Vsu

bsi

den

ce/V

exca

vate

d (

%)

No injections

Injections

NaturalFills

Quaternary

A. GraniteV

A. GraniteI V

A. anit III

NaturalFills

Quaternary

A. Granite V

A. Granite IV

A. Granite II–III

No injectionsInjections

Figure 19. Volume loss analysis.

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50

Distance to the cross measured on the ATLL (m)

Ver

tical

dis

plac

emen

t (m

m)

-0.200

-0.175

-0.150

-0.125

-0.100

-0.075

-0.050

-0.025

0.000

0.025

0.050

Str

ain

(mm

/m)

Injections No Injections Strain I. Strain N.I.

Figure 20. Horizontal strains in the ATLL gallery bottom.

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9

An automated procedure for 3-dimensional mesh generation

A.K. ChughBureau of Reclamation, Denver, CO, USA

T.D. StarkUniversity of Illinois, Urbana, IL, USA

ABSTRACT: An automated procedure is presented to generate a 3-dimensional mesh for numerical analysisof engineering problems. The procedure is simple, effective and efficient, and can be applied to represent complexgeometries and material distributions. A listing of the program that was used for the sample problem of a landfillslide is included.

1 INTRODUCTION

One of the essential tasks in a 3-dimensional (3-D)numerical analysis is to represent the geometry and dis-tribution of materials in the numerical model. FLAC3D

provides means to facilitate mesh generation and thebuilt-in programming language FISH can be used todevelop and implement additional program instruc-tions during execution of a data file.

In geotechnical engineering, surface geometry,distribution of materials, and water table conditionsusually vary from one location to the next and pose adifficult set of conditions to represent in a numericalmodel. In order to facilitate the analysis of landslides,a simple procedure was devised to represent complexsurface geometry, subsurface material horizons, andwater table conditions. The objectives of this paper areto present:

1 a simple method to describe field geometry andconditions for a 3-D numerical model of a slopeproblem;

2 a simple procedure for automatic generation of a 3-D mesh; and

3 an illustration of the use of the procedure for analy-sis of a large slide in a landfill.

A listing of the program for the landfill slide is includedin the paper. This program listing is in the FISH lan-guage and uses some of the functions available in theFISH library.

2 CONCEPTUAL MODEL

The conceptual model for the generation of a 3-D mesh follows the conventional procedure of portrayingspatial variations of materials in 3-D via a series of 2-dimensional (2-D) cross-sections. This techniqueis commonly used by engineers and geologists inconstructing visual models of complex geologic siteswhere a number of 2-D cross-sections are used to repre-sent the field conditions. In these representations, linearvariations between material horizons in consecutive 2-D cross-sections are used to depict the 3-D spatial vari-ability of a site. The accuracy of the representation isimproved by using closely spaced 2-D cross-sections.

The 3-D mesh generation procedure presentedherein follows the conventional practices used by engi-neers in constructing 2-D numerical meshes by handfor geotechnical problems to be solved using methodsother than FLAC3D. For example, in the creation of a2-D numerical model of a slope to be analyzed usinga limit-equilibrium based procedure, it is a commonpractice to define profile lines via a set of data pointsfollowed by specifications of their connectivities.Also, in the creation of a 2-D model of a continuum tobe solved by a finite-element based procedure, it is acommon practice to discretize the continuum into a network of zones; assign identification numbers to the grid points; define the coordinates of the grid points; and then specify the connectivity of grid points.

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Thus, in the conceptual model for the generation ofa 3-D mesh in FLAC3D, use is made of defining aseries of 2-D cross-sections at representative locationsof a site; defining each of the 2-D sections as anassemblage of data points with line-segment connec-tions; and organizing the data for an efficient andeffective discretization of the volume.

3 WATER TABLE

The water table surface is specified using the watertable data of individual 2-D cross-sections and throughthe use of 3-point planar polygons between consecutive2-D cross-sections. This scheme allows incorporationof non-coplanar variations in the water table surfacein the entire 3-D model.

4 DESCRIPTION OF THE PROCEDURE

In geotechnical engineering, the ground-surface geo-metry is obtained using contour maps that areprepared from land or aerial survey of the area. Thesubsurface material horizons are estimated from geo-logic data and information obtained from exploratoryboring logs. The subsurface water conditions are esti-mated from field observations, piezometers installedat various depths, and/or from water levels in borings.Subsurface data are used to develop contour maps ofthe subsurface geology and water conditions.

From these contour maps, the region-of-interest,and the locations of significant cross-sections areidentified; information for 2-D cross-sections areread and tabulated; and 2-D cross-sections are drawnfor an understanding of the site details and preparationof input data for a 2-D analysis. In general, the cross-sectional data for a site varies from one location to the next. These variations may be caused by changesin the ground surface and (or) in subsurface materialhorizons, discontinuity of some materials, or a com-bination of these or some other variations.

In the proposed procedure, the following steps arefollowed: (For ease of presentation, 2-D cross-sectionsare assumed to lie in x-z plane and the x,y,z coordi-nate system follow the right hand rule.)

1 The following steps are used for creating anorderly assemblage of field data for 3-D discretiza-tion of the continuum of the region-of-interest:a On the site map, select values of x, y, and z coor-

dinates that completely circumscribe the 3-Dregion-of-interest;

b Mark locations of all significant 2-D cross-sections oriented in the same and preferably par-allel direction;

c For each 2-D cross-section, tabulate (x,y,z) coor-dinates of end-points of all line segments for

each profile line and the water table (for parallel2-D cross-sections, y-coordinate shall have sameconstant value between two consecutive cross-sections).

2 The following steps are used for creating similarsets of data at each of the 2-D cross-sections:a From the data in step 1(c) above, select control

points that are of significance in defining theprofile lines in all of the 2-D cross-sections.Tabulate the x-coordinates of these control pointsin increasing order. For reference purposes, thistable is referred to as Table 100.

b Use of the “Interpolate” function expands the 2-Dcross-sectional data of step 1(c) by linear inter-polation for all of the control points listed in Table 100 for all of the profile lines and storesthe data in separate tables; assigns Table num-bers in increasing order starting with the userspecified starting number and incrementing it by1; assigns an identification number to eachpoint; and positions the points in the 3-D modelspace. These tables contain the (x,z) coordinatesof expanded 2-D cross-sectional data. A samplelisting of the “Interpolate” function and itsdependency function “zz” in FISH language isgiven in Figure 1. The starting table number usedin the sample problem data file is 200.

3 The following steps are used for creating zones inthe 3-D model space:

10

def zz zz=table(t_n,xx) end

def interpolate loop j (js,je); profile line #s - ; js is for the bottom, je is for top dt_n=dt_n_s+j; dt_n is destination table number loop i (is,ie); is is the first interpolation #, ; ie is the last interpolation # xx=xtable (100,i); x-coordinate of the ;interpolation point command set t_n=j end_command table(dt_n,xx)=zz id_pt=id_pt+1 x_pt=xtable(dt_n,i) y_pt=y_pt z_pt=ytable(dt_n,i) command generate point id id_pt x_pt y_pt z_pt end_command endloop endloop end

Figure 1. Listing of the “Interpolate” function and itsdependency function “zz” in FISH language.

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a Tabulate the y-coordinates of the 2-D cross-sections in increasing y-direction. For referencepurposes, this table is referred to as Table 101.The number of entries in Table 101 should equalthe number of 2-D cross-sections marked in step 1(b).

b Considering the spacing of x-coordinates of thecontrol points in step 2(a), select the number of

zones desired for each interval in the x-direction.Tabulate these values for all of the intervals inthe increasing x-direction. For reference purposes,this table is referred to as Table 102. The numberof entries in Table 102 should be one less thanthose in Table 100.

c Considering the spacing between the 2-D cross-sections in the y-direction, select the number ofzones desired for each interval in the y-direction.Tabulate these values for all of the intervals in theincreasing y-direction. For reference purposes,this table is referred to as Table 103. The numberof entries in Table 103 should be one less thanthe number of 2-D cross-sections.

d Considering the spacing of the profile lines inthe z-direction, select the number of zones desiredfor each material horizon in the z-direction.Tabulate these values for all of the intervals inthe increasing z-direction. For reference purposes,this table is referred to as Table 104. The numberof entries in Table 104 should be one less than thenumber of profile lines.

e Use of the “Fill_grid” function generates a brick mesh and assigns a group name to each 3-D volume zone. A sample listing of the“Fill_grid” function in FISH language is givenin Figure 2.

5 COMMENTS

1 Use of a Brick mesh with an 8-point description isversatile and allows for creation of degenerated brickforms through the use of multiple points with dif-ferent identification numbers occupying the same(x,y,z) coordinate location in the 3-D model space.

2 During the development of the grid, it is possible toassign group names to different segments of themodel. This information can be useful in modifyingthe generated grid.

3 Expanding the (x,y,z) location data for all 2-D cross-sections to a common control number of locationsvia interpolations facilitates the programming ofthe automatic grid-generation procedure.

4 In engineering practice, it is generally desirable to analyze a few 2-D cross-sections at select loca-tions prior to conducting a 3-D analysis. Becausedevelopment of data for 2-D cross-sections is one of the steps for use of the proposed procedure,it is relatively easy to conduct a 2-D analysis usingthe 2-D cross-sectional data and the programFLAC.

5 The program instructions listed in Figures 1 and 2can be modified to accommodate geometry andother problem details that are different or morecomplex than those encountered in the sampleproblem described in Section 6.

11

def fill_grid i_n=table_size(102) j_n=table_size(103) k_n=table_size(104) loop jy (1,j_n) ny=xtable(103,jy) p0_d=(jy-1)*(i_n+1)*(k_n+1) loop kz (1,k_n) nz=xtable(104,kz) if kz=1 then material='shale' endif if kz=2 then material='ns'; native soil endif if kz=3 then material='msw'; municipal solid waste x_toe=xtable(105,jy) endif loop ix (1,i_n) if kz=3 then xx_toe=xtable(100,ix) if xx_toe < x_toe then material='mswt' endif endif nx=xtable(102,ix) p0_d=p0_d+1 p3_d=(p0_d+i_n+1) p6_d=(p3_d+1) p1_d=(p0_d+1) p2_d=((i_n+1)*(k_n+1)+p0_d) p5_d=(p2_d+(i_n+1)) p7_d=(p5_d+1) p4_d=(p2_d+1) command generate zone brick size nx,ny,nz ratio 1,1,1 & p0=point (p0_d) p3=point (p3_d) & p6=point (p6_d) p1=point (p1_d) & p2=point (p2_d) p5=point (p5_d) & p7=point (p7_d) p4= point(p4_d) group material end_command if kz=3 then material='msw' endif end_loop p0_d=p0_d+1 end_loop end_loop end

Figure 2. Listing of “FILL_GRID” function in FISHlanguage.

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6 SAMPLE PROBLEM

The problem used to illustrate the proposed 3-D meshgeneration procedure is the 1996 slide in a waste con-tainment facility near Cincinnati, Ohio (Stark & Eid1998, Eid et al. 2000). Figure 3 is an aerial view of theslide. Figure 4 is the plan view of the landfill andshows the location of the sixteen cross-sections usedto construct a FLAC3D model of the site (the projectdata shown are in Imperial units). There are threematerial horizons bounded by four profile lines, and aliquid level present at this site. Figure 5 shows the 2-Dcross-sectional views of the site at the 16-locationsprior to failure (the available project data were con-verted to SI units and this conversion lead to nume-rical values with fractional parts). Figure 6 shows apartial listing of the data file for the sample problemwith the following details:

– Table 100 lists the x-coordinates of the 22 controlpoints considered significant from the sixteen 2-Dcross-sectional data.

– Table 101 lists the y-coordinates of the sixteen 2-Dcross-section locations.

– Table 102 lists the number of zones desired in eachof the 21 segments in the x-direction.

– Table 103 lists the number of zones desired in eachof the 15 segments in the y-direction.

– Table 104 lists the number of zones desired in eachof the 3 material horizons at the site.

– Table 105 lists the x-coordinates of the toe loca-tions of the top profile line in the 2-D cross-sectionsin the increasing y-direction.

For each cross-section, x- and z-coordinates fordata points defining the profile lines are recorded inindividual tables numbered as Table 1 for profile line1 data, Table 2 for profile line 2 data, Table 3 for pro-file line 3 data, and Table 4 for profile line 4 data inthe data file shown in Figure 6. Profile lines are num-bered from 1 to 4 in the increasing z-direction and eachprofile line uses a different number of data points todefine the line. For cross-sections where the top pro-file line terminates in a vertical cut at the toe, the topprofile line was extended to x � 0.

For each cross-section and for each of the four pro-file lines, the x-coordinate locations identified in Table100 are used to create data by interpolation at each ofthe 22 control points. For the sample problem, thisamounts to 88 pairs of (x,z) coordinates per cross-section, and the y-coordinate of the data points is readfrom Table 101. Thus, the x-,y-, and z-coordinates for all of the points defined and (or) interpolated are known. Each point is assigned a numeric identitynumber (id #) starting with one and incrementing byone. The data points are located in the 3-D modelspace using their id # and x-,y-, z-coordinates. Thistask is accomplished using the “Interpolate” function

and its listing in FISH language is given in Figure 1.At the end of this task, all of the defined and (or)interpolated points with an assigned id # have beenlocated in the 3-D model space.

The connectivity of data points to define volumediscretization is accomplished in the function named

12

Figure 3. Sample problem – aerial view of Cincinnatilandfill failure (from Eid et al. 2000). (Reproduced by per-mission of the publisher, ASCE).

Figure 4. Plan view of the sample problem showing locationsof selected 2-D sections.

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13

Figure 5. 2-D cross-sectional views of the sample problem.

“Fill_grid”. For each interval in the location of cross-sections in the y-direction (Table 103), and for eachmaterial horizon between the profile lines in the z-direction (Table 104), and for each interval in the

x-direction (Table 102), the values of number ofzones desired in the x, y, and z-direction and the id #sof points in the 3-D model space are used in the“GENERATE zone brick p0, p1, … p8” command of

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14

; Rumpke landfill site; Data are in metric units set g=0,0,-9.81

; table 100 is for the x-coordinates of ; the desired 3-D grid table 100 0,1 13.11,2 15.54,3 22.86,4 34.75,5 table 100 42.67,6 49.07,7 57.61,8 63.70,9 table 100 64.92,10 72.54,11 78.94,12 92.66,13 table 100 100.89,14 107.90,15 115.21,16 table 100 158.50,17 199.64,18 284.38,19 table 100 318.52,20 337.72,21 348.08,22

; table 101 is for y-coordinates of the ; 2-D cross-section locations table 101 0,1 15.24,2 20.73,3 28.96,4 42.06,5 table 101 62.48,6 96.93,7 138.07,8 164.29,9 table 101 201.47,10 234.09,11 253.29,12 table 101 268.83,13 287.43,14 293.83,15 table 101 307.85,16

; table 102 is for the number of zones ; desired in the x-direction table 102 2,1 1,2 1,3 2,4 1,5 1,6 1,7 1,8 1,9 table 102 1,10 1,11 2,12 1,13 1,14 1,15 5,16 table 102 5,17 10,18 4,19 2,20 2,21

; table 103 is for the number of zones ; desired in the y-direction table 103 2,1 1,2 1,3 2,4 2,5 3,6 4,7 3,8 4,9 table 103 3,10 2,11 2,12 2,13 1,14 2,15

; table 104 is for the number of zones ; desired in the z-direction table 104 5,1 3,2 10,3

; table 105 is for the x-coordinates of the ; receding toe table 105 0,1 0,2 15.54,3 22.86,4 34.75,5 table 105 49.07,6 57.61,7 64.92,8 78.94,9 table 105 92.66,10 100.89,11 107.90,12 table 105 115.21,13 63.70,14 0,15

set is=1 ie=22 set js=1 je=4 set id_pt=0 set dt_n_s=200

; Station at y=0 set y_pt=0 table 1 -100,200 500,200 table 2 0,223.60 154.23,223.60 307.24,238.84 table 2 348.08,239.14 table 3 0,228.60 154.23,228.60 307.24,243.84 table 3 348.08,244.14 table 4 0,260.00 66.45,280.42 98.15,283.46 table 4 156.67,286.51 187.15,289.56 table 4 348.08,332.54 interpolate

; station at y=15.24 m set y_pt=15.24 table 2 erase table 3 erase table 4 erase set dt_n_s=dt_n table 2 0,223.60 163.07,223.60 306.02,238.84 table 2 348.08,240.67 table 3 0,228.60 163.07,228.60 306.02,243.84 table 3 348.08,245.67

table 4 0,251.46 91.14,280.42 107.90,283.46 table 4 144.48,286.51 169.77,289.56 table 4 194.46,292.61 332.54,338.33 table 4 348.08,338.33 interpolate . . . ; station at y=307.85 m set y_pt=307.85 table 2 erase table 3 erase table 4 erase set dt_n_s=dt_n table 2 0,254.08 348.08,254.08 table 3 0,259.08 348.08,259.08 table 4 0,261.08 29.87,265.18 185.93,268.22 table 4 348.08,307.24 interpolate fill_grid delete range group mswt ; water surface water den=1 table & face 0,0,228.60 0,15.24,228.60 & 332.54,15.24,268.22 & face 0,0,228.60 332.54,15.24,268.22 & 348.08,15.24,268.22 & face 0,0,228.60 348.08,15.24,268.22 & 348.08,0,268.22 & ;interval # 1 face 0,15.24,228.60 0,20.73,228.60 & 340.77,20.73,268.22 & face 0,15.24,228.60 340.77,20.73,268.22 & 348.08,20.73,268.22 & face 0,15.24,228.60 348.08,20.73,268.22 & 332.54,15.24,268.22 & face 332.54,15.24,268.22 348.08,20.73,268.22 & 348.08,15.24,268.22 &;interval # 2 . . . face 0,293.83,259.08 0,307.85,259.08 & 63.70,307.85,259.08 & face 0,293.83,259.08 63.70,307.85,259.08 & 348.08,307.85,268.22 & face 0,293.83,259.08 348.08,307.85,268.22 & 63.70,293.83,259.08 & face 63.70,293.83,259.08 348.08,307.85,268.22 & 348.08,293.83,268.22;interval # 15 save cin_3D_grid.sav

Figure 6. Partial listing of the data file for the sample problem for FLAC3D.

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FLAC3D for a regular 8-noded brick mesh. The materialbetween the profile lines is assigned a group name forease of modifying the grid and for convenience inassigning material properties and/or addressing themfor some other reason. This task is also accomplishedin the function named “Fill_grid” and its listing in FISHlanguage is given in Figure 2. Table 105 data are usedto assign a group name “mswt” to the zones past thevertical cut which are later deleted using the DELETEcommand with the range defined by the group name“mswt”. At the end of this task, a 3-D grid of specifi-cation exists in the region-of-interest. For the sampleproblem, the generated 3-D grid is shown in Figure 7.The representation of continuity of the vertical cut atthe toe of the slope (as seen in 2-D cross-sections,Figure 5) in the 3-D model can be improved byincreasing the number of 2-D cross-sections.

7 ADVANTAGES OF THE PROPOSEDPROCEDURE

1 The proposed procedure for describing 3-D fieldconditions utilizes 2-D cross-sections, which areessentially the same as commonly used by geolo-gists and engineers to describe the field conditions.

Linear variation in geometry, material horizons,and groundwater descriptions between known datapoints is generally accepted.

2 Changes in field data can be incorporated in the numerical model by updating the affectedtables.

3 New cross-sections can be introduced or old cross-sections deleted and a new discretization of thecontinuum made quickly.

4 Describing the spatial location of data in a 3-Dspace followed by descriptions of their connectivityis a simple yet powerful way of constructing a 3-Dnumerical model for analysis purposes.

5 The proposed procedure produces regions withacceptable geometries, i.e. no conflicts in connec-tivity.

6 Changes in discretization due to changes in fielddata or due to numerical considerations can beincluded in the proposed procedure efficiently anda new discretization accomplished.

7 Number of discretized volume units in differentparts of the numerical model is estimated at thestart of the problem solving effort. If it becomesnecessary to change or refine the discretization,very little effort is needed to change the tabulardata and the procedure is then rerun to obtain anupdated 3-D mesh.

8 A complete brick element is used to generate otherdegenerated volume element shapes.

9 Because the proposed procedure is based on sim-ple and commonly used ideas, it should be adapt-able when using computer programs or proceduresother than FLAC3D to perform numerical analysiswork. The program instructions can be rewritten inother programming languages.

8 SUMMARY

To facilitate 3-D analyses using FLAC3D or other soft-ware, an automated procedure is presented to create a 3-D mesh. The procedure utilizes commonly used techniques for drawing 2-D cross-sections and interpolation between 2-D cross-sections to portray spatial variations of geometry and distribu-tion of materials in 3-D.

REFERENCES

Eid, H.T., Stark, T.D., Evans, W.D. & Sherry, P.E. 2000.Municipal solid waste slope failure. II Stability analyses.Journal of Geotechnical and Geoenvironmental Engi-neering 126(5): 408–419.

Stark, T.D. & Eid, H.T. 1998. Performance of three-dimensional slope stability methods in practice. Journalof Geotechnical and Geoenvironmental Engineering124(11): 1049–1060.

15

Figure 7. 3-D mesh for the sample problem.

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17

A new constitutive model based on the Hoek-Brown criterion

P. Cundall, C. Carranza-Torres & R. HartItasca Consulting Group, Inc., Minneapolis, MN, USA

ABSTRACT: A new constitutive model is proposed based on the Hoek-Brown failure criterion. This modelincorporates a plasticity flow rule that varies as a function of the confining stress level. For a low confiningstress, at which a large rate of volumetric expansion at yield is anticipated, an associated flow rule is applied.For high confining stress, at which the material no longer dilates at failure, a constant-volume flow rule is pre-scribed. A composite flow rule, which provides a linear variation from associated to constant-volume limits, isused between the low and high confining stress states. Using an appropriate softening relation, the model canalso represent the transition between brittle and ductile rock behavior. The new model is programmed inCand compiled as a DLL file (dynamic link library) that can be loaded directly into either FLAC orFLAC3D. This paper describes the model and its implementation as a DLL. Physical justification is provided forthe formulation and, specifically, the representation of the volumetric behavior during yield, which depends onconfining stress. A verification example is provided.

1 INTRODUCTION

The Hoek-Brown failure criterion is an empirical rela-tion that characterizes the stress conditions that leadto failure in intact rock and rock masses. It has beenused very successfully in design approaches that uselimit equilibrium solutions, but there has been littledirect use in numerical solution schemes. Alternatively,equivalent friction and cohesion values have beenused with a Mohr-Coulomb model that is matched tothe nonlinear Hoek-Brown strength envelope at partic-ular stress levels. Numerical solution methods requirefull constitutive models, which relate stress to strain in ageneral way; in addition to a failure (or yield) criterion,a flow rule is also necessary, in order to provide a rela-tion between the components of strain rate at failure.

There have been several attempts to develop a fullconstitutive model from the Hoek-Brown criterion: e.g.Pan & Hudson (1988), Carter et al. (1993) and Shah(1992). These formulations assume that the flow rulehas some fixed relation to the failure criterion, and thatthe flow rule is isotropic, whereas the Hoek-Brown cri-terion is not. In the formulation described here, there isno fixed form for the flow rule; it is assumed to dependon the stress level, and possibly on some measure ofdamage.

In what follows, the failure criterion is taken as ayield surface, using the terminology of plasticity theory.Usually, a failure criterion is assumed to be a fixed,limiting stress condition that corresponds to ultimate

failure of the material. However, numerical simula-tions of elasto-plastic problems allow continuing thesolution after failure has taken place, and the failurecondition itself may change as the simulation pro-gresses (by either hardening or softening). In this event,it is more reasonable to speak of yielding rather thanfailure. There is no implied restriction on the type ofbehavior that is modelled – both ductile and brittlebehavior may be represented, depending on the soft-ening relation used.

2 GENERAL FORMULATION

The generalized Hoek-Brown criterion (Hoek & Brown1998), adopting the convention of positive compressivestress, is

(1)

where �1 and �3 are the major and minor effectiveprincipal stresses, and �ci, mb, s and a are material con-stants that can be related to the Geological StrengthIndex (GSI) and rock damage (Hoek et al. 2002). Forinterest, the unconfined compressive strength is givenby �c � �ci sa and the tensile strength by �t ��ci s/mb. Equation (1) and the stresses �c and �t are represented in Figure 1.

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It should be noted that the failure criterion (equa-tion 1) does not depend on the intermediate principalstress, �2; thus, the failure envelope is not isotropic.

Assume that the current principal stresses are (�1,�2, �3) and that initial trial stresses (�t

1, �t2, �

t3) are

calculated by using incremental elasticity, i.e.,

(2)

where E1 � K 4G/3 and E2 � K � 2G/3 and (�e1,�e2, �e3) is the set of principal strain increments. Ifthe yield criterion (equation 1) is violated by this setof stresses, then the strain increments (prescribed asindependent inputs to the model) are assumed to becomposed of elastic and plastic parts, i.e.,

(3)

Note that plastic flow does not occur in the inter-mediate principal stress direction. The final stresses(�f

1, �f2, �

f3) output from the model, are related to the

elastic components of the strain increments; hence,

(4)

Eliminating the current stresses, using equations(2) and (4),

(5)

We assume the following flow rule,

(6)

where the factor � depends on stress, and is recom-puted at each time step. Eliminating �ep

1 from equa-tion (5)

(7)

At yield, equation (1) is satisfied by the finalstresses; that is,

(8)

18

Figure 1. Graphical representation of the generalizedHoek-Brown failure criterion (equation 1) in the (a) com-pressive and (b) tensile region of the principal stress space(�1, �3).

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By substituting values of �f1 and �f

2 from equation(7), equation (8) can be solved iteratively for �ep

3,which is then substituted in equation (7) to give the finalstresses. The method of solution is described later, butfirst the evaluation of � is discussed.

3 FLOW RULES

We need to consider an appropriate flow rule, whichdescribes the volumetric behavior of the material dur-ing yield. In general, the flow parameter � will dependon stress, and possibly history. It is not meaningful tospeak of a dilation angle for a material when its confin-ing stress is low or tensile, because the mode of failureis typically by axial splitting, not shearing. Althoughthe volumetric strain depends in a complicated way onstress level, we consider certain specific cases for whichbehavior is well known, and determine the behavior forintermediate conditions by interpolation.

Three cases are considered below.

3.1 Associated flow rule

It is known that many rocks under unconfined com-pression exhibit large rates of volumetric expansionat yield, associated with axial splitting and wedgingeffects. The associated flow rule provides the largestvolumetric strain rate that may be justified theoretically.This flow rule is expected to apply in the vicinity ofthe uniaxial stress condition (�3 ≈ 0). An associatedflow rule is one in which the vector of plastic strainrate is normal to the yield surface (when both areplotted on similar axes). Thus,

(9)

where the subscripts denote the components in theprincipal stress directions, and F is defined by equa-tion (8). Differentiating this expression, and usingequation (6),

(10)

The associated flow rule used in the constitutivemodel is graphically represented in Figure 1a. The normal to the plastic strain-rate vector e�p at pointA is tangent to the yield envelope (equation 1) at�3 � 0. The slope of the normal to e�p, denoted as k0 in the figure, is inversely related to the coefficient�af defined by equation (10) – i.e., k0 � 1/�af at�3 � 0.

3.2 Constant-volume flow rule

As the confining stress is increased, a point is reachedat which the material no longer dilates during yield. A constant-volume flow rule is therefore appropriatewhen the confining stress is above some user-prescribed level, �3 � �cv

3. This flow rule is given by

(11)

The constant-volume flow rule defined by equation(11) is represented graphically by point C in Figure 1a.The normal to the vector e�p at point C has a slope equalto unity, and therefore the rate of volumetric expansionin the plastic regime is null.

3.3 Radial flow rule

Under the condition of uniaxial tension, we mightexpect that the material would yield in the direction of the tensile traction. If the tension is isotropi-cally applied, we imagine (since the test is practi-cally impossible to perform) that the material woulddeform isotropically. Both of these conditions are fulfilled by the radial flow rule, which is assumed to apply when all principal stresses are tensile. For aflow-rate vector to be coaxial with the principal stressvector, we obtain

(12)

The radial flow rule defined by equation (12) isrepresented graphically by points D1, D2 and D3 inFigure 1b. The directions of vectors e�p at these pointsintercept all the origin of the diagram.

3.4 Composite flow rule

We propose to assign the flow rule (and thus, a valuefor �) according to the stress condition. In the fully ten-sile region, the radial flow rule, �rf, will be used. Forcompressive �1 and tensile or zero �3 the associatedflow rule, �af, is applied. For the interval 0 � �3 � �cv

3,the value of � is linearly interpolated between the asso-ciated and constant-volume limits, i.e.,

(13)

Finally, when �3 �cv3, the constant-volume value,

� � �cv, is used. It is noted that if �cv3 is set equal to

zero, then the model uses a non-associated flow rulewith a zero dilation angle, for �3 0. If �cv

3 is set to a very high value relative to �ci, the model uses anassociated-flow rule for �3 0.

19

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The composite flow rule defined by equation (13)in the case of compressive stresses is representedgraphically by point B in Figure 1a. The slope of thenormal to e�p at point C is the linear interpolation of theslopes at points A and B.

4 IMPLEMENTATION

The equations presented above are implemented in aDLL (dynamic link library) written in C, with the model name hoekbrown. One difficulty with thefailure criterion (equation 8) is that real values for Fdo not exist if �3 � �s�ci/mb. During an iterationprocess, this condition is likely to be encountered, soit is necessary that the expression for F, and its firstderivatives, be continuous everywhere in stress space.This is fulfilled by adapting the following compositeexpressions:

• if �3 � �s�ci/mb then

(14)

• if �3 � �s�ci/mb then

(15)

To initialize the iteration, a starting value for, �ep3

is taken as the absolute maximum of all the strainincrement components. This value, denoted as ��1, isinserted into equation (7), together with the value for� found from the flow-rule equations, and the result-ing stress values inserted into equations (14) and (15).The resulting value of F is denoted by F1. Taking theoriginal value of F as F0 (and the corresponding plas-tic strain increment of zero as ��0), we can estimate a new value of the plastic strain increment, using avariant of Newton’s method,

(16)

From this, we find a new value of F (that we call F2),and if it is sufficiently close to zero, the iteration stops.Otherwise, we set F0 � F1, F1 � F2, ��0 � ��1 and��1 � ��2, and apply equation (16) again.

Tests show that the iteration scheme converges for allstress paths tried so far, including cases in which s � 0(material with zero unconfined compressive strength),which led to problems in previous implementations. For

high confining stresses, the iteration converges in onestep, but at low confining stresses, up to ten steps arenecessary (the limit built into the code is presently 15).

5 MATERIAL SOFTENING

In the Hoek-Brown model, the material properties,�ci, mb, s and a, are assumed to remain constant, bydefault. Material softening, after the onset of plasticyield, can be simulated by specifying that thesemechanical properties change (i.e., reduce the overallmaterial strength) according a softening parameter.The softening parameter selected for the Hoek-Brownmodel is the plastic confining strain component, e

p3.

The choice of ep3 is based on physical grounds. For

yield near the unconfined state, the damage in brittlerock is mainly by splitting (not by shearing) with cracknormals oriented in the �3 direction. The parameter ep

3is expected to correlate with the microcrack damagein the �3 direction.

The value of ep3 is calculated by summing the strain

increment values for �ep3 calculated by equation (16).

Softening behavior is provided by specifying tablesthat relate each of the properties, �ci, mb, s and a, to ep

3.Each table contains pairs of values: one for the ep

3value and one for the corresponding property value. Itis assumed that the property varies linearly betweentwo consecutive parameter entries in the table.

A multiplier, � (denoted as mult in FLAC andFLAC3D), can also be specified to relate the softeningbehavior to the confining stress, �3. The relationbetween � and �3 is also given in the form of a table.

To illustrate the definition of softening parametersin the constitutive model proposed in this paper, weanalyze the idealized response of a cylindrical sampleof homogeneous-isotropic material in a typical tri-axial experiment – as represented in Figure 2a.

For example, Figure 2b shows a piecewise-linearstress–strain relationship expressed in terms of thedeviator �1 � �3 and the shear strain � � e1 � e3.The different curves in the diagram correspond toincreasing values of confinement �3 in the triaxialexperiment of Figure 2a.

Two cases of practical interest will be consideredhere. The first case assumes that the slope of the soft-ening branch is maintained for increasing values ofconfinement �3. In Figure 2b the case is represented by continuous curves (e.g., the line OPR). The secondcase assumes that the slope of the softening branchdecreases (in absolute value) as confinement increases,and that the material behaves in a ductile manner (i.e.,the slope of the softening branch becomes zero) for aconfinement level �3 � �dc

3. In Figure 2b this case isrepresented by the dashed curves (e.g., the line OPR).

To illustrate the definition of input parameters inthe constitutive model we need to consider in some

20

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detail the relationships that govern the response of thematerial represented in Figure 2.

We assume that the peak and residual strength ofthe material are given by the following equations,

(17)

(18)

The residual parameters �Rci and mR

b in equation (18)are decreased in the same proportion, multiplying theinitial (peak) parameters by the factor (1 � �), i.e.,

(19)

(20)

The parameter � in the equations above, that lie in therange 0 � � �1, controls the jump of strength fromthe peak to residual stages. If � � 0, then the peakand residual strength are the same and the materialbehaves in a ductile manner (see Figure 2b). If � � 1then the material behaves in a brittle manner, with theminimum possible value for the residual strength(i.e., �1 � �3 in equation 18).

In the simplest case we can consider that the loss ofstrength in the softening branch in Figure 2b is lin-early related to the plastic shear-strain �p as follows,

(21)

Note that in the equation above, �crp is the critical value

of plastic shear-strain for which the residual stage isreached (see point R in Figure 2b).

The loss of strength can also be expressed in termsof the drop modulus 2G�. indicated in Figure 2b (thedefinition of drop modulus used here is as in Linkov,1992). This parameter controls the ductile/brittlebehavior of the material. For example, when � � 0 thematerial behaves in a perfectly-plastic manner andwhen � → � the material behaves in a perfectly-brittle manner. The relationship between �cr

p and � is,

(22)

In the constitutive model discussed in this paper, theplastic strain ep

3 (rather than the plastic shear-strain�p) is taken as a softening parameter. The relationshipbetween ep

3 and �p can be constructed from the analytical-solution of the triaxial experiment ofFigure 2a. This relationship, that is represented in

21

Figure 2. (a) Idealized triaxial experiment of a cylindricalsample of isotropic-homogeneous Hoek-Brown material.The diagrams (b) and (c) represent an idealized piecewiselinear response obtained from the triaxial experiment.

09069-03.qxd 08/11/2003 20:14 PM Page 21


Figure 2c, depends on the flow rule assumed for thematerial as follows,

(23)

In the equation above the parameter K� is related tothe instantaneous dilation angle � as

(24)

For interest, we list here the expressions for theslopes corresponding to the elastic, softening and resid-ual branches in the e3 vs. e1 – e3 diagram of Figure 2c,

(25)

(26)

(27)

We consider now a practical case of definition ofsoftening parameters in a FLAC model.

Let us assume the following values for the param-eters that control the response of the material inFigure 2:

�ci � 0.1 MPamb � 5s � 1a � 0.5� � 0.5� � 0.2 (for �3 � 0)E � 100 � �ci� � 0.3� � 0o

[Note that the condition � � 0o implies that thematerial does not dilate in the plastic regime; in theFLAC model this condition is satisfied by specifying�cv

3 � 0.]For the value of � defined above, the residual

parameters �Rci and mR

b are computed with equations(19) and (20) and result to be,

�Rci � 0.05 MPa

mRb � 2.5

From equation (23), and considering �3 � 0 the criticalvalue of plastic shear-strain for which the residualstage is achieved is,

�crp (0) � 0.039

From equation (23), again considering �3 � 0, thecritical value of plastic strain is,

ep3

cr (0) � �0.013

In the FLAC model, the tables for the softeningparameters should be defined as follows:

In addition to the table above, a table defining therelationship between the multiplier � and the confin-ing stress �3 will be normally defined. The type ofrelationship to consider depends on how the drop mod-ulus of the softening branch is assumed to vary withthe level of confinement.

To illustrate the definition of the multiplier � weconsider first the case in which the drop modulus ofthe softening branch, 2G�, is maintained for increasingvalues of confinement �3 (see line OPR in Figure 2b).For this case, the multiplier � is defined as follows,

(28)

Assuming an upper limit for the confining stressequal to 10 � �ci, and taking 5 points to represent thisrelationship, the definition of the multiplier � inFLAC will be as follows:

Note that in the table above, the second column iscomputed using equation (28).

As a second example of the definition of the mul-tiplier � we consider now the case for which the dropmodulus of the softening branch, 2G�, decreases (inabsolute value) for increasing values of confinement�3 (see line OPR in Figure 2b). To achieve the ductilebehavior (� � 0) at the confinement level �3 � �dc

3,we can use the following relationship between themultiplier � and the confining stress �3,

(29)

22

ep3 �ci [MPa] mb s a

0.000 0.10 5.0 1.0 0.50.013 0.05 2.5 1.0 0.5� 0.05 2.5 1.0 0.5

�3 [MPa] �

0.00 1.00000.25 3.67420.50 5.09900.75 6.20481.00 7.1414

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(30)

Assuming then a value of �dc3 � �ci, and an upper

limit for the confining stress equal to 10 � �ci, the def-inition of the multiplier � in FLAC will be as follows:

Note that in the table above, the second column iscomputed using equations (29) and (30).

6 VERIFICATION

Stresses and displacements are calculated for the caseof a cylindrical tunnel in an infinite brittle Hoek-Brownmedium subjected to an in-situ stress field. A uniformcompressive stress of �0 � 15 MPa is assigned as thefar-field stress, and an internal pressure pi � 2.5 MPais applied inside the tunnel (see Figure 3a).

The problem is based on an example posed byHoek & Brown (1980). The closed-form solution inthat example only provided the stress distribution cal-culation, and is extended here to include the displace-ment solutions for both associated and non-associatedplastic flow. (A description of the equations that sum-marize the solution is provided in the Appendix A.)

The properties and conditions selected for this testare also listed in Figure 3a. Both initial rock and bro-ken rock properties for the Hoek-Brown model arespecified.

The brittle behavior of the rock is simulated byinstantaneous softening – i.e. the Hoek-Brown prop-erties are changed from initial values at ep

3 � 0 to bro-ken values at a small value of ep

3 � 0 (an arbitrarilysmall value for ep

3 equal to 10�20 is assumed).The comparison of the results from the FLAC

model using model hoekbrown to the analytical solu-tion (given in the Appendix A) is shown in Figure 3bfor the calculation of hoop stress and radial stressaround the tunnel, and in Figure 3c for the calculationof radial displacements for both the associated andnon-associated flow cases. In all cases, the agreementbetween FLAC and analytical results is characterizedby an error of less than 1%.

7 CONCLUDING REMARKS

A full constitutive model based on the Hoek-Browncriterion has been implemented for use in FLAC and

FLAC3D. The flow rule is based on general knowledgeof the volumetric behavior of rock, which usuallyexhibits large dilation at low confining stresses andsmall or zero dilation at large confining stresses, asthe failure condition is approached. Although thisassumption conforms to practical experience, it will be

23

�3 [MPa] �

0.000 1.00.033 1.50.067 3.00.100 �1.000 �

Figure 3. Elasto-plastic solution for excavation of a cylin-drical tunnel in a brittle generalized Hoek-Brown material.

09069-03.qxd 08/11/2003 20:14 PM Page 23


necessary to compare the response of the new modelwith actual measurements of rock behavior, both in thelaboratory and in the field, and calibrate the parameter�cv

3 from observations of volumetric strain. Further, thesoftening behavior is assumed to depend on the confin-ing stress, not on deviatoric stress, which is the moreusual assumption. This decision was made on a gen-eral knowledge of rock behavior, but it will need to beverified (or falsified) by comparing model predictionswith measurements of rock response under post-peakconditions.

REFERENCES

Carranza-Torres, C. & Fairhurst, C. 1999. The elasto-plasticresponse of underground excavations in rock masses thatsatisfy the Hoek-Brown failure criterion. InternationalJournal of Rock Mechanics and Mining Sciences 36(6),777–809.

Carter, T., Carvalho, J. & Swan, G. 1993. Towards the prac-tical application of ground reaction curves. In W.F.Bawden & J.F. Archibald (Eds), Innovative mine designfor the 21st century, pp. 151–171. Rotterdam: Balkema.

Hoek, E. & Brown, E.T. 1980. Underground Excavations inRock. London: The Institute of Mining and Metallurgy.

Hoek, E. & Brown, E.T. 1997. Practical estimates of rockmass strength. International Journal of Rock Mechanicsand Mining Sciences 34(8), 1165–1186.

Hoek, E., Carranza-Torres, C. & Corkum, B. 2002. Hoek-Brown failure criterion – 2002 edition. In H.R.W. Bawden,J. Curran & M. Telesnicki (Eds), Proceedings of the 5thNorth American Rock Mechanics Symposium and the 17thTunnelling Association of Canada Conference: NARMS-TAC 2002. Mining Innovation and Technology. Toronto – 10July 2002, pp. 267–273. University of Toronto.

Linkov, A.M. 1992. Dynamic phenomena in mines and theproblem of stability. MTS System corporation. 14000Technology Drive, Eden Praire, MN 55344, USA. Notesfrom a course of lectures presented as MTS visiting pro-fessor of Geomechanics at the University of Minnesota,Minneapolis, MN, USA.

Pan, X.D. & Hudson, J.A. 1988.A simplified three dimen-sional Hoek-Brown yield condition. In M. Romana (Ed.),Rock Mechanics and Power Plants. Proc. ISRM Symp.,pp. 95–103. Balkema. Rotterdam.

Shah, S. 1992. A study of the behaviour of jointed rock masses.Ph. D. thesis, Dept. Civil Engineering, University ofToronto.

APPENDIX A. CLOSED-FORM SOLUTION FOR A CYLINDRICAL HOLE IN AN INFINITEBRITTLE HOEK-BROWN MEDIUM

The solution presented in thisAppendix is based on ascaled solution for cylindrical tunnels in Hoek-Brownmedia discussed in Carranza-Torres and Fairhurst(1999). Analytical expressions to compute the fieldquantities �r, �� and ur are presented here for theplastic and elastic regions around the tunnel.

A.1 Plastic region, r RplThe critical internal pressure below which the failurezone develops is computed from the following tran-scendental equation,

(A.1)

The extent Rpl of the failure zone is,

(A.2)

The solution for the radial stress field is

(A.3)

The solution for the hoop stress field is

(A.4)

For the case of non-associated flow rule (with dila-tion angle equal to zero) the solution for the radialstress field is computed from integration of the fol-lowing second-order differential equation,

(A.5)

where is defined as,

(A.6)

24

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The functions d�r/d and d��/d in the differentialequation above are

(A.7)

and

(A.8)

while the coefficients A1, A2 and A3 are

(A.9)

(A.10)

(A.11)

The boundary conditions to integrate the differentialequation (A.5) above are

(A.12)

and

(A.13)

In the case of associated flow rule, the solution forthe radial stress field is obtained from integration of

the following non-linear second-order differentialequation,

(A.14)

The only difference with the case of non-associatedflow rule (zero dilation angle) is that the coefficientsA1, A2 and A3 depend now on the solution of the stressfield �r as follows,

(A.15)

(A.16)

(A.17)

A.2 Elastic region, r � RplThe solution for the radial stress field is

(A.18)

The solution for the hoop stress field is

(A.19)

The solution for the radial displacement field is

(A.20)

The plastic and elastic solutions for the field quanti-ties �r and ur presented above are continuous at theelasto-plastic boundary (i.e., at r � Rpl). The solutionfor the field quantity �� is discontinuous when there isa jump of strength from peak values (�ci, mb, s and a) toresidual values (�R

ci, mRb, sR and aR) – see Figure 3b.

25

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27

A study of compaction band formation with the Double-Yield model

C. Detournay & P. CundallItasca Consulting Group, Inc., Minneapolis, MN, USA

J. ParraPDVSA – Intevep, S.A., Los Teques, Venezuela

ABSTRACT: The occurrence of thin localized bands associated with concentration of compressive strain hasrecently been reported in very porous rocks, both in field and laboratory settings. These structures exhibit areduction of porosity, and are of importance to the petroleum industry because they can impact reservoir perme-ability. Compaction bands have been the object of both theoretical and experimental studies by Olsson (1999),Issen & Rudnicki (2000), Bésuelle (2001) and others. In this paper, we examine, in a simple theoretical framework,the basic conditions for a band to appear. We consider the case of the Double-Yield model, identify conditions forlocalization related to the volumetric cap, and give examples of numerical simulations that illustrate band formation.

1 INTRODUCTION

Mollema & Antonellini (1996) recently identified thepresence of thin compacted bands in porous sandstone,and made reference to these features as “compactionbands”. Although these structures are sometimes asso-ciated with the presence of shear bands, they haveindividual characteristics, which are outlined in thesedefinitions, found in the literature:

– Compaction bands are narrow planar zones oflocalized compressive deformation perpendicularto the maximum compressive stress (Issen &Rudnicki 2000).

– A compaction band is a tabular zone that exhibitsnormal closure but no shear offset (Olsson 1999).

– Pure compaction bands are bands that exhibit anormal compacting strain and a zero shear strain(Bésuelle 2001).

Compaction bands have attracted attention becauseof the potential impact that the reduced porosity ofthese features may have on oil reservoir exploitation.The authors cited above are among those who haveinvestigated the condition for their formation in theoret-ical, laboratory and field settings.

In this paper, we work in a basic theoretical frame-work. The change of porosity localized in the band isinterpreted as an inelastic volume deformation, whichcan occur as an alternative to the homogeneous mode.Irrecoverable volumetric deformations are associated

with the presence of a cap in the yield surface. Weconsider the case of a strain softening/hardening cap,normal to the mean pressure axis in effective stressspace, and examine the conditions on the cap forcompaction bands to appear.

For numerical investigation with FLAC, we use theDouble-Yield constitutive model. This model is charac-terized by a strain softening Mohr Coulomb behaviorfor shear yielding, and by an independent strain harden-ing cap behavior for volumetric yielding.

The theoretical conditions for compaction bandformation associated with stress states on the volu-metric cap of the Double-Yield model are derived insection 2. The results of numerical experiments arepresented in section 3. Conclusions for the work aregiven in section 4.

2 EXISTENCE CRITERIA FOR COMPACTION BAND

As a convention in this paper, tension and extensionare positive for stress and strain, respectively, compres-sion is positive for pressure, and effective stresses aredenoted without a dash. Stresses are denoted as �ij,and strains as �ij with i � 1, 2, 3 and j � 1, 2, 3.Volumetric strain, ev, is defined by ev � �11 �22 �33, and mean pressure, by p � �(�11 �22 �33)/3. Rates are denoted by a superscript dot.First, we derive the stress-rate/strain-rate relations

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for evolution of a stress point on the Double-Yieldvolumetric cap, and then, we express the condition forvolumetric localization.

2.1 Cap constitutive relations

In the Double-Yield model, the volumetric yieldfunction is:

(1)

where pc is the cap pressure, and Fv 0 for elasticconditions.

The cap Fv � 0 is represented by a straight line inthe plane of shear stress, q, versus mean pressure, p,shown in Figure 1.The flow rule for volumetric yielding is associated;thus, the potential function is:

(2)

The total strain rates are partitioned into elastic andplastic parts:

(3)

The stress-strain relations are, in rate form:

where

(5)

and Kc, Gc are current values of tangent bulk andshear modulus.

The cap pressure is a function of plastic volumetricstrain, ev

p, and the hardening rule is:

(6)

The coefficient a is the hardening modulus (positivefor softening) which is a function of total plastic volu-metric strain. An example of volumetric hardeningbehavior is represented in Figure 2.

The flow rule gives the direction of plastic strain rate,which is parallel to the gradient of Gv in stress space:

(7)

The plastic multiplier ., gives the magnitude of plas-

tic strain rate. It may be found from the consistencycondition:

(8)

Substitution of the expression 2 for plastic potentialin Equation 7 gives, after differentiation:

(9)

Using Equation 9 for the plastic strain rate, thehardening rule in Equation 6 takes the form:

(10)

28

Figure 1. Volumetric cap for the Double-Yield model. Figure 2. Example of volumetric hardening rule.

(4)

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After substitution of this expression in the consis-tency condition, we obtain:

(11)

The plastic strain increments in Equation 9 maynow be expressed as:

(12)

From Equation 3, elastic strain rate may beexpressed as total rate minus plastic rate:

(13)

Finally, after substitution of Equation 13 inEquation 4, using Equation 12 and some manipulations,the cap constitutive behavior may be written in the form:

(14)

where

(15)

A symbolic way to write this expression is:

(16)

where Lijkl is the stiffness matrix, and Einstein summa-tion convention on repeated indices is used.

2.2 Condition for localization

We consider the deformation of a homogeneous sampleof material under gradual application of prescribed uni-form stresses, as shown in Figure 3. We will assume thatthe major compressive stress, �1 is vertical. Prior tothe occurrence of a band, the sample deforms uni-formly. After appearance of the band, stress and strainrates will be uniform inside and outside the band, butthey will be different from each other. A bifurcationfrom homogeneous deformation has occurred. Thenon-homogeneous solution can only exist provided thatsome continuity requirements on stress and strain are

met within the material sample. First, equilibrium atthe interface of the band only allows a stress disconti-nuity for the direct stress parallel to the band. Second,the direct strain parallel to the band must remain con-tinuous.

We analyze the situation at the onset of band forma-tion, and denote as ni the unit normal to the potentialplanar band in which localized deformation occurs.Mathematically, the condition for non-uniquenesstranslates as (see e.g. Issen & Rudnicki 2000):

(17)

where the components of the stiffness matrix may befound in Equation 14.

We look at the case when the out of plane componentof the normal to the band is zero, or n3 � 0. Band for-mation is predicted to occur when the condition

(18)

is first met in a program of deformation. By using therelations

for unit length of the vector ni, and �2 � �1 � 2Gc inEquation 5, the condition may be expressed as:

(19)

29

Figure 3. Material sample, prescribed stresses, and potentialcompaction band.

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The possible band orientations, given by the rootsof a quadratic equation, are given by:

(20)

where

(21)

For a real solution, we must have:

(22)

Using the definition of b, c, and �1 given above,the condition takes the form:

(23)

This condition can only be satisfied if a 0, thatis when softening of the cap occurs, see Figure 4.Physically, cap softening can correspond to grain col-lapse or breakage of cemented grains.

By definition, a pure compaction band is orientedperpendicular to the maximum compressive stress.According to our convention, we must have: n1 � 0,in which case Equation 19 implies:

(24)

Finally, using Equation 5 for �1, and Equation 15for b, the localization condition may be expressed as:

(25)

From the above consideration it follows that soften-ing of the cap is a necessary but not sufficient condition

for the occurrence of compaction band. A relationbetween hardening modulus and stiffness propertiesmust also be satisfied for the bands to appear: accordingto Equation 25, the hardening modulus, (which is posi-tive for softening of the cap) must be equal or largerthan 4Kc/(4 3Kc/Gc).

In a uniaxial compression test (oedometric test),with �

.11 � �

.33 � 0, the constitutive relation,

Equation 14, yields �.

22 (�1 b) �. 22, so the localizationcondition, Equation 24, which may also simply bederived using the compliance approach of Vermeer(1982), corresponds to the first occurrence of a plateauin the plot of vertical stress versus strain, see Figure 5.

It is interesting to note that, according to Equation 19,the condition for band formation in the direction paral-lel to the maximum compressive stress is also givenby Equation 24. So the same condition predicts bandformation in two perpendicular directions.

3 NUMERICAL EXPERIMENTS

Our theoretical derivation shows that cap softening is a necessary but not a sufficient condition for the formation of compaction bands. A relation betweenhardening modulus (a) and stiffness properties (Kc,Gc) must also be satisfied for the bands to appear. In anoedometric test, two sets of bands (one horizontal, andone vertical) are predicted to occur for

(26)

Numerical experiments are carried out to validatethis prediction, and illustrate band formation.

Numerical simulations of an oedometric test are per-formed using the finite difference code FLAC. The con-figuration is axi-symmetric. The grid contains a total of400 elements. The boundary conditions correspond to

30

Figure 5. Plateau corresponding to pore collapse in aschematic porous material stress–strain curve.

Figure 4. Example of cap softening behavior.

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roller boundaries at the bottom and lateral side of themodel. A Double-Yield constitutive model is assignedto the zones in the grid. Friction is zero, and cohesionis assigned a large value (compared to maximum meanstress in the simulation). We consider the state of themodel at the onset of band formation. The initialstress field is isotropic, and the material is normallyconsolidated. Several cases are considered, corre-sponding to hypothetical values of current hardeningmodulus. The test is strain-controlled: a compressivevelocity is applied at the top of the model.

The stiffness properties for the simulation are cho-sen such that Kc/Gc � 2, and Equation 26 translates toa � 0.4 Kc. To trigger the localization process, thematerial bulk modulus is given a random deviation of1%. The simulations are performed using the data filecb.dat, listed in the Appendix. In cases when the ini-tial hardening modulus is equal to 0.2Kc or 0.3Kc,(softening of the cap occurs, but the criteria for bandformation is not met), no band is observed. In the casewhen a � 0.405Kc (a value slightly larger than thethreshold for band formation), two sets of bandsdevelop in the model: one normal, and one parallel tothe major compressive stress. Simulation results atthree different stages, and for two different randomseeds (used for assignment of a small deviationaround an average value for the bulk property) arepresented below.

Figure 6 show the results obtained for seed 1: a firsthorizontal band appears, then a second one starts togrow, and a third one develops. At each step, the addi-tional deformation is seen to localize in the new band.The behavior of the normal stress parallel to the bands,at the end of the simulation is shown in Figure 7.

When the simulation was repeated, this time withanother seed, the results in Figure 8 were obtained. Thefirst band to appear is vertical and it grows across the

31

Figure 6. Seed 1 – Contours of volumetric strain incre-ments and displacement vectors at: a) 4000, b) 5000 and c) 7000 steps.

Figure 7. Seed 1 – Contours of �11 at 7000 steps.

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grid before a second horizontal compaction bandappears. Contours of vertical stress at the end of thesimulation are shown in Figure 9. Apparently it is ran-dom whether horizontal or vertical bands appear first.

4 CONCLUSIONS

A simple theoretical framework has been adopted toderive the basic conditions for compaction band forma-tion. The Double-Yield constitutive model in FLAC wasconsidered, and conditions for localization related tothe volumetric cap were identified using the approachof Issen & Rudnicki (2000). It was found that soften-ing of the cap, which can correspond to grain collapseor breakage of cemented grains, was a necessary con-dition for the occurrence of compaction band. But thecondition is not sufficient; in addition, the hardeningmodulus (positive for softening of the cap) must exceeda critical value, which is a function of material bulkand shear moduli. The critical value, which may alsobe derived using the compliance approach of Vermeer(1982), corresponds to the first occurrence of a plateauin a plot of major compressive stress versus strain.The analysis predicts the occurrence of two sets ofbands, normal and parallel to the direction of majorcompressive stress. Examples of numerical simulationsare given that illustrate band formation.

ACKNOWLEDGEMENTS

The work related in this paper was performed as part ofa research project carried out for INTEVEP. ChadSylvain is thanked for editing of the figures and MicheleNelson for her help in formatting the manuscript.

32

Figure 8. Seed 2 – Contours of volumetric strain incre-ments and displacement vectors at: a) 4000, b) 5000 and c) 7000 steps.

Figure 9. Seed 2 – Contours of �22 at 7000 steps.

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REFERENCES

Bésuelle, P. 2001. Compacting and dilating shear bands inporous rock: Theoretical and experimental conditions.Journal of geophysical Research, 106(B7): 13,435–13,442.

Issen, K.A. & Rudnicki, J.W. 2000. Conditions for compactionbands in porous rock. Journal of Geophysical Research105(B9): 21,529–21,536.

Itasca Consulting Group, Inc. 2000. FLAC Ver. 4.0 User’sGuide. Minneapolis: Itasca.

Mollema, P.N. & Antonellini, M.A. 1996. Compaction bands:A structural analog for anti-mode I cracks in Aeoliansandstone. Tectonophysics 267: 209–228.

Olsson, W.A. Theoretical and experimental investigation ofcompaction bands in porous rock. Journal of GeophysicalResearch 104(B4): 7219–7228.

Vermeer, P.A. 1982. A simple shear-band analysis using com-pliances. IUTAM Conference on Deformation and Failureof Granular Materials, Delft. 31Aug–3 Sept, 1982.

APPENDIX A: DATA FILE

newtitleOedometric test with DY model

config axig 20 20gen 0 0 0 1 1 1 1 0mo dypro bu 1110e6 sh 555e6 cap_pressure 5e6 cptable 1

mul 10pro den 1000 coh 1e10 ten 1e10

prop bul 1110e6 rdev 1110e4 ;�--- 1% deviation; case 1 a�0.2K;table 1 0 5e6 1e-3 47.78e5 1e-2 2.78e6 1 0.5e5 ; no cb;ini yvel 0 var 0 -1e-6; case 2 a�0.3K;table 1 0 5e6 1e-3 46.67e5 1e-2 1.70e6 1 0.5e5 ; no cb;ini yvel 0 var 0 -1e-6; case 3 a�0.405Ktable 1 0 5e6 1e-3 45.5e5 1e-2 0.5e6 1 0.5e5 ; cbini yvel 0 var 0 -1e-7; case 4 a�0.5K;table 1 0 5e6 1e-3 44.45e5 0.8e-2 0.56e6 1 0.5e5 ; cb;ini yvel 0 var 0 -1e-7

fix x i�21fix y j�1fix y j�21ini sxx -5e6 syy -5e6 szz -5e6step 4000

save cb1.savplot hold vsi fillstep 1000save cb2.savplot hold vsi fill

step 2000save cb3.savplot hold cap_pressure fillplot hold vsi fillplot hold sxx fillret

33

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35

A new viscoplastic model for rocks: application to the Mine-by-test ofAECL-URL

F. LaigleElectricité de France, Hydro Engineering Centre, France

ABSTRACT: A new viscoplastic constitutive model has been developed by EDF-CIH. Its aim is able to takeinto account delayed behavior of rock materials in the framework of nuclear waste repository studies. In thiscase, it’s important to predict the rock damage evolution in time in the neighboring storage tunnels. The mainassumptions of the constitutive model are presented in this paper. One application to the Mine-by-Test done bythe AECL in the Lac de Bonnet granite is shown. The low field strength of the granite in comparison with laboratory measurements may be justified by the delayed behavior of this granite. Failure with v-shape notchesis well shown by the simulation. A prediction of the hydraulic permeability increasing around the tunnel versustime is presented.

1 INTRODUCTION

In the framework of studies of underground nuclearwaste storage, it’s important to predict the evolutionin time and at very long term of the ground surround-ing the excavations. One objective of these studies is to assess the evolution of the EDZ (ExcavationDamage Zone) in the time. This EDZ is assumed to be the zone where rock is fractured and where the permeability increasing is large.

In this aim, a viscoplastic constitutive model hasbeen developed by EDF and integrated in FLAC. Thismodel will be briefly presented in the paper.

The application is about the Mine-by-Test done inthe AECL-URL. The evolution in time of the failurezone in the roof and invert of the gallery is well sim-ulated. Assumptions of this model allow to assess thedamage zone (fissured rock) and the fractured zone(continuous fissure) associated with a strong perme-ability increasing.

2 THE MINE-BY TEST EXPERIMENT

The Underground Research Laboratory (URL) of the Atomic Energy of Canada Laboratory (AECL) hasbeen dug in the framework of the Canadian nuclearwaste management program launched in the 70’s. Thisunderground laboratory is located in the state ofManitoba. It’s composed of a main shaft of 443 mdepth, reaching two experimental levels excavated at240 m depth and 420 m depth.

The Mine-by test has been done at the level – 420, inan undamaged granite mass (Lac de Bonnet Granite).This experiment consists of digging a gallery in well-known conditions in a previously monitored part ofthe rock mass. The major aim is to observe the behav-ior of the granite during the excavation phase and atlong-term.

The direction of the gallery has been defined inaccordance with initial stresses in the ground. At thisdepth, the major compressive principal stress is moreor less horizontal, and is about 55 MPa. This stress is 3.9 times the vertical stress corresponding at theweight of overburden. The intermediate principalstress is about 48 MPa.

Figure 1. Location and view of the URL.

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2.1 Observed and monitored behavior

In these conditions, during digging process and afterthat, a stress-induced fractured zone has been observedabove the crown and below the floor of the gallery,corresponding to a brittle failure mechanism generat-ing a classical v-shape notches.

Outside of these highly stressed zones, some Micro-seismic events could be monitored (Cai et al. 2001)but no major damage could be observed.

2.2 Laboratory tests

Some usual unconfined compressive tests have beendone by Read et al. (1998) on the Lac du Bonnetgranite. These tests provided following mechanicalcharacteristics (Fig. 3):

– short term UCS strength: �f � 213 MPa– compressive stress corresponding to the volumetric

strain reversal: �cd �160 MPa– compressive stress corresponding to the initiation

of the crack growth: �ci � 90 MPa.

If we only consider the short term strength, it’simpossible to justify the appearance of the brokenzone in highly compressive zones, considering initialstate of stresses. Martin shows if we want to find bysimulation the occurring of the failure, it’s necessaryto consider a limit strength threshold around 100 MPa.So some reasons have to be found to explain thisstrong decreasing of the strength in field in compari-son with laboratory measurements. These explana-tions could be:

– The well-known “scale effect”. A decrease instrength is generally observed with increasing elementary size of rock. In the case of the Lac de

Bonnet Granite, it seems that the cracking andexisting damage is not enough to justify a suffi-cient decrease in the strength.

– A “structural effect” as suggested by Diederich(2002), taking into account some geometrical andshape differences between laboratory and fieldconditions.

– A softening effect due to the stress path generatedby the digging process. Some 3D simulations(Eberhardt 2001) show that there is a stress rota-tion forward the face heading which can induceadditional damage in rock not taken into account ina 2D approach.

– A progressive damage of the rock structure in time.Some creep tests and UCS tests have been done bySchmidtke & Lajtai (1985) showing that this granitepresents an apparent “viscous” behavior. This phe-nomenon corresponds to a decrease in the strengthin accordance with the loading rate. So, Martinshows that the UCS can decrease 30% if the loadingrate reduces from 0.75 MPa/s to 0.0075 MPa/s. Atthis low loading rate, the measured strength isabout 150 MPa. This does not seem enough to jus-tify the field failure, however, we have to be con-scious that field loading rates are much lower thanthose applied in laboratory conditions. The previ-ous strength measured at 0.0075 MPa/s is still notrepresentative of the in situ characteristic.

Some other observed phenomena on site suggestthat there is a significant time behavior of the ground.As we see on the Figure 2, the roof spalling failureappeared progressively during several months. Simi-larly, some acoustic emissions have been registeredseveral years after the digging of this gallery. Timebehavior seems to be the major phenomenon, whichcan explain and justify the reduction of laboratory

36

Figure 2. Location of v-shaped notches in the Mine-by testtunnel.

Figure 3. Stress–strain curve for Lac de Bonnet granite.

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strength from 213 MPa to a field strength less than100 MPa.

2.3 Definition of the in situ strength

In the case of an undamaged rock like the Lac deBonnet Granite, the apparent in situ strength under aspecific confinement state is function of the loadingrate. As this loading rate is much lower than theseapplied in laboratory conditions, it seems reasonableto assume that the in situ strength could be assimi-lated to the long term strength of the rock. From lab-oratory tests, this “long term strength” could haveseveral definitions:

– From Sangha et al. (1972), the long term strengthcorresponds to the volumetric strain reversal.Above this threshold, the crack growth is assumedto be “unstable”. This notion of “instability” seemstotally subjective because is related to the delayallowed by experimental testing in laboratory conditions.

– From Morlier (1966) or Wiid (1970), the long termstrength of rocks is assumed to be correlated to thebeginning of the crack initiation. Assuming thatthe time behavior of cohesive materials like rocksis associated with a crack growth, this definitionappears more physical. It’s in accordance withexperimental results carried out by Schmidtkewhich show that the long term strength of the Lacde Bonnet granite could be estimated to 90 to100 MPa. This threshold corresponds to the crackinitiation under unconfined conditions.

3 CONSTITUTIVE MODEL

3.1 General principles

A new constitutive model has been suggested byLaigle (2003) aiming at accurate simulation of therock behavior in the averaged and large strainsdomain. A first version of the model has been initiallydeveloped in the framework of the elastoplastic theory.In this case, the yield surface corresponds to a gener-alized form of the Hoek-Brown criterion. Internalparameters “m”, “a” and “s” change according to anhardening variable �p:

(1)

(2)

The elastoplastic strain component ensues from thefollowing non-associated rule function:

(3)

The hardening of the elastoplastic mechanism is onlynegative. Some specific hardening laws were definedfor each of the internal parameters m, s and a, allow-ing to describe the evolution of the rock samplestrength from maximum peak value to the residualstate. The softening behavior domain reached afterthe peak strength, is assumed to be divided in threephases:

1. The first phase of softening corresponds to a dete-rioration of the rock’s cementation illustrated by a progressive disappearance of the cohesion at themacroscopic scale. This first phase is associatedwith an increasing of the dilatancy.

2. The second phase corresponds to the shear of aninduced fracture. It’ s associated with a decreasingof the dilatancy at the macroscopic scale.

3. Finally, the last domain corresponds to a purelyfrictional behavior, which defines the residualstrength. The shear occurs without any volumetricstrain.

A viscoplastic version of the model has been devel-oped after that. This version is based on the Perzyna’stheory, which assumes that the viscoplastic strain rate is a function of the distance between the loadingpoint representative of the state of stresses and a yieldviscoplastic surface, in accordance with the followingflow rule:

(4)

O�

(F) is a flow function and F is the overstress func-tion. Their expressions are followings:

(5)

(6)

37

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An important assumption is to assume that the hard-ening parameter rate ��p corresponds to the total irre-versible strain as follows:

(7)

and the global strain rate is the following:

(8)

��el is the elastic strain rate.��ep is the elastoplastic strain component.��vp is the viscoplastic strain component.

3.2 Identification of parameters

Three main sets of rheological parameters are neededfor the constitutive model:

1. Parameters describing the elastic reversible behav-ior which is assumed to not be time dependent.

2. Parameters affected to the elastoplastic instanta-neous mechanism: Four variables �c, mpeak, speakand apeak are needed to describe the initial elasto-plastic loading surface. This yield surface charac-terizes the rock strength for a very large loadingrate. An additional parameter defines the residualstrength criterion. Some few additional parametersdescribe the hardening kinetic of the loading surface from its initial position to the residual state.Another set of variables describes an intermediatecriterion corresponding to the stress thresholdwhen the apparent cohesion vanishes.

3. Parameters affected to the viscoplastic timedependent mechanism: Two parameters are neededto describe the yield viscoplastic surface. Thisyield corresponds to the damage criterion, which isthe initiation of cracking and so of the dilatancy.Up to now, in this constitutive model, it has beenassumed to be a Tresca surface described by theparameters �c, m0, and s0.

The kinetic of viscous behavior is adjusted by 2parameters “n” and “A” intervening in the flow rule(equation 5). These 2 parameters are identified using2 types of experimental results:

– Evolution of the strength in accordance with theloading rate (Martin & Read 1992). Comparison ofthese experimental results and simulations isshown in Figure 5.

– Evolution of the creep time to failure versus thedeviatoric creep level (Schmidtke & Lajtai 1985).The adjustment of the theoretical curve is shown inFigure 6.

Figure 7 shows simulations of a triaxial test with10 MPa of confinement at several strain rates. Inaccordance with the increasing of the strain rate, wemay observe both an increasing of the peak strengthand a changing of the behavior. It appears that thelower the rate, the more the rock behaves like a duc-tile material.

38

0

50

100

150

200

250

300

UC

S (M

Pa)

1,E-12 1,E-10 1,E-08

Strain rate (s-1)

1,E-06 1,E-04 1,E-02 1,E+00

Experimental results

Simulation

Figure 5. UCS vs. strain rate – comparison simulation–experience.

0

10

20

30

40

50

60

70

80

90

100

1 100 1000 10000 100000 1000000 10000000

Dev

iato

ric

stre

ss le

vel

(%)

1year1hour 1day 1month

10 1E+08 1E+09

Time to failure (s)


Simulation

Figure 6. Time to failure vs. deviatoric stress level –comparison simulation–experience.

00 20

1000

Dev

iato

ric

stre

ss (

MPa

)

900

800

700

600500

400

300

200

100

Minimal principal stress (MPa)

40 60 80 100

Viscoplastic yield surface

Residual strength criterionStrength criterionfor aStrain rate of 10-5/s

Initial elastoplastic loading surface

Figure 4. Stress criteria held for the Lac de Bonnet granite.

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3.3 Results interpretation

The formulation of the constitutive model and someinitial assumptions allow interpretation of computa-tional results in accordance with physical criteria. Forexample, it’s not always interesting and useful to estimate accurately ground displacements induced bya tunnel excavation in a hard rock. In this case, thecollapse mechanism is a brittle failure, which occursviolently and rapidly without any significant advancemovements.

In the framework of studies for undergroundnuclear waste repositories in hard rock masses, itseems more accurate to interpret numerical simula-tions using 2 kinds of results:

1. The physical state of the rock.2. The evolution and increasing of the rock mass per-

meability.

Depending on the hardening level, which is char-acterized by the hardening variable �p, it’s possible toestimate qualitatively the local damage of the rock(see Table 1).

– If �p � 0, the rock is assumed to be intact.– During the first phase of softening, as long as the

parameter s(�p), so the cohesion, is not null, therock is assumed to be fissured. This cracking maybe generated by a stress variation (activating of theplastic mechanism) or/and the delayed behavior(activating of the viscoplastic mechanism).

– As soon as the parameter s(�p) becomes equal to zero,it’s assumed that the cohesion at a macroscopic scaleis null. Physically, this corresponds to the creation ofa continuous fracture crossing the elementary vol-ume of rock. The global mechanical behavior is gov-erned by the mechanical response of the fractureunder a shear loading. During the fracture slide, thedilatancy will involve until a residual state is reached.

The main difference between these two last physicalconfigurations of the rock is the associated evolution

of the dilatancy. As long as the rock stays only fis-sured, the dilatancy increases. As soon as the rock isfractured, the dilatancy starts to decrease. It’s animportant aspect because we assume that the evolu-tion of the rock mass water permeability is a functionof this irreversible volumetric strain, in accordancewith the following equation:

(9)

where k is the current permeability, k0 is the ground’sinitial permeability, �p

v is the volumetric plastic straininduced by the load, and � is an adjustment parameterfor the model.

Figure 8 shows the theoretical evolution of the relative permeability in accordance with the deviatoricstress for a triaxial test under 10 MPa of confinement.The strain rate is assumed to be 1.5 � 10�5/s.

4 SIMULATION OF THE GALLERY

The excavation of the gallery is simulated in two phases:

– Phase 1: Simulation of the short term behaviorduring excavation phase using the elastoplastic

39

0

50

100

0

ε =1.5.10-6/s

ε =1.5.10-4/s

ε =1.5.10-5/s

ε =1.5.10-7/sε =1.5.10-8/sε =1.5.10-9/s

Dev

iato

ric

stre

ss (

MPa

)

300

250

200

150

0,1 0,2 0,3 0,4 0,5

Axial strain(%)

.

..

.

.

.

.

Figure 7. Simulation of triaxial tests at several strain rates(confinement: 10 MPa).

Table 1. Definition of domains in accordance with therock damage.

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version. The excavation is simulated by a progres-sive reducing of initial stresses existing on thegallery perimeter.

– Phase 2: Simulation of the long term behavior ofthe gallery, using viscoplastic version of the consti-tutive model. Initial state of stresses and hardeningparameters at the beginning of this phase are issuedfrom the previous elastoplastic step. This long termanalysis is continued until a stabilization of mechan-ical phenomena.

4.1 Damage of rock at short term

The Figure 9 shows the state of the rock around thegallery at the end of the excavation process.

Two domains may be observed:

1. An intact zone rock: In this area, the stress level is enough low to be under the damage criterion. No viscoplastic strains will appear in this domainwithout significant change of the stress diagramaround the tunnel.

2. A fissured rock zone: At the roof and under theinvert, one part of the rock mass is fissured. In thisdomain, the state of stresses is sufficient to beabove the damage criterion. Some viscoplasticstrains may be created in time in these zones. Withthe time, viscoplastic strains will appear in thesezones, generating a negative hardening of the peakcriterion. If this hardening is sufficient, the rockwill locally loose its cohesion. At this moment, itcould assume that a macroscopic induced fracturehas been created.

4.2 Evolution of the damage in time

Figures 10–13 show the increasing of a fractured zone above and below the gallery in time. There is an

40

0

50

100

150

200

250

300

350

1 10 100 1000 10000

Dev

iato

ric

stre

ss (

MP

a)

Relative permeability k/k0

0,1

Strain rate: 1.5e-5/sConfinement: 10MPa


Simulation

Figure 8. Lac de Bonnet granite – evolution of the per-meability during a triaxial test in the pre-peak domain (confinement: 10 MPa).

Fissured rock

Intact rock

Figure 9. Damage state of rock around Mine-by test tunnelat the end of the excavation.

Fractured rock

Tension zone

2 months

Fissured rock

Figure 10. Theoretical damage state of rock after 2 months.

Fractured rock

Tension zone

2 years

Fissured rock

Figure 11. Theoretical damage state of rock after 2 years.

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expansion of the fracturing in the vicinity of the tunnel in time. This expansion is located in highlystressed zone. Table 2 presents the depth of the frac-tured zone at the crown versus time.

In accordance with rheological parameters esti-mated before, the simulation shows an evolution of

the fracturing process over several years. From thesetheoretical results, it’s only after several hundredyears that this fracturing will stop.

From available information, it seems that theexpansion rate of the fractured zone is too slow andthat this evolution has been more rapid (apparently,few months). However, we have to precise that onlyone simulation has been done using set of parametercoming from an adjustment with few laboratory tests.It could be now possible to do a back analysis and to adjust some parameters like “n” and “A” which govern the kinematic viscoplastic strains creation.Despite this, after several years, when the stabilizationis reached, the shape of the fractured zone is similar tothese which has been observed on site (Fig. 14).

Figure 15 shows the evolution of the parameter “s”versus time. This variable represents the cohesion of

41

Fractured rock

Tension zone

4 years

Fissured rock


Fissured rock

10 years

Tension zone

Fractured rock


Table 2. Depth of the fracture zone versus time.

Thickness of theTime fractured zone in roof (cm)

2 months 151 year 202 years 254 years 35

10 years 43100 years 54500 years 65

1,000 years 6510,000 years 65

Figure 14. Theoretical damage state of rock at long term.

0

1

0 100 200 300 400 500 600 700 800 900 1000

Para

met

ers

⇔ D

amag

e in

dica

tor

Years

Distance to the crown 71cm

Distance to the crown 60cm50cm

0,9

0,8

0,7

0,5

0,4

0,3

0,2

0,1

0,6

Figure 15. Evolution of the damage indicator versus timeabove the crown.

09069-05.qxd 08/11/2003 20:16 PM Page 41


the rock. As long as it isn’t null, the rock is intact oronly fissured. So, this value is a damage indicator.According to rheological parameters retained, Figure 15show that the damage continues to involve during along time. This is the case at a distance of 71 cmabove the crown. Even if the kinematic is not correctin our simulations, the results are similar to monitor-ing results. Several years after the end of excavation,some acoustic emissions have still been registeredwhile fracturing process is stopped.

4.3 Evolution of the permeability in time

Damage of the rock results in a local increasing of therock macroscopic hydraulic permeability. This increas-ing is in accordance with the suggested Equation (9).Considering the parameter �, which has been adjustedon triaxial test results, this increasing could be aboutseveral order of magnitude in the fractured zone.Figures 16–19 show increasing of the permeabilityaround the mine-by test tunnel.

42

20 days

Log(k/k0)

Figure 16. Increasing of the hydraulic permeability after20 days.

2 months

Log(k/k0)

Figure 17. Increasing of the hydraulic permeability after 2 months.

Log(k/k0)

2 ans

Figure 18. Increasing of the hydraulic permeability after 2 years.

1000 years

Log(k/k0)

Figure 19. Increasing of the hydraulic permeability after1000 years.

09069-05.qxd 08/11/2003 20:16 PM Page 42


Figure 20 shows that the increasing of hydraulicpermeability is not only located in the fractured zone,but also in the fissured zone. Possible permeabilitychanges in tension zones are not taken into account inthe presented approach.

5 CONCLUSION

Even in very hard rock like granites, a delayed behav-ior could exist. This phenomenon could result in aprogressive damage of rock and delayed failure.Several microscopic theories have been suggested tophysically justify this delayed behavior in cohesiverocks. One explanation could be a “stress corrosion”in high stressed zones around existing and inducedcracks. The aim of the presented work here was tosuggest a macroscopic and phenomenological modeltaken into account this behavior. This model has beenintegrated in EDF’s local version of FLAC.

Considering there is little published informationabout creep behavior of the Lac de Bonnet granite, asimulation has been done using this model. It seemsthat this time behavior could justify the apparent lowfield strength of the granite in comparison with thelaboratory strength. Only one computation has beendone and parameters of the model have not beenadjusted after this first simulation. However, even ifthe kinetic seems too slow, this computation allows us to find the observed failure mechanism in highstressed zones, associated with an increase ofhydraulic permeability.

REFERENCES

Cai, M., Kaiser, P.K. & Martin, C.D. 2001. Quantification of rock mass damage in underground excavation frommicroseismic event monitoring. Int. J. Rock Mech. &Min. Sci. 38, 1135–1145.

Diederichs, M.S. 2002. Stress induced damage accumula-tion and implications for hard rock engineering. InHammah et al. (eds), NARMS-TAC 2002, 7–10 July 2002.University of Toronto press.

Eberhardt, E, 2001. Numerical modelling of three-dimensionstress rotation ahead of an advancing tunnel face. Int. J.Rock Mech. & Min. Sci. 38, 499–518.

Laigle, F. 2003. Modélisation rhéologique des roches adap-tée à la conception des ouvrages souterrains. Ph.D. EcoleCentrale de Lyon, in prep.

Morlier, P. 1966. Le fluage des roches. Annales de l’instituttechnique du bâtiment et des travaux publics: 80–111.

Read, R.S., Chandler, N.A. & Dzik, E.J. 1998. In situstrength criteria for tunnel design in highly-stressed rockmasses. Int. J. Rock Mech. & Min. Sci. 35, 261–278.

Sangha, C.M. & Dhir, R.K. 1972. Influence of time on thestrength, deformation and fracture properties of a lowerDevonian sandstone. Int. J. Rock Mech. & Min. Sci. 9,343–354.

Schmidtke, R.H. & Lajtai, E.Z. 1985. The long-term strengthof Lac du Bonnet Granite. Int. J. Rock Mech. Min. Sci. &Geomech. Abstr. Vol. 22, N°6, 461–465.

Wiid, B.L. 1970. The influence of moisture on the pre-rupture fracturing of two rock types. Proc. 2nd Cong. Int.Soc. Rock Mech., Belgrade. 239–245.

43

k/k0=10

k/k0=100Fractured zone

Figure 20. Increasing of the hydraulic permeability at verylong term.

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45

Prediction of deformations induced by tunneling using a time-dependent model

A. Purwodihardjo & B. CambouLaboratoire de Tribologie et Dynamique des Systèmes, Ecole Centrale de Lyon, France

ABSTRACT: Since the past 30 years, the research for estimating an accurate prediction of deformationsinduced by tunneling has been a major engineering challenge all around the world. The in situ measurementshave shown that deformations of the soil on the vicinity of a tunnel show a strong evolution with time. Threeessential phenomena, actually, can be related to this evolution: the evolution with time of the distance to theworking face, the distance of the lining to the working face and the viscous effects occurring in the soil.

The objective of this paper is to propose a procedure for predicting the deformations induced by tunneling,by taking into account these three essential phenomena, particularly the third phenomenon. Therefore, in thisstudy, a constitutive model for the time-dependent behavior of cohesive soil has been developed within theframework of elastoplasticity–viscoplasticity and critical state soil mechanics. The consideration of viscoplas-tic characteristic sets the current model apart from the CJS model, and introduces an additional viscous mech-anism. The evolution of the viscous yield surface is governed by a particular hardening called “viscoushardening” with a bounding surface.

To describe this procedure and the capability of the model, a comparison between numerical calculations andmonitoring the progressive closure of tunnel conducted in the TGV tunnel of Tartaiguille, is performed. Thefinite difference software, Fast Lagrangian Analysis of Continua (FLAC), has been used for the numerical sim-ulation of the problems. The comparison results show that the observed deformations could have been reason-ably predicted by using the proposed excavation model.

1 INTRODUCTION

The behavior of a tunnel is greatly influenced by thecharacteristics of the soils and the tunneling procedure.They will give a strong influence to the initial and longterm deformations on the vicinity of a tunnel and on theground surface, particularly when the ground traversedby tunnels has poor geotechnical characteristics: littleor no cohesion, medium-high deformability and highviscosity. In this area, more considerations should betaken because deformations of the soil on the groundsurface and on the vicinity of a tunnel show generally astrong evolution with time. This evolution is essentiallyrelated to three phenomena, i.e. the evolution with timeto the distance from the working face (the advance rateof tunneling), the distance of the lining to the workingface and the viscous effects occurring in the soil.

To predict the deformations induced by tunnelingby considering these three phenomena, a better under-standing of these influences and proper considerationsof their effects on the support design and installationare required. Therefore a time-dependent model has

been developed in the Ecole Centrale de Lyon, to ana-lyze the influence of these essential phenomena in theprediction of deformations induced by tunneling byusing numerical methods. This model is within theframework of elastoplasticity–viscoplasticity from the basic elastoplastic model (CJS model) includingan additional viscous mechanism.

2 DESCRIPTION OF THE MODEL

The CJS model is a constitutive model with differenthierarchical levels which has been developed 16 yearsago in the Ecole Centrale de Lyon (Cambou & Jafari1987, Maleki 1998). This model is based on nonlinearelasticity and two mechanisms of plasticity. It alsotakes into account the dependency on density of geo-materials through the critical state. The rate of thestrain tensor can be decomposed into an elastic partand a plastic part. The plastic deformations consist ofan isotropic and a deviatoric mechanism. Figure 1shows the two plastic mechanisms in the CJS model.

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The total strain of the model is decomposed in fourparts:

(1)

The first part is an elastic mechanism, the second partis an isotropic plastic mechanism, the third part is adeviatoric plastic mechanism and the last part is con-cerned with an added viscous mechanism.

2.1 Brief description of the basic elastoplasticmodel of CJS

2.1.1 Elastic mechanismThe elastic law is given by the following incrementalnonlinear relation:

(2)

where I1 and S are the first invariant and the devia-toric part of stress tensor while K and G are the bulkand shear modulus, respectively, which depend on thestress state through a power law:

(3)

Koe, Go and n are parameters while Pa is a reference

pressure which equals to 100 kPa.

2.1.2 Isotropic plastic mechanismThe yield surface associated to this mechanism is aplane perpendicular to the hydrostatic axis. The yieldsurface is given by:

(4)

The yield surface’s evolution is defined by anisotropic hardening mechanism depending on a scalarvariable Q and Tr is a parameter of the model to takeinto account the cohesion. The hardening rule has theform:

(5)

The isotropic flow rule is described as:

(6)

Kp is the plastic bulk modulus and n is a parameter ofthe model which can be determined by experimentaltest. i is a plastic multiplication for the isotropicplastic mechanism.

2.1.3 Deviatoric plastic mechanismIn the deviatoric plastic mechanism, for the sake ofsimplicity no kinematic hardening but only isotropichardening is taken into account (CJS level 2). Theyield surface can be written as:

where � is a parameter of the model and Tr is a param-eter of the model to take into account the cohesion.

The evolution of the yield surface is characterizedby the evolution of R with the internal variable p. Therelationship between R and p is written as:

(8)

where Rm is a parameter that corresponds to a radiusof the rupture surface and A is a parameter of themodel. The evolution of p is defined by:

(9)

The deviatoric flow rule is given by:

(10)

46

Figure 1. Plastic deviatoric mechanism and plasticisotropic mechanism in CJS.

(7)

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where d is a plastic multiplication for the deviatoricplastic mechanism. The deviatoric potential function(gd) used in relation (10) corresponds to a non-associ-ated plastic mechanism. Tensor nij is a symmetricaltensor so that tr(n2) � 1 and is a tangential tensor tothe surface corresponding to the potential function. Itis defined by:

where SIIc represents the second invariant of the devia-

toric part of stress in the characteristic state and � is aparameter of the model. The characteristic surface isdefined by:

(12)

where Rc represents the radius of the characteristicsurface. Figure 2. shows the deviatoric mechanism inthe CJS.

2.1.4 Critical stateTwo important phenomena can be noted from thedrained triaxial tests:

– an increase of the peak resistance with the initialdensity of material

– the material tends to the state called the critical statecharacterized by a null volume variation and a ratioq/p constant independent to the initial density.

To take into account these phenomena and beinspired by the formulation developed by theCambridge University (Roscoe et al. 1968), in thismodel, the radius of rupture surface varies as a func-tion of the mean effective stress and the density ofmaterial. For simplifying the problem, we take thecritical state similar to the characteristic state. Hence,the evolution of rupture surface is defined by:

(13)

where � is a parameter of the model, and pc is a criti-cal pressure which is defined by:

(14)

where c is a parameter of the model, pco is a criticalpressure corresponding to the initial density and �v isan accumulated volume strain.

2.1.5 Strain softening modelThe CJS model takes into account the strain softeningbehavior of the soil which depends on the accumu-lated deviatoric strain. This model is made up of threeportions, an elastoplastic portion up to the peakstrength, a softening portion in which the strength (Rcand Tr) reduces from the peak to residual, and finally,a constant residual strength portion. Figure 3 showsthe strain softening behavior in the CJS model.

2.2 Viscous hardening with a bounding surface

The viscous effect of the soil is connected with aninternal characteristic. This internal characteristic isrepresented by a creep surface which is bounded by a(current) state of stress surface defined by Equation20. It means that the creep surface can evolve but theevolution is limited by the state of stress surface. Sowe call this function as a viscous hardening with abounding surface, where the bounding surface in thiscase is the state of stress surface. Meanwhile, the evo-lution of state of stress surface is limited by the yieldsurface (elastoplastic concept). The evolution of theyield surface is limited by the rupture surface. Figure 4shows an illustration of viscous evolution conceptwith a bounding surface.

The basic formulation for this viscous mechanismis inspired by the overstress model of Perzyna (1966).

47

characteristic

rupture surface (Rm)

surface (Rc)

yield surface (R)

S11

S22 S33

Figure 2. Different surfaces in the deviatoric mechanismof the CJS model.

eIIe fe°

SII

IIII

Figure 3. Strain softening behavior in the CJS model.

(11)

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To keep on near to the framework of elastoplasticity isthe reason of the use of this formulation. The idea is,then, starting from the general framework of elasto-plasticity and introducing the viscosity of materialand the retardation function.

Many authors, (Katona 1984, Adachi 1982,Sekiguchi 1984, etc.), have employed this formula-tion, and they have shown that this model is incapableto introduce the acceleration deformation phenome-non in the case of tertiary creep. Therefore the ambi-tion of the proposed model is to take into account thetertiary creep.

Thus, three important terms have to be defined inthe framework of this model. The first one is the vis-cosity of the material, the second one is the functionof retardation and the last one is the direction of theviscoplastic strain. The function is as follows:

(15)

where 1/� is the viscosity of the material, (�v/�r) isthe function of retardation and Gij

vd is the direction ofthe viscoplastic strain.

The viscosity of the material in this model is afunction of the distance of the state of stress surface(Re) to the rupture surface (Rm). This function isdefined by:

(16)

where �0 is a parameter of the model, Re is the radiusof the current state of stress surface, Rm is the radius ofthe rupture surface and k is a parameter of the model.

The function of retardation, (�v/�r), is inspired bythe bounding surface theory (Kaliakin & Dafalias1990). This function will drive the evolution of thethree types of creep. The secondary creep will be

reduced as a passing point between the primary creepand the tertiary creep. This idea has been selected forthe sake of simplicity in the measurement of the dis-tance in the stress space. This function is defined by:

(17)

where Rv corresponds to the radius of the creep sur-face. The power m in relation (16) is defined as:

(18)

where m1 and m2 are parameters of the model.Lade (1998) has shown in his laboratory test

results that the potential plastic surfaces for the elasto-viscoplastic and the elastoplastic are homothetic.Based on this idea, it means that the direction of theplastic strain in the elastoplastic is similar to the vis-coplastic one. Thus, the direction of viscoplasticstrain is defined as:

(19)

where f e is the artificial state of stress surface whichis homothetic to the yield surface for the deviatoricmechanism. It is defined by:

(20)

The rupture surface is defined by:

(21)

The creep surface is defined by:

(22)

The evolution of the creep surface is given by:

(23)

Av is a parameter of the model and eIIvd is an accu-

mulated deviatoric viscoplastic strain, which isdefined by:

(24)

where eII is the deviatoric viscoplastic strain rate.

48

creep surface

artificial state of stress state of stress

S33

S11

rupture surface (Rm)

yield surface (R)

surface (Re)

S22

Figure 4. Illustration of viscous evolution concept with abounding surface.

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2.3 Parameter identification

The identification of elastoplastic mechanism param-eters can be determined by using classical laboratorytests. The procedure for calibrating the model param-eters is briefly defined by Maleki (1998). Drained orundrained creep tests with different levels of stresseshave to be achieved, for identifying the viscoplasticmechanism parameters.

3 CALCULATIONS

3.1 Presentation of the case studied

The tunnel of Tartaiguille is located on the newMéditerranée high-speed line between Valence andMontélimar (France). All these structures are doubletrack single tube, allowing a speed of 300 km/h. Itcrosses fractured limestones on the north sides, stiffmarls and sandstones in the south and stampian claysin the central parts.

Convergence measurement devices had beeninstalled by CETU (Centre d’Etudes des Tunnel) dur-ing construction processes in this tunnel. PM 1168has been selected as the simulation data in the defor-mation analysis. From the cross section of the tunnel,we can see that the support of the tunnel is a shotcrete(thickness � 300 mm) and a steel frame every 1.5 m.Five samples of soil blocks have been obtained andthese blocks have been studied in detail by Serratrice(1999) (Laboratoire Regional des Ponts et Chausséesd’Aix en Provence). From the five samples of soilblocks, three layers of soil can be concluded at thatsection. The upper one is the black marl, the middleone is the calcareous marl and the lower one is thegrey marl. The soil characteristics for the black marland the grey marl are almost similar, on the otherhand the soil characteristics of the calcareous marl issignificantly different. The calcareous marl is stifferthan the black marl and the grey marl. For the sake ofsimplicity, only two types of soil will be considered,for the upper one and the lower one, we will use thesame parameters of soil.

Figure 5 shows the dimensions of the tunnel andFigure 6 shows the soil stratigraphy at section PM 1168.

3.2 Model parameters

Elastoplastic and viscoplastic model parameters iden-tified in this analysis can be seen in Table 1. Figure 7shows the simulation results of triaxial tests by usingelastoplastic soil parameters. Figure 8 shows the sim-ulation results of creep tests by using elastovisco-plastic soil parameters.

It can be seen from Figures 7 & 8 that the simula-tion results are closely match to the experimentalresults. It means that this model can take into account

the elastoplastic and viscoplastic behavior of the soilsquite satisfactory.

3.3 Plane strain calculations

The convergence curves of the unsupported tunnelare derived by using virtual support pressures in planestrain calculations (Panet et al. 1982, AFTES 2002).

Two types of calculation have been achieved. Thefirst one is by using the circular shape of tunnel withR � 7.65 m, Ko � 1.0, only used one soil layer (blackand grey marl parameters) and the second one is byusing the actual shape of the tunnel, the actual valueof Ko � 1.2 and the actual soil layers. The objective ofthis calculation is to get the result comparisonbetween those two shapes for the reason that the cal-culation taking into account the distance from theworking face and the progressing of the tunneling

49

Figure 5. Dimensions of tunnel and the measurementposition.

Figure 6. Soil stratigraphy at section PM 1168 (Lunardi2000).

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will be performed using an axisymmetric calculation(see section 3.4). The result comparison obtained inthe plane strain calculation allows the results obtainedin the axisymmetric condition to be transformed totake into account the actual conditions of the tunnelsection. This approximation has been taken becausethose two shapes are almost similar and the calcare-ous marl is not dominant.

In the first calculation, a quarter of tunnel geome-try has been modeled, and for the soil parameters, theblack and grey marl parameters have been employed.

On the other hand, in the second calculation, the twotype parameters of soil have been employed. Theoverburden pressure height is 100 m from the crownof the tunnel and the ground water table is 6.6 m fromthe ground level.

3.4 Axisymmetric calculation

This computation is performed by using the sequen-tial excavation method (SEM) in the axisymmetriccondition. Distance to the working face is defined by dand the advance rate of the excavation is defined by p.Figure 12 shows the geometry of the tunnel in theaxisymmetric calculation and Figure 13 shows themesh used in the axisymmetric calculation.

Figures 9 and 10 show the mesh used in the planestrain calculations and Figure 11 shows the compari-son results of the convergence analysis between thetwo shapes. From that figure, the ratio between theactual shape and the circular shape can be deter-mined, and we get the ratio of convergence (RT) atposition F-G equal to 0.885. This ratio will be usedfor adjusting the axisymmetric calculations, since inthe axisymmetric calculations, we can only use thecircular shape, Ko � 1.0 and one layer of the soil.

The lining support in this tunnel is a combinationof shotcrete ring and steel frame. So for simplifyingthe analysis, the equivalent stiffness of the lining of

50

Table 1. Parameters of the model.

Parameter Black and grey marl Calcareous marl

Density (kN/m3) 22.15 24.34

Elastoplastic parametersGo (MPa) 27 96.15Ko

e (GPa) 139 208.33Rm 0.103 0.2661� 0.3616 0.7852Rc pic(Rc res 0.0784 0.213A (kPa�1) 2 25n 0.7 0.6Ko

p(MPa) 139 208.33

c 60 75� 0.033 0.05pco (MPa) 17 40� �0.005 �0.38Tr pic (MPa) 11.825 8.768Tr res (MPa) 7.112 5.273e°

II 0.02 0.02e f

II0.065 0.065

Viscoplastic parametersAv 125 450�0 108 106.69

k 6.0 30.673m1 0.3 0.4083m2 0.0 8.0214

Figure 7. Simulation results of triaxial tests.

Figure 8. Simulation results of drained creep tests.

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the combined lining support has been used. The elas-tic model is used for this lining.

For modeling the ground anchor on the workingface, the equivalent pressure on the working face hasbeen used (Peila 1994). This pressure is determined by:

(25)

where n is the number of the anchor, Tb is the tensilestrength of the anchor, Sb is the shear strength of the anchor, and S is the working surface area. In thistunnel, 120 fiberglass anchors with 800 kN tensilestrength have been installed on the working face tostabilize the working face.

In this calculation, the black and grey marl param-eters have been used because they are more dominantthan the calcareous marl in the soil stratigraphy. Forthe SEM analysis, we use d � 1.5 m and p � 1.5 m(the advance length of tunneling).

The convergence of the tunnel is determined by:

(26)

where U(x) is the deformation of the tunnel as a func-tion of the distance from the working face, U(o) is thedeformation of the tunnel on the working face and RTis the shape ratio of the tunnel.

In the first simulation, the influence of the advancerate of the tunneling is illustrated. Three types of theadvance rate are used, i.e. 1.5 m per 0.5 day, 1.5 m per1.0 day and 1.5 m per 2.0 day. The elastoplastic calcu-lation is used to represent the infinite advance rate of tunneling. The tunneling simulation results can beseen in Figure 14.

From that figure, we can see that the convergenceof the tunnel can be reduced by increasing the

51

Figure 9. Mesh used in the circular shape.

Figure 10. Mesh used in the actual shape.

Figure 11. Comparison of convergence analysis.

Figure 12. Geometry of the tunnel in the axisymmetriccalculation.

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advance rate of tunneling. Actually, in this case, weprevent the evolution of the creep deformation. If weonly use the elastoplastic constitutive model, we can-not illustrate this phenomenon. The total elastovis-coplastic deformation of the tunnel could be two orthree times bigger than the elastoplastic deformation.

In the second simulation, the actual advance rate oftunneling at section PM 1168 is used. The sequencesof the actual excavation are as follows:

1. Excavating the upper section with the advancerate: 1.5 m per 0.5 day until 6 m.

2. Stopping for one day (to represent the excavationof the lower section).

3. Continuing the excavation of the upper sectionwith the advance rate: 1.5 m per 0.5 day until 6 m.

4. Stopping for three days (to represent the excava-tion of the lower section and the installation timeof the anchors on the working face).

5. Continuing the four sequences above.

The measurements at position F-G (see Figure 5)are started after 6 m from the working face. Figure 15shows the comparison results between the calcula-tions and the measurements started from the firstmeasurement (6 m from the working face).

From those figures we can see that the model pro-posed can model acceptably the viscous behavior ofthe soil in the tunneling. It has been demonstrated

also in the case of viscous materials, a time-dependentmodel is very essential.

4 CONCLUSIONS

Analysis of deformations due to tunneling using theelastoplastic–viscoplastic constitutive model has beenperformed in this study.

It has been demonstrated that the influence of vis-cous effects cannot be neglected in the soil which hasbeen analyzed. It means that the role of the time-dependent model in this case is very important and anecessity. The influence of viscous effects can bereduced by increasing the advanced rate of tunnelingbut an attention to the lining should be taken becausethe load transfer to the lining will be higher. Thisbecomes significant when there is a large distancebetween the installation point of the lining and theworking face, and could induce plastic deformationaround the tunnel.

The calculation procedure proposed has provided aneffective approach for analyzing the ground-structureinteraction situation and offers a systematic way ofoptimizing lining design. This kind of calculation canbe improved by using a complete 3D approach.However, this is a rather difficult calculation and thecomputation time will be long. In practice, the 2D andthe axisymmetric analysis can be successfully used todevelop a pragmatic solution.

The constitutive model, which has been presented,is quite satisfactory to model the elastoplastic–viscoplastic behavior of the soils. The parameters ofthe model can be identified by using the classical lab-oratory tests such as, triaxial tests and creep tests.

ACKNOWLEDGEMENTS

The authors gratefully acknowledge the information onthe geotechnical data and the convergence data provided

52

Figure 13. Mesh used in the axisymmetric calculation.

Figure 14. Tunneling simulation results in elastoplastic–viscoplastic calculation.

Figure 15. Convergences of the tunnel as a function of thedistance from the working face.

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by Mr. Alain Robert and Mr. Adrien Saïtta fromCETU (Centre d’Etudes des Tunnel), Lyon, France.

REFERENCES

Adachi, T. & Oka, F. 1982. Constitutive equations for nor-mally consolidated clays based on elasto-viscoplasticity,Soils and foundations, Vol. 22, No. 4: 57–70.

AFTES. 2002. La méthode convergence-confinement,Tunnels et ouvrages souterrains, No 170: 79–89.

Cambou, B. & Jafari, K. 1987. A constitutive model for gran-ular materials based on two plasticity mechanisms.Constitutive equations for granular non-cohesive soils,Saada & Bianchini (Eds), Balkema, Rotterdam: 149–167.

Kaliakin, N. & Dafalias, F. 1990. Theoretical aspects of theelastoplastic-viscoplastic Bounding surface model forcohesive soils, Soils and foundations, Vol. 30, No. 3: 11–24.

Katona, M. G. 1985. Evaluation of viscoplastic cap model,Journal of Geotechnical Engineering, Vol. 110, No. 8:1107–1125.

Lade, P. V. 1998. Experimental Study of Drained CreepBehavior of Sand, Journal of Engineering Mechanics,Vol. 124, No. 8, August: 912–920.

Lunardi, P. 2000. The design and construction of tunnelsusing the approach based on the analysis of controlleddeformation in rocks and soils, ADECO-RS.

Maleki, M. 1998. Modélisation hiérarchisée du com-portement des sols, Phd. Thesis, École Centrale de Lyon.

Panet, M. & Guenot, A. 1982. Analysis of convergencebehind the face of a tunnel, Tunneling’ 82: 197–204.

Peila, D. 1994. A theoretical study of reinforcement influ-ence on the stability of a tunnel face, Geotechnical andGeological Engineering, 12.

Perzyna, P. 1966. Fundamental Problems in viscoplasticity.Advances in Applied Mechanics, Vol. 9: 243–377.

Roscoe, K. H. & Burland, J. B. 1968. On the GeneralisedStress-Strain Behavior of ‘Wet Clay’, EngineeringPlasticity, J. Heyman and F. A. Leckie (Eds). Cambridge:Cambridge University Press: 535–609.

Sekiguchi, H. 1984. Theory of undrained creep rupture of normally consolidated clays based on elasto-viscoplasticity, Soils and foundations, Vol. 24, No. 1:129–147.

Serratrice, J.F. 1999. Tunnel de Tartaiguille (Drôme) TGVMéditerranée, Essais de laboratoire sur la marne, LRPCd’Aix en Provence.

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55

Modeling of anhydrite swelling with FLAC

J.M. Rodríguez-OrtizGamma Geotécnica SL, Madrid, Spain

P. Varona & P. VelascoItasca Consultores SL, Asturias, Spain

ABSTRACT: Anhydrite and rocks containing argillaceous minerals experience swelling phenomena whenthey come into contact with water. In tunneling, this can lead to a strong heave of the floor and to a high levelof stresses in the lining. Although characterization of swelling potential, monitoring of swelling process, and a lotof relevant case histories of tunnel construction in swelling rocks are currently available, the design of the supportin swelling rocks usually do not consider an accurate stress–strain relationship for the swelling. Current trendin tunneling design considers numerical modeling of the rock-support interaction, but the available geotechni-cal codes do not include the swelling formulation. This paper presents the implementation in FLAC (via FISHroutines) of the analytical stress–strain formulation for the swelling presented by Wittke (1999) and the valida-tion of this algorithm against the swelling tests carried out by different authors and presented by Wittke (1999).

1 DESCRIPTION OF THE ANHYDRITESWELLING PHENOMENA

1.1 Chemical description

Calcium sulphate appears naturally as two differentminerals: gypsum (CaSO4.H2O), in which waterappears within the crystalline structure, and anhydrite(CaSO4). The hydration of anhydrite is a complexprocess that depends on the pressure and the temper-ature. For ambient conditions the chemical reaction isillustrated in Table 1.

With an external inflow of water, the volumetricincrement associated to this process is presented inEquation 1:

(1)

Transformation of anhydrite into gypsum can beinhibited at 20°C with a pressure of 1.6 MPa; this process is reversible, being necessary a pressure of 80 MPaat 58°C to transform gypsum into anhydrite.

Previous data refer to pure anhydrite, but in case ofinterbedded mudstone-anhydrite the maximum swellingvolume is lower but the swelling stress is larger (in theorder of 2 to 5 MPa).

In the case of pure anhydrite, as the hydration process begins, the thin layer of impervious gypsum

created at the surface of the grains inhibits the waterpenetration stopping the process. In the case of inter-bedded anhydrite-mudstone the swelling processleads into the disintegration of the rock, reducing itsstrength. Steiner (1993) quantifies this reduction ofstrength with an angle of friction of 20°.

1.2 Characterization of the swelling behavior

The International Society of Rock Mechanics has pro-posed a set of tests to quantify the swelling of argilla-ceous rocks: the Maximum Axial Swelling Stress test,the Axial and Radial Free Swelling Strain test, and theAxial Swelling Law test (axial swelling stress as afunction of axial swelling strain, or Huder-Ambergswelling test).

An illustration of the results from the Huder-Amberg swelling test (total vertical strain of the sample,�z in %, versus vertical load, �z in kPa) is presented inFigure 1, taken from Wittke (1999). Stages 1, 2 and 3correspond to the initial loading phase with 2 loadcycles; stage 4 corresponds to the watering of thesample (no stress increment but strain increment),and finally stage 5 corresponds to the different pointsof the unload-swelling process. The swelling strainequals the total strain (stage 5 of the test) minus theelastic strain (stages 2 and 3 of the test).

If the strain due to swelling is plotted against the stress in a semi-logarithmic scale (Figure 2) a

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straight line is obtained which may be described bythe relationship presented in Equation 2.

(2)

where �qz� � final axial strain due to swelling (final

deformation means deformation at the end of the

swelling process); Kq � swelling deformation parame-ter; �z � axial effective stress; and �0 � axial swell-ing stress (as showed in Figure 2, it is intersection ofthe straight line with �q

z� � 0).

2 ANHYDRITE SWELLING LAW

The following description of swelling law is taken fromWittke (1999), and starts from the axial stress–strainrelationship previously presented (Equation 2). Theswelling law only applies for compressive effectivestresses, where �c (Figure 2) is a minimum stress rep-resenting the lower limit of validity of the swelling law.As the swelling strain equals to zero for compressivestresses larger than the swelling stress, the swelling lawcan be finally formulated as Equation 3:

(3)

where �qi� � final axial strain due to swelling in the

direction i; Kq � swelling deformation parameter;�i � axial stress in the direction i; �0 � axialswelling stress; and �c � minimum limit for the axialstress.

Equation 3 represents the 3D (i � 1,2,3, means the 3 directions in the space) isotropic (the same Kqparameter is considered for the 3 directions) swellinglaw, where the final axial strain is reached at the endof the swelling process.

This swelling law considers that the principaldirections of swelling �q

i� (i � 1,2,3) are coaxial withthe principal stresses �i (i � 1,2,3), and therefore thevalue of the swelling principal strain depends only onthe value of its coaxial principal stress.

Previous relations refer to the strain reached at the end of the swelling process. Furthermore, for thekinetics of the process Wittke (1999) presents the fol-lowing Equation 4 for the swelling strain rates at time t:

(4)

where �q � swelling time parameter; �qi � principal

swelling strains for t � �; and �qi(t) � principal

swelling strains which already occurred until time t.According to Wittke (1999), the time dependence

of swelling is adequately described by Equation 4 aslong as the strength of rock is not exceeded. The plas-tic deformations occurring if the rock strength isexceeded lead to a volume increase and to an increaseof permeability that accelerates the penetration ofwater, increasing the swelling strain rate.

56

Figure 1. Swelling test of an interbedded anhydrite-mudstone sample (Wittke 1984, in Wittke 1999).

Figure 2. Axial swelling law (Grob 1972, in Wittke 1999).

Table 1. Transformation of anhydrite into gypsum.

Anhydrite � water Gypsum

Equation CaSO4 � H2O CaSO4.2H2OMass (gr) 136.14 � 36 172.14Density (gr/cm3) 2.96 � 1 2.32Volume (cm3) 46.2 � 36 74.3

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To consider this effect in the formulation, Wittke(1999) includes the following relationship for theswelling time parameter �q, Equation 5:

(5)

where a0, ael, avp � constant values. The parameter a0represents the dependence of the swelling velocity onthe anhydrite content, regardless of whether a strainoccurred before or not; �el

v is the elastic volumetricstrain occurred prior to the beginning of swelling thatalso influence the permeability; �pl

v is the volumetricplastic strain; and maxEVP represents an upper limitof the plastic volume strain with regard to an eventualacceleration of swelling. According to Wittke (1999),plastic volumetric strains larger than maxEVP do notlead to a further increase of the swelling velocitybecause the penetration of water into the rockmasscannot be further accelerated by these.

Following Equation 5 the swelling time parameteris no longer constant but dependent on time as elastic –plastic volumetric strains varies during the swellingprocess.

3 FINITE DIFFERENCES CALCULATIONALGORITHM

The swelling law presented in previous paragraphshas been implemented in FLAC, coupling the swellingphenomena with the built-in elastic–plastic constitu-tive relationships via FISH routines. The principalconcept of this algorithm is that the volumetric strain isreached in the zones of the model introducing ofsmall increments of isotropic stress within them,Noorany et al. (1999).

The sketch of the algorithm is to calculate the finalswelling strain tensor for all the zones of the model,transform the strains into an increment of stresses,and then “inject” the stresses in small increments intothe zones. The flowchart of the calculation algorithmis presented in Figure 3, and can be resumed in thefollowing points:

– Determination of the principal effective stresses.�1 and �2 are principal stresses in the calculationplane, and �3 is the out-of-plane stress.

– Determination of the swelling principal strains and of the stress increments associated with thesestrains.

– Determination of the swelling time parameter.– Determination of the minimum timestep necessary

for numerical convergence of the algorithm and to synchronize the swelling rate of all the elementsin the model.

– “Injection” of a fraction of the stress incrementassociated to the swelling strain.

– Solve to mechanical equilibrium of the currenttimestep.

– Accumulation of swelling strains and time.– Repetition of the algorithm until the expected age

of the simulation is reached.

After initiation of swelling time to zero, the princi-pal stress tensor for all the elements is calculatedfrom the current stress state. As the routine has beenimplemented in a 2D model, the out-of-plane stress,�zz in FLAC, is a principal stress (�3 in the formula-tion) but not strictly the minor principal stress.

Then, a loop is performed until the swelling timereaches the expected simulation age. Within this loop,the strain tensor due to complete swelling is calcu-lated according to Equation 3. The stress increments

57

END

t< T_fin

t = t + ∆tmin

i`ii iεq = εq + (εq − εq) ∆tmin

ηq

∆tmin = min{∆t}

Mechanical equilibrium

∆tmin

ηq

σi = σi + ∆σi

σzz = σ3

σxx ,σyy , τxy = f(σi=1,2,θ)

∆σi∆t = ηq

mini=1,2,3{ {10% σi. .

1ηq

. .=a0 +a

el εel +a

vp min{εpl , maxEVP}vv

∆σi = f(εq ;λ,G)i`

i`εq = Kq log

σi

σ0

σi=1,2 = f(σxx ,σyy ,τxy,θ)

σ3 = σZZ

t = 0

Principalstresses

Swelling strainsAssociated stressincrement

Swelling timeparameter

Minimumtimestep

“Injection” of afraction of thestress increment

Solve tomechanicalequilibrium

Accumulation ofswelling time andswelling strains

Figure 3. Flowchart of the calculation algorithm.

09069-07.qxd 08/11/2003 20:43 PM Page 57


associated to these strains are calculated with the fol-lowing lineal elastic relationship, Equation 6:

(6)

where and G are constants known as Lamé’s parameters ( � K 2/3G; K is the bulk modulus andG is the shear modulus).

Previously to the calculation of the swelling timeparameter the plastic component of the volumetricstrain has to be determined.

The total volumetric strain, addressed in FLAC witha FISH variable, is the sum of the following compo-nents, Equation 7:

(7)

where �totv � total volumetric strain; �el0

v � elastic vol-umetric strain produced in the model previous to anycalculation; �el

v � elastic volumetric strain producedduring the calculation; it can be calculated as�el

v � (�1 � �2 � �3)/(3 K), being K the bulk modulus;�q

v � swelling volumetric strain accumulated duringcalculation, �q

v � �q1 � �q

2 � �q3; therefore, the plastic

volumetric strain, �plv, can be calculated with the fol-

lowing Equation 8:

(8)

The swelling strain does not occur instantaneouslybut following the kinetics formulated with Equation 4.Expressing this differential equation in finite differ-ences we obtain the following Equation 9:

(9)

and therefore,

(10)

As in all finite difference algorithm schemes, thisequation applies only for values of �t that are signifi-cantly low. This means that the swelling strain at timet that still remains to produce, [�q

i�(t) – �qi�(t)] cannot

be induced in the model instantaneously because themodel would degenerate. Thus, the next phase is todetermine a critical value of �t to use in the finite difference scheme.

To determine a value of �t small enough, only afraction of the stress increment associated to theremaining swelling strain should be “injected” in theelements of the model.

A criterion of a maximum of 1% of the currentstress state has been adopted to determine de frac-tion of ��i to “inject”. The minimum fraction obtainedfrom the 3 principal directions in each element isadopted. These relationships are illustrated inEquation 11 for every element in the model.

(11)

where ri � fraction of the stress increment ��i;�i � stress state; and r � minimum fraction of the 3 principal directions.

From Equations 6 and 11 the fraction r of the stressincrement that are going to be “injected” in the ele-ments of the model can be expressed with the follow-ing Equation 12:

(12)

and therefore, the timestep for each element can beobtained as, Equation 13,

(13)

It is necessary to synchronize the rate of swellingfor all the elements of the model adopting the sametimestep for all; the minimum timestep of all the elements is the searched,

(14)

The “injection” of stresses associated to the swelling behavior can be expressed with the followingEquation 15,

(15)

that gives, the relationship between the current stressstate – �i(t), the total increment of stresses due toswelling – ��i, the minimum timestep – �tmin, andthe swelling time parameter – �q.

58

09069-07.qxd 08/11/2003 20:43 PM Page 58


These stress increments have to be transformed fromthe principal axes reference to the coordinated axes ref-erence, assuming that the principal stresses have notrotated during the swelling processes, and the anglebetween the principal stresses and the horizontalremains the same.

Finally, once the mechanical equilibrium has been reached for this fraction of stresses injected intothe elements of the model, it is necessary to actualizethe accumulated swelling strain of each element of themodel, Equation 16:

and for the swelling time, Equation 17.

(17)

This procedure is repeated until the accumulatedswelling time reaches the expected age of the swellingsimulation.

4 VALIDATION OF THE ALGORITHM

The algorithm introduced in previous paragraphs hasbeen implemented in FLAC via FISH routines. Now,the validation of this algorithm against the swellingtests presented in Wittke (1999) is presented.

4.1 Swelling pressure test

The first validation test is a swelling pressure test of a cubic sample of swelling mudstone carried out in atriaxial test apparatus. After a load–unload cycle thesample was flooded. Preventing the strains in the 3directions the swelling pressures were measured infunction of time. Figure 4 presents the evolution ofthe swelling pressure, in the 3 directions, versus time,and the parameters for the swelling law.

To simulate this test, a FLAC model has been setup; the constitutive model is elastic with the sameproperties presented in Figure 4. The results fromFLAC simulation are presented in Figure 5 (stress inkPa versus time in hrs).

As the model is isotropic, the 3 components of thestress are identical in the simulation.

4.2 Huder-Amberg swelling test in elasticity

Figure 6 shows the swelling strain-time curves obtainedfor the different stages of loading in a Huder-Ambergswelling test on an anhydritic mudstone from deGypsum Keuper.

A FLAC model with 1 element has been set up; theconstitutive model is elastic with E � 1000 MPa and

v � 0.33. The results from FLAC simulation are pre-sented in Figure 7 (strain in % versus time in days)against the results of the test.

4.3 Huder-Amberg swelling test in plasticity

The objective now is to validate the algorithm againsta test in which the strength of the sample is exceededand therefore, plastic strains develop in addition tothe elastic and swelling deformations.

The test was carried out following the Huder-Amberg procedure. The initial vertical load (appliedin two cycles) is 15 MPa; the sample is then floodedand unloaded to a vertical pressure of 6.5 MPa. Theresults for this test are presented in Figure 8.

The elastic constants of the material areE � 2800 MPa and v � 0.33, and swelling parame-ters are Kq � 6.4% and �0 � 89.2 MPa.

59

(16)

Figure 4. Swelling pressure test (in Wittke 1999).

0

100

200

300

400

500

0 5 10 15t (horas)

� (k

N/m

2 )

sx sy sz FLAC

Figure 5. Swelling pressure test simulation with FLAC.

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Wittke (1999) suggests that it is necessary to con-sider the plastic strain of the sample to reproduce thistest, as it is shown in Figure 8 where the back-analysiswith elastic–plastic stress–strain relationship fits bet-ter with the measured values than the back-analysiswith elastic stress–strain relationship. The plasticconstants are c� � 0, � 11° and � � 5.5°.

Figure 9 presents the results from the FLAC model,also for elastic and elastic–plastic behaviors, together

with the results from the tests and the values fitted byWittke (1999).

4.4 Combined swelling pressure and swelling strain test

This is a swelling test on a sample taken from GypsumKeuper. The test was carried out in a confined compression test apparatus (horizontal strains of thesample were prevented during the test) with boundaryconditions, which were variable with time for a periodof more than 14 years. The test sequence and the testresults are presented in Figure 10.

The description of the test, taken from Wittke (1999)is as follows. Phase 1 may be divided into 4 partialstages, form 1a to 1d. In phase 1a strains in the verticaldirection were also prevented and the vertical stress wasmonitored. After 2.7 years, a vertical stress of 4.2 MPa

60

Figure 6. Huder-Amberg swelling test in elasticity (inWittke 1999).

-5

-4

-3

-2

-1

00 5 10 15

t (días)

εzq (

%)

sz = 520 kN/m2 sz = 260 kN/m2 sz = 130 kN/m2 sz = 65 kN/m2 sz = 32.2 kN/m2FLAC sz=520 kPa FLAC sz=260 kPa FLAC sz=130 kPa FLAC sz=65 kPa FLAC sz=32.2 kPa

Figure 7. Huder-Amberg swelling test in elasticity simulatedwith FLAC.

Figure 8. Huder-Amberg swelling test in plasticity (inWittke 1999).

-24

-20

-16

-12

-8

-4

0

40.1 1 10 100

sz (MPa)

ez (

%)

Ensayo: Carga-descarga inicial Ensayo: hinchamientosWittke elástico Wittke plásticoFLAC elástico FLAC plástico

Figure 9. Huder-Amberg swelling test in plasticity simu-lated with FLAC.

09069-07.qxd 08/11/2003 20:44 PM Page 60


was reached. Starting phase 1b it was allowed for asmall vertical strain (that is not recognizable in Figure10 because of the chosen scale) that results in a reduc-tion of the vertical stress to approximately 3.8 MPa.

Following to this, the vertical deformation of thesample was again prevented, and consequently, thevertical stress increased again to the same value of4.2 MPa.

The course of phase 1c was equivalent to the one ofphase 1b. During phase 1d it was allowed for a verticalstrain slightly larger than during the preceding phases.At the beginning this led to decrease the vertical stressto less than 0.5 MPa. Subsequently, the vertical stresswas increased to 2.5 MPa over a period of 0.3 yearwithout stabilization of the vertical stress.

During phase 2 of the test, the vertical stress waslowered to 0.5 MPa keeping it constant for more than5 years. The vertical strain was measured as a functionof time. Phase 2 was stopped after a vertical strain ofapproximately 28% had occurred without stabilizationof the deformations.

During the phase 3 of the test, a further increase of the vertical strain was prevented and the increaseof the vertical stress was registered as a function oftime. The slope of the stress–strain curve decreasecontinuously with time and after a period of 5.7 years avertical stress of 4 MPa was measured.

Wittke (1999) reproduced this test with the elastic–plastic properties presented in Table 2.

For the swelling parameters, Wittke (1999) uses the following values, �0 � 16 MPa and Kq � 15%.Nevertheless, regarding on the kinetics of the swelling,Wittke found necessary to change the swelling timeparameter during the course of the test. The parametersproposed are presented in Table 3.

As Wittke (1999) refers, to reproduce accurately thephase 3 of the test it is necessary to reduce the value

of the coefficient avp from 40 year1 to 2 year1, thatis equivalent to a reduction of the permeability of thesample during phase 3 due to the increment of thevertical stress in this phase of the test.

Changes in permeability of the sample during the load process are not taken into account in the formulation of the kinetic, thus Wittke (1999) suggeststhat the coefficient avp should vary during the calcula-tion for an accurate simulation of the swelling process.

The test described in this paragraph has been sim-ulated with FLAC, considering the same parameters(in Table 2 and Table 3). Figure 11 presents the evo-lution of the vertical stress (in MPa) versus time (inyears), comparing the results from FLAC simulationagainst the test.

Figure 12 presents the evolution of the verticalstrain (in %) versus time (in years), comparing theresults from FLAC model against the test.

61

Figure 10. Combined swelling pressure and swelling straintest on a sample from Gypsum keuper (in Wittke 1999).

Table 2. Elastic–plastic parameters used in Wittke (1999)to reproduce the test.

Elastic Plastic

Parameter E (MPa) v c (MPa) (°) �(°)

Gypsum keuper 4000 0.2 0.65 30 30

Table 3. Swelling kinetics parameters used in Wittke(1999) to reproduce the test.

a0 ael avp maxEVP(year1) (year1) (year1) (%)

1a2 0.0018 0.0 40.0 0.13 0.0018 0.0 2.0 0.1

0

1

2

3

4

5

0 5 10 15t (años)

�z(

MP

a)

Ensayo FLAC

Figure 11. Combined swelling pressure–strain test simu-lated with FLAC; comparison of stresses.

09069-07.qxd 08/11/2003 20:44 PM Page 61


5 CONCLUSIONS

The formulation for the swelling behavior presentedby Wittke (1999) has been reviewed and a calculationalgorithm, based in this formulation, has been imple-mented in FLAC, via FISH routines. This algorithmallows the simulation of the swelling behavior withFLAC code.

The algorithm has been checked against differentswelling tests presented by Wittke (1999), and theresults from the model fit quite well to the results ofthe different tests. Therefore, these routines can be usedto simulate the swelling behavior of expansive groundsin real engineering problems.

Nevertheless, when using these routines to simu-late a swelling behavior, the following limitations ofthe formulation have to be remembered:

– The direction of the principal stresses does notchange during the swelling process.

– All the elements of the model are susceptible toswell; this means that the whole rockmass is satu-rated and the penetration of water is enough to per-mit the complete swelling of the anhydrite.

– The proposed kinetics describe adequately theswelling process when the strength of the rock isnot exceeded, but the parameters of the formula-tion need to be changed in case of large plasticdeformations.

REFERENCES

ISRM 1989. Suggested Methods for Laboratory Testing ofArgillaceous Swelling Rocks. In Int. J. Rock Mech. Min.Sci. & Geomech. Abstr, Vol. 26, No. 5: 414–426.

Huder J. & Amberg G.1970. Quellung in Mergel, Opalinustonund Anhydrit. Schweizer, Bauzeit, 83: 975–980.

Noorany I., Frydman S. & Detournay C. 1999. Prediction ofsoil slope deformation due to wetting, In Detournay & Hart(eds), FLAC and Numerical Modeling in Geomechanics:101–107. Rotterdam: Balkema.

Saïta A., Robert A. & Le Bissonnais H. 1999. A SimplifiedFinite Element Approach to Modeling Swelling Effectsin Tunnels. In Alten et al. (eds), Challenges for the 21stCentury: 171–178. Rotterdam: Balkema.

Steiner W. 1993. Swelling Rock in Tunnels: Rock Charac-terization, Effect of Horizontal Stresses and ConstructionProcedures. In Int. J. Rock Mech. Min. Sci. & Geomech.Abstr. Vol. 30. No. 4: 361–380.

Wittke W. 1999. Stability Analysis for Tunnels. Fundamentals.Geotechnical Engineering in Research and Practice.WBI-Print 4. Ed. WBI Prf.Dr.Ing. W. Wittke. Consultingengineers for Foundation and Construction in Rock Ltd.Verlag Glückauf GmbH. Essen.

62

0

10

20

30

40

0 5 10 15t (años)

εz (%

)

Ensayo Cálculo FLAC

Figure 12. Combined swelling pressure–strain test simu-lated with FLAC; comparison of strains.

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63

Scenario testing of fluid-flow and deformation during mineralization: fromsimple to complex geometries

P.M. Schaubs, A. Ord & G.H. GermanCSIRO Exploration and Mining, Bentley, Western Australia, Australia

ABSTRACT: We present the use of FLAC3D in conjunction with Gocad and the CSIRO-developed software3DMACS to model deformation – fluid processes during mineralization. Simple idealized geological modelscontaining one planar fault are used to determine the effects of a number of deformation scenarios on volumestrain, pore pressure and resultant fluid flow patterns. Our results show that whether fluid flows up or down thefault is strongly controlled by the dip of the fault; steep faults cause fluids to flow up the fault, while more shal-low faults dilate and draw in fluid from the overlying sedimentary unit as well as the surrounding host rocks.Geometrically complex models, which more closely resemble the geology surrounding an ore deposit, areaimed at determining how the shape of a doubly plunging dome affects fluid flow patterns and the location ofsites of dilation. Areas of maximum dilation occur on the flanks of the dome near its crest. Complex meshes areconstructed using Gocad, which is then translated into FLAC3D using 3DMACS. This software may also be usedto set model parameters and properties, and for coupling FLAC3D to other numerical codes.

1 INTRODUCTION

An understanding of the relationship between fluidflow and deformation is important for determining howhydrothermal ore deposits form. Deformation maylead to the development, or reactivation, of structuressuch as faults, fractures and veins which may host oredeposits or may act as conduits for mineralizing flu-ids. Deformation may also induce volume changes(dilatancy) that further affect rock permeability andpore pressure gradients.

Here we present two methodologies for determin-ing the relative importance of certain parameters and processes during deformation and mineralization.The first method involves simplifying the geometryof the structures involved and is aimed at determiningthe effects of various parameters. In this way we areable to narrow down the effects of one parameter andreduce the uncertainty caused by geometrically com-plex models. For this reason the geometry of the modelis rather simple and number of zones in the model islow (12500). This allows us to run a large number ofmodels with different parameters in a short period of time.

The second type of model is aimed at testing theeffects of complex geometry, which more closelyapproximates that of the geology we see in the field.Here we are not concerned with changing a large

number of parameters and only require one or twomodels. Simple models are aimed at determiningwhat causes fluid to flow up or down faults and whereareas of high positive volume strain and low fluidpressure occur. This has implications for the locationof fluid mixing and mineralization if it is assumedthat there are two distinct fluid reservoirs within themodel. The geometrical complex model is concernedmore with the effect of the irregular shape of the geo-logical units on fluid flow patterns and the location ofsites of dilatancy.

2 MODEL BUILDING AND VISUALIZATIONTHROUGH THE USE OF ADDITIONALSOFTWARE

2.1 Simple and complex model generation

Simple single fault models are constructed usingFLAC3D “generate” commands. The use of FISH allows for the rapid construction of models with dif-ferent fault dips and strikes. Geometrically complexmeshes are constructed using Gocad. This is donefirst by building or importing tri-surfaces, which represent the contacts of the various geologic units.These surfaces are then used to “distort” the initiallyorthogonal and regular Gocad mesh (stratigraphic grid)

09069-08.qxd 08/11/2003 20:18 PM Page 63


so that the zones become parallel to the surfaces. TheCSIRO-developed software 3DMACS (Fig. 1) is thenused to import this model, along with a set of proper-ties, into FLAC3D.

2.2 The 3DMACS software suite

Primarily, 3DMACS is used for the importation,parameter-selection/editing and running of 3D geolog-ical models. It is a suite of software modules that at its core, leverages an XML data model. It harnessesvarious vendor-provided software, such as FLAC3D,to provide the background simulation capabilities.Overall, it provides the following functionality:

– Allows for the importation of 3D models such asthose produced by Gocad or FracSIS.

– Can “couple” a simulation across 4 phenomeno-logically distinct domains: mechanical/deformation,thermal, fluid and chemical. Currently FLAC3D

and FastFlo (a CSIRO package for partial differentialequation solving) are used to provide modelingacross these domains.

– Allows the user to set group properties importedfrom an external properties database, which canthen be edited by the user. These properties can befrom any of the 4 domains above.

– Allows the user to set model parameters and choosevisualization outputs.

– Due to its underlying XML character, users canuse a web browser (or the built-in 3DMACS GUI)from any machine connected to the internet and runtheir simulations remotely via 3DMACS. Multipleprocesses can be distributed amongst variousmachines.

– Allows for the storage of all user parameters andselected properties within a nominated repository,so that the user can re-run prior defined problems.

The above functionality allows the simulation to befully specified within the user-domain, rather than theprocess domain, which normally requires specializedknowledge of syntax and macro languages such asFISH. By providing basic problem “templates” forscenarios such as mechanical/fluid, mechanical/fluid/thermal and mechanical/fluid/chemical modeling,

64

Figure 1. Screen shot of web-browser interface of 3DMACS showing how properties and boundary conditions are applied.

09069-08.qxd 08/11/2003 20:18 PM Page 64


users without particular expertise in FLAC3D can stillbuild and run a model. However, expert users caninteract directly with the underlying processes andoverride any presets set in the templates.

As 3DMACS has the ability to couple FLAC3D toother software packages, we are able to create modelswhich simulate deformation, fluid flow, thermal andchemical processes, all of which may be important formineralization.

2.3 Visualizing results

3DMACS provides for the visualization of FLAC3D

results in Gocad or the commercial software FracSIS,via the export of scalar and vector data as 3D point-cloud sets. Gocad allows for the creation of isosur-faces from scalar point data. Both Gocad and FracSIScan be used for volume rendering of scalar data.FracSIS also allows the user to control the opacity ofcertain color values in both scalar and vector data. By“hiding” certain values we are able to see inside theFLAC3D model more easily and are not required touse cross-sections or cut planes.

Using FISH from within FLAC3D, we are also ableto create VTK files of scalar and vector data, whichare used by the freeware software MayaVi. MayaVi isable to visualize isosurfaces, and scalar and vector cutplanes as well as fluid flow vectors so that their colorvaries with magnitude.

All of these software packages are able to createVRML files which, given the appropriate plug-in,allows one to use a web-browser to view results.

3 SIMPLE FAULT MODELS

3.1 Model setup, properties and boundaryconditions

In this group of models we present a number of sce-narios with a simple geometry. The initial model ismade up of a simple fault region bounded by steeplydipping hangingwall and footwall rocks. These rocksare truncated by a horizontal interface and flat-lyingsedimentary unit (Fig. 2). We test different orientationsof far-field stresses, various dip and strike angles, forthe fault and different hanging wall and footwall per-meabilities (Fig. 3). The types of deformation appliedinclude (Fig. 4):

– compression and extension, where the bottomboundary is fixed and initial velocities are horizontaland perpendicular to the left and right boundaries,

– strike slip, where initial velocities are horizontaland parallel to the left and right boundaries but inopposite directions,

– reverse and normal movement, where initial veloc-ities are parallel to the dip of the fault and the base

65

Figure 2. Typical simple fault model. Arrows indicateapplied fluid discharge.

Figure 3. Examples of initial geometries of simple faultmodels.

Figure 4. Different styles of deformation applied to simplefault models. a) compression, b) extension, c) dextral strike-slip, d) sinistral strike-slip, e) reverse movement, f) normalmovement, g) transpression, and h) transtension.

09069-08.qxd 08/11/2003 20:18 PM Page 65


of the model is allowed to move in the vertical direc-tion, and

– transpression and transtension, which are similarto the reverse and normal models but contain astrike slip component of movement.

By changing the dip and strike of the fault we havea number of scenarios which range from a model witha shallow dipping faults with a dip of 30° and com-pression at right angles to the strike of the fault to amodel with a steeply dipping fault (60°) where thecompression direction is at 45° to the strike of the fault.

Constant fluid fluxes of 1m/yr are applied at thebase of the fault and the left boundary of the sedi-mentary unit. Permeability is isotropic and remainsconstant during deformation. Mechanical anisotropyis modeled using the ubiquitous joints constitutivemodel. In the sedimentary unit these are oriented hor-izontally and represent bedding, while in the base-ment units they are oriented roughly parallel to thecontacts of the units and represent a pervasive cleavage.These fabrics are given 90% of the strength (cohe-sion, tensile strength) of the rock type. Mechanicalproperties are listed in Table 1. The size of the modelvaries depending on the dip of the fault. All modelsare 2 km tall (z-direction) and 2 km deep (y-direction)but the width (x-direction) changes. In all cases thebottom of the fault is a minimum of 1500 m awayfrom either boundary. All models are deformed to 5%shortening or the equivalent amount of displacementfor those models with a strike-slip component.

3.2 Results

3.2.1 Pore pressureModels which have an extensional component of deformation (extension – pure shear, normal faulting,transtension) cause the greatest decrease in pore pressures because they have the greatest dilation(positive volume strain). The normal and transtensionmodels also have the steepest pore pressure gradientsand therefore fluid flow rates in the fault are highest inthese models. Pore pressure at the bottom of all modelsis similar; however, in the models with a component

of extension pore pressure is 30 MPa lower than those,which are essentially compressional. In all models,contours of pore pressure in the sedimentary unit slopetowards the right due to the application of a fluid flux(discharge) at the left boundary.

3.2.2 Volume strainIn all models the fault region is an area of high positivevolume strain (dilation) and a zone of significant dila-tion propagates from the tip of the fault into the sedi-mentary (Fig. 5). In the compression, reverse andtranspression models this zone is oriented roughlyparallel to the strike of the fault. In the extension, nor-mal and transtension models this zone is much steeper,and in the strike-slip models it is close to vertical.

Positive volume strain in the fault is greatest inmodels with an extensional component. In the reversemodel only the fault is a region of significant positivevolume strain and therefore is also a region of lowpore pressure relative to the other basement units. Thetranspression model is similar; however, it containsregions of dilation in both the hangingwall and foot-wall. In the normal and transtension models the faultis also a region of significant dilation (higher than thereverse and transpression models). As with volumestrain (dilation/contraction) the fault region recordsthe highest shear strain in all models. The locationand orientation of the zones of high shear strain arecoincident with those of significant dilation in allmodels.

66

Table 1. Mechanical and fluid flow properties used in simple fault models.

Property Units Sandstone Granite Fault Pelitic gneiss

Density kg/m3 2400 2700 2600 2600Bulk modulus Pa 2.40E 10 5.0E 10 9.5E 09 1.9E 10Shear modulus Pa 2.60E 10 3.0E 10 9.6E 09 2.0E 10Cohesion Pa 2.70E 07 4.0E 07 1.8E 07 3.5E 07Tensile strength Pa 1.20E 06 2.0E 07 2.8E 06 5.5E 06Friction angle deg 28 30 15 20Dilation angle deg 4 6 5 5Permeability m2 1.00E � 14 1.01E � 16 2.02E � 15 2.02E � 15

Figure 5. Volume strain increment and fluid flow vectors incompression model (cross-section view through middle ofmodel). Maximum fluid flow velocities are 2.02 � 10�8m/s.

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3.2.3 Fluid flow vectorsFluid flows towards the center of the fault in the com-pression, reverse and transpression models (Fig. 6).This is a result of the fault being an area of significantdilation, and low fluid pressure, relative to the sur-rounding rocks. In the extension, normal and transten-sion models, fluid flows up and out of the fault intothe hanging wall. This occurs despite the fault beingan area of higher positive volume strain (dilation)than the surrounding rocks. Fluid flows up becausepore pressure gradients are much steeper than in thecompressional models. In the strike-slip models fluidflows up and parallel to the boundaries of the fault. Inthe extension, normal and transtension models fluidon the right side of the sedimentary unit flows to theright towards the zone of significant dilation whichpropagates from the tip of the fault.

3.2.4 Dip and strike of faultThe dip angle of the fault strongly controls whetherfluid flows up and parallel to the fault boundaries.Models with the fault dipping 60° allow fluid to flowup the fault. Fluid will flow up a steep fault whetherthe footwall rocks are quartzite or pelitic gneiss.Similarly when the fault dips 45°, fluid flows into thefault from the foot and hanging walls whether thefootwall rocks are quartzite or pelitic gneiss. In mod-els where the fault is steep (60°), the fault does notdilate as much and therefore pore pressure does notdecrease in the fault as much as it does in modelswhere the fault has a shallower angle. Decreasing thedip angle of the fault to 30° causes the fault to dilatemore than models where the fault dips at 45°. Thishowever does not significantly affect the pore pressurevalues in the fault and fluid still flows towards thecenter of the fault from the hanging and footwalls.Moderate to shallowly dipping faults are able to dilate

more because they are oriented at an angle which isparallel to the direction of maximum compression.Steeply dipping faults and those oriented normal tothe maximum compression direction are more likelyto contract.

In models where the compression direction is nor-mal to the strike of the fault, the dip of the fault haslittle affect on the orientation of the high strain zone,which propagates into the sedimentary unit. In allmodels where the compression direction is normal tothe strike of the fault, this zone of high strain takes ona dip of 45°. In models where the fault dips 60° andthe hanging and footwall are both pelitic gneiss theorientation of the high strain zone in the basement isalso 45°. When the footwall rock type is made forrigid, the high strain zone in the basement is nearlyparallel to the dip of the fault. In models where thefault dips 30° and the high strain zone and the foot-wall is more rigid the high strain zone in the basementis parallel to the fault. This high strain zone becomessteeper (close to 45°) in the sedimentary unit.

Changing the orientation of the fault with respectto the model boundaries and direction of compressionresults in different orientations for the high strainzones. In the models where the strike of the fault is ori-ented at 45° to the maximum compression direction(and the dip is 60°), close to the fault the high strainzone is parallel to both the dip and strike of the faultin both the basement units and the sedimentary unit.Away from the fault zones, high strain zones formwith a strike normal to the maximum compressiondirection and a dip of 45°. When the angle betweenthe strike of the fault and the maximum compressiondirection is increased to 67.5°, the orientation of thehigh strain zone in the basement is nearly parallel tothe fault. As this zone propagates into the sedimen-tary unit its orientation rotates towards a strike whichis normal to the maximum compression direction anda dip closer to 45°. Therefore, both the strike and dipof the fault, with respect to the maximum compres-sion direction as well as the strength of the rocks, maycontrol the orientation of the high volume and shearstrain zones.

In models where the direction of maximum com-pression is oriented less than 90° to the strike of thefault, fluid flow vectors change along a line stretchingfrom the top west end to the bottom east end of thefault. On the west side flow is directed up and out intothe hanging wall, while on the east side fluid flowsdown and into the footwall side of the fault.

Increasing the permeability of the fault marginallydoes not change the values of volume strain or porepressure in a significant manner. Fluid flow patternsremain the same however fluid flow velocities areincreased slightly.

Decreasing the strength of bedding and cleavagefabrics from 90 to 75% of the strength of the host

67

Figure 6. Fluid flow vectors in and around the fault (compression model).

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rocks has little effect on the fluid flow patterns. In themodel where the joint strength is 90% of the host rocksstrength, slip along bedding planes occurs within thesedimentary unit above the fault and may indicate thatsome flexural slip has occurred. In the model wherejoint strength is 75% of the host rocks strength, slipalso occurs along bedding planes in the sedimentaryunit as well as along cleavage planes within the fault.

4 COMPLEX GEOMETRIES

4.1 Model setup, properties and boundaryconditions

The geometry of the model is reasonably complex(Fig. 7) and contacts between different units aresmooth curved surfaces. The objective of this modelis to determine what affect the shape of a basalt domehas on fluid flow patterns and the position of regionsof dilation in relationship to the formation of golddeposits. The model is made up of rigid doubly plung-ing basalt dome which is blanketed by a thin weakaltered metasedimentary unit and surrounded by amoderately stiff metamorphic rocks (Fig. 8). Mechan-ical properties are listed in Table 2. Deformation isapplied so as to simulate horizontal compression per-pendicular to the long axis of the dome.

4.2 Results

The altered metasedimentary unit contains regions ofnegative volume strain (contraction) on the flanks ofthe basalt dome where the dip is steep and at a highangle to the compression direction. Towards the top of the dome (but not at the crest) the weak alteredmetasedimentary unit contains regions of high posi-tive volume strain (dilation) above the areas of con-traction (Fig. 9).

This causes fluid flow rates to be highest close tothe top of the dome where areas of contraction andmaximum dilation are in close proximity (Figs. 10 &11). Contraction occurs within the matrix above thehighest point of the dome. Regions of high positivevolume strain are also regions that have failed in tension. These areas are more likely to have formedquartz veins, which commonly host gold.

5 CONCLUSIONS

FLAC3D has been used to test the effects of fault andfar-field stress orientation and the shape of irregularlyshaped bodies on fluid flow in regions of mineraliza-tion. In geometrically simple models with a single pla-nar fault the results of the models show that a lowangle fault with permeability similar to the surroundinghost rocks causes the fault to dilate and fluid to flow

down from the sandstones into the fault. Steeply ori-ented faults, strike-slip deformation and high perme-ability faults cause fluid to flow up the fault. This has implications for the location of fluid mixing and

68

Figure 7. Outline of basalt unit in geometrically complexmodel (in FLAC3D).

Figure 8. Cross-section through center of model showingoutline of the main basalt dome in light grey, the thin alteredmetasedimentary unit in dark grey and the surroundingmetamorphic matrix in white.

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mineralization, if it is assumed that the horizontalinterface represents a boundary between two distinctfluid reservoirs.

In the geometrically complex model of a basaltdome, the area of maximum dilation occurs on theflanks of the dome near its crest. Areas of maximumpositive volume strain are coincident with maximumfluid flow velocities and occur within the weak alteredmetasedimentary unit which blankets the dome. Sites

69

Table 2. Mechanical and fluid flow properties used in geometrically complex model.

Property Units Basalt Matrix Altered unit

Density kg/m3 2700 2700 2700Bulk modulus Pa 5.00E 10 4.0E 10 3.0E 10Shear modulus Pa 3.00E 10 2.0E 10 1.0E 10Cohesion Pa 4.00E 07 3.0E 07 2.0E 07Tensile strength Pa 2.00E 07 1.0E 07 9.0E 06Friction angle deg 30 25 20Dilation angle deg 2 3 3Permeability m2 1.00E � 16 1.00E � 15 5.00E � 15

Figure 9. Isosurface of high positive volume strain (black)occurs on flanks of the basalt within the weak alteredmetasedimentary unit. Surface of basalt exported fromGocad is shown in grey. Visualized in MayaVi.

Figure 10. Fluid flow vectors as visualized in FracSIS.Vectors are shaded using greyscale where black is highestand white is lowest fluid velocity. Only the highest valuesare shown (others remain transparent). The highest valuesare coincident with areas of high positive volume strain onflanks of the basalt within the weak altered metasedimentaryunit. Surface of basalt is shown in grey.

Figure 11. Cut plane of fluid flow vectors through highestportion of the basalt dome. Surface of basalt is shown ingrey. Visualized in MayaVi.

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of high volume strain or dilation are likely sites ofquartz vein formation and gold mineralization.

GoCAD has been used to construct models of significant geometrical complexity and the CSIRO-developed software 3DMACS has been used to trans-late the resultant mesh to FLAC3D. Numerical model-ing results are visualized in either Gocad, MayaVi orFracSIS any of which allow for the creation of isosur-faces of scalar data and the export of VRML files.

ACKNOWLEDGEMENTS

We would like to thank Irvine Annesley, MichelCuney, Jon Dugdale, Nick Fox, Peter Hornby,Catherine Madore, Phillipe Portella, Dave Quirt, TimRawling, Dave Thomas, Chris Wilson and RobWoodcock for their advice and input into the modelspresented here and with help visualizing the results.

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71

Constitutive models for rock mass: numerical implementation,verification and validation

M. SouleyINERIS, Ecole de Mines de Nancy, Nancy Cedex, France

K. SuANDRA, Châtenay-Malabry, France

M. GhoreychiINERIS, Parc Technologique ALATA, Verneuil-en-Halatte, France

G. ArmandANDRA, Laboratoire de Recherche Souterrain Meuse/Haute-Marne, Bure, France

ABSTRACT: This paper deals with numerical implementation of non-linear constitutive models of rock massand its verifications and validations. In the 3-dimensionnal code, FLAC3D, an elasto-damage-plastic model(damage is approached through the theory of plasticity) for Hoek-Brown media has been implemented.Simulations of triaxial compression tests provide a verification of the numerical implementation with a goodagreement between predictions and theoretical values of peak and residual strengths. The applicability of theimplemented model to predict the damage and/or failure development around a circular opening is checked.Finally a validation of poroplastic calculations based on the drainage of a cylindrical hole in poroplastic mediais achieved by comparison to an existing semi-analytical solution.

1 INTRODUCTION

Analysis of stresses and displacements around under-ground openings in rock mass is required in a widevariety of civil and geotechnical, petroleum and min-ing engineering problems such as tunnels, boreholes,shafts, disposal of radioactive waste and mines. Inaddition, an excavation damaged zone (EDZ) is gen-erally formed around underground openings exca-vated in rocks in relation to high in situ stresses and/orhigh anisotropic stress ratios even without blasting.The mechanical and hydraulic properties are thenchanged within EDZ. The failure mechanism in thedamaged zone is the initiation and growth of cracksand fractures, and is directly related to the constitu-tive behavior of the rock mass. Several experimentalstudies on rocks have shown that there are many dif-ferent mechanisms through which cracks can be initi-ated and grown under compressive stresses (Wong1982, Steif 1984, Martin & Chandler 1994, etc.).Indeed, irreversible deformations and failure of rockssubjected to compressive stresses occur through

progressive damage as microcracks initiate and growat small scale and coalesce to form large-scale fracturesand faults. The involved mechanisms include slidingalong pre-existing cracks and grain boundaries, porecollapse, elastic mismatch between mineral grains,dislocation movement, etc.

In the model considered in this study, the initiationand growth of cracks as well as failure and the post-peak behavior are approached through the theory ofplasticity. Furthermore, the transition between thebrittle failure and the ductile behavior depending on themean stress is generally observed on rock samples.

The purpose of this paper is to present: (a) a numer-ical implementation of an elasto-damage–plastic modelobeying to the Hoek-Brown criterion and taking intoaccount the brittle/ductile transition, (b) the corre-sponding verification based on simulation of triaxialcompression tests and the prediction of the extent ofdamaged/failed zone around a hypothetical circularopening, (c) validation of poroplastic calculations basedon a variant of the previous implemented model andan existing semi-analytical solution.

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2 MECHANCICAL SHORT TERM BEHAVIOR

2.1 Brief mathematical description

Based on several triaxial laboratory tests performedon the argillite rock samples, the typical characteris-tics of stress–strain curves are displayed in Figure 1:

Phase 1: linear isotropic and elastic behavior aftera short non-linear phase corresponding to the closureof microcracks;

Phase 2: strain-hardening in the pre-peak regioncorresponding to the initiation and the growth ofmicrocracks assumed to be described by plasticity,contrary to the concept of effective stress and thehypothesis of strain equivalence (Lemaitre 1995, Ju1989) that is generally used;

Phase 3: softening after the peak (failure) associatedwith a progressive loss in material cohesion and then adecrease in strength;

Phase 4: residual phase where the rock strengthremains practically constant.

Based on these observations, a constitutive modelfor this material was firstly developed in the frameworkof the European project: EURATOM MODEX-REPand recently compiled by Su (2003).

The features of this model are:

(a) linear elasticity to model the Phase 1;(b) damage initiation and growth are approached by a

strain-hardening based on Hoek-Brown criterionwhere the Hoek-Brown constants and the uniaxialcompressive strength are plastic strain dependent;

(c) the peak, post-peak (Phase 3) and residual (Phase4) are also based on Hoek-Brown criterion withrespect to brittle/ductile transition in accordancewith the experimental data.

For instance, the initiation of damage (Fsend), the

peak (Fsrup) and residual (Fs

res) strengths are given by:

(1)

where mend, send and mrup, srup are Hoek-Brown con-stants respectively corresponding to onset of damageand the peak; �c

end and �crup are uniaxial compressive

strength at the onset of damage and peak; � � uniaxialresidual strength; �3

b�d � confining pressure for brittle/ductile transition; �1 and �3 � major and minorprincipal stresses (compressive stress is negative and�1 � �2 � �3).

2.2 Constitutive equations

In order to obtain a simple but general constitutivemodel, an extended Hoek-Brown yield function is used.The general form of the yield function is expressed inthe following equation:

(2)

where � � softening flow function (parabolic formwith respect to the internal plastic variable, � in phase3, and null elsewhere); m, s � Hoek-Brown constants(linearly varying with � in phase 2, and constants in phase 3 and 4); �c � uniaxial compressive strength(linearly varying with � in phase 2, and constant inphase 3 and 4).

It is assumed that the material damage (hardening)and failure (up to the peak) depend on the generalizedplastic strain, �:

(3)

where d��

p � increment of total damage/plastic straintensor in phases 2, 3 and 4.

For simplicity, an associated flow rule is used (theplastic potential is identical to the yield function givenin Equation 2). In addition, in order to take into accountthe geometry of stresses (compression differing toextension), the previous yield function is generalized

72

41 2 3

Axial strain( ε1)

ResidualOnset of damage

Peak

Dev

iato

ric

stre

ss (

σ 1-σ

3)

Figure 1. A typical stress–strain curve provided by a triaxialtest.

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in terms of the three stress invariants (J1, J2, J3).Then, principal stresses are expressed in terms of meanstress (p), generalized deviatoric stress (q) and Lode’sangle (�) according to:

where

��

� stress tensor.Assuming that only small strain occurred, the total

strain increment, d��

, can be subdivided in elastic part,d�

�e and damage/plastic part, d�

�p:

(4c)

(5)

where � � plastic multiplier. This leads to:

(6)

The incremental expression of Hooke’s laws in termsof generalized stress and strain tensors has the form:

(7)

where C�

� isotropic linear tensor.The consistency condition, dFs(��) � 0 leads to:

(8)

By substituting Equations (4), (7) and (5) in Equation(8), we can express the plastic multiplier:

(9)

and then, the elasto-damage-plastic behavior:

(10)

2.3 Numerical implementation

In the three-dimensional explicit finite-difference code,FLAC3D, we have implemented the elasto-damage-plastic model described above. The main procedure issummarized below.

– The first approximation of stress tensor ��

I, is eval-uated by adding to the previous stress tensor thestress increments computed from the total strainincrements and the Hooke’s law.

– Computation of the corresponding mean stress pI,deviatoric stress qI and Lode’s angle �I correspond-ing to �

�I.

– Compute the generalized yield function, Fs(pI, qI,

�I). If �I verifies the yield function (Fs(��I) 0),

the derivatives of Fs with respect to ��

and � of m, sand �c (phase 2) or � (phase 3) with respect to �,are evaluated, and then Equation 10 is used tocompute the current increment of stress tensor.

– Current stress tensor, generalized plastic strain andflow functions are updated. It should be noted that,in FLAC3D, zones are internally discretized intotetrahedra and the current flow functions (damage/plastic) and, stress and strain tensors for each zoneare evaluated as a volumetric average for the zone.

This routine has been written in C and compiledas DLL file (dynamic link library) that can be loadedwhenever it is needed.

2.4 Verification and validation

In order to verify the implemented model, seven triaxialcompression tests with confining pressures of 2, 5,10, 12, 16, 20 and 25 MPa have been simulated. Theyare the part of the wide number of triaxial compressiontests used to characterize the non-linear behavior ofthe studied materials.

73

(4a)

(4b)

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The input parameters needed are derived fromstandard laboratory tests and based on the identificationof model parameters. These parameters are summarizedin Table 1. The isotropic elastic characteristics usedare E � 4500 MPa and � � 0.3. The generalized peakand residual plastic strain were also identified fromtriaxial laboratory tests. These are: �rup � 0.0063 and�res � 0.0175.

FLAC3D simulations are carried out on a singlezone of unit dimensions: fixed normal displacementsare applied to 3 perpendicular planes (bottom and twoperpendicular vertical faces). First, the three otherfaces of model are subjected to an isotropic stress statecorresponding to the given confining pressure.Secondly, the deviatoric stress is exerted by applyinga constant displacement rate at the top of model.

Figure 2 presents the deviatoric stress–axial and lat-eral strain curves for different confining pressure. Fromthis figure, we note that the post-peak behavior is con-fining pressure dependent: the transition stress betweenbrittle failure and ductile behavior is clearly markedand the numerical transition stress, �b�d

3, is approxi-mately about 20 MPa. These curves are qualitativelysimilar to the experimental ones (not reported herein).

Figure 3 shows a comparison in terms of the damagethreshold, the peak and residual strengths between thepredictions (corresponding values in Fig. 2) and theory(Eq. 1).

The match is very good as may be seen in this fig-ure, where numerical and analytical solutions coincide.More precisely, the relative error for peak and residualstrengths is less than 0.3%, and 0.9% for the onset ofdamage (dependent on the magnitude of loading at

the beginning of phase 2). This validates the numericalimplementation of the elasto-damage-plastic modelin FLAC3D.

2.5 Application to a circular opening

The aim of this section is to provide a verification ofthe implementation for non-triaxial stress paths and toshow numerically the ability of the implementedmodel to evaluate the extent of damaged and/or failedzones around a circular underground excavation.

We then consider an infinite circular opening in aninfinite elasto-damage-plastic medium initially sub-jected to an anisotropic initial stress in order to maxi-mize the deviatoric stress and then, the risk of damageand/or failure. The axis of gallery is parallel to thehorizontal minor stress leading to a maximum devia-toric stress in the gallery section.

The 2D-plane strain geometry as well as the initialin situ stresses and model geometry including a cir-cular gallery are plotted in Figure 4. The modelingsequence was performed as follows:

(1) the model without excavation was consolidatedunder the previous in situ stresses, and

(2) the circular excavation was carried out usingroller boundaries to the model sides respectivelyparallel to x- and z-axis for seeking symmetry.

74

05

10152025303540455055

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Dev

iato

ric

stre

ss �

1��

3(M

Pa)

Axial strain [-]Lateral strain [-]

25 MPa

10 MPa

5 MPa

2 MPa

20 MPa

16 MPa

12 MPa

Figure 2. Numerical result of triaxial compression tests.

010

20

30

40

50

60

70

80

-10 0 10 20 30

Principal minor stress �3 (MPa)

Prin

cipa

l maj

or �

1(M

Pa)

Peak [Eq. 1]Residual [Eq. 1]Onset of damage [Eq. 1]Peak - Flac3dResidual - Flac3dOnset of damage - Flac3d

Figure 3. Onset of damage, peak and residual strengths:numerical and analytical solutions.

Table 1. Values of input parameters.

Onset of damage Peak Residual

mend send �cend (MPa) mrup srup �c

rup (MPa) �3b�d � (MPa)

1 0.9 15 0.43 2.5 33.5 20.01 3

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Figure 5 shows the extent of damaged and failedzones around the circular opening. Damaged zonescorrespond to the set of elements where the behavioris associated within the pre-peak region; the peakstrength is not yet reached. Failed zones correspondto the model region where the peak strength wasreached: it should be noted that, in the direction of theinitial major principal stress (Ox), the extent of damageis about 17 cm: the radius of damaged zone in thisdirection is 1.06 times greater than the gallery radius.

The extent of failed areas is concentrated in thecompressive region where the maximum deviatoricstress is located. Then, in the direction of the initialminor principal stress (Oz), the maximum extent offailed zone reaches 26 cm, whereas the damaged areasare ranged between the failed and elastic regions withan extension about 1.5 m.

Finally, in the case of a circular opening created inan infinite elasto-damage-plastic medium initiallysubjected to an anisotropic initial stress, the damagedzone has an elliptical form (big axis is parallel to thedirection of the initial minor principal stress) similarly

to damage models based on the concept of effectivestress and assuming the strain equivalence (Shao et al.1998, Souley et al. 1998, Homand et al. 1998, Souleyet al. 1999, etc.).

Figure 6 shows the profiles of radial, orthoradialand axial stresses along two radial lines at 4.5° and85.5° with respect to x-axis as a function of the adi-mensional radial distance (r* � r/a; where r � radialdistance and a � gallery radius). In addition, the corre-sponding stresses for elastic calculations are also plot-ted. From the profile of orthoradial and axial stresses,one can distinguish three different regions (elastic,damaged and failed) through the slopes of curves.

At � � 4.5°, only one loss of slope can be noticedalong the profile of orthoradial stress: the correpond-ing radial distance (approximately 3.2 m) is in accor-dance with the previous investigation of damage extent.Up to this radial distance, the orthoradial stress pro-file is qualitatively similar to the elastic ones.

At � � 85.5°, the first failure of curve slopes isnoted at a radial distance of 3.3 m from the gallery wall,as well as for orthoradial stress profile (major principalstress) than axial stress (intermediate principal stress).This radial distance corresponds to the extent of failedzones in the direction of initial principal minor stress.The second failure of orthoradial and axial stress slopescan be shown at a radial distance of 1.5 m from thegallery wall. This corresponds to the damaged regionlocated between the failed zone and the elastic zone.

75

v

H

v

H

�v �v

�H

radius of gallery: 3 mmodel length : 30 mmodel heigth : 30 mmodel thickness : 1 mgallery axis : // à �h(// à Oy)

�v= h= 10.8 MPa

�H= 15.1 MPa

Figure 4. Model geometry, initial stress state and boundaryconditions.

Failed

Damaged

Figure 5. Extent of damaged and failed zones around cir-cular opening.

-20

-18

-16

-14

-12

-10

-8

-6

-4

-20

20 4 6 8 10

Stresses (MPa)

r*

orthoradial

radial

axial

(a)

-35

-30

-25

-20

-15

-10

-5

00 10

(b)

r*

Stresses (MPa)

radial

orthoradial

axial

2 4 6 8

Figure 6. Radial, orthoradial and axial stresses alongradial lines (a) at 4.5° (b) at 85.5° (elastic � lines; elasto-plastic � circles).

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3 HYDROMECHANICAL VALIDATION

Generally, validation of poroelastic calculations withFLAC3D for which the formulation of hydromechanicalcoupling is used within the framework of the quasi-static Biot theory, has been undertaken by severalexamples and studies:

– one-dimensional filling of an initially dry porousmedia compared with the analytical solution devel-oped by Voller et al. (1996);

– one-dimensional consolidation compared with theanalytical solution developed by Detournay &Cheng (1993);

– two-dimensional consolidation of a borehole in anelastic medium compared with the analytical solu-tion developed by Detournay & Cheng (1988).

Validation of hydromechanical coupling in theframework of poroplasticity is specific to each non-linear model and each numerical code. The aim of thissection is to provide an example of validation of poro-plastic calculations in Hoek-Brown media. This exam-ple concerns the drainage of an infinite medium by acylindrical hole for which a semi-analytical solution isdeveloped by Vouille et al. (2001). In this solution, themechanical model is based on the Hoek-Brown yieldfunction and can be viewed as an extension ofCarranza-Torres and Fairhust works and, also a partic-ular case of the previously described constitutive modelin the sense that neither damage (hardening in pre-peakregion) nor brittle/ductile transition are considered.

3.1 Brief mathematical description of mechanicalmodel

The mechanical behavior is described by an elasto-plastic model with a post failure softening phase. Themodel is based on the Hoek-Brown criterion withassociated plastic potential. The main characteristicsof this model are: (a) linear and isotropic behavior inthe pre-peak region; (b) peak strength governed by theHoek-Brown criterion; (c) a softening phase based on aHoek-Brown yield function and an associated flow rule;(d) a perfectly plastic behavior in the residual phase.

Assuming that compressive stress is negative and�1 � �2 � �3, the peak strength and residual strengthare given by:

(11)

(12)

where �c � uniaxial compressive effective strengthof the intact rock; m � peak value of Hoek-Brownconstant; � � residual strength parameter.

For softening phase, the yield function is assumedto be:

(13)

where � � softening internal variable, representingthe opposite value of the plastic strain �p

1 associatedwith the major principal stress �1; �R (0 � � ��R) � value of the softening internal variable forwhich residual phase is reached.

Finally, the potential function is given by:

(14)

This formulation slightly differs from the elasto-damage-plastic model detailed in section 2 by theabsence of hardening in the pre-peak region and brit-tle/ductile transition. Based on the previous imple-mentation, this variant of the elasto-damage-plasticmodel is implemented in FLAC3D. As verification, tri-axial com-pression tests were simulated. The resultsare shown in Figure 7. In addition, the correspondingnumerical residual and peak strengths are representedin Figure 8 and compared with the analytical expres-sions (Eq. 12 & 13). From Figure 7, it should be notedthat for a given level of confining pressure, the threephases (elastic before failure, softening for post-peakbehavior and perfect plastic for residual behavior) areclearly distinguished.

The match is very good as may be seen in Figure 8,where numerical and analytical solutions coincide.The relative error for strengths is less than 0.5%.

3.2 Definition of hydromechanical problem

Problem definition consists of a cylindrical hole cre-ated in an infinite poroelastoplastic medium initially

76

0

5

10

15

20

25

30

35

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Dev

iato

ric

stre

ss �

1��

3(M

Pa)

Axial strain [-]Lateral strain [-]

�3 = 20 MPa�3 = 15 MPa

�3 = 10 MPa

�3 = 5 MPa

�3 = 1 MPa

�3 = 0.5 MPa

Figure 7. Verification – simulated triaxial compression tests.

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subjected to a uniform and isotropic stress state and auniform pore pressure. The induced mechanical andhydraulic perturbations are examined during and afterexcavating. The main assumptions are:

– gravity forces are neglected;– mechanically, the medium behaves as an isotropic

and elastoplastic material according to the modeldescribed in §3.1;

– hydromechanical coupling process is expressed byBiot’s theory;

– hydraulic and mechanical boundary conditions at thehole walls are time-dependent: continuous reduc-tion of normal stress and pore pressure at the holeboundaries from their initial values to zero.

The geometry of this 1D problem is shown inFigure 9. It consists of a thick wall cylinder withinternal radius of 3 m and external radius of 30 m. Theinitial and boundary conditions are summarized inTable 2.

As previously mentioned, the hydraulic and mechan-ical boundary conditions along the inner wall are time-dependent. More precisely, the total radial stress �r

and pore pressure p, along the inner wall are expressedas follows:

(15)

(16)

where t � time; T � 1.5 � 106s represents the exca-vation duration.

A semi-analytical solution of this H-M 1D problemhas been developed in the framework of the Europeanproject: EURATOM MODEX-REP (Su 2002).

Finally, the geometry shown in Figure 9, initial andboundary conditions reported in Table 2 are used inour FLAC3D model. Hydromechanical properties areshown in Table 3, where E0 and �0 denote the drainedelastic properties; �h is the hydraulic conductivity; b the Biot coefficient; M the Biot modulus and �wthe specific weight of water.

For both semi-analytical and numerical solutionsthe required results are:

– the radial displacement;– the pore pressure;– the radial; orthoradial and axial effective stresses.

as a function of radial distance from the hole center (r ranged from 3 to 30 m) and time (ranged from 0 to 100 Ms) in this paper.

77

00

10

20

30

40

50

60

-10 -5 5 10 15 20 25 30

Peak [Eq. 11]

Residual [Eq. 12]

Peak - Flac3d

Residual - Flac3dMaj

or p

rinc

ipal

str

ess

�1

(MPa

)

Minor principal stress �3 (MPa)

Figure 8. Peak and residual strengths: numerical and ana-lytical solutions.

P9

PY

PZ0

PZ1

r3

r309º

3 m

0,1 m

30 m

Figure 9. FLAC3D geometry of the 1D problem.

Table 2. Initial and boundary conditions of the 1D problem.

Initial conditionsTotal stresses (MPa) �11.5 �ijPore pressure (MPa) 4.7

Boundary conditionsNormal displacement (P9; PY; PZ0; PZ1) nullHydraulic flux (P9; PY; PZ0; PZ1) nullRadial total stress at the outer radius r30 (MPa) �11.5Pore pressure at the outer radius r30 (MPa) 4.7Radial total stress at the inner radius Eq. 15Pore pressure at the inner radius Eq. 16

Table 3. Hydromechanical properties used in poroplastic-ity validation.

E0 (MPa) �0 �c (MPa) m �R �

5800 0.3 14.8 2.62 0.015 0.01

�h (m/s) b M (MPa) �w (kN/m3)

10� 12 0.8 6000 10

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Height radial distances are considered for output.They are: 3, 3.05, 3.1, 3.2, 3.5, 3.7, 5, and 10 m. Tentime periods are also considered for result output. Theinvolved times are: 1.2, 1.5, 1.6, 2.5, 10, 50, and100 Ms (million of seconds).

In the case of the semi-analytical, all of theserequired results are given at the previous radial distances. Because of displacements and pore pres-sure are gridpoint variables whereas stresses are zonevariables and evaluated at the zone centroid, numericalsolutions are checked at the following set of radialdistances:

– 3, 3.05, 3.1, 3.2, 3.5, 3.7, 5, and 10 m for radial dis-placement and pore pressure;

– 3.0125, 3.0625, 3.1125, 3.2125, 3.5625, 3.725,5.05, and 10.05 m (centroid of the closest zone) forstresses.

Therefore, small differences in the results of stressescompared to the semi-analytical solution are to beexpected.

3.3 Comparison with the semi-analytical solution

Figure 10 presents a comparison of normal displace-ment between the semi-analytical solution and thenumerical ones. This shows a very good agreementbetween both the solutions. In particular, the maximumof relative error between semi-analytical solution andFLAC3D results is about 0.7% and corresponds toradial distance inferior to 3.2 m and t 5 Ms. In theother cases, the relative error is about 0.2%.

Figure 11 illustrates the comparison of pore pressurebetween the semi-analytical solution and the numericalones. For a given radial distance, both numerical andsemi-analytical solutions are quantitatively and qualita-tively similar. In particular, it should be noted that someunderpressures (i.e. “unsaturated” zones corresponding

to negative pore pressure) are well reproduced bynumerical results for radial distance and time rangedrespectively from 3.05 to 3.2 m, and from 1.5 to1.6 Ms (corresponding to the start of full drainage).

Due to null and negligible values of pore pressurein the vicinity of the inner radius; relative errorsbetween semi-analytical and numerical solutions arenot evaluated for radial distance inferior to 3.7 m; sothe difference in results of pore pressure does notexceed 0.02 MPa. For radial distance superior to 3.7,the maximum relative error between semi-analyticaland numerical solutions is about 1.2%.

Comparison of radial and orthoradial effectivestresses between the semi-analytical and numericalsolutions is plotted in Figures 12 & 13. It should benoted that the profiles of principal effective stressesare qualitatively returned.

From a quantitative point of view and for a radialdistance superior to 3.1 m; the absolute error on the

78

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0

0 20 40 20 20 100r=3 - Analr=3,05 - Analr=3,1 - Analr=3,2 - Analr=3,55 - Analr=3,7 - Analr=5 - Analr=10 - Analr=3 - Flac3Dr=3,05 - Flac3Dr=3,1 - Flac3Dr=3,2 - Flac3Dr=3,55 - Flac3Dr=3,7 - Flac3Dr=5 - Flac3Dr=10 - Flac3D

Time (Ms)

Rad

ial d

ispl

acem

ent (

mm

)

Figure 10. Numerical and semi-analytical solutions: radialdisplacement.

-1

0

1

2

3

4

5

0 20 40 60 80 100

r=3 - Analr=3,05 - Analr=3,1 - Analr=3,2 - Analr=3,55 - Analr=3,7 - Analr=5 - Analr=10 - Analr=3 - Flac3Dr=3,05 - Flac3Dr=3,1 - Flac3Dr=3,2 - Flac3Dr=3,55 - Flac3Dr=3,7 - Flac3Dr=5 - Flac3Dr=10 - Flac3D

Time (Ms)

Pore

pre

ssur

e (M

Pa)

Figure 11. Numerical and semi-analytical solutions: porepressure.

-10

-8

-6

-4

-2

00 20 40 60 80 100

r=3 - Anal

r=3,05 - Analr=3,1 - Anal

r=3,2 - Anal

r=3,55 - Analr=3,7 - Anal

r=5 - Analr=10 - Anal

r=3,013 - Flac3D

r=3,063 - Flac3Dr=3,113 - Flac3D

r=3,213 - Flac3D

r=3,563 - Flac3Dr=3,725 - Flac3D

r=5,05 - Flac3Dr=10,05 - Flac3D

Rad

ial e

ffec

tive

stre

ss (

MPa

)

Figure 12. Numerical and semi-analytical solutions: radialeffective stress.

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orthoradial and radial is respectively about 0.3 and0.2 MPa; that corresponds to a relative error of 2%.However in the vicinity of the inner wall (3 m for thesemi-analytical solution and 3.0125 m in FLAC3D), themaximum difference between both solutions is 0.6 MPa.

In order to capture the magnitude of error in termsof stresses due to the difference in the radial distanceswhere the principal effective stresses were computedrespectively for semi-analytical and numerical solu-tions, the closed-form solution for prediction dis-placements and stresses around circular openings inelasto-brittle-plastic rock (based on Hoek-Brown cri-terion) recently developed by Sharan (2003) is used.This closed-form solution is only valid for the mechan-ical configuration.

For Sharan closed-form solution, the previous holegeometry, mechanical properties, mechanical initialand boundary conditions are used. In addition, it wasassumed that the 3 m-radius hole is instantaneouslyexcavated. Under these conditions, the orthoradial andradial stresses are calculated based on the closed-formsolution for these pairs of radial distances (in meters):3–3.0125, 3.05–3.0625, 3.1–3.1125, 3.2–3.2125, 3.5–3.5625, 3.7–3.725, 5–5.05 and 10–10.05. The maxi-mum of difference for each pair is about 0.2 MPa.

In relation to the previous discussion on the radialand orthoradial effective stresses, we can say that thedifference between the semi-analytical and numericalsolutions for radial distance superior to 3.1 m remainsin an acceptable order of magnitude while in thevicinity of the inner radius, numerical results derivedfrom FLAC3D can be ameliorated by increasing themesh density (unfortunately, this will considerablydecrease the FLAC3D hydraulic characteristic time, andthen increase the calculations duration).

Finally, the investigation of orthoradial and radialplastic strains (not reported herein) between the semi-analytical and numerical solutions leads to the fol-lowing comments.

– The plastic radius rp for the semi-analytical solutionis about 3.55 m whereas rp equals 3.7 in numericalmodel. However, for numerical solution the ortho-radial and radial plastic strain are respectively � 5and 12 �, and reached at the beginning of fulldrainage (t � 1.5 Ms). Note that similarly to thestress tensor, strain tensor and principal plasticstrains are zone variable.

– For radial distance ranged between the hole walland the elastic/plastic transition region, the maxi-mum relative error between the semi-analyticaland numerical solution is about 8%.

4 CONCLUSION

This paper presents numerical implementation ofnon-linear constitutive model of rock mass in thethree-dimensional code FLAC3D, as well as its verifi-cation and validation. Firstly, a non-linear elasto-damage-plastic model based on the Hoek-Brownfailure criterion and for which hardening in pre-peak(characterizing the material damage), softening (char-acterizing the post-peak behavior and the failure ofsample) is implemented in FLAC3D.

Simulation of triaxial compression tests at differ-ent level of confining pressure provides a verificationof the implemented model. The resulting curves dis-play four regions (elastic, damage in pre-peak, soften-ing in post-peak and residual phase) when the confiningpressure is below the transition stress, and three regions(elastic, damage and perfect plastic phase) under highconfining pressure. In addition, the onset of damage(limit between elastic/damage region), the peak andresidual strengths derived from these simulations arecompared with the theoretical envelops: the corre-sponding relative error does not exceed 0.3%.

The ability of the implemented model to predictthe damaged and failed regions around an undergroundexcavation is successfully tested. In this verification,a circular and an initial anisotropic stress (in order tomaximize the extent of damage and failure) are con-sidered. The extent of failed areas is concentrated at the gallery wall in the compressive region where themaximum deviatoric stress is prescribed (direction ofthe initial minor principal stress), whereas the damagedareas are ranged between the failed and the elasticregions. As a result, the damaged zone has an ellipticalform similarly to the prediction of damage models basedon the concept of effective stress (damage theory).

Secondly, a variant of the elasto-damage-plasticmodel, for which a semi-analytical solution of drainageof an infinite medium by a cylindrical hole exists, isused in order to validate the poroplastic calculationsin FLAC3D. The previous implementation has beenslightly modified for the variant version, and firstlytested on triaxial compression tests with a good

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r=3 - Analr=3,05 - Analr=3,1 - Analr=3,2 - Analr=3,55 - Analr=3,7 - Analr=5 - Analr=10 - Analr=3 - Flac3Dr=3,05 - Flac3Dr=3,1 - Flac3Dr=3,2 - Flac3Dr=3,55 - Flac3Dr=3,7 - Flac3Dr=5 - Flac3Dr=10 - Flac3D

Ort

hora

dial

eff

ectiv

e st

ress

(M

Pa)

Time (Ms)

Figure 13. Numerical and semi-analytical solutions:orthoradial effective stress.

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agreement between predicted peak and residualstrengths and theoretical ones.

Comparison of normal displacement between thesemi-analytical solution and the numerical ones indi-cates a very good agreement: the relative error is glob-ally about 0.2%. Also, for a given radial distance, both numerical and semi-analytical solutions arequantitatively and qualitatively similar in terms of pore pressure results with a neglected difference(0.02 MPa) compared to the initial field of pore pres-sure (4.7 MPa).

In terms of radial and orthoradial effective stresses,the semi-analytical and numerical solutions are qual-itatively the same. Far to the gallery wall, the stressdifference does not exceed 0.3 MPa (that correspondsto a relative error of 2%).

REFERENCES

Carranza-Torres, C. & Fairhurst, C. 1999. The elasto-plasticresponse of underground excavations in rocks massesthat satisfy the Hoek-Brown failure criterion. Interna-tional Journal of Rocks Mechanics and Mining Sciences.36(6): 777–809.

Detournay, E. & Cheng, A.H.-D. 1993. Comprehensive RockEngineering. Pergamon Press Ltd.

Detournay, E. & Cheng, A.H.-D. 1988. Poroelastic Responseof a Borehole in a Non-Hydrostatic Stress Field. Interna-tional Journal of Rocks Mechanics and Mining Sciences.25(3): 171–182.

Homand-Etienne, F., Hoxha, D. & Shao, J.F. 1998. A contin-uum damage constitutive law for brittle rocks. Computersand Geotechnics. 22(2): 135–151.

Ju, J.W. 1989. On the energy based on coupled elastoplasticdamage theories: constitutive modeling and computa-tional aspects. International Journal of Solids Structures.25(7): 803–833.

Lemaitre, J. 1985. A course on damage mechanics. 2nd edi-tion. Springer.

Martin, C.D. & Chandler, N.A. 1994. The progressive fail-ure of Lac du Bonnet granite. International Journal ofRocks Mechanics and Mining Sciences. 31(6): 643–659.

Shao, J.F., Chiarelli, A.S. & Hoteit, N. 1998. Modeling ofcoupled elastoplastic damage in rock materials. Interna-tional Journal of Rocks Mechanics and Mining Sciences.35(4–5): Paper No. 115.

Sharan, S.K. 2003. Elastic-brittle-plastic analysis of circu-lar openings in Hoek-Brown media. to appear inInternational Journal of Rocks Mechanics and MiningSciences & Geomechanics Abstracts.

Souley, M., Homand, F., Hoxha, D. & Chibout, M. 1999.Damage around a keyed URL excavation: change in per-meability induced by microcracks growth. In Detournay& Hart (eds), FLAC and Numerical Modeling in Geo-mechanics: 205–213. Rotterdam: Balkema.

Souley, M., Hoxha, D. & Homand-Etienne, F. 1998. Distinctelement modelling of an underground excavation using acontinuum damage model. International Journal of RocksMechanics and Mining Sciences. 35(4–5): Paper No. 6.

Steif, P.S. 1984. Crack extension under compressive load-ing. Engineering Fracture Mechanics. 20(3): 463–473.

Su, K. 2002. Analysis of the capacity of numerical models to simulate excavation in deep argillaceous rock, 5th EURATOM framework programme, MODEX-REPproject contract FIKW-CT2000-00029 – Deliverable 1,August 2002.

Su, K. 2003. Constitutive models of the Meuse/Haute-Marne Argilites, MODEX-REP project contract FIKW-CT2000-00029 – Deliverable 2&3, February 2003.

Voller, V., Peng, S. & Chen, Y. 1996. Numerical Solution ofTransient, Free Surface Problems in Porous Media.International Journal of Numerical Methods in Engi-neering. 2889–2906.

Vouille, G., Tijani, M. & Miehe, B. 2001. Hydro-mechanicaltheoretical problem: Drainage of an infinite medium by a cylindrical hole. In EC-5th EURATOM frameworkprogramme 1998–2000 MODEX-REP project: contractFIKW-CT-200-00029, NOT-EMP-01-02, Technical Note,fevrier 08.

Wong, T.F. 1982. Micromechanics of faulting in Westerlygranite. International Journal of Rocks Mechanics andMining Sciences. 19(1): 49–62.

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Slope stability

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83

A parametric study of slope stability under circular failure condition by a numerical method

M. Aksoy & G. OnceOsmangazi University, Mining Engineering Dept., Eskisehir, Turkey

ABSTRACT: Slope failures can cause delay in the production schedule and the loss of life and equipment. Inthis study, slopes excavated in very weak rock masses where expected failure mode is circular failure have beenstudied. The effects of slope height, slope angle, water saturation, cohesion, internal friction angle and densityon slope stability under circular failure conditions have been investigated by three methods: Hoek and Bray stabil-ity diagrams, Bishop’s simplified method of slices, and finite difference numerical code, FLAC3D (Itasca 1997).Safety factor calculations have been carried out for the various values of parameters and obtained values arecompared with each other. However, the main focus is on the results of the numerical modeling. The presenceof correlation between the studied parameters and the factors of safety obtained from numerical models hasbeen searched and the fitted equation has been given.

1 INTRODUCTION

Slope stability is one of the most important subjectsin mining and civil applications. In open pit mining,especially, the design of a stable slope has becomeimportant to meet the safety regulations in addition tothe profitable extraction of the deposit. This can beachieved by the proper selection of slope angle, shapeand height.

The factors governing the stability of an open pitslope can be listed as follows (Stacey 1968):

– Geological structure– Rock stresses and ground water conditions– Strength of discontinuities and intact rock– Pit geometry including both slope angles and slope

curvature– Vibrations from blasting or seismic events– Climatic conditions– Time

The failure mode of a pit slope is also determinedby these factors. It can be said that a pit slope isdesigned according to the failure mode expected tooccur (Sjöberg 1999). The main failure modes observedin slopes can be listed as:

– Plane failure– Wedge failure– Circular failure– Toppling failure

Factor of safety is used as an index to define theslope stability and it can be simply described as the ratioof the total resisting force to the total inducing force.

In this study, the investigation of circular (rotationalshear) failure usually observed in the altered rock orsoil slopes has been based on the effects of geomechan-ical properties of rock or soil and the shape of theslope on the slope stability. How the safety factor valuesare affected with the variation of the parameters valueshave been searched by three methods and calculatedsafety factors have been compared.

2 METHODS APPLIED IN THE STUDY

As mentioned before, safety factors have been calcu-lated by three different approaches:

1. Hoek and Bray stability diagrams2. Bishop’s simplified method of slices3. Numerical modeling in FLAC3D

It should be emphasized that the assumptions ofeach of these three methods are quite distinct andclearly stated in the literature. In fact, one of the maindifferences of these methods is that Hoek and Brayand Bishop’s simplified method of slices are basedon 2-dimension limit equilibrium analysis whereasFLAC3D is based on 3-dimension numerical analysis.Therefore, in order to compare the results of thesemethods, the location of critical failure surface

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determined from the chart given by Hoek and Bray(1981) was chosen as a basis for comparison purpose.In other words, safety factors have been calculated forthis critical failure surface by these three methods.However, the main focus of this study is on the resultsof numerical modeling.

Geomechanical properties of rock and the shape ofthe slope used in this study can be listed as:

– Cohesion– Internal friction angle– Density– Water condition– Slope angle– Slope height

All these factors have been taken as parametersand safety factors have been calculated. For all threemethods, the value of the parameter whose effect onsafety factor will be investigated has been changedwhile the other parameters have been kept constant.The parameters and values used are given in Table 1.

2.1 Hoek and Bray stability diagrams

Hoek and Bray have adopted an approach in which aseries of the slope stability charts have been presentedfor circular failure (Hoek & Bray 1981). These stabilitydiagrams have been used to find safety factor values.

2.2 Bishop’s simplified method of slices

For the safety factor calculations, models have beenformed in SLOPE/W program (Geo-soft, student edi-tion). The search for the critical failure surface couldbe carried out in the program. But as mentionedbefore, instead of this, these calculations have beendone for the critical failure surfaces whose locationshave been determined from the Hoek’s chart. A modelused in the analysis is shown in the Figure 1.

2.3 Numerical modeling in FLAC3D

According to the methodology proposed by Starfield &Cundall (1998), numerical modeling can be used todetermine how different variables affect the slope sta-bility. In this study, FLAC3D, a commercial finite differ-ence code by Itasca, was selected for the purpose of numerical modeling and analysis. It is a three-dimensional explicit finite difference program for engi-neering mechanics computations and it offers an idealanalysis tool for the solution of three-dimensionalproblems in geotechnical engineering (Itasca 1997).

2.3.1 Safety factor calculations in FLAC3D

FLAC3D does not calculate factor of safety directly (inversion 2.0). However, it can be done by writing a fishfunction. In this study, safety factor calculations for

numerical models have been performed by means of afish function written for this purpose (Aksoy 2001). Thesafety factor definition used has been based on the firstapproach proposed by Kourdey et al. (2001), but it hasbeen modified and these modifications are as follows:

– The mohr-coulomb failure criteria is directly used– The state of stress of zones are obtained from elastic,

isotropic models– Normal and shear stresses are calculated on the

critical plane of each zone

The stresses developed on the any zone in thenumerical model can be expressed in terms of �1 and �3and these stresses can be plotted on the mohr diagramas seen in Figure 2.

To make safety factor definition clear, it is explainedbelow in detail for the case in which the value ofcohesion is changed while the value of internal frictionangle is kept constant.

There will be a critical plane on which the avail-able shear strength will be first reached as �1 isincreased. The orientation of this critical plane for

84

Table 1. Parameter values used in all approaches.

Parameters Values Parameters Values

Cohesion (kPa) 50 Cohesion fully 5090* saturated (kPa) 90130 130170 170

Internal friction 20 Slope angle (°) 30angle (°) 25* 40

30 5035 60*40 70

Density (t/m3) 1.6 Slope height (m) 201.9 50*2.2 802.5* 1402.8 200

*Constant values.

Figure 1. A model formed in the SLOPE/W.

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each zone can be calculated from the Equation 1(Brady & Brown 1993):

(1)

where � � angle between critical plane and the hori-zontal; � � internal friction angle.

Normal and shear stresses developed in this planecan be expressed as follows:

(2)

(3)

where �n � normal stress; �1 � major principal stress;�3 � minor principal stress; � � angle between criticalplane and the horizontal; �st � shear stress.

For the critical plane, these equations are rewrittendue to sin2� � cos� and cos2� � �sin� (Brady &Brown, 1993):

(4)

(5)

where �st � shear stress; �1 � major principal stress;�3 � minor principal stress; � � internal frictionangle; �n � normal stress.

As it can be seen on Figure 3, mohr failure envelopesthat have different cohesion values with the same inter-nal friction angle are drawn. Shear stress values at theintersection points (A1, A2) of mohr failure envelopeswith the A2 � D line are accepted as the shear strengthvalues (�s1, �s2) of the zone depending on the value ofcohesion and internal friction angle. And these shearstrengths can be calculated from the Equation 6:

(6)

where �si � shear strength; �n � normal stress;� � internal friction angle.

Local safety factor is described as the ratio of shearstrength to shear stress developed on the critical planefor each zone:

(7)

where F1 � local safety factor; �si � shear strength;�st � shear stress.

For the calculation of general safety factor Fg, thezones on the critical failure surfaces whose locationshave been determined from the Hoek’s chart are usedand general safety factor defined as:

(8)

where Fg � general safety factor; Fli � local safetyfactor of the zone I; and vi � volume of the zone i.

This approach is used for all parametric studies.But, in the case of different internal friction angle val-ues, the orientation of critical plane for which shearand normal stresses are calculated is taken as constantat the value found for internal friction angle 25°. Inother words, it is assumed that the orientation of criticalplane has not been affected by the change of internalfriction angle value. The reason for this is to comparesafety factor values at the same normal stress level.

3 NUMERICAL MODEL STUDIES ANDPARAMETRIC ANALYSIS

Rock mass has been assumed as isotropic and homoge-neous material through the study and the stresses in thenumerical models have been initialized by taking theslope geometry into consideration. The general slopemodel and initial stress state is given in the Figure 3.In addition to cohesion, internal friction angle anddensity properties given in Table 1, the other material

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Figure 2. Mohr diagram.Figure 3. The general slope model and initial verticalstress state in FLAC3D.

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properties used in numerical modeling are given inTable 2. k ratio, the ratio of horizontal stress to verticalstress, has been found from the Equation 9:

(9)

where �h � horizontal stress; �v � vertical stress;� � Poisson ratio.

The procedure followed in numerical modelingcan be described as follows; first, material model forall numerical models has been selected as elastic,isotropic model. Safety factors have been calculatedby using the results of these model runs. After safetyfactor calculations, all numerical models have beenmodified in such way that their material models havebeen changed from elastic, isotropic model into themohr-coulomb plasticity model. Modified modelshave been run again and evaluated to determine thefailure condition (Aksoy 2001).

As it can be seen in Table 2, it has been assumedthat rock mass has no tensile strength. However, inorder to observe the effect of tensile strength on thesafety factor values of numerical models, new modelshave been formed. In these models, tensile strength ofrock mass has been calculated from the Equation 10(Brady & Brown 1993).

(10)

where �t � tensile strength; c � cohesion; � � internalfriction angle.

It has been also considered that changing thePoisson ratio taken as 0.25 for all models will changethe magnitude of the horizontal stress and this willdifferentiate the stress state developed within theslope. As a result, this will affect the safety factor val-ues of the slopes and to observe this effect, new mod-els having different Poisson ratios have been run fordifferent cohesion values.

At the final stage of this study, the presence of arelationship between the parameters with the additionof Poisson ratio and safety factors in the numerical

models has been searched. For this purpose, a R factorhas been proposed and calculated from the Equation 11:

(11)

where c � cohesion; � � internal friction angle;H � slope height; � � slope angle; � � rock massdensity; and v � Poisson ratio.

4 RESULTS

Results of three methods and failure conditions ofnumerical models are summarized in Tables 3 & 4.

During the evaluation of the numerical models interms of failure, it should be noted that FLAC3D doesnot produce a solution at the end of its calculation.However, several indicators such as unbalanced force,gridpoint velocities, plastic indicators and historiesare used to asses the state of the numerical model interms of stable, unstable, or in steady-state plasticflow (Itasca 1997).

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Table 2. Material properties.

Properties Values

Elastic modulus 1.70 GPaBulk modulus 1.13 GPaShear modulus 0.68 GPaTensile strength 0.00 PaPoisson ratio 0.25k Ratio 0.33Dilation angle 0.0

Table 3. Safety factors by Bishop and Hoek and Bray.

Safety factor

Parameters Values Bishop Hoek and Bray

Cohesion 50 0.752 0.695(kPa) 90 0.952 0.895

130 1.156 1.070170 1.361 1.236

Internal fric- 20 0.849 0.800tion angle (°) 25 0.952 0.895

30 1.088 1.01435 1.222 1.15240 1.371 1.263

Density (t/m3) 1.6 1.210 1.1081.9 1.097 0.9902.2 1.015 0.9602.5 0.952 0.8952.8 0.904 0.859

Cohesion 50 0.364 0.498(fully satu- 90 0.561 0.695rated) 130 0.751 0.848(kPa) 170 0.943 0.990Slope angle 30 1.866 1.630(°) 40 1.375 1.306

50 1.114 1.07060 0.952 0.89570 0.802 0.760

Slope height 20 1.589 1.485(m) 50 0.952 0.895

80 0.781 0.733140 0.666 0.593200 0.614 0.522

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These indicators are shown in Figure 4 for one of thenumerical models. The displacement vectors and thecontours of shear strain increment of the same numeri-cal model are given in Figure 5.

In the light of the evaluation of the numerical modelshaving the mohr-coulomb plasticity model, safety fac-tors of numerical models have been classified in termsof failure, and given in Figure 6.

Safety factor values obtained from the models hav-ing different Poisson ratios for cohesion parameterare given in Figure 7.

Results of the safety factor calculation for numeri-cal models with different tensile strengths calculatedaccording to Equation 10 are shown in Figure 8 forthe cohesion parameter.

A correlation has been established between R fac-tor and safety factors of numerical models. Accordingto this, the obtained linear regression model and cor-relation coefficient are as follows:

(12)

where F � safety factor; r � correlation coefficient.It can be said that there is a very strong positive

linear relationship between R factor and safety factorand it is shown in the Figure 9.

5 CONCLUSIONS

For all parameters high safety factor values have beengiven by Bishop Approach and it has been followedby Hoek and Bray and FLAC3D approaches. But infull saturated condition, high safety factor values aregiven by Hoek and Bray approach and it is followedby Bishop and FLAC3D approaches.

It was considered that the reason for low safetyfactor values in numerical models was the assumptionof no tensile strength. Then new numerical models inwhich tensile strength was calculated depending on thecohesion and internal friction angle have been run and

87

Table 4. Safety factors and failure conditions of numericalmodels in FLAC3D.

Parameters Values Safety factor Failure

Cohesion 50 0.551 YES(kPa) 90 0.705 YES

130 0.859 NO170 1.013 NO

Internal fric- 20 0.641 YEStion angle (°) 25 0.705 YES

30 0.750 NO35 0.788 NO40 0.821 NO

Density (t/m3) 1.6 0.900 NO1.9 0.815 NO2.2 0.753 YES2.5 0.705 YES2.8 0.668 YES

Cohesion 50 0.103 YES(fully satu- 90 0.141 YESrated) 130 0.180 YES(kPa) 170 0.218 YES

Slope angle 30 1.450 NO(°) 40 1.038 NO

50 0.856 NO60 0.705 YES70 0.551 YES

Slope height 20 1.094 NO(m) 50 0.705 YES

80 0.597 YES140 0.533 YES200 0.472 YES

Figure 4. Indicators used to assess the state of the numerical model.

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safety factor values have been calculated. As it can beseen in Figure 8, the new values are higher.

It can be also said that another reason for low safetyfactor values in numerical models can be the numericaldiscretization chosen in this study.

When safety factor values of numerical modelsgiven in Figure 6 are considered, no failure has beenobserved in the models whose safety factors are higherthan 0.8.

Finally, a preliminary estimate value of the safetyfactor can be obtained before numerical modeling byusing proposed regression model (equation 12). It can

88

Figure 5. The displacement vectors and the contours of shear strain increment of the numerical model.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1 4 10 11 12 13

Safe

ty F

acto

r

Failure

No failure

2 3 5 6 7 8 9

Figure 6. Safety factor values classified in terms of failure.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

50 90 130 170Cohesion (kPa)

Safe

ty F

acto

r

� 0.15

� 0.20

� 0.25

� 0.30

� 0.35

Figure 7. Safety factors calculated for different cohesionvalues with the different Poisson ratio.

0.00.20.40.60.81.01.21.41.61.8

50 90 130 170Cohesion (kPa)

Safe

ty F

acto

r

στ = 0

στ # 0

Figure 8. Safety factors at two different tensile strengthconditions.

0

0.5

1

1.5

2

0 0.005 0.01 0.015 0.02

R Factor

Safe

ty F

acto

r

Figure 9. The relationship between R factor and safety factor.

09069-10.qxd 08/11/2003 20:20 PM Page 88


be useful during the design stage. But it should benoted that this regression model should be used withgreat caution. The reason for this is that safety factorscalculated by using equation 12 will depend on numer-ical models formed in this study. Because, equation 12has been obtained from regression analysis carried outon the results of numerical modeling studies. Theseresults are greatly affected by numerical discretizationchosen for numerical models in this study.

REFERENCES

Aksoy, M. 2001. A Study on the Effect of ParametersAffecting safety Factor of Slopes under Circular FailureCondition, MSc Thesis, Osmangazi University, Turkey.

Brady, B.H.G. and Brown, E.T. 1993. Rock Mechanics forUnderground Mining. London Second Edition, Chapman &Hall.

Hoek, E. and Bray, J.W. 1981. Rock Slope Engineering.London Institution of Mining and Metallurgy, 358 p.

Itasca Consulting Group, Inc. 1997. FLAC3D – FastLagrangian Analysis of Continua in 3 Dimensions, Version2.0 User’s Manual. Minneapolis, MN: Itasca.

Kourdey, A., Alheib, M. and Piguet, J.P. 2001. Evaluation ofSlope Stability by Numerical Methods, 17th Int. MiningCongress and Exhibition of Turkey, IMCET 2001. Ankara.

Sjöberg, J. 1999. Analysis of Large Scale Rock Slopes,Doctoral Thesis, Lulea University of Technology.

Stacey, T.R. 1968. Stability of Rock Slopes in Open Pit Mines.National Mechanical Engineering Research Institute.Council for Scientific and Industrial Research, CSIRReport MEG 737, Pretoria, South Africa, 66 p.

Starfield, A.M. and Cundall, P.A. 1988. Towards a Method-ology for Rock Mechanics Modeling. Int. J. Rock Mech.Min. Sci. & Geomech. Abstr. 25(3): 99–106.

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91

Numerical modeling of seepage-induced liquefaction and slope failure

S.A. BastaniLeighton Consulting, Inc., Irvine, California, USA

B.L. KutterUniversity of California, Davis, California, USA

ABSTRACT: Several earth dams, tailings dams, and slopes failed or were severely damaged due to liquefactionduring or after earthquakes. In seismic areas, earth structures such as embankments may be subjected to two forces:the static loads due to gravity and the inertia forces caused by earthquakes. In a significant number of cases, lique-faction-induced failure of embankments occurred from seconds to hours after an earthquake. In these cases, lique-faction reduced the material strength and the failure occurred only under static loads. A finite slope was modeled atUC Davis National Geotechnical Centrifuge to evaluate post-earthquake deformations with an injection-inducedliquefaction system. A constitutive model was developed to capture the behavior of sands with a minimum numberof physically meaningful parameters to enable prediction of post-earthquake liquefaction and/or seepage-inducedliquefaction. This constitutive model is based on the Mohr-Coulomb constitutive model and the Critical State con-cept by adding three parameters to the conventional Mohr-Coulomb model. The constitutive model performed ade-quately for modeling the sand behavior under monotonic drained and undrained triaxial loading and water injectionfor a simple shear test under a constant shear stress. Using the new constitutive model, the failure mode of the cen-trifuge model due to seepage-induced liquefaction was studied utilizing FLAC. Stress and strain paths for specificelements in the embankment are studied and presented in this paper.

1 INTRODUCTION

Examples of post-earthquake liquefaction-induced fail-ures of embankments are reported by Dobry & Alvarez(1967), Seed et al. (1975), Okusa et al. (1978), and Finn(1980). In these cases, liquefaction reduced the materialstrength and the failure occurred under static forcesafter the earthquake shaking. One mechanism for thedelayed failure is the softening associated with redistri-bution of void ratio caused by gradients of pore waterpressure in sloping ground with non-uniform perme-ability. This mechanism has been studied by Malvick et al. (2003) and Kokusho & Kojima (2002).

For the present study, the post-earthquake liquefaction-induced failure of granular embankmentswas investigated by a static centrifuge test in which thewater that might be produced during an earthquake dueto densification of deep saturated soil was simulated byinjecting a similar volume of water at the base of themodel as presented in detail by Bastani (2003). Thiscentrifuge model consisted of a coarse sand layer witha constant thickness at its base to spread the injectedwater beneath an embankment composed of a finesand capped by a layer of low permeability clayey silt.

The centrifuge test was modeled by Fast LagrangianAnalysis of Continua (FLAC) computer code utilizinga new constitutive model as presented in this paper. Formore details on the centrifuge and numerical modelsrefer to Bastani (2003).

2 CENTRIFUGE MODEL

The centrifuge model consisted of three layers:

1. A uniform 51 mm thick layer of Monterey Sand(mean grain size � 1.25 mm);

2. A fine sand (Nevada Sand, mean grain size �0.12 mm) embankment with a minimum thicknessof 102 mm at its toe and a maximum thickness of356 mm at the slope crest; and

3. A uniform 51 mm thick layer of Yolo Loam thatcapped the Nevada Sand embankment.

The horizontal lengths of the embankment toe, theslope, and the crest were 356, 584, and 533 mm, respec-tively. The slope angle was 23.5 degrees. The averagevoid ratio of the Nevada Sand was 0.77 correspondingto a relative density of 33 percent; at this density,

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the sand was highly dilative at the confining pressuresexperienced in the experiment. The Yolo Loam had anaverage undrained shear strength of about 10 kPa and a water content of 33 percent. An additional overbur-den pressure equivalent to 90 mm of water head wasapplied over a plastic membrane on the Yolo Loamlayer. All dimensions are provided in the model scaleand the embankment’s configuration is presented inFigure 1. The centrifuge model was consolidated inseveral stages as the centrifuge speed was increased upto 37.9 g.

3 CONSTITUTIVE MODEL

A constitutive model was developed to simulatebehavior of Nevada Sand in the FLAC programframework, and it was incorporated in the numericalmodeling of the centrifuge test. The purpose of thismodel was to predict the principal behavior of NevadaSand with a minimum number of parameters that arephysically defined and measurable.

The failure envelope for this constitutive model cor-responds to the Mohr-Coulomb constitutive model(shear yield function) with tension cutoff (tensile yieldfunction). The shear flow rule is non-associated and thetensile flow rule is associated. The shear potential func-tion corresponds to a non-associated flow rule. Detailsof Mohr-Coulomb model implementation are explainedin the FLAC manual published by Itasca (2001).

Several modifications are made to the Mohr-Coulomb model. The mobilized friction angle (�mobilized � �cs �dilation) is represented as a sum of the critical state friction angle (�cs) and the dilationangle (�dilation) as described by Bolton (1991). �cs isconsidered constant, while, the �dilation is assumed tobe variable depending on the distance of the materialstate from the critical state line in e-log(p) space,defined by the state parameter:

(1)

where e is the void ratio, (ecs)a is the critical state voidratio at one atmosphere, is the virgin compressionslope, p is the mean effective stress, and pa is theatmospheric pressure.

As explained by Been & Jefferies (1985), the grad-ual change from dilative to contractive behavior can bequantified in terms of the state parameter �. The dila-tion angle (�dilation) was modified based on the stateparameter � and its changes according to the equation:

(2)

where �dilation is the updated dilation angle, (�dilation)0is the initial dilation angle, � is the state parameter, and�� is the variation of the state parameter. The soil’sbulk and shear moduli are also modified in the model.The bulk modulus (K) is evaluated using the relation:

(3)

where � is the unloading slope and p and e aredefined as above. The shear modulus (G) was conse-quently determined based on the bulk modulus (K)and their elastic relationship:

(4)

where v is Poisson’s ratio.The behavior of the constitutive model under

selected load paths are presented in Figure 2 conven-tional drained triaxial compression, conventionalundrained triaxial compression, and a simple shearelement subjected to a constant applied shear stressand water injection. Results are shown for deviatorstress (q), p, volumetric strain (�v), void ratio (e), andshear strain (�).

This constitutive model predicted the strain hard-ening behavior of the Nevada Sand during undrainedshearing until cavitation occurred prior to reachingthe critical state line. The undrained path in Figure 2approximately simulated the triaxial test data. Bastani(2003) compared the calculated undrained stresspaths with experimental data (not shown here). Themodel behaved more stiffly under the undrained con-dition, approximately 2 times more than what wasobserved in the triaxial experiments for the NevadaSand with a relative density of 26%; but the modelreasonably matched test results for the Nevada Sandwith a relative density of 39.4%.

The model behavior exhibited elastic contractionunder the drained condition up to the peak shearstress. Dilation started after the peak shear stress and

92

Grid plot0 0.5

(13,9)

(18,4)(20,13)

(40,16)(34,12)(15,16)

(15,19)

(18,6)

(28,16)

Monterey Sand

Nevada Sand (47,6)

Yolo Loam

Figure 1. FLAC grid.

09069-11.qxd 08/11/2003 20:20 PM Page 92


continued up to the critical state condition. Finally, themodel behavior was studied under a constant shearstress and pore water pressure increase, modeling asimple shear test with pore fluid injection. The consti-tutive model slightly dilated prior to reaching the fail-ure envelope; thereafter, the sample dilated with theincrease of pore water pressure and the stress pathapproached the origin along the failure envelope in thep-q space until it reached the critical state conditionsimilar to the stress path suggested by Boulanger(1990). The dilation rate was less than that shown byhis experiment (Boulanger, 1990); however, the stresspath, boundary condition, and initial condition of theexperiments performed by Boulanger prior to waterinjection into his simple shear tests were not known,and therefore were not completely simulated by thiscalibration. As expected, the water injection to theelement led to an unstable condition when the strength of the element dropped below the applied shear stress. Continued softening caused the stress path todrop toward the origin while the sample collapsed

dynamically under the unbalanced external loads.Some oscillation is observed in the q–� curves at shearstrains greater than 0.12, but the softening behavior canstill be clearly observed during the dynamic collapse.

The parameters used for this calibration and laterin the numerical modeling based on this constitutivemodel are provided in Table 1.

93

Model Behavior:

Drained Triaxial Test

Undrained Triaxial Test

Simple Shear (Khc =0.6, τxy

=62 kPa)

0 50 100 150 200 250 300p'(kPa)

p'(kPa)

0

50

100

150

200

250

300

q (k

Pa)

q (k

Pa)

(�')cs

(�')cs+(�')dilation

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40

50

100

150

200

250

300

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-100

-50

0

50

100

150

Pore

Wat

er P

ress

ure

(kPa

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4-0.01

0

0.01

0.02

0.03

0.04

1 10 1000.76

0.8

0.84

0.88

0.92

e

Cavitation

ε v

γ γ

γ

Figure 2. Behavior of the new constitutive model.

Table 1. Model parameters.

Parameters Value

* 0.022� (� /5) 0.0044�cs (degree) 32(�dilation)max (degree) 10� 0.25e0 (initial void ratio) 0.77(ecs)a* 0.809Atmospheric pressure, pa (kPa) 101.2

* Archilleas et al. 2001.

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4 FLAC MODEL BEHAVIOR

FLAC version 4.0 was utilized to model the centrifugetest. The numerical model was run twice. The firstrun used the conventional Mohr-Coulomb constitu-tive model, while the second run utilized the new con-stitutive model (discussed in Section 3) to model theNevada Sand behavior. This numerical model wasbounded with its and the constitutive model’s limita-tions; however, it was successfully used to observe thegeneral mechanism of localized increase in void ratiojust beneath the less permeable clayey silt layer, andthe failure mechanism; exact predictions were notexpected. The FLAC runs were performed with thelarge-strain mode.

Figure 1 shows the grid utilized in this model. Thegrid nodes and elements are identified in the subse-quent figures with their column and row numbers(i,j). The column and row numbers increase from leftto right and bottom to top, respectively.

Contours of mobilized friction angle, volumetricstrain, and shear strain and grid deformation patterns

for the two runs are plotted on Figure 3. This figureindicates the following behaviors:

1. The mobilized friction angle was reduced alongthe Nevada Sand interface elements by the newconstitutive model and along a deeper seated fail-ure plane as shown by the new constitutive model;

2. Volumetric strains were concentrated along theinterface of Nevada Sand and Yolo Loam in bothnumerical models. However, deeper volumetricstrains were observed in the modified constitutivemodel, which coincided with the friction angle andshear strain patterns;

3. Shear strains were also concentrated at the slopeinterface within the Nevada Sand layer. Similarly adeeper shear zone was predicted by the new consti-tutive model matching the volumetric strain andmobilized friction angle reduction patterns;

4. Sand and clay layers moved downward at the slope,which was translated to vertical uplift at the toe.

It is worthwhile to mention that the pore water pres-sure was mainly increased from the slope toe within

94

Figure 3. Friction angle, volumetric and shear strains, and deformation patterns at 13 seconds of seepage.

09069-11.qxd 08/11/2003 20:20 PM Page 94


the Nevada Sand layer toward the slope crest and witha slower rate from the back of the slope crest towardthe slope.

It should also be noted that the development of adeep failure mechanism, or not, was affected by therate at which the water was injected relative to thepermeability of the soils. For somewhat slower injec-tion, the deeper mechanism would disappear and slid-ing along the bottom interface of the Yolo Loamwould be apparent. For much greater injection rates, afailure mechanism at the interface between the coarseMonterey Sand and the fine Nevada Sand wasobserved (Bastani 2003).

Stress/strain paths of several elements at the toe,along the slope, and at the slope crest are plotted onFigure 4. Effective stresses of slope/leaning elementsreduced while oscillating around constant shear stressesup to the failure envelope. However, shear stresses ofcarrying elements along the slope and its toe increasedduring the failure of leaning elements until reachingthe failure envelope. Stress paths moved toward theorigin after reaching the failure envelope and strainsoftening was observed. In general, the elements at thetoe and along the slope showed higher shear strengthsprior to their stress paths diving toward the origin in

the second run due to the ability of the modified con-stitutive model to withstand a mobilized friction anglegreater than the critical state friction angle during dila-tion. The majority of volumetric strains of elementswere induced when the mean effective stress (p)became less than 10 kPa and close to zero. The volu-metric strains were stabilized wherever the injectiondid not cause the strength to fall below the appliedshear stress.

Predicted pore water pressures and deformationsare compared with the experimental results in Figures5 & 6, respectively. In general the predicted pore waterpressures are in good agreement with the experimen-tal results for both constitutive models. However, theinitial rate of pore water pressure increase is in betteragreement with the new constitutive model. Similartrends were obtained by the numerical model, such asstabilization of pore water pressure at the beginningand its further increase for PPT# 5296 (Fig. 5). Otherthan deformation at the middle of the slope (LVDT#3), where the experimental result indicated bulging,deformation rates and magnitudes were predicted verywell by both models.

The numerical model successfully predicted thedilatancy to cause a very loose layer of sand below

95

Figure 4. Stress path and behavior of elements. Solid and dashed lines refer to the results of the new and Mohr-Coulombconstitutive models, respectively.

09069-11.qxd 08/11/2003 20:20 PM Page 95


the less permeable layer of Yolo Loam. The calculatedvolumetric strains of the dilated sand indicate a negli-gible residual strength after dilation.

5 CONCLUSION

A modified Mohr-Coulomb constitutive model wasdeveloped based upon critical state theory in conjunc-tion with a new expression for dilatancy that dependson the state parameter (the distance between the stateand the critical state). The constitutive model wasshown to enable calculation of strain-softening paths,and dilation due to water injection.

The constitutive model was implemented in FLACand used to analyze results of centrifuge model tests oflayered sloping ground subject to pore fluid injection.The injection was intended to simulate the upwardflow of water that might be generated by densificationof deep soil deposits during earthquake shaking.

In the past, embankments made of dilative materialwere considered to be safe, because the undrainedstrength is greater than the driving stress (Poulos et al.1985). The centrifuge tests and the FLAC analyses

96

Figure 5. Pore water pressure time histories.

Figure 6. Deformation time histories.

09069-11.qxd 08/11/2003 20:20 PM Page 96


presented here clearly demonstrate the possibility thatlayers that impede drainage may cause a significantlocalized zone of softened material that should beconsidered a possibility in seismic design. To deter-mine induced deformations due to local drainage of asystem, a material model that captures this processshould be incorporated in the numerical model. Themode of failure and local drainage of the centrifugetest presented here was successfully predicted utiliz-ing the modified Mohr-Coulomb constitutive modelin conjunction with FLAC numerical framework.

ACKNOWLEDGEMENT

The authors would like to thank Dr. Ben Hushmand,James Ward, and Vivian Cheng for reviewing thispaper and providing constructive comments.

REFERENCES

Archilleas, G.P., Bouckovalas, G.D. & Dafalias, Y.F. 2001.Plasticity Model for Sand Under Small and Large Cyc-lic Strains. Journal of Geotechnical Engineering, ASCE127(11):.973–983.

Bastani, S.A. 2003. Evaluation of Deformations of EarthStructures due to Earthquakes. Dissertation presented toUniversity of California, at Davis, in partial fulfillment ofthe requirements for the degree of Doctor of Philosophy.

Been, K. & Jefferies, M.G. 1985. A State Parameter forSands. Geotechnique 35(2): 99–112.

Bolton, M. 1991. A Guide to Soil Mechanics. Published by M D & K Bolton, Printed by Chung Hwa Book Company,pp. 63–92.

Boulanger, R.W. 1990. Liquefaction Behavior of SaturatedCohesionless Soils Subjected to Uni-Directional and Bi-Directional Static and Cyclic Simple Shear Stresses.Dissertation presented to University of California, atBerkeley, in partial fulfillment of the requirements forthe degree of Doctor of Philosophy.

Castro & Poulos (ASCE paper circa 1984).Dobry, R. & Alvarez, L. 1967. Seismic Failure of Chilean

Tailing Dams. Journal of Soil Mechanics and Founda-tions Division, Proceeding of the American Society ofCivil Engineers 93(SM6): 237–260.

Finn, W.D. 1980. Seismic Response of Tailing Dams.Presented at Seminar on Design and Construction ofTailing Dams, Colorado School of Mines, Denver,Colorado, pp. 76–97.

Itasca Consulting Group, Inc. 2001. FLAC – Fast Lagran-gian Analysis of Continua, Ver. 4.0 User’s Manual.Minneapolis, MN: Itasca.

Kokusko, T. & Kojima, T. 2002. Mechanism for Postlique-faction Water Film Generation in Layered Sand. Journalof Geotechnical Engineering, ASCE 128(2): 129–137.

Malvick, E.J., Kulasingam, R., Boulanger, R.W. & Kutter, B.L.2003. Analysis of a Void Ratio Redistribution Mech-anism in Liquefied Soil. To be Published in Proceedingsof the June 2003 Soil and Rock America Conference.

Okusa, S., Anma, S. & Maikuma, H. 1978. Liquefaction ofMine Tailing in the 1978 Izu-Ohshima-Kihkai Earth-quake, Central Japan. Engineering Geology Vol. 16, pp.195–224, Elsevier Scientific Publishing Co.

Poulos, S.J., Castro, G. & France, J.W. 1985. LiquefactionEvaluation Procedure. Journal of the Geotechnical Engi-neering Division, ASCE 111(6): 772–792.

Seed, H.B., Lee, K.L., Idriss, I.M. & Makdisi, F.I. 1975. TheSlides in the San Fernando Dams during the Earthquakeof February 9, 1971. Journal of the Geotechnical Engi-neering Division, ASCE 101(GT7): 651.

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99

Complex geology slope stability analysis by shear strength reduction

M. Cala & J. FlisiakDept. of Geomechanics, Civil Engineering & Geotechnics, AGH University of Science & Technology, Poland

ABSTRACT: The stability of slopes may be estimated using 2D limit equilibrium methods (LEM) or numericalmethods. Due to the rapid development of computing efficiency, several numerical methods are gaining increasingpopularity in slope stability engineering. A very popular numerical method of slope stability estimation is the shearstrength reduction technique (SSR). It’s a well known fact that for a simple slope factor of safety (FS) obtained fromSSR is usually the same as FS obtained from LEM. However for slopes of complex geology, considerable differ-ences between FS values may be expected. Application of SSR for such slopes is usually restricted to the weakestlink estimation – that part of the slope with the lowest FS. Finite Difference Method code, FLAC (Itasca 2000), givesthe opportunity to analyze several slip surfaces by using the modified SSR technique (MSSR). The method is basedon reducing shear properties of soils after identification of the first slip surface. MSSR allows a complete estima-tion of stability for any type of slope.

1 INTRODUCTION

The stability of slopes may be estimated using 2D limitequilibrium methods (LEM) or numerical methods.Due to the rapid development of computing efficiency,several numerical methods are gaining increasing pop-ularity in slope stability engineering. A very popularnumerical method of slope stability estimation is shearstrength reduction technique (SSR). In that procedure,the factor of safety (FS) of a soil slope is defined as thenumber by which the original shear strength parame-ters must be divided in order to bring the slope to thepoint of failure (Dawson & Roth 1999).

It’s a well known fact that for simple slopes FSobtained from SSR is usually the same as FS obtainedfrom LEM (Griffiths & Lane 1999, Cala & Flisiak2001). However for complex geology slopes consid-erable differences between FS values from LEM andSSR may be expected (Cala & Flisiak 2001). Severalanalyses for the slope with weak stratum were per-formed to study the differences between LEM and SSR.

It must be also stated that classical SSR techniquehas several limitations. Application of SSR requiresadvanced numerical modeling skills. Calculation time,in case of complicated models, can last as long as sev-eral hours.

However, the most fundamental limitation of SSR isidentification of only one failure surface (in some casesit may identify more than one surface, but with the sameFS value). This is not a significant limitation in caseof simple geometry slope. But in case with complex

geometry (and geology) it’s not possible to analyze FSfor other parts of the slope. This may sometimes leadto serious mistakes.

2 STABILITY OF SLOPE WITH WEAKSTRATUM

To investigate the influence of a weak stratum on FSsome 350 models were analyzed. The thickness of theweak stratum was changed from 1.0 to 10.0 m and itwas localized from 0 to 50 m from the top of the slope(Fig. 1).

All slopes in this paper were simulated with FLAC/Slope (Itasca 2002) or FLAC in plane strain, usingsmall-strain mode.

It was assumed that embankment is 25 m high andhas a slope angle of 45°. It consists of two different geo-logical units. The soil was given friction angle � � 30°

25 m

25 m

hg

45°

Figure 1. Slope with weak stratum.

09069-12.qxd 08/11/2003 20:21 PM


and cohesion c � 75 kPa. The weak, thin layer hadfriction angle � � 10° and cohesion c � 25 kPa. Bothsoils had unit weight � � 20 kN/m3. The thickness “g”of the horizontal weak layer was changed from 1.0 m to10.0 m and its distance “h” from the top of the slopechanged from 0 to 50 m.

Figure 2 shows the FS values for a 1.0 m thick weaklayer and Figure 3 for a 5.0 thick one. The decrease ofFS is quite small if the thin weak layer is located closeto the top of the slope.

Increasing the weak layer thickness produces con-siderable decrease of FS. The differences in FS valuesare significant especially in case of small thickness(1 m–3 m) of weak stratum

Increase of weak layer thickness (irrespectively of itslocalization) reduces differences between FS valuesfrom LEM and SSR. Especially FS values estimatedwith Bishop’s are within 8 % of the FS obtained fromSSR.

For the thickness of the weak layer less than or equalto 5 m SSR produces lower FS values than any of theLEM methods. For the weak layer 5 m thick Bishop’smethod produces FS � 1.114 and SSR shows FS �1.07.

Further increase of weak layer thickness (7.5 m and10 m) produces lowest FS values from Bishop’s method(FS � 0.926 and FS � 0.811 respectively). SSR tech-nique shows respectively FS � 0.95 and FS � 0.87 inthis case.

It seems that application of Bishop’s method pro-duces the most reliable results among LEM. Theseresults are simultaneously closest to the FS valuesobtained from SSR. Application of Fellenius’s methodproduces unreliable FS values in case of weak layer

localization below slope toe. It shows the influence ofweak layer on FS values even if the roof of the stratumlays 15 m below the slope toe.

It must be also pointed out that failure surfaces iden-tified by SSR technique are sometimes considerablydifferent than surfaces identified by LEM (Fig. 4).Figure 4 shows the situation when FS computed by SSRis considerably lower and unit volume of failed slopeis significantly higher than estimated from LEM.

3 MODIFIED SHEAR STRENGTHREDUCTION TECHNIQUE (MSSR)

3.1 Benched slope stability case

Application of SSR for complex geology slopes is usu-ally restricted to the weakest “link” estimation – part

100

0 10 20 30 40

1.4

1.5

1.6

1.7

1.8

1.9

2F

S

Weak layer 1 m thickFLACFelleniusBishopJanbu

Distance of weak layer from slope crest

Figure 2. FS values for a 1.0 m thick weak layer.

0 10 20 30 40 50

Distance of weak layer from slope crest

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

FS

Weak layer 5 m thick

FLACFelleniusBishopJanbu

Figure 3. FS values for a 5.0 m thick weak layer.

BishopFS = 1.731

FLACFS = 1.54

20 m

1 m

Figure 4. Critical slip surfaces identified by SSR and LEM.

09069-12.qxd 08/11/2003 20:21 PM


of the slope with the lowest FS. However the FiniteDifference Method code FLAC gives the opportunityto analyze several slip surfaces using modified shearstrength reduction technique – MSSR (Cala & Flisiak2003a, b).

This method is based on reducing shear properties ofsoils after identification of first slip surface (FS1). Itis simply the continuation of classic SSR, but afterfirst instability occurrence. It is possible only usingFinite Difference Method. The FLAC program usesthe explicit, Lagrangian calculation scheme. The fulldynamic equations of motion are used, even whenmodeling systems that are essentially static. Thisenables FLAC to follow physically unstable processes(i.e. several processes simultaneously) without numer-ical distress. In fact, FLAC is most effective whenapplied to nonlinear or large-strain problems, or to sit-uations in which physical instability may occur. Thismay lead to identification of several other slip sur-faces. The same criterion is used to identify secondary(and further) failure surfaces. The primary and the fol-lowing identified failure modes are constantly active(not suppressed) during entire calculation process.Let’s consider benched slope stability (Fig. 5).

Figure 6 shows the slip surfaces identified inbenched slope by MSSR and LEM. Failure of the lowerpart of the slope was detected first. FS1 � 0.90 cal-culated by SSR is very close to FS � 0.921 given byBishop’s method. And precisely here ends the range of

classical SSR technique – especially with applicationof any Finite Element Method code. However FLAC iscreated especially for modeling physical instability (inthis case – physical instabilities would be better term).

This allows to continue shear strength reductionand to identify another possible slip surfaces. In ana-lyzed case, next identified failure surface is located in the upper part of the slope. FS2 � 1.00 calculatedby MSSR is again very close to FS � 1.008 given byBishop’s method. And finally application of MSSRallowed to evaluate FS for entire slope – FS3 � 1.24 isalso very close to FS � 1.228 given by Bishop’smethod.

It seems that FS calculated with MSSR are within afew percent of the FS obtained from LEM for simplecases. It must be however underlined that effectivenessof MSSR must be verified on real cases.

3.2 Large scale, complex geology slope stability case

Let’s consider a slope consisted of eight different geo-logical units (from a Polish lignite open pit mine).The mechanical properties of the soil units involvedin the slope are given in Table 1.

Figure 7 shows geometry and geology of the ana-lyzed slope. The overall sloping angle was equal ! �7.477°.

Figure 8 presents the slip surface identified byMSSR and LEM. Again SSR finds the location of thelowest safety factor FS1 � 0.67. Application of MSSRidentifies four new slip surfaces in several parts of theslope. FS2 �0.87 also shows the local failure surfacewhich, in fact, does not affect the overall slope stabil-ity (precisely like previous one). Another possiblefailure surface with FS3 �1.02 is based on layer 5(very thin and weak one) and broken line upward.

Further analysis showed development of previousfailure surface with FS4 � 1.17 occurring mainly inlayer 5. Bishop’s method applied to the upper part of theslope shows cylindrical failure surface with FS �1.351.

It must be noted that due to cylindrical shapeBishop’s slip surface covers a little more soil volume.

101

10 m

10 m

15 m

20.918 m15 m15 m

45º

40º

� = 20 kN/m3

� = 20ºC = 10 kPa

Figure 5. Benched slope case geometry.

Bishop

Bishop

Bishop

FS1 = 0.90

FS = 0.921

FS3 = 1.24

FS = 1.228 FS2 = 1.00

FS = 1.008

Figure 6. Several slip surfaces identified in benched slopeby MSSR and LEM.

Table 1. Mechanical properties of soil units.

Cohesion Friction angle Unit weightUnit c, kPa �, deg �, kN/m3

1 14.0 6.5 18.32 90.0 10.9 19.53 11.4 7.9 19.54 90.0 10.9 19.55 11.4 7.9 19.56 90.0 10.9 19.57 28.0 8.5 20.08 1000 30.0 23.0

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FS � 1.351 is however considerably higher thanFS4 � 1.17 from MSSR.

And finally an overall slope failure surface withFS5 � 1.29 is identified. Bishop’s method showsFS � 1.255, but it covers considerably lower soil vol-ume. Generally, the results obtained from LEM are notthat close to MSSR as in the simple case discussedbefore.

It’s a well-known fact that application of LEMrequires assumption about shape and location of slipsurface. Circular failure surfaces were assumed here forcalculation purposes. Critical slip surface with lowestFS value was estimated from 20,000 circles.

In MSSR there is no need for such assumptions.Stress and strain field in analyzed soil determines theshape and location of the slip surfaces.

4 CONCLUSIONS

For a simple, homogeneous slope FS calculated withSSR are usually the same as FS obtained from LEM.In the case of a simple geometry slope consisting oftwo geological units, FS calculated with SSR may beconsiderably different than FS from LEM.

In the case of complex geometry and geologyslopes SSR technique is much more “sensitive” thanLEM. Another step forward is the modified shearstrength reduction technique – MSSR. Application ofSSR with FLAC may be recommended for the large-scale slopes of complex geometry.

Such a powerful tool as MSSR with FLAC givesthe opportunity for the complete stability analysis forany slope.

ACKNOWLEDGEMENTS

Support for this research by the State Committee forScientific Research (Project No. 5 T12A 022 24) isgratefully acknowledged.

REFERENCES

Cala M. & Flisiak J. 2001. Slope stability analysis withFLAC and limit equilibrium methods. In Billaux, Rachez, Detournay & Hart (eds) FLAC and NumericalModeling in Geomechanics; Proc. Intern. Symp., Lyon,France, 29–31 October 2001: 111–114. Rotterdam:Balkema.

Cala M. & Flisiak J. 2003a. Analysis of slope stability withmodified shear strength reduction technique. XXVIWinter School of Rock Mechanics: 348–355. Wroclaw.IGiH, (in polish).

Cala M. & Flisiak J. 2003b. Slope stability analysis withnumerical and limit equilibrium methods. ComputationalMethods in Mechanics; Proc. Intern. Symp., 3–6 June2003 (in press).

Dawson E.M. & Roth W.H. 1999. Slope stability analysiswith FLAC. In Detournay & Hart (eds) FLAC andNumerical Modeling in Geomechanics; Proc. intern.symp., Minneapolis, MN, 1–3 September 1999: 3–9.Rotterdam: Balkema.

Itasca Consulting Group. 2000. FLAC – Fast LagrangianAnalysis of Continua, Ver. 4.0 User’s Manual.Minneapolis, Minnesota: Itasca.

Itasca Consulting Group. 2002. FLAC/Slope Ver. 4.0 User’sManual. Minneapolis, Minnesota: Itasca.

Griffiths D.V. & Lane P.A. 1999. Slope stability analysis byfinite elements. Geotechnique. 49(3): 387–403.

102

168

m

800 m

63 m

1

3 4

5

8

6

7

2

Figure 7. Slope geometry and geology.

FS1 = 0.67FS2 = 0.87

FS3 = 1.02 FS4 = 1.17

FS5 = 1.29

BishopFS = 1.255

BishopFS = 1.351

Figure 8. FS values and critical slip surfaces identified with MSSR and LEM.

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103

Analysis of hydraulic fracture risk in a zoned dam with FLAC3D

C. PeybernesElectricité de France, Centre d’Ingénierie Hydraulique, France

ABSTRACT: The construction and first filling of a 150 m high zoned dam are modeled with FLAC. The site isa deep and curved canyon under the dam. The aim of this study is the understanding of the dam behavior and theassessment of the dam safety. A lot of attention is put on the hydraulic fracturing risk during construction or duringfirst filling due to the core arching in the deep canyon. 2D and 3D models are compared. The 2D model is unableto explain the monitoring data, but the 3D model fits more accurately the monitoring measurement. Although ahigh contrast of modulus exists between core and shell, no hydraulic fracturing is observed in the core.

1 DESCRIPTION OF STRUCTURE

The zoned dam has a clayey core with vertical down-stream face, gravely downstream and upstream filtersand shells. The canyon in the bottom of the valley is60 m deep, narrow, and turns under the dam. Main fea-tures are (Fig. 1):

– Maximum height of dam: 137.00 m,– Maximum elevation: 1000.00 m,– Minimum elevation: 863.00 m,– Slopes of the faces: H/V � 2/1.

The downstream toe is submerged by the reservoirof the downstream dam from elevation 890.0 to906.0 m.

2 TWO GEOMETRICAL MODELS

Two geometrical models were meshed by ItascaConsultant Spain office. The strategy was to use the2D model to set the characteristics of the materials,the loading scenario and the boundary conditions,and to use the 3D model to analyze the arching effectscaused by the stiff banks.

2.1 Two-dimensional model

The section is the deepest one in the canyon perpen-dicular to the dam axis (Fig. 2). It is 1 m wide mod-eled by FLAC3D. The mesh has 1861 3D elements and3890 nodes. The constitutive equations of materialare programmed for both 2D and 3D files.

2.2 Three-dimensional model

The mesh has 66,054 3D elements and 71,828 nodes.The geometry of the contact dam foundation, in par-ticular the canyon is rather faithful to reality.

This model has great advantages over the 2D. Ittakes into account the following items:

– load transfer from dam body to the canyon,– turn on the left of the valley (Fig. 3),– dissymmetry between the banks (Fig. 4),– plating of the filter against left bank (Fig. 5).

3 SETTING OF MATERIAL PROPERTIES

The model is fitted on several indicators from themonitoring measurement.

Figure 1. Standard profile. Figure 2. Two-dimensional mesh (“section”).

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– Settlements in the core measured at the end of theconstruction (up to 80 cm).

– total stresses in the core– pore water pressures in the core– deformations in the downstream shoulder recorded

by three tassometers and two extensometers.

4 CONSTITUTIVE EQUATIONS FOR SOIL

Two different constitutive models are used:

1. elastic model, for first parameter setting,2. plastic law: Mohr-Coulomb.

5 SCENARIO OF LOADING

5.1 Initialization of the model

The initial stress state is calculated in the foundation,alluvium filling and excavation at the core location.The initial equilibrium calculation is only mechanical(zero pore pressure).

5.2 Construction

The construction period is 4 years and the embankmentis placed in 4 m layers.

5.2.1 Hydraulic boundary conditionsAt the boundary of the core, the water pressure (Pw)is fixed at zero to dissipate pore pressures in the coreto allow drainage due to the filter and shell.

5.2.2 Mechanical boundary conditionsWith every placed layer, the increase of vertical andhorizontal stresses is given by the weight of the layer,�zz � 0.5 * h * � and �xx � 0.5 * �zz.

5.2.3 Hydraulic-mechanic couplingThe pore pressure is assumed to be generated bySkempton’s B coefficient via the water modulus,Kw � (B * n * Kcore)/(1 – B).

5.3 First filling and steady state

5.3.1 Hydraulic boundary conditionsThe pore pressure is fixed by the value of the hydraulicload caused by the reservoir filling, Pw � (Hw – h) *�w, if Hw h where Hw � storage level; h � nodelevel; and �w � unit weight of water.

104

Figure 3. Global sight of the model.

Figure 4. Sight of top of the valley, with studied sections.

Figure 5. Section right bank – left bank. Figure 6. Boundary conditions to construction.

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Two storage versus time curves are used: one forthe downstream shell and one for the upstream storage(Fig. 7).

5.3.2 Mechanical boundary conditionsOn the upstream and downstream faces of the dam,mechanical pressure caused by the impounding ismodeled by normal stresses on the dam faces. Thespecific weights of materials are modified when theyare saturated.

6 COMPARISON OF 2D AND 3D MODELS

In this section, the results of the model analyses arecompared to measurements at the end of construction(Fig. 9).

6.1 2D Model

The calculated stresses are too high at the base of the core compared to the measured ones. A parametricstudy of mechanical properties could not solve the dis-crepancy. It is speculated that the arching effect unloadsthe central section and transfers stresses to the banks.No realistic calculation could be reached using the 2Dmodel.

6.2 3D Model

For this model, the calculated settlements in the coreare in good agreement with the measured ones. Thestresses are smaller than the 2D problem, because ofarching. Nevertheless, the deformation of the shells isstill larger than measured. A parametric study of coreand shell moduli was undertaken to reconcile the discrepancy.

7 PARAMETRIC STUDY OF THE 3D MODELDURING CONSTRUCTION

Several simulations were studied varying themechanical properties (moduli) and flow parameters(saturation). The initial and final mechanical proper-ties are presented in Table 1. A comparison of resultsis made at the end of construction.

105

880

900

920

940

960

1979

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

First filling dates

cote

(m

)

1000

980

1980

1981

1982

1983

1984

1985

1986

upstream downstream

Figure 7. Curves of upstream and downstream fillings.

Figure 8. Boundary conditions during the filling.

Figure 9. Comparison between effective stresses with 2D and 3D models.

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7.1 Displacements

Vertical and horizontal displacements are in good agree-ment with the measurements. The shell modulus showngave results that agreed with the deformations measureddownstream, but decreased the core settlements. Thesimulation carried out using a decreased modulus of thecore gave the closest values to the measurements.

7.2 Pore pressures

Pore pressures are very low, like the measurements atthe end of construction (Fig. 10). The clay compaction

carried out (dry or very dry of optimum) results invery low pore pressure generation. This lowers the riskof hydraulic fracturing during first filling.

7.3 Stresses

The results that best agree with cell measurementswere obtained with a relatively high shell modulus,and low core modulus. On the other hand, the cal-culated total vertical stresses near the left bank werefar from the measured stresses. This phenomenon isnot clearly understood. The most important conclu-sion is that hydraulic fracture was not indicated in anycalculated case, even though the base of the core is heavily unloaded in the narrow canyon (Fig. 11Area 1). The load transfer is clearly observed in the core section parallel to the dam crest (Fig. 11 Area 2).

After completing the parametric study at the end ofconstruction, calculation was continued for the firstfilling case.

106

Figure 10. Pore pressure at the end of construction.

Figure 11. Vertical effective stresses at the end of construction.

Table 1. Initial and final mechanical properties.

C Phi EMaterial (kPa) (°) (Mpa) v B

Core initial 20 25 40 0.35 0.6Core final 20 25 20 0.35 0.1Shell initial 0 40 120 0.30 –Shell final 0 40 240 0.30 –

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8 3D MODEL FOR FIRST FILLING CASE

8.1 Pore pressures

The pore pressures calculated in the core at the level925 are close to the measured values (Fig. 13). It was difficult to get agreement between the measuredand calculated values below this level. Variations of permeability in the core were not represented in themodeling. Spatial variation of the hydraulic propertiesin the core may improve the correlation, but this wasnot done.

8.2 Stresses

Generally the shapes of the calculated and monitoredstresses versus time were similar. The constructionphase is apparent in the plots, then the filling of the

two reservoirs, and finally a steady state was reached(Fig. 14).

The values from the simulation and from measure-ments are rather close until the date 1981 for the cells to the level 925, and 1983 for the cells of level905 (Figs. 15 & 16). These dates correspond to thesudden drop of measured stress values. Then the sim-ulations and measured values disagree. It is specu-lated that this phenomenon was induced by waterinfiltration at the cell level, collapsing the clay andlowering the stress. This phenomenon should be inte-grated in the future and modeled by the clay collapseafter wetting.

Some hydraulic fracturing can be observed in theupstream shell, but this is not of concern. The shell isdrained and the water tightness of the core is notaltered.

107

-20

20

60

100

140

180

220

260

300

340

380

420

01/01/76 31/12/77 01/01/80 31/12/81 01/01/84 31/12/85 01/01/88 31/12/89 01/01/92 31/12/93 01/01/96 31/12/97 01/01/00 31/12/01

First filling125-E11-905

125-E15-914125-E17-925

E11-calcul elastiqueE15-calcul elastique

E17-calcul elastique

Figure 13. Pore pressures in kPa in the core (calculation: full features).

Figure 12. Pore pressures at steady state.

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108

9 CONCLUSION

Detection of hydraulic fracturing in the core is the objective of the analysis of this zoned dam. Thethree-dimensional modeling appears to be the onlyway to model the problem. The real geometry of thefoundation has to be carefully modeled to representthe phenomenon of stresses transferred to the banksand the unloading of the core in the canyon.

The parametric study of the mechanical propertiesof the materials was required to fit the monitoringdata and ensure the accuracy of the analysis. Finally,

according to this calibration, hydraulic fracturing ofthe core was not indicated.

REFERENCES

Varona P. 2001. Curso de FLAC3D, Itasca Consultant Spain,January 15–19, 2001.

Laigle, F. & Boymond, B. 2001. CERN-LHC Project –Design and excavation of Large-Span Caverns at Point 1,EDF-CIH France.

Figure 14. Vertical effective stresses during steady state.

�10e2 kPa11

10

9

8

7

6

5

4

3

2

1

0

-1

126-T2-905T2 Szz calcul

126-T3-905127-T5-905

126-T4-905T4 Srd-rg calcul

126-T6-905T6 Sami-avl calcul

01/0

1/76

31/1

2/77

31/1

2/81

01/0

1/84

31/1

2/85

01/0

1/88

31/1

2/89

01/0

1/92

31/1

2/93

01/0

1/96

31/1

2/97

01/0

1/00

31/1

2/01

01/0

1/80

Figure 15. Comparison between measured and calculatedtotal stresses at level 905.

1

2

3

4

5

6

7

8

9

10

-1

0

�10e2 kPa

01/0

1/76

31/1

2/77

01/0

1/80

31/1

2/81

01/0

1/84

31/1

2/85

01/0

1/88

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2/89

01/0

1/92

31/1

2/93

01/0

1/96

31/1

2/97

01/0

1/00

31/1

2/01

125-T7-925

T7 Szz calcul

125.5 T8-925

T8 Srd-rg calcul

124.5-T9-925

T9 Samt-avl cal cul

126-T10-925

125-T11-925

Figure 16. Comparison between measured and calculatedtotal stresses at level 925.

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109

Mesh geometry effects on slope stability calculation by FLAC strengthreduction method – linear and non-linear failure criteria

R. Shukha & R. BakerFaculty of Civil and Environmental Engineering, Technion I.I.T., Haifa, Israel

ABSTRACT: Results of FLAC’s strength reduction technique are compared with conventional limit equilib-rium analysis for both linear and non-linear strength criteria. The comparison includes both safety factors andfailure modes (critical slip surfaces and normal stress functions). The collection of FLAC’s plastic points is nota reasonable criterion for estimating the potential failure zone and it is necessary to establish this zone by post-processing FLAC’s results. It is shown that failure modes implied by FLAC analysis are sensitive to mesh geom-etry effects and, in order to obtain reasonable results, it is necessary to use meshes consisting of nearly squareelements. Safety factors are much less sensitive to mesh geometry effects than failure modes. FLAC’s mesh sen-sitivity is more pronounced for non-linear failure criterion than in the linear case. Using acceptable mesh geom-etry, FLAC’s strength reduction technique and limit equilibrium procedures yield comparable results (failuremodes and safety factors) for both linear and non-linear strength criteria. Engineering implications of linear andnon-linear failure criteria in the context of slope stability analysis are presented and discussed. It is shown thatequally valid interpretations of the same experimental information may, under certain conditions (e.g. steepslopes), lead to very different engineering implications. Under such conditions the choice between alternativestrength models must be based on the practical implications of these laws.

1 INTRODUCTION

Almost all practical slope stability calculations quan-tify the stability of a given slope using the notion ofsafety factor with respect to shear strength. This quan-tity is commonly defined as a reduction constant bywhich the available shear strength function of the soilneeds to be factored down in order to bring the slopeto failure. In conventional limit equilibrium (L-E)calculations, safety factors are associated with “testbodies” and it is necessary to search for the criticaltest body that yields the minimal safety factor for agiven slope. Incorporation of safety factor, with respectto strength, in a general continuum mechanics frame-work results in a class of slope stability proceduresknown as strength reduction (S-R) methods. Thisapproach was used as early as 1975 by Zienkiewics et al. (1975) and has since been applied by Naylor(1982), Donald & Giam (1988), Matsui & San (1992),Ugai (1989), Ugai & Leshchinsky (1995) and others.

2 COMPARISON OF THE L-E ANDS-R FRAMEWORKS

Both L-E and S-R techniques analyze an equivalentmaterial characterized by a mobilized strength function

Strm(�f) which is defined as:

(1)

where �f is the normal (in general effective) stressacting at failure on the failure plane, Str(�f) is thestrength function (Mohr envelope) of the material,and F is the slope’s safety factor.

Equation 1 is used by both the S-R technique andconventional L-E procedures. However, the conceptualframework employed in these two approaches is notequivalent. In particular:

1. Application of the S-R technique requires com-plete specification of the soil’s constitutive relationwhile the L-E framework does not depend ondeformation parameters.

2. L-E procedures include a minimization stage, whichestablishes the critical slip surface and its associ-ated minimal safety factor. In S-R techniques onthe other hand, existence of a unique slip surfacesis not a priori assumed, but such a surface can beestablished after completion of the basic analysis.

3. S-R methods define failure at a point (element).However yielding of a single element does not

09069-14.qxd 08/11/2003 20:25 PM


mean global failure of the slope. The local defini-tion of failure embedded in all S-R methods isprobably the main disadvantage of these procedurescompared with the inherently global L-E approach.

The L-E and S-R techniques have their strengthsand weaknesses. A number of previous studies (e.g.Naylor 1982, Dawson et al. 1999, Griffith & Lane1999) showed that both methods yield approximatelythe same safety factors. The present work extends thiscomparison to critical slip surfaces and distribution ofnormal stress acting along such surfaces.

3 STRENGTH FUNCTIONS FOR A GIVENSTATE OF INFORMATION

Most practical slope stability calculations are based onthe linear Mohr-Coulomb (M-C) strength function:

(2)

where {c, �} are the conventional M-C strengthparameters cohesion and angle of internal frictionrespectively. Experimental studies by Penman (1953),Bishop et al. (1965), Day & Axten (1989), andMaximovic (1989) have indicated that actual failureenvelopes of most soils are not linear, particularly inrange of small normal stresses. There exists a numberof studies incorporating non-linear (N-L) failure crite-ria in conventional L-E calculations such as Maximovic(1979), Charles & Soares (1984), and Perry (1994).The N-L strength function used in most of these stud-ies is the Mohr form of the Hoek-Brown (H-B) empir-ical failure criterion (Hoek & Brown 1980). Thiscriterion can be written as:

(3)

where Pa stands for atmospheric pressure and {A,n,T}are non-dimensional strength parameters. This non-dimensional form was introduced by Jiang et al.(2003), where it was shown that the parameters {A,n,T}must satisfy the requirements {A 0, 1⁄2 � n �1,T � 0} and T represents a non-dimensional tensilestrength. Baker (2003a) demonstrated that Equation 3provides a reasonable representation of experimen-tal results for a wide range of different geological materials.

It is important to realize that a physically signifi-cant assessment of the effect of strength functionsnon-linearity of the results of slope stability calcula-tions is possible only if the linear and N-L strengthfunctions are fitted to the same data set. Stated differ-ently, in order to asses the effect of different strength

functions (strength models) on results of slope stabil-ity calculations it is necessary to consider a givenstate of experimental information (given data set).Jiang et al. (2003) and Baker (2003b) performed suchstudies using approximate L-E procedures andshowed that, under certain conditions, the strengthfunctions non-linearity may have very significanteffect on results of slope stability calculations. One ofthe purposes of the present work is to study the sameeffect using a FLAC based S-R slope stability analy-sis. The points in Figure 1 show results of 103 consol-idated undrained triaxial tests with pore pressure.

Measurements were performed on compacted Israeliclays. These tests were done as part of routine test-ing programs for design of small water reservoirs.Additional information about the clays and tests isgiven by Frydman & Samoocha (1984). The lines inFigure 1 are the M-C and H-B strength envelopes fit-ted to the experimental data set, using the iterativeleast square procedure described by Baker (2003a).The fitting process resulted in � � 25°, c � 11.7 kPa,SOSMC � 31908 kPa2 and A � 0.58, n � 0.86, T � 0,SOSHB � 30263 kPa2, where SOSMC and SOSHB arethe sum of squares associated with the M-C and H-Bmodels respectively.

The following comments are relevant with respectto the above results:

1. T � 0 is a result of the estimation process, not an a priori assumption. This result implies that theoptimal H-B model represents a zero tensilestrength material. The M-C model on the otherhand predicts a non-negligible tensile strength oft � c/Tan(�) � 25 kPa.

2. The sum of squares associated with the M-C andH-B models are nearly equal SOSHB/SOSMC �0.95. This result implies that the M-C and H-Bmodels provide equally valid descriptions of theexperimental information (data points).

3. Inspection of Figure 1 shows that the M-C and N-L criteria predicts almost identical strength val-ues in the range of experimental normal stresses

110

Figure 1. Experimental strength functions for compactedIsraeli clays.

09069-14.qxd 08/11/2003 20:25 PM


33.4 � �f � 351.2 kPa. The predictions of thesemodels differ from each other only at very low andvery high normal stresses. In both of these rangesthe H-B model predicts smaller strength valuesthan M-C (convexity of the H-B criterion guar-anties that this will always be the case). In fact themain practical significance of the H-B criterion isthat it delivers conservative (compared with M-C)strength estimates in normal stress ranges in whichthere are no direct experimental information.

4 FLAC IMPLEMENTATION OFH-B CRITERION

FLAC has a feature allowing a direct use of the H-Bcriterion in slope stability calculations. However inFLAC this criterion is formulated in the principlestress space and for the present purpose it is conven-ient to use the Mohr form of this criterion (Eq. 3).Formally this is done by considering a M-C modelwith the following stress dependent tangentialstrength parameters �t and ct:

(4.1)

(4.2)

Equations 4.1 & 4.2 were programmed as a simpleFISH routine, and using the “whilestepping” optionembedded in FLAC, this routine updates the tangen-tial M-C parameters in each FLAC’s time step. The S-Rtechnique was applied using the definition of mobi-lized strength function in Equation 1, i.e. FLAC wasrun with a sequence of progressively increasing trialsafety factors until the slope failed, (i.e. until FLACfails to converge to a static equilibrium configura-tion). Attempting to apply FLAC’s SOLVE FOS com-mand with the mobilized H-B criterion we haveencountered convergence difficulties, and all the fol-lowing results were obtained by manual change of trialsafety factors.

5 EFFECT OF MASH GEOMETRY ONRESULTS OF THE S-R TECHNIQUE

The calculation framework presented above wasapplied to a simple homogeneous slope stability prob-lem without pore pressure or external loads. Theslope is defined by an inclination � � 30°, slopeheight H � 6 m, unit weight � � 18 kN/m3, and the twostrength functions shown in Figure 1. Figures 2a, bshow FLAC results the for the M-C criterion usingtwo different mesh geometries; a mesh includingessentially square elements (Fig. 2a), and the inclined

mesh shown in Figure 2b. Figures 3a, b show the cor-responding results for the H-B model.

The following observations are relevant withrespect to the results in Figures 2 & 3:

1. The H-B model resulted in significantly lower safetyfactors then the linear M-C model (FHB � 1.4

111

Figure 2. FLAC results for the M-C criterion. (a) Squaremesh. (b) Inclined mesh.

Figure 3. FLAC results for the H-B criterion. (a) Squaremesh. (b) Inclined mesh.

09069-14.qxd 08/11/2003 20:25 PM


compared with FMC � 1.95). The previous discus-sion showed that these two strength models aresupported equally well by the available experimen-tal information (Fig. 1). Faced with a situation inwhich two material models are equally justified bythe data, the choice between these models must bebased on their engineering consequences. In thepresent problem the H-B model delivered smallersafety factors than M-C and this model should beused in order to ensure a safe design.

2. For the M-C criterion the two meshes result in prac-tically the same safety factor. The correspondingdifference for the H-B model (�F � 0.1) is small,but it is not negligible.

3. Figures 2 & 3 show that the square and inclinedmeshes resulted in very different failure mecha-nisms. This difference is seen in terms of both dis-tribution of FLAC’s plastic point, and contours ofshear strain increments. Plastic points identify ele-ments, which are at yield (failure). However yield-ing of a particular element does not imply that theslope as a whole is at failure. Consequently, thecollection of FLAC’s plastic points does not pro-vide a clear indication of the global failure mecha-nism. In particular, some of the plastic points forthe inclined mesh are located deep in the interior ofthe slope, and they are obviously not relevant for thepurpose of identifying the slope’s failure mecha-nism. The distribution of plastic points and shearstrain increments in the square meshes appears toindicate that very large area of the slope is in astate of simultaneous failure. The inclined meshesimply failure mechanisms of the type postulated inconventional L-E calculations; namely an essentiallyrigid body sliding along a narrow transition zone.It is noted however that the “critical slip surface”implied by Figure 3b emerges above the toe of theslope. From a L-E perspective, such a surface can-not be critical, corresponding essentially to a slopewith a “reduced height”. Yet, this “unreasonable”slip surface is associated with a smaller safety fac-tor than the reasonable (but ill-defined) critical slipsurface in Figure 3a.

6 FAILURE MODES IMPLIED BY FLAC’S

S-R TECHNIQUE

Following a FLAC run, the state of stress (Mohr circles)is known at each element of the mesh. The state ofstress in failed elements satisfies Equation 1, and suchstress states are represented by Mohr circles, whichare tangential to the mobilized strength envelope. Thetangency requirement has to be satisfied with a cer-tain tolerance in order to prevent exclusion of all ele-ments. Each tangential Mohr circle is associated witha certain mesh element, which can be identified; and

the collection of all such elements represents a L-Edefinition of the critical slip surface function impliedby the S-R technique. Mesh elements defined by theabove process are shown as the open circles inFigures 2 & 3, and critical slip surfaces defined bythis process are shown as the heavy dashed lines inthose figures. In principle, the above identification offailed elements is not different from FLAC’s defini-tion of plastic points. Nevertheless the set of failedelements shown in Figure 3b is quite different fromthe set of FLAC plastic points. The source of this difference is probably related to an internal pro-gramming detail in the FLAC program. More detailedinvestigation appears to indicate that the internalFLAC criterion used for definition of plastic pointsemploys a too-large tolerance in the definition ofthese points, resulting therefore with inclusion of ele-ments which are not really at failure. Controlling theaccuracy with which the tangency requirement isenforced provides a convenient numerical mechanismeliminating at least some failed elements, which arenot relevant for definition of global slope failure (crit-ical slip surface).

The following comments are relevant with respectto the process of identifying failed elements:

1. Inferred critical slip surfaces defined by the aboveprocess are consistent with the shear strain incre-ment contours shown in Figures 2 & 3, but they pro-vide a clearer definition of the global failure mode.

2. In some cases the set of failed elements includes agroup of elements located in the vicinity of thehigh entry point of the critical slip surface. Thisgroup represents elements failing in tension ratherthan shear. The L-E critical slip surface is not welldefined in such zones.

3. Tangency points between Mohr circles and themobilized strength envelope define the normalstress acting on the critical slip surface passingthrough a given element. Consequently, the aboveprocess results with L-E definition of both criticalslip surfaces and normal stress functions.

The inferred L-E critical slip surfaces and normalstress functions resulting from the above process areshown in Figures 4 & 5, which correspond to Figures2 & 3 respectively. In those figures we have super-imposed also critical slip surfaces and normal stressesfunctions resulting from the following approximateL-E analyses:

1. Simplified Bishop’s method. The original formula-tion of this procedure was based on the linear M-Cstrength functions. For the present purpose wehave modified this classical procedure incorporat-ing in it also the H-B criterion.

2. The local linear approximation (LLA) techniquepresented by Baker (2003b). This approximation is

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based on classical Taylor analysis in which the effectof strength functions non-linearity is accounted forby an iterative procedure which utilizes Janbu’sapproximation (Janbu 1957) of the normal stressfunction. When applied to M-C material the LLAtechnique is identical to Taylor’s analysis.

It is noted that both of the above L-E proceduresare based on the a priori assumption that critical slipsurfaces can be approximated by a circular arc. Criticalslip surface inferred based on FLAC’s results do notinclude such a restriction. The following commentsare relevant with respect to Figures 4 & 5:

1. Using the square mesh; FLAC’s S-R techniqueresults with safety factors which are very close tothose based on both Bishop’s analysis and the LLAtechnique. Critical slip surfaces and normal stressfunctions inferred based on the S-R technique aresimilar but not identical, to the corresponding L-Efunctions. The difference between the critical slipsurfaces inferred based on FLAC analysis and thecorresponding L-E surfaces is mainly due to thecircular arc restriction used in the present approxi-mate L-E methods. It is frequently stated that thecircular arc restriction provides a reasonable approx-imation for homogeneous slopes. The results inFigures 4 & 5 do not support such a far-reachingconclusion. The variational formulation of L-Eproblems (Baker & Garber 1978, Baker 2003c)provides a means of avoiding a priori assumptionswith respect to the form of critical slip surfaces. Suchadvanced L-E procedures are not widely used, andthey are not considered in the present work. BothJanbu’s normal stress approximation and the normalstress function implied by Bishop’s analysis appearto be consistent with FLAC’s results. Those obser-vations are valid for both M-C and H-B criteria.

2. Using the inclined mesh; there is a small but notnegligible difference between safety factors based on the S-R technique and L-E safety factors.However the failure mechanisms implied by thesetwo approaches are significantly different. Figure 6illustrates the extent of FLAC’s mesh sensitivitywith respect to inferred critical slip surfaces andnormal stress functions for the case of H-B failurecriterion. It is noted again that the inclined meshresulted with an unreasonable critical slip surfaceemerging above the toe, while the squared mesh isconsistent with the L-E based argument that slipsurfaces emerging about the toe correspond in effectto a slope having a reduced height, and such sur-faces cannot be critical.

Using an inclined mesh the discrepancies betweenresults based on S-R and L-E techniques are morepronounced for the H-B model than the M-C crite-rion, but they exist in both cases. Based on the above

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Figure 4. Normalized critical slip surfaces and normalstress functions for the M-C criterion. (a) Square mesh. (b) Inclined mesh.

Figure 5. Normalized critical slip surfaces and normalstress functions for the H-B criterion. (a) Square mesh. (b) Inclined mesh.

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discussion it is clear that FLAC results are sensitive tomesh geometries. Using a square mesh yields moreconsistent results than use of the inclined mesh. Thisis not really surprising; some elements in the inclinedmesh have relatively high aspect ratios, and it is wellknown that results based on such meshes should beviewed with suspicion. Safety factors are relativelyinsensitive to mesh geometries. However mesh geome-tries have quite a significant effect on failure mecha-nisms. In the following we restrict attention to resultsobtained using only square meshes of the type shownin Figure 2a.

Figure 7 compares FLAC’s failure mechanisms(critical slip surfaces and normal stress functions)obtained for the M-C and H-B strength functions inFigure 1. It is seen that the critical slip surface associ-ated with the H-B strength function is shallower thanthe one associated with the M-C criterion. As a result,normal stresses acting on the H-B slip surface aresmaller than those operating along the M-C slip sur-face. Inspection of Figure 1 shows that, in the rangeof small normal stresses, the H-B criterion predictssmaller strength than M-C. It is not surprising there-fore that the safety factor FHB � 1.45 obtained using

the H-B strength law is significantly smaller than thesafety factor FMC � 1.96 associated with the M-C cri-terion. This result depends however on the slope sta-bility problem under consideration and in the followingsection we establish a more general perspective forinvestigating the effect of strength functions on resultsof slope stability calculations.

7 THE CRITICAL HEIGHT FUNCTION

Safety factors are practically useful abstractions.However, the physical significance of results obtainedby S-R or conventional L-E techniques is clear only atfailure when F � 1. At any other value of F such cal-culations deal with failure conditions of an equivalentmaterial with a reduced strength, rather than theactual physical problem. In order to avoid this con-ceptual difficulty it is convenient to study the effect ofstrength criteria on results of slope stability calcula-tions in terms of critical heights rather than safetyfactors. The critical height of a slope is defined as aheight for which the minimal safety factor is equal toone. Critical heights depend on the inclination of theslope, i.e. Hcr � Hcr(�). Figure 8 show critical heightfunctions resulting from following analyses:

1. FLAC S-R technique based on the H-B criterion(triangles).

2. Bishop analysis based on the H-B criterion (crosses).3. The LLA technique (Baker 2003b) (solid heavy

line) based on the H-B criterion.4. FLAC S-R technique based on M-C criterion.

(open circles).5. The LLA technique based on the linear M-C crite-

rion (light solid line). It is noted that for homoge-neous slopes and a M-C strength criterion thistechnique is reduced to classical Taylor analysis.

It is noted that both the H-B and M-C criteria are fit-ted to the same experimental data set obtained forcompacted Israeli clays (Fig. 1), and both models rep-resent this data equally well (sum of squares ratio isequal to 0.95).

The following comments are relevant with respectto the results in Figure 8:

1. In the linear M-C framework the safety factor ofslopes with � � � is always larger than 1.Consequently a critical height function based onthis criterion is asymptotic to a vertical line locatedat � � �. It can be verified that the non-linear H-Bcriterion implies finite critical heights for all non-zero slope inclinations.

2. Using the linear M-C criterion corresponding to thedata set in Figure 1, Taylor’s analysis shows the crit-ical height of a vertical slope is 3.9 m. On the otherhand, analysis based on the H-B criterion fitted tothe same data set implies that slopes steeper than

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Figure 7. Critical failure mechanisms associated with theM-C and H-B criteria.

Figure 6. Mesh effect on inferred critical slip surfacesimplied by the S-R technique (H-B failure criterion).

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approximately 55° are not stable. The very signifi-cant difference between prediction of the two mod-els illustrate very clearly the importance of usingnon-linear failure criteria in stability calculationsof steep slopes.

3. Recalling that the M-C and H-B failure criteriaprovide equally valid descriptions of the compactedIsraeli clay’s data set, the practical implications of Figure 8 are worrying. The figure shows thatequally valid interpretations of the same experimen-tal information may under certain conditions lead to diverging practical implications (i.e. HCrHB(� 60°) → 0, HCrMC (� � 90°) � 3.9 m.

4. The physical basis of the above result is related tothe following observations:

a) Critical slip surfaces for steep slopes areshallow, resulting therefore with small normalstresses acting on this surface.

b) A T � 0 H-B model predicts significantlysmaller shear strength at small normal stressesthan a M-C criterion with a non-zero cohesion.

5. There is a range of slope inclinations(28° � � � 34°) in which the critical height pre-dicted by the M-C criterion is slightly larger thenthose predicted on the basis of the H-B law. Thephysical reason for this behavior is discussed byBaker (2003b).

6. The open circles in Figure 8 show results of FLACanalysis based on the M-C criterion. Those resultsare almost identical with results based on the clas-sical Taylor analysis (light solid line).

7. For the non-linear H-B criterion, the critical heightfunctions based on Bishop analysis, and FLAC’s S-Rtechnique are practically identical with resultsbased on Baker’s (2003b) L.L.A. technique. Thisobservation supports the validities of all three cal-culation methods. It is noted that Bishop’s analysisis restricted to circular slip surfaces and it is notexpected to yield good results in non-homogeneous problems. Both FLAC and LLA canbe applied to non-homogeneous problems.

8. It is important to realize that the critical heightfunctions shown in Figure 8 are relevant for thecompacted Israeli clays data set (Fig. 1), and thisfigure does not represent a general relations.However, qualitatively similar results were obtainedalso for a number of other data sets.

8 SUMMARY AND CONCLUSIONS

Two general approaches (FLAC’s strength reductiontechnique and conventional limit equilibrium calcula-tions) for analysis of slope stability are discussed andcompared. The present work extends previous presen-tations on this subject in number of respects:

1. The comparison includes failure mechanisms (crit-ical slip surfaces and normal stress functions) inaddition to safety factors.

2. The comparison is done for both the linear Mohr-Coulomb failure criterion and the non-linear Hoekand Brown strength function.

Mesh geometry effects on FLAC’s results are pre-sented and discussed. It is shown that a mesh consist-ing of essentially square elements results in moreconsistent results than meshes including relativelyslender elements. Safety factors are relatively insensi-tive to mesh geometries, however failure mechanismsdepend very strongly on mesh geometry, and using a mesh which includes slender elements may lead to wrong conclusions with respect to critical slip surfaces and normal stress functions inferred on thebasis of FLAC’s S-R technique. FLAC’s mesh sensi-tivity is more pronounced for non-linear strengthfunctions than in the linear M-C case. It is noted that FLAC’s plastic points do not provide a reason-able measure of the potential failure zone. A proce-dure which identifies the critical slip surface based on FLAC’s calculated stresses is presented and dis-cussed. Considering results obtained using squaremeshes, FLAC’s strength reduction technique andconventional limit equilibrium procedures yield similar failure mechanisms and safety factors. Thisconclusion is valid for both linear and non-linear failure criteria.

Engineering implications of linear and non-linearfailure criteria in the context of slope stability analy-sis are presented and discussed. In particular it isshown that equally valid interpretations of the sameexperimental information may under certain condi-tions (steep slopes) lead to very different engineer-ing implications. Under such conditions the choicebetween strength functions must be based on thepractical implications of these laws, which in mostcases means using the Hoek-Brown criterion ratherthan the conventional linear Mohr-Coulomb strengthfunction.

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Figure 8. Critical slope heights for compacted Israeli clays.

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REFERENCES

Baker, R. & Garber, M. 1978. Theoretical analysis of the sta-bility of slopes. Geotechnique, 28(4): 395–411.

Baker, R. 2003a. Inter-Relation between experimental andcomputational aspects of slope stability analysis. Inter.Jour. Numer. Anal. Meth. Geamech. 27(5): 379–401.

Baker, R. 2003b. Non-linear strength envelopes based on tri-axial data. Accepted for publication in J. Geotech. AndGeoenvir Engrg., ASCE.

Baker, R. 2003c. Sufficient conditions for existence of phys-ically significant solutions in limiting equilibrium slopestability analysis. Accepted for publication in Inter. Jour.of Solids and structures.

Bishop, A.W., Webb, D.L. & Lewin, P.I. 1965. Undisturbedsamples of London clay from the Ashford common shaft:strength-effective normal stress relationship. Geotech-nique, 15(1): 1–31.

Charles, J.A. & Soares, M.M. 1984. The stability of slopeswith nonlinear failure criterion. Cand. Geoth. J., 21(3):397–406.

Day, R.W. & Axten, G.W. 1989. Surficial stability of com-pacted clay slopes. J. Geoth. Eng. ASCE, 115(4): 577–580.

Dawson, B.M., Roth, W.H. & Drescher, A. 1999. Slope sta-bility factors of safety by strength reduction. Geotechnique,49(6): 835–840.

Donald, I.B. & Giam, S.K. 1988. Application of nodal dis-placement method to slope stability analysis. Proc. 5thAustralia-New Zealand Conf. on Geomech., Sydney,Australia, 456–460.

Frydman, S. & Samoocha, Y. 1984. Laboratory studies onIsraeli clays for reservoir embankment design. Proc. 5thInter. Conf. on Expansive soils, Adelaide, SouthAustralia, 94–98.

Griffith, D.V. & Lane, P.A. 1999. Slope Stability analysis byfinite elements. Geotechnique, 49(3): 387–403.

Hoek, E. & Brown, E.T. 1980. Empirical strength criterionfor rock masses. ASCE Jour. Geotech. Eng., 106(9):1013–1035.

Jiang, J.C., Baker, R. & Yamagami, T. 2003. The effect ofstrength envelope nonlinearity on slope stability compu-tations. Can. Geoteh. J., 40(2): 308–325.

Matsui, T. & San, K.C. 1992. Finite element slope stabilityanalysis by shear reduction technique. Soils andFoundations, 32(1): 59–70.

Maximovic, M. 1979. Limit equilibrium for non-linear fail-ure envelope and arbitrary slip surface. Proc. 3rd Intr.Conf. on Numerical Methods in Geomechanics, 769–777.

Maximovic, M. 1989. Nonlinear failure criterion for soils.J. Geoth. Eng. ASCE, 115(4): 581–586.

Naylor, D.J. 1962. Finite element and slope stability. Nume,Meth. un Geomech., Proc. NATO advanced study institute.Lisbon, Portugal, 229–244.

Penman, A. 1953. Shear characteristics of saturated silt intriaxial compression. Geotechnique. 15(1): 79–93.

Perry, J.A. 1994. A technique for defining non-linear shearstrength envelopes and their incorporation in slope stabil-ity method of analysis. Quart. J. of Eng. Geology, 27(5):231–241.

Ugai, K. 1989. A method of calculation of total factor ofsafety slopes by elasto-plastic FEM. Soils and Foundations,29(2): 190–195.

Ugai, K. & Leshchinsky, D. 1995. Three-dimensional limitequilibrium and finite element analyses: a comparison ofresults, Soils and Foundations, 35(4): 1–7.

Zienkiewicz, O.C., Humpheson, C. & Lewis, R.W. 1975.Associated and non-associated visco-plasticity and plas-ticity in soil mechanics. Geotechnique, 25(4): 671–689.

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3D slope stability analysis at Boinás East gold mine

A. Varela Suárez & L.I. Alonso GonzálezRío Narcea Gold Mines, S.A., Belmonte de Miranda, Principado de Asturias, Spain

ABSTRACT: The Boinás East open pit mine has been exploited by the gold producer Río Narcea Gold Minesin Northwest of Spain. Considering the influence of the radius of curvature on the factor of safety for slopeangle, and taking into account that a small increase in the overall angle will result in a very high increment inthe total amount of the ore mined, the slope stability analysis was made using the finite difference code FLAC3D

to calculate the factor of safety by reducing the rock shear strength. Due to the existing complicated geologyand the complexity of the 3D geometrical modeling, a “FISH routine” was used to import the block model ofthe mine into the FLAC3D program. This block model is the database commonly used in the mine works and wasgenerated with Datamine. This method is a very good tool to generate a complex model in FLAC3D.

1 INTRODUCTION

Río Narcea Gold Mines has been operating from1997 the gold deposit El Valle-Boinás, located in theNorthwest of the Iberian Peninsula, within the well-known Rio Narcea Gold Belt (Fig. 1).

The deposit is in the environs of the town ofBoinás, Belmonte de Miranda, in the Principality ofAsturias. It’s formed by five separated bodies locatedaround a granitic stock. The mineralization consistsof various skarn types and zones with silicificationsand significant epithermal oxidation.

Of these five ore bodies, three have been operated(El Valle, Boinás East and Boinás west) by open pit

mining techniques, being at the present time solely inproduction the deposit of El Valle. The present articletries to develop the methodology used for the designof stable slopes in the deposit of Boinás East, doinganalysis of stability by means of FLAC3D software.

2 SITE GEOLOGY

The Rio Narcea Gold Belt structure of 17 km in lengthhas an approximate direction N 35° E and includes, inaddition to the mentioned deposit Valle-Boinás, at leastfive other gold mineralizations, some of them widelyoperated during the rule of the Roman Empire.

Geologically the gold belt consists of an anticlineof Hercynian age, in the core of which there are thecarbonate materials of the Láncara Formation (MiddleCambrian), above which are shale and sandstone of theOville Formation (Middle-Upper Cambrian). Goldmineralization was initially deposited as calcic andmagnesic copper-gold skarns at the contact betweenthe Boinas granodiorite and limestone and dolomiteof the Lancara Formation (Martin-Izard et al. 1998,Cepedal 2001).

The auriferous mineralization mainly occurred dur-ing the phase of retro-gradation of the metamorphicprocess, to temperatures between 450°C and 250°C,separated in two stages (Cepedal 1998, 2001). Duringthe Lower Permian, after an important dismantling ofthe hercynic relief, takes place the location of subvol-canic and porphyritic dikes that originate hydrothermalalterations with important silicification of pre-existingFigure 1. 3D diagram of E1 Valle-Boinás deposit.

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rocks, to temperatures between 150°C and 250°C andsmaller pressure of 0.2 kbar. This process gives rise tothe formation of oxidized and very brecciated materi-als, with bad geotechnical quality and that approxi-mately constitute 90% of the operated material in theopen pits.

The gold mineralization remained hidden by Tertiarylacustrine deposits, which as well were partially hiddenby Alpine thrusts that placed an important repetitionof limestone of the Láncara Formation above theTertiary and the sandstone of the Oville Formation.Figure 1 shows a scheme of the zone of the deposit ElValle-Boinás and Figure 2 a scheme of the whole RioNarcea Gold Belt.

3 PRELIMINARY ANALYSIS

From the geotechnical data of the different materialstaken in from the geological exploration holes, theshear strength of the different lithologies were deter-mined following the Bieniawski (1989) classificationand the Hoek-Brown 99 methodology.

With the obtained values SRK Ltd. carried out thefeasibility study in October 1996, updating it in 1999.

The analyses were carried out using the XSTABLsoftware over sections with simplified geology andconsidering different overall slope angles. Consideringthe slope totally drained, 330 meters of vertical height,the Bishop’s method and circular surfaces of failure,were obtained the factor of safety showed in the Table 1(SRK 1996, 1999).

The geology and the rock mass properties used bySRK in their analysis are summarized in Table 2.

4 NUMERICAL MODELING WITH FLAC

The holes drilled during the year 1998 defined a newmineralized zone amenable to extraction by open pit

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Figure 2. Geological scheme of the Rio Narcea Gold Belt.

Table 1. Feasibility study data.

Overall slope angle(°) Factor of safety

55 0.9150 1.0345 1.15

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provided it would be possible to increase the over allangle of the East slope of the pit.

In the following paragraph will be described indetail the stability analysis carried out using moredetailed geological models and two and three dimen-sional finite-difference programs.

4.1 Preliminary analysis

Considering the effect of the radius of curvature ofthe pit in the global stability (Lorig 1999), differentanalysis were carried out using the program FLAC(Itasca 1998).

In collaboration with Itasca Consultores S.L., wehave made different analysis over a representativesection, contemplating plane-strain and axisymmetricconditions.

In first of them one assumes that the slope extendsindefinitely in the perpendicular direction of the analy-sis plane and that deformations in the perpendiculardirection of this plane do not exist. This type would becompatible with the one made in the feasibility studywith the XSTABL software. In the case of axisym-metric analysis one assumes that the slope has truncated

cone form. The graphical representation of both casesis indicated in Figure 3.

Table 3 shows the factor of safety obtained for thesame section and using the same strength parameters.In that table it shows also the factor of safety obtainedin a back analysis of the slope already excavated inthe pit as Phase 1. The strength parameters are thosegiven in Table 4, and the slope was considered to befully drained.

The breakage mechanism that takes place is verysimilar in both types of analysis. It consists of theshear failure of the materials located below the mainalpine thrust and tensile failure in the dolomite abovethat thrust. In the case of the axisymmetric analysis,the tensile failure is cushioned, increasing in this waythe factor of safety of the slope.

In following figures show a schematic of the geol-ogy (Fig. 4), the failure mechanism in the case ofplane-strain geometry (Fig. 5), and the failure in thecase of axisymmetric analysis (Fig. 6).

Studying at the values of the obtained factors ofsafety it is clear that the analyses of three-dimensionalgeometry provide higher and more realistic factors ofsafety, considering the influence of the radii of curva-ture previously mentioned. In any case, the axisym-metric considerations are too optimistic, since in thereality the analyzed topography will not be totally

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Table 2. Rock mass properties in the SRK model.

� Cohesion DensityLithology (°) (kPa) (t/m3)

Good Quality Lancara Limestone 37 293 2.70Fair Quality Lancara Limestone 31 258 2.70Lower Oville Sandstone 33 275 2.50Fair Quality Granite 34 290 2.60Ore (MIN) 32 265 2.50Marble 32 265 2.70

Figure 3. Plane-strain and axisymmetric analysis withFLAC.

Table 3. Factor of safety obtained with FLAC.

Type of Slope angle

Factor of Phase analysis Upper* Lower* safety

Final Plane 65 45 1.00–1.05Final Axisymmetric 65 45 1.65–1.701 Plane 50 50 1.20–1.30

* Upper/lower means above or below the main Alpine thrust.

Table 4. Rock-mass properties.

K G � Cohesion DensityLithology (GPa) (GPa) (°) (kPa) (t/m3)

Upper Sandstone 3.75 2.17 49 720 2.65Upper dolomite 7.92 5.09 43 1120 2.70Lower Sandstone 1.32 0.75 21 270 2.29Alterd. Granite 1.17 0.66 20 250 2.28Ore (MIN) 1.11 0.62 19 250 2.21Fresh granite 16.99 11.52 65 4770 2.67Fresh skarn 10.63 6.79 56 2430 3.17Marble 2.45 1.41 33 500 2.43Hornfels 23.24 16.39 62 11330 2.77Tertiary 4.64 2.96 38 650 2.25Black skarn 10.63 6.79 56 2430 3.17

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conical and the different lithological units have a verymarked dip towards the East.

For the final design of the stable slope in theBoinás East pit, the FLAC3D software (Itasca 1997)

was used. This program allows define a more accu-rate topographical and geological model.

4.2 FLAC3D modeling

Three-dimensional model construction of a geologi-cally complex deposit such as El Valle-Boinás can bean arduous task. In this case a “FISH routine” wasdesigned that allows the geological block model to beimported directly into FLAC3D.

In a first step, using the DATAMINE program, dif-ferent block models were created deactivating thoseblocks located over the topography that we try to ana-lyze. The X, Y and Z coordinate of the center of eachblock is exported into a data file, as well as its lithology.

In our particular case the mining blocks were 4meters side bricks, that were rebuilt to 16 meters sidebricks, with the intention of making the model usable.The blocks thus obtained were placed in the own localcoordinates, and more blocks were defined around themodel in order to avoid that the artificial contour con-dition do not affect the results of the stability analysis(Lorig et al 2000). Figure 7 shows the geology on thesurface of the final pit plotting the DATAMINE blockmodel, and the Figure 8 shows the geological modelloaded into FLAC3D.

The FISH routine was used to generate the 16meter side blocks using the coordinates of the centerof the blocks taken from DATAMINE (file“BE16X16A.txt”). The first line of this file has theidentification of the rest of the parameters, and it willnot be imported to the FLAC3D program. The wholeroutine is shown in the appendix.

4.3 Rock-mass properties

The rock-mass properties used in all the stabilityanalysis are indicated in Table 4. The tensile strengthhas been considered to be a tenth of the cohesion ofeach material. To obtain the factor of safety of theproposed model the shear-strength reduction tech-nique was used. To perform slope-stability analysis,simulations are run for a series of increasing trial fac-tors of safety, f, until the slope fails. At failure, thesafety factor equals the trial safety factor (i.e. f � F)(Lorig et al. 2000).

Actual shear strength properties, cohesion (c) andfriction (�), are reduced for each trial according to thefollowing equations:

(1)

(2)

The reduction in the shear strength properties is madesimultaneously for all materials.

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Figure 4. Summarized geology.

Figure 5. Mode of failure in a plane-strain analysis.

Figure 6. Mode of failure in an axisymmetric analysis.

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Figure 7. Datamine block model.

Figure 8. FLAC3D block model.

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5 RESULTS AND CONCLUSIONS

Once analyzed the different proposed models, it wasproposed an open pit with lower level at 340 metersASL, a maximum slope height (in the East wall) of345 meters. A factor of safety of 1.45 was obtainedwith slope angles of 70° in the dolomite above the mainthrust, 60° in upper sandstone, fresh granite, freshskarn, tertiary, and black skarn, and 50° in the rest.

A comparison between the factors of safetyobtained for the different models and programs usedare provided in Table 5.

The minimum factor of safety values are obtainedin the East wall of the pit. Figure 9 shows the FLAC3D

model with the zone of maximum displacement. Thesezone correspond with a convex geometry in the slope,therefore steeper slopes could have been consideredin areas where the pit slopes were concaves, but thesewas not our case. We only have changed the face angleaccording with the geology and not with its geometry.

As we can see in the factor of safety obtained in thedifferent model, the influence of the radii of curvaturein the global stability of a pit is very considerable,specially taking into account that a small increasingin the overall face angle results in a very large amountof ore recovered, as it was in our case.

On the other hand, the hard work required to designa model for a complex deposit in three dimensions isavoided when we import the block model into FLAC3D.With the routine described above, it is easy to create ablock model with DATAMINE, or whatever otherprogram, and delete the block above the surface wewant to analyze, and them import all the model toFLAC3D and obtain a factor of safety.

It must be take into account that with this systemwe are going to have free faces at 90º and the high ofthe block size, so we must confirm that the factor ofsafety obtained corresponds to the slope factor andnot to the brick face.

ACKNOWLEDGEMENTS

The authors would like to thank Mr. Manuel G.Fernández of Río Narcea Gold Mines for his greatwork with Datamine, and Mr. Alan Riles, COO of RíoNarcea Gold Mines Ltd. for the valuable help in thetranslation of the paper. Finally the authors are grate-ful to Mr. Pedro Varona of Itasca Consultores, S.L. forhis technical support. Thanks are extended also to Mr.Pedro Velasco and Ms. Montserrat Senís for their help.

REFERENCES

Cepedal, A., Martin-Izard, A., Fuertes, M., Pevida, L.,Maldonado, C., Spiering, E., Gonzalez, S. & Varela, A.1998. Fluid Inclusions and Hydrothermal Evolution ofthe El Valle-Boinas Copper-gold Deposits. In Arias, A.,Martin-Izard, A. & Paniagua, A. (eds), Gold Explorationand mining in NW Spain: 50–58. Oviedo.

Cepedal, M.A. 2001. Geología, Mineralogía, Evolución yModelo Genético del yacimiento de Au-Cu de El Valle-Boinas. Belmonte (Asturias). Ph.D. thesis, University ofOviedo.

Itasca Consulting Group, Inc. 1997. FLAC3D (Fast LagrangianAnalysis of Continua in 3 Dimensions), Version 2.0.Minneapolis: Itasca.

Itasca Consulting Group, Inc. 1998. FLAC (Fast LagrangianAnalysis of Continua), Version 3.4. Minneapolis: Itasca.

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Table 5. Factor of safety for different slope angles and withdifferent types of analysis.

Face angleDeepest Type of Factor

Phase A1 B2 level Program analysis of safety

1 50 50 404 FLAC Plane 1.256 65 45 340 FLAC Plane 1.056 65 45 340 FLAC Axis 1.706 70 45 340 FLAC Axis 1.506 50 50 350 FLAC3D 1.456 70 50 340 FLAC3D 1.503

6 70 50 300 FLAC3D 1.35

1 A represents the face angle of the slope located above themain Alpine Thrust. It represents the slope angle in theUpper Limestones.2 B represents the face angle in the materials located belowthe main Alpine Thrust.3 The higher value of FoS in the steeper design is due to achange in the lithology present on that slope.

Figure 9. Zone with the maximum displacement in thefinal Boinás East pit.

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Lorig, L. 1999. Lessons learned from slope stability studies.In Detournay & Hart (eds), FLAC and NumericalModeling in Geomechanics: 17–21. Rotterdam: Balkema.

Lorig, L. & Varona, P. 2000. Practical Slope-StabilityAnálisis Using Finite-Difference Codes. In Hustrulid, W.A.,McCarter, M.K. & Van Zyl, D.J.A. (eds), Slope Stabilityin Surface Mining: 115–124. Colorado: Society forMining, Metallurgy and Exploration, Inc.

Martin-Izard, A., Cepedal, A., Fuertes, M., Pevida, L.R.,Maldonado, C., Spiering, E., Varela, A. & Gonzalez, S.1998. The El Valle Deposit: an example of Koper-goldSkarn Mineralization overprinted by late epithermalevents. Cantabrian Mountains, Spain. In Arias, A.,Martin-Izard, A. & Paniagua, A. (eds), Gold Explorationand mining in NW Spain: 43–50. Oviedo.

Steffen, Robertson & Kirsten (UK) Ltd. 1996. Investigationinto the Stability of Proposed Excavated Slopes andExcavatability of Materials at El Valle, Boinas West andBoinas East. Report to Rio Narcea Gold Mines, S.A.Report no. ADM/752MH001.REP, October 1996.

Steffen, Robertson & Kirsten (UK) Ltd. 1999. Boinas EastOpen Pit Verification of Overall Slope Angles for pirOptimisation Studis. Report to Rio Narcea Gold Mines,S.A., January 1999.

APPENDIX – FISH ROUTINE

newdef creamallaarray aa(11488);(number of lines in file *.txt)status � open(‘BE16x16a.txt’,0,1)status � read(aa, 11488)loop k(2, 11488); no lee la primera líneaxx � parse(aa(k),1)yy � parse(aa(k),2)zz � parse(aa(k),3)xxmax � max(xx,xxmax)mat � parse(aa(k),4)xc � (xx-1)*16 662;put blocks in x localyc � (yy-1)*16 9745;put blocks in y localzc � (zz-1)*16x0 � xcx1 � xc 16x2 � xcx3 � xcx4 � xc 16x5 � xcx6 � xc 16x7 � xc 16y0 � ycy1 � ycy2 � yc 16y3 � ycy4 � yc 16y5 � yc 16y6 � ycy7 � yc 16z0 � zc

z1 � zcz2 � zcz3 � zc 16z4 � zcz5 � zc 16z6 � zc 16z7 � zc 16commandgen zon bri p0 x0 y0 z0 p1 x1 y1 z1 p2 x2 y2 z2

p3 x3 y3 z3 p4 x4 y4 z4 &p5 x5 y5 z5 p6 x6 y6 z6 p7 x7 y7 z7 size 1 1 1

group matend_command

end_loopstatus � close

endcreamalla

123

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Underground cavity design

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127

The effect of tunnel inclination and “k” ratio on the behavior of surrounding rock mass

M. Iphar, M. Aksoy, M. Yavuz & G. OnceOsmangazi University, Mining Engineering Dept., Eskisehir, Turkey

ABSTRACT: Rock behavior around tunnels excavated in the same rock with various inclinations and “k” ratioshas been investigated by numerical analysis employing the FLAC3D finite difference code. Stress distributions anddisplacements at the critical points of the tunnels in underground mining have been examined. Observed stress anddisplacement values with respect to the change in tunnel inclination and “k” ratio have been analyzed by using sta-tistical methods employing “multiple regression analysis” in order to find out a meaningful correlation between thestress, displacement values and the inclination and “k” ratios. Statistical analyses have presented meaningful corre-lations giving mathematical equations whose dependent variable is displacement or stress and independent variablesare tunnel inclination and “k” ratio.

1 INTRODUCTION

Numerical modeling is a very powerful and usefultool used widely in designing underground structuressuch as tunnels, roadways, caverns etc. The displace-ments and stresses around the underground openingscan be predicted by employing numerical modeling inadvance.

During a project carried out for the GLI (WesternLignite Company in Turkey), a main roadway inclinedat 8° dip has been designed down to a depth of 500 min the underground coal colliery (Once et al. 2001a).The geomechanical properties of the rock mass wherethe roadway will be driven have been obtained fromlaboratory tests (Çekilmez 1988). In the light of thisproject, a new study has been carried out to investi-gate the effect of “k” ratio (the ratio of horizontalstress to vertical stress) and tunnel inclination on therock mass behavior in terms of displacements andstresses. To achieve this goal, the FLAC3D finite dif-ference code has been used.

After the numerical modeling, statistical analyseshave been performed to find out a meaningful cor-relation explaining the effects of “k” ratios and theinclination on the stresses and displacements.

2 NUMERICAL MODELING STUDIES

2.1 Applied method

Forty different numerical models have been formed inFLAC3D (Itasca 1997). During the model formation, the

tunnel inclinations have been varied between 0 and 45degrees in 5 degrees intervals while the “k” ratios havebeen varied between 0.5 and 2 in 0.5 intervals.

Three points have been selected to observe the dis-placements around the tunnel. These history pointshave been located on the center-line of the tunnel, oneat the roof and one at the floor, and one at axis level inthe sidewall. The stresses in the zones adjacent to thehistory points have also been monitored. To evaluatethe rock behavior properly, the coordinates of the his-tory points have been kept at the same coordinates ineach of the models although the tunnel inclination hasbeen changed.

Displacement values and maximum (�1) and mini-mum (�3) principal stresses observed in three direc-tions at the roof, floor and sidewall of the tunnel havebeen subjected to the statistical analysis.

After carrying out multiple regression analysis fordisplacements and �1, �3 principal stresses at the roof,floor and sidewall in x, y and z directions, regressionmodels have been proposed to predict the displace-ments and stresses.

Displacements in the x-direction at the historypoints in the roof and the floor have not been includedbecause these points lie on the plane of the symmetryfor the models.

2.2 Rock mass properties

The rock mass in which the tunnel will be driven ismarl formation. The geomechanical properties of themarl were determined from MTA (General Directorate

09069-16.qxd 08/11/2003 20:26 PM Page 127


of Mineral Research & Exploration Institution) drillingsand discontinuity spacing were obtained using theapproach proposed by Priest & Hudson (1976) becauseof the lack of information about discontinuities in theMTA report (Çekilmez 1987).

The RMR value was calculated using the RMR clas-sification system described by Bieniawski (1979) andtheir ratings are shown in Table 1.

2.3 Numerical models in FLAC3D

The rock mass has been assumed to be an isotropic,homogenous material. It has been modeled as a Mohr-Coulomb material through the study.

The geomechanical properties used in the numericalmodeling have been taken from the GLI project as men-tioned before and the values of these properties aregiven in Table 2 (Once et al. 2001b).

In-situ stresses have been calculated by using thefollowing equations.

�z � � � h (ton/m2)�x � �y � k � �z (ton/m2)

where �z � vertical stress; � � density of marl;h � depth, �x and �y � lateral stresses, k � “k” ratio(Hoek & Brown 1980).

During the modeling, the presence of groundwaterwas ignored. The tunnel, whose dimensions are given inFigure 1, has been simulated as a single step excavation.

One of the models with fixity condition and coor-dinate system is shown in Figure 2.

3 RESULTS

Displacement and stress values obtained from themodeling studies have been evaluated separately con-sidering “k” ratio and inclination of the tunnel.

3.1 Evaluation of models in terms of displacements

The y and z displacements at the roof have been cate-gorized in terms of “k” ratio and tunnel inclination. InFigure 3, it is observed that the magnitude of the y displacements was not significantly affected by vary-ing the “k” ratio but was greatly affected by varyingthe tunnel inclination.

It can be seen from the Figure 4 that displacements inthe roof in the z-direction generally increase as the“k” ratio increases for the case where the inclination

128

Table 1. RMR calculation for marl.

Geomechanical properties Values Rating

Uniaxial 11.2 MPa 2compressive strength

RQD 61.4% 13Spacing of joints 80–200 mm 8Condition of joints Slightly rough surfaces, 20

separation �1 mm,soft joint wall rock

Ground water 115.2 lt/min 4Joint orientation Unfavorable �10RMR 37

3.5 m

4.60 m

Figure 1. Tunnel geometry.

Table 2. Rock mass properties used inmodeling (Çekilmez 1988).

Property Values

Poisson’s ratio 0.25Bulk modulus (MPa) 790Shear modulus (MPa) 475Tensile strength (MPa) 0.28Internal friction angle (°) 37Cohesion (kPa) 70Density (kg/m3) 2500

Figure 2. The FLAC3D model with fixity condition andcoordinate system.

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is constant. On the contrary, displacements in the roofin z-direction generally decrease as the inclinationincreases for the case where the “k” ratio is constant.Moreover, it should be noted that increasing the inclination angle has a great effect on the amount ofvertical displacement in the case of a high “k” ratio.

The proposed regression equation for the displace-ments in the y-direction at the roof utilizing the datafrom the models studied is given in Equation 1:

(1)

where Ydisprf � y-displacement in the roof; k � “k”ratio and I � inclination of the tunnel (°).

The predicted values obtained from the proposedequation are plotted against the values from thenumerical models in Figure 5. This regression modelshows a strong correlation between the observed andpredicted values (r � 0.99).

The proposed regression equation for the displace-ments in the z-direction in the roof is presentedbelow:

where Zdisprf � z-displacement at the roof; k � “k”ratio and I � inclination of the tunnel (°).

The predicted values are plotted versus the observedvalues in Figure 6. This regression model also showsa strong correlation between the observed and pre-dicted values (r � 0.98).

The magnitude of the y displacements for the historypoint in the floor was greatly affected by the variation

129

k-ratioIncline (˚)

Zdisprf(m)

0

5045403530252015105

0.5

1.0

1.5

2.0

0.0

-0.3

-0.6

-0.9

-1.2

-1.5

-1.8

Figure 4. z displacements vs “k” ratio and inclination (roof ).

Ydisprf(m)

k-ratio Incline (o)

05101520253035

4550

40

0.2

-0.2

-0.4

-0.6

-0.8

-1.0

0.0

1.5

1.0

2.0

0.5

Figure 3. y displacements vs “k” ratio and inclination (roof).

-1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2

Observed Values (m)

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Pred

icte

d V

alue

s (m

)

Figure 5. Observed vs predicted values in y-direction (roof ).

-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

Observed Values (m)

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0Pr

edic

ted

Val

ues

(m)

Figure 6. Observed vs predicted values in z-direction (roof ).

(2)

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of “k” ratio and tunnel inclination. As can be seenfrom Figure 7, the horizontal (y) displacementsincrease with an increase in both the “k” ratio andtunnel inclination.

Figure 8 shows that the magnitude of the verticaldisplacements in the floor increased with an increasein the “k” ratio. On the contrary, the vertical displace-ments decrease as the tunnel inclination increases.

The regression equation proposed for predictingthe displacements in the y-direction in the floor isgiven below:

(3)

where Ydispfl � y-displacement at the floor; k � “k”ratio and I � inclination of the tunnel (°).

The predicted values versus observed values in thefloor in the y-direction are shown in Figure 9. Thisregression model shows a strong correlation betweenthe observed and predicted values (r � 0.99).

The regression equation proposed for predictingthe z displacements in the floor is as follows:

where Zdispfl � z-displacement at the floor; k � “k”ratio and I � inclination of the tunnel (°).

The predicted values versus observed values in thefloor in the z-direction are shown in Figure 10. Thisregression model also shows a strong correlationbetween the observed and predicted values (r � 0.99).

130

Ydispfl(m)

Incline (˚) k-ratio

0.9

0.6

0.3

0.0

-0.3

-0.6

2.0

1.5

1.0

0.5

5045

35

25

15

40

30

20

105

0

Figure 7. y displacements vs “k” ratio and inclination (floor).

k-ratioIncline (˚)

Zdispfl(m)

1.8

1.5

1.2

0.9

0.6

0.3

2.0

1.5

1.0

5.0

50

3545

3040

2520

1510

50

Figure 8. z displacements vs “k” ratio and inclination (floor).

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Observed Values (m)

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Pred

icte

d V

alue

s (m

)

Figure 9. Observed vs predicted values in y-direction (floor).

0.1 0.2 0.3 0.4 0.5 0.6 0.7Observed Values (m)

0.1

0.2

0.3

0.4

0.5

0.6

0.7Pr

edic

ted

Val

ues

(m)

Figure 10. Observed vs predicted values in z-direction(floor).

(4)

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The x displacements in the sidewall are not affectedby the change in the tunnel inclination. But they areaffected by the change in the value of “k” ratio as seenfrom Figure 11.

In the case of the y displacements in the sidewallshown in Figure 12, they tend to increase as the tunnelinclination increases. When the change in the “k”ratio is considered, it should be noted that there is nosignificant increase in the y displacements up to 20°tunnel inclination. After this inclination, y displace-ments have increased as “k” ratio increased.

Figure 13 shows that vertical displacements at thesidewall increase due to an increase in the “k” ratio.However, the vertical displacements decrease due toan increase in the tunnel inclination.

The proposed regression equation for the displace-ments in the x-direction at the sidewall is given inEquation 5:

(5)

where Xdispsdw � x-displacement at the sidewall;k � “k” ratio and I � inclination of the tunnel (°).

The predicted values versus observed values in the sidewall in the x-direction are shown in Figure 14.This regression model also shows a strong correlationbetween the observed and predicted values (r � 0.99).

131

k-ratio Incline (˚)

Xdispsdw(m)

0.5

1.0

1.5

2.0

5040

3020

10

4535

2515

50

0.0

-0.2

-0.4

-0.6

-0.8

Figure 11. x displacements vs “k” ratio and inclination(sidewall).

k-ratioIncline (˚)

Zdispsdw(m)

0.60

0.45

0.30

0.15

2.01.5

0.5

1.0

50

40

30

20

100

45

35

25

155

Figure 13. z displacements vs “k” ratio and inclination(sidewall).

k-ratioIncline (˚)

Ydispsdw(m)

2.0

1.5

1.0

0.5

5045

35

25

155

40

30

20

100

0.20

0.15

0.10

0.05

0.00

-0.05

Figure 12. y displacements vs “k” ratio and inclination(sidewall).

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0

Observed Values (m)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

Pred

icte

d V

alue

s (m

)

Figure 14. Observed vs predicted values in x-direction(sidewall).

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The regression model recommended to predict they displacements in the sidewall is given below:

(6)

where Ydispsdw � y-displacement at the sidewall;k � “k” ratio and I � inclination of the tunnel (°).

The predicted values versus observed values in thesidewall in the y-direction are shown in Figure 15.This regression model shows a strong correlationbetween the observed and predicted values (r � 0.99).

The regression model proposed for predicting the z displacements in the sidewall is as follows:

(7)

where Zdispsdw � z-displacement at the sidewall;k � “k” ratio and I � inclination of the tunnel (°).

The predicted values versus observed values in thesidewall in the z-direction are shown in Figure 16.This regression model also shows a strong correlationbetween the observed and predicted values (r � 0.99).

The correlation coefficients of all the proposedregression models for displacements have been sum-marized in Table 3.

3.2 Evaluation of models in terms of stresses

As the observed principal stresses are examined, nostrong relationship between the principal stresses andthe “k” ratio or the tunnel inclination has been foundexcept for the principal stresses in the tunnel floor.Therefore, only principal stresses in the floor are takeninto consideration.

The graphs of maximum and minimum principalstresses are given in Figures 17 and 18 respectively.

The regression equation proposed for predictingthe magnitude of �1 in the floor and the correlationcoefficient are given below:

(8)

r � 0.95

132

Table 3. Correlation coefficients for displacements.

Correlation coefficient (r)

Direction Roof Floor Sidewall

x – – 0.99y 0.99 0.99 0.99z 0.98 0.99 0.99

0.0 0.1 0.2 0.3 0.4 0.5

Observed Values (m)

0.0

0.1

0.2

0.3

0.4

0.5

Pred

icte

d V

alue

s (m

)

Figure 16. Observed vs predicted values in z-direction(sidewall).

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20

Observed Values (m)

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

Pred

icte

d V

alue

s (m

)

Figure 15. Observed vs predicted values in y-direction(sidewall).

Incline (˚)k-ratio

�1 fl (kPa)

2.0

1.5

1.0

0.5 5045

3525

155

0

40

2030

10

-315

-330

-345

-360

-375

Figure 17. �1 vs “k” ratio and inclination (floor).

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where �1fl � max principal stress at the floor;k � “k” ratio and I � inclination of the tunnel (°).

The predicted values versus observed values of �1at the floor are shown in Figure 19. This regressionmodel shows a strong correlation between the observedand predicted values.

The proposed regression equation and the correla-tion coefficient for �3 at the floor are as follows:

(9)

r � 0.96where �3fl � min principal stress at the floor; k � “k”ratio and I � inclination of the tunnel (°).

The predicted values versus observed values of �3in the floor are shown in Figure 20.

This regression model also shows a strong correla-tion between the observed and predicted values.

The correlation coefficients of all proposed regres-sion models have been summarized in Table 4 formaximum and minimum principal stresses. As seen,there is only a meaningful correlation for the historypoint in the floor.

4 CONCLUSIONS

When the displacements and stresses in the modelstudies have been examined in detail, the results canbe stated as follows:

1. The y-components of the movements in the roofand the floor increase radially into the tunnel asthe tunnel is inclined. This trend becomes moreobvious with the increase of the horizontal stresses.

2. The z-components of the movements in the roofand the floor decrease as the tunnel is inclined. Onthe contrary, they increase as the horizontal stressesincrease.

3. Displacements in the x-direction in the sidewallare not affected by the change of tunnel inclination.But, they increase towards the tunnel inside as thehorizontal stresses increase.

4. The y-component of the movement in the sidewallincreases due to the increase of tunnel inclination.This trend becomes more obvious with the increaseof horizontal stresses.

5. The z-component of the movement in the side-wall tends to decrease with the increase of tunnel

133

Incline (˚)

k-ratio

�3 fl (kPa)0

510

2030

4050

15

35

25

45

2.0

1.5

1.0

0.5

-20

-16

-24

-28

-32

-36

-40

Figure 18. �3 vs “k” ratio and inclination (floor).

-400 -380 -360 -340 -320 -300

Observed Values (kPa)

-400

-380

-360

-340

-320

-300

Pred

icte

d V

alue

s (k

Pa)

Figure 19. Observed vs predicted �1 (floor).

-45 -40 -35 -30 -25 -20 -15

Observed Values (kPa)

-45

-40

-35

-30

-25

-20

-15

-10

Pred

icte

d V

alue

s (k

Pa)

Figure 20. Observed vs predicted �3 (floor).

Table 4. Correlation coefficients for stresses.

Correlation coefficient (r)

Stress Roof Floor Sidewall

�1 0.75 0.95 0.68�3 0.63 0.96 0.68

09069-16.qxd 08/11/2003 20:26 PM Page 133


inclination. The magnitude of this componentincreases as the horizontal stresses increase.

6. The maximum and minimum principal stresses inthe floor generally increase as the tunnel is inclined.But, it should be noted that some model results arenot consistent with this general trend.

The results of the numerical model studies have beenanalyzed to find out the presence of correlation forthe displacements and stresses at the tunnel boundarywith the “k” ratio and tunnel inclination. As a result ofmultiple regression analyses carried out, strong rela-tionships have been found between the “k” ratio, tunnelinclination and displacements. But the relationshipbetween the “k” ratio, tunnel inclination and principalstresses are not as strong as those of displacementsexcept for those of principal stresses in the floor.

These proposed regression models can be usedespecially to predict the displacements around theboundary of tunnels which will be excavated in simi-lar rock masses (marl formation). It should be notedthat these predicted values should be used with greatcaution. Because they will be preliminary and roughestimates of displacements and principal stresses.Proposed regression equations can easily be affectedby the chance of rock properties, size and shape oftunnel, groundwater etc.

Similar analyses can be caried out for different rockmasses. These regression models can be developedand they can be used for various rock mass conditions.

REFERENCES

Bieniawski, Z.T. 1979. The geomechanics classification inrock engineering applications. Proc. Xth. Congress Int.Soc. Rock Mech. Vol. 2: 41–48. Montreux.

Çekilmez, V., Koç, S. & Alemdarog�lu, T. 1987. The geo-technical research of the drills in Kütahya-Tavsanl1-Tunçbilek District. M.T.A. Institute, Ankara/Turkey.

Çekilmez, V. 1988. The geotechnical research of the JT4 drillin Kütahya-Tavsanl1-Tunçbilek district. M.T.A. Institute,Ankara.

Hoek, E. & Brown, E.T. 1980. Underground excavations inrock. Institution of Mining Metallurgy, London.

Itasca Consulting Group, Inc. 1997. FLAC3D – FastLagrangian Analysis of Continua in 3 Dimensions,Version 2.0 User’s Manual. Minneapolis, MN: Itasca.

Once, G., Iphar, M. & Yavuz, M. 2001a. Design of the maintransport road of the deep coal seam panels of GLITunçbilek mine in Turkey, Osmangazi University ResearchFund Project, Eskisehir, Turkey.

Once, G., Iphar, M. & Yavuz, M. 2001b. Study of groundcontrol of the main transport road of the deep coal seampanels of GLI Tunçbilek mine in Turkey, FLAC andNumerical Modeling in Geomechanics, Lyon, France,29–31 October 2001. Rotterdam, Balkema.

Priest, S.D. & Hudson, L. 1976. Discontinuity spacings inrock. International Journal of Rock Mechanics and MiningSciences. Vol. 13: 135–148.

134

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135

Numerical analysis of the volume loss influence on building during tunnel excavation

O. Jenck & D. DiasINSA Lyon, URGC Géotechnique, Villeurbanne, France

ABSTRACT: Shallow tunneling performed by a Tunnel Boring Machine (TBM) induces volume loss, mainlydue to the conical shape of the machine and the consolidation of the injected grout. This excavation volume losscauses ground movements at the surface that can induce damages to surrounding structures. However, to knowthe influence on structures, it is not sufficient to apply the Greenfield strains because of the influence of thestructure’s stiffness. Therefore a computational three-dimensional soil-structure interaction analysis is requiredto take into account all the complexity of the problem. This paper presents a FLAC3D analysis of the interactionbetween shallow tunnel excavation and surface buildings, applied to the case of the Lisboa subway. A simpli-fied simulation of TBM tunneling is adopted by imposing volume loss. To highlight the influence of settlementson a six-floor structure, different cases of volume loss are studied from 0.5 to 5 % of the total excavated volume.

1 INTRODUCTION

During the construction of a tunnel at shallow depthin urban areas, prediction of the effects induced by theexcavation on surrounding buildings is very important.In fact, the volume loss in tunnel generates soil dis-placements on surface that can cause damages toexisting structures. The traditional design of the con-structions doesn’t take into account this type of loadingconditions. In order to control the volume loss in thetunnel and to limit the damages, the tunnel is exca-vated, when it is possible, with a TBM.

A first approximation to predict the damagescaused to surrounding structures is done by applyingthe soil deformations without any structure on surface,called Greenfield deformations, to the structure’sfoundations. This method is recommended by AFTES(1999). The Greenfield deformations can be calcu-lated by an empirical (O’Reilly & New 1982, Peck1969), analytical (Panet 1995, Sagaseta 1987) ornumerical method (Oteo & Sagaseta 1982, Swobodaet al. 1989).

However, it is important to consider the structureto estimate the soil movements because it contributesto stiffen the ground and consequently to reduce thesoil displacements. Then, the determination of theunderground works influence on surrounding structuresbecomes very difficult with empirical or analyticalmethods. Only the numerical method is able to take

account of all the complexity of this type of soil-structure interaction problem.

Potts & Addenbrooke (1997) used two-dimensionalnumerical calculations considering the structure as anequivalent weightless beam with variable stiffness.They showed the structure’s rigidity influence on sur-face ground movements induced by tunneling. Franzius& Addenbrooke (2002) have then analyzed the influ-ence of the structure’s weight. They showed that theweight has very low influence on ground movementswhen rigidity increases. Another two-dimensional cal-culation coupling the soil with a masonry buildingwas performed by Miliziano et al. (2002). The three-dimensional building is taken into account by anequivalent two-dimensional wall. They demonstratedthe significant effect of the relative structure stiffnessin reducing differential displacements, and on thepredicted damage.

Nevertheless, with 2D simulations it is worth notingthat an empirical parameter such as the deconfinementratio or the volume loss in tunnel has to be consideredas remarked by Benmebarek et al. (1998). Dias et al.(1999) have compared results from 2D and 3Dnumerical simulations with experimental data. Theyshowed that the surface settlement trough obtainedwith the 3D calculation is more realistic than thetrough obtained with the 2D calculation, even with asimple constitutive model for the soil. Moreover, it isimpossible to study the damages induced on the

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structure in the tunnel axis direction when using 2Dsimulations.

Some authors have already used three-dimensionalnumerical modeling. For instance Mroueh & Shahrour(2003) have compared the results of a soil-structureinteraction calculation of tunnel excavation below astructure with the results obtained by imposing theGreenfield movements upon the structure. Theyshowed that this last method is very severe in terms ofinduced forces in the structure. Netzel & Kaalberg(2000) have modeled the interaction between TBMdigging and masonry structures in order to obtainspecific damage criteria.

This article presents a three-dimensional numericalanalysis of the soil-structure interaction phenomenonduring shallow tunneling. The tunnel excavation is asimplified simulation of the real phases of a TBMbased on the concept of volume loss. The soil behavioris elastic perfectly plastic. The structure is composedof columns and floors founded on a raft. The para-metrical study deals with the influence of volume lossin tunnel. Results are analyzed in terms of groundsurface displacements and of stresses induced in thestructure.

2 EXPERIMENTAL DATA

2.1 Experimental section

The studied model is based on the case of the Lisboasubway. The experimental section (Fig. 1) is locatednear the new Ameixoeira station. The geotechnicalproperties (Table 1) are given by Ribeiro e Sousa et al.(2003).

The excavation is 26 m deep. The tunnel diameteris equal to D � 9.8 m. The section is located in thesilty sand layer, where the mechanical properties arerelatively poor.

2.2 Measured surface settlements

Measured settlements have been obtained on severalsections near the Ameixoeira station (Ribeiro e Sousaet al. 2003). Results are shown in Figures 2 & 3.

Figure 2 presents the settlements measured abovethe tunnel axis in three different sections, for differentpositions of the TBM. Figure 2 illustrates experimen-tal longitudinal settlement troughs. The maximumobserved settlement is about 0.4 cm settlements areobserved even when the TBM has not already reachedthe instrumented section. On this underground works,an earth pressure shield is used.

Figure 3 shows the observed surface settlements in atransversal section (S31), at the final state of the exca-vation (when the TBM is far away). This curve is calledtransverse settlements trough. In this instrumented

section, the maximum measured settlement is equalto 0.3 cm.

3 NUMERICAL MODEL ADOPTED

3.1 Ground mass

Figure 4 presents the numerical model of the ground-mass. Due to the symmetry conditions, only half of the

136

26 m

4 m

7 m

9 m

14 m

17 m

C1

C2

C3

C4

C5

Figure 1. Experimental section.

Table 1. Geotechnical properties.

Name: C1 C2 C3 C4 C5Soil type Clay Clay Limestone Silty sand Clay

E [MPa] 15 15 266.5 44.7 180� 0.35 0.4 0.35 0.35 0.37c [kPa] 5 5 10 0 250" [°] 30 32 37 35 30� [kN/m3] 20.5 20.7 20.5 20.65 20.4Ko 0.6 0.7 0.7 0.8 1.05

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-40 -20 0 20 40 60 80 100 120Distance to the section (m)

Surf

ace

sett

lem

ents

(cm

)

Section S28Section S29Section S30

S28 S29 S30

Figure 2. Longitudinal settlement troughs.

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ground mass is modeled (plane of symmetry Y–Z).The model is 100 m wide in the X direction, 90 mlong in the Y direction (parallel to tunnel axis) and51 m high (Z direction). The numerical model con-sists of approximately 85,000 nodes.

3.2 Ground behavior

The behavior of the soil is set as elastic perfectly plas-tic with a Mohr-Coulomb failure criterion. The flowrule is non-associated and the dilatancy angle is set as# � " � 30°.

Due to the very fast pore pressure dissipationobserved, the calculation is done in drained conditions.

3.3 Simulation of excavation

The adopted excavation process for the calculation isa simplification of the confinement-deconfinementphases induced by the boring machine. The hypothe-ses are as follows: the soil displacements at the tunnelface are blocked, simulating a perfect equilibrium ofconfinement pressures. The lateral soil displacements

have a linear variation on the distance of 20 m. Afterthis distance, the lateral displacements are constant,as shown on Figure 5. This process simulates the prin-cipal excavation phases:

– Conical shape of TBM– Grout injection– Grout consolidation– Setting of the concrete rings

The initial position of the tunnel is Y � 0 m andthe numerical phases of excavation are as follows:

– excavation on one element length,– fixation of the tunnel face nodes,– convergence of tunnel walls until reaching the given

displacement shape,– if a node reaches the limit, it will be fixed,– when the model equilibrium is reached, all the

nodes are freed,– translation of the loading system of one element

length.

Hence, there are as many excavation steps as there areelements on the model length.

The displacement field imposed at the tunnel wallscorresponds to a volume loss. This volume loss intunnel normalized by the total excavated volume iscalled Vt. The excavation is ended when the model isentirely bored. Then the tunnel face is at Y � 90 20 � 110 m which corresponds to the entire modellength added with the distance between the tunnelface and the position where the lateral displacement isconstant.

137

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

-20 -15 -10 -5 0 5 10 15 20Distance to tunnel axis (m)

Surf

ace

sett

lem

ents

(cm

)

Figure 3. Transverse settlement trough.

X

Z

Y

Figure 4. Numerical model.

Z

Y

Tunnel face

One element length

Step n

20 m

20 m Step n+1

Figure 5. Longitudinal tunnel section – excavation process.

09069-17.qxd 08/11/2003 20:27 PM Page 137


4 REFERENCE CASE: EXCAVATIONWITHOUT STRUCTURE ON SURFACE

The reference case corresponds to an excavation ofthe numerical model with Vt � 5%.

According to Benmabarek et al. (1998), this valueseems to be the higher value obtained with the TBMexcavation method and do not correspond to theobserved settlements.

4.1 Surface settlements

The surface settlement distribution in a transversesection at the final state is shown on Figure 6. The dis-tance to tunnel axis is normalized by the tunnel diame-ter D. The maximum settlement is equal to 4 cm; it isabout ten times higher than the observed settlements(see section 2.2).

The adopted numerical process for simulating thetunnel excavation with a TBM is able to reproduce asurface displacement trough in agreement with theGaussian distribution, which matches very closelyexperimental observations. This settlement distribu-tion is given by Peck (1969) equation:

(1)

Where S � settlement at distance x of the tunnelaxis, Smax � maximum settlement obtained in thetrough center, i � distance from the inflection pointof the trough to tunnel axis. In this case, i � 1.8 D.

4.2 Horizontal surface displacement

Figure 7 presents the horizontal surface soil strains ina transverse section, at the final state. In this figure,two distinct zones are observed. In the center of thetrough, the soil is in compression and on the edges thesoil is in extension. The limit between these two zones

is the point of inflection of the transverse settlementtrough.

5 INFLUENCE OF VOLUME LOSS IN TUNNELON GREENFIELD DISPLACEMENTS

In order to study the influence of volume loss in tun-nel on Greenfield soil displacements, several calcula-tions are done with Vt from 0.5% to 5%.

5.1 Surface settlements

Figure 8 compares the transversal settlement troughsfor the different values of Vt, at the final state.

All the curves show a Gaussian distribution withthe same value of i. The maximum surface settle-ments are:

– for Vt � 3%, Smax � 2.5 cm– for Vt � 1%, Smax � 1 cm– for Vt � 0.5%, Smax � 0.5 cm

These results are reported in Figure 9. A quasi-linearrelation between tunnel volume loss and maximumsurface settlement is observed.

138

-0.20

-0.15

-0.10

-0.05

0.00

0.050 1 5 7 8 9

Distance to tunnel axis (x/D)

Hor

izon

tal s

trai

n (%

)

compression extension

2 3 4 6

Figure 7. Horizontal strain in a transverse section.

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

00 1

0.5%

1%

3%

5%

Vt

2 4 6

Surf

ace

sett

lem

ents

(cm

)


0.5%

1%

3%

5%

3 5 7


-4

-3

-2

-1

00 1 2 4 6


Surf

ace

sett

lem

ents

(cm

) i

3 5 7


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The tunnel volume loss of Vt � 0.5% correspondsto the observed surface settlement (Figs. 2 & 3).

5.2 Horizontal surface displacement

Figure 10 compares the horizontal surface dis-placements for the values of Vt, at the final state. The limit between the two zones is the same for allvalues of Vt. As in the previous paragraph, a linearrelation is observed between Vt and the maximumsoil compression.

6 COUPLED CALCULATION: MODELING OFTHE STRUCTURE

In order to study the soil-structure interaction duringshallow tunneling, three-dimensional calculations cou-pling ground mass and structure are presented.

6.1 Geometry of the structure

The studied structure is a simplification of existingbuildings. It is composed of columns of square section

(0.4 m � 0.4 m), and slabs of 0.3 m thickness, foundedon a 0.3 m raft, which size is 12 m � 36 m. The build-ing has seven levels of 4 m height.

Figure 11 shows the column’s position. There is noeccentricity considered between the structure and thetunnel; therefore, only the half of the structure ismodeled.

6.2 Numerical model coupling soil and structure

Figure 12 shows the numerical model couplingground mass and structure. The structure is located inthe middle of the model length (Y-axis). Columns aretaken into account as beams, slabs and raft as shells.The structure behavior is elastic with properties of areinforced concrete: E � 19 GPa and � � 0.2. Thesoil nodes are bound with the structure nodes.

The structure is disposed on the numerical modelin one phase. Then equilibrium of the model is reached.The structure is only loaded with its own weight(� � 25 kN/m3). After that the soil displacements areinitialized in order to study only the tunneling effect.

139

Smax = 0.82Vt

0

1

2

3

4

5

0 1 2

Max

imum

set

tlem

ent

(cm

)

Tunnel volume loss (%)3 4 5

Figure 9. Liner relation between maximum settlement and Vt.

-0.20

-0.15

-0.10

-0.05

0.00

0.050 1 2


Hor

izon

tal s

trai

n (%

)

5%

3%

1%

0.5%

Vt

3 4 5 6 7 8

Figure 10. Horizontal strain in a transverse section.

Y

X

Symmetry axis

4m

4m

12m

18mTunnel

D/2

Figure 11. Column position.

XYZ

Figure 12. Numerical model coupling soil and structure.

09069-17.qxd 08/11/2003 20:27 PM Page 139


The forces in the structural elements are not initial-ized: this step represents the initial state in terms ofinternal forces in the structure. Finally, the obtainednumerical model is excavated.

7 COUPLED CALCULATION: REFERENCECASE, Vt � 5%

The reference case of the calculation coupling soiland structure is compared with the reference casewithout structure, presented in section 4.

7.1 Surface soil displacements

Figure 13 compares the surface settlement troughs inthe middle of the model length, at the final state. Witha structure on surface, the settlements increase thanwith the Greenfield calculation. The maximum surfacesettlement increase from 4.1 cm to 4.8 cm is equal to15%; and the volume trough increase is equal to 5%.Nevertheless, the settlement trough remains a Gaussiancurve in the two cases, with appreciatively the samevalue for i.

Figure 14 compares the horizontal surface dis-placements in a transverse section with and withoutstructure, at the final state. In the first case, the hori-zontal soil displacements under the structure are neg-ligible compared to the Greenfield case. This is due tothe high axial stiffness of the raft and due to the factthat the raft is bound with the soil.

This is a significant result, which has also beenobserved in field monitoring (Standing et al. 2002).Hence, it is very severe to apply the Greenfield move-ments of soil to this structure to estimate the induceddamages, as recommended in first approximation byAFTES (1999).

7.2 Induced forces in the structure columns

During the excavation, the forces in the structurecolumns are analyzed. Figure 15 defines the shearforces and the bending moments in a column section.Figure 16 shows the maximum values of the internalforces normalized by the maximum initial values F/Fini,for different excavation length.

The transverse shear forces increase considerablywhen the TBM passes below the structure. Thenimportant values are kept until the end of the excava-tion. Longitudinal shear forces increase when TBM islocated below and just next to the building, anddecrease to their initial values when TBM moves off.The axial forces are very lightly affected by diggingcompared to the others forces. Then, they are not rep-resented here.

The forces induced in the columns are studiedmore in detail for four specific columns specified onFigure 17. These columns are chosen because theyundergo a great stress increase or are representativeof the general behavior. Only the evolution of the

140

-5

-4

-3

-2

-1

00 1 2


Surf

ace

sett

lem

ent

(cm

)

Greenfield

With structure

Building

543

Figure 13. Surface settlement trough with and withoutstructure, Vt � 5%.

TY : longitudinal shear forceTx : transversal shear forceMY : transversal bending momentMx : longitudinal ending moment

Tunnel axis Tx Mx

X

Y

TY

MY

Figure 15. Forces in a column section.

-2

-1.5

-1

-0.5

00 1 2 8


Hor

izon

tal d

ispl

acem

ent

(cm

)

Greenfield

With structure

3 4 5 6 7 9

Figure 14. Horizontal displacement with and withoutstructure, Vt � 5%.

09069-17.qxd 08/11/2003 20:27 PM Page 140


bending moment is studied, because shear forces varyin the same way.

Figure 18 presents the evolution of the maximumvalue of MY for the different studied columns. It can benoted that more columns are far from the tunnel axis,more columns are affected in the transverse direction.

Figure 19 presents the evolution of the maximumvalue of Mx for the different studied columns. Itseems that the longitudinal stresses are similar in allthe building’s columns.

The most affected column in both directions is col-umn C. Bending moments and shear forces distributionin these columns are analyzed for different excavationlengths.

Figures 19 and 20 present the repartition of thelongitudinal forces in column C. The most prejudicialexcavation step corresponds to the excavation lengthof about 80 m. The maximum longitudinal bendingmoment is 19 kN.m, reached between levels 3 and 4(Fig. 19). For a length bored of 45 m, figure 19 showsthat the column is affected in the opposite direction,with a value of 7 kN.m between levels 2 and 3. Themaximum longitudinal shear force is equal to 2.8 kN,reached at level 5. Level 3 is also affected withTY � 2.6 kN (Fig. 20).

Figures 21 and 22 present the repartition of thetransversal forces on column C. The most prejudicialexcavation step is at the final state. The initial state isnot represented because of negligible values. This fig-ure show that the most affected levels in the trans-verse direction are levels 1 and 2. The maximumvalue of MY is 44 kN.m (Fig. 21) and the maximumvalue of TX is 11 kN (Fig. 22).

141

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100 110Excavation length (m)

F/F

ini

My Tx

Mx Ty

Figure 16. Maximum beam forces.

Tunnel axis

Y

X A

B

C

D

Figure 17. Position of studied columns.

0

10

20

30

40

50

0 2 40 60 80 100Excavation length (m)

Tra

nsve

rsal

ben

ding

mom

ent

(kN

.m) A

BCD

Column

Figure 18. Maximum transversal bending moment in thefour columns.

0

5

10

15

20

0 20 40 60 80 100Excavation length (m)

Lon

gitu

dina

l ben

ding

mom

ent

(kN

.m)

ABCD

Column

Figure 19. Maximum longitudinal bending moment on thefour columns.

0

4

8

12

16

20

24

28

-20 -15 -10 -5 0 5 10Longitudinal bending moment (kN.m)

Col

umn

heig

ht (

m)

Initial stateExcavation of 45mExcavation of 77mFinal state

Figure 20. Longitudinal bending moment (Mx) on column C.

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8 COUPLED CALCULATION: INFLUENCE OFVOLUME LOSS IN TUNNEL

Influence of tunnel volume loss on a structure is alsostudied. Calculations coupling soil and structure aredone using different volume loss in tunnel.

8.1 Influence of volume loss on surface settlements

Figure 23 compares the surface settlement troughsobtained with a structure on surface for different vol-ume loss imposed in tunnel. This figure can be com-pared with Figure 8. For a volume loss more than 1%,the surface settlements are greater in the coupled cal-culation than in the Greenfield case. All curves pres-ent a Gaussian distribution of surface settlements.

Figure 24 presents the ratio between settlementswith structure and without structure according to theposition on the model length, above the tunnel axis, atthe final state. A ratio greater than 1 corresponds tosettlements higher than values without structure. It isalways the case, except with a very low tunnel volumeloss (Vt � 0.5%). This figure illustrates that greateris the volume loss in tunnel, greater is the settlementincrease compared to the Greenfield case.

8.2 Influence of volume loss on induced forces incolumns

Figure 25 presents the evolution of the maximumvalue of TX (same evolution for maximum MY) fordifferent values of Vt. For each value of Vt, the trans-versal forces increase when TBM passes under thestructure and keep important values at the final state.

Figure 26 presents the evolution of the maximumvalue of TY (same evolution for maximum MX) for

142

0

4

8

12

16

20

24

28

-4,0 -3,0 -2,0 -1,0 0,0 1,0 2,0 3,0Longitudinal shear force (kN)

Col

umn

heig

ht (

m)

Initial stateExcavation of 77.5mFinal state

Figure 21. Longitudinal shear force (TY) on column C.

0

4

8

12

16

20

24

28

-10 0 10 20 30 40 50Transverse bending moment (kN.m)

Col

umn

heig

ht (

m) Excavation of 45m

Final state

Figure 22. Transversal bending moment (MY) on column C.

0

4

8

12

16

20

24

28

-12,0 -8,0 -4,0 0,0 4,0 8,0 12,0Transversal shear force (kN)

Col

umn

heig

ht (m

)

Excavation of 45m

Final state

Figure 23. Transversal shear force (TX) on column C.

-5-4.5

-4-3.5

-3-2.5

-2-1.5

-1-0.5

00 1 2 4 6


Surf

ace

sett

lem

ent

(cm

)

0.5%

1%

3%

5%

Vt

Building

3 5 7

Figure 24. Surface settlement trough with structure on sur-face for different volume loss in tunnel.

0.95

1

1.05

1.1

1.15

1.2

0 10 20 30 40 50 60 70 80 90

0,5%1%3%5%

Rat

io =

Sw

ith s

tr./S

with

out

Numerical model length (m)

Figure 25. Longitudinal settlements ratio.

09069-17.qxd 08/11/2003 20:27 PM Page 142


different values of Vt. For each Vt, the shape of theF/Fini evolution is similar but with higher values forhigher Vt.

When reporting the maximum values for the trans-versal and longitudinal forces according to the volumeloss in tunnel, Figure 27 shows a linear relation.Maximum values for the transversal forces are obtainedat the final state, whereas maximum values for the

longitudinal forces are obtained for a length bored ofabout 80 m.

9 CONCLUSIONS

This numerical analysis highlights the soil/structureinteraction during shallow tunneling with TBM. Theattention was focused on the influence of the volumeloss in tunnel.

The presence of the structure increases the surfacesettlement for a volume loss in tunnel higher than 1%,in comparison to the Greenfield case.

Due to the presence of a raft with a great axial stiff-ness and the fact that the raft is bound with the soil,the horizontal displacements are negligible under thestructure in comparison with the Greenfield case.

This study has showed that the more affected direc-tion of the building during tunneling is the transversedirection. Moreover, transversal loads are keepingimportant values at the final state whereas longitudi-nal loads are coming back to their initial values.

The analysis of the volume loss influence showed alinear relation between volume loss in tunnel and forcesinduced in the structure columns during tunneling.

REFERENCES

AFTES 1999. Recommandations pour les tassements liés aucreusement des ouvrages en souterrain. Tunnels etOuvrages Souterrains: 106–128.

Benmebarek, S., Kastner, R. & Ollier, C. 1998. Auscultationet modélisation numérique du processus de creusement àl’aide d’un tunnelier. Géotechnique 48 (6): 801–818.

Dias, D., Kastner, R. & Maghazi, M. 1999. Three dimensionalsimulation of slurry shield tunneling. In InternationalSymposium on Geotechnical Aspects of Underground Con-struction in Soft Ground, Tokyo, Japan, 6p.

Franzius, J.N. & Addenbrooke, T.I. 2002. The influence ofbuilding weight on the relative stiffness method of pre-dicting tunnelling-induced building deformation. In 4thSymposium Geotechnical Aspects of Underground Con-struction in Soft Ground, Toulouse, France, 1, 53–58.

Miliziano, S., Soccodato, F.M. & Burghignoli, A. 2002.Evaluation of damage in masonry buildings due to tun-nelling in clayey soils. In 4th Symposium GeotechnicalAspects of Underground Construction in Soft Ground,Toulouse, 3, 49–54.

Mroueh, H. & Shahrour, I. 2003. A full 3-D finite elementanalysis of tunneling-adjacent structures interaction. Com-puters and Geotechnics 30: 245–253.

Netzel, H. & Kaalberg, F.J. 2000. Numerical damage riskassessment studies on masonry structures due to TBM-Tunnelling in Amsterdam. In GeoEng 2000, Melbourne,Australia, 235–244.

O’Reilly, M.P. & New, B.M. 1982. Settlements above tunnelin the United Kingdom – their magnitudes and prediction.In Tunelling 82’, London, IMM, 173–181.

143

0

10

20

30

40

50

60


F/F

ini

5%

3%

1%

0.5%

Vt

Figure 26. Evolution of maximum transversal forces incolumns.

0

1

2

3

4

5

6

7

8

9


F/F

ini

5%

3%

1%

0.5%

Vt

Figure 27. Evolution of maximum longitudinal forces incolumns.

y = 171x

y = 1032x

0

10

20

30

40

50

60

0 1 2 3 4 5Volume loss in tunnel (%)

F/F

ini

Tx - final state

Ty - excavation of 77 m

Figure 28. Maximum forces increase according to Vt.

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Oteo, C.S. & Sagaseta, C. 1982. Prediction of settlementsdue to underground openings. In Int. Symp. On numeri-cal Models in Geomechanics, Zurich, 653–659.

Panet, M. 1995. Le calcul des tunnels par la méthode convergence-confinement. Paris: Presses de l’ENPC.

Peck, R.B. 1969. Deep excavation and tunnelling in softground, State of the art report. In 7th InternationalConference on Soil Mechanics and Foundation Engi-neering, Mexico, 225–290.

Potts, D.M. & Addenbrooke, T.I. 1997. A structure’s influenceon tunnelling induced ground movements. In Instn CivilEngineers in Geotechnical Engineering. 125, 109–125.

Ribeiro e Sousa, L., Dias, D. & Barreto, J. 2003. LisbonMetro Yellow Line extension. Structural behaviour of theAmeixoeira Station. In 12ª Conferência Panamerican onSoil Mechanics and Geotechnical Engineering, Boston.

Sagaseta, C. 1987. Evaluation of surface movements abovetunnels, a new approach. In Colloque International ENPCInteractions sol/structure, Paris, Presses ENPC, 445–452.

Standing, J.R., Gras, M., Taylor, G.R., Gupta, S.C., Nyren,R.J. & Burland, J.B. 2002. Building response to tunnelstep-plate junction construction – the former LloydsBank building, St James’s, London. In 4th SymposiumGeotechnical Aspects of Underground Construction inSoft Ground, Toulouse, France, 3.

Swoboda, G., Mertz, W. & Schmid, A. 1989. Three dimen-sional numerical models to simulate tunnel excavation.Numerical Models in Geomechanics NUMOG III.Elsevier. 581–586.

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FLAC and Numerical Modeling in Geomechanics, Brummer et al. (eds) © 2003 Swets & Zeitling, Lisse, ISBN 90 5809 581 9

145

Application of FLAC3D on HLW underground repository concept development

S. Kwon, J.H. Park, J.W. Choi & W.J. ChoKorea Atomic Energy Research Institute, Korea

ABSTRACT: For the safe design of a deep underground high-level radioactive waste (HLW) repository, it isimportant to understand the thermal–mechanical behavior of the engineering barriers and rock mass around therepository influenced by the high stress and the heat generated from the waste. In this study, thermal–mechani-cal coupling analysis was carried out to investigate the reliability of the Korean HLW repository concept usingFLAC3D with the thermal and mechanical properties of rock and rock mass measured at two drilling sites. Foreffective thermal–mechanical coupling, a FISH routine was developed and used for the modeling of differentconditions. By using FLAC3D with the FISH routine, the evaluation of the thermal–mechanical stability of thepreliminary disposal concept could be done successfully.

1 INTRODUCTION

The Republic of Korea began operating commercialnuclear power plants in 1978. Now there are 17 operat-ing plants, 4 CANDU (Canadian Deuterium UraniumReactor) and 13 PWR (Pressurized Water Reactor).The current generating capacity is 14,720 MWe witha share of 39.3% of the total production of electricity.The total generating capacity is expected to be about26.05 GWe by 2015. The cumulative amount of spentfuel from existing nuclear power plants reached 5,641MTU by June 2002. It is expected that approximately11,000 MTU and 19,000 MTU will be accumulatedby the years 2010 and 2020, respectively.

In Korea, a reference HLW disposal system is underdevelopment. According to the previously determineddisposal concept, the PWR and CANDU spent fuel incorrosion resistant canister will be emplaced in a deepunderground repository constructed in crystalline rocksuch as granite. To confirm whether the disposal con-cept is reliable or not under certain geological condi-tions, waste type, and operation procedure, computersimulations need to be carried out. FLAC3D had beenwidely applied in radioactive waste repository projectsrelated to different rock types by many researchers(Johansson & Hakala 1995, Berge & Wang 1999,Fairhurst 1999, Francke et al. 2001, and Patchet et al.2001). In Korea, FLAC3D had already been used forthe thermal analysis (Park et al. 1998) and mechanicalanalysis (Park et al. 2001) for Korean reference reposi-tory design.

In this study, FLAC3D was used for investigatingthe thermal–mechanical coupling behavior of rock,buffer, backfill, and canister. In order to carry out the thermal–mechanical coupling analysis, a FISHroutine was developed and used for the modeling toinvestigate the coupling behavior of the rock massaround the disposal tunnel and deposition hole.

2 HLW REPOSITORY CONCEPT IN KOREA

The Korea Atomic Energy Research Institute has beendeveloping a reference HLW disposal system since1997. According to the preliminary disposal concept,the repository is located in a crystalline rock mass atseveral hundred meters below surface. Like many other countries such as Sweden, Canada, Finland,Switzerland, and Japan, a multibarrier system consist-ing of canister, buffer, and backfill is supposed to beapplied for safe containment of the radioactive waste.The buffer acts as a barrier to suppress the detrimentaleffects of the corrosive water in the host rock and toenhance the life of the container and serves as a geo-chemical filter for the sorption of radionuclides. Thebuffer dissipates the decay heat from the waste intothe surrounding rock to avoid the possibility of thermalstress on the container. It also provides the mechanicalstrength to support the canisters and isolates the con-tainers from detrimental rock mass movements(Selvadurai & Pang 1990). In many countries includingKorea, bentonite is now considered as the buffer

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material because of its low hydraulic conductivity, highsorption capacity, self-sealing characteristics, anddurability in nature.

In the Korean repository concept, the mixture ofbentonite and crushed rock will be used as the back-filling material. The deposition tunnels are 6 m wideand 7 m high. The canister containing spent fuel isassumed to be emplaced in the vertical boreholesdrilled along the center line on the floor as shown inFigure 1.

3 FLAC3D MODELING

3.1 Materials in the model

3.1.1 Fuel part and outshellFour PWR assemblies are inserted in a canister withoutshell thickness of 5 cm. The mechanical and thermalproperties of the fuel part, which represents the partinside of the outershell, were determined with theassumption of that the fuel and cast iron were uniformlymixed. The average properties of the fuel part werecalculated based on volume ratio and listed in Table 1.Among the candidate material types for the outshell,stainless steel was considered in this study. The diam-eter of the canister is 1.22 m and the length is 4.78 m.The thermal–mechanical properties of stainless steelare also listed in Table 1.

3.1.2 Buffer and backfillSome of the thermal and mechanical properties ofbuffer and backfill material could be determined from

laboratory tests using Korean bentonite (Kyungjubentonite), which is considered as a candidate buffermaterial for the Korean repository. The other materialproperties, which could not be determined from tests,were chosen from literature review and listed in Table 2.In this study, the buffer and backfill materials weremodeled with a Drucker-Prager plastic model. TheDrucker-Prager parameters in Eq. (1) for buffer andbackfill could be determined from the triaxial com-pression tests under different confining pressures.

(1)

where, ! and kshear are material parameters, J1 is the first invariant of the stress tensor, and J2D is thesecond invariant of the deviatoric stress tensor (Desai & Siriwardane 1984).

146

7m6m

2.2m

TB

DepositionHole

40m

6m

BT

FT

Spent Fuel

Buffer Outshell

Backfill

TB: Backfill thickness(1.5m)BT: Upper buffer thickness(1m)FT: Bottom & side buffer

thickness(0.5m)

Canister

Figure 1. Schematic drawing of the reference Korean repository design.

Table 1. Material properties of fuel part and outshell.

Unit Fuel part Outshell

Material type Fuel Stainlesscast iron steel

Model type Elastic ElasticE GPa 190 200� 0.3 0.3Density Kg/m3 6500 8000Thermal conductivity W/m°K 43 15.2Specific heat J/Kg°K 424 504Thermal expansion /°K 1.2e-5 8.2e-6

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3.1.3 Rock propertiesIt is assumed that the underground repository is con-structed in a granite body at 500 m below surface. Themechanical and thermal properties of the granitesfrom two drilling sites, Kosung and Yusung, are listedin Table 2. The two sites are representing the east andwest sides of Korean Peninsula. NX size rock coreswere retrieved from the drill holes reached up to 500 mbelow surface.

The influence of discontinuities is considered indi-rectly using the equations proposed by Fossum (1985)to calculated the modulus of randomly jointed rockmass. The effective bulk and shear moduli can be writ-ten in terms of the intact and joint properties. Effectivebulk and shear moduli are

(2)

(3)

where, E is Elastic modulus of rock, � is Poisson’sratio, S is joint spacing, and kn and ks are normal andshear stiffness of joint. Thermal logging was carried outto find the geothermal gradients at the two sites (Parket al. 2001).

3.2 Modeling method

Thermal–mechanical coupling is important due to thethermal stress developed by the decay heat from the

waste. Subsequent heating of the rock mass by theheat-generating waste would increase the stresses inthe buffer, canister, and rock mass because of thermalexpansion (Simmons & Baumgartner 1994). Thethermal stress due to the thermal expansion can becalculated as follows:

(4)

where, �� is increase in stress due to the expansion of rock, � is thermal expansion coefficient, �Tis temperature increase, E is Young’s modulus, and � is Poisson’s ratio. FLAC3D has functions for coup-ling behaviors such as hydraulic–mechanical, thermal–mechanical, and thermal–hydraulic couplings.In FLAC3D, the thermal–mechanical coupling occursonly in one direction: temperature changes cause ther-mal strains to occur which influence the stresses, whilethe thermal calculation is unaffected by the mechanicalchanges taking place (Itasca 1996). As normal in mostmodeling situations, the initial mechanical conditionscorrespond to a state of equilibrium which must firstbe achieved before the coupled analysis is started.There are the following three suggestions for ther-mal–mechanical coupling in the FLAC3D manual.

1. A thermal only calculation is performed until thedesired time and then the thermal calculation is tobe turned off and the mechanical calculation is per-formed. When the mechanical equilibrium isreached, thermal calculation is performed again.

2. For each thermal time step, several mechanicalsteps are taken until detecting equilibrium condition.

3. The STEP command is used while both mechanicaland thermal modules are on. In this approach, onemechanical step will be taken for each thermal step.

147

Table 2. Material properties of buffer and back fill.

Rock

Unit Buffer Backfill Kosung Yusung

Material type Bentonite Crushed rock bentonite Granite GraniteModel type Drucker-Prager Drucker-Prager Mohr-Coulomb Mohr-CoulombModulus GPa Bulk � 0.345 Bulk � 0.038 E � 56.6 E � 46.8

Shear � 0.258 Shear � 0.029 � � 0.25 � � 0.28Density Kg/m3 Dry 1800 Dry 1800 2650 2660T. Conductivity W/m°K 1.47 2.04 2.523 3.541Specific heat J/Kg°K 888 900 1576 1212T. Expansion /°K 3.1e-4 3.1e-4 19.244e-6 19.312e-6UCS MPa 7.66 0.93 149.55 132.5Cohesion MPa 1.1 1.1 22.5 30.4Friction angle Degree 50 17 61 51Drucker-Prager Qvol � 1.23 Qvol � 0.24parameters

Kshear � 944 Kshear � 1472Geothermal gradient °C/km 37.5 25

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The first approach is useful for thermal–mechanicalanalysis of an elastic model. For non-linear modelssuch as plastic models, the thermal change must becommunicated to the mechanical module at closertime intervals to respect the path dependency of thesystem. In this case, a certain number of mechanicalsteps are taken for each thermal step to allow the sys-tem to adjust. In this approach, the transition fromthermal to mechanical calculation is based on timeinstead of temperature variation. Since the heat gen-eration is varying with time, the transition based ontemperature variation is more reasonable in the earlystage of the repository. The second and third approachesmay be more accurate than the first approach, but theproblem is that the calculation will take a long time tomodel the long-term behavior of repository. In orderto overcome the disadvantages of the three approaches,another technique for thermal–mechanical couplingwas developed. In the new approach, the transitionfrom thermal to mechanical calculation is based ontemperature change. A FISH program for the newapproach was developed for PWR spent fuel. Figure 2shows the flow chart of the thermal–mechanical cou-pling adapted in this study.

3.3 Model mesh and boundary conditions

3.3.1 Model meshFigure 3 shows the model mesh around the depositiontunnel and deposition hole. The model mesh aroundthe disposal tunnel and deposition hole located at thecenter of the whole model mesh, which covers fromsurface to 1000 m level. The backfilling material inthe disposal tunnel is not shown in the figure to clearlyshow the model mesh of the floor and deposition hole.

In the model, 5 different materials, rock, buffer, back-fill, outshell, and fuel part, were included. In theKorean preliminary disposal concept, the backfillthickness L1 � 1 m, upper buffer thickness L2 � 1.5 m,bottom buffer thickness and side buffer thicknessL3 � 0.5 m.

3.3.2 Initial and boundary conditionsThe in situ stress was assumed to be hydrostatic inthis study based on the fact that the stress ratios inYusung and Kosung sites are more or less 1.0 at 500 mdepth. The initial temperature in the model was calcu-lated with the geothermal gradients of Yusung andKosung sites. It was assumed that the average surfacetemperature is 20°C.

3.4 Decay heat

Decay heat is the thermal energy resulting from theradioactive decay of the radioactive materials in thespent fuel discharged from reactors. In Korea, the PWRspent fuel with 45,000 MWd/tHM is now consideredas the reference PWR spent fuel, because that type of

148

Figure 2. Flow chart of the TM coupling. Figure 3. Model mesh around the tunnel and deposition hole.

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spent fuel occupies 64% of all spent fuel from Koreanreactors. Even though significant amount of CANDUspent fuel with 7500 MWd/tHM is generated fromCANDU reactors, CANDU spent fuel is not consideredin this study, because of its much lower burnup com-pared to PWR spent fuel. For the spent fuel with 45,000MWd/tHM, the heat decays exponentially with timeas following:

(5)

where, t is time (year) after discharge from reactors.

4 FLAC3D MODELING RESULTS

4.1 Temperature

In order not to lose the required properties of ben-tonite buffer, the maximum temperature at the canistersurface and throughout the buffer must not exceed100°C. As shown in Figure 4, the highest temperaturein the Kosung case was 92.97°C at 15 years afteremplacement, while it was about 80.71°C in 20 yearsafter emplacement in Yusung case. Since the highesttemperatures in the model in both cases are lowerthan 100°C, the peak temperature in the buffer cannot

be higher than 100°C and thus the disposal design cansatisfy the thermal criteria.

Because of the higher geothermal gradient inKosung area, the temperature around the repository inthe Kosung case is higher than in the Yusung case.From Figure 4, it is possible to see when the transitionsbetween mechanical and thermal steps had happened.At the 500 m deep location, the initial temperature inthe Kosung case was 38.75°C while the temperaturein the Yusung case was 32.5°C. The initial tempera-ture difference due to the difference in geothermalgradient was about 6°C. Table 3 lists the temperaturesat the checking points at 20 years and 200 years afterthe emplacement of canister. The difference in tem-perature after 20 years in the Kosung and Yusungcases ranges from 8 to 12.4°C, which is higher thanthe initial temperature difference. The increase oftemperature difference is due to the lower thermalconduction in Kosung case, which has lower thermalconductivity than that in Yusung case. With increasein time, the temperature difference between the twocases decreased and it was about 7–9°C at 200 yearsafter emplacement.

4.2 Displacement

In the deposition hole, the heat from the waste willlead to thermal expansion of the canister, buffer, andbackfill. The displacements around the tunnel will alsobe influenced by the heat generation from the deposi-tion hole. Since the thermal and mechanical propertiesof rock are different in the Yusung and Kosung cases,the displacements around the disposal tunnel anddeposition hole are different. Figure 5 shows the dis-placement plot around the deposition tunnel at 200years after the emplacement of the canister and buffer.In Yusung case, the maximum displacement, which isrecorded at the upper backfill, was about 19 cm, whileit was about 23 cm in Kosung case. The upward dis-placement from the deposition hole to the tunnel is

149

50556065707580859095

100

0 50 100 150 200 250Time after emplacement (year)

Tem

pera

ture

(de

g. C

)

Yusung case

Kosung case

Figure 4. Variation of maximum temperature with time forKosung and Yusung case.

Table 3. Temperatures (°C) at the checking points and different time for Kosung and Yusungcases.

20 years 200 years

Check points Kosung Yusung Difference Kosung Yusung Difference

1 92.84 80.72 12.12 81.95 72.68 9.272 92.51 80.39 12.13 81.83 72.56 9.273 80.62 68.19 12.43 77.70 68.32 9.384 72.77 62.45 10.32 74.97 66.33 8.655 65.72 57.60 8.12 72.32 64.52 7.806 69.42 60.35 9.07 73.66 65.51 8.157 89.46 77.73 11.74 80.78 71.65 9.138 89.44 77.53 11.91 80.79 71.59 9.209 76.42 64.76 11.66 76.29 67.17 9.12

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thought to be mainly from the thermal expansion ofthe materials inside of the deposition hole.

Buffer movements can cause the canister to movein the deposition hole. In order to check the displace-ment pattern in the canister, the displacements at theoutshell were plotted in Figure 6. The displacementsin 20 years after emplacement were relatively smallerthan those in 200 years. One interesting thing in thefigure is that the displacement direction at the bottom

of the outshell in 20 years is downward while the otherparts show upward displacement. This can be explainedby the tensile stresses developed in the outshell. Thedirection change of displacement at the outshell bottomto upward in 200 years does not mean that the tensilestresses disappeared, but the difference of displace-ment along the outshell shows that there are tensilestresses in the canister. In the Kosung case, the maxi-mum displacement at the outshell is about 1.1 cm,which is a little larger than that in Yusung case.

4.3 Stress distribution

Figure 7 shows the principal stress distribution at thecanister in the Yusung case. The minimum principalstress was compressive and the magnitude was up to18 MPa. The maximum principal stress was tensile andit was up to 28 MPa. The tensile stress needs to beconsidered as an important factor in the disposal con-cept design, since it may cause mechanical failure ofcanisters. It is important to check von-Mises stress,because the distribution of von-Mises stress is closelyrelated to the mechanical stability of rock opening.

Figure 8 shows the von-Mises stresses at the check-ing points. At the checking points, the calculated von-Mises stresses from the Kosung case are higher thanthose from Yusung case. In the case of checking point 3,which represents the borehole surface, the von-Misesin Kosung case is highest up to 75 MPa, while that inthe Yusung case is about 40 MPa. The higher stressdistribution in the Kosung case might be due to thehigher temperature as well as more stiff rock proper-ties at the Kosung site.

5 CONCLUSIONS

In this study, thermal–mechanical coupling analysisfor the preliminary Korean disposal concept had beencarried out using FLAC3D. In order to overcome thedisadvantages of the previous approaches for ther-mal–mechanical coupling, a new method based ontemperature variation was suggested and a FISH rou-tine was developed. From the studies, the followingconclusions could be drawn:

– In both Kosung and Yusung cases, the maximumbuffer temperature was found to be lower than100°C, which is the most critical criteria for disposalconcept design.

– When using the geological information from theKosung drilling site, it was found that the maximumtemperature was 92.97°C in 15 years after emplace-ment, while it was 80.71°C in 20 years when theYusung data were used. This could be explained withthe higher geothermal gradient and lower thermalconductivity in Kosung site.

150

Figure 5. Displacement plot around the excavation in Yusungand Kosung sites.

Figure 6. Displacement plot at outshell.

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– From the fact that the peak temperature around therepository is reached in several tens of years afterthe emplacement of canister, modeling up to sev-eral tens of years are good enough for the sensitiv-ity analysis, which is mainly for investigation therelative influence of design parameters.

– The stress distribution on the canister surfaceshows that the tensile stress is highest at lower partof the canister. The maximum principal stresseswere tensile and it was up to 26 MPa and 28 MPa inYusung and Kosung cases, respectively. Such atensile stress may induce catastrophic failure of theoutshell and thus needs to be carefully analyzed.

– FLAC3D with FISH function could be successfullyapplied to evaluate the thermal–mechanical stabil-ity of the Korean preliminary repository design indeep underground rock.

REFERENCES

Berge, P.A. & Wang, H.F. 1999. Thermomechanical Effectson Permeability for a 3-D Model of YM Rock, BernardAmadei et al. (eds), Proceedings of the 37th U.S. RockMech. Symp., Vail, Colorado, June 1999, Vol. 2: 729–749.Rotterdam: Balkema.

151

Figure 7. Principal stress contours at the canister for Yusung case, 200 years after emplacement.

0

10

20

30

40

50

60

70

80

1 4 5 9Check Points

Mis

es s

tres

s (M

Pa)

KosungYusung

2 3 6 7 8

Figure 8. Comparison of von-Mises stress at different locations for Kosung and Yusung cases.

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Desai, C.S. & Siriwardane, H.J. 1984. Constitutive laws forengineering materials, Prentice-Hall, Inc., EnglewoodCliffs, NJ.

Fairhurst, C. 1999. Rock Mechanics and Nuclear WasteRepositories. S. Saeb and C. Franke (eds), Proceedingsof the International Workshop on the Rock Mechanics ofNuclear Waste Repositories, Vail, Colorado, June 1999:1–43. Alexandria, Virginia: American Rock MechanicsAssociation.

Fookes, P.G. 1995. Aggregates: a review of prediction andperformance, Proceedings of STATS 21st Anniversaryconference, London, UK: 91–170.

Francke, C.T., Saeb, S. & Carrasco, R.C. 2001. Three-Dimensional Analysis of Nuclear Waste Disposal inHorizontal Boreholes, Proceedings of the 38th U.S. RockMechanics Symposium, Washington, D.C., July 2001, Vol. 1:497–503. Lisse, The Netherlands: Swets & Zeitlinger B.V.

Itasca Consulting Group, Inc. 1996. FLAC3D – FastLagrangian Analysis of Continua in Three-Dimensions,Ver 1.1 User’s Manual. Minneapolis, MN: Itasca.

Johansson, E. & Hakala, M. 1995. Rock Mechanical Aspecton the Critical Depth for a KBS-3 Type Repository Basedon Brittle Rock Strength Criterion Developed at URL inCanada, SKB, AR D-95-014, SKB.

Kwon, Y.J., Kang, S.W. & Ha, J.Y. 2001. Mechanical struc-tural stability analysis of spent nuclear fuel disposalcanister under the internal/external pressure variation,KAERI/CM-440/2000, KAERI.

Park, J.H., Kuh, J.E. & Kang, C.H. 1998. An examination ofthermal analysis capability of FLAC3D on the near fieldof high level radioactive waste repository, KAERI/TR-1187/98, KAERI

Park, B.Y., Bae, D.S., Kim, C., Kim, K.S., Koh, Y.K. & Jeon,S.W. 2001. Evaluation of the Basic Mechanical andThermal Properties of Deep Crystalline Rocks, KAERI/TR-1828/2001, KAERI.

Park, J.H., Kwon, S., Choi, J.W. & Kang, C.H. 2001.Sensitivity analysis on mechanical stability of the under-ground excavations for a high-level radioactive wasterepository, KAERI/TR-1749/2001, KAERI.

Patchet, S.J., Carrasco, R.C., Francke, C.T., Salari, R. &Saeb, S. 2001. Interaction Between Two Adjacent Panelsat WIPP,” in Rock Mechanics in the National Interest,Proceedings of the 38th U.S. Rock Mechanics Symposium,Washington, D.C., July 2001), Vol.: 517–523. Lisse, TheNetherlands: Swets & Zeitlinger B.V.

Selvadurai, P.S. & Pang, S. 1990. Mechanics of the interac-tion between a nuclear waste disposal container and abuffer during discontinuous rock movement, EngineeringGeology, Vol. 28: 405–417.

Simmons, G.R. & Baumgartner, P. 1994. The disposal ofCanada’s nuclear fuel waste: Engineering for a disposalfacility, AECL Research, AECL-10715, AECL.

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153

Numerical simulation of radial bolting: Application to the Tartaiguille railway tunnel

F. LaigleElectricité de France, Hydro Engineering Centre, France

A. SaïttaCentre d’Etudes des Tunnels, Lyon, France

ABSTRACT: In this paper, a numerical model is used to predict radial bolting performance in soft rocks crossedby the Tartaiguille TGV (high speed railway) tunnel located on the new TGV French southeast line. First, a briefdescription of results of field investigations and monitored displacements during excavation of this under-ground tunnel are done. The support system is constituted by shotcrete and radial grouted bars. Because of toohigh monitored displacements in some sections in these marls, a modification of the support system has beendecided on site. This modification consists in an increasing of the density of bars in a specific zone. Back analysisconfirmed the very strong efficiency of these additional bars in this case. However, conventional continuousmodeling of the tunnel done up to now, strongly underestimated the real contribution of these extra bolts. Somenew numerical simulations have been done using a constitutive model proposed by Laigle. This constitutive modelintegrated in FLAC focuses on the post-peak behavior of rocks. It’s based on a simple and physical description ofthe behavior of ground in this domain, with accurate evolutions of the cohesion and the dilatancy. This paperdescribes numerical results obtained using this new constitutive model applied to the Tartaiguille tunnel case. Thesignificant monitored effect of additional grouted bars is well shown by this computation.

1 INTRODUCTION

Bolting corresponding to grouted bars or friction boltsis a frequently used component in light support systemswhen driving underground galleries. This techniqueentails reinforcing a ring of ground around an excava-tion by introducing stiffer linear elements. This method,both effective and inexpensive, is the basis for methodsof tunnel driving such as the new Austrian tunnelingmethod.

Bolt design has been based for a long time on empir-ical rules and on an optimization during the worksthemselves. At the present time, we notice a very clearevolution in design practices toward the frequent useof numerical methods to the detriment of empiricalones. However, there is considerable doubt about theability of models now used to correctly simulate the effect of bolting. So we wanted to contribute to this reflection reporting, in an applicable way, theresults obtained during the works on the Tartaiguilletunnel. The support system installed in this Aptianmarls mainly consisted of grouted bars. In an initialstage, this study has made it possible to accurately

quantify the influence of bolts on the deformations ofthe tunnel wall. Beginning with these conclusions, itthen became possible to make a comparison with theresults of a numerical modeling.

In a second part of this paper, some numerical sim-ulations are presented. These simulations are appliedto the Tartaiguille tunnel. Using a new elastoplasticconstitutive model developed by Laigle (2003), thesesimulations allow to find by computation the strongcontribution of a grouted bars system on stabilityconditions of the tunnel.

2 EXPERIMENT FEEDBACK FROM THETARTAIGUILLE TUNNEL

2.1 The new TGV southeast (high speed railway)line and the Tartaiguille tunnel

After the first “short” Paris-provinces lines, the high-speed train network has been extended in France andfirst provincial towns will soon be connected to eachother. At the present time, the first line being completed

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is that of the TGV Southeast which should make itpossible to go from Paris to Marseilles in three hours.After Valence, the Tartaiguille tunnel is the first ofunderground structures in the southerly direction. It’sa 2340 m long tunnel which has been driven from thenorth and south extremities (Fig. 1).

Excavations began in February 1996. In the Aptianmarl geological formation, digging method retainedwas the upper half-bench cut method (Fig. 2).

The support system is composed of radial bolts,associated with shotcrete and sometimes with yieldingarches.

The tunnel driving cycle was broken down into theexcavation of the upper half-section and the laying ofthe support system. Then, about a hundred meters inback of the working face, a second station excavated thelower half-section. Finally, further in back of the face,a reinforced concrete invert, then the final concretelining was poured.

From the beginning of the driving in the Aptianmarls, major convergences of the tunnel wall weremeasured. The alert thresholds were quickly exceededand the phenomenon grew with the passing of the lowersection. Strains of the wall resulted in a loading of theshotcrete shell that was greater than its breaking limitand a consistent cracking in crown, which generatedsafety problems for the worksite.

Complementary tests then made it possible to esti-mate the ratio between horizontal and vertical initial

stresses. This ratio has been estimated from 1.2 to 1.7while the project had concluded on lower values. Themajor stress is horizontal which explains the crownbreaking mechanism of the shotcrete shell, subjectedto a lateral thrust.

The support system had to be modified to controlstrains. The improvements, which made it possible tosignificantly reduce convergences, are an increase inthe density of the radial bolting and their integral useat the face. Convergences were brought to a tolerablethreshold for a double density of bolts when com-pared with the initial plan.

Our study of radial bolting is restricted to the geo-logical formation composed of Aptian marls. This isbecause of two advantages presented by this facies, one being the homogeneity of the geology and the otherbeing the presence of a zone, the support system ofwhich is composed only of shotcrete and bolts. Thisgeological description comes from the geological andgeotechnical wrap-up paper, prepared by the engi-neering firm Coyne and Bellier (1995).

2.2 Study zone and measures carried out

For our study, the support zone is 335 meters lengthwithin which the support system is composed only ofshotcrete and radial bolts composed of grouted bars 4 meters in length. The reinforcement of the bolting ofthe upper half-section takes place in several phases:

– Zone 1: One ring of bars every 2 meters (Fig. 2).– Zone 2: 2 sets of bars inserted at the springing of

sidewalls.– Zone 3: Return to the initial density (idem zone 1).– Zone 4: Re-establishing interposed ring (idem

zone 2).

In order to monitor the evolution of the ground andthe efficiency of the support system, measurements ofwall displacements were performed. Five measurementtargets were used for each section, a target A, positionedat the crown, two targets B and C at the spring lines ofthe side walls of the upper half-section, and two targetsD and E at the side walls of the lower half-section(Charmetton 2001).

2.3 Results of the measurements

The effectiveness of the support system during the driv-ing was essentially monitored beginning with two val-ues out of the three targets, which comprise each testingsection. These are the measurements of the leveling ofpoint A and of the convergence of cord BC. The mea-surements sections called S07, S08, S09 and S10 wereincluded in the Zone 1. The sections S11, S12, S13,S14, S15, S16 and S17 were in the Zone 2. The sec-tions S18, S19 and S20 measured displacements ofthe Zone 3 while S21 and S22 were inside the Zone 4.

154

Figure 1. Layout of the Tartaiguille tunnel.

Figure 2. Cross section in Aptian marls.

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So we interested ourselves, for each of the mea-surement sections, in these values, but also in the con-text in which they were obtained, that is, everythingthat could have an influence on the results of themeasurements.

The following graphs present the results, whichcome directly from the worksite of the convergencesof the cords BC for sections S07 to S22 (Fig. 3).

2.4 Study of the results of the upper half-section

The calculation of the average convergence at 30 m foreach of the zones (see Fig. 4) shows a reduction of 37%between Zone 1 and Zone 2, and 46% between Zone 3and Zone 4. The increase in the number of bolts at thesidewalls (from 1 bolt/3.3 m2 to 1 bolt/1.6 m2) thereforeresulted in a reduction of at least 37% in short-term hor-izontal convergences of the BC cord. We may show thatthis reduction is directly due to the bolting. All the other

parameters (excavation speeds, steps of advancement,cover height…) varied, but without our being able toestablish direct links between their evolution and thatof the two groups of curves of convergences.

The alternation of the two bolt densities along ourstudy zone is an argument which confirms the role ofthe bolts because this assures that the evolution of theconvergences does not result from a modification of theground.

3 NUMERICAL SIMULATIONS

The aim of these numerical simulations was to simulatein the framework of usual continuous modeling thestrong effect of a grouted bars system on the stabilityconditions of the Tartaiguille tunnel. The particularityof this work is to use a new constitutive model devel-oped by Laigle. These numerical simulations are doneusing the elastoplastic version. A more general elasto-plastic–viscoplastic version has been developed andpresented in another paper (2003).

Following phenomena are studied in the frame-work of these simulations:

– What would be the behavior of the tunnel with anunder-estimated support and what was the potentialfailure mechanism?

– Was a support system needed?– What is the effect of a delay in the installation of the

support system?– What is the effect of a local failure of the shotcrete

layer?

3.1 The constitutive model

In general, at least for deep and no-urban tunnels, thegoal of a support system composed with shotcrete andgrouted bars is to prevent mechanical failure within asufficient safety level. In cohesive rocks, which can beconsidered as a continuous material, this failure is asso-ciated with the development of a fracturing processinduced by the excavation. So, it seems necessary tofocus the simulation of the mechanical behavior bothon the pre-peak behavior and the post-peak behavior.

This new constitutive model has been written in theframework of the elastoplastic theory The expressionof the yield surface corresponds to the generalizedHoek and Brown criterion. This surface is governedby 4 parameters, which are the unconfined compressivestrength and three other parameters “m”, “s” and “a”.These 3 last parameters change in accordance with aninternal variable �p, which is the irreversible shearstrain defined below:

(1)

155

-140

-120

-100

-80

-60

-40

-20

00 50 100 150 200 250 300

Number of days

Con

verg

ence

s (m

m)

S07

S08

S09

S10

S11

S12

S13

S14

S15

S16

S17

S18

S19

S20

S21

S22

Figure 3. Measurements of convergence of cords BC.

0

10

20

30

40

50

60

70

S07

S08

S09

S10

S11

S12

S13

S14

S15

S16

S18

S19

S20

S17

S21

S22

Measurement sections

Con

verg

ence

s (m

m)

Zone 1

Zone 2

Zone 3

Zone 4

: Average convergence

-37%

-46%

Figure 4. Measurements of the convergences at 30 m fromthe face.

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(2)

Several thresholds for the yield surface are proposed.

– A first threshold corresponds to the damage crite-rion. This criterion is assimilated to the crack initi-ation, so to the beginning of the dilatancy.

– The second threshold corresponds to the peakstrength criterion.

– The third threshold characterizes the strength of adamage rock sample crossed by an induced shearfracture. In these conditions, cohesion of the rockat a macroscopic scale is assumed to be null.

– The last criterion corresponds to the residual strengthcriterion, which is reached at very large shear strains.

Some specific hardening laws are suggested for eachparameter “m”, “s” and “a” allowing to describe theevolution of the yield surface from one threshold toanother (Laigle 2003). In the softening domain, thenegative hardening is assumed to be divided into threephases:

– The first phase of softening corresponds to a dete-rioration of the rock’s cementation illustrated by aprogressive disappearance of the cohesion at themacroscopic scale. This first phase is associatedwith an increasing of the dilatancy.

– The second phase corresponds to the shear of aninduced fracture. It’s associated with a decreasing ofthe dilatancy at the macroscopic scale.

– Finally, the last domain corresponds to a purely fric-tional behavior, which defines the residual strength.The shear occurs without any volumetric strain.

Figure 5 shows schematically various domainsdescribing the physical state of a rock sample under amechanical triaxial loading. In the domain 1, thebehavior is elastic linear.

Figure 6 presents thresholds retained for the Aptianmarl of Tartaiguille. Major mechanical properties are asfollowing:

– UCS � 0.85 MPa– Young’s modulus: E � 1000 MPa– Poisson ratio: � � 0.36

This constitutive model has been integrated in the EDF’s local version of FLAC V3.4, using FISHprocedures.

3.2 Behavior of the tunnel without support

A first simulation has been done without consideringany support system. The excavation is simulated by

decreasing initial internal stresses applied to the tunnelperimeter. Excavation is simulated in two phases: thevault and the bench. Figure 7 shows the state of the rockmass at a decrease of 97.8% of these stresses duringthe vault excavation (100% corresponds to the end ofexcavation process of the vault). We may observe animportant damage zone near the foot of the tunnel,progressing behind sidewalls towards the roof.Without any support, this mechanism will generate aglobal instability of the gallery in the short term.

3.3 Behavior of the tunnel with initial support

A second simulation has been done considering the ini-tial support system (Zone 1, ring of grouted bars every

156

Dev

iato

ric

stre

ss

Axial strain

Axial strain

Domain 5: Fractured rock in aresidual state

Domain 4: Fractured rock

Dom

ain1

: Int

act r

ock

Dom

ain

2: F

issu

red

rock

inpr

e-pe

ak d

omai

n

Dom

ain

3: F

issu

red

rock

inpo

st-p

eak

dom

ain

Vol

umet

ric

stra

in

Figure 5. Schematic behavior of a rock sample during atriaxial test.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.5 1 1.5 2 2.5 3

Minimal principal stress (MPa)

Max

imal

prin

cipa

l str

ess

(MP

a)

Peak criterion

Fractured

rock cr

iterion Damage criterionResidual crite

rion

Figure 6. Threshold criteria for the Aptian marl.

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2 meters). Bars are simulated using structural cableelements and a shotcrete layer using structural beamelements. These bars and beams are installed after adeconfinement of 70%, so approximately in the first2 meters behind the face heading.

Bars are linked to some beam nodes with the goalto simulate face plates. Despite this, we will observethat the maximum tensile stain is located in theground and not near the wall (Fig. 8)

With this support, the stability of the tunnel duringthe upper-half excavation can be theoretically justi-fied, even if a damage zone exists near tunnel the footof the tunnel (Fig. 8). Figure 9 presents horizontal

convergences versus time, assuming an advancing rateof 2 m/day. These simulations don’t take into accountthe time. Convergences are drawn versus time only tocompare with monitored values.

Without any modification of this initial support, theexcavation of the bench can’t be finalized. Because ofthe bench excavation, the fracturing phenomenon takesoff again toward the roof. The initial support is not suf-ficient to stop the development of this damage zone.The stability cannot be demonstrated from this numer-ical simulation (Fig. 10).

These results seem in accordance with the siteengineer’s decision to adapt the support system. Con-sidering high measured displacements and the devel-opment of a local failure of the shotcrete layer at theroof, he decided to reinforce this initial support system.

157

Fissured rock (Domain 2)

Rock in tension

Fractured rock (Domain 4)

Fissured rock in post-peak domain (Domain 3)

Deconfinement: 97,8 %

Intact rock (Domain 1)

Figure 7. Physical state of rock without any support.

Fractured rock( Domain 4)

Strains ofbolts

Fissured rock( Domain 3)

Fissured rock inthe pre-peak domain( Domain 2)

Intact rock (Domain1)

Figure 8. Physical state of rock at the end of the upper-halfexcavation.

-20-19-18-17-16-15-14-13-12-11-10

-9-8-7-6-5-4-3-2-10

0 1 2 3 4 5 6 7 8 9 10 11 141312

Without any support

Hor

izon

tal c

onve

rgen

ce o

f th

e tu

nnel

(m

m)

FailureWith initial support

Time (days)

Figure 9. Horizontal convergence of the vault without andwith an initial support system.

Fractured rock(Domain 4)

Fissured rock inthe pre-peak

domain(Domain 2)

Fissured rock in thepost-peak domain

(Domain 3)

Development of theinduced fractured

zone

Intact rock(Domain 1)

Figure 10. Physical state of rock during the bench excava-tion and failure mechanism.

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3.4 Behavior of the tunnel with a reinforcedsupport

The reinforcement of the support system correspondsto the installation of additional bolts on sidewalls ofthe tunnel. In this zone, the bolting density is double.

In the framework of numerical simulations, severalpatterns are considered:

– 2 additional grouted bars are installed on each side-wall at 6 meters behind the face heading (90% of thevault deconfinement).

– 1 additional grouted bar is installed on each side wallas soon as possible, about 2 meters from the faceheading (70% of the vault deconfinement).

In these 2 last cases, it appears that theoretically,the stability is demonstrated both during the vault andthe bench excavation. The local increase of the bolt-ing density prevents the development of induced frac-tures, which appear on the sidewall during the vaultexcavation. The precise location of these additionalbolts is essential. We understand that an increase ofthe bolting density on the roof is not useful for thegoal to delay and stop the observed mechanics on site.

A second interesting aspect is the effect of the timeat which additional bolts are installed. Figure 12shows that the final horizontal convergence is smallerwith only one additional grouted bar installed earlierrather than 2 bars added later.

These simulations highlight very well what projectengineers already knew but which has never been

shown by numerical computations. With the aim ofreinforcing the rock mass, the grouted bars systemhas to be installed as early as possible, before any crit-ical increasing of monitoring displacements. This isbecause the goal of this type of support is to protectand help the rock to keep a sufficient shear strength toensure the global stability.

3.5 Effect of a local failure of the shotcrete lining

During excavation of the vault, a crack appeared anddeveloped in the shotcrete at the crown of theTartaiguille tunnel. This same phenomenon has beenobserved during excavation of one large cavern of the CERN-LHC project in Geneva (Laigle 2002).Depending on the support design, a shotcrete failurecould be critical for the global stability of the tunnel.However, in these two previous cases, the support hasbeen designed with the consideration that one majorcomponent is the grouted bars system and not onlythe shotcrete layer. If the shotcrete keeps an essentialfunction, it can’t be assimilated in these cases to acontinuous shell like in the SCL approach.

In the case of Tartaiguille tunnels, it was interest-ing to know if this crack in the shotcrete was reallycritical from a global stability point of view. A simu-lation has been done, considering the reinforced bolt-ing system on sidewalls. Cracking of the shotcretelayer has been simulated by deleting some structuralelements near the crown. This deletion is done at 90%of the vault deconfinement.

Figure 13 shows the physical state of the rock massat the end of the tunnel excavation. The local failureof the shotcrete generates new damage and a frac-tured zone above the tunnel roof. The growth of thisnew fractured zone is stopped by grouted bars.

Figure 14 presents the evolution of the horizontalconvergence in accordance with time, with and with-out failure of the shotcrete. An increase of displace-ment appears at the time of the failure but a newstable configuration is reached after that.

158


Fissured rockin the pre-peak

domain(Domain 2)

Fissured rock inpost-peak domain

(Domain 3)

Rock at residual state(Domain 5)


Figure 11. Physical state of rock at the end of the benchexcavation, considering a reinforced support system.

-25

-20

-15

-10

-5

00 10 20 30 40 50 60 70 80

Without additional bolt

Excavation of vault Excavation of bench

Failure

With 1 additionalbolt at 70%

With 2 additionalbolts at 90%

Time (days)

Hor

izon

tal c

onve

rgen

ce (

mm

)

Figure 12. Influence of reinforced bolting pattern of hori-zontal convergences.

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3.6 Justification of the length of bars

Some computations have been done considering sev-eral lengths for grouted bars. Figure 15 shows hori-zontal convergences versus time, in accordance withthese lengths. If the length is greater than 4 m, globalstability is assured. On the contrary, if this length is2 m or less, a stable configuration can’t be obtainedand stability of the tunnel can not be justified duringthe bench excavation.

From these results, we may conclude that there isan optimal length for grouted bars, depending on thepotential failure mechanism of the tunnel during theexcavation process. These numerical results confirmusual formulas, which provide an estimation of the

length in accordance with the span of the tunnel:

L � 2 0.2D � 4.7 mL � 0.30D � 4.0 m

where D is the tunnel span in meters.

4 CONCLUSIONS

The study of monitored convergences in the Aptianmarls of the Tartaiguille Tunnel has made it possibleto approach quantitatively the effect of bolting on thestructure’s stability. We were able to confirm the veryconsiderable efficiency that a few extra bolts bring toa mass of non-fractured soft rock. This back analysishas made it possible to quantify the effect of reinforc-ing the sidewalls on the reduction of convergences inthe Tartaiguille tunnel.

In the framework of a back analysis, some numeri-cal computations have been done considering a newconstitutive model well adapted to underground engi-neering expectations. Goals of these simulations wereto find with a suited numerical tool major behaviorsobserved and monitored during the Tartaiguille tunneldigging.

These simulations allow us to identify:

– The potential major failure mechanism of the tun-nel. The knowledge of this mechanism is essentialboth during the design phase and during excava-tion process.

– The significant efficiency of a grouted bar systemon tunnel stability conditions.

– The effect of a local reinforcement of the boltingsystem on displacements and safety level duringthe digging.

– The limited effect of a local shotcrete failure if thegrouted bar system is sufficient and if the stabilityis not only ensured by a shotcrete shell.

159


Fissured rock in thepost-peak domain

(Domain 3)


Crack in the shotcrete

Fissured rock in thepre-peak domain

(Domain 2)

Figure 13. Physical state of rock at the end of the benchexcavation, considering a local failure of the shotcrete.

-40

-35

-30

-25

-20

-15

-10

-5

00 10 20 30 40 50 60 70 80

Without failure of the shotcrete layer

With failure of the shotcrete layer

Time (days)

Hor

izon

tal c

onve

rgen

ce (

mm

)

Figure 14. Influence of a local shotcrete failure on horizontalconvergences.

Length: 2m

Length: 4mLength: 6m

Length 8m

Failure-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

00 10 20 30 40 50 60 70 80

Hor

izon

tal c

onve

rgen

ce (

mm

)

Time (days)

Figure 15. Influence of bolt lengths on horizontal convergences.

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REFERENCES

Charmetton, S. 2001. Reinforcement des parois d’un tunnelpar boulons expansifs – retour d’expérience et étudenumérique. Ecole Centrale de Lyon. Ph.D. Thesis, 2001(In French).

Coyne and Bellier. 1995. Geological and geotechnical wrap-up paper. Mediterranean TGV. Tartaiguille tunnel (InFrench).

Laigle, F. 2001. CERN-LHC Project – Design and excavationof Large-Span Caverns at point 1. Proc. of the IRSM

regional Symposium Euorock 2001 – Rock Mechanics – achallenge for Society – Espoo – Särkkä & Eloranda (eds).Balkema Publishers.

Laigle, F. 2003. Modélisation rhéologique des roches adaptéeà la conception des ouvrages souterrains. Ph.D. EcoleCentrale de Lyon, in prep.

Laigle, F. 2003. A new viscoplastic model for rocks – Appli-cation to the Mine-by-test of AECL-URL. Proc. Intern.Symp., Sudbury, Canada. To be published.

160

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161

Recent experiences of the prediction of tunneling induced groundmovements

C. Pound & J.P. BeveridgeMott MacDonald Ltd, Croydon, Surrey, UK

ABSTRACT: The ability to predict ground movements caused by tunneling is becoming increasingly impor-tant as more tunnels are constructed in urban areas. It is generally recognized that the ground surface settlementtrough above a tunnel is well represented by a gaussian curve. Data is available from many projects around theworld, which provides guidance on the values to use in the gaussian curve. However, if novel forms of tunnel-ing are used, if unusual ground conditions are present or if horizontal or subsurface ground movement predic-tions are required, then these empirical methods are not suitable.

This paper presents the results of a suite of numerical analyses carried out to identify the most appropriate soilmodel to use for the prediction of surface settlement troughs. As shown by many other authors linear elastic orlinear elastic perfectly plastic soil models are unsuited to the prediction of realistic surface settlements. Most ofthe analyses carried out in the study predict a settlement trough that is wider than observed despite modifica-tions to the size of the model, the boundary conditions, the in-situ stress conditions and the initial small strainstiffness. The constitutive model that predicts the most realistic settlement trough was a non-linear anisotropicsoil model with a higher horizontal than vertical stiffness. The soil non-linearity was based on the approach suggested by Jardine but modified for anisotropy. The shape of the trough was found to be sensitive to the valueof the vertical to horizontal shear modulus and the ratio of the horizontal and vertical Young’s moduli.

1 INTRODUCTION

The prediction of ground movements is very importantduring the planning and design phase of any tunnelconstruction project in an urban area. This predictionis used to identify the risk of damage to adjacentstructures and utilities and to assess whether the pro-posed construction method needs to be modified. Itcan also be used to highlight where mitigation meas-ures may be necessary in advance or during tunnelconstruction.

Surface settlements caused by tunneling are nor-mally assessed using empirical methods (O’Reilly &New 1982, Macklin 1999). The method was developedfrom review of settlement data from a large numberof tunneling projects around the world. However, themethod is difficult to apply when the ground condi-tions or construction method is unusual or where morethan one tunnel is present. Many attempts have beenmade to use numerical methods to predict groundmovements due to tunneling but almost without excep-tion the analyses have predicted unrealistic surfacesettlement troughs.

This paper presents the results of a numerical mod-eling study to identify the factors affecting the predic-tion of surface settlements above tunnels.

2 NUMERICAL ANALYSIS

Numerical analysis is often used to predict the loadson tunnel linings using a variety of finite element andfinite difference programs. However, unless the groundmovements are predicted accurately it is difficult tobe confident that the predicted ground load acting onthe linings is correct. The prediction of surface settle-ment troughs caused by tunneling is difficult and eventhe adoption of sophisticated constitutive models forthe soil rarely results in a realistic surface settlementtrough.

The following sections present a series of analysescarried out in an attempt to match the surface settle-ment troughs observed above the bored tunnels on the Heathrow Express project (Pound & Beveridge, inpress). A section of single bored tunnel was consideredwhere the volume loss was typically 0.8% with a

09069-20.qxd 08/11/2003 20:29 PM Page 161


trough width factor of 0.5. The back analysis was carried out using the finite difference program FLAC.

The ground conditions comprise 4.4 m of TerraceGravel overlying London Clay. The base of the LondonClay is at a depth of about 60 m below ground level.The water table was taken to be 2.1 m above the baseof the Terrace Gravel and initial water pressures weretaken to be hydrostatic through the Terrace Graveland the London Clay. The tunnel axis was taken to beat a depth of 22.5 m below ground level and the tunneldiameter was taken as 6.115 m.

The basic geotechnical parameters used in the analysis are given in Table 1. The strength properties forthe London Clay represent the fissured undrainedshear strength and the values vary with depth, z, belowground level. The stiffness adopted for the TerraceGravel and London Clay is described in detail.

The variation of coefficient of earth pressure atrest, ko, with depth is shown in Figure 1. The values in the London Clay were derived from assessment ofthe results of self-boring pressuremeter tests and poresuction measurements made on undisturbed samples.

Only the short-term ground movements were modeled. Throughout the analysis the response of theTerrace Gravel was taken as drained whereas the res-ponse of the London Clay was taken to be undrained.This was achieved by setting the bulk modulus of thepore fluid to be zero in the Terrace Gravel and 2 GPa inthe London Clay.

The mesh for the modeling is shown in Figure 2 andcomprises over 5000 elements. Advantage was takenof symmetry about a vertical plane through the tunnelaxis. The far boundary is located 90 m from the tunnelcenterline, which represents a distance of 4 times thetunnel depth. The base of the model was located at thebase of the London Clay and was fixed against move-ment in both directions while in most analyses the ver-tical boundaries were fixed only in the horizontaldirections. Tunnel excavation was modeled by firstreplacing the elements within the profile of the tunnelby equivalent grid-point forces and then by progres-sively reducing these grid-point forces. The volumeloss was determined by integrating the vertical dis-placements at the ground surface. Once a volume lossof 0.8% was achieved the segmental concrete liningwas installed and the remaining grid-point forcesremoved. The segmental lining was taken to have the

properties given in Table 2. The moment of inertia wasreduced to allow for the number of joints in the liningin accordance with Muir-Wood (1975).

Soils are known to have a non-linear stress–strainbehavior prior to peak with a high initial tangent stiffness at very small strains and reducing stiffnesswith strain. One set of equations often used to representthis decay of stiffness with strain was developed by

162

Table 1. Basic geotechnical parameters.

Parameter Terrace gravel London clay

Bulk Unit Weight (kN/m3) 20.0 20.0Porosity (%) 35 50Cohesion (kPa) 0 37.5 6zFriction (°) 38 0

0

10

20

30

40

50

60

0 1 2 3

Dep

th b

elow

gro

und

leve

l (m

)

Earth pressure at rest, ko

Test dataMayne and Kulhawy

0.5 1.5 2.5

Figure 1. K0 profile.

Table 2. Segmental lining properties.

Young’s modulus Thickness Moment of (Gpa) (mm) inertia (m4)

20.0 225 3.1 � 10�4

Figure 2. Mesh.

09069-20.qxd 08/11/2003 20:29 PM Page 162


Jardine et al. (1986). The approximate tangential version of these equations were presented by Potts &Zdravkovic (1999) and are given below:

where �s is a generalized shear strain related to theoctahedral shear strain, �oct, by the following equation:

and p is the current mean effective stress.Throughout the analysis the stiffness was continu-

ally updated. Up to a specified minimum strain (�sminor �vmin), the stiffness varies only with p, but thereafterthe stiffness depends both on the current strain (�) andthe mean effective stress (p). It is considered that theseequations lead to unrealistically low elastic moduli atvery low stresses and therefore the minimum meaneffective stress used in calculating the elastic moduliwas 50 kPa.

The constants used in the Jardine equations aregiven in Table 3.

2.1 Linear elastic analyses

Initial analyses were carried out using linear elastic andlinear elastic perfectly plastic soil models (analyses t1and t2). The elastic moduli were taken as multiples ofthe mean effective stress in order to give a load in thelining of between 35 and 40% of overburden which isconsidered to be a typical short-term load on a boredtunnel lining in London Clay. To achieve this criterionthe elastic model was taken to be 20% of the smallstrain stiffness for the linear elastic model and 35% ofthe small strain stiffness for the plastic analysis. Thusfor the elastoplastic analysis the shear modulus for theLondon Clay was given by the following equation.

This is significantly higher than is conventionallyused in tunnel analyses even in overconsolidatedmaterials.

Figure 3 shows the surface settlement troughs forthese two analyses and the corresponding gaussiancurve for a volume loss of 0.8% and a trough widthfactor, K, of 0.5. The surface settlement troughs fromthe two analyses are clearly unrepresentative of theobserved ground settlement showing a maximum settlement around 15 m from the tunnel centerline. Theanalysis with the elastoplastic model is worse becauseof the ground yielding that is predicted between thetunnel crown and the ground surface resulting from thehigh in-situ horizontal stresses. These results clearlyshow the limitations of using linear elastic groundmodels for the prediction of ground movementaround tunnels.

2.2 Non-linear elastic analyses

Figure 4 shows the surface settlement trough whenthe non-linear behavior given by the Jardine equa-tions is adopted in the analysis (t3). The maximum

163

Table 3. Jardine constants.

Parameter Terrace gravel London clay

A 1104 1260B 1035 1143C 5.00E-06 1.00E-06R 275 618S 225 570T 2.00E-05 1.00E-05! 0.974 1.335� 0.94 0.617� 1.044 2.069� 0.98 0.42�smin 8.80E-06 1.40E-05�smax 3.50E-03 2.00E-03�vmin 2.10E-05 1.00E-04�vmax 2.00E-03 2.00E-03

0

1

0

Set

tlem

ent /

Max

imum

Set

tlem

ent .

Distance from centre-line (m)

10 20 30 40 50

0.2

0.4

0.6

0.8

1.2

t1t2Gauss Curve

Figure 3. Linear elastic/elastoplastic analyses.

09069-20.qxd 08/11/2003 20:29 PM Page 163


settlement is still not located on the tunnel centerline,but is offset by about 10 m. The trough is significantlynarrower than the linear elastic case, but is still muchbroader than the gaussian curve. To investigate theinfluence of the high horizontal stress on the shape ofthe settlement trough, an analysis was run using a k0profile based on the approach suggested by Mayne &Kulhawy (1982). The k0 profile assumed that 170 mof overburden had been removed from the top of theLondon Clay prior to the deposition of the TerraceGravel. The k0 was taken as 0.4 in the Terrace Gravel.The shape of the k0 profile is given in Figure 1 andshows lower k0 values particularly in the top tenmeters of the London Clay than the profile used inanalysis t3. The resulting settlement trough fromanalysis t4 is shown in Figure 4. Although the lowpoint of the settlement trough is nearer to the tunnelcenterline and the trough is generally narrower, theoverall shape of the settlement trough is only slightlydifferent.

To consider the effect that fixity conditions on the far boundary have on the shape of the settlementtrough, analyses were run with the far boundary fixedboth horizontally and vertically and also with a stressboundary condition. Neither analysis gave an improvedshape of settlement trough.

Analyses were also carried out with wider meshes tosee if a boundary width of 4 tunnel depths was inade-quate. Analyses were carried out with a mesh width of150 m and 1000 m. The effect of an increased meshwidth was minor with a small reduction in the settle-ment at 50 m from the tunnel centerline, but a corre-sponding increase in the settlement 5 m from the tunnelcenterline.

Analyses were carried out to investigate the effectof modifying the shape of the non-linear model andthe results are presented in Figure 5. In the first analy-sis (t5) the stiffness was increased by 50% at allstrains compared to the model prediction. In the nextanalysis (t6) the strain limit for the plateau region of the model was extended to a higher strain level.

In both of these analyses the increase in the soil stiff-ness made the shape of the settlement trough worse.In the third analysis (t7) the small strain stiffness wasincreased by 50%, but the shape of the stress–straincurve was the same after the end of the initial plateauregion as in analysis t3. The modifications to thesmall strain stiffness had only a modest influence onthe shape of the settlement trough.

2.3 Anisotropic soil model

A number of authors have indicated that only with ananisotropic soil model can a realistic shaped settlementtrough be obtained (Simpson et al. 1996, Addenbrookeet al. 1997). There is good evidence that the behaviorof London Clay is anisotropic with a higher hori-zontal than vertical Young’s modulus (Bishop et al.1965). Data also exists for other overconsolidatedclays (Lings et al. 2000).

The anisotropic elastic model was modified to allowinput of non-linear elastic behavior. No anisotropicelastoplastic soil model currently exists in FLAC,however with a volume loss of only 0.8%, the strainsin the ground surrounding the tunnel are only suffi-cient for very local plastic yielding of the ground tooccur and therefore there should be only a small errorin the adoption of an elastic model. In the absence of any definitive anisotropic constitutive soil modelfor the London Clay, the basic Jardine equation wasmodified as follows:

where X is defined as above. The values of the constants in the above equations are given in Table 3.

164

0

1

0 20

Set

tlem

ent /

Cen

tre-

line

Set

tlem

ent .


10 30 40 50

0.2

0.4

0.6

0.8

1.2

t1t3t4Gauss Curve

Figure 4. Non-linear elastic analyses.

0

1

0

Set

tlem

ent /

Cen

tre-

line

Set

tlem

ent .


10 20 30 40 50

0.2

0.4

0.6

0.8

1.2

1.4

t3t5t6t7Gauss Curve

Figure 5. Small strain model.

09069-20.qxd 08/11/2003 20:29 PM Page 164


The small strain stiffnesses are slightly lower as theynow relate to the vertical stiffness rather than anisotropic stiffness given previously. The adoption ofthe equations given above relate the variation of thestiffnesses only to shear strain and not to volumetricstrain. This model is therefore only suitable for mod-eling shear deformations and would need to be modi-fied to consider swelling or consolidation. The twoindependent Poisson’s ratios were taken as follows:

The analysis was first run with a pseudo-isotropicanalysis with both Emul and Gmul set to 1.0. The resultsof the analysis are shown in Figure 6 (analysis t8).The shape of the curve is somewhat improved compared to the previous isotropic analyses. Two fur-ther analyses (t9 and t10) were carried out with higherhorizontal stiffnesses by setting Emul to 1.6 and 2.0.The settlement troughs are also given in Figure 6. The shape of the settlement trough is significantlyimproved as the horizontal to vertical stiffness ratio is increased. The data by Bishop et al. (1965) and byAtkinson (1975) suggested that for London Clay theratio of horizontal to vertical Young’s modulus isaround 1.6.

To investigate the effect of the value of the shearmodulus on the shape of the settlement trough, a setof analyses were carried out with the value of Emul setto 1.6 and with values of Gmul of 0.5, 0.8 and 1.6(Analyses t11, t12, t13). The results are comparedagainst the analysis (t9) with a Gmul of 1.0 in Figure 7.The shape of the settlement trough is very sensitive to the value of the shear modulus. Generally as theshear modulus is increased the width of the settlementtrough is also increased. The best fit to the middlepart of the settlement trough is achieved when theshear modulus is only one third of the vertical Young’smodulus. However, this shape of settlement trough

can also be achieved by setting the horizontal Young’smodulus to be equal to twice the vertical Young’smodulus.

The settlement at the edges of the settlementtrough is much greater than that suggested by thegaussian curve and is also greater than observed inpractice. The settlement towards the boundary of themesh results from the horizontal ground movementsand a corresponding Poisson’s ratio effect. To reducethe vertical strains due to horizontal displacements,the Poisson’s ratio in the vertical plane was set to zero.Analyses t9 and t12 were repeated with a Poisson’sratio of 0.0 as analyses t14 and t15 and the results areshown in Figure 8. There is a significant narrowing ofthe trough as well as a significant reduction in the far-field settlement. However, the settlement 50 m fromthe tunnel centerline is still 10% of the centerline settlement.

Due to the assumed undrained response of theLondon Clay any horizontal movement will result inan equivalent vertical settlement. To prevent this surface settlement would require a volumetric change inthe soil and thus a drained soil response. To investigatethis effect, analysis t14 was repeated as analysis t16with the soil more than 20 m from the tunnel centerline

165

0

1

0 5 10 15 20 25 30 35 40 45 50

Set

tlem

ent /

Cen

tre-

line

Set

tlem

ent .


0.2

0.4

0.6

0.8

1.2

t8t9t10Gauss Curve

Figure 6. Anisotropic model.

0

1

0 5 10 15 20 25 30 35 40 45 50

Set

tlem

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assumed to be fully drained. The predicted settlementtrough is shown in Figure 9. As can be seen, the shapeof the surface settlement trough predicted by thisanalysis is very similar to the shape of a gaussian distribution with only a very small surface settlement50 m from the tunnel centerline. It is considered that theshape of this predicted curve is probably as good a match to observed tunnel settlements as a gaussiancurve.

3 DISCUSSION

The results of the series of analyses show that only withthe adoption of an anisotropic model can a realisticsettlement trough be predicted. This is similar toresults found by others (Simpson et al. 1996). Three-dimensional numerical analyses, which model the full construction sequence, have also been carried outand these also demonstrate that an isotropic soil modelleads to wider surface settlement troughs than thoseobserved in practice. Adoption of an anisotropic soil model results in surface settlement troughs verysimilar to those predicted by the two-dimensionalanalyses.

The anisotropic behavior of stiff overconsolidatedclays can be explained on the basis of the preferentialalignment of clay particles. There is also some datafrom field and laboratory testing to indicate that stiff overconsolidated clays are anisotropic. Howeversands have an even narrower settlement trough thanthose of clays as indicated by the trough width factornormally adopted. There is less justification for theadoption of an anisotropic soil model for sands fromfield and laboratory testing data. Sands of course will not respond in an undrained manner during tunneling and it may be that the different pattern ofgroundwater pressures around the tunnel during exca-vation will result in a different pattern of surface settlements predicted by the isotropic soil model.

The stress–strain behavior of most rocks is con-trolled not by the elastic behavior of the intact mate-rial but by the orientation and properties of thediscontinuities. The presence of the discontinuity setswill inevitably impose an anisotropic response to themass behavior of the rock which it is logical to sup-pose will influence the shape of the settlement trough.It is also reasonable to assume that where ground settlements are large, slip on discontinuities will occur.This could explain the narrow settlement troughsobserved over many tunnels in rock. The in-situ stressconditions could also influence this behavior.

The only way found to prevent the prediction ofsignificant settlements at the boundary of the modelwas to assume drained behavior for the soil at a dis-tance from the tunnel. The ratio of bulk stiffness ofthe water to that of the soil controls the drained orundrained behavior even where there is no groundwaterflow. Because of the increased strains near to the tunneland the formulation of the anisotropic model, the bulkstiffness of the soil model nearer to the tunnel is lowerthan that further from the tunnel. With a bulk stiffnessfor the water of 2 GPa, the water is at least one orderof magnitude stiffer than even the small strain stiff-ness of the soil. Reducing the bulk modulus of thewater has the effect of making the response of the soilapparently partially drained far from the tunnel andessentially undrained near to the tunnel. The effectwas found to be modest with a bulk modulus of0.2 GPa (analysis t17) but resulted in a realistic shapedsettlement trough with a bulk modulus of 0.02 GPa(analysis t18). The results of these two analyses areplotted in Figure 9. Unfortunately a bulk modulus forthe water of 0.02 GPa, is unrealistically small. It ispossible that the apparent drained response of theground far from the tunnel is due to a combination of a lower bulk modulus of water, a higher bulk stiffness of the soil than currently assumed and theeffect of some drainage of the soil due to the slowsmall stress changes occurring in the soil far from thetunnel.

4 CONCLUSIONS

The numerical analyses show that traditional linearelastic analyses with or without a yield criterion cannot predict settlement troughs similar to thoseobserved. Even non-linear elastoplastic analyses withisotropic soil stiffnesses overpredict the width of thesurface settlement trough. Only by adopting a non-linear anisotropic elastic soil model can surface set-tlement troughs similar to those observed be predicted.To reduce the predicted settlements at the edges of thetrough it is necessary to assume partially drainedbehavior of the soil. It is suggested that this couldresult from a lower bulk stiffness of the water.

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REFERENCES

Addenbrooke, T.I., Potts, D.M. & Puzrin, A.M. 1997. Theinfluence of pre-failure soil stiffness on the numericalanalysis of tunnel construction. Géotechnique 47(3):693–712.

Atkinson, J.H. 1975. Anisotropic elastic deformations in laboratory tests on undisturbed London Clay Géotechnique25(2): 357–374.

Bishop, A.W., Webb, D.L. & Lewin, P.I. 1965. Undisturbedsamples of London Clay from the Ashford Common Shaft:strength-strain relationships. Géotechnique 15(1): 1–31.

Jardine, R.J., Potts, D.M., Fourie, A.B. & Burland, J.B. 1986.Studies of the influence of non-linear stress–strain characteristics in soil-structure interaction. Géotechnique36(3): 377–396.

Lings, M., Pennington, L., Nash, D.S. & Poisson, D.F.T.2000. Anisotropic stiffness parameters and their meas-urements in a stiff natural clay. Géotechnique 50(2):109–125.

Macklin, S.R. 1999. The prediction of volume loss due to tunneling in overconsolidated clay based on heading

geometry and stability number. Ground Engineering,32(4).

Mayne, P.W. & Kulhawy, F.H. 1982. K0-OCR relation-ships in soil. Proc. ASCE, Journal of the GeotechnicalEngineering Division, Vol. 108, No. GT6, 851–872.

Muir-Wood, A.M. 1975. The circular tunnel in elasticground. Géotechnique 25(1): 115–127.

O’Reilly, M.P. & New, B.M. 1982. Settlements above tunnels in the United Kingdom–their magnitude and prediction. Tunnelling ’82, The Institution of Mining andMetalllurgy, 1982 pp. 173–181.

Potts, D.M. & Zdravkovic, L. 1999. Finite element analysis ingeotechnical engineering: theory. London Thomas: Telford.

Pound, C. & Beveridge, J.P. 2002. Recent experiences of the measurement of ground movements around tunnels.In press.

Simpson, B., Atkinson, J.H. & Jovicic, J.H. 1996. Geotech-nical aspects of underground construction in soft ground.pp. 591–594. Balkema.

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169

Numerical modeling of remedial measures in a failed tunnel

Y. Sun & P.J.N. PellsPells Sullivan Meynink Pty Ltd, Sydney, Australia

ABSTRACT: A FLAC3D analysis was conducted for the investigation of the failure and the design of remedialworks for one tunnel in Melbourne Australia. Failure of an approximately 8 m length of the sidewall of the formedconcrete arch occurred in mid February 2000. The original design of the un-reinforced concrete lining was suchthat it just sits on the flat upper surface of the approximately 1.8 m-thick concrete invert. It was generally believedthat the failure was primarily due to the compressive stresses across the arch/invert joint being substantially low,which means that the compressive stresses in the arch lining induced by the groundwater pressure at time of failuremust have been transferred by 3D action to the west and east of the failure. The purpose of the numerical mod-eling is to return the failed section to a fully functional arch/invert concept. Key factors in the remediation arethe width and sequence of removal of the panels, which were investigated in details in this paper.

1 INTRODUCTION

The remedial concept is to remove the formed archconcrete within and immediately around the failure areaand reinstate the original design. As a precursor to the3D analysis, 2D analyses using Phase II was carriedout in PSM office to assess the likely compressiveand tensile stresses generated above and around panelcut-outs, and the effects of flat jack stressing. Theresults were used as a guide in selecting the 1.5 mpanel width and the excavation sequence proposed inthe design.

A FLAC3D model includes a 36 m length of the tun-nel and the surrounding rock, which contains interfaceelements between the arch lining and rock, betweenthe arch lining and floor, and between arch pours Aand B. The model allows an initial 3 mm gap to existat the arch/invert interface within the modeled failurearea, prior to application of groundwater pressures.

2 NUMERICAL MODELING

2.1 Geometry of model

The model includes a 36 m length of the tunnel in thelongitudinal direction that is divided into two equalparts, named as arches A and B. The depth of the tun-nel is 60 m below the surface. Five vertical panelswith a width of 1.5 m each from the contact betweenarches A and B were designed and named as Panels 1through 5 sequentially along the arch A side. Panels 1

and 2 are 4 m high, Panels 3 and 4 are 3.2 m high andPanel 5 is 1.6 m high. The model geometry is shownin Figures 1a, b & 2.

2.2 Interfaces

Interfaces are planes within a FLAC3D model alongwhich sub-grids can interact, slip and/or separation isallowed. A total of eight interfaces shown in Figure 3are modeled as:

– Interface 1: between arch and invert.– Interface 2: between concrete arch/invert and rock

simulating the membrane.– Interface 3: between arches A and B.– Interface 4: between panel 1 and arch B.– Interface 5: between panel tops and concrete arch A.– Interface 6: between back of panels 1, 3 and 5 and

rock surface simulating the membrane.– Interface 7: between back of panels 2 and 4 and

rock surface simulating the membrane.– Interface 8: between panels and concrete arch A.

2.3 Initial conditions

The initial stresses sxx, syy and szz of 60 m of rockload were applied to the model boundary with a gra-dient zero as required. A 3 mm gap from arch contactextending 6 m along arch A side and tapering off at 7 m was modeled to replicate the field observedinitial stress conditions.

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2.4 Modeling sequence

The following stages were developed:

– Stage 0: Initial condition. A 3 mm gap was modeled.The pore pressure of pre-leak value of 470 KPa

was initially applied at the interface between theconcrete arch/invert and rock surface. To do so, avirtual interface inside the tunnel surface has to beset up in order to store the face list for applying theequivalent normal stress to the rock surface.

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Figure 1b. Model geometry showing arch and invert.

Figure 1a. Model geometry showing entire model.

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– Stage 1: Reduce the pore pressure around the tunnelto pre-repairing condition that is zero behind thefailure area and increases linearly to 470 KPa at 30 maway from the failure.

– Stage 2: Remove the Panels 1, 3 and 5. The corre-sponding interface element at the back of the pan-els should be removed from the list and the sameprincipal applies to the corresponding rock surface

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Figure 2. Model geometry showing layout of panels.

Figure 3. Plot showing the interfaces.

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(virtual interface) that was in contact previouslywith the back of removed panels.

– Stage 3b: Cast concrete panels 1, 3 and 5 in place.The interfaces 1, 5, 6 and 8 for panels 1, 3 and 5 areestablished in the model in addition to the interface 4for panel 1, and the bottom gap between new pan-els and invert is set to zero. Interface 6 has to beadded into the face list for applying the normalstresses to the back of new panel concrete. Thegravity stresses are initialized in the new panels and low strength properties were used for the inter-face 8 first (Stage 3a), and then the properties werereturned to normal values.

– Stage 3d: apply flat jack load of 2 MPa at the top ofPanels 1, 3 and 5. High strength properties wereused (Stage 3c) for the arch including the new pan-els to get rid of the possible dynamic effect, andreturned to normal afterwards.

– Stage 4: Remove Panels 2 and 4. Follow the similarprocedure as described in Stage 2.

– Stage 5a: Cast concrete panels 2 and 4 in place.Follow the similar procedure as described in Stage 3b.

– Stage 5b: apply flat jack load of 2 MPa at the top of Panels 2 and 4. Follow the similar procedure asdescribed in Stage 3d.

– Stage 6: Increase all flat jack loads equivalent topressure of 4 MPa.

– Stage 7: Increase the hydrostatic load to 470 KPa.

2.5 Pore pressure

The pore pressures are modeled explicitly by apply-ing two opposite normal pressures that are equivalentto the pore pressures to the interface between the con-crete arch/invert and rock (interfaces 2, 6 and 7) and tothe corresponding rock surface. For the pre-repairingcondition, it is assumed that the drain center islocated at the top center of the panel 3. The regionwith a distance of less than 4 m from the drain centerhas zero pore pressure, while the region with a dis-tance of more than 30 m from the drain center has afull pore pressure of 470 KPa. The region that falls inbetween has a linear distribution of pore pressure.

2.6 Parameters

The concrete is assumed to have a Young’s modulus of 32000 MPa, an unconfined compressive strengthof 50 MPa and a tensile strength of 2.5 MPa. The totalzone elements of rock and concrete are 10624 and12104, respectively. The surrounding rock is modeledas elastic material and the concrete arch and invert are modeled as Mohr-Coulomb material. The shearstrength parameters adopted for the various interfacesare summarized in Table 1.

3 FISH CODING

Various FISH codes were developed to perform thefollowing functions as:

– Storing all zone faces connected to the concrete/rockinterfaces (2, 6 and 7) to create a list of all faces for “app nstr” late. Generally, there are two inter-face elements that are associated with one zone.We can pick up the first element and skip the sec-ond one to set up a list where the address of thezone to which the interface element is attached,and the corresponding face ID number are storedin a 2-dimensional array.

– Removing faces from the list if zones are changedto a different model (here anisotropic) prior to beingmade “null”.

– Applying the equivalent pore pressure to the zonefaces in the current list.

– Shifting the solid back to the tunnel for quickmanipulation.

– Adjusting the contact between the arch and theinvert slab.

– Setting the gap between arch A and the invert slab.– Calculating the pore pressure distribution at the

pre-repairing condition.

4 RESULTS AND DISCUSSIONS

Figure 4 shows contours of smin at the initial stage,where the 3 mm gap between the concrete arch A and invert slab was maintained. The majority areaimmediately adjacent to the gap has a compressivestress up to 5 MPa, while elsewhere has a notablyhigher compressive stress.

Figures 5a & b show contours of major and minorprincipal stresses when panels 1, 3 and 5 were excavated. Figure 6 shows the tensile crack at the topof panel 1 and bottom between panel 3 and 5.

Figures 7a & b show contours of major and minorprincipal stresses when panels 2 and 4 were excavatedand the flat jacks above panels 1, 3 and 5 are stressedto 2 MPa. Tensile stresses above these panels weredropped from 1 MPa to less than 500 KPa. The maxi-mum compressive stresses show quite a complex distribution with a maximum less than 8 MPa. Tensilefailure (Fig. 8) remains at the top corner of panel 1

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Table 1. Interface properties.

Interfaces Cohesion FrictionType ID No. (KPa) (deg.)

Concrete/concrete 1, 3, 4 & 8 0 35Concrete/membrane 2, 6 & 7 0 10Concrete/flat jack 5 0 40

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Figure 5a. Contour of smax at stage 2 where panels 1, 3 and 5 (from the right) were excavated.

Figure 4. Contour of smin at stage 0 showing the arching effect due to 3 mm gap between arch and invert.

and develops between panels 4 and 5, where a stressconcentration is noticed due to the difference inheight. It should be noted that tensile cracking inthese locations is not of a particular concern.

Figure 9 shows the cracking pattern after all flat jacks are stressed to 4 MPa with groundwater pressure at low values, corresponding to the pro-cess during the repair work. In general, the stresses

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174

are benign and there is no new cracking. Figure 10shows the crack pattern when the groundwater pressures are returned to a high value at 470 KPa.A tensile crack is predicted on the rock side of the

arch about 2.5 m above the panels vertically. Thisindicates the need to increase the flat jack pressuresprogressively as the groundwater pressures areallowed to recover.

Figure 5b. Contour of smin at stage 2 where panels 1, 3 and 5 (from the right) were excavated.

Figure 6. Plasticity plot at stage 2.

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As an alternative, one more model was run fromthe end of stage 6. Instead of increasing the hydro-static load to 470 KPa in one go, a progressiveapproach was adopted this time. First adjust the

hydrostatic load around the tunnel to a lower and uni-form load of 400 KPa. Then increase the hydrostaticload to 425 KPa, 450 KPa and finally to 470 KPaprogressively. An improved cracking was noticed as

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Figure 7a. Contour of smax at stage 4 where panels 2 and 4 (from the right) were excavated.

Figure 7b. Contour of smin at stage 4 where panels 2 and 4 (from the right) were excavated.

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shown in Figure 11. Clearly it indicates that bothslowly and uniformly recovering of the groundwaterpressure will reduce the final cracking on the con-crete arch remarkably.

5 CONCLUSIONS

The results provide confirmation that 1.5 m panelwidth, and the sequence of excavation of panels

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(excavate panels 1, 3 & 5 first and then panels 2 & 4)is reasonable design assumptions. Tensile cracking ispredicted at the eastern top corner of Panel 1 adjacentto the frictional joint between arches pours A and B. It

is recommended that the measures of progressivelyincreasing the flat jack pressures as well as slowlyrecovering the groundwater pressure to a full uniformvalue of 470 KPa are to be taken to minimize the

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Figure 11. Plasticity plot at stage 7 for the alternative approach.

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tensile cracks in the concrete arch region above therepaired section.

It also demonstrates that the FLAC3D is a usefultool and can be well applied to solve the complicatedengineering problem.

ACKNOWLEDGEMENT

The authors would like to thank Dr. Mike Coulthardfrom M.A. Coulthard & Associates Pty Ltd for his

assistance in developing FISH coding. The authors alsobenefited from many discussions with him as well.

REFERENCES

Itasca Consulting Group, Inc. 1997. FLAC3D – FastLagrangian Analysis of Continua in 3 Dimensions,Version 2.0 User’s Manual. Minneapolis: Itasca.

Internal Report, PSM500.R3, April 2001. Design report forstructural and water inflow remediation at CH 11945m,Appendix C, Three-dimensional analysis.

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Mining applications

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181

Sill pillar design at the Niobec mine using FLAC3D

P. Frenette & R. CorthésyDépartement des Génies Civil, Géologique et des Mines, École Polytechnique, Montréal, Canada

ABSTRACT: The paper presents the numerical analyses performed with FLAC3D to study the stability of therock mass surrounding the stopes at the Niobec mine in Chicoutimi, Québec. Since the mine expansion is doneat depth, the stability of the planned stopes had to be evaluated in order to determine the support requirementslinked to an increase of the in situ stresses. The paper focuses on the dimensioning of the sill pillar between mining blocks 3 and 4 using FLAC3D. A rock mass characterization of the site has been made prior to the numer-ical modeling. The characterization consisted of structural geological mapping, laboratory testing of rock sam-ples and in situ stress measurement. All the stopes were then modeled and the parameters obtained from the sitecharacterization were used in the FLAC3D model. Various alternatives have been analyzed, including the use ofbackfill and variations of pillar thicknesses for the third pillar.

1 INTRODUCTION

Safe and economical dimensioning of undergroundexcavations is often hard to achieve because of thenumerous parameters involved. These parametersinclude rock mass characteristics, orientation and mag-nitude of stresses, excavation method and sequencing.Any combination of these factors may change fromone point to another, requiring a reevaluation of themine design. This is the case at the Niobec mine,located near Chicoutimi, Québec, were undergroundproduction is soon reaching the fourth mining block.The increase in stresses with depth requires calculat-ing the dimension of the sill pillar between the thirdand fourth mining blocks. At the present time, Niobecmine has 3 horizontal pillars. The crown pillar with a thickness of 90 m (300 feet), the pillar between mining blocks 1 and 2 with a 30 m (100 feet) thick-ness and the pillar between mining blocks 2 and 3being 45 m (150 feet) thick. These pillars are neces-sary for the stability of the excavations and absorbpart of the stresses caused by the mining of the stopeswhich remain open after being mine out.

As for any rock mechanics design, there is no directmethod for dimensioning horizontal pillars in hardrock mines as each mine has its own geometric andgeomechanical settings, which make it difficult tohave a universal recipe that allows an optimal pillardesign. Consequently, numerical modeling was con-sidered the best tool for the project.

Although the overall quality of the rock mass at theNiobec mine is good, the increasing stress levels with

depth, as confirmed by in situ stress measurements,will increase the potential for failure which has to beinvestigated. Moreover, the stope geometry being rela-tively massive, a two-dimensional model was not con-sidered realistic for the Niobec mine. Although theauthors did not find applications of FLAC3D for themodeling of a complete mine in the literature, theyfound it would be interesting to use the software forthat purpose, since it could efficiently model rock massfailure and, if required, the use of backfill in the openstopes.

In order to gather the data for the numerical model,a rock mass characterization program including structural geological mapping, laboratory testing andin situ stress measurements was conducted. The workwas facilitated by the fact that the rock mass includ-ing the ore bearing zones and host rock are relativelyhomogeneous and can be considered as a single zone.

2 SITE INVESTIGATION

2.1 Niobec mine

Niobec mine is located near Chicoutimi in Québecand has been producing niobium since 1976. It isowned in equal part by Cambior and Mazarin. Themine produces 3500 tons of ore each day by long holestoping. Each stope is 45 m deep, 25 m wide and 90 mhigh. Mining is done using primary and secondarystopes, creating openings up to 200 m wide. The minehas 3 mining blocks and 8 levels, the lowest production

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level being 1450 feet deep. A fourth mining block isscheduled to open in 2011.

2.2 In situ stress measurements

Stresses were measured at the 1450 level (Corthésy2000) using the modified doorstopper method (Leiteet al. 1996). Unlike the conventional doorstoppermethod, the modified method allows continuousreading of the strains at the bottom of the hole includ-ing temperature readings. These continuous readingsallow evaluating the quality of each measurement.These are performed in three differently oriented holesin order to obtain the three-dimensional stress tensor.Table 1 shows the principal stress tensor obtained bycombining the data obtained from the three holesusing the least squares approach. The stress calculationprocedure allows considering both local anisotropyand heterogeneity.

Those results were compared with another in situstress measurement campaign made by Canmet(Arjang 1986) using the CSIR triaxial cell (Leeman1967). In this earlier campaign, stress tensors werecalculated on levels 850 and 1000. Two holes were usedon level 850 and one on level 1000. Principal stressgradients on levels 1450 and 1000 are similar, butresults from the 850 level are not, probably becausethe measurements were made in the influence zone ofa stope. Table 2 shows the stress gradients for level850, 1000 and 1450.

2.3 Structural geological mapping

Structural geological mapping was conducted on levels 1150 and 1450. Over 8000 joints were identified

along 1370 m of drift using the scanline method. Majorjoints of over 1m were plotted for both level 1150 and1450. The results were compared with two other stud-ies made on the previous levels and the comparisonshowed the persistence of two major families of jointson all levels with the appearance of a third family withincreasing depth.

2.4 Rock mass classification

Once all the required parameters were obtained, therock mass was classified according to the RMR andQ indexes. The RMR value was found to be 77 and theQ index was estimated at 40, which corresponds to agood rock mass in both cases. These results were alsoused to obtain the failure envelope of the rock massfor the Mohr–Coulomb criterion.

2.5 Laboratory testing

Laboratory tests were conducted on rock samples.Seventeen samples were tested to obtain the unconfinedcompressive strength of the rock while seventeen othersamples were tested to determine the tensile strengthof the rock. Three triaxial compression tests were alsoconducted to verify the adjustment of the data to empir-ical strength criterion. The deformability parameterswere obtained indirectly from the stress measure-ment campaign as they were required to interpret theresults following a procedure suggested by Corthésyet al. (1993). Table 3 shows the mechanical parametersobtained for the rock substance.


3.1 Model geometry

Autocad files representing all the stopes minedbefore 2000 were used as a database to build thegeometry of the model. Unfortunately, no interfaceallowing the importation of dxf files is available with FLAC3D, so the dxf file containing all the stopescoordinates was used to build the model (see Fig. 1).

The three existing mining blocks were divided intoseparate entities that were later merged. Each block

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Table 1. Principal stresses tensor.

Stress Strike Plunge

�1 29.5 MPa 45° 04°�2 16.0 MPa 138° 38°�3 9.1 MPa 310° 51°

Table 2. Stress gradients.

850

Level (Mpa/m) (hole #1) (hole #2) 1000 1450

Mean horizontal 0.0642 0.0454 0.0466 0.0484stress gradient

Vertical stress 0.0386 0.0194 0.0274 0.0267gradient

Table 3. Mohr–Coulomb parameters for therock substance.

ParameterUnconfined compressive strength 124 MPaTensile strength 8.1 MPaYoung’s modulus 55 GPaPoisson’s ratio 0.254

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was subdivided into smaller sections to model theexcavations using a 20 m � 20 m � 20 m mesh at the outer edge of the sections (Fig. 2). These sectionswere made from 7 different parts, the 4 walls alongwith the floor, the roof and the stope itself. Thismethod allowed modeling all the excavations, butsome simplifications were necessary in order to keepthe number of zones and time spent to building themodel geometry to reasonable values (Fig. 3). All thestopes were modeled this way and adjacent sectionswere merged to obtain a uniform mesh. A transitionzone had to be inserted between each mining block tomerge them together without creating a discontinuityin the model.

3.2 Boundary conditions

Some problems were encountered for applying thestresses on the model. Since means of applying shearstress gradients on the boundary of the model werenot found, the principal stresses with their orienta-tions as shown in Table 1 could not be applied to

the model. The solution was to make a simplifyingassumption stating that the principal stresses �1 and�2 were horizontal with an azimuth of 45° and 135°respectively and that �3 was vertical (Fig. 4). The cen-ter part of Figure 4 (the small square) is the area containing the stopes while the rest of the modelallows the boundaries to have the required orientationfor applying the principal stress gradients and alsoinsure these boundaries are not in the zone of influenceof the excavations. This buffer zone was considered tohave an elastic behavior since no failure around theexcavations should extend that far and this would also

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Figure 1. Isometric view of the three mining blocks.

Figure 2. View of part of the first mining block showing the20 m � 20 m grid and the simplifications made to the stopes.

Figure 4. View showing the zone added to the model so thestresses can be applied at 45°.

Figure 3. Perspective view showing the modelling of thewalls surrounding a stope.

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speed up the run time of the analyses. In order to eval-uate the influence of the principal stress tensor reori-entation, two analyses were done using the Examine 3Dboundary element program developed by Rocscience.This software allows us to apply the in situ majorprincipal stresses with any orientation relative to themodel. In one analysis, the measured in situ stresstensor (Table 1) was input and in the other, the modi-fied stress tensor with �1 and �2 in the horizontalplane was used. As expected, the second model gavelower strength factor values, which confirms the sim-plifying assumption related to the reorientation of thestresses puts the model on the safe side. The strengthfactor given by Examine 3D compares the stress stateat every point of the model with the strength enve-lope. A strength factor of 1.0 means the stress state ison the strength envelope while a lower value meansthe stress state exceeds the strength of the rock mass.These strength factors must be considered with caresince they tend to underestimate the extension ofpotentially failed zones, since the elastic analysis thesoftware performs does not consider the post failurestress redistribution.

3.3 Constitutive laws and material properties

In the analyses shown in this paper, the rock mass wasassumed to present an elastic perfectly plastic behav-ior, so no post peak strength values are given. Table 4shows the strength and deformability parameters ofthe rock mass used in the model. These parameterswere obtained by combining the laboratory test resultswith the rock mass classification parameters presentedin section 2.4.

The authors are aware that for fragile hard rocksuch as the one found at the Niobec Mine, perfectplasticity is a not realistic assumption, but using astrain softening constitutive law would have sloweddown the runtime of the analyses which already tookover 5 days to run on a 1.0 GHz Pentium PC. Nonethe-less, now that the model is built, it would be a simplematter to implement the strain softening parametersand perform a sensitivity analysis by varying the post-peak strength parameters. The authors are also awarethat perfect plasticity will underestimate the exten-sion of eventual failure zones.

4 FLAC3D SIMULATIONS

Before running the analyses used to estimate what sillpillar size would be optimal, various scenarios werestudied in order to perform the numerical analysesmore efficiently and to verify if the use of certain support elements such as backfill would have aneffect on the local and overall stability of the mine. Itshould also be stated that a validation of the model bycomparing its results with in situ observations wasdifficult for various reasons. First, no in situ monitoringof displacements was available. Secondly, as the rareinstabilities around the excavations in the mine aremostly controlled by the presence of discontinuitieswhich are not considered in FLAC3D, it is difficult toperform a direct comparison between the extent ofinstabilities in the numerical model and the onesobserved in the field. This only emphasizes the factthat in the absence of field monitoring and in the presence of a good quality rock mass, validation ofnumerical models is difficult.

4.1 Mining sequence

The influence of the mining sequence (excavating the stopes in the same sequence they were mined outin blocks 1 and 2) on the results of the analyses wasstudied. This was an important point to verify since themining sequence for the new mining blocks (3 and 4)was unknown and excavating the stopes all at once inthe numerical model was an interesting alternative asit would allow important time savings. Consequently,two analyses were run, one by excavating the stopesone after the other and waiting for the unbalancedforces to stabilize in between and the other by nullingthe elements in the stopes all at once. The comparisonbetween the two runs is done by taking the number offailed elements in each simulation as shown in Table 5.

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Table 4. Material parameters used in themodel to simulate the rock mass behavior.

Uniaxial compressive strength 34.4 MPaTensile strength 2.1 MPaCohesion 10.5 MPaFriction angle 38.5°Young’s modulus 47.3 GPaPoisson’s ratio 0.254

Table 5. Comparison of the number of failed zones ondifferent sections of the model for the analyses with andwithout sequential mining.

Mining all at Sequentialonce (failed mining (failed

Section elements) elements) Difference

4410 36 37 14465 74 80 64530 65 64 –14575 70 81 114625 58 71 134675 45 35 –104730 41 37 –44795 87 92 54830 37 43 64900 14 15 1

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Most of the failed elements are located around thestopes and their dimensions are small compared to the 20 m � 20 m � 20 m mesh used in areas remotefrom the excavations. The section heading refers to var-ious sections in the model that cut through the stopes inthe model. It is considered that the excavation sequencehas a negligible influence and that no clear pattern isobserved. Consequently, it was concluded that for thefollowing simulations, the stopes could be excavated allat once without much effect on the outcome of the runs.

4.2 Influence of backfill

As the mine had never used backfill, it was decided tosee if the use of such support would allow minimizingthe occurrence of local failure around certain stopes.To estimate the influence of backfill on the stability ofthe openings, the same methodology as used to evalu-ate the influence of the mining sequence was adopted.In these simulations, after a stope was mined out andthe unbalanced forces had stabilized, the null elementswere replaced by zones having properties matching a backfill with 8% cement. The next stope was thenexcavated and the sequence was repeated for all theopenings. The number of failed zones with and withoutbackfill are presented in Table 6. These simulationsshowed the fill to have no significant influence on thestability of the mine. It is believed that the very lowstiffness of the backfill compared to that of a good qual-ity rock mass makes it almost impossible for it to absorb

any significant stresses, so it would not serve the pur-pose of reducing the size of the third horizontal pillar.

4.3 Design of the third horizontal pillar

The main objective of this project was to find theoptimal thickness of the third horizontal pillar. Thefirst pillar between mining blocks 1 and 2 is 30 mthick, the second between mining blocks 2 and 3 is45 m thick and the third one between mining blocks 3and 4 was also planned to be 45 m thick. Since the useof backfill was found of little use, only three simula-tions were made. One optimistic analysis with a 30 mthick pillar, another with a 45 m thick pillar identical

185

Table 6. Comparison of the number of failed zones on dif-ferent sections for the simulations with and without backfill.

Without WithSection backfilling backfilling Difference

4410 37 37 04465 80 77 34530 64 61 34575 81 79 24625 71 67 44675 35 33 24730 37 34 34795 92 88 44830 43 43 04900 15 15 0

Figure 5. Failure zones for the 30 m (100�) pillar for section 4795.

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to the one between blocks 2 and 3 and finally a lastsimulation with a 60 m pillar.

As the stope layouts for the third and fourth miningblocks were not available at the time the simulationswere run, the geometry of these blocks was assumedto be similar to that of blocks 1 and 2.

The simulation with a 30 m pillar showed that thepillar would be stable, but that only a 20 m thicknesswould remain intact (Fig. 5). The two other simula-tions showed that a 40 m thick zone would remainintact with the 45 m pillar (Fig. 6) and 53 m would befree from failed zones for the 60 m pillar (Fig. 7).

The number of failed zones in the pillars were thencompared. The direct comparison can be done sincethe number of zones remained constant between sim-ulations and only the zone thicknesses were changedto modify the pillar thickness. The analyses showedthe 30 m pillar to have 657 failed zones on a total of4425 zones in the pillar, while the 45 m pillar had 603failed zones and the 60 m pillar showed 583 failedzones. Although the 30 m sill pillar showed an overallstability, the intact thickness is considered too smallas the presence of planes of weakness not consideredin the analyses may cause important instabilities.Bearing this in mind, the 45 m pillar would leave anintact rock section considered more adequate. Theresults show the 60 m thick pillar would not increasethe overall safety factor significantly and the sideeffects of having a pillar which is too thick, is the oreloss and also the fact that a thicker pillar will exposethe stopes in the fourth mining block to higher in situstresses (due to their increased depth) causingunwanted dilution.

5 DISCUSSION AND CONCLUSIONS

The proposed pillar design presented in this papershould, prior to accepting it, be analyzed using a morerealistic constitutive law than perfect plasticity for hardrock. The strain-softening model available in FLAC3D

should be tested with various post-peak strengthparameters in order to perform a sensitivity analysis ofthe excavation response to these parameters.

Also, if one wishes to fine-tune the model, an opti-mization of the element size around the excavationscould be made.

In conclusion, the work presented in this papershowed that modeling a complete mine with FLAC3D isquite an undertaking since there are no simple ways tocreate the geometry, although there is now an interfacewith Ansys�CivilFEM which should facilitate thistask (it was not available at the time the project started).

There are also difficulties in the application of theboundary conditions as mentioned in section 3.2 sincethe authors we unable to apply shear stress gradientsto the model boundaries. Besides these difficulties,once the model is built, it is interesting to be able toperform sophisticated sensitivity analyses by modify-ing the parameters of various constitutive laws.

ACKNOWLEDGEMENTS

The authors wish to acknowledge Martin Lancet andthe personnel of the Niobec Mine who have contributedto the success of this M.A.Sc project. They also want toacknowledge the National Research Council of Canadafor its financial support (grant # OGP0089752).

REFERENCES

Arjang, J. 1986. In situ stress measurement at Niobec Mine,Canmet Laboratory Report.

Corthésy, R. 2000. Mesure des contraintes in situ, mineNiobec. CDT report, Ecole Polytechnique.

Corthésy, R., Gill, D.E., Leite, M.H. 1993. An integratedapproach to rock stress measurement in anisotropic nonlinear elastic rock, Int. J. Rock Mech. Min. Sci., Vol. 30,no. 3, pp. 395–411.

Leeman, E.R., 1967. The doorstopper and triaxial rock stressmeasuring instruments developed by the CSIR, J. of theSouth Afr. Inst. of Mining and Metall., Vol. 69, no. 7,1967, pp. 305–339.

Leite, M.H., Corthésy, R., Gill, D.E., St-Onge, M., Nguyen, D.1996. The IAM – A down-the-hole data logger condi-tioner for the modified doorstopper technique. 2nd NorthAmerican Rock Mechanics Symposium, Montréal, pp. 897–904.

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189

Stability analyses of undermined sill mats for base metal mining

R.K. Brummer, P.P. Andrieux & C.P. O’ConnorItasca Consulting Canada Inc., Sudbury, Ontario, Canada

ABSTRACT: Mines are often faced with sill extraction situations, and one technique that can be used to extracta sill is to leave a consolidated fill mat in the sill cut. Several Canadian mines employ this sill extraction technique,and in this paper the stability of these sills was modeled using two-dimensional FLAC numerical simulations. Theobjective of this parametric study was to derive relationships between the required strength of the sill mats andthe maximum stable unsupported undercut span for various orebody dips. The footwall-to-hangingwall spansdescribed in this paper were 1.2 m (4 ft), 2.4 m (8 ft), 3.6 m (12 ft), 4.8 m (16 ft), 6.0 m (20 ft) and 10.5 m (35 ft), withmining dips of 60°, 70° and 80°. The range of fill cohesive strength used was from 100 to 500 kPa, a typical rangefor most hydraulic or paste fills. Stability charts were derived (one per ore body dip), that can be used to selectthe minimum fill strengths required (in terms of cohesion) to maintain stability for different combinations ofspans and dips.

1 INTRODUCTION

FLAC simulations were set up to examine the behaviorof a typical backfill sill mat for sill extraction. Theobjective of this parametric study was to derive a rela-tionship between the strength of the sill mat and itsmaximum stable unsupported span, for various differ-ent orebody dips.

The footwall-to-hanging wall spans described herewere 1.2 m (4 ft), 2.4 m (8 ft), 3.6 m (12 ft), 4.8 m (16 ft),6.0 m (20 ft) and 10.5 m (35 ft).

2 FLAC MODEL

2.1 FLAC model geometry

The stability of the fill mats was investigated by carry-ing out several FLAC analyses, using orebody widthsof 1.2 m (4 ft), 2.4 m (8 ft), 3.6 m (12 ft), 4.8 m (16 ft),6.0 m (20 ft) and 10.5 m (35 ft). The dips consideredwere 60°, 70°and 80° to give a realistic range of dips. Itwas assumed that 45° is too flat to allow for a stableunsupported backfill sill mat to be built because of slipon the hangingwall.

For each combination of dip and span, the objectivewas to determine the cohesive strength required fromthe fill to ensure the stability of the sill mat, withoutadditional support, when fully undercut.

The range of cohesion values used was 100 to500 kPa, which corresponds to cement contents in the

range 5% to 12% for most fills in common use. A typi-cal FLAC geometry is shown schematically in Figure 1.

The thickness of the sill mat was assumed to be3.3 m (10 ft), which is a typical thickness for a sill mat.A surcharge loading of up to 33.3 m (100 ft) above themat was used.

In order to reproduce this geometry, a 60 by 120 ele-ment grid was generated and the appropriate coordi-nates applied to its four corner nodes. As a result, eachelement, or individual zone, was 0.3 m by 0.3 m (1 ft by1 ft) in size, which is sufficiently detailed for the prob-lem considered. Interfaces were defined between the

Figure 1. Typical FLAC layout for stability analyses.(Schematic cross-section. Not to scale.)

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fill and each wall, in order to allow movement of thebackfill along this surface.

2.2 FLAC model run sequence

The sequence of each FLAC run was as follows:

1. Define the grid to reproduce the required stopespan and dip. At this point, consider the ore underthe mat to still be in place (i.e. consider the sill matto rest on solid rock).

2. Set up eight history points in the center of thebackfill column, at 0.7 m (2 ft), 1.3 m (4 ft), 2.3 m(7 ft), 3.3 m (10 ft), 5.0 m (15 ft), 6.7 m (20 ft),10.0 m (30 ft) and 13.3 m (40 ft) behind the backof the next cut.

3. Define materials properties with real elastic prop-erties, but very high strengths for the fill, and zerofriction and cohesion along the interfaces betweenthe backfill and the two rock walls.

4. Cycle the model to equilibrium. This first part ofthe run is required to allow the fill to settle under itsown weight, as would happen in real life. The arti-ficially high strength of the fill ensures its elasticbehavior, while the null friction at the interfacesprevents the development of artificial stresses during this gravity-driven compaction process.

5. Once at equilibrium, reset the fill material andwall contacts to realistic strength properties(these are discussed later).

6. Reset all the displacements tracked by the historypoints, in order to reflect only the changes subse-quent to equilibrium.

7. Remove the restraint below the fill mat by “min-ing” the stope so that the fill mat takes load fromits own weight and the waste fill column above.

8. Apply some convergence to the stope walls as aresult of mining. Because the act of mining willinvolve some wall convergence a 10 mm incre-mental convergence was assumed to take place.Due to the explicit time-marching scheme used inFLAC, this movement had to be indirectly appliedto the walls by applying a horizontal velocity tothe model boundaries. To obtain the desired10 mm closure, a horizontal velocity of 0.001 mmper time step was applied inwards on both the leftand right model boundaries for 5,000 steps.

9. Remove the horizontal velocity applied on themodel boundaries (as the desired closure has beenreached), and cycle the model to equilibrium.

10. Check the history points and displacement resultsto see if the configuration is stable.

2.3 Constitutive models and material properties

The constitutive model used for the host rock waselastic, while a strain-softening behavior was retainedfor the backfill.

Elastic–plastic strain softening constitutive lawsallow specifying a transition zone between the peakand residual mechanical properties of a material. In thecases where these mechanical properties decrease asthe material yields (which is the case with typical back-fills), a strain softening behavior was used to describehow the material’s strength is progressively decreasedfrom its peak value to its residual one as irreversible/plastic strain accumulates in it.

2.4 Host rock

The properties retained for the rock mass (both the hostrock and the ore) are shown in Table 1. The exact elas-tic and strength properties of a typical host rock are notimportant, because the behavior of a sill mat is not verysensitive to these properties, as long as they are ordersof magnitude larger than those of the backfill.

2.5 Fill properties

The fill properties used in the FLAC analyses werebased on a large in-house database of fill properties.The main variable for the fill is its cohesive strength,which, as mentioned, was varied between 100 and500 kPa. As the cohesive strength was changed, so werethe elastic properties (even though not critical, thisrefinement was useful as a certain degree of conver-gence between the footwall and hanging wall was con-sidered, which, in turn, induced stresses in the sill mat).In other words, as the cohesive strength of the fill wasincreased, so was its stiffness. The overall fill propertysetting process was carried out in the followingmethodology:

1. set the cohesive strength, for example 200 kPa;2. set the tensile strength at half the cohesion –

100 kPa for our example;3. multiply the cohesive strength by 4 (assuming a

friction angle in the range 30° to 33°) to obtain thecorresponding unconfined compressive strength –800 kPa for the example considered;

4. derive the corresponding cement content usingFigure 2 (to obtain an 800 kPa UCS, the requiredcement content would be around 6.4%);

5. derive the corresponding elastic modulus usingFigure 3 (for a 6.4% cement content, the elasticmodulus would be near 0.53 GPa);

190

Table 1. Mechanical properties used for the host rock andthe ore.

Elastic Poisson’s Shear Bulkmodulus ratio modulus modulus Density(GPa) ( ) (GPa) (GPa) (kg/m3)

40 0.25 16.0 26.7 2700

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6. calculate the corresponding shear modulus G andbulk modulus K, assuming a Poisson’s ratio of0.30 (for our example G would be 206 MPa whileK would be 442 MPa); and,

7. use these values as input to the FLAC model. Forthe example, these inputs would be: cohesion �200 kPa, tensile strength � 100 kPa, G � 206 MPa,K � 442 MPa.

The nominal friction angle was chosen at 30° (frictionangle values will be discussed in more detail later).Table 2 summarizes the properties used for the variousfill strengths considered. The density of the backfill wasassumed to be 2.0 t/m3.

The mechanical properties of the surcharge fillplaced on top of the sill mats were lowered by 10% toaccount for the lower quality of fill typically placed on top of sill mats. The properties affected were thecohesion, tensile strength, shear modulus and bulkmodulus – the friction angle was kept the same.

As mentioned, the constitutive model used for theconsolidated fill material was a strain-softening one. Inorder to simulate this behavior, the decrease in strength

as a function of the plastic strain accumulated in theyielding material needs to be explicitly described. Forthe purpose of this study, it was assumed that the cohe-sion and, hence, tensile strength, would decrease lin-early from their maximum value at zero plastic strain,down to zero at a cumulative plastic strain of 1.5%.The internal angle of friction was set to also vary lin-early, but from its maximum value of 33° at zero plas-tic strain, down to 30° at a cumulative plastic strain of1.5% and beyond. Neither the shear nor the bulkmoduli are affected by plastic strain and were thus leftunchanged.

The older waste fill above the sill mat was subjectedto the same plastic strain-dependent weakening process.Cohesion and tensile strength were also decreased lin-early from their maximum value (set, as mentioned pre-viously, at 90% of those of the sill mat) at zero plasticstrain, down to zero, also at a cumulative plastic strainof 1.5%. Similarly to the sill mat, the internal angle offriction of the weaker fill material was set to decreaselinearly from its maximum value of 33° at zero plasticstrain, down to 30° at a cumulative plastic strain of1.5% and beyond.

2.6 Interface between rock and fill

As previously discussed, interfaces between the hostrock and the fill, along both the footwall and hangingwall, had to be specified due to the continuum natureof the finite difference approach used in FLAC. Duringthe initial compaction stage of each run, both the inter-nal angle of friction and the cohesive strength of theseinterfaces were set to zero in order to prevent artifi-cial stresses from developing along them as the back-fill settled. During the later stages of the runs thesevalues were reset – the internal angle of friction wasset equal to the internal angle of friction of the sillmat, and the cohesion was set equal to the cohesion ofthe sill mat.

191

0

200

400

600

800

1,000

1,200

1,400

1,600

1,800

0.0 2.0 4.0 6.0 8.0 10.0 12.0

Cement Content (%)

UC

S (

kPa)

Figure 2. Typical relationship between UCS and cementcontent for backfill.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0Cement Content (%)

Ela

stic

Mo

du

lus

(GP

a)

800400Average

Confinement during test(kPa)

PASTE FILL PROJECTPaste Fill Moduli vs. Cement Content

Figure 3. Typical stiffness properties for backfill (based ontriaxial lab tests carried out on a typical paste fill).

Table 2. Strength and elastic properties used for the back-fill sill mat.

Cement Elastic Shear BulkCohesion UCS1 content modulus modulus2 modulus2

(kPa) (kPa) (%) (GPa) (MPa) (MPa)

100 400 4.2 0.24 92 200150 600 5.5 0.40 154 333200 800 6.4 0.53 204 442300 1200 8.3 0.84 323 700400 1600 10.0 0.90* 346 750500 2000 11.2 0.90* 346 750

1Assuming a 30° internal angle of friction.2Assuming a Poisson’s ratio of 0.30.

* Value outside of data range – elastic modulus fixed at0.90 GPa.

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3 INTERPRETATION OF ANALYSES

Several characteristics of the FLAC analyses were usedto determine whether the various sill mats were stableor had failed.

Figures 4 & 5 show the unbalanced force history fora stable layout and an unstable layout, respectively. Theunbalanced force is a key element of the time-marchingalgorithm used in FLAC as it indicates the degree ofstatic equilibrium reached within the model at any givencycle (as the unbalanced force diminishes, the degree of

equilibrium increases). As can be seen, the unbalancedforces converge to zero for stable configurations, butcontinue to oscillate, or even increase, for unstable ones.

Figures 6 & 7 show examples of the vertical dis-placement history for the control points locatedwithin the waste fill or backfill column, as describedearlier. As can be seen, stable spans displace verticallyby only a finite amount, whereas unstable spans continue to deform vertically as they fail.

More crudely, the deformation of the FLAC gridcan be examined, as can be seen in Figure 8. Unstable

192

FLAC (Version 3.30)

LEGEND

step 9000

HISTORY PLOT Y-axis :

Max. unbal. force X-axis :

Number of steps

1 2 3 4 5 6 7 8 9

(10+03 )

1.000

2.000

3.000

4.000

5.000

(10+05 )

JOB TITLE : Undermined Sill Mat Stability Analyses

Itasca Consulting Canada Inc.

Figure 4. Example of a FLAC unbalanced force history plotfor a stable sill mat configuration. Note that the maximumunbalanced force stabilizes at zero, since the fill panel is stable.

FLAC (Version 3.30)

LEGEND

step 9000

HISTORY PLOT Y-axis :

Max. unbal. force

X-axis :Number of steps

1 2 3 4 5 6 7 8 9(10+03)

1.000

2.000

3.000

4.000

5.000

(10+05)

JOB TITLE : Undermined Sill Mat Stability Analyses

Itasca Consulting Canada Inc.

Figure 5. Example of a FLAC unbalanced force historyplot for an unstable configuration. Note that the maximumunbalanced force is not zero and increases without bound asthe fill panel fails.

Figure 6. Example of a FLAC “y-displacement” (vertical) history plot at the various control points located within the back-fill column, for a stable configuration. Note that the maximum displacement is finite at about 45 mm.

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configurations exhibit severe deformation of thegrids. Stable spans also will deform to some extentdue to settlement, as can be seen in Figure 9, but willstabilize and not continue to deform as the runs arefurther cycled.

4 SUMMARY OF RESULTS

Figures 10 & 11 show the FLAC results obtained for a60° dip and a 1.2 m (4 ft) mining width, with a 200 kPacohesion backfill sill mat.

193

Figure 7. Example of a FLAC “y-displacement” (vertical) history plot at the various control points located within the back-fill column, for an unstable configuration. Note that the maximum displacement increases without bound (up to 1.4 m at theend of 12,000 cycles in this case).

Figure 8. Example of a FLAC grid plot showing failure of the sill mat at mid span, and especially at the hanging wall contact. Note that the fill displaces downward by up to 500 mm, indicating failure.

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Back support in the form of mat reinforcing (e.g.,screen placed on the floor of the stope to fill, togetherwith vertical bolts) will stabilize the local back whenthe sill stopes are extracted and the fill mat is under-mined. The aim of this study is limited to the overall sta-bility of the backfill sill mats, and excludes minor fallsof fill from the back that must be expected to occurunless appropriate mat reinforcing techniques are used.It is understood that the support of the immediate backwill be ensured by mines through the appropriate use ofthis type of reinforcing.

These charts can be used to select fill strengths (interms of cohesion) for different combinations of spans

and dips. Since fill strength is normally measured interms of uniaxial compressive strength, this can beestimated by multiplying the cohesion by a factor of4, i.e. for a cohesion of 200 kPa, a uniaxial compres-sive strength of 800 kPa will be necessary. Note thatthe FLAC analyses as presented do not incorporateany Factors of Safety – appropriate Factors of Safetymust therefore be applied to the fill strength for designpurposes. This will depend on the quality and degree ofuniformity of the fill as placed. Note also that the chartssuggest that it is not possible (for the fill strengths considered) to undermine a fill panel of 10.5 m widthwithout some form of additional reinforcement.

194

Figure 9. FLAC results for a 70° dip, a 400 kPa cohesion sill mat and a 6.7 m (20 ft) span. Most of the movement occurs inthe lower part of the sill mat, and the whole arrangement remains stable.

0

100

200

300

400

500

600

0 5 10 15Stope Span (m)

Ave

rag

e C

oh

esio

n (

kPa)

Stable 60 Failed 60

Failed

Stable

Figure 10. Stability Chart for 60° dipping ore showingfailed cases and stope spans. Also shown is a crude contourseparating the failed cases from the stable cases.

0

100

200

300

400

500

600

0 5 10 15Stope Span (m)

Ave

rag

e C

oh

esio

n (

kPa)

Stable 80 Failed 80

Failed

Stable

Figure 11. Stability Chart for 80° dipping ore showingfailed cases and stope spans. Also shown is a crude contourseparating the failed cases from the stable cases.

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5 CONCLUSIONS

The FLAC model is capable of modeling both a “sag”mode of failure as well as a “rotational” mode of failurefor a sill mat. For near-vertical or steep stope walls, thesag mode is observed to occur and is dominant. For flat-ter dips, it is possible to observe a rotational mode offailure, as the fill falls away from the hangingwall, andeventually rotates about the footwall support.

The objective of this study was limited to the overallstability of the sill mats, and excluded consideration ofminor falls of fill from the back that must be expectedto occur unless appropriate mat reinforcing techniques(e.g. properly anchored screen or shotcrete) are used.For practical reasons, it is understood that the supportof the immediate back will be ensured through theappropriate use of this type of reinforcing.

All of the analyses presented show that it is usuallypossible, with sufficient binder, to create a stable mat

back under a variety of geometric and loading conditions. However, this applies to the overall mat –not the immediate back. Even with a very strong mat,it is still possible to have falls of fill from the imme-diate back, unless some form of back support (e.g.screen on the back, with bolts or Splitsets) is used.This is equivalent to the screen commonly used on arock back.

This raises the possibility of incorporating the screenwith the mat reinforcing (e.g., screen placed on the floorof the stope prior to filling). This screen will then beexposed on the fill back as the panel is undermined. Ifthe screen is tied to the fill mat reinforcing, this willtake the place of the bolts and will eliminate the sup-port cycle when the mat is undermined. This has beensuccessfully done at a number of mines, and is aneconomical way to reinforce the fill mat (and has thepotential to save binder), as well as provide supportfor the back of the undermined fill panel.

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197

FLAC numerical simulations of tunneling through paste backfill atBrunswick Mine

P. Andrieux* & R. BrummerItasca Consulting Canada, Inc., Sudbury, Ontario, Canada

A. MortazaviPreviously with Itasca Consulting Canada, Inc., Sudbury, Ontario, Canada

B. Simser*Falconbridge Ltd., Sudbury Mines/Mill Business Unit – Craig Mine, Onaping, Ontario, Canada

P. George*New Brunswick, Canada

*previously with Noranda, Inc., Brunswick Mine, Bathurst, New Brunswick, Canada

ABSTRACT: In early 2001 Itasca Consulting Canada Inc. was contracted by Noranda Inc. to assist in thedesign of the first two drifts that were going to be driven through paste backfill at Brunswick Mine in the southend of the 1000 m Level in order to create alternate accesses to the western ore zones. A numerical stabilityanalysis of the proposed tunnels was carried out by means of two-dimensional FLAC simulations, which tookinto account different fill strengths, alternate tunnel geometries, various floor conditions and the presence ofunconsolidated plugs of waste rock within the paste backfill at close proximity to the tunnel in one area. Themain objectives of this work were to investigate the self-standing characteristics of the exposed paste material,evaluate the deformations expected as a result of tunneling through it and recommend adequate ground supportalternatives. This paper describes the modeling approach used, the results obtained and how they correspondedto the behaviors later encountered underground during the excavation of the tunnels.

1 INTRODUCTION AND BACKGROUND

Itasca Consulting Canada Inc. (ICCI) of Sudbury,Ontario, was contracted to assist in the design of twodrifts that were going to be tunneled through pastebackfill in the south end of the 1000 m Level at theNoranda Inc. Brunswick Mine operation near Bathurst,New Brunswick. Drifting through the paste backfillwas required in order to create alternate accesses to the western ore zones because some of the existingaccesses were either in highly stressed ground thatcould burst, or were planned to be removed when futurestopes were going to be mined. Tunneling through pastebackfill being then a new procedure at BrunswickMine, it was decided by senior engineering personnelat the site that a thorough numerical investigation wasnecessary to identify possible design limitations.

The numerical analyses of the process of drivingthrough paste backfill were carried out at the ICCI

offices in Sudbury by means of two-dimensionalFLAC simulations that used actual fill strengths,geometry, floor conditions and other expected fieldconditions. The main objectives of this work were toinvestigate the self-standing characteristics of theexposed paste material and the deformationsexpected as a result of tunneling through it. A seriesof numerical exercises were completed with theFLAC code to address these objectives for drifts ofvarious shapes driven in paste materials of varyingcohesive strength. Two situations were simulated: (1) the situation in the 236-8 Access on 1000-2 sub, where failed waste rock (which had caved fromthe back of the drift) and unconsolidated rockfillmaterial (which had run from the 235-8 and 237-8stopes above) ended below the paste material with anear-45° angle of repose; and, (2) the situation in the129-7 Access on 1000-1 sub, where only paste mate-rial was present.

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2 GENERAL APPROACH

As mentioned, the FLAC numerical analyses performedfocused primarily on investigating the free-standingcharacteristics of the paste backfill as it is excavated,without any support. This was deemed important forsafety reasons (i.e. for assessing the level of risk theunderground crews performing the excavation workwere going to be exposed to), and for defining the long-term support requirements of the excavation.

A sensitivity analysis was carried out for thisaspect of the work, whereby the cohesive strength ofthe paste material was varied over a reasonable range,in order to identify threshold values with regard toinstability. These threshold values were then com-pared to the values measured in the paste backfill inboth the 129-7 and 236-8 excavation areas.

This numerical sensitivity analysis also examinedthe effect the tunnel shape has on the transfer of theloads around it. Two profiles, a flat back and a pro-nounced horseshoe shape, were modeled, to determinethe impact of shape on the self-standing stability ofthe tunnel. This analysis also produced deformationand convergence data for all the cases examined,

which were important to derive adequate long-termsupport requirements. The two-dimensional approachwas deemed adequate based upon the geometry of thetunnels, which were much longer in the third dimen-sion. It however did not allow the examination of theactual driving process, whereby local stresses redis-tribute around and ahead of the tunneling front.

The dimensions used in the FLAC model were basedon measurements made on site at the beginning of theexcavation process. Photographs were taken under-ground from which precise scaling was done in orderto generate a very representative numerical mesh.Figure 1 shows a front view of the drift that was used toconstruct the numerical model.

The FLAC strain-softening/hardening model wasused to capture the non-linear behavior of the pastebackfill material in its post-elastic range. This particu-lar model considers the cohesion, tensile strength andfriction angle to change as a function of the cumula-tive plastic strain within the material. In the numericalanalyses performed, it was assumed that the cohesionand tensile strength of the paste backfill dropped to 25% of their original values after the material hadexperienced a cumulative plastic strain of 1.5%. (Thesesettings were based upon previous Itasca modelingexperience.)

The simulations were designed such that they rep-resented the actual sequence of events leading to thedrift excavation. The models were initially cycled toequilibrium in order to simulate the various cured and hardened backfilling materials present. The driftwas then excavated and the models cycled to equilib-rium, with stresses and displacements being moni-tored throughout the cycling process.

The failure mechanisms within the paste backfill andthe stability of the drift were investigated as a functionof the strength of the paste itself and of the variousmaterials surrounding it. Table 1 shows a brief sum-mary of the model input data used in these parametricanalyses.

The cohesive strength can be used to get an idea of the compressive strength of the material – assuming a 30° friction angle, cohesion is about 25% of theunconfined compressive strength (UCS). Hence, a400 kPa cohesive strength paste material would havea UCS of about 1.6 MPa.

198

N

9.00

10.00

11.00

12.00

13.00

14.00

15.00

16.00

8 9 10 11 12 13 14 15 16 17 18 19

Figure 1. Front view looking from the footwall into the236-8 Access on 1000-2 sub. This photograph was used tobuild the numerical model. The mesh and drift outline visi-ble in the foreground were generated by Microsoft Excel™,using the “digitizer mode”. The scale was obtained from thelines painted on the face visible on the background.

Table 1. Input property data used in the FLAC numerical analyses.

Bulk Shear Material Frictionmodulus modulus density Cohesion Tensile strength angle

Material type (MPa) (MPa) (kg/m3) (kPa) (kPa) (degrees)

Paste backfill 400 240 2000 50 to 400 Half of cohesion 32Caved waste 100 60 2700 zero Half of cohesion 35Rockfill 120 70 2700 0.0 and 50.0 Half of cohesion 35

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3 FLAC SIMULATIONS

Simulations were run for the following cases: (1) a horseshoe shape drift driven with loose materialon both sides; (2) a flat back drift also developed with loose material on both sides; (3) a horseshoeshape drift driven without any loose material on eitherside; and, (4) a horseshoe shape drift driven withoutany loose material on either side, but with a horizon-tal discontinuity in the paste material directly above it(in order to investigate potential large-scale slabbingof the paste into the tunnel).

3.1 Case 1 – arched back drift, with loose wastematerial on both sides

This situation prevailed in the 236-8 Access on 1000-2sub, where caving had occurred along a band of weakwaste rock inside the original drift, and some of the

dry rockfill placed between 1000-3 sub and 1000-2 subin the secondary stopes on both sides of this accesshad run into the area due to local caving on the 1000-2sub horizon. Despite repeated attempts to remove thiswaste material, uncontrolled runs of fill had resultedin significant amounts of loose material being presentabove the 1000-2 sub elevation at the time the pastebackfill was poured in the 236-8 Access. This, asshown in Figure 1, resulted in loose material beinglocated on both sides of the future drift, at a reposeangle of about 45°.

Figure 1 also shows the numerical grid, geometryand boundary conditions of the model constructed forthis first series of runs. Paste material cohesion valuesof 50, 100 and 400 kPa were investigated – the resultsindicated stable self-standing conditions for a pastematerial with a cohesive strength greater than 50 kPa.

As shown in Figure 2, a 50 kPa cohesion resulted inthe prediction of a maximum displacement of over31 cm 12,000 cycles into the simulation. Further stepping of the model (to 15,760 cycles) confirmedthe complete failure of the paste backfill material andclearly described the paste failure mechanism takingplace under the simulated conditions.

It is interesting to note that the back failure was notas pronounced as that of the walls. As intuitivelyexpected, the simulation confirmed that, under vertical(gravity) loading conditions, most of the vertical loadaround the excavation is deflected and concentrated inthe drift walls.

The existence of weak contacts between the pastebackfill and the loose waste material towards the bot-tom and on both sides of the drift initiated a deforma-tion of the paste along this contact. This, in turn, led tothe shearing of paste material on both sides of the drift.After the side wall failure, the process propagatedupwards and led to the shearing of the paste materialabove the drift, as shown in Figure 3. A maximum dis-placement of as much as 1.20 m was observed at thislater stage. Moreover the unbalanced force historywithin the model showed that after the initiation of fail-ure the unbalanced force continued to increase, indicat-ing that a progressive failure kept on evolving withinthe model as no state of equilibrium was being reached.

3.2 Case 2 – flat back drift, with loose wastematerial on both sides

Following this first set of analyses it was decided to fur-ther investigate the failure mechanisms by consideringa worst-case drift geometry, which would correspondto a flat back profile. In this case, one would intuitivelyexpect significant roof deformation and failure. All theFLAC simulations done for this case were conductedusing identical boundary conditions and input data asfor the previous case, except for the geometry of thetunnel itself.

199

Rock fill

Loose Waste

Paste Fill

Figure 1. Material regions (top), and numerical grid,geometry and boundary conditions (bottom) of the modelconstructed for the first series of simulations.

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Comparing Figure 4 and Figure 2, which both cor-respond to the same paste material cohesive strengthof 50 kPa and the 12,000th analysis step, shows thatthe maximum displacement predicted is significantlylarger in the case of the flat back geometry (1.31 m vs.only about 0.31 m) – this does highlight the improvedstability the arched back geometry provides.

The same failure mechanism (i.e. a side walls failurefirst, followed by a shearing effect through the overly-ing paste material) is however seen in both the flat andarched back arrangements.

As in the case of the arched back profile, 100 and400 kPa paste backfill cohesive strengths resulted in the

material maintaining its integrity and the drift remain-ing stable regardless of its profile. With a 100 kPa pastematerial cohesion the same overall results were obtainedas with the arched back case, but more displacementwas predicted.

The analyses also showed that if the area back-filled with paste material is damaged, due to dynamicloading from blasting, for example, then the underly-ing loose waste material does not offer much supportagainst vertical movement in the paste backfill, whichcould potentially lead to large and even catastrophicfailures in the paste material.

200

Figure 2. Displacement vectors at step 12,000. A maximum displacement of over 31 cm was predicted in this case.

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3.3 Case 3 – arched back drift, with no loose wastematerial on the sides

The aim of this set of analyses was to investigate thestability of a drift driven through paste backfill under“normal” conditions, i.e., with no loose waste materialin the pasted region and with the paste material poureddirectly on compacted rockfill. This situation, illus-trated in Figure 5, corresponded to the situation in the129-7 Access in the south end of the 1000 m Level #1

sub-level elevation. Three different cohesive strengthvalues were again considered for the paste backfill,which were 25, 50 and 100 kPa in this case.

No major displacement was predicted to occur inthe drift for the cases of the 50 and 100 kPa cohesion.For the 50 kPa cohesion case a maximum displacementof just under 3.7 cm was predicted, whereas this maxi-mum displacement was predicted at just over 3 cm forthe 100 kPa cohesion case.

201

Figure 3. Deformed model geometry at step 15,760.

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However, complete failure of the drift was predictedto occur when driven in 25 kPa cohesive strength pastebackfill. Figure 6 shows the displacement vectors inthis case, 8000 cycles into the simulation.

Overall, and as intuitively expected, indicationswere that the absence of 45° piles of loose materialunderneath the paste fill helped significantly withrespect to the stability of a tunnel driven through it.

3.4 Case 4 – arched back drift with no loose wastematerial on the sides, but with a horizontaldiscontinuity in the paste fill above it

The objective of this analysis was to simulate the effectsof a weak horizontal cold joint within the paste back-fill, which could have been caused, for example, byinterruptions during the pouring process. If sufficientlylong interruptions occur in the normally continuousfilling process, the previously placed material can curesufficiently, eventually resulting in a strength disconti-nuity at the contact with more recently poured material.As shown in Figure 7, a horizontal interface elementwas thus incorporated at a distance of 1.5 m above thedrift back, which represented a very adverse situationwith regard to a potential layer of material in the backof the drift developing instability. Zero cohesion andzero tensile strength were assigned to the interface, inorder to consider the worst-case scenario. The modelwas run using a 50 kPa cohesive strength paste material.

The displacement results are presented in Figure 8.As shown, the presence of the horizontal discon-tinuity did not affect the overall behavior of the drift.Comparing Case 3 for a 50 kPa cohesion (whichshowed an identical situation, but without the hori-zontal discontinuity) and Case 4, the maximum dis-placement observed remains small (3.7 cm withoutthe discontinuity vs. 3.6 cm with it).

It should be noted that a fairly weak paste backfill(with only a 50 kPa cohesive strength) was used inCase 4. For the “ordinary” strength paste fill used atBrunswick Mine (with a 400 kPa cohesive strength), the

202

Figure 4. Displacement vectors at step 12,000 (50 kPa cohesion, flat back drift profile).

Paste Fill

Rock fill

Figure 5. View of the material regions modeled in FLAC forthe case of the arched back drift, with no loose waste materialaround the drift.

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effect of a horizontal discontinuity can, for all intendedpurposes, be ignored.

4 GENERAL OBSERVATIONS DURING THE FIELD WORK

Two different excavation methods were tried under-ground during the development of these tunnels: a

mechanical one, and a drilling and blasting one. Bothmethods showed promise, but initially required a subse-quent finishing step to be carried out to smooth the finalpaste arch. As expected, the Brunswick Mine 400 kPacohesive strength paste backfill stood well during thedevelopment phase. Based upon the observationsmade underground during a three-week period, the mostsignificant improvements that could be made to thedevelopment procedure in paste backfill would be: (1) the development of a mechanical scrapingmachine, which would remove and trim all ridges andundulations along the initial excavation boundaries;or, (2) the adjustment of the drilling and blastingpractice, in order to precisely cut the proper shape andeliminate damage to the excavation surface.

A combination of both could potentially yield thebest results, such as the rough mechanical excavation ofa center cut, followed by the trimming to the propershape using controlled blasting. This approach wasimplemented with success in February 2001. It con-sisted of excavating a center plug with a scooptram,and of trimming the tunnel to its final dimensions usinglightly charged (with B-line detonating cord only)blastholes, including a series of trim blastholes drilledon a 20 cm (8 in) spacing directly along the plannedperiphery of the tunnel. Good results have also beenreported when using a purely mechanical excavationapproach, without any subsequent blasting. In thesetests, a scooptram-mounted scaler normally used toscale unstable areas was used to trim the excavation toits final shape, after a center cut had first been exca-

203

Figure 6. Displacement vectors for a cohesion of 25 kPa. This situation evolved into the complete collapse of the tunnel.

Interface element

Paste filled region

Rock fill region

Figure 7. Addition of an interface element to model a hor-izontal discontinuity in the paste fill material.

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vated with an 8 yd3 scooptram. Certain precautionswere however required in order to attain these results,such as carefully leveling the floor beforehand, andensuring no ground support elements were sticking out

above the tunnel surface. This approach also report-edly provided an overall quick cycle time.

Significant cycle time improvements can beachieved if a smooth initial arch profile is obtained ona “first pass” since this potentially allows one to deferthe application of the required second layer of shot-crete until the end of the excavation process. The post-ponement of the second layer of shotcrete can howeveronly be considered if a proper arch is created, if nocracks develop in the initial layer of steel fiber-rein-forced shotcrete, and if no abnormal inclusions areencountered in the surrounding paste material duringthe bolting cycle.

Figure 9 shows the type of results that were obtainedwith the blasting approach after it was optimized. Theeffective advance achieved with this particular shotwas 4 m (13 ft), and, as can be seen, the blast was quitesuccessful and produced a uniform fragmentation.

ACKNOWLEDGEMENTS

The authors would like to thank Noranda Inc. for thepermission to present these data and publish this paper.

204

Figure 8. Displacement vectors for a cohesion of 50 kPa. A maximum displacement of just over 3.6 cm was predicted in this case.

N

Figure 9. Photograph looking west in the 236-8 Access on1000-2 sub on the footwall side showing the results of thethird round blasted there in January 2001.

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205

FLAC3D numerical simulations of ore pillars at Laronde Mine

R.K. Brummer & C.P. O’ConnorItasca Consulting Canada Inc., Sudbury, Ontario, Canada

J. Bastien, L. Bourguignon & A. CossetteAgnico-Eagle Mines – Laronde Mine, Cadillac, Quebec, Canada

ABSTRACT: On November 27, 2002 Agnico Eagle’s Laronde Mine experienced a magnitude 2.6 seismicevent. As part of the investigation into the cause of the burst, a high-resolution FLAC3D model was created todetermine the effect of the mining sequence on stresses in the area of the burst. At this mine, secondary stopesare intended to fail following primary stope extraction. The numerical simulations showed that where remnantswere left with non-ideal geometry (through unfortunate but necessary mining decisions), these remnants couldbe too strong to yield as intended.

The FLAC3D model showed that one such 3-wide pillar centered at the location of the burst was subject to alocal high stress concentration. This provided a unique opportunity to confirm calibration of the FLAC3D model.Further modeling also highlighted other areas of the mine where pillars were in a high stress state and recom-mendations were made to alter the mining sequence to prevent future events.

1 INTRODUCTION

Agnico Eagle’s Laronde Mine is a high-tonnageunderground mining operation in the Abitibi miningdistrict in Northern Quebec. Currently the majority ofthe mining takes place at a depth of 1500 meters but anew mining horizon starting on 2150 meters has beenin production since early 2002 and will become themajor producing area of the mine as the upper levelsbecome depleted.

On November 27, 2002 the mine experienced amagnitude 2.6 rockburst between the main levels of149 and 152 and centered along the main access intothese levels. Damage on the 149 Level was light tomoderate with some floor heaving and spalling alongthe footwall. On the 152 Level the damage was muchmore extensive and resulted in a large failure in themain intersection of the level.

The burst occurred approximately 2 hours after asmall slot blast in stope 146-20-62, a secondary stopeexpected to be carrying little stress. The slot blast wasquite small and was not a likely trigger for the eventalthough the timing of the burst in close proximity tothe slot blast leaves this as a possibility. Fortunately noone was in the area at the time. Typically the mines inthe area are seismically quiet which made the event thatmuch more troubling and the cause of the burst needed

to be found so similar situations could be avoided inthe future.

As part of the investigation into the cause of theburst, a high resolution FLAC3D model was built inorder to examine the stresses throughout the region andthe role of the mining geometry on the event. Otherwork being conducted at the mine provided a well-calibrated set of properties to be used for this purpose.

2 MINE LAYOUT AND GEOLOGY

Laronde Mine is currently producing 7000 tpd. Mainproduction comes from the stopes in the 152 horizonwhile new production levels down on the 215 Levelcome online. In the coming years, the bulk of produc-tion is expected to come from the deeper levels as theupper levels become depleted.

The geology of Laronde Mine is quite complexwith multiple ore bodies spaced parallel to each other(of which 20-Zone is the major producer). The ore-body is a gold-zinc massive sulphide with a thicknessranging from 10 to 30 meters. Along each contact ofthe orebody is a region of highly sheared schist mate-rial that can be up to 5 meters in thickness which can,at times, present hangingwall stability problems. Thereis also a regular banding of highly sheared material

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throughout the footwall that is more prevalent in thedeeper sections of the mine. Figure 1 shows a typicalcross section of the Laronde mine as constructed in theFLAC3D model.

The mine uses a primary-secondary blasthole stop-ing mining method. Stopes are all sized at 15 metersalong strike, 30 meters high, and the thickness of theorebody, which typically ranges from 15 to 25 meters.This stope dimension was chosen specifically so thatsecondary stopes will be in a post-failure state afterprimary mining. This helps to reduce stress problemsby forcing the stresses to the abutments instead ofsecondary stopes and generally makes secondarymining easier.

Pastefill is currently the backfill of choice for pri-mary stopes in the upper levels. Rockfill is used insecondary stopes that will not be mined against. Thedeeper levels of the mine currently use cemented rock-fill until the paste system is extended into this area.

The mining sequence is based upon an expandingchevron extending upwards from the 149 Level withsecondary stopes being mined first on the 152 Level.Secondary stopes typically lag behind the primariesby 2 stopes. For several reasons, the mining sequencehad some instances in which the ideal mining shapecould not be maintained. First, the main accesses tothe 21-Zone run directly through the 64 and 66 seriesof stopes (refer to Fig. 2). This presented some stabilityconcerns for these accesses if the 65 stope was broughtup to its ideal position in the sequence. Additionally,on the 152 Level, the secondary (i.e. even numberedstopes) are mined first and were set up in a retreatingfashion from each abutment back towards the mainentrance of the level. These two scenarios combinedto make a series of pillars three stopes wide. Unlike asingle secondary stope, a three-wide stope is expectedto be too large to fail.

The burst appears to have been caused by a slip-page along a foliated zone that runs parallel to the ore-body and right through the back of the 152 Level.This same foliated zone also passes through the lowerfootwall of the 149 Level. The damage seen on both lev-els occurred along this contact. It was hypothesized

that the high stresses being forced through the three-stope-wide pillar on 152 level was the driving forcefor the slippage along the foliated zone, but without anumerical model to determine the stresses in the region,no solid answers could be gleaned.

3 EVENT

The seismic event measured 2.6 on the Nuttli scaleand was centered on a foliated zone running betweenthe 149 Level and the 152 Level. An investigationrevealed what was thought to be a probably cause forthe event. The three stope-wide pillar centered in thelower abutment on the 152 Level had been createdthrough the mining sequence which likely providedthe driving force for the event by concentrating stressthrough this region. At the same time, the foliatedzone intersecting a long strike distance along bothupper and lower levels provided a method of releaseby which the foliated zone was free to move. It wasbelieved that the intense stress concentration wrap-ping around the lower abutment and through thethree-wide pillar on the 152 Level provided enough ofa driving force to cause a slip along the foliated zone.

Most of the damage on 149 Level occurred on thefootwall side along the floor with lots of displaced

206

Figure 1. A typical cross-section of the Laronde orebodies looking East showing the different rock elements.

Figure 2. Long section looking North of Laronde showingthe current mining sequence.

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slabbing and some bagging of material in the screen (seeFig. 4). On the 152 Level, the most damage occurredright in the back of the stope with the large intersec-tion failure being the dominant feature. Accessibilityto the east was limited but most of the damageseemed to occur towards this direction (see Fig. 5).

One of the other questions that arose in the after-math of the burst was how to ensure that a similar sit-uation did not occur again. Since the burst appearedto have been caused by a combination of the miningsequence and the unfortunate location of the foliatedzone running directly through two main drifts, thisparticular mechanism might be a one-time event.However, this does not eliminate the possibility thatpinch points in other areas could not cause seismicitythrough another mechanism.

In order to try and determine the validity of thistheory, a high resolution FLAC3D model of the area wascreated in order to determine the anticipated stressesand failure zones passing through the three-wide pillar.

4 FLAC3D MODEL

At the time of the event, Itasca was actively involvedin modeling using FLAC3D at Laronde Mine onanother project. From this other work and previousprojects at the mine going back to 1997, a well-calibrated set of material properties and stresses wereavailable for the model. Previous model work how-ever did not have sufficient resolution to be useful insuch a specific case and so a more detailed model ofthe region was created.

The model generated for the burst investigationfocused on the 152 mining horizon with a block sizethrough the area of interest of 3 meters on a side. Withthe stopes 15 meters wide this provided 5 blocks alongthe strike of the stope which was deemed important toensure a proper modeling of pillar and confinementeffects.

5 RESULTS

The FLAC3D model provided evidence that theassumption of the three-wide pillar being createdthrough the retreating extraction sequence resulted ina pillar that was too large to fail and hence became astress concentrator. Figure 6 shows a principal stressplot on a long section looking North through the ore-body. The location of the burst matches nearly perfectlywith the high stress concentration predicted in theFLAC3D model.

Figure 7 (which is the section marked as A-A inFigure 6 through the 64 stope) gives an indication ofhow the stress concentration in the pillar acts on thefoliated zone some 30 meters into the footwall, causingit to slip. The stresses are deflected under the lowerabutment and concentrated through the three-widepillar, resulting in a vertical stress component. This

207

20-North Ore-body. Mined andBackfilled.

149 Level

152 Level

Foliated Zone

Shear stress onthe foliations

152-20-64

Stope

Stress trajectories

Figure 3. Simplified cartoon view of the intersection ofthe foliated zone with the levels involved in the burst.

Figure 4. Damage on the 149 Level. Most of the debriscame from the base of the footwall (right side) with someadditional secondary bagging of material in the screen(upper left).

Figure 5. Looking East on the 152 Level. The damage ismore severe especially to the east. A large amount of mate-rial up to 2 meters deep is seen in the main entrance to thelevel on the lower left.

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vertical component appears to have provided enoughlocalized stress on the foliated zone to cause the slip.

From the mining plan, there are two other three-wide pillars that appear to be concentrating stresses.A series of modeling runs was conducted in order todetermine how the short-term mining plan needed tobe adjusted to prevent additional stress building up inthese areas. Figure 6 shows these areas above and to theEast of the location of the burst. These two stopes,

although not as critical as the one that caused theburst, were cause for concern. The short term miningplan did not include these particular stopes althoughafter the modeling, recommendations were made tomine these stopes as early as possible.

The additional scenarios showed that the mining of adjacent stopes created incremental increases inthe stresses in these areas, so although they need notbe mined immediately, a rapid development and

208

Figure 6. Maximum principal stress plot looking North showing the location of the three high stress pillars and the locationof the burst.

Figure 7. A cross section looking East through the region of the burst (section A-A in Fig. 6).

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production schedule should be implemented in orderto avoid further seismic events in the region.

Using the simulations for the short term miningplan of the area, it was determined that the best sce-nario would involve mining the 143-20-65 pillar first.Figure 8 shows the stresses around the 152 mininghorizon after mining out of 143-20-65 stope. The deci-sion to recommend mining this stope first was it was

made because it was postulated that the burst hadlikely dissipated some of the stored energy in 152-20-64 stope and it was therefore unlikely that a secondseismic event would occur in the short term. Alsosome significant rehabilitation on 152-level woulddelay the development and production of this stope bya couple of months. The mining of 143-20-65 wouldnot shed much additional stress on the other stopes but

209

Figure 9. Maximum principal stress along a long section looking North following the mining of 155-20-59 stope in the bottomleft.

Figure 8. Maximum principal stress plot showing the changes caused by mining 143-20-65 stope.

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would take care of the most worrisome of the remain-ing three-wide areas.

The mining of 143-20-65 stope would shed a smallamount of additional stress on 143-20-69 but this stopewas partially protected by the lead primary stopebetween them. By the time 143-20-65 was mined andfilled, rehabilitation of the 152 Level should be nearlycompleted which would then be immediately devel-oped to allow the mining of 152-20-64 (the locationof the burst). This should push the entire lower abut-ment stress below the 152 level and relieve what stresswas left after the burst. Finally, 143-20-69 does notappear to be critical in the short term as most of themining in this area is to the west and this stope is wellshielded from these stopes.

In order to provide some short-term productionwhile the 152 Level was closed for cleanup operations,the possibility of taking a stope down on 155 Level wasexamined. Figure 9 shows the results of the modelingof this particular scenario. A concentration of stressesappears to occur two stopes away on either side of themined stope. The proximity of this increased stress tothe location involved in the burst resulted in a recom-mendation not to mine this particular stope until 152-20-64 was mined, relieving this stress concentration.

6 CONCLUSION

Based on the on-site investigation and the FLAC3D

modeling, the mechanism responsible for the burstappears to have been well established. A three stopewide pillar on the 152 Level resulted in a large stressconcentration. This stress concentration resulted in anincreased vertical stress component in the footwall ofthe orebody, allowing the foliated zone to slipbetween the 149 Level and the 152 Level.

The FLAC3D model was also able to provide rec-ommendation on which other areas might be of con-cern as well as the best sequence in which to take careof these problem areas. Finally, the modeling showedthat mining of a stope down on the 155 Level wouldnot be prudent at this stage until the three-wide pillarresponsible for the burst was removed.

ACKNOWLEDGEMENTS

The authors thank Agnico Eagle Mines for permis-sion to publish this paper.

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211

Modeling arching effects in narrow backfilled stopes with FLAC

L. Li, M. Aubertin & R. SimonÉcole Polytechnique de Montréal, Quebec, Canada

B. Bussière & T. BelemUniversité du Québec en Abitibi-Témiscamingue, Quebec, Canada

ABSTRACT: Numerical tools can be very useful to investigate the mechanical response of backfilled stopes. Inthis paper, the authors show preliminary results from calculations made with FLAC. Its use is illustrated by showingthe influence of stope geometry. The results are compared with analytical solutions developed to evaluate archingeffects in backfill placed in narrow stopes. Some common trends are obtained with the two approaches, but thereare also some differences regarding the magnitude of the stress redistribution induced by fill yielding.

1 INTRODUCTION

Even though backfill has been placed in undergroundstoping areas for many decades, it can be said that back-filling still remains a growing trend in mining opera-tions around the world. This is particularly the case inCanada where significant efforts have been devoted,over the last 25 years or so, to improve our understand-ing of mining with backfill (e.g. Nantel 1983, Udd1989, Hassani & Archibald 1998, Ouellet & Servant2000, Belem et al. 2000, 2002).

In recent years, the increased use of backfill in min-ing has been fuelled by environmental considerations(e.g. Aubertin et al. 2002). Many regulations now favor(and sometimes require) that the maximum quantity ofwastes from the mine and the mill be returned to under-ground workings. This practice may induce significantadvantages, as it can reduce the environmental impactof surface disposal and the costs of waste manage-ment during mine operation and upon closure.

The first purpose of mine backfill is nevertheless toimprove ground control conditions around stopes.Various types of fills can be used to reach this goal,each with its own characteristics. Backfill is oftenrequired to offer some self support properties, so it gen-erally includes a significant proportion of binder such asPortland cement and slag. But even the strongest back-fill is “soft” when compared to the mechanical proper-ties of the adjacent rock mass. This difference instiffness and yielding characteristics between the twomaterials can be the source of a stress redistribution inthe backfill and surrounding walls, as deformation ofthe backfill under its own weight may create shearstresses along the interface. For relatively narrow

stopes, the load transfer to the stiff abutments inducesarching effects. When this phenomenon occurs, the ver-tical stress below the main arching area tends to becomesmaller than the backfill overburden pressure, as shownby in situ measurements (e.g. Knutsson 1981, Hustrulidet al. 1989).

The same type of arching behavior is also knownto occur in other types of structural systems, where arelatively soft material (like soil and grain) is placedbetween stiff walls; examples include silos and bins(Richards 1966, Cowin 1977, Blight 1986), ditches(Spangler & Handy 1984), and retaining walls (Hunt1986, Take & Valsangkar 2001).

Arching effects and load redistribution can be inves-tigated using physical models, in situ measurements,analytical solutions, and numerical methods. The lattertwo approaches are particularly attractive to identify themain influence factors, and to evaluate how these mayaffect the load distribution in and around backfilledstopes.

In a recent paper (Aubertin et al. 2003), the authorshave proposed simple equations based on the Marston(1930) theory to evaluate the load distribution in stopebackfill. The results of analytical calculations have beencompared to numerical modeling performed with acommercially available finite element code. The calcu-lation results highlighted some important differencesbetween the two approaches, for the specific set ofassumptions adopted.

In this paper, the authors use FLAC (Itasca 2002) tofurther advance our understanding of the load transferprocess in and around narrow backfilled stopes. Inthese calculations, some of the assumptions and inputconditions differ from the previous FEM calculations,

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including the use of a somewhat more representa-tive constitutive model for the backfill. The miningsequence is also taken into account. It is shown that forspecific cases amenable to analytical solutions, the cal-culated results from both approaches are fairly close toeach other.

2 ARCHING EFFECTS

Arching conditions can occur in geomaterials such assoil, jointed rock mass and backfill, when differentialstraining mobilizes shear strength while transferringpart of the overburden stress to stiffer structural components.

Arching typically occurs when portions of a fric-tional material yield while the neighboring materialstays in place. As the yielding material moves betweenstable zones, the relative movement within the formeris opposed by shear resistance along the interface withthe latter. The shear stress generated along the contactarea tends to retain the yielding material in its originalposition. This is accompanied by a pressure reductionwithin the yielding mass and by increased pressure onthe adjacent stiffer material. Above the position of themain arch, a large fraction of the total overburdenweight in the yielding mass is transferred by frictionalforces to the unyielding ground on both sides.

Investigations on models and in situ measurementshave shown that the magnitude of the stress redistribu-tion depends to a large extent on the deformation of thewalls confining the soft material (e.g. Bjerrum 1972,Hunt 1986).

A few analytical solutions have been developed toanalyze the pressure distribution since the pioneeringwork of Janssen (1895) (see Terzaghi 1943 for earlygeotechnical applications). Among these, the Marston(1930) theory was proposed to calculate the loads onconduits in ditches (see also McCarthy 1988). Theauthors have used this theory to develop an analyticalsolution for arching effects in narrow backfilled stopes(Aubertin 1999).

Figure 1 shows the loading components for a verti-cal stope. On this figure, H is the backfill height, B thestope width, and dh the size of the layer element; Wrepresents the backfill weight in the unit thicknesslayer. At position h, the horizontal layer element is sub-jected to a lateral compressive force C, a shearing forceS, and the vertical forces V and V dV.

The force equilibrium equations for the layer ele-ment provide an estimate of the stresses acting acrossthe stope (Aubertin et al. 2003). From these, the verticalstress can be written as follow:

(1)

with

(2)

where �vh and �hh are the vertical and horizontalstresses at depth h, respectively; � represents the unitweight of the backfill; � is the effective friction anglebetween the wall and backfill, which is often taken asthe friction angle of the backfill, �bf. Equations 1 and2 constitute the Marston theory solution. In this repre-sentation, K is the reaction coefficient corresponding tothe ratio of the horizontal stress �hh to vertical stress�vh. This reaction coefficient depends on the horizontalwall movement and material properties. When there isno relative displacement of the walls, the fill is said tobe at rest, and the reaction coefficient is usually givenby (Jaky 1948):

(3)

where �bf is the friction angle of the backfill. For typ-ical fill properties (�bf ≅ 30° to 35°), K0 is muchsmaller than unity.

If the walls move outward from the opening, thehorizontal pressure might be relieved, and the reactioncoefficient tends toward the active pressure coefficient,which can be expressed as (Bowles 1988):

(4)

with Ka � K0. If an inward movement of the wallscompress the fill, it increases the internal pressure.Then, the reaction coefficient tends toward the passivecondition, which becomes (Bowles 1988):

(5)

212

H

B

V

V + dV

SS

CC W

h

dh

B

backfillstope

void space

rock mass

rock mass

layer element

dh

Figure 1. Acting forces on an isolated layer in a verticalstope.

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with Kp 1 K0.In the above equations, it is assumed that cohesion is

low in the backfill; the fill then behaves as a granularmaterial. Based on limit equilibrium, it can be antici-pated that a cohesion would increase Kp but decreaseKa. However, more work is needed to investigate itsinfluence on arching effects and stress distribution.

Figure 2 shows values of �vh and �hh calculated withEquations 1 and 2 (with K � K0 � 0.5), and calculatedwith the overburden pressure (i.e. �vh � �h and�hh � K0 �vh). It can be seen that the overburden stressrepresents the upper-bound condition, which is appli-cable for low fill thickness (or for wide stopes). Typi-cally, when H � 2 to 3B, the pressure near the bottomof the stope becomes more or less independent of thedepth of the fill, in accordance with measurements inbins (Cowin 1977).

3 NUMERICAL CALCULATIONS

3.1 Vertical stope

Recently, the authors have shown some preliminarycalculation results obtained with a finite elementcode (Aubertin et al. 2003). Significant differenceshave been revealed between the Marston theory andthese numerical calculations, which may beexplained, in part, by different assumptions associ-ated to the two approaches. In this paper, the samegeometry and material properties (Fig. 3a) are used forthe basic calculations made with FLAC. The dimen-sions of the opening are H � 45 m and B � 6 m, witha void of 0.5 m left at the top of the stope. The naturalin situ vertical stress �v in the rock mass is obtained

by considering the overburden weight (for an overalldepth of 250 m). The natural in situ horizontal stress �his taken as twice the vertical stress �v, which is a typi-cal situation encountered in the Canadian Shield. Therock mass is homogeneous, isotropic and linear elastic,while the granular backfill obeys a Coulomb criterion.Figure 3b shows the stress-strain relation used with theCoulomb plasticity model available in FLAC. This con-stitutive behavior is different from the one used for thefinite element calculations presented in Aubertin et al.(2003), which was of the elastic-brittle type. There are

213

0

0.1

0.2

0.3

0.4

0.5

h/B

stre

ss (

MPa

)

for B = 6m

overburden

Marston theory

0 1 2 3 4

svh

shh

svh

shh

Figure 2. Overburden pressures are compared to vertical(�vh) and horizontal (�hh) stresses calculated with the Marstontheory (Eqs. 1–2), with �bf � 30°, � � 0.02 MN/m3, andK � K0 � 0.5.

backfill

E = 300 MPa

� = 0.2

ã = 1800kg/m3

�' = 30°

c = 0 kPa

rock mass

(linear elastic)

E = 30 GPa

� = 0.3

� = 2700 kg/m3

�v

�h = 2�v

H =

45

m

B = 6 m

depth = 250 m0.

5m

x

y

void spacenatural stresses

rock mass(a)

1

�1

(b)

Figure 3. (a) Narrow stope with backfill (not to scale) used for modeled with FLAC; the main properties for the rockmass and backfill are given using classical geomechanicalnotations; (b) Schematic stress-strain behavior of the backfill(available as a material model in FLAC).

09069-26.qxd 08/11/2003 20:33 PM Page 213


no interface elements in the calculations made withFLAC (see discussion).

The mining and filling sequence is considered as follow in the numerical modeling. The whole stope is

first excavated, and calculations are then performedwith FLAC to an equilibrium state. Backfill is placed inthe mined stope afterward, with the initial displacementfield set to zero when the calculation is performed. In this manner, wall convergence before backfilling isnot included in the calculations (this assumption is discussed in Section 4).

Figure 4 shows the vertical stress (Fig. 4a) and hor-izontal stress (Fig. 4b) distribution in the backfilledstope. As can be seen, the vertical and horizontalstresses show a non-uniform distribution. At a givenelevation, both stresses tend to be lower along the wallthan at the center. The stresses along the central lineincrease more slowly than the overburden pressureswith depth. This indicates that arching does take placein this backfilled stope.

Figures 5 and 6 present modeling results for stressesalong the full height, with the overburden and the

214

Figure 4. Stress distribution in the backfilled stope calcu-lated with FLAC: (a) vertical stress �yy; (b) horizontal stress�xx.

0

0.2

0.4

0.6

0.8

0 9 18 27 36 45

h (m)

�yy

(M

Pa)

modeling with FLAC-2D

overburden stress

Marston theory

K = 1/3

K = 1/2

K = 3

(a)

0

0.1

0.2

0.3

0 9 18 27 36 45h (m)

�xx

(M

Pa)


overburden stress

Marston theory

K = 1/3

K = 1/2

K = 3

(b)

Figure 5. Comparison of the stresses calculated along thevertical central line, at different elevations h, with the analyt-ical and numerical solutions: (a) vertical stress �yy; (b) hori-zontal stress �xx.

09069-26.qxd 08/11/2003 20:33 PM Page 214


Marston theory solutions. As expected, the overburdenstress is fairly close to analytical and numerical resultswhen the backfill depth is small. At larger depth, arch-ing effects become important and the vertical and hori-zontal stresses tend to be lower than those due to theoverburden weight of the fill. However, the numericalresults indicate that the Marston theory typically over-estimates the amount of stress transfer, hence underesti-mating the magnitude of the vertical stress �yy and ofthe horizontal stress �xx along the stope central verticalline (Fig. 5). Along the walls (Fig. 6), the horizontalstress is also underestimated by the Marston theory,while the vertical stress component �yy would be over-estimated for the active and at rest cases, with K � 1/2or 1/3, respectively (and underestimated with K � 3,but the passive case is not representative of this systembehavior).

Figure 7 shows the stress distribution on the floor ofthe stope, as obtained from the numerical and analyticalsolutions. It can be seen that the overburden pressure

exceeds the stress magnitudes given by the Marston theory (with K � 1/2 and 1/3), which is in fair agree-ment with the numerical simulations.

3.2 Inclined stope

Mining stopes are rarely vertical. The inclination of thefoot-wall and hanging-wall may have a non-negligibleeffect on the load distribution.

Figure 8 shows the geometry of an inclined back-filled stope modeled with FLAC (a similar stope wasalso modeled with the FEM code – see Aubertin et al.2003). The rock mass and fill properties as well as thein situ natural stresses are identical to the previous case(see Fig. 3).

Figure 9 shows numerical calculations and resultsbased on overburden pressure and on the Marston the-ory solution (without modification for inclination).The horizontal stress calculated with FLAC along the

215

0

0.2

0.4

0.6

0.8

0 9 18 27 36 45

h (m)

σyy

(MPa

)


over burden stress

Marston theory

K = 1/3K = 1/2

K = 3

(a)

0

0.1

0.2

0 9 18 27 36 45

h (m)

σxx

(MPa

)


overburden stress

Marston theory

K = 3

K = 1/3

K = 1/2

(b)

Figure 6. Comparison of the stresses on the wall calculatedat different elevations h, with the analytical and numericalsolutions: (a) vertical stress �yy; (b) horizontal stress �xx.

0

0.4

0.8

1.2

0 2 4 6x (m)

σyy

(MPa

)σx

x (M

Pa)


overburden stresses

Marston theory

K = 3

K = 1/3

K = 1/2

(a)

0

0.1

0.2

0.3

0 2 4 6x (m)


overburden stresses

Marston theory

K = 3K = 1/2

K = 1/3

(b)

Figure 7. Stresses calculated at the bottom of the verticalstope, with the analytical and numerical solutions; (a) verti-cal stress �yy; (b) horizontal stress �xx.

09069-26.qxd 08/11/2003 20:33 PM Page 215


inclined central line of the stope is fairly close to theanalytical solution (Fig. 9a), but the vertical stress isunderestimated by the Marston theory (see Fig. 9b).Hence, modifications could be required to apply suchanalytical approach to the case of inclined stopes.

4 DISCUSSION

4.1 Influence of mining sequence

In the numerical calculations presented in Aubertin et al. (2003), the mining sequence was not taken intoaccount, so the wall convergence due to elastic strain-ing of the rock mass was imposed on the fill. This created an increase of the mean stress in the fill, whilevertical and horizontal stresses locally exceeded theoverburden pressure and the Marston theory solution(near mid-height of the stope).

Modeling in this manner implies that the backfill isplaced in the stope before wall displacement takesplace. For a single excavation stope, this is not a real-istic representation (at least for hard rock masses).However, with a cut-and-fill mining method where themining slices (or layers) are relatively small comparedto the whole height of the stope, filling is usuallymade quickly after each cut. In this case, wall conver-gence after each cut compress the fill already in place(Knutsson 1981, Hustrulid et al. 1989). The inwardmovement of the walls may then create a conditioncloser to the passive pressure case.

When a stope is excavated in a single step, wall con-vergence essentially takes place before any backfilling.If the rock mass creep deformation is negligible, the numerical modeling approach presented here ismore appropriate. In this case, the Marston theory,with the “at rest” reaction coefficient (K � K0) can beused to estimate the induced stresses in a narrow verti-cal backfill (see Figs. 5–7), at least for preliminarydesign calculations.

4.2 Marston theory limitations

Analytical solutions can be useful engineering tools asthey are generally quick, direct and economic whencompared to numerical methods. However, analyticalsolutions are only available for relatively simple situa-tions and may involve strong simplifying hypotheses.For instance, with the Marston theory, the shear stressalong the interface between the rock and fill is deducedfrom the Coulomb criterion (see details in Aubertin et al. 2003). Its value then corresponds to the maxi-mum stress sustained by the fill material, as postulatedin the limit analysis approach (e.g. Chen & Liu 1990).However, numerical simulations indicate that thisassumption is not fully applicable. Figure 10 showsthat for the vertical stope analyzed here the maximumshear stress is only reached near the bottom part of the

216

�v�h = 2�v

H =

45

m

depth = 250 m

0.5m

x

y

B = 6 m

60˚

h

void space

backfill

rock mass

rock mass

stope

Figure 8. The inclined backfilled stope modeled withFLAC (properties are given in Fig. 3).

0

0.1

0.2

0 9 18 27 36 45

h (m)

�xx

(M

Pa)

�yy

(M

Pa)


overburden

Marston theory

K = 1/2

K = 3

K = 1/3

(a)

0

0.2

0.4

0 9 18 27 36 45h (m)

modeling without FLAC-2D

overburden

Marston theory

K = 1/2

K = 3

K = 1/3

(b)

Figure 9. Comparison between stresses obtained withnumerical and analytical solutions along the central line of theinclined stope: (a) horizontal stress �xx; (b) vertical stress �yy.

09069-26.qxd 08/11/2003 20:33 PM Page 216


stope. Hence, arching effect and stress redistributionare thus exaggerated.

Another important assumption in the Marston theoryis that both the horizontal and vertical stresses are

uniformly distributed across the full width of the stope.Results shown in Figure 11 indicate that this is in accor-dance with numerical calculations for the horizontalstress component (Fig. 11a), but not for the verticalstress which shows a less uniform distribution (Fig.11b). Also, this simplified theory considers that thereaction coefficient, K, depends exclusively on the fillproperty and not on the position in the stope. Resultsshown in Figure 12 indicate that this hypothesis is nottoo far from the numerical results. Near the boundary,the value of K would nevertheless be better described bya K value between Ka and K0.

Work is underway to modify the analytical solutionto extend the use of the Marston theory to more general cases.

4.3 Constitutive behavior

The reliability of any numerical calculations depends,to a large extent, on the representativity of the constitu-tive models used for the different materials (and on thecorresponding parameter values). In this paper, aCoulomb plasticity model (see Fig. 3) was employedfor the fill material. This model is representative ofsome aspects of the mechanical behavior of backfill,such as the nonlinear relationship between the stressand strain (e.g. Belem et al. 2000, 2002). However, thissimplified model neglects some important characteris-tics of the media, including its pressure dependentbehavior under relatively large mean stresses. More rep-resentative models, such as the modified Cam-Claymodel, are built in FLAC (e.g. Detournay & Hart,1999). However, the application of such model is notstraightforward because of the difficulties in obtainingthe relevant material parameters. The influence of cohe-sion due to cementation and possible oxidation of thefill material may also be relevant to include in theanalyses.

217

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0 9 18 27 36 45

h (m)

σxy

(MPa

)


Marston theory

K = 3K = 1/3

K = 1/2

Figure 10. Comparison of shear stress distribution along the wall.

(a)

0.04

0.06

0.08

0.1

0.12

0x (m)

�xx

(M

Pa)


at 1/4H

at 3/4H

at 1/2H

(b)

0

0.1

0.2

0.3

�yy

(M

Pa) at 1/4H

at 3/4H

at 1/2H

2 64

0x (m)

2 64


Figure 11. Distribution of (a) lateral pressure �xx and (b)vertical stress �yy obtained with FLAC across the full widthat different elevations of the vertical stope.

0.1

0.3

0.5

0 x (m)

K

at floor

at 1/2Hat 1/2H

at 3/4H

modeling with FLAC

at rest

active

2 4 6

Figure 12. Reaction coefficient K obtained with analyticaland numerical solutions across the full width of the verticalstope at different elevations h.

09069-26.qxd 08/11/2003 20:33 PM Page 217


An interesting aspect of FLAC is that it allows user-defined models, which can be introduced with the language FISH. The authors are now working on intro-ducing in FLAC a multiaxial, porosity dependent crite-rion (Aubertin et al. 2000, Li & Aubertin 2003) for theyielding and failure conditions of geomaterials. Thisaspect will be presented in upcoming publications.

4.4 Interface elements along the walls

As was done with a finite element code in a previousinvestigation (Aubertin et al. 2003), some calculationswere also performed with interfaces included in FLAC,to represent the contact between backfill and rock mass.

Preliminary results (not shown here) indicate that thepresence of interfaces along the walls and floor of thestope, which allow localized shear displacements, hasrelatively little influence on the stress distribution in thestope and at its boundary. Some differences between thecases shown here and models with interfaces neverthe-less appear near the bottom and top of the stope wheresome stress reorientation and concentration seem totake place. This aspect however requires further investi-gation. The applicability of the (Coulomb) strength cri-terion and the numerical stability of the calculationsalong these elements also need more study.

5 CONCLUSION

In this paper, numerical simulations have been per-formed with FLAC for a vertical and an inclined back-filled stope geometry. The results are compared to theMarston theory solutions. It is shown that the resultsobtained with the Marston theory can be considered asacceptable, especially for preliminary calculations.Nevertheless, the numerical results also reveal that theMarston theory tends to overestimate arching effect,and thus underestimate the stress magnitude near thebottom of backfilled stope. Also, the influence of themining sequence can not be introduced in the Marstontheory. The numerical results indicate that the fillingsequence can significantly influence the stress distri-bution in and around filled stopes. For inclined stopes,the Marston theory is of limited use to estimate thestress magnitude. Additional work is underway intoboth analytical and numerical solutions to betterdescribe the behavior of backfilled stope. More workis also needed to study the rock-fill interface behaviorand the actual field response of backfill in stopes.

Other important issues also remain to be resolved,including the possible degradation of the arch due tolow pressure (and tensile stresses), the influence ofwater flow and distribution in backfilled stopes, theevolving properties of the fill material (particularlyconsidering the action of cement in the presence ofsulfide minerals), the dynamic response of the back-fill, and the forces generated on retaining structures.

ACKNOWLEDGEMENT

Part of this work has been financed through grantsfrom IRSST and from an NSERC Industrial Chair(http://www.polymtl.ca/enviro-geremi/). The authorswould also like to thank the anonymous reviewerswho provided valuable comments to improve themanuscript.

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Aubertin, M., Bussière, B. & Bernier, L. 2002. Environnementet Gestion des Rejets Miniers. Manual on CD-ROM,Presses Internationales Polytechniques.

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Aubertin, M., Li, L. & Simon, R. 2000. A multiaxial stresscriterion for short term and long term strength of isotropicrock media. International Journal of Rock Mechanics andMining Sciences, 37: 1169–1193.

Belem, T., Benzaazoua, M. & Bussière, B. 2000. Mechanicalbehavior of cemented paste backfill. Proceedings of the55th Canadian Geotechnical Conference, 1: 373–380.Canadian Geotechnical Society.

Belem, T., Benzaazoua, M., Bussière, B. & Dagenais, A.M.2002. Effects of settlement and drainage on strength devel-opment within mine paste backfill. Tailings and MineWaste ’02: 139–148. Swets & Zeitlinger.

Blight, G.E. 1986. Pressure exerted by materials stored insilos. Part I: coarse materials. Géotechnique, 36(1): 33–46.

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Bowles, J.E. 1988. Foundation Analysis and Design. McGraw-Hill.

Chen, W.F. & Liu, X.L. 1990. Limit Analysis in Soil Mechan-ics. Amsterdam: Elsevier.

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Detournay, C. & Hart, R. (eds) 1999. FLAC and NumericalModeling in Geomechanics – Proceedings of the Inter-national FLAC Symposium, Minneapolis, Minnesota, 1–3September 1999. Rotterdam: Balkema.

Hassani, F. & Archibald, J.H. 1998. Mine Backfill. CIM, CD-ROM.

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Janssen, H.A. 1895. Versuche über Getreidedruck inSilozellen. Zeitschrift Verein Ingenieure, 39: 1045–1049.

Knutsson, S. 1981. Stresses in the hydraulic backfill fromanalytical calculations and in-situ measurements. In O.Stephansson & M.J. Jones (eds), Proceedings of theConference on the Application of Rock Mechanics to Cutand Fill Mining: 261–268. Institution of Mining andMetallurgy.

Li, L. & Aubertin, M. 2003. A general relationship betweenporosity and uniaxial strength of engineering materials.Canadian Journal of Civil Engineering (in press).

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Nantel, J.H. 1983. A review of the backfill practices in themines of the Noranda Group. In S. Granholm (ed), Miningwith Backfill: Proceedings of the International Symposiumon Mining with Backfill: 173–178. Rotterdam: Balkema.

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221

FLAC3D numerical simulations of deep mining at Laronde Mine

C.P. O’Connor, R.K. Brummer & P.P. AndrieuxItasca Consulting Canada Inc., Sudbury, Ontario, Canada

R. Emond & B. McLaughlinAgnico-Eagle Mines – Laronde Mine, Cadillac, Quebec, Canada

ABSTRACT: Agnico-Eagle’s Laronde Mine is currently investigating mining options down to 3000 metersbelow surface. FLAC3D was used to simulate the entire current mine from 1340 m to 2150 m below surface aswell as the potential future expansion. FLAC3D allowed each of the three main ore lenses to be modeled together,providing information on the interaction between lenses which had never been available before. The model providesinformation useful for determining the ideal stope dimensions, mining method, mining sequence, support optionsfor large excavations, as well as the best option for the shaft location.

1 INTRODUCTION

Agnico Eagle’s Laronde Mine is a 7000 tpd under-ground operation located in Northern Quebec in theAbitibi Mining district near Rouyn-Noranda. Drillingfrom the bottom levels of the current mine have shownthat there are significant reserves down to at least3000 m below surface. Most of the tonnage has histori-cally come from above the 152 Level but with a shiftcurrently taking place to turn the 215 Level into themajor source of ore as the upper levels are progressivelybecoming depleted.

As part of the feasibility into the potential expansiondown to 3000 m, a geomechanical review of the pro-posed expansion was conducted using FLAC andFLAC3D as numerical modeling tools to determine theanticipated response to mining at extreme depths.

FLAC3D was used in several forms. First, it was usedto model the entire mine from the top of the currentmining horizon down to a depth of 3000 m. From thismodel, the in-situ and post-mining stresses were tracedalong with an analysis of the interaction between thedifferent ore lenses in the upper levels. The secondmodel was a high resolution mining method model usedto determine the stresses and failure zones for differentsized stopes and different ore thicknesses at extremedepths.

FLAC was used in order to check shaft stresses at depth including post-mining stress changes. FLAC was also used to model the stresses around a hypothetical large excavation (conveyor drift) in order to investigate some of the issues that could

be encountered when placing infrastructure at depth.

2 MINE GEOMETRY

Laronde Mine is located in the Abitibi Mining district innorthern Quebec. The orebody is a gold-zinc depositthat is part of an extensive intrusive complex that runsthroughout the region. The 20-North deposit, which isthe major producer, runs from a depth of 900 m down toat least 3000 m but currently mining is only taking placedown to 2150 m. The orebody is steeply dipping to theSouth and raking towards the West.

The current mining method used is primary-secondary stoping with high quality backfill (mainlypastefill). This method has been used since the miningof 20-North began and has proved to be successful inmaintaining good stability in the hangingwall and min-imizing stress related problems.

Secondary stopes are designed to fail with theextraction of primary stopes, which results in a rela-tively low stress environment in which to mine (the sec-ondary stopes are not large enough to carry significantstresses in post failure). Current stopes are 15 m alongstrike, 30 m high and the thickness of the orebody.With a few exceptions, this system has produced rela-tively trouble free mining at the current depths.

The progression to greater depths however will resultin greater stresses and more extensive failure zones thatwill cause greater difficulties than have been experi-enced to-date.

09069-27.qxd 8/26/03 10:44 AM Page 221


3 FLAC3D MODELING

The use of FLAC3D in this project was a logical choicebased on the geometry of the orebody and work previ-ously performed by Itasca for the mine (a reasonablygood calibration of the model, material properties, andstresses had already been performed). To extend uponthis base information, new stress data and core testingby CANMET resulted in a more refined picture of thecomplex interactions between the different rock struc-tures and stresses.

Around each ore lens is a layer of highly shearedschist that varies in thickness up to 5 m on both thehangingwall and footwall. Surrounding the schist is agarnetiferous tuff material that is relatively strong andstiff with a uniaxial compressive strength (UCS) of

180 MPa. The orebody itself is competent, also with aUCS of 180 MPa. Both of these values come fromCANMET test results (Labrie 2000a).

The stresses at Laronde were measured by CANMET on the 146 level and the 150 level (Labrie2000b). The stress gradients are shown in Figure 2.

3.1 Overall mine model

The overall mine model was used for a number of pur-poses. First, it was used as a calibration of the stressand material properties based upon information col-lected from site visits and previous experience at themine. Secondly, it was used to determine how much ofan interaction between ore zones was likely takingplace. Finally, it was used to determine the in-situ and post-mining stresses along the potential shaft loca-tions and the overall stress regime throughout theentire mine.

3.1.1 CalibrationThe calibration of the material properties became easierto perform after a seismic event occurred on the 152Level of the mine. Using the existing model as a frame-work, an investigation showed – and the model con-firmed – that there was a large stress concentrationcentered right around the location of the burst causedby a 3-wide pillar being formed by retreating stopestowards the central access for the level.

In March 2003, a large fall of ground occurred in astope on the 215 Level where a double width stopehad been taken. Again, the model was able to show asimilar pattern in the stresses and failure zones in thisarea. Between these two events, a comfortable degreeof confidence was gained that the model reflects realis-tic stresses and failure zones based upon the knowngeology and geometry of the orebody at depth.

3.1.2 Ore lens interactionsIn all of the previous work performed, the investigationof the impact of the different ore lenses on each otherwas not considered relevant because of the distance

222

CANMET Stress Model Used to Obtain Gradients for Flac 3D Model

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CANMET @ 146S1 Average = 59.31S2 Average = 48.13S3 Average = 25.08

CANMET @ 150S1 Average = 70.2S2 Average = 61.18S3 Average = 47.61

100

Figure 2. Stress gradients used in the FLAC modeling atLaronde Mine based upon CANMET stress measurements.

Figure 1. Cross-section looking East of the Laronde orebody showing the different rock units included in the modeling.

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between the mining fronts. As part of this project, it wasdecided to include the multiple ore lenses in order todetermine just how small or large an interaction waslikely to occur between these zones.

From the model, it appeared that there is an interac-tion between the different ore zones as mining pro-gresses into the future. Figure 3 shows a surface contourplot on the 215 Level with all three lenses being minedto the end of the five-year plan (as of November 2002).The mined out stopes show up as depressions whilethe abutment stresses appear as peaks in this perspec-tive view of a surface plot. There is a definite bridgingeffect between the middle and upper zone abutmentareas (shown as the raised region), as well as somestress shadowing occurring between the middle andlower zones (shown as a depression).

3.1.3 Shaft stressesThe location of the shaft was one of the most importantaspects of the project. In order to determine the best

location for the shaft, a series of history points weretaken along the length of five potential locations. Eachshaft location represented a suitable surface location atwhich a shaft could be placed within the existing limitsof the property.

Tracking of the major and minor principal stressesalong the length of each possible shaft location, a com-parison between these locations became possiblethrough the use of the stress ratio (SR). The SR (shownin Fig. 4) is calculated based upon the anticipated pointat which damage will occur in an ideal circular openingand is defined as:

(1)

The stress ratio plots and an analysis of the maxi-mum and minimum stresses showed that there was nosingle shaft location that stood out as being signifi-cantly better or worse than any other – rather it appearedthat all of the shafts were likely going to experiencesimilar stress levels with some variation in the timingand location of peak stresses depending on their prox-imity to the orebody.

Based upon several factors, shaft location #3 wasproposed as the best. It was located near the centroid ofthe orebody at depth, which will reduce haulage dis-tances, and had the benefit of enjoying some level ofstress shadowing from the orebody at the deepest levels.The geology of the shaft location was unknown at thetime, as drilling had focused on delineating the orebodyand not so much on investigating the footwall materials.Future drilling of this region could change the ideallocation to avoid adverse geology.

3.1.4 Overall mine stressesThe final purpose of the overall mine model was toexamine the overall mine stresses over the entire lifeof the mine. In order to make a model that could berun within a reasonable amount of time, the resolu-tion in the upper regions was reduced to allow for ahigher resolution in the 2150 to 3000 m depth region.

The mine stress model showed a number of interest-ing things. Firstly, the failure zones around a fullyformed mining front are quite extensive depending onthe ore thickness, and can even be greater than an entirestope width into the abutment. (see Fig. 5.) Stresses inthe abutments and sill regions can exceed 250 MPa (asseen in Fig. 6). Also, even in failed ground very highstresses can be seen due to heavy confinement levelsthat will likely cause some significant issues in areassuch as sills in which post-failure ground is subjected tovery high stresses.

3.2 High resolution mining method analysis

In order to take a closer look at the stope level stresses,another model was built with the sole purpose of run-ning high resolution simulations at the maximum depth

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Figure 3. Surface contour plot generated by Surfer( on the215 Level in the final year of the five-year plan. The upperright region is Zone 7, the middle is the main Zone 20, and thelower region shows Zone 21.

Stress Ratio Versus Depth1400

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Little to no Damage Moderate Damage

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Figure 4. Stress ratio plot for all shaft locations based uponFLAC3D results.

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of the mine. The model used a simplified representationof the orebody to give a constant strike and thickness ofthe ore and schist zones to eliminate geometrical effectsin the comparison. A total of four simulations were run,three with different ore thicknesses ranging from 10 mup to 30 m, and a fourth simulation in which the stopesize was reduced to determine the impact on the stopestability.

To simplify the running of multiple scenarios withsimilar geometry, a FISH function was written in which

the thicknesses of the different units and their locationin the model could be defined, as well as the depth atwhich the simulation was to occur. This automationreduced the turnaround time between model runs toonly a few minutes.

3.2.1 ResultsSome results from the different ore thicknesses are shown below, in each case, the early stages ofmining are shown when only four stopes have been

224

Figure 6. Maximum principal stress plot on a long section looking North through the orebody. Peak stresses in a couple ofareas exceed 250 MPa.

Figure 5. Failure plot on a long section looking North of the overall mine model as mining approaches a sill. Blue blocks areintact, red and light blue blocks have failed in shear and green blocks also have a tensile failure component.

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mined.A 10 m and 30 m ore thickness are shown forcomparison.

Looking at the principal stress plots it can be seen that there is a definite pinching of the stresseswith the narrower ore geometry Fig. 7) due to the

stronger secondary stopes carrying more loading and the smaller failure zones. In contrast, the 30 m ore zones (Fig. 8) result in a very large and smoothstress distribution with lower peak stress levels. This pattern of pinched stresses and higher peak

225

Figure 7. Maximum principal stress plot on a long section looking North through the orebody with a 10 meter ore thick-nessshowing some pinching of high stresses above the trailing primary stopes. Secondary stopes are in post-failure even in narrow ore.

Figure 8. Maximum principal stress plot on a long section looking North through the orebody with a 30 meter ore thicknessshowing that the destressed zone is much larger due to the extra freedom provided by the larger stoping spans. Overall, thestresses are more spread out and peak stresses are predicted to be much lower in this case.

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226

levels continued throughout the simulation as miningproceeds.

Looking at the failure plots (Figs. 9 & 10) the rea-son for the pinching of the stresses from the pre-vious plots (Figs. 7 & 8) can be seen. The failurezones are much larger with the thicker ore-body; this creates a more even shell of failure aroundthe stopes which becomes spherical in shape. The

impact of a single stope is lost in the overall picture.In the narrower orebody, the impact of individualstopes on the overall shape of the failure zone is stillapparent.

From these results it can seen that there is a signifi-cant impact on the stresses and failure zones withincreasing ore thickness as would be expected with thechange in the pillar width-to-height ratio.

Figure 9. Cross-section looking North showing failed regions in the model with 10 meter ore thickness. The impact of indi-vidual stopes can be seen by the irregular shape of the failure zone. Most of the blocks are failed in shear (light blue and redblocks, although some tensile failure is evident along the stope boundaries (green blocks).

Figure 10. Cross-section looking North showing failed regions in the model with a 30 meter ore thickness. The failure regionis much larger and more even with the wider ore at these great depths. The color coding is the same as in Figure 9.

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227

4 FLAC MODELS

Two other models were created in FLAC in order to givesome measure to the stresses and anticipated failurezones around excavations at extreme depths. The firstmodel was used to model the shaft in both in-situ andpost-mining situations, whereas the second was used tomodel a hypothetical large excavation at 3000 m(9840 ft).

4.1 Shaft model

The shaft model created in FLAC used the double donutFISH function provided with FLAC, which was modi-

fied to provide a single circular opening with a linercomponent added. The advantage of using this schemeis that it allows for the modeling of both in-situ and post-mining stresses in the same model, by adding the rota-tion to the stress tensor and observing the effects. Figure11 shows the FLAC grid used for the shaft modeling.

The liner was set up as a 12 inch concrete layer,which was added after the in-situ stresses had reachedequilibrium in order to properly mimic the trueresponse of the liner. The liner was assumed to respondonly to post-mining stresses. The maximum principalstress was set to run North–South. The in-situ stressesand failure zones the shaft at a depth of 2000 meters areshown in Figures 12 & 13 respectively.

Figure 11. Grid used in the FLAC modeling of a 9 meter diameter shaft with liner support.

Figure 12. Maximum principal stress plot around a 9 meter diameter shaft at a depth of 2000 meters.

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228

In order to determine the stresses around a largeselection of shaft locations and depth/stress condi-tions, the stress ratio plot (see Fig. 4) was used todetermine the best and worst conditions that may beexpected from each shaft location. Six shaft modelswere run – this included three generic in-situ runs at2000, 2500, and 3000 m. Another set of three modelswere used to represent post-mining conditions wherethe largest increases, decreases and rotations of thestress tensor were occurring.

Figure 14 shows an example of the #3 shaft loca-tion at a depth of 2100 m (which corresponds to thelargest increase in stress ratio). It can seen that thestresses have rotated clockwise about 45 degrees,which is shown in both the stress plot and the plastic-ity plot (Fig. 15). These results correspond very wellwith what was intuitively expected based on thegeometry of the region. At this elevation, the shaft isjust passing through the western abutment stress ofthe 215 mining horizon, which is reflected by the

Figure 13. Predicted failure envelope around a 9 meter diameter shaft at 2000 meters subjected to in-situ stresses. The depthof failure predicted from this model is around 3 m.

Figure 14. Maximum principal stress plot at shaft location #3 at a depth of 2100 meters when subjected to mining inducedstress changes.

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slight increase in stresses and the general rotation ofthe stress tensor.

Looking at the stresses in the liner (see Fig. 16), it can be seen that the stresses in the liner due to mining-induced stress are quite small with a peak stressof around only 4 MPa, which is well below the strengthof the concrete to be used in the liner. From this analy-sis it was determined that there should be no excessivestresses or failure zones that cannot be designed forwith current technology. Barring any poor geologicalhorizons through the shaft locations, no significant

difficulties are anticipated beyond those expectedwith mining at extreme depth.

4.2 Generic large excavation model

The large excavation model provided some generalguidelines that can be used in the design of infrastruc-ture in the mine. To do this, an arched back drift wascreated in FLAC with a span of 11 m and a height of6.5 m, as shown in Figure 17. The drift was set up at a3000 meter depth using the in-situ stress. Post-mining

229

Figure 15. plot for shaft location #3 at a depth of 2100 m at the post-mining stage. The rotation of the stresses at this locationhas had a significant impact on the failure zones with an increased depth of failure along a line running North-West to South-East.

Figure 16. Stresses in the 12 inch concrete liner for post-mining stresses of shaft #3 at 2100 m. Peak stresses approach 4 MPain the liner in this case.

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stresses were not considered since they are location-dependant and no information was available for place-ment of infrastructure in the deep mine. The first run ofthe model assumed that the drift ran parallel to themaximum principal stress (North–South), while thesecond model ran perpendicular to the maximum prin-cipal stress (East–West) in order to provide informa-tion on these two extreme situations.

With the drift set up to run parallel to �1, the back isshielded from the effect of the highest stresses. As aresult, the stress seen in the drift was a combination of�2 and the overburden-related vertical stress. This

resulted in a peak stress of around 200 MPa and a fail-ure zone that extended up to 5 meters into the back(Fig. 18). This represented a very extensive shell offailed material, which, depending on geology, could bedifficult to support. The ground at Laronde tends toinvolve some significant displacements which makestiff support such as shotcrete a less attractive supportsystem as it cannot accommodate much displacement.

With the alternative scenario, which had the driftrunning perpendicular to �1, the drift was exposed tothe full impact of the highest stress component (see Fig.19). As a result, the peak stresses across the back of the

230

Figure 17. Grid used to create the large excavation model.

Figure 18. Large excavation running parallel to the maximum principal stress. Peak stresses approach 200 MPa with a failurezone that extends around 5 m into the back of the drift.

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stope reached 250 MPa (which is about 25% higherthan the previous case). The failure zones extend about15% farther into the back that in the previous case,making for an even more difficult ground supportrequirement.

5 CONCLUSIONS

The numerical modeling exercise using FLAC andFLAC3D enabled the known behavior of the mine inshallower areas to be extended to the planned deepermining. With the knowledge thus gained, it is possibleto estimate with some degree of confidence some of theissues that may come into play at the extreme depthsinvolved in this mine expansion.

Based upon experience gained in the upper levels ofthe mine, several important recommendations weremade. Among other things, the mining sequence is crit-ical to the stability of the mine at these depths. Ininstances in upper levels where deviations from theoriginal plan created unfavorable geometry, problems

were encountered. These problems will be magnifiedat depth.

ACKNOWLEDGEMENTS

The authors thank Agnico Eagle Mines for permissionto publish this paper.

REFERENCES

Labrie, D. December 2000a. Strength and Elastic Modulus asDetermined on the Drill Core HQ5 at #3 Shaft – Agnico-Eagle Mines, Laronde Division, Cadillac, Quebec. (InFrench.), Technical Note from Laboratoires des mines etdes sciences minérales, CANMET to Agnico EagleLaronde Mine. Nepean, Ontario, Canada.

Labrie, D. December 2000b. Laronde Mine (Project 610 660,Task B) – Evolution of the Stress Field as a Function of theNumber of Mathematical Iterations Executed. (In French.),Technical Note from Laboratoires des mines et des sciencesminérales, CANMET to Agnico Eagle Laronde Mine.Nepean, Ontario, Canada.

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Figure 19. Large excavation running perpendicular to the maximum principal stresses. In this case the peak stresses are about25% higher and the failure zone extends nearly 6 m into the back.

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233

Three-dimensional strain softening modeling of deep longwall coal mine layouts

S. Badr, U. Ozbay, S. Kieffer & M. SalamonColorado School of Mines, Golden, Colorado, USA

ABSTRACT: This paper describes a FLAC3D model for a typical deep two-entry longwall coal mine. The coalseam is modeled as a strain softening material to attain a representative analysis of stresses and deformationsexperienced by the coal ribs and yielding chain pillars corresponding to various loading stages. The strain soft-ening parameters are established by calibrating separate test pillar models to common empirical pillar strengthformulas. The test pillar models showed that strain softening material behavior results in lower pillar strengthsthan the traditional Mohr–Coulomb models based on constant peak cohesion and friction values. The longwallmodel incorporates compaction simulations of the gob material in the back area. Two algorithms for representinggob compaction are described.

1 INTRODUCTION

In mining practices, it is common for the induced load-ing to exceed the strength of the rock mass. Realisticrepresentation of stresses and deformations in suchsituations requires use of constitutive laws that canaccount for the response of the rock mass in the post-peak state. Mohr–Coulomb (MC) and Hoek & Brown(HB) plasticity models are commonly used in thesesituations. Considering the brittle nature of many rockmasses, strain softening type models, such as theMohr–Coulomb Strain Softening (MCSS) option inFLAC3D (Itasca 2002), allow more realistic modelingof rock mass failure.

A typical mining situation where the modeling ofbrittle behavior becomes important is the analysis ofyielding chain pillars in deep longwall mines. At depthsmore than about 300 m, the vertical stress exceeds the strength of unconfined coal, resulting in failure ofthe excavation walls while they are being exposed.This can result in the sides of entry pillars failing beforethe pillars are fully isolated. Realistic estimation of theloads carried by these pillars during subsequent miningrequires the use of a softening model.

The longwall mining geometry and the sequenceof excavation considered in this study are illustrated ina plan view in Figure 1. Three longwall panels areshown in this illustration. The upper panel is alreadyextracted. The panel at the bottom of the illustration hasbeen developed, but extraction has not yet commenced.As the longwall face in the middle panel moves from

right to left as indicated, the chain pillars undergo fivestages of loading. These stages are indicated in thediagram; the first three affect the pillars next to the headgate and the last two affect the pillars next to the tail-gate. Stage 1 corresponds to the situation where theentry-pillar system is fully developed, but the extrac-tion of the longwall panels has not yet affected the load-ing of the pillar. Stage 2 refers to the situation where thefront and side abutments contribute to the pillar loadingdue to the approaching longwall face.

In Stage 3, the gob on one side, and an unminedpanel on the opposing side, affect the loading. The gobin the vicinity of the development is not fully com-pacted so it does not support the full weight of the

(1 ( (3)

(4) (

GobCo

adv

(1) (2) (3)

(4) (5)

GobCoal

adva

nce

Figure 1. Simplified plan view of a two-entry longwallmine layout showing pillar loading stages.

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overburden. In Stage 4 on the tailgate side, as the faceapproaches, the front abutment increasingly contributesto loading of the pillar; hence the conditions aroundthe tailgate pillars become progressively more adverse.Stage 5 corresponds to the situation where the influenceof the face is no longer detectable and the chain pillarsare surrounded on both sides by gobs.

This paper describes a numerical model for assess-ing the longwall mining scenario described above. Thecoal seam is modeled as a MCSS material. A separateseries of numerical analyses was carried out on a sin-gle pillar (test pillar model) to determine representa-tive MCSS strength parameters for the coal seam. Thetest pillar model analysis was also performed withMC materials to permit comparison of the pillar res-ponse based on MC and MCSS behavior.

Compaction of the fractured, particulate material,called the “gob”, created by the caving of the roof in thearea from where the coal has been extracted, requiresattention in the numerical modeling of longwall mining.With continuing extraction, the upper strata and thefloor converge and gradually the vertical load on thegob material increases. Representation of this processrequires consideration of the deformations of both thegob materials and the surrounding strata. This paperdescribes two alternative algorithms to simulate gobcompaction.

2 LONGWALL MODEL

The modeled longwall layout is similar to that shown inFigure 1. It represents a two-entry longwall mine locatedat a depth of 680 m below surface. The panel length is220 m and the mining height is 3 m. The width of theentries and cross cut is 6.5 m. The chain pillars betweenthe entries are 3 m high, 8 m wide and 26 m long.

The mining geometry is built in a 1000 m long,240 m high, and 240 m wide block with graded mesh,as shown in Figure 2. The bottom layer in this figurerepresents half of the 3 m thick coal seam. The meshingat the central portion of the base of the block is madefiner in order to represent the entries and chain pillarsin detail (Fig. 3). Within the fine meshed region, MCinterface separates the coal seam from the roof strata.The roof and floor strata are assumed to remain elasticthroughout all stages of mining. The vertical planesbounding the block are free of shear stresses and hor-izontal displacement. The horizontal plane at the baseof the model, which is a plane of symmetry, is alsofree of shear stresses and subject to zero vertical dis-placement. The model is loaded at the top with a uni-form vertical stress of 11 MPa to give a total overburdenpressure of 17 MPa at the coal seam level. As seen inFigure 4, the element size in the chain pillars withinthe fine meshed central region is 1 m � 3 m � 0.5 min the x, y and z-directions, respectively.

2.1 Determination of material properties

In addition to the peak cohesion, friction angle, anddilation angle in the MC model, the MCSS modelalso requires parameters describing the rate of cohesion

234

1000 m

240 m

240 m

1000 m

240 m

240 m

Figure 2. The FLAC3D block model developed for longwallmining simulations.

Figure 3. Bottom view of the FLAC3D block model showingthe fine mesh at the central area.

1.5 m6.5 m

6.5 m

8 m

26 m

1.5 m6.5 m

6.5 m

8 m

26 m

Figure 4. The entry system dimensions.

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and/or friction drop as a function of plastic strain in the post-peak region. The determination of the MCand MCSS parameters for a rock mass is a difficulttask, but can be carried out empirically by performingback-analyses. In this study, the parameter determina-tion is based on the two most commonly used empiricalpillar strength formulas given by Salamon (1967) andBieniawski (1984).

A FLAC3D model of a single test pillar was developedto establish the most suitable combination of coalMCSS parameters for replicating pillar strength val-ues based on empirical formulas. Figure 5 shows theFLAC3D model of the test pillar in a room and pillarenvironment. By considering symmetry conditions, onequarter of the pillar is modeled. The vertical walls ofthe model are set as frictionless by fixing the normaldisplacements on them, except for pillar sides whenthey are formed. The model is loaded along the topboundary using a constant displacement of 2 �10�7mper FLAC step.

The floor material is modeled as an elastic layerhaving a 20 GPa elastic modulus. The MC interfacebetween the pillar and floor has strength parametersof 0.5 MPa cohesion and friction angle of 23 degrees.For all pillar test simulations, the friction and dila-tion angles are held constant at 30 and 15 degrees,respectively.

Four pillar width-to-height (w/h) ratios (1, 2, 3,and 4) were modeled. For each w/h ratio, the numericalmodel was run with different combinations of a peakcohesion and cohesion drop rate.

The strengths established from the test pillar modelsare plotted against the empirical pillar strength for-mulas in Figure 6 for the cohesion drop rates of 35,50, and 100 MPa per plastic strain (�p) increment.Based on the trends of these plots, a peak cohesion

of 2.2 MPa and cohesion drop rate of 50 MPa/�p isconsidered suitable for modeling yielding of thechain pillars.

The test pillar models were repeated using the MCfailure criterion with the same peak cohesion, friction

235

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Figure 6. Model pillar strength versus empirical pillarstrength at cohesion drop rates of 35,50,100 MPa/�p(Strength formulas: Salamon: 9(w0.46/h0.66), Bieniawski:9(0.64 0.36 w/h) in MPa; assuming a coal cubic strengthvalue of 9 MPa).

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and dilation angle values as for the MCSS model. Byaveraging vertical stress and the vertical deformationhistories across the top of the pillar, an overallstress–strain curve for an individual pillar could beobtained. Figure 7 shows such curves for pillar w/hratios of 1, 2 and 3, using MC and MCSS criteria.

The difference in pillar response is obvious; MCdoes not allow the true softening (no peak strength andno strength drop) and pillars maintain high residualstrengths. On the other hand, MCSS models yield andreach much lower residual strengths. The pillar strengthvalues, corresponding to both MC and MCSS materi-als, are plotted against the empirical pillar strengthformulas of Salamon (1967) and Bieniawski (1984)

in Figure 8. The MC model strengths tend to increaserapidly while MCSS model strengths follow the empir-ical strength trends, indicating that MCSS models givemore realistic pillar stress–deformation curves thanMC models.

2.2 Gob compaction

The gob compaction process is an essential part of thelongwalling process since it can alter the pillar andabutment loads by acting as an additional support forthe system. The gob behavior is based on the following“compaction” model: vertical stress (�v) in the gobincreases with increasing vertical strain (�v) accordingto the relationship given by Salamon (1990),

(1)

where “a” is gob initial deformation modulus; and “b”is the limiting vertical strain. Based on studies carriedout at the USBM on gob behavior, the values for theconstants were taken as a � 3.5 MPa and b � 0.5(Deno & Mark 1993).

Two different algorithms are considered for imple-mentation of the gob behavior of Equation 1 in theFLAC3D model. In the first algorithm, referred to asthe “nodal force”, the compaction load is modeled asthe sum of vertical forces applied at the grid points ofthe roof elements in the back area after mining. Aftereach mining step, the vertical strain in a particularzone within the gob area is used to calculate the verti-cal stress according to Equation 1. Grid reaction forcesare then calculated by multiplying vertical stress by thecorresponding area of the roof element. In the secondmethod, the gob is modeled as a non-linear elasticmaterial. Its bulk modulus is continually increased asfunction of vertical strain within the gob area. Thealgorithm for this “modulus updating” method uses thebulk modulus K for each gob element:

(2)

where �z is the vertical strain in the element (Badr et al. 2002).

Implementation of these two methods makes use ofthe “linked list” concept in FLAC3D. The nodes (orzones) that will be replaced by gob material are definedby their addresses in a particular linked list. Then, usingthe FLAC3D programming language “FISH”, a functionupdates the forces (or bulk modulus) of each node (orzone) using Equation 1 or 2. After each mining step, thealgorithm is executed in 50 step intervals until themodel is brought to equilibrium (Badr 2003).

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1 / MCSS

2 / MCSS

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3 / MC

Figure 7. The vertical stress–strain curves of MC andMCSS pillars.

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MC models peak strength could not be defined beyond this point

874321

Figure 8. Pillar strength determination from numericalmodeling and empirical formulas (refer to Figure 6 forempirical strength formulas).

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The gob compaction curves for the analytic solu-tion (Salamon 1990) and the two FLAC3D algorithmsare compared in Figure 9. As shown, both nodal forceand modulus updating algorithms compare well withthe analytical model. Since the nodal force algorithmrequires longer running time, the modulus updatingmethod was embraced as the gob model for theFLAC3D longwall simulations performed in this study.

3 RESULTS

Figure 10 defines the MCSS material parameters usedin the model, which are also summarized in Table 1.For the coal seam, these parameters correspond to anMCSS material having a cubic strength of about9 MPa, friction angle of 30 degrees, and cohesiondrop rate of about 50 MPa/�p.

The model of the longwall layout described inSection 2 is brought to equilibrium elastically to hori-zontal and vertical virgin stress conditions of 17 MPa at the coal seam level. The elastic coal seam is thenreplaced by a MCSS material prior to development. Theentries are developed with the right entry leading theleft entry by 9 m. The entries advance by 3 m in eachmining step. A cross-cut is then mined when the trailingentry is 9 m ahead. Mining of the longwalls is carriedout starting at the right panel. The longwall advancesinitially in steps of 50 m and then the steps are reducedto 10 m in the fine-meshed central region of the model.After each longwall advance the area behind the long-wall face is changed to “gob material” and the modelis brought to equilibrium. The pillar response to miningis monitored using a FISH algorithm. The algorithmkeeps a record of the vertical stress and vertical strainhistories of all zones comprising the top of the pillar,

and then averages these values to produce an averagevertical pillar stress–strain curve.

Figure 11 shows a typical pillar stress–strain curveobtained from the FLAC3D simulation. The verticaldashed line on the left shows the pillar loading at theend of entry development.

At this stage, the pillar is at or close to its peakcapacity. The pre-peak stress drops indicate sidewallfailures experienced by the pillar during entry devel-opment. As the longwall approaches, the pillar initiallysheds load slowly and subsequently rapidly, eventuallyreaching eight per cent compression. At its residualstrength, the pillar carries a vertical stress of 4 MPa,

237

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40% closure

Gob

str

ess

(Mpa

)

Analytical solutionNodel force methodModulus updating method

50

Figure 9. The gob stress-closure results from the analyticalsolution and two FLAC3D algorithms.

0

0.5

1

1.5

2

2.5

0 0.02 0.04 0.06

Plastic strain

Coh

esio

n (M

Pa)

0

5

10

15

20

25

30

35

40

45

50

Fric

tion

and

dila

tion

(deg

rees

)

CohesionFrictionDilation

Figure 10. MCSS parameters used for modeling of thecoal material.

Table 1. Material properties used in longwall simulations.

Property Values

MiscellaneousSeam depth 680 mStress gradient 0.025 MPa/m�x, �y and �z 17 MPa

Coal propertiesCoal elastic modulus 3 GPaCoal Poisson’s ratio 0.25Coal strength 7.6 MPaCoal density 1313 Kg/m3

Roof propertiesElastic modulus 20 GPaPoisson’s ratio 0.25Density 2500 Kg/m3

Interface propertiesType Mohr–CoulombCohesion 0.5 MPaFriction angle 20°

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which is considered sufficient for supporting the roofin two entry systems.

The pillar strength in the longwall model is morethan that estimated by the test pillar model and empir-ical strength formulas. Further refinement of thestrength parameters could be achieved by iterating onthe contact and coal seam properties through parametricstudies, which would involve six independent variables,not including parameters for the roof material. As wasthe case with the test pillar model, this iterative processwould likely provide more than one set of parame-ters giving strength values similar to those predictedby the empirical strength formulas. Further studies inthis area are needed to fine-tune the optimum param-eter combination.

Figure 12 shows the gob compaction as mining pro-gresses, referenced to a point at the center of the firstpanel. After mining of the first panel, the verticalstress in the gob is 1.8 MPa. The gob stress increasesto the virgin stress level of 17 MPa after the secondpanel is mined.

The results from the longwall model are comparedto in-situ measurements using borehole pressure cells(BPCs) from a mine with similar conditions (Schissler2002). The FLAC3D model shows that the pillar hard-ens to 22 MPa while the in-situ pillar monitoring showed16 MPa during entry development. This difference isprobably partly due to the selection of the modelparameters as discussed above, and partly due to theinstallation sequence of the BPCs, which occurredafter the pillar was developed, and thus did not com-pletely capture the side wall loading by the approach-ing development faces. When the pillar yielded in themodel, the longwall face was approximately 150 mfrom the pillar centerline. Although there is no in-situload measurement available in pillars under similar

situations, the authors’ observations of intense pillarscaling in similar face positions in deep coal minessupport the finding of the model.

4 CONCLUSIONS

A three dimensional model of a coal longwall mine isdeveloped using FLAC3D. The model incorporates min-ing stages, softening behavior of the coal seam, and gobcompaction in the mined out area. The model resultsindicate that FLAC3D is a suitable tool to aid in thedesign, evaluation, and performance assessments forcomplex longwall layouts.

The test pillar studies show that the Mohr–CoulombStrain Softening model is more realistic than the tradi-tional Mohr–Coulomb constitutive law for estimatingthe strength and post peak behavior of coal pillars.

The strain softening parameters developed in thisstudy could be used as a starting point for modeling ofcoal seams. However, due to more than one combinationof strength parameters giving the same rock massstrength value and also mesh size dependency of theprogram, it is advised that the strength parameters fora particular coal seam be developed on a case bases,using a back-analysis process similar to that describedin the paper.

ACKNOWLEDGMENT

This publication was supported by CooperativeAgreement number U60/CCU816929-02 from theDepartment of Health and Human Services, the Centerfor Disease Control and Prevention (CDC). Its contentsare solely the responsibility of the authors and do notnecessarily represent the official views of the Department

238

0

5

10

15

20

25

0 10 20 30 40 50

% CLOSURE

Gob

Str

ess

(MP

a) 17

After firstlongwall

After second longwall

Figure 12. Vertical stress and closure induced at a point inthe gob.

0

2

4

6

8

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22

24

0 0.02 0.04 0.06 0.08

Strain

Ave

rage

pill

ar s

tres

s (M

Pa) Average Pillar Stress (MPa)

LongwallingDev.

Figure 11. Complete average vertical stress–strain curveof the yielding chain pillar in modeled longwall layout.

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of Health and Human Services, CDC. Support providedby Department of Health and Human Services, CDC, isgreatly acknowledged. The work presented is part of theHealth and Safety research activities currently carriedout at Western Mining Resource Center (WMRC) at theColorado School of Mines.

REFERENCES

Badr, S.A., Schissler, A., Salamon, M.D.G. & Ozbay, U.2002. Numerical Modeling of Yielding Chain Pillars inLongwall Mines. Proc. of the 5th North American RockMechanics Symposium, Toronto, Canada, pp 99–107.

Badr, S.A. 2003. Numerical Analysis of coal yield pillars atdeep longwall mines. Ph.D. Thesis in preparation.Department of Mining Engineering, Colorado School ofMines, Golden, Colorado (To be submitted.).

Bieniawski, Z.T. 1984. Rock Mechanics Design in Miningand Tunneling. A.A. Balkema, p. 1–272.

Deno, M.P. & Mark, C. 1993. Behavior of Simulated LongwallGob material. United States Department of the Interior,Bureau of mines, Report of investigation No. 9458.

Itasca Consulting Group, Inc. 2002. FLAC3D – FastLagrangian Analysis of Continua in Three Dimensions,Ver. 2.1. Minnesota: Itasca.

Salamon, M.D.G. 1990. Mechanism of caving in longwall coalmining. Paper in Rock Mechanics Contributions andChallenges Proceedings of the 31st US Symposium, Ed.W. Hustrulid and G. A. Johnson. Denver, Colorado, June18–20, 1990. A.A. Balkema, 1990, p. 161–168.

Salamon, M.D.G. 1967. A study of the strength of coal pillars.Journal of South Africa Institute of Mining and Metallurgy,v. 68, p. 55–67.

Schissler, A. 2002. Yield pillar design in non-homogenousand isotropic stress fields for soft minerals. Ph.D. Thesis.Department of Mining Engineering, Colorado School ofMines, Golden, Colorado.

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241

FISH functions for FLAC3D analyses of irregular narrow vein mining

H. Zhu & P.P. AndrieuxItasca Consulting Canada Inc., Sudbury, Ontario, Canada

ABSTRACT: FISH functions were developed to generate FLAC3D elements and to accurately present numer-ical simulation results for underground mining situations whereby the ore lenses have complex geometries anderratic distributions. The element-generating FISH functions for FLAC3D proved flexible and provided signifi-cant timesavings during the model construction stage. Furthermore, these functions made it easier to modify a model, to achieve a high degree of resolution in the domains of interest and to create a minimum number ofelements in order to minimize the computational power required to run the model.

Long, narrow and winding ore bodies with complex geometries do not lend themselves well to the represen-tation on longitudinal sections of the modeling results, mainly because the rendering planes wander in and outof the ore body. Such ore bodies are however often visualized and managed based on their longitudinal appear-ance, which is typically projected and simplified on an idealized plane. This difficulty of longitudinally show-ing the predicted stresses and displacements within the ore lens can impair the full and clear understanding ofthe modeling results. This paper describes two means of solving this problem based on FISH functions. TheFISH functions presented in this paper have been applied to, and validated by, a FLAC3D modeling exercise carried out at the Falconbridge Thayer Lindsley Mine near Sudbury, Ontario, Canada.

1 GENERAL INSTRUCTIONS

There are essentially two ways to build a FLAC3D

numerical model: one is to generate regular elementsover the entire domain and then structure the desiredgeometry and geology around them, the other is to setFLAC-provided blocks for specific objects to simulateand to assemble these blocks into the model. In under-ground mining numerical modeling applications thefirst approach is generally used because there is usuallyno need to account for topographically irregular groundsurfaces or very complex and precise excavations, as isoften the case in civil engineering applications. Thismethod, although quite versatile, can however result inthe creation of a large number of elements in order toachieve the desired degree of resolution, particularlywhen the geometry of the ore lenses is complex, orwhen multiple independent ore lenses are present. Alarge number of elements can, in turn, result in exces-sively long running times and even prevent a modelfrom running if the computer platform is insufficientlypowerful. In such cases, the second strategy may not beadequate either, due to the irregularity of the geometryof the ore lenses. Furthermore, it is usually more time-consuming to build a model using the second approach.

Narrow and undulating ore lenses also make it dif-ficult to represent the simulation results on longitudi-nal views. The existing FLAC3D “plot” command canprove inadequate to illustrate load and deformationresults because longitudinal sections generated throughthe approximate center of a given narrow and undu-lating ore lens typically wanders in and out of it. Thismade it difficult to visualize the stress redistributionand deformation everywhere within the ore lens itself.

Two approaches can be used to solve this problem.One consists of extracting from save files the stressand deformation data at each point along a curvedsurface centered in the middle of the undulating nar-row body of interest, and to generate iso-contour plotswith specialized software (such as Goldsoft Surfer®,for example). This approach has the advantage ofallowing the user to extract and plot any desiredparameter or criterion, such as factors of safety orcustom-defined stress ratios, and to clearly representtheir variation. Another way is to define a thin centralzone in the middle of the undulating narrow body as aFLAC3D Group or a FLAC3D Range, which can subse-quently be used to represent a true longitudinal sec-tion. This approach allows use of existing FLAC3D

commands and functions to generate the plots. Both

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methods require FISH functions that are quite similar.The first approach is however more flexible and moreprecise, but requires additional third-party software.

2 GENERATION OF THE ELEMENTS FOR AFLAC3D MODEL

As shown in Figure 1, the geometry of the #2 Zoneore lens at the Falconbridge Thayer Lindsley (T.L.)Mine near Sudbury, Ontario, Canada, is quite compli-cated and resulted in difficulties being encounteredwhen constructing a representative FLAC3D model. Aseries of FISH functions was therefore developed togenerate the elements throughout the entire model.Also, the narrow and undulating geometry of this lensmade it difficult to represent the simulation results ona true longitudinal section, i.e. on a longitudinal sec-tion that did not wander in and out of the ore material.

The T.L. ore body is up to 500 meters in strike(East–West) and occurs as several distinct lensesbelow Level 13-2 as shown in Figure 1. The narrow-est lens width encountered is approximately 5 meters.The maximum width is of the order of 20 meters. A400 m-high, 240 m-thick (in the North–South direc-tion) and 300 m-long (in the East–West direction)section of the mine, centered on the #2 Zone, repre-sented the region of interest for the FLAC3D simu-lations. A resolution of 5 meters in the East–Westdirection, by 2 meters in the North–South direction,by 5 meters in the vertical direction was considered aminimum requirement within the domain of interest.Such a resolution would require as many as 960,000elements for the inner domain, and close to 1.2 mil-lion elements for the entire model. This would make italmost impossible to run the model on even the mostpowerful personal computers currently available.

From a geomechanics perspective it is not neces-sary to generate elements with the same resolutionwithin the entire internal region of a model, as illus-trated in Figure 2. As a result, this internal model canbe divided into several blocks with different elementresolutions, in order to end up with a reasonably sizedmodel. This rationale lead to the development of a FISHfunctions-based approach for the construction of theFLAC3D model, which would be broadly applicable tomany other geometries.

For the T.L. Mine analyses, four different elementresolutions were used, as shown in Figure 2. Block #1,which contained the #2 Zone of interest, was assignedthe finest resolution of 5 meters in the East–Westdirection, by 2 meters in the North–South direction,by 5 meters in the vertical direction. Block #2, whichhad a different panel height, was assigned a coarser10 m � 2 m � 10 m resolution. Block #3, which cov-ers the satellite ore lenses and previously mined-outvoids, was assigned a yet coarser 10 m � 4 m � 10 m

resolution. Finally, Block #4, which encompassed therest of the internal model, was fitted with the coars-est 20 m � 8 m �20 m resolution. As a result, thenumber of elements inside the internal modeldropped from 960,000 to 248,500, which allowed the model to be run on a personal (albeit upper end)computer.

The FISH function developed and used to generatethe elements in the five blocks within the internalmodel, as well as in the outer model, is shown inAppendix I.

This FISH function can be used to modify themodel and to create other models for situations with agenerally similar geometry – different values simplyhave to be assigned to the variables. The function as itstands allows the user to break the internal model intoup to ten blocks – this maximum number of blockscan also be customized if the user is familiar with theFISH language. A separate file was prepared to invokeeach sub-routine in the function in order to assign initialvalues to the parameters.

242

Ore lens of#2 Zone

Ore lens of#3 Zone

Ore lens of#4 Zone

(a) Plan view of Level 11-0.

(c) Plan view fo Level 13-1.(b) Cross-section looking East, showingthe vertical extent of the ore bodies.

Figure 1. Various views of the Thayer Lindsley Mine orelenses showing the relatively narrow and winding #2 Zone,which was the ore lens of interest in the numerical exercise.The figure also shows the satellite lenses of the #3 and#4 zones, which had to be considered in the model. (Thefigures are at different scales.)

Boundaryblock (outermodel)

Internal model

Coordinates at point O:dxyz (i, j)Index j (1, 2, and 3) cor-responding to x, y, and z.

Illustrationof the initialparametersof a block

Coordinates at the point O: dxyz(i,J)Index j (1,2,and3) corresponds to x,y, and z.

dxyz

(1,3

)

dayz(i,1)

Figure 2. Schematic sketch showing the model structureand block parameters.

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3 CREATION OF A NON-PLANARLONGITUDINAL SECTION THATFOLLOWS THE CENTRE OF A NARROWAND UNDULATING ZONE

Longitudinal sections are well-suited to, and widelyused for, the planning and sequencing of full-widthopen stopes in narrow ore lenses that are extracted bymeans of retreating mining methods. When longitudi-nal sections are used for planning and sequencing it isadvantageous to also use them to display the stressand deformation results from numerical analyses. Notonly do they represent a viewpoint familiar to the minepersonnel, they also clearly show how stress is redis-tributed in workings ahead of the mining front as theextraction sequence progresses. Such longitudinal sec-tion views are being widely used at T.L. Mine. FLAC3D

results would ideally have been shown on them. Thedifficulty, as described previously, is that – as is the caseat T.L. Mine – it is impossible when the ore lens is nar-row and undulates over an amplitude greater than itswidth for a true (planar) longitudinal section to remainentirely within it. (In general under these circumstances,the longitudinal sections used for planning purposes arecomposite simplified views, not true sections.) As aresult, FISH functions had to be developed to createstress and displacement plots along the geometricalcenter of the lens. The approach is illustrated in Figure 3.

The procedure can be summarized as follows:

1. select the groups that encase the area to be examined;2. search the footwall and handing wall boundaries by

element along the strike of the ore lens;3. identify the ID of the elements where the bound-

aries are;4. calculate the coordinates of the point between the

two boundaries. This point needs to keep thedesired hanging wall-to-footwall distance ratio tothese boundaries;

5. if this point lies in an edge element along the strikedirection, extend the extraction further outwards

(into the surround rock mass) by a predetermineddistance (represented by the lines of A—A, B—B,and C—C in Fig. 3);

6. trace the element ID to which this point belongs;7. output the element stresses to a file for further analy-

sis, or name this element in a new Range/Group; and,8. repeat the procedure for each level.

A continuous surface entirely comprised withinthe ore lens and following its center will be obtainedby connecting all the points generated in this manner.The desired longitudinal section can thus be con-structed either by projecting this surface onto a longi-tudinal plan, or by ignoring the y coordinate (as wasdone in the case of T.L. Mine).

The FISH function developed to generate a longi-tudinal section that follows the center of a narrow andundulating zone is shown in Appendix II. The optionof naming a new Group is recommended in order toavoid the need for external software packages, such asSurfer™ for example, to present the FLAC3D results.However, the alternate approach of extracting the ele-ment stresses from the middle of the ore zone wasemployed for the T.L. Mine, due to its higher precisionand the need in this particular case to examine a user-defined stress ratio.

4 FURTHER APPLICATION OF CUSTOMISEDFISH FUNCTIONS AT THE FALCONBRIDGETHAYER LINDSLEY MINE

A user-defined stress ratio was also used at T.L. toevaluate the state of the rock mass throughout the #2Zone of interest. The objective of this work was to notonly assess which elements had started to undergofailure, but also determine how far elastic state ele-ments were from. The stress ratio retained is definedas �1/�1, and is illustrated Figure 4.

243

Hanging wall

Footwall

Element

A A

BB

C C

Figure 3. Sketch illustrating the concept of constructing anon-planar longitudinal section from an arbitrary narrowand undulating ore lens.

�3 �1

τ

�1' �

Strength envelope

Figure 4. Illustration of the user-defined stress ratio where�1 and �3 are the major and minor principal stress of rockmass, respectively, at a given point as computed by FLAC3D.�1

is the major principal stress at this point assuming that therock mass is undergoing yielding under the same confine-ment conditions.

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244

Figure 5. Longitudinal section looking North illustration of the Surfer™-generated contours of a user-defined stress ratiobased on the customized FISH functions described in the previous sections.

Figure 5 shows a simplified composite longitudinalsection of the T.L. Mine #2 Zone. The small squaresalong the edge of the excavations represent the stateof the local rock mass at a given mining step, asinferred from the FLAC3D results. The overlaid contoursof the user-defined stress ratio were generated basedon the stresses at each element extracted with the FISHfunction mentioned previously. In the areas where therock mass is still in its elastic state, which are beyondthe small squares as shown in Figure 5, the low valuesof the stress ratio refer to a low likelihood that stress-induced problems will arise at this mining stage. Withthe help of the FISH function, the state of the rockmass can be illustrated quite precisely.

In Figure 5, all the numerical elements from whichthe stresses were extracted are those located the clos-est to the middle of the #2 Zone ore lens at T.L. Mine,or those extending into the surrounding rock massaway from the east and west boundaries of the lens.These contours present quite a bit of informationabout the state of the rock mass, both in the main lensand the surrounding rock. Furthermore, how far awaythe elastic state rock is from the onset of yielding canbe readily estimated and displayed by the contoursgenerated by the customized FISH functions.Currently, this cannot be achieved with any built-inFLAC command.

5 CONCLUSIONS

User-defined FISH functions can be a powerful toolto solve various FLAC or FLAC3D numerical modelingproblems. The FISH function presented in this paperfor the generation of regular elements is applicable tomany scenarios where similar geometrical issues arepresent. The other FISH function discussed in thispaper is a good example of how experienced FLAC

and FLAC3D users can develop very specific functionsto solve specific problems.

ACKNOWLEDGEMENTS

The authors would like to thank Scott Carlisle1 forreviewing, and Falconbridge Limited for grantingpermission to publish this paper and for the use ofThayer Lindsley data.

REFERENCE

Itasca Consulting Group, Inc. (1997) FLAC3D – FastLagrangian Analysis of Continua in 3 Dimensions,Version 2.0. Minneapolis, MN: Itasca.

APPENDIX I – FISH FUNCTION FOR THEGENERATION OF THE ELEMENTS OF AFLAC3D MODEL

; Define blocks in the internal region of the model; num_box�10 blocks currently limiteddef Ore_boxarray xyz(10,3),dxyz(10,3),p_xyz(10,3)

loop i (1,num_box)P0_x�xyz(i,1)P0_y�xyz(i,2)P0_z�xyz(i,3)

p_xyz(1,1)�p0_xp_xyz(1,2)�p0_yp_xyz(1,3)�p0_z

1Falconbridge Limited, Sudbury Mines/Mill Business Unit –Mining Services, Onaping, Ontario, P0M 2R0, Canada.

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P_xyz(2,1)�p0_xdxyz(i,1)P_xyz(2,2)�p0_yP_xyz(2,3)�p0_zP_xyz(3,1)�p0_xP_xyz(3,2)�p0_ydxyz(i,2)P_xyz(3,3)�p0_zP_xyz(4,1)�p0_xP_xyz(4,2)�p0_yP_xyz(4,3)�p0_zdxyz(i,3)P_xyz(5,1)�p0_xdxyz(i,1)P_xyz(5,2)�p0_ydxyz(i,2)P_xyz(5,3)�p0_zP_xyz(6,1)�p0_xP_xyz(6,2)�p0_ydxyz(i,2)P_xyz(6,3)�p0_zdxyz(i,3)P_xyz(7,1)�p0_xdxyz(i,1)P_xyz(7,2)�p0_yP_xyz(7,3)�p0_zdxyz(i,3)P_xyz(8,1)�p0_xdxyz(i,1)P_xyz(8,2)�p0_ydxyz(i,2)P_xyz(8,3)�p0_zdxyz(i,3)

loop n (1,8)id_p�n8*(i-1)P_x�p_xyz(n,1)P_y�P_xyz(n,2)P_z�P_xyz(n,3)

commandgen po id id_p p_x p_y p_zend_command

end_loopend_loop

end

; Building boundary blocksdef right_box; calculate the dimensions of this box in x, y and z; xyz_ratio � grid length ratio; n_grid � number of grids; x_1, y_1 and z_1 � element length along; corresponding directions in boundary blocks; x0, y0 and z0 � coordinates of point O of the; internal model as shown in Figure 2.

x_len�x_1*(1-xyz_ratio^n_grid)/(1-xyz_ratio)y_len�y_1*(1-xyz_ratio^n_grid)/(1-xyz_ratio)z_len�z_1*(1-xyz_ratio^n_grid)/(1-xyz_ratio)

P0_x�x0dxP0_y�y0P0_z�z0P1_x�x0dxx_lenP1_y�y0-y_lenP1_z�z0-z_lenP2_x�x0dxP2_y�y0dy0P2_z�z0P3_x�x0dxP3_y�y0P3_z�z0dz

P4_x�x0dxx_lenP4_y�y0dy0y_lenP4_z�z0-z_lenP5_x�x0dxP5_y�y0dy0P5_z�z0dzP6_x�x0dxx_lenP6_y�y0-y_lenP6_z�z0dzz_lenP7_x�x0dxx_lenP7_y�y0dy0y_lenP7_z�z0dzz_lenY_S_boun � p1_yZ_B_boun � p1_zcommandgen po id 1011 p0_x p0_y p0_zgen po id 1012 p1_x p1_y p1_zgen po id 1013 p2_x p2_y p2_zgen po id 1014 p3_x p3_y p3_zgen po id 1015 p4_x p4_y p4_zgen po id 1016 p5_x p5_y p5_zgen po id 1017 p6_x p6_y p6_zgen po id 1018 p7_x p7_y p7_zend_commandend

def Back_boxP0_x�x0P0_y�y0dy0P0_z�z0P1_x�x0dxP1_y�y0dy0P1_z�z0P2_x�x0-x_lenP2_y�y0dy0y_lenP2_z�z0-z_lenP3_x�x0P3_y�y0dy0P3_z�z0dzP4_x�x0dxx_lenP4_y�y0dy0y_lenP4_z�z0-z_lenP5_x�x0-x_lenP5_y�y0dy0y_lenP5_z�z0dzz_lenP6_x�x0dxP6_y�y0dy0P6_z�z0dzP7_x�x0dxx_lenP7_y�y0dy0y_lenP7_z�z0dzz_lenX_E_boun�p7_xY_N_boun�p7_yZ_T_boun�p7_zcommandgen po id 1021 p0_x p0_y p0_zgen po id 1022 p1_x p1_y p1_zgen po id 1023 p2_x p2_y p2_z

245

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gen po id 1024 p3_x p3_y p3_zgen po id 1025 p4_x p4_y p4_zgen po id 1026 p5_x p5_y p5_zgen po id 1027 p6_x p6_y p6_zgen po id 1028 p7_x p7_y p7_zend_commandend

def Top_boxP0_x�x0P0_y�y0P0_z�z0dzP1_x�x0dxP1_y�y0P1_z�z0dzP2_x�x0P2_y�y0dy0P2_z�z0dzP3_x�x0-x_lenP3_y�y0-y_lenP3_z�z0dzz_lenP4_x�x0dxP4_y�y0dy0P4_z�z0dzP5_x�x0-x_lenP5_y�y0dy0y_lenP5_z�z0dzz_lenP6_x�x0dxx_lenP6_y�y0-y_lenP6_z�z0dzz_lenP7_x�x0dxx_lenP7_y�y0dy0y_lenP7_z�z0dzz_lenX_W_boun�p3_xcommandgen po id 1031 p0_x p0_y p0_zgen po id 1032 p1_x p1_y p1_zgen po id 1033 p2_x p2_y p2_zgen po id 1034 p3_x p3_y p3_zgen po id 1035 p4_x p4_y p4_zgen po id 1036 p5_x p5_y p5_zgen po id 1037 p6_x p6_y p6_zgen po id 1038 p7_x p7_y p7_zend_commandend

; generate elements in the internal regions of the; modeldef gen_elearray Len_xyz(10,3); reference point for reflected boundary boxes

x_ref�x0dx/2.0y_ref�y0dy0/2.0z_ref�z0dz/2.0num_x1�dx/x_1num_y1�dy0/y_1num_z1�dz/z_1

loop n (1,num_box)

id_p1�18*(n-1)id_p2�28*(n-1)id_p3�38*(n-1)id_p4�48*(n-1)id_p5�58*(n-1)id_p6�68*(n-1)id_p7�78*(n-1)id_p8�88*(n-1)num_x�dxyz(n,1)/Len_xyz(n,1)num_y�dxyz(n,2)/Len_xyz(n,2)num_z�dxyz(n,3)/Len_xyz(n,3)commandgen zone bri &p0 po id_p1 p1 po id_p2 p2 po id_p3 p3 po id_p4 &p4 po id_p5 p5 po id_p6 p6 po id_p7 p7 po id_p8 &size num_x num_y num_z &group waste &rat 1 1 1end_commandend_loop

; generate elements in boundary blocks; generate elements in one boundary block and then; reflect to the opposite side

command; Left-hand and right-hand sidesgen z brick &p0 po 1011 p1 po 1012 p2 po 1013 p3 po 1014 &p4 po 1015 p5 po 1016 p6 po 1017 p7 po 1018 &size n_grid num_y1 num_z1 &group right_side &ratio xyz_ratio 1 1gen zone reflect dip 90 dd 90 ori x_ref y_ref z_ref &range group right_side

; Back and frontgen z brick &p0 po 1021 p1 po 1022 p2 po 1023 p3 po 1024 &p4 po 1025 p5 po 1026 p6 po 1027 p7 po 1028 &size num_x1 n_grid num_z1 &group Back_side &ratio 1.0 xyz_ratio 1gen zone reflect dip 90 dd 180 ori x_ref y_ref z_ref &range group back_side

; Top and bottomgen z brick &p0 po 1031 p1 po 1032 p2 po 1033 p3 po 1034 &p4 po 1035 p5 po 1036 p6 po 1037 p7 po 1038 &size num_x1 num_y1 n_grid &group Top_side &ratio 1.0 1.0 xyz_ratiogen zone reflect dip 0 dd 0 ori x_ref y_ref z_ref &range group top_sideend_commandcommand

group external range group waste notend_commandend

246

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APPENDIX II – FISH FUNCTION FOR THECREATION OF A LONGITUDINAL SECTIONIN THE CENTRE OF THE UNDULATING OREZONE

def get_profilex_0�x_west; x coordinates on west edge of area to be searchedy_0�y_southx_1�x_easty_1�y_north; element length in the x directionx_d�x_lengthy_d�y_length; group name of the main zone to be examinedname_g�group_name; referring to the next FISH function for z_t and z_bz_po�(z_tz_b)/2.0n_x�(x_1-x_0)/x_dn_y�(y_1-y_0)/y_dif flag_boun�0 then; top element. Extend search 25m upwardsz_1�z_t25.0z_0�z_bend_if; bottom elementif flag_boun�1 then; extend search 25m downwards to next levelz_1�z_tz_0�z_b-25.0end_if; either top or bottom levelif flag_boun�2 thenz_1�z_tz_0�z_bend_ifkkn�0loop n1 (1,n_x);start point in the x directionx_po�x_0x_d*(n1-0.5)

x_w�x_po-0.5*x_dx_e�x_po0.5*x_da_min�y_northa_max�y_southkn�0; searching FW and HW

loop n2 (1,n_y)y_po�y_0y_d*(n2-0.5)p_z�z_near(x_po,y_po,z_po)z_gr�z_group(p_z)if z_gr�name_g then

g_p1�z_gp(p_z,1)g_p2�z_gp(p_z,3)y_min�gp_ypos(g_p1)y_max�gp_ypos(g_p2)

if y_min��a_min thena_min�y_min

end_ifif y_max>�a_max then

a_max�y_maxkn�kn1

end_ifend_if

end_loop; main zone has been found

if kn # 0 thenkkn�kkn1x_kkn�x_ey_s�(a_min+a_max)/2.0-0.55*y_dy_n�(a_min+a_max)/2.00.55*y_d

end_if; on west edge of the main zone

if kkn�1 thenx_w�x_0

end_ifif kn # 0 thencommandgroup profile ra x x_w x_e y y_s y_n z z_0

z_1end_commandend_if; on the east edge of the main zone

if n1�n_x thencommandgroup profile ra x x_kkn x_1 y y_s y_n z z_0 z_1

end_commandend_if

end_loopend; search loop on z direction in the next FISH functiondef get_paraloop n (1,60)

if n��40 thenz_interval�5elsez_interval�10end_ifz_b�2310-z_interval*(n-1)z_t�z_bz_intervalflag_boun�2if n�1 then

flag_boun�0 ;top elementend_ifif n�10 then

flag_boun � 1 ;bottom elementend_if

get_profile ; invoke the above FISH functionend_loopend

247

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Soil structure interaction

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251

A calibrated FLAC model for geosynthetic reinforced soil modular block walls at end of construction

K. Hatami & R.J. BathurstGeoEngineering Centre at Queen’s-RMC, Royal Military College of Canada, Kingston, Ontario

T. AllenWashington State Department of Transportation, Washington, USA

ABSTRACT: The paper describes a FLAC numerical model that was developed to simulate the constructionand measured response of large-scale geosynthetic reinforced soil walls that were constructed at the RoyalMilitary College of Canada (RMC). The reinforced soil structures were constructed with three different poly-meric reinforcement configurations. The backfill strength properties and reinforcement material propertieswere determined from conventional laboratory tests. The soil elastic modulus values were back calculated fromsurcharge loading tests on the wall backfill. The numerical models were able to capture the observed differencesin wall behavior due to different reinforcement configurations to within the accuracy of the measurements.Reinforcement strain magnitudes and distribution were more accurately predicted using a stress-dependentmodel for the soil backfill compared to a linear elastic model.

1 BACKGROUND

A recent study by Allen et al. (2002) of the design,analysis and performance of instrumented full-scalegeosynthetic reinforced soil walls constructed in thefield has demonstrated that current design practice isexcessively conservative. For example, they showedthat most walls constructed to date could be expectedto perform satisfactorily with as little as 50% of the reinforcement that has been used in the past.Nevertheless, the number of instrumented field wallsreported in the literature is sparse and there is arequirement for better data and a wider range of casestudies in order to refine current design methodolo-gies that are based on conventional geotechnical limitequilibrium approaches. To fill this requirement, theGeoEngineering Centre at Queen’s-RMC at the RoyalMilitary College of Canada (RMC) has been engagedin the construction, surcharge loading and monitoringof carefully instrumented large-scale geosynthetic rein-forced soil retaining walls built within a controlledlaboratory environment (Bathurst et al. 2001). Thison-going research program has also been conceivedto generate high-quality and comprehensive data thatcan be used to calibrate advanced numerical modelsof geosynthetic reinforced soil walls. The calibratedmodels can then be used to extend the database of

physical tests to a wider range of reinforced soil walltypes and configurations. The combination of physi-cal and numerical test results can then be used tocheck or refine recently proposed analytical designmethods for geosynthetic reinforced soil wall struc-tures that hold promise to make these systems morecost effective (e.g. Allen et al. 2003).

This paper is focused on the second step in thislong-term research program, i.e. calibration of aFLAC numerical model for three recent large-scaletest walls at RMC that were constructed with apolypropylene geogrid reinforcement material. Thispaper extends the results of an earlier paper byHatami & Bathurst (2001) that was focused on aFLAC (Itasca 1998) numerical model for a single wallin the physical test program that was constructed usinga polyester geogrid reinforcement material. In the cur-rent paper, the numerical model is calibrated againstthe end-of-construction stage for each of the walls,which represents a working stress condition. This is theoperational condition that is of most interest to design-ers as opposed to an ultimate limit state or failurecondition. The paper reports details of the constitu-tive models used for the component materials in thewalls and compares selected measured and predictedresponses for the three walls including facing horizon-tal displacements, horizontal and vertical toe boundary

09069-30.qxd 08/11/2003 20:38 PM Page 251


reactions and reinforcement strain distributions. Thecomparisons are based on both linear elastic and non-linear hyperbolic models for the soil backfill.

2 PHYSICAL TEST MODELS

Figure 1 shows a front view of an RMC test wall witha modular block (segmental) facing. The walls were3.6 m high with a facing batter of 8° from the vertical.The first wall (Wall 1 – control) was built with sixlayers of weak polypropylene geogrid (PP) reinforce-ment placed at a vertical spacing of 0.6 m. The second

wall was a nominally identical structure except thatthe reinforcement stiffness and strength of the geogridwere reduced by 50% by removing every other longi-tudinal member in each layer. Wall 3 was nominallyidentical to Wall 1 except that only four reinforce-ment layers were used in the wall at a vertical spacingof 0.9 m. In each structure, the wall facing consistedof a column of discrete, dry-stacked, solid masonryconcrete blocks with continuous concrete shear keys.The wall facing was built with three discontinuousvertical sections with separate reinforcement layers inplan view. The width of the instrumented middle section was 1 m. The backfill was a clean uniformsize rounded beach sand (SP) with a flat compactioncurve. The sand was compacted to a unit weight of16.7 kN/m3 using a lightweight vibrating mechanicalplate compactor. The friction between the backfillsoil and sides of the test facility was minimized byplacing a composite arrangement of plywood, Plexiglasand lubricated polyethylene sheets over the sidewalls.The discontinuous wall arrangement and sidewalltreatment were used to minimize the frictional effectof the lateral boundaries of the test facility and tothereby approach, as far as practical, a plane-straintest condition for the instrumented middle section ofthe wall structure. The reinforcement layers were rigidlyattached to the facing using mechanical connectionsto simplify the interpretation of connection perform-ance (i.e. this arrangement prevented any possibilityof reinforcement slippage between the blocks).

Figure 2 illustrates the test configuration for Walls1 and 2 and the instrumentation that was used torecord wall response. The horizontal movement of thewall facing was measured using displacement poten-tiometers mounted at different elevations against the

252

Instrumented middle section of the wall

1.15 m 1.00 m 1.15 m

8°

24 r

ow

s o

f se

gm

enta

l blo

cks

Reinforcement layers

3.60

m

Figure 1. Large-scale instrumented geosynthetic rein-forced soil modular block retaining wall constructed in theRMC Retaining Wall Test Facility.

Facingpotentiometer

Connection loadrings

Horizontal toeload ring

Vertical toe loadcells

Facing blocks

Reinforcement layer

0.3 m

0.15 m

3.6 m

2.52 m

Strain gauge

Extensometer

1

2

3

4

5

6

Figure 2. Schematic instrumentation layout of the test walls used in calibrating the numerical model (Walls 1 and 2). Note:Wall 3 is constructed with four reinforcement layers with a vertical spacing Sv � 0.9 m.

09069-30.qxd 08/11/2003 20:38 PM Page 252


facing column. Horizontal toe loads were measuredusing load rings (a horizontal restrained toe boundarycondition). Vertical toe loads were measured usingload cells supporting a double row of steel plates,which were used in turn to seat the first course ofmodular block units. A set of steel rollers was locatedbetween the steel plates to de-couple the horizontaland vertical toe load reactions. Reinforcement strainsin the wall were measured using strain gauges thatwere bonded directly to the polypropylene geogridlongitudinal members and extensometers attached toselected geogrid junctions. Backfill settlements weremeasured using tell-tales and settlement plates.

Further details of the construction and monitoringtechniques used in the RMC test walls have beenreported by Bathurst et al. (2001).


3.1 General

The finite difference-based computer program FLAC(Itasca 1998) was used to simulate the response of thereinforced soil test walls up to the end of construc-tion. Figure 3 shows the numerical grid used for thesegmental retaining walls.

3.2 Material mechanical models and properties

3.2.1 SoilThe backfill in all simulations was modeled as a cohesionless granular soil with Mohr–Coulomb failure criterion and dilation angle. The backfill elastic response was simulated using two different

approaches: 1) a linear elastic (perfectly plastic)model, and 2) the stress-dependent hyperbolic modelproposed by Duncan et al. (1980; also see Itasca1998). Bathurst & Hatami (2001) and Hatami &Bathurst (2002) reviewed previous attempts reportedin the literature to numerically model the response ofreinforced soil structures. Their survey showed thatthe stress-dependent nonlinear elastic model (hyper-bolic model) proposed by Duncan et al. (1980), orvariants, was the most common constitutive modelused to simulate the backfill response during con-struction and under surcharge loading. However, nocomparisons have been reported for simulationsusing other models including a simple linear elasticmodel. The backfill material properties used in thecurrent study are reported in Table 1. The values ofsoil hyperbolic parameters were determined by

253

Sand backfill

Reinforcement

Concrete facing blocks0.3 m

24 x

0.1

5 =

3.6

m

0.6

m

3.6

m

° °

2.5 m

5.5 m

°

°

Interfaces

Figure 3. Numerical model of the segmental retaining walls (Walls 1 and 2). Note: Wall 3 is constructed with four reinforcement layers with a vertical spacing Sv � 0.9 m.

Table 1. Material properties for sand used in the numericalmodel.

Value

Stiffness properties (Hyperbolic model)Kc (elastic modulus number) 2000Kb (bulk modulus number) 2000n (elastic modulus exponent) 0.5m (bulk modulus exponents) 0.5Rf (failure ratio) 0.73� (range of permissible Poisson’s ratio values) 0–0.49

Strength properties� (peak friction angle) (deg) 44c (cohesion) (kPa) 0� (dilation angle) (deg) 11 (density) (Kg/m3) 1730

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adjusting initial values taken from the results of triax-ial compression tests on backfill sand specimens. Thevalue of Poisson’s ratio for each soil zone duringanalyses was determined from the calculated valuesof the soil elastic modulus and bulk modulus from thehyperbolic model and hence was allowed to varybetween values of 0 and 0.49 as noted in Table 1. Themodulus numbers Ke and Kb were increased to matchthe measured settlement response of the backfill inthe retained zone behind the reinforced soil zone dur-ing uniform surcharge loading. The backfill peakplane-strain friction angle value was taken as� � 44° (Bathurst et al. 2001). The backfill dilationangle value from direct shear tests was found to varyfrom � � 9° to 12° for the range of confining soilpressures in the test walls. The value � � 11° wasused in the numerical models.

3.2.2 ReinforcementThe reinforcement layers were modeled with two-noded elastic-plastic cable elements with a strain-dependent tensile stiffness, J(�), tensile yield strength,Ty and no compressive strength. The reinforcementload–strain response was modeled in parabolic form as:

(1)

where T is axial load and � is axial strain. This equa-tion is valid for � � 2.5% which captures the range ofin-situ measured strains that correspond to the end-of-construction working stress levels for the experi-mental walls and is well below the reinforcementstrain at yield. The strain-dependent, secant tensilestiffness of the reinforcement, Js (�), was calculatedfrom Equation 1 as:

(2)

Parameter A in Equation 2 is the initial stiffness mod-ulus and parameter B is the strain-softening coeffi-cient, which is a positive value for polypropylenereinforcement prior to yield. The stiffness of thepolypropylene geogrid reinforcement was determinedfrom the constant rate of strain tests on virgin geo-grid specimens tested in-isolation at a strain rate of0.01%/min. The reinforcement material propertiesused in numerical simulations are presented in Table 2.The structural nodes of the reinforcement cable ele-ments were rigidly attached to the gridpoints of thebackfill numerical mesh. This was done to ensurecompatibility of displacements between reinforce-ment structural nodes and backfill gridpoints. Withthis approach, the grout interface was not utilized in the numerical model. Therefore, pullout of the

reinforcement from the backfill was prevented, whichwas consistent with measurements recorded in thephysical tests.

3.2.3 InterfacesThe concrete facing units in the test walls were mod-eled as linear elastic continuum zones separated bynulled zones of zero thickness that contained inter-faces (Fig. 4). Table 3 summarizes the values for the

254

Table 2. Reinforcement stiffness and strength properties.

Polymer Number Stiffness Jt (�) TyWall type of layers (kN/m)* (kN/m)

W1 PP 6 138–1698 14W2 PP 6 69–845 7W3 PP 4 138–1698 14

* Equations valid for � � 2.5%

Nulled Zone(magnified)

BackfillNumericalGrid

Backfill(Continuum)Zones

Block-BlockInterfaces

Soil-BlockInterfaces

FacingModularBlocks

Two-nodedReinforcementElements

ConnectionBeamElements

Soil ColumnBehind Facing

Figure 4. Details of facing-backfill-reinforcement connection.

Table 3. Interface properties.

Value

Soil–Block�sb (friction angle) (deg) 44�sb (dilation angle)(deg) 11knsb (normal stiffness) (kN/m/m) 0.1 � 106

kssb (shear stiffness) (kN/m/m) 103

Block–Block�bb (friction angle) (deg) 57c (cohesion) (kPa) 45.7knbb (normal stiffness) (kN/m/m) 106

ksbb (shear stiffness) (MN/m/m) 50

09069-30.qxd 08/11/2003 20:38 PM Page 254


interface properties used in the wall simulations. Theinterfaces were modeled as a spring-slider systemwith constant strength and stiffness properties. Thevalues of interface normal (kn) and tangential (ks)stiffness parameters were chosen after a parametricanalysis to minimize computation time. The valuesreported in Table 3 were found to be smaller than thedefault values recommended in the FLAC manual(Itasca 1998) that were used as the starting point inthe parametric analysis. The magnitude of the normalinterface stiffness value was made as large as possibleto avoid the intrusion of adjacent zones but not tocause excessive computation time. The wall deforma-tion response was found to be relatively insensitive tothe value of inter-block shear stiffness for ksbb 50 MN/m/m. Smaller values of ksbb together withmaterial properties reported in Tables 1 to 3 wereshown to over-predict measured wall deformationresults. The value ksbb � 50 MN/m/m gave the bestoverall agreement with the measured data. This valueis also within the range of shear stiffness values back-calculated from load-displacement results of labora-tory interface shear tests on the block units (Hatamiet al. 2002). The interface shear strength was modeledwith the Mohr–Coulomb failure criterion defined by interface cohesion and friction angle. The block–block interface peak friction angle and equivalentcohesion values were determined from the laboratoryinterface shear tests as �bb � 57° and cbb � 45.7 kPa,respectively (Hatami and Bathurst 2001). The inter-face friction angle, �sb, between the backfill and fac-ing blocks was back-calculated from measured toereactions and the sum of measured connection forcesusing the facing equilibrium analysis described byHatami and Bathurst (2001). Their analyses demon-strated that the soil-facing interface friction anglevalue in the test walls was close to the magnitude ofthe backfill peak plane-strain friction angle (i.e. �sb �� 44°).

3.2.4 Construction and boundary conditionsFixed boundary conditions in horizontal and verticaldirections were assumed in the numerical model forgridpoints at the rigid foundation level, and in thehorizontal direction at the backfill far-end boundary.The toe boundary condition in the physical and num-erical models is a reasonable approximation to therestraint that can be expected for the typical field caseof a buried footing. The backfill and facing units wereplaced in lifts of 150 mm (i.e. the height of one mod-ular block) and the reinforcement layers were numer-ically installed as each reinforcement elevation wasreached. Backfill compaction during constructionwas modeled by applying a horizontal stress compo-nent on the back of the facing units as the soil layersand facing blocks were put in place and the modelsolved to reach equilibrium. However, as described in

Section 3.2.1, greater soil modulus values than thoseobtained from laboratory triaxial compression testswere used for the backfill model. With this approach,negligible horizontal stress was needed behind thefacing panel to simulate backfill compaction in thewalls reported in this paper.

4 RESULTS

4.1 Calibration results

The response results for each of the three test walls inthis investigation were obtained by changing the rein-forcement stiffness (Table 2) or number of layers inthe numerical model to match the physical test. Thematerial properties for all other wall componentswere kept the same.

4.1.1 Facing displacementsFigure 5 shows the measured and numerically calcu-lated facing lateral displacement at potentiometer levels at the end of construction. The measured dis-placement results are readings from the potentiome-ters that were mounted against the facing blocks atreinforcement layer levels during construction. Thepredicted results are obtained using the material prop-erties shown in Tables 1–3. The results of Figure 5show satisfactory agreement between recorded andpredicted facing lateral displacements for all three

255

0 2 4 6 8 2 4 6 8 2 4 6 810

Ele

vatio

n (m

)

0

1

2

3

4

0 10Facing displacement (mm)

0 10

measured predicted

a) Wall 1 b) Wall 2 c) Wall 3

Figure 5. Measured and predicted facing displacements atend of construction.

09069-30.qxd 08/11/2003 20:38 PM Page 255


test walls. Both experimental and numerical resultsshow greater facing displacement magnitudes forWalls 2 and 3 constructed with lower stiffness rein-forcement and fewer layers, respectively, compared tothe control wall (Wall 1).

4.1.2 Reinforcement strainsFigures 6, 7 & 8 show the measured and predictedreinforcement strain distributions in the test walls atend of construction. The measured results are the datafrom the strain gauge readings. The predicted straindistributions for test walls show overall satisfactoryagreement with the experimental results. The strainmagnitudes at end of construction for all test walls aretypically less than 1%. Measured strains of this mag-nitude for the polymeric reinforcement used in thesewalls have been calculated to have a standard devia-tion as large as $0.3% strain (Bathurst et al. 2003).Hence, within the accuracy of the physical measure-ments, the results shown in Figures 6–8 capture boththe magnitude and distribution of strains in the meas-ured data. The data show that strain magnitudes anddistributions are similar for all three walls. This canbe explained by the contribution of the very stiff con-crete facing column that carries a large portion of the lateral earth forces at the end of construction.

However, while not reported in this paper, the magni-tudes of strain are very different between the threewalls under surcharge loading at which time largerwall lateral deformations have occurred and the

256

0.00.10.2

0.00.20.4

0.00.20.4

0.0

0.4

0.8

0.0

0.4

0.8

Distance (m)

0.0 0.5 1.0 1.5 2.0

Str

ain

(%)

0.0

0.4

0.8

Layer 6

Layer 5

Layer 4

Layer 3

Layer 2

Layer 1

predictedmeasured

Figure 6. Measured and predicted strain distributions atend of construction using hyperbolic soil model (Wall 1).

Distance (m)0.0 0.5 1.0 1.5 2.0

Str

ain

(%)

0.0

0.4

0.8

0.00.20.4

0.00.20.4

0.00.20.40.6

0.0

0.5

1.0

0.0

0.4

0.8

measured predicted

Layer 6

Layer 5

Layer 4

Layer 3

Layer 2

Layer 1


0.0

0.4

0.8

0.0

0.4

0.8

0.0

0.4

0.8

0.0 0.5 1.0 1.5 2.0

0.0

0.4

0.8

measured predicted

Layer 4

Layer 3

Layer 2

Layer 1

Distance (m)

Str

ain

(%)


09069-30.qxd 08/11/2003 20:38 PM Page 256


tensile load capacity of the reinforcement layers ismobilized (Bathurst et al. 2001). The calculated max-imum reinforcement load in all wall models at end ofconstruction was less than 1 kN/m, which was wellbelow the yield strength of the reinforcement materi-als (Table 2).

4.1.3 Toe reactionsFigure 9 shows the histories of the measured and cal-culated horizontal and vertical toe loads for the testwalls during construction. The figure shows a satis-factory agreement between the predicted and mea-sured horizontal and vertical toe reactions for the wallsat the end of construction. The plots of measured hor-izontal toe load during construction of Walls 2 and 3deviate from a smooth curve at early stages duringconstruction. This is thought to be due to a localgreater soil compaction effort at the back of the fac-ing units during construction. Otherwise the plots ofpredicted and recorded horizontal toe loads are inclose agreement. The results shown in Figure 9 indi-cate that wall reinforcement stiffness or number oflayers has a negligible effect on the magnitude of toeloads during construction for the wall height and rein-forcement configurations examined. The reason forthis consistent response, particularly with respect to

horizontal loads is the strong influence of the heavyfacing column as noted previously.

4.2 Influence of soil model on predicted wall response

Selected response features of Wall 2 were examinedusing a linear elastic soil model with the values ofYoung’s modulus and Poisson’s ratio taken asE � 48 MPa and v � 0.2, respectively. The value ofYoung’s modulus was back calculated from the mea-sured pressure-settlement results of the backfill behindthe reinforced soil zone during surcharge loading.Figure 10 shows the measured response and the pre-dicted facing displacement results using the linearelastic and hyperbolic soil models for Wall 2. Theplotted values are deformations with respect to thetime of installation of each displacement device.Hence, these plots should not be confused with theactual wall deformation profiles at the end of con-struction. Both predicted curves capture the range ofwall deformations recorded at the end of construc-tion. The close agreement between the predicted fac-ing displacement results in Figure 10 indicates thatthe values of soil hyperbolic model parameters reportedin Table 1 are consistent with a constant soil modulusvalue that was determined from the measured load-settlement response of the backfill. This result maynot be unexpected since the hyperbolic modulus num-bers Ke and Kb in Table 1 were independently

257

0

10

20

30

40

50

0

10

20

30

40

50

0 5 10 15 20 250

10

20

30

40

50

a) Wall 1

b) Wall 2

c) Wall 3

Number of facing units (blocks)

Toe

rea

ctio

n (k

N/m

)

measured

predicted

vertical

horizontal

vertical

horizontal

vertical

horizontal

Figure 9. Measured and predicted toe reaction forces during wall construction.

Facing displacement (mm)0 2 64 8 10

Ele

vatio

n (m

)

0

1

2

3

4measuredhyperbolic soil model linear elastic soil model

Figure 10. Measured and predicted facing displacementsof Wall 2 at end of construction.

09069-30.qxd 08/11/2003 20:38 PM Page 257


obtained by matching the load-settlement response ofthe backfill retained zone under surcharge loading asexplained in Section 3.2.1. Figure 11 also shows thatthe predicted toe reaction responses were essentiallyidentical using both soil models. Taken together, thedata in Figures 10 and 11 suggest that the simplerelastic soil model is sufficient to model these perfor-mance features of the wall. However, the results usingthe two soil models shown in Figure 12 illustrate asubtle but important difference in the distribution and

magnitude of predicted strains in the reinforcementlayers at the end of construction for Wall 2.Specifically, the predicted peak reinforcement strainsusing the soil linear elastic soil model are located far-ther back from the facing compared to the peaks fromthe hyperbolic model. Comparison with Figure 7shows that the measured predicted peak strains arelocated close to the back of the facing column and notwithin the reinforced soil mass as predicted for alllayers in Figure 12 with the exception of layer 1. Theabsence of peak strain values within the reinforcedsoil mass (as predicted using the linear elastic soilmodel) was corroborated by the lack of a visible shearzone in the backfill at the time of careful soil excava-tion of the wall. On the other hand, both the hyper-bolic model results and the measured data showrelatively high reinforcement strain magnitudes at theconnections with the facing panel at end of construc-tion, which are not captured using the linear elasticmodel. It can be argued that horizontal stresses in thesoil decrease locally behind the facing due to the out-ward horizontal movement of the facing column dur-ing construction. As a result, the stress-dependenthyperbolic model predicts smaller soil stiffness val-ues behind the facing compared to the constant stiff-ness model. Therefore, the predicted strain magnitudesat the reinforcement connections with the facing canbe expected to be greater (and hence more accurate)using the hyperbolic model rather than the constantstiffness (linear elastic) soil model.

Finally, the better match between the predicted andmeasured wall response using back-fitted modulusvalues from the measured load-settlement response ofthe backfill in the actual physical tests highlights theinability of conventional triaxial compression tests tocapture the backfill plane strain stiffness in the large-scale wall tests.

5 CONCLUSIONS

A numerical model has been developed using FLACto predict the measured response of carefully instru-mented, large-scale geosynthetic reinforced soil mod-ular block retaining walls during construction. Thenumerical model accounts for staged construction ofthe retaining walls and incremental lateral displace-ment of the modular facing using FISH functions.Additional subroutines are included in the program tomodel the backfill stress-dependent stiffness proper-ties and the nonlinear reinforcement strain-dependentaxial stiffness.

The measured and numerical results for the con-struction stage of each wall showed satisfactory agree-ment for different response parameters includingfacing displacements, reinforcement strains and his-tory of toe forces. In particular, reinforcement strain

258

0 5 10 15 20 250

10

20

30

40

50

Number of facing courses placed

Toe

rea

ctio

n (k

N/m

)

vertical

horizontal

linear elastic

hyperbolic

Figure 11. Comparison of predicted toe reaction forcesusing linear elastic and hyperbolic soil models during construction (Wall 2).

Distance (m)0.0 0.5 1.0 1.5 2.0

Str

ain

(%)

0.00.40.81.2

0.00.10.2

0.00.20.4

0.00.20.40.6

0.0

0.5

1.0

0.0

0.5

1.0

Layer 6

Layer 5

Layer 4

Layer 3

Layer 2

Layer 1

hyperboliclinear elastic

Figure 12. Comparison of predicted strain distributions atend of construction using linear elastic and hyperbolic soilmodels (Wall 2).

09069-30.qxd 08/11/2003 20:38 PM Page 258


distributions using a hyperbolic soil model were foundto be in good agreement with the measured data.

ACKNOWLEDGEMENTS

The financial support for this study has been providedby the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada, 11 Departments ofTransportation in the USA, and grants from theDepartment of National Defence of Canada.

REFERENCES

Allen, T.M., Bathurst, R.J. & Berg, R.R. 2002. Global Levelof Safety and Performance of Geosynthetic Walls: AHistorical Perspective. Geosynthetics International, (9):395–450.

Allen, T.M., Bathurst, R.J., Lee, W.F., Holtz, R.D. & Walters,D.L. 2003. A New Working Stress Method for Predictionof Reinforcement Loads in Geosynthetic Walls, CanadianGeotechnical Journal, (in press).

Bathurst R.J. & Hatami, K. 2001. Review of numerical mod-eling of geosynthetic reinforced-soil walls. Proc. 10th

Inter. Conf. Comp. Meth. Adv. Geomech. Invited ThemePaper, Tucson, AZ, USA, January 2001: (2) 1223–1232.

Bathurst, R.J., Walters, D.L., Hatami, K. & Allen, T.M. 2001.Full-scale performance testing and modeling of rein-forced soil retaining walls. Special Lecture, IS-Kyushu2001. Fukuoka, Japan, November 2001.

Duncan, J.M., Byrne, P., Wong, K.S. & Mabry, P. 1980.Strength, stress-strain and bulk modulus parameters forfinite-element analysis of stresses and movements in soilmasses. Report No. UCB/GT/80-01. University ofCalifornia, Berkeley: Department of Civil Engineering.

Hatami, K. & Bathurst, R.J. 2001. Modeling static responseof a segmental geosynthetic reinforced soil retaining wallusing FLAC. Proc. 2nd Int. FLAC Symp. NumericalModeling in Geomechanics, Lyon, October 2001,223–231.

Hatami, K. Blatz, J.A. & Bathurst, R.J. 2002. Numericalmodeling of geosynthetic reinforced soil retaining wallsand embankments. Proc. 2nd Can. Spec. Conf. Comp.Appl. Geotech., Winnipeg, MB, Canada, April 2002.

Hatami, K. & Bathurst, R.J. 2002. Numerical simulation ofa segmental retaining wall under uniform surcharge load-ing. Proc. 55th Can. Geotech. Conf. Niagara Falls, ON,Canada, October 2002.

Itasca Consulting Group, Inc. 1998. FLAC – FastLagrangian Analysis of Continua, Ver. 3.40. Minneapolis,MN: Itasca.

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Three-dimensional modeling of an excavation adjacent to a major structure

J.P. HsiSMEC Australia Pty Ltd, Sydney, Australia

M.A. CoulthardM.A. Coulthard & Associates Pty Ltd, Melbourne, Australia

ABSTRACT: An excavation adjacent to a major bridge was carried out for the construction of a cut and covertunnel, Hawthorne Street Tunnel, as part of the South East Transit Project Section 2 in Brisbane, Australia. Thebridge was founded on shallow foundations whilst the excavation extended to below the foundation level. Therewas limited tolerance for the bridge foundations to deflect resulting from the excavation, due to its structuralarticulation. The support system for the excavation consisted of contiguous and scallop bored piles and groundanchors. The excavation was carried out in stages taking advantage of the 3D effects to minimize ground defor-mation. To predict the ground performance during excavation and to optimize the design of the ground support,FLAC3D was employed to simulate the 3D effects, the construction sequence and the soil-structure interaction.Field monitoring results showed performance comparable with that predicted by FLAC3D.

1 INTRODUCTION

The South East Transit Project Section 2 (SETP2) in Brisbane, Australia was recently constructed to pro-vide a dedicated traffic corridor for public buses andemergency services vehicles. The project route, of atotal length of 2.1 km, traversed well-developed areas,which imposed significant constraints on the con-struction work.

One of the major challenges of the project was toprovide a design for a transport corridor that passesthrough the inner urban zones of Brisbane, whilst mini-mizing the impact on adjacent properties, heavily tra-fficked arterial roads, public utility services and otherinfrastructure. The design therefore made substantialuse of tunnels, bridges and retaining walls to minimizesuch impacts.

A critical component of the project was to constructa cut and cover tunnel, Hawthorne Street Tunnel,below Hawthorne Street and closely adjacent toHawthorne Street Bridge, which was an importantbridge carrying through traffic between major roads.The bridge was supported on shallow foundations andwas very sensitive to ground movement. Excavationfor the tunnel construction in close proximity to thebridge was a major concern.

A robust support system for the excavation wasadopted to control ground movement and preventdamage to the existing bridge. This system involved

installation of contiguous and scallop bored pilewalls. As the excavation proceeded ground anchorswere installed through the bored piles next to thebridge abutment. The excavation was staged and thebridge deflection was monitored to ensure that groundmovements fell within the design limit.

Due to the critical nature and complexity of thework detailed numerical modeling using FLAC3D

(Itasca 1997) was carried out. The modeling consid-ered the excavation and construction sequence, andthe interaction between the ground, the bridge foun-dations, the bored piles, and the ground anchors. Anoptimized design of the support system was achievedvia the use of FLAC3D.

This paper presents the project overview, the con-struction constraints, the site geology, the design cri-teria, the work performance and, particularly, theFLAC3D modeling.

2 PROJECT DESCRIPTION

The South East Transit Project (SETP), an initiativeof Queensland Department of Transport, was devel-oped to provide a state-of-the-art busway for publictransport and emergency services vehicles. The routestretches from Brisbane’s Central Business District toLogan City, about 20 km to the south-east.

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In December 1998 Thiess Contractors Pty Ltd wasappointed to design and construct a 2.1 km section ofthe Busway from Water Street in Woolloongabba to O’Keefe Street in Buranda, Brisbane. This sectionof the Busway is known as South East Transit Pro-ject Section 2 (SETP2); its route plan is shown inFigure 1. SMEC Australia Pty Ltd was appointed byThiess as the Principal Designer of the project to pro-vide detailed design of all civil engineering works.

The SETP2 contract, valued at approximately$A70 million, included the construction of three busstations, seven underpass structures, a 150 m long cut and cover tunnel, a 230 m long driven tunnel, athree span Super T bridge, and substantial retaining structures.

Construction of the project commenced in April1999 and was completed in November 2000. The Wool-loongabba section of the project was completed in earlyAugust 2000 in readiness for the Olympic events to beheld at the Woolloongabba Cricket Ground.

One of the fundamental challenges of the projectwas to provide a design for a transport corridor thatpasses through the inner urban zones of Brisbane,whilst minimizing the impact on adjacent properties,heavily trafficked arterial roads, public utility ser-vices and other infrastructure (including the ClevelandRailway Line). The design therefore made substantialuse of tunnels, bridges and retaining walls to reducesuch impacts.

3 SITE GEOLOGY

3.1 General

The SETP2 busway route passes through several geological formations, the oldest of which is theDevonian-Carboniferous low grade metasedimentsbelonging to the Bunya Phyllites and NeranleighFernvale Beds. These rocks are overlain by theyounger Tertiary volcanics of the Brisbane Tuff, andthe sedimentary rocks of the Tingalpa Formation. Inthe northern part of the alignment the metasedimentsof the Bunya Phyllites and Neranleigh Fernvale form

prominent topographic highs; the lows are generallyinfilled with Quaternary alluvial deposits and somefill. To the south the Brisbane Tuff trends north-east tosouth-west adjacent to the Neranleigh Fernvale Beds,both of which form undulating topographic highs.

The metasediments mainly comprise fine andmedium grained strongly foliated interbedded phyllites,argillites and greywacke with some thin quartzites,while the poorly bedded volcanics comprise weldedash flow, bedded tuffs, and breccia with some inter-bedded conglomerate and sandstone. The Tertiary sedi-mentary rocks mainly consist of conglomerate andsandstone.

During construction these rocks were found to bemainly highly weathered or moderately weathered, ofvery low to low strength, with extremely weatheredseams throughout. The extremely weathered seamscontained extremely to very low strength material.Some slightly weathered rock was also encountered.

3.2 Hawthorne Street tunnel site

The area is underlain by Bunya Phyllite Formationcomprising weathered fine grained phyllite. The soilprofile consists of residual soil described as sandysilty clay and gravelly clay overlain by topsoil or a thinlayer of fill in parts. The residual soil thins towards thesouth and west as the thickness of extremely and highlyweathered rock increases. Moderately weathered rockwas expected near the design level of the busway onthe north-east side of Hawthorne Street Bridge butwas expected to dip below this level towards the southand west. This change in level of moderately weatheredrock is related to the contact with Brisbane Tuff about15 m south of Hawthorne Street. The groundwater hadbeen measured about 1 m below the busway designlevel.

4 SITE CONDITIONS

The SEPT2 route intersected Hawthorne Street at anangle of approximately 75° (see Figure 2). A cut andcover tunnel (Figure 3) was proposed to be builtbelow Hawthorne Street to provide grade separatedthrough traffics. The construction of this cut and covertunnel was in the immediate proximity of the existingHawthorne Street Bridge, which is a four lane, singlespan arch bridge supported on a strip footing at eachabutment.

Excavation for the cut and cover tunnel mightcause movement of the foundation resulting in dam-age to the bridge structure. Structural analysis ofHawthorne Street Bridge indicated that the structurewas very sensitive to any movement, as the bridge hada hinged mid-span, and therefore was held up by thehorizontal support at the foundation level.

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Figure 1. SETP2 route plan.

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A horizontal movement of the footing of 10 mmwould result in a mid-span vertical movement ofapproximately 18 mm. Damage to the bridge struc-ture was expected to occur should the foundationmovement exceed 10 mm in the horizontal direction.The design of the support system therefore adopted amaximum lateral movement of the bridge foundationof 5 mm for conservative reasons.

5 STRUCTURAL SYSTEM

The structural system for the cut and cover tunnelconsisted of contiguous and scallop pile walls and apre-stress concrete plank roof with a reinforced con-crete deck slab on top. The planks were placed on a slope to match the slope of Hawthorne Street toreduce the amount of fill on top thereby reducing the

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Figure 2. Site plan.

Figure 3. Site elevation.

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structural depth required. Fill has been maintained onthe structure for the location of services.

The western bored piles are located very close tothe existing abutment of the Hawthorne StreetBridge. Each of the piles next to the bridge was to befitted with permanent ground anchors to maintain therequired horizontal bearing pressure and reduce foun-dation movements for the Hawthorne Street Bridge.Utilizing permanent ground anchors, the anchors canbe stressed to counteract any predicted movement,and re-stressed if the measured movement is greaterthan that predicted.

The eastern wall consisted of 0.9 m diameter piles at1.5 m c/c spacing whilst the western wall comprised1.2 m diameter piles at 1.77 m c/c spacing, except forthe section within 3 m from the existing bridge foun-dation where the piles were at 1.25 m c/c spacing. Allthe piles were socketed 0.5 m into slightly weatheredphyllite. The gaps between the piles were shotcretedwith fibrecrete. Two VSL permanent ground anchorswere installed on each of the piles adjacent to the bridgeabutment. Each anchor was socketed 10 m into slightlyweathered phyllite and prestressed to 1000 kN.

During construction the horizontal movement ofthe abutment and vertical mid-span movement of thebridge were monitored. The measurements were com-pared to the estimated values and adjustments madeto the construction method and program if required.Temporary struts and hydraulic jacks were specifiedas part of the contingency plan to help control pilemovement at the most critical section.

6 CONSTRUCTION SEQUENCE

To minimize ground movement associated with theconstruction work, excavation was carried out instages, as follows:

1. Constructed bored pile walls and headstocks.2. Installed instruments, with minimum accuracy of

1 mm, for monitoring deflection of headstocks,and abutments and mid-span joints of HawthorneStreet Bridge.

3. Undertook baseline readings of the instrumentswithout live load on the bridge and at relativelysimilar climatic temperature during construction.

4. Excavated southern half of the tunnel to the levelof the first (upper) row of ground anchors, withfibrecrete applied between piles progressively.

5. Installed and stressed the first (upper) row of roundanchors on the western wall of the southern halfof the tunnel.

6. As for 4, but for northern half of the tunnel.7. As for 5, but for northern half of the tunnel.8. Excavated southern half of the tunnel to the level

of second (lower) row of ground anchors withfibrecrete applied between piles progressively.

9. Installed and stressed the second (lower) row ofground anchors on the western wall of the south-ern half of the tunnel.

10. As for 8, but for northern half of the tunnel.11. As for 9, but for northern half of the tunnel.12. Excavated to the tunnel floor level.13. Installed the pre-stressed concrete planks over

the headstocks.14. Constructed cast in situ concrete slab over the

tunnel floor.15. Took deflection measurements frequently dur-

ing the above construction stages with record ofcorresponding temperature and time.

16. At any time if the measured lateral deflection of the bridge foundation was greater than 5 mm,contingency measures including further stressingthe ground anchors would be implemented.


7.1 Approach

The complex three-dimensional nature of the prob-lem and the need to allow for possible yield of thevarious rock units and to account for a range of struc-tural elements suggested that FLAC3D would be wellsuited to the modeling. The work was performed in1999, using version 2.0 of that program.

The existing bridge was included only as a loadedfoundation on one side of the new tunnel then theconstruction sequence outlined above was repre-sented in the numerical model. Coding in the in-builtprogramming language FISH was used to managegrid generation, excavation stages, installation andlinking of piles, struts, crossbeams and anchors, andmany other aspects of the modeling. The task ofdevelopment and testing would have been much moredifficult without this powerful feature.

7.2 Geotechnical model

The geotechnical model adopted for the numericalmodeling included the subsurface stratigraphy andgeotechnical parameters. Boreholes in this area indi-cated that the subsurface consisted of residual soils to 5–6 m depth underlain by extremely to highly wea-thered (EW/HW) phyllite to depths ranging from 9 mnear the eastern wall to 12 m near the western wall, over-lying moderately weathered (MW) phyllite. Slightlyweathered (SW) phyllite occurred at depths between14 m at the eastern wall and 16 m at the western wall.

A uniform subsurface profile based on the morecritical profile on the western side of the excavationwas assumed and the Mohr-Coulomb soil/rock con-stitutive model was adopted for each rock and soilunit. The assumed geotechnical model and material

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parameters are given in Table 1. The tensile strengthfor each rock unit was taken to be 25% of the cohe-sion, and zero for the soil.

In situ horizontal stresses were assumed to be halfthe vertical in the soil layer and equal to the vertical inthe rock units, where an initial approximation to thevertical stress was computed from the above layeringand densities.

7.3 Bridge footing

The Hawthorne Street Bridge was represented simplyvia loads applied to the footing shown in the central partof the grid in Figure 4. Two loading cases were con-sidered, based on (a) SMEC’s independent structural

analysis of the existing bridge and (b) the bearingpressure specified on the original bridge drawing.Loading (a) was significantly smaller than (b), whichis understandable as the foundation pressure shown inthe drawing would generally include a factor of safety.For prudent and conservative reasons, the design of thesupporting structures to the excavation was based oncase (b). The total applied vertical and horizontal loadswere 31.55 MN and 21.86 MN respectively, where thehorizontal load was taken to act in the direction of the short axis of the footing, i.e. at 25° to the normalto the busway walls. These loads were converted toequivalent Cartesian stress components, which wereapplied to the upper surface of the footing.

7.4 Retaining system

The retaining system consists of a line of piles oneach side of the busway and two rows of groundanchors attached to the piles on the western side tosupport the existing bridge footing. Temporary strutson the northern half of the busway were also modeledin some cases but proved to be ineffective.

The piles on the eastern side are 0.9 m diameter,installed at 1.5 m center spacing. Those on the western

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Table 1. Geotechnical model adopted for analysis.

Soil/Rock Depth �t c � EType (m) (kN/m3) (kPa) (deg.) (MPa) �

Residual 0–6 18 10 30 40 0.35EW/HW 6–12 20 500 35 50 0.30MW 12–16 22 750 40 200 0.25SW 16 25 2500 45 400 0.20

Figure 4. Rock units and zoning in inner section of FLAC3D grid, before tunnel construction.

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side are of 1.2 m diameter, at 1.25 m c/c spacing adja-cent to the nearest corner of the bridge footing and1.77 m c/c away from that corner. All piles are sock-eted 0.5 m into SW phyllite.

The ground anchors for the tunnel as built are 8-strand cables of 15.2 mm diameter, installed in0.145 m diameter holes and pre-tensioned to 1 MNafter installation. They dip at 45° and are socketed10 m into SW phyllite but are ungrouted over theremainder of their lengths. Other forms of anchorwere considered in several of the numerical models,e.g. varying numbers of strand, hole diameter andpre-tension force.

Non-geometric property values for the FLAC3D

structural elements representing the piles and groundanchors, are given in Tables 2 and 3 respectively.

7.5 Finite difference grid

As shown in Figure 2, the tunnel was to run obliquelyunder the existing bridge, with a bridge foundationimmediately adjacent. In addition, a soil slope aroundthe bridge foundation was to be replaced, as the tun-nel was excavated, by a reverse-angle slope that wasnot constant in profile along the excavation. The gridwas generated in sections, some of which had to bejoined via “attach” commands, then the entire modelwas transformed to create the correct skew anglebetween the bridge and tunnel. A view of the innerpart of the pre-construction grid is given in Figure 4.The cut-and-cover tunnel was to be constructedwithin the finer-zoned region to the left of, and paral-lel to, the slope shown in Figure 4. Note that the griddid not conform precisely to the assumed horizontalboundaries on either side of the MW rock unit, and

that a coarsely zoned region was attached at the baseof the finely-zoned part shown, to provide a betterrepresentation of the rock mass response at depth.

7.6 Modeling strategy

The excavation and support sequence as modeled inFLAC3D was as follows, where the computation wasstepped to equilibrium at each stage:

0. Apply boundary conditions to far boundaries(fixed horizontal displacements on vertical sidesand fixed all displacement components on base),initialize approximate in situ stresses, apply grav-ity and footing load.

1. Excavate entire busway to 3 m depth.2. Install both sets of piles then excavate more distant

half of busway (relative to view in Figure 4) a fur-ther 3 m.

3. Install upper ground anchors, attached to pilesbetween previous excavation and bridge footing,and excavate 2 m from nearer half of busway.

4. Install crossbeams along lines of piles and strutsacross excavation (not in all models), and exca-vate more distant half of busway a further 3 m.

5. Install lower ground anchors attached to pilesbetween stage 2 excavation and bridge footing,and excavate further 3 m from nearer half ofbusway.

6. Remove struts (if installed at stage 4), installupper ground anchors attached to piles betweenstage 3 excavation and bridge footing, and exca-vate 2 m more from further half of busway.

7. Excavate another 2 m from nearer half of busway.8. Install lower ground anchors attached to piles

between stage 3 excavation and bridge footing and excavate final 1 m from nearer half of busway.The final model configuration is shown in Figure 5,where the view is as in Figure 4.

Implementation of this construction sequence in aFLAC3D model was largely straightforward, withFISH routines controlling the various excavationstages and the placement of all types of structural ele-ments. However, one aspect of the modeling provedto be unexpectedly complex, viz. the setting of linksbetween the many structural nodes and rock zones orother structural nodes.

The key constraint (D. Potyondy, private communi-cation) is that each structural node in a FLAC3D

model can only be the source of one link. This linkmay provide either a node-to-zone connection or anode-to-node connection. This means that, wherecables or beams are linked to piles at sub-surface nodes,then multiple nodes must be created to manage thevarious links, and the direction of those links must be carefully controlled. For example, creation of a pileautomatically creates links from each sub-surface

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Table 3. Ground anchors (cable structural elements).

Parameter 8-strand 27-strand

Young’s modulus 200 200 GPaDensity 7860 7860 kg/m3

Tensile/compressive yield strength 1.19 4.0 MNGrout stiffness 10 9 GPaGrout cohesive strength 1.13 1.7 MN/mGrout friction angle 0° 0°

Table 2. Piles (pile structural elements).

Parameter Value

Young’s modulus 31 GPaPoisson’s ratio 0.2Density 2400 kg/m3

Shear/normal coupling spring stiffness 50 MPaShear/normal coupling spring cohesion 1 MN/mShear/normal coupling spring friction angle 30°

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structural node in the pile to the rock zone that con-tains it. If a ground anchor is to be connected to thatpile then the pile must be formed in such a way thatthere is a node at the intended connection point.When the ground anchor is created as a series of cableelements, a duplicate node will exist at the same coor-dinates as the pile node to which it is to be connected.By default, each cable node will also have a node-to-zone link. These must be deleted at all nodes above thegrouted section of the cable, then a new node-to-nodelink must be created, emanating from the node at thetop end of the cable and ending on the correspond-ing pile node (so that the pile node is the target of thenode-to-node link from the cable node and the sourceof the node-to-zone link to the rock).

A similar procedure had to be implemented to han-dle links between piles and crossbeams and betweencrossbeams and struts, in cases where the latter weremodeled. In that case, it was critical that the final setof links be ordered thus: strut node → beam node →pile node → zone. Rigid links were used for all con-nections between structural elements.

Further, when structural elements were deletedfrom a model, such as when struts were removed atcomputational stage 6 above, then the links from the

associated structural nodes were not automaticallydeleted by FLAC3D; this also had to be done explicitlyin the data and FISH files.

A final complication arose from the fact that mul-tiple links at a point in space can only be distin-guished via the link number that is set within FLAC3D

when each particular link is created. Management ofthe links therefore required careful monitoring of thenumbers of active structural nodes, elements andlinks and the highest id number for each. Again, thiswas handled effectively using FISH coding.

7.7 Cases modeled

After a great deal of development and testing severalproduction analyses were performed. As indicated insection 7.4 above, temporary struts were predictednot to be effective, so the final analyses only includedpiles and ground anchors, thus:

– bbhsu9c: 8-strand cables with 1 MN pre-tension;case (a) footing loading from section 7.3;

– bbhsu9d: as for 9c except for higher case(b) foot-ing loading;

– bbhsu9e: as for 9d except 27-strand cables with3 MN pre-tension.

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Figure 5. Inner section of FLAC3D model, showing structural elements at completion of excavation and support.

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Figure 6 shows the predicted horizontal displace-ments of the bridge footing at the end of runs 9d and 9erespectively. These results suggested that the heavierground anchors, with 3 MN pre-tension, would actuallypull the footing away from the busway excavation,

i.e. they would overdo the support. In contrast, theanchors in model 9d allowed the footing to relaxtowards the excavation, but the maximum horizontaldisplacement was constrained to be less than 5 mm, as required. Histories of x-, y- and z-components of

Figure 6. Induced horizontal displacements in footing at end of construction, from run bbhsu9d (upper plot) and bbhsu9e(lower plot). Piles and ground anchors are shown as solid and dashed lines respectively; busway is to left of piles.

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displacement of several points on the footing, for thesame model 9d, showed that they would vary throughthe various computational stages but were also alwayspredicted to be less than about 5 mm (see Fig. 7).Some further representative results from run bbhsu9dare presented and discussed below.

The axial forces acting within one set of groundanchors at the completion of construction are shownin Figure 8; the vertical axis represents the verticalcoordinate (RL) along the cables and the horizontalaxis the axial force (note that the sign convention forforces in structural elements in FLAC3D depends

Figure 7. Histories of induced x-, y- and z-displacements at two points on footing, during the stages of construction, as computed in model bbhsu9d.

Figure 8. Axial forces in one set of ground anchors at end of construction, as computed in model bbhsu9d.

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Figure 9. Axial forces in some of the piles on the footing side of the excavation, at the end of construction, as computed inmodel bbhsu9d.

Figure 10. Moments (my – upper and mz – lower) in the same set of piles as in Figure 9.

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upon the relative orientation of the nodes). In thegrouted (lower) section, the developed forces areclose to the pre-tension force of 1 MN but they areabout 8% smaller in the ungrouted (upper) section.This is consistent with the response expected as thesystem is re-equilibrated after pre-tensioning of theanchors. Examples of forces and moments generatedin piles adjacent to the bridge footing are given inFigures 9 and 10. In each case the vertical axis givesthe RL (z-coordinate) of pile elements and the hori-zontal axis the force or moment, in SI units. The axialforces in Figure 9 clearly show the effects of the con-nection of ground anchors at two points in the uppersections of some of the piles.

8 FIELD PERFORMANCE

During the entire excavation process the measuredlateral deflection of the abutment of HawthorneStreet Bridge was less than 5 mm. There was no dis-tress of Hawthorne Street Bridge during and afterconstruction of the Hawthorne Street Tunnel.

9 SUMMARY

Design and construction of the section of the BrisbaneBusway had to ensure that the adjacent footings of an

existing bridge were not disturbed. Analysis of thebridge indicated that lateral displacements of the foot-ing must be constrained not to exceed 5 mm. ProgramFLAC3D was used to simulate the complex construc-tion sequence, including the placement of piles andground anchors. FISH programming was used exten-sively to assist in generating the grid and in managingthe links between the various structural elements. The results from a series of production analyses indi-cated that a design based on 8-strand ground anchors,pre-tensioned to 1 MN, would be satisfactory. Thepredictions of FLAC3D have been confirmed by mon-itoring during construction.

ACKNOWLEDGEMENTS

The authors wish to acknowledge many valuablecommunications with Dr. D. Potyondy, Itasca, partic-ularly about the linking of structural elements inFLAC3D.

REFERENCES

Itasca Consulting Group, Inc. 1997. FLAC3D – FastLagrangian Analysis of Continua in 3 Dimensions,Version 2.0. Minneapolis, MN: Itasca.

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273

Pile installation using FLAC

A. KlarTechnion – Israel Institute of Technology, Haifa, Israel

I. EinavCentre for Offshore Foundation Systems, UWA, Australia

ABSTRACT: This paper presents a numerical simulation of pile installation using FLAC. A new contact for-mulation between rigid and deformable bodies is employed. This formulation utilizes equations of motion todescribe the behavior of the deformable nodal point along the contact surface. Unlike FLAC’s own embeddedinterface formulation, the new formulation does not encounter discontinuities problem along nonlinear orpiecewise linear surfaces.

1 INTRODUCTION

The evaluation of pile installation has great signifi-cant in design, for two main reasons:

1. In saturated clay soils, a considerable change inpore pressure takes place due to the pile installa-tion. This change of pore pressure and its subse-quent dissipation process affect the pile capacity.

2. Simulation of pile installation allows for moreaccurate evaluation of the end bearing capacity.

Over the last three decades, the problem of pileinstallation has been extensively researched by differ-ent analytical/numerical methods. These methods canbe, generally, categorized into five groups:

1. Limit analysis approach,2. cavity expansion solution,3. strain path method,4. Eulerian large strain finite element analysis, and5. Lagrangian large strain analysis.

The problem of pile installation is strongly related tothe problem of cone penetration. As a result, advancesin understanding were, and still are, strongly connectedto research of cone penetrations.

In the present work, simulation of pile installationis presented using the Lagrangian large strain analysiscode FLAC. To understand the importance of usingLagrangian analysis, the following section overviewsthe different methods and their limitations.

Since the problem of pile installation involvesinteraction between two bodies (pile and soil), there isa need to employ some kind of interface formulation.

FLAC’s own interface formulation is associated withundesirable behaviors, especially along nonlinearconvex surfaces. As a result, an alternative and simpleapproach is suggested.

2 REVIEW OF METHODS

Generally, it may be said that five numerical/analyti-cal methods exist for the evaluation of pore-pressuregeneration and/or end bearing capacity:

2.1 Limit analysis approach

This group of methods includes lower bound solution(or slip line method), upper bound solution and limitequilibrium analysis. Results obtained by this approachcorrespond to collapse mechanism. In these methodsonly the strength parameters of the soil are introduced;i.e. the stiffness of the soil has no influence on theresults. One may refer to Durgunoglu & Mitchell(1975) for some examples of failure mechanisms.

2.2 Cavity expansion solutions

In this group, the stresses along the pile are related tosolutions of cavity expansion. It is commonly assumedthat the solution of cylindrical cavity expansion mayrepresent the deformations and stresses along points,which are far from the end of the pile, and that solu-tion of spherical cavity expansion may be used forapproximation of field quantities near the pile lowertip. Figure 1 shows the different zones. Zone II and III

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are the ones that may be represented by the cavityexpansion solutions. The behavior in Zone I is highlyaffected by the soil surface. One of the purposes ofthe work presented here is to evaluate the surfaceeffect on the solution, and to discover the requireddepth for which the evolution of the end bearingcapacity factor Nc is redundant. To learn more aboutthe use of cavity expansions in the solution of pile andcone penetration, one may refer to the excellent bookby Yu (2000).

2.3 Strain path method

The strain path method was first suggested by Baligh(1985). In this method, a flow field of soil is assumedto exist around the pile. From this assumed flow field,strains are derived, while stresses can then be deter-mined according to a particular constitutive relation.Two main drawbacks are associated with this method:(a) equilibrium will not necessarily exist, and (b) theeffect of the surface and/or changing properties withdepth cannot be included, i.e. it can represent onlydeep steady penetrations. The first limitation may bepartly overcome by the use of the iterative proceduressuggested by Teh & Houlsby (1988).

2.4 Large strain Eulerian finite element analysis

In the Eulerian large strain finite element analysis, thematerial (soil) streams through fixed points in space[e.g. van den Berg 1994]. This method possesses thesame limitation of the last methods; i.e. inability tomodel the soil free surface.

2.5 Large strain Lagrangian finite elementanalysis

This method is the one employed in FLAC, in whichthe mesh is updated throughout the pile penetrationprocess. This method is the only one that in theory cancapture the installation process as it is, while includingthe influence of both soil surface and changing prop-erties with depth. It should be noted, however, that thismethod is associated with numerous numerical prob-lems, when simulating pile or cone penetrations, dueto the great deformations involved in the problem.

3 PENETRATION DIFFCULTIES IN FLAC

At this current stage, if one chooses to utilize FLAC’sembedded interface formulation to simulate the soil-pile contact, he should acknowledge that on top of theproblems due to the great deformations, he introducesnew problem. As in many other codes, in FLAC’s inter-face formulation the two bodies are prevented fromcrossing each other. This leads to discontinuities in thecontact between the bodies (Itasca 2000), if nonlinear orpiecewise linear surfaces are involved. Figure 2 showsan example of the problem for piecewise linear rigidcontact surface. In Figure 2a the contact formulationcorresponds to that of FLAC; i.e. it does not allow forthe deformable body (represented by the quadrilateralelements) to overlap the rigid body (represented by thethick black line), and therefore gaps between these twoare developed near discontinuity points along of therigid body. There are two kinds of gaps that maydevelop between the rigid body and the deformable one.The first kind, (noted as type I in Fig. 2), is a gap thatwill always result when the deformable body is in con-tact with a concave surface of a rigid body, and is fic-tional because the lamped grid points are still in contact.The second kind (noted as type II in Fig. 2) is true gapassociated with zero forces acting on the grid points.

274

Zon

e II

IZ

one

IIZ

one

I

Figure 1. Cavity expansion zones along the pile. Figure 2. Contact problems along piecewise linear surface.

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This behavior, shown in Fig. 2(a) for piecewise lin-ear surface, will be more pronounced in nonlinear sur-faces, since, at least, every second grid point will beout of contact with the rigid body in a convex surfaces.To overcome this problem of undesirable gaps, thecontact formulation must allow the deformable bodyto overlap the rigid body, as shown in Figure 2b. In this“overlapping” formulation, the grid points, fromwhich the strain increments are derived, travel alongthe rigid body surface, unless some sort of contactlogic that allows separation is included. In the follow-ing section, such a contact formulation is presented,and is used later for the analysis of pile installation.

4 NEW CONTACT FORMULATION FOR FLAC

The following contact formulation is applicable to theinteraction between rigid and deformable bodies. Ingeneral, both the rigid and the deformable bodies arefree to move in space. In the present formulation therigid body motion is prescribed. However, it can easilybe extended to a more general case where the motionof the rigid body is determined by the solution of itsmotion equations; this feature is studied these daysand is being employed for the simulation of anchorinstallation. In the most degenerate way, the presentformulation can also be used to create roller fixing inan inclined angle; an option that is absent from FLAC.

4.1 Formulation

A body may be defined as rigid if the distance betweenany two points of it is constant with time. The motionof a non-rotating rigid body can be described by twocomponents, a velocity vector, vR and an accelerationvector aR. The motion of each lumped mass located ona grid point that represents the deformable body canalso be described by two vectors, vD and aD for veloc-ity and acceleration respectively. vD and aD are here

defined in the fixed coordinate system. If consideringthe moving coordinate system then the deformablebody motion is defined by a velocity vector ofvL � vD � vR and acceleration vector of aL � aD � aR.Note the rigid body is stationary in the moving coordi-nate system. Figure 3 shows velocity and accelerationdiagrams of a grid point located on the rigid body. Thisbody can be represented by a shape function x� � f (z�)(x� and z� are the coordinates of the moving system).

Since the deformable body cannot enter the rigidbody nor departure from it (unless tensile failure is con-sidered as will be discussed later) the motion of it canonly be tangential to the rigid body; i.e. only the tangen-tial components of both velocity vector and accelerationneed to be introduced in to the equations of motion. Ifwe consider an explicit time marching numericalscheme, the following expression can be written:

(1)

where vx�L , vLz�and aL

x�, aLz�are the components of veloc-

ity and acceleration vectors in the moving coordinatesystem (x�, z�) at time t. vx

D, vzD, ax

D, azD, vx

R, vzR, ax

R, azR

are the components of the motion vectors of thedeformable and rigid body in the fixed coordinatesystem (x, z) at time t. If the motion of the rigid bodyis prescribed (i.e. know a priori) then the motion ofthe deformable body in the fixed coordinate system isas follows:

(2)

Note, that in explicit numerical scheme it is assumedthat state variable are frozen at each step (dt); i.e. foreach time step the rigid and deformable body are fixedin space, and therefore, all values in the right side ofEquation 1 are known. The value of aD in Equation 1 isobtained from the assumption that the deformable bodyis not in contact with the rigid body; i.e. the accelera-tion is obtained from the forces acting on the grid pointdue to the deformation of the deformable body. To

275

Velocity Diagram Acceleration Diagram

X'

Z'

Shape function X’ = f(Z’)

aD

a L a R

ν R

νD

ν L

Figure 3. Velocity and acceleration diagrams.

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introduce some friction between the rigid and thedeformable body one can add frictional force in thetangential direction and re-obtain from it the value ofaD. If a rigid-plastic tangential interface is desired thenone can define the friction forces direction simplyaccording to the relative velocity between the soil andthe pile. If an elastic plastic tangential interface isdesired a slightly more complicated formulation isrequired. Since in the current paper only smooth pilesare considered, this kind of formulation is not pre-sented, although written and verified by the writers.

To consider possible separation between the rigidbody and the deformable body, a contact logic mustbe introduced. If, for example, the contact logic con-siders zero tensile forces between the rigid and thedeformable bodies as condition for separation, then itwill occur once 180 � a � � � a. Whenever thiscondition is satisfied, the grid point is solved accord-ing to aD; i.e. vD(t � dt) � vD(t) � aD(t)dt. If duringone of the following steps the grid point comes in contact with the rigid body, Equations 1 & 2 areapplied. Some small changes need to be introducedinto Equation 1 if it is desired to apply one of FLAC’sdamping schemes which operates on grid points mass.If damping results only from the constitutive model,then Equation 1 is satisfactory. One may refer toEinav & Klar (2003) where the above formulation isextended to a more general case of three-dimensionalrotating rigid-deformable bodies in space.

The described procedure is easily implemented inFLAC using a FISH function, which is called duringeach of the calculation cycles. Generally, since themotion of the contact grid points is solved indepen-dently (i.e. using Equations 1 & 2 rather than by FLAC),they need to be fixed in both directions. Quantitiesrelated to aD are extracted from FLAC’s gridpoint vari-able xforce and yforce. Quantities related to velocities,both readable and writeable, are manipulated usingFLAC’s gridpoint variable xvel and yvel.

5 NUMERICAL ANALYSIS

5.1 Pile shape

In the present numerical analysis the pile tip is repre-sented by a continuous nonlinear shape functionaccording to the following expression:

(3)

where r0 is the radius of the pile and � and bc areparameters that define the shape of the pile tip. Figure 4shows three examples of tip shapes once � is set togive 95% of r0 at z� � 3r0. Due to space limitation, inthe current paper we present results only for pile tipwith bc equal to 2.

5.2 Assumptions

The numerical analysis was conducted under the fol-lowing assumptions:

1. The material behaves elastic perfectly plastic andsatisfies the von-Mises failure criteria. Since theanalysis is associated with undrained loading, andthe volumetric stresses are decoupled from thedeviatoric ones in the considered constitutivemodel, it is possible to perform a “Dry” simulation;i.e. to obtain the excess pore pressure value usingSkempton’s parameter, B � 1 � 1/(1 � (Kw/n)/ Ks),utilizing the formula �u � B(�ii � �ii

0)/3, wherethe superscript 0 denotes initial state.

2. The undrained strength of the soil is definedaccording to the relation Cu � 0.25�v�0OCR0.95,�v�0 is the initial vertical effective stress and OCRis the over consolidation ratio.

3. The shear modulus is taken proportional to theundrained strength Cu, and the bulk modulus washigh enough so the material can be considered asincompressible.

4. The analysis presented herein considered a con-stant OCR with value that equals 2.

5. Initial stress condition corresponded to K0 of 0.7.6. To avoid the kinematic constrains of a fixed bound-

ary, and to allow approximate simulation of an infi-nite soil layer, a prescribed boundary condition wasapplied to the outer radius of the grid. The boundarycondition corresponded to the cylindrical cavityexpansion solution of an incompressible elasticmaterial; i.e. the external pressure acting on the meshwas defined by the analytical solution of the internal

276

bc = 1 bc = 2 bc = 3

Figure 4. Different pile tip shapes.

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pressure of cavity expansion with an identical radiusto that of the outer boundary. It was verified that theplastic zone did not reach the outer boundary, thusthe elastic cavity expansion solution was suited.

6 RESULTS

Figure 5 shows typical distortion of the mesh associ-ated with steady penetration of the pile. It should benoted that analyses with much coarser mesh resultedwith almost identical stress distributions and response,and that it was impossible for the pile to penetrate thesurface without causing a bad geometry, unless morecoarser mesh, than that shown in Figure 5, was pre-scribed near the soil surface. As a rough rule of a thumb,it was found that a contact soil element near the surfaceshould have a radial dimension of about one pile radius,and this can be rapidly become finer as elements aredeeper. Figure 6 shows normalized excess pore pressureassociated with the state of Figure 5.

Figure 7 shows the development of excess pore pres-sure at depth of 25 radiuses for different radial distances(xi is the initial distance from the axis-symmetric line).Figure 8 shows the changes of the second invariant ofthe stress during the installation of the pile for pointslocated a depth of 25 radiuses. The y-axis is normalizedsuch that it gives maximum value of one, in accordanceto the von-Mises yield surface radius. Initial K0 condi-tions create initial value that is different than zero.Clearly, as the tip advances towards the checkpoints, thevalue of the second invariant increases until failure isreached; failure is reached quicker when the points arecloser to the axis.

As discussed in section 2, only Lagrangian largestrain analyses can simulate the penetration of the pilethrough the surface, and therefore are the only onesthat can evaluate the influence of the surface on thecone penetration. Figure 9 shows the cone resistancefactor, Nc, for different rigidity indexes (G/Cu). TheNc factor was calculated according to:

(4)

277

Figure 5. Distorted mesh in steady penetration(G/Cu � 100).

Figure 7. Development of excess pore pressure(G/Cu � 100).

Figure 6. Normalized excess pore pressures (G/Cu � 100).

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where, Ftot is the total vertical force acting on the pile(considering a smooth pile), �v(Zc) is the total verti-cal stress at depth Zc, and defined as the distancebetween the surface and the middle of the pile’s tip(and in our shapes, 1.5r0 above its tip). Note that inFigure 9 the horizontal axis Zc/D does not start atzero. This is due to the fact that the cone must becompletely positioned inside the soil in order for theNc factor to have a proper meaning, if it is obtainedfrom the net vertical force,

Fnet � Nc · Cu · �r20.

As can be seen from Figure 9, as the rigidity indexincreases, both the Nc value and depth in which itbecomes constant increase. The dashed line in Figure 9represents a required depth to obtain 95% of the max-imum Nc values. This depth is referred herein as thedepth of the steady state front, Zss. It is obvious fromFigure 9 that Zone III (see Fig. 3) can be associatedwith spherical cavity expansion solution, as was sug-gested by Yu (2000), only if some minimal pile slen-derness ratio (L/D) is satisfied. It seems that for pilesinstalled in soil with rigidity G/Cu 100 the require-ments for minimum slenderness are irrelevant, sincethe normalized steady state penetration depth is

smaller than any reasonable slenderness ratio (L/D)associated with piles foundations. However, forG/Cu � 100 the normalized steady state depthbecomes in the same order of the piles slendernessratios. For example, for G/Cu � 1118 the pile slen-derness ratio must be greater than 25 in order for thesoil surface to have no effect. In case the pile slender-ness is smaller than that value, the soil surface influ-ences the Nc factor. In such case it is not legitimate touse cavity expansion solutions for obtaining the Ncfactors, as they assume that there is no influence fromsurface. The same may be regarded to the solutionsbased on the strain path method, which also does notconsider the soil surface.


A contact formulation for interaction between rigid anddeformable bodies is presented. This approach over-comes some of the problems associated with FLAC’sown built-in interface formulation when it is applied tononlinear or piecewise linear surfaces. The proposedformulation can easily be used to create rolling fixingalong any line inclination, an option that is currentlyabsent from FLAC. The contact formulation isemployed in the large strain simulation of pile installa-tion. A study on the generation of pore pressures and onsurface effects is presented. There is a strong indicationfrom the analysis results, that the use of cavity expan-sion or strain path method solutions should be carefullyexamined before employed in the estimation of endbearing capacity, specially for low slenderness drivenpiles in soils with high rigidity index.

REFERENCES

Baligh, M.M. 1985. Strain path method, J. Soil Mech. andFound. Div., ASCE, 111(9): 1108–1136.

Durgunoglu, H.T. & Mitchell, J.K. 1975. Static PenetrationResistance of soil, I: Analysis. Proc. ASCE Spec. Conf. onIn Situ Measurement of Soil Properties, New York, Vol. 1151–171.

Einav, I & Klar, A. 2003. An approach for nonlinear contactsurface analysis and application to pile installation. BGAInt. Conf. On Foundations: “Innovations, Observations,Design and Practice”, Dundee, Scotland, Sept 2003.

Itasca Consulting Group, Inc. 2000. FLAC (Fast LagrangianAnalysis of Continua) Ver. 4.0 User’s Manual, Minnea-polis Minnesota: Itasca.

Teh, C.I. & Houlsby, G.T. 1988. Analysis of the ConePenetration Test by the Strain Path Method. Proceedingsof the 6th International Conference on Numerical andAnalytical Methods in Geomechanics, Innsbruck, April,Vol. 1, ISBN 90-6191-810-3, pp 397–402.

van den Berg P. 1994. Analysis of soil penetration. Ph.D. the-sis. The Netherlands: Delft University Press.

Yu, H.S. 2000. Cavity Expansion Methods in Geomechanics.London: Kluwer Academic Publisher.

278

0

0.4

0.8

1.2

0 10 20 30 40Penetration/r0

J 2D

0.5 /(

2/30.

5 Cu

)

xi/r0=0.31xi/r0=1.0xi/r0=1.65xi/r0=3.15xi/r0=5.5xi/r0=10.5

Points dignedwith cone tip

0

0.4

0.8

1.2

0 10 20 30 40Penetration/r0

J 2D

0.5 /(

2/30.

5 Cu

)

xi/r0=0.31xi/r0=1.0xi/r0=1.65xi/r0=3.15xi/r0=5.5xi/r0=10.5

Points dignedwith cone tip

456789

101112131415

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Zc/D

Nc

G/Cu=20

G/Cu=100

G/Cu=500

G/Cu=44.7

G/Cu=223ZcD

G/Cu=1118

Steady state front

Figure 8. Shear behavior (G/Cu) � 100.

Figure 9. Development of Nc factor with depth.

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279

Axial tension development in the liner of a proposed Cedar Hills regionalmunicipal solid waste landfill expansion

F. MaWashington State Department of Ecology Solid Waste Program, Eastern Regional Office, Washington, USA

ABSTRACT: A finite difference analysis using the computer code FLAC was conducted of a municipal land-fill expansion proposal at the Cedar Hills Regional Landfill (CHRL), King County, Washington State. The mainobjective of the modeling efforts was to assess whether a standard design of a liner system would be adequateto withstand typical loading conditions for municipal solid waste landfills. The loading conditions were: (1) thegradual layered waste dumping up to 38 m (125�) over the High Density Polyethylene (HDPE) liner; (2) dynamicloadings caused by a shallow earthquake and a deep subduction zone earthquake; and (3) the simulation of acavity development in the old existing waste underneath the HDPE liner due to collapsing of some bulky items.The FLAC (2D) analyses have revealed (1) the developments of the axial tensile stress and displacement in theHDPE liner; (2) the stress and deformation developments in the municipal solid wastes; and, (3) the accumulativeand separate developments of stress and displacement of the landfill system under waste dumping, earthquakesand cavity collapsing. The main conclusion was that the maximum axial tension in the 60 mil HDPE liner is higherthan the yielding strength of a GSE 60 mil HDPE liner (HDR/Golder 2001) under the proposed site, operationaland loading conditions. Thus some local reinforcement or stronger geomembrane liners will be needed.

1 INTRODUCTION

A finite difference analysis using the computer codeFLAC (Fast Lagrangian Analysis of Continua) wasconducted of a municipal solid waste landfill expansionproposal at the Cedar Hills Regional Landfill (CHRL),King County, Washington State. The work was doneto independently verify FLAC modeling conducted by King County Solid Waste Division’s ConsultantHDR/Golder (2001). The main focus of the analyseswas to predict axial tensions that could develop in the60 mil HDPE geomembrane liner sandwiched betweenexisting wastes and a future 38 m (125 foot) high wastepile. The analyses involved three loading conditions:

1. the future emplacement of wastes;2. earthquakes; and3. a cavity opening up in the existing waste at a shal-

low depth below the proposed liner due to the col-lapsing of some bulky items.

2 SITE CONDITIONS

Landfilling commenced at the CHRL site in the early 60s. Those portions of the landfill started before

1986 lacked a bottom liner (Fig. 1). A portion of theproposed expansion at the CHRL will be located overthe existing wastes. This portion of the expansion foot-print was called liner-over-refuse (HDR/Golder 2001,Fig. 1). The largest axial stresses were expected todevelop in the liner-over-refuse area due to anticipatedexcessive overall and differential settlements of theunderlying wastes. The remainder of the expansion willbe founded on a highly competent glacial till subgradewhere settlements of the liner are anticipated to beminimal and thus settlement induced tensile stresseswould not be of concern.

Figure 1. Cross section of the landfill including location ofexisting wastes, liner, new wastes to be disposed and foun-dation soil.

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3 MODEL CONFIGURATIONS

3.1 Consultant’s study

HDR/Golder (2001) predicted the axial tension in thegeomembrane in the liner-over-refuse area using theFLAC code. This tension was induced by the overallsettlement in the existing and future wastes. The geom-etry of the proposed landfill expansion was used in thisload Case 1 modeling. Load Cases 2 and 3, i.e. the axialstresses induced by earthquake loading and the collaps-ing of a cavity in the existing waste, were analyzedparametrically using the simplifying assumption of ahorizontally stratified site. The modeling assumed a rectangular mesh. The effects of the simplifyingassumptions were unknown and were considered minorand probably conservative. Therefore it is probably rea-sonable to consider the predictions for load Cases 2 and3 as upper bounds of the field behavior of the geomem-brane. This will be further evaluated later in the paper.

3.2 The new models

To further refine the understanding of the overall axialstress in the liner-over-refuse and settlements of theexisting and future wastes, a single model (Fig. 1) isused in these analyses for the three different stages ofloading as stated in the introduction. To make thecomparison easier, the properties of existing, newwastes and interface used in the HDR/Golder (2001)study were adopted.

i. The existing waste is modeled using FLAC’s Mohr-Coulomb option. Using the field measurement ofthe existing waste, stiffness parameters were devel-oped by HDR/Golder (2001) as shown in Table 1.The relationship between unit weight and depth formunicipal solid waste by Kavazanjian et al. (1995)is adopted. The friction angle of 35 degree isassumed.

ii. The new waste is modeled using FLAC’s ModifiedCam-Clay option. The model parameters are elasticshear modulus G � 5.12 � 104kPa (7430.6 psi),maximum elastic bulk modulus Kmax � 1.53 �105kPa (22222.2 psi), density � a variable withdepth (Kavazanjian et al. 1995), slope of elasticswelling liner � � 0.03, slope of normal consolida-tion line � � 0.13, frictional constant M � 1.418,preconsolidation pressure pc � 71.8 kPa (10.4 psi),reference pressure p1 � 71.8 kPa (10.4 psi), andspecific volume at reference pressure on normalconsolidation lien v� � 1.75.

iii. The 60 mil HDPE liner is connected with newwaste above and existing waste or foundation soilbelow by FLAC interfaces. The interface allows therelative slip movements between the liner andwastes or foundation soil. The interface input para-meters are normal stiffness kn � 5.12 � 104kPa

(7430.6 psi), shear stiffness ks � 2.26 � 105kPa(32777.8 psi), and friction angle � � 24°. The60 mil HDPE liner is modeled with isotropic-elastic beam segments; its material properties perunit length are area A � 1.524 mm (0.06 inch), andelastic modulus 9.29 � 105kPa (134722.2 psi) .The unit weight of the 60 mil HDPE geomembraneis 9.26 kN/m3 (59 pcf).

iv. The foundation layer was modeled using FLAC’sMohr-Coulomb option. The material parame-tersare elastic shear modulus G � 5.12 � 104kPa(7430.6 psi), bulk modulus Kmax � 1.53 � 105kPa(22222.2 psi) and frictional angle � 35 degree.

For static analyses, the left and right boundariesare constrained from horizontal movements and thebottom boundary is constrained from both horizontaland vertical movements.

4 MODELING RESULTS

4.1 Case 1 loading from up to 38 m (125�) of solid waste

The increase of tensile forces in the geomembrane in the liner-over-refuse area was modeled by simulatingthe time-history of waste emplacement as a sequence ofsome 3 m (10�) thick layers. The axial stress in the liner-over-refuse and the overall displacement in the solidwaste are shown in Figures 2 & 3, respectively.

280

Table 1. Stiffness of existing waste*.

Distance from left boundary of the Shear modulus Bulk modulusmodel G Km (ft) kPa (psi) kPa (psi)

0–73 (0–240) 1966.9 (285.2) 5245.4 (760.6)73–88 (240–290) 1985.5 (287.9) 5294.4 (767.7)88–104 (290–340) 1924.8 (281.7) 5181.1 (751.3)

104–119 (340–390) 1829.0 (265.2) 4877.4 (707.2)119–134 (390–440) 1818.9 (263.7) 4850.3 (703.3)134–149 (440–490) 1737.5 (251.9) 4633.4 (671.8)149–165 (490–540) 1678.3 (243.4) 4475.5 (649.0)165–180 (540–590) 1562.1 (226.5) 4165.6 (604.0)180–195 (590–640) 1532.4 (222.2) 4086.4 (592.5)195–210 (640–690) 1590.9 (230.7) 4242.3 (615.1)210–226 (690–740) 1640.4 (237.9) 3474.3 (634.3)226–241 (740–790) 1684.7 (244.3) 4492.6 (651.4)241–256 (790–840) 1771.7 (256.9) 4724.5 (685.0)256–271 (840–890) 1948.3 (282.5) 5196.0 (753.4)271–287 (890–940) 2511.1 (364.1) 6696.1 (970.9)287–293 (940–960) 2670.3 (387.2) 7120.8 (1032.5)

* Note: the values are obtained and deduced fromHDR/Golder (2001).

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As shown in Figure 2, the maximum axial tensionsin the geomembrane liner are predicted to occur at theleft anchor trench (El 1 of Fig. 1) and the transitionarea (El 75 of Fig. 1) between the liner-over-refuseand regular liner. The axial tension for the portion ofliner bearing on the competent foundation is minimal.The tension spike at the toe (El 75 of Fig. 1) of theliner-over-refuse is likely caused by the slope transi-tion form 5H : 1V to 3H : 1V.

Figure 3 shows that the center portion around the let-ter A of the new wastes physically displaced the mostat approximate 3 m (10�).

4.2 Representative earthquakes, damping andboundary conditions

The USGS Probabilistic Seismic Hazard Deaggrega-tion website (USGS 1996 maps) identifies two sourcesfor the CHRLF site as the principal contributors to theearthquake hazard. The sources are:

i. a crustal Moment Magnitude (Mw) 6.5–7.0 earth-quake at a hypocentral distance within 20 km; and

ii. a subduction zone earthquake Mw 8.3–9.0 at a hypo-central distance of approximately 135 km.

The near field crustal earthquake has been associatedwith the potential rupture of the Seattle fault; and, thefarther and larger quake represents an interface eventon the Cascadia Subduction Zone along the PacificNorthwest Coast. In this analysis the near field quakewas represented by the velocity time-history derivedfrom the acceleration time-history recorded from the M7.3 Landers earthquake on June 28, 1992 inCalifornia (HDR/Golder 2001); and, the June 23, 2001M7.9–8.4 Peru Earthquake was considered represen-tative of the larger subduction earthquake. The acceler-ation time-history for this subduction earthquake was

recorded at Moquegua, Peru, which is approximately190 km southwest of the epicenter. The peak acceler-ations for the east-west and north-south time-historiesare approximately 30 and 20% g, respectively. Theseaccelerations fall within the ranges of predicted peakaccelerations (mean plus one standard deviation) fromattenuation relationships developed for the PacificNorthwest (Crouse 1991, Youngs et al. 1997).

The Moquegua record was 200 seconds long. Afterthe first 120 seconds the shaking produced minimalchanges in axial tension. To facilitate further model-ing, only the first 120 seconds of the record were used.As the site is asymmetric and the plastic nature of thewastes modeled, the time history was reversed (multi-plied by �1) to account for directivity effects. Every-thing else remained the same. As the results showedalmost no impact, further analyses were done usingonly the unmodified earthquake time history. Sincepeak acceleration in the E-W direction is about 50%higher that of the N-S direction, both records of theMoquegua time history were used in the analyses.

Only 5% Raleigh damping is used for the dynamicanalyses. Before running the dynamic analyses, thefree field boundary condition of FLAC is applied tothe numerical model. Then, either a velocity-time oracceleration-time history is applied from the bottomof the numerical model.

4.3 Dynamic analyses under the subduction 2001Moquegua, Peru, earthquake time-histories

Using the N-S record, the cumulative development ofaxial tension from the dynamic loading on top of thestatic results (Fig. 2) is summarized in Figure 4.

Under dynamic loading, axial tension increasedapproximately 70% in the geomembrane in the lowerhalf of the liner-over-refuse segment, while the upperhalf stayed almost the same. As a result the maximumtension is predicted to shift to the break in slope (tran-sition from the 5H : 1V to 3H : 1V grade). To furtherillustrate this effect, the axial tension histories atthese two locations are shown in Figures 5 & 6.

The very different predicted responses of the twolocations are likely due to the different overburdenconditions, the slope or slope change of the liner, etc.For example, the overburden for El 1 is only a couple

281

Figure 2. Distributions of axial tension in the geomem-brane liner.

Figure 3. Vertical-displacement contour.

Figure 4. Distributions of axial tension in the geomem-brane liner due to static and dynamic loadings.

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of meters; but El 75 is under approximately 23 m (75�)of solid waste and the slope of the liner at this locationchanged from 5H : 1V to 3H : 1V. Thus, due to theplastic nature of the solid wastes, as they were mod-eled by the Cam-Clay Elasto-Plasticity Model, the axialtensions in El 75, were not released when the earth-quake wave reversed its direction for the cases whenthe ground accelerations were relatively large.

Similarly, the axial tensions predicted in the linerfrom the Moquegua, Peru 2001, E-W accelerationtime history are summarized in Figures 7–9.

As shown in Figures 7–9, the maximum axial tension occurred at the same location but is projectedto be 70% larger when using the Moquegua, Peru E-W acceleration record rather than the N-S record.

282

Figure 5. History of axial tension development in the liner-over-refuse due to static and dynamic loadings at EL 75.

Figure 6. History of axial tensions at EL 1 in response tothe static and dynamic loadings.

Figure 7. Distributions of axial tension in the geomem-brane liner due to static and dynamic loadings.

Figure 8. History of axial tension development in the liner-over-refuse due to static and dynamic loadings at El 75.

Figure 9. History of axial tension development in the liner-over-refuse due to static and dynamic loadings at El 1.

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The increase reflects the larger peak ground acceler-ation of the E-W record.

4.4 Dynamic analyses using the 1995 Landers,California, earthquake

The cumulative development of axial tension on topof the static results (Fig. 2) in the liner from the 1995Landers quake is summarized in Figure 10. Similarlythe axial tension history at the two locations of El 1and El 75 are shown in Figures 11 & 12.

4.5 Loading from a cavity collapsing in the existing wastes

Since local different settlement can be very detrimen-tal to the integrity of liner system, the effects of a voidin the waste caused by deterioration of a large metallicobject were investigated. Although the existing wasteis covered by a 4- to 7-foot-thick interim soil cover, ageophysical survey was conducted and did not indi-cate any large metallic objects at depths down to about10 feet. However, it is theoretically possible that col-lapse of such a void could cause a potential local set-tlement up to 0.9 m (3�) (HDR/Golder 2001). It ismore likely that a collapsing bulky item would cause

a local settlement up-to 0.9 m (3�) after an earth-quake. Such localized collapsed items were assumedto occur at varying depths below the geomembrane inthe liner-over-refuse area. The worst case scenario wasassumed to be the collapse of the cavity following amajor earthquake. Therefore, the local settlement inthe old wastes was modeled after the dynamic analyses,i.e. after subjecting the model to the E-W Moqueguaacceleration time-history. The original wastes arescheduled to be capped with a minimum of 3 m (10�)of sand to act as a cushion. Thus, the cavity was placedat a minimum 3 m (10�) below the liner. Cavities weresimulated at greater depths in the existing waste atlocations below the upper-middle part of the liner-over-refuse as pointed in the Case B of Figure 13.However, the maximum tensile stress in the geomem-brane resulted from the shallowest assumed cavityposition. Thus, the results of the deeper cavities werenot included here. Also, since the cavity collapsing wasstress-controlled, the deformation at locations A, B andC in Figure 13 were only approximately 0.9 m (3�).This deficiency would not have substantial impact onthe results summarized in Figure 13.

In Figure 13 as the location of the cavity changesfrom the toe of the existing waste (Case A) to the mid-dle of the waste (Case B), the predicted axial tensionsincrease above the void by 3114 to 4448 N (700 to1000 lbs). When the void was placed near the top ofthe slope (Case C of Fig. 13), the axial tension insteaddecreased by approximately one half of the originalvalue. The reason for the decrease was that the voidcaused the anchor trench of the liner to move towardthe collapsed hole and therefore relaxed the axial ten-sion in the liner. Figure 13 shows that the maximumaccumulated axial tension from the three loading sce-narios will occur near the toe of the existing waste

283

Figure 10. Predicted distribution of axial tension in thegeomembrane liner due to static and dynamic loadings.

Figure 11. History of axial tension development in the liner-over-refuse due to static and dynamic loadings at EL 75.

Figure 12. History of axial tension development in theliner-over-refuse due to static and dynamic loadings at EL 1.

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and the maximum axial tension in the geomembranewas only slightly impacted by the location of the col-lapsed void as in Case A of Fig. 13.

5 COMPARING CURRENT PREDICTIONS TOTHOSE OF MODELING BASED ONSIMPLIFIED ASSUMPTIONS(HDR/GOLDER, 2001)

The contributions to the maximum axial tension inthe geomembrane liner from the three loading scenar-ios by current and simplified models (HDR/Golder2001) are compiled in Table 2.

Table 2 showed that the maximum tension at thelower edge of liner-over-refuse based on current model(Case 11) are very similar to that based on the simpli-fied assumptions (Case 20, HDR/Golder 2001); andthe tension caused by the subduction Moqueguaquake represented by the E-W acceleration time-history is about 50% larger than that based on thecrustal near field quake using the Landers velocitytime-history (Case 11 vs. 13).

CONCLUSIONS

First, the maximum axial tensions occur at differ-ent locations in the liner-over-refuse segment of the

geomembrane depending on the loading condition. Themaximum axial tension under three loading conditionsat El 75 is 8301 N (1867 lb) per foot. This is higher thanthe yielding strength of 6938 N (1560 lb) per foot fora GSE 60 mil HDPE liner (HDR/Golder 2001). Thussome local reinforcement or stronger geomembraneliners will be needed. Although the current modelingresults of maximum tensions are very similar to theinterpretations of the HDR/Golder’s work (2001), theadvantage of analyzing the three loading conditionson the same numerical model is that it provides theauthor with a clearer understanding of where and howthe axial tensions in the liner developed as the wastepile rises and is subject to strong shaking and possiblythe development of shallow voids. At the same time italso confirmed that the approach of superimposingtensions from individual, simplified loading mecha-nisms (HDR/Golder 2001) can yield reasonable resultscomparing with more complex modeling efforts.

Secondly, for locations like Cedar Hills in westernWashington State, both the crustal earthquake as rep-resented by the Landers velocity records (HDR/Golder2001); and, the subduction earthquake along the PacificNorthwest Coast need to be considered, since the ten-sion caused by the subduction quake (Moquegua E-Wacceleration time-history) was almost 50% larger thanthat caused by the near field crustal quake (Table 2).The selection of appropriate source zones is necessaryto envelop the likely seismic response. This process is greatly aided by the USGS Probabilistic Seis-mic Hazard Deaggregation website. Further, the moresevere responses to the strong distant subductionMoquegua E-W acceleration time-history may be dueto its low frequency and long duration. A similar phe-nomenon has been observed by Matasovic et al. (1998).

284

Figure 13. Predicted axial tension in the geomembraneliner due to a cavity collapsing variously at locations A, Band C at a depth of 3 m (10�) under the liner in addition tostatic and dynamic loadings.

Table 2. Maximum tensions in the geomembrane linerfrom FLAC models.

Static Dynamic Cavity TotalCase tension tension tension tensionNo. N (lb) N (lb) N (lb) N (lb)

11 3018(678) 5175(1165)1 108(24) 8301 (1867)12 3018(678) 3038(683)2 N/A 6056 (1361)13 3018(678) 3519(791)3 N/A 6537 (1469)204 2893(650) 890(200) 4448(1000) 8231 (1850)

1Note: Modeling results using Moquegua, Peru E-W Acc.-time history;2Note: Modeling results using Moquegua, Peru E-W Acc.-time history;3Note: Modeling results using Landers, California Vel.-time history; and4Note: Modeling results recommended for 100-foot-widetransition zone along lower edge of liner-over-refuse area(HDR/Golder, 2001) using Landers, California Vel.-timehistory and based on simplified assumption.

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ACKNOWLEDGMENTS

The author wishes to sincerely thank the King CountySolid Waste Division (KCSWD) for its permission topublish the paper. Special thanks to the reviews andcomments of Dr. Victor Okereke of KCSWD and Mr. Frank Shuri of Golder Associates Inc. It is worthnoting that the current work is a follow up to theHDR/Golder’s earlier work (HDR/Golder 2001).

Special thanks also to Mr. Jerald LaVassor ofWashington State Department of Ecology WaterResources Program Dam Safety Section. The author isindebted deeply to his direction, support and invalu-able technical and editorial revisions of the paper.

REFERENCES

Crouse, C. 1991. Ground-motion attenuation equations for Cascadia subduction zone earthquakes. EarthquakeSpectra, 7, 201–236.

HDR Engineering, Inc. & Golder Associates Inc. 2001.Cedar Hills Regional Landfill Area 6 Development DraftPreliminary Design Technical Memorandum Lining Sys-tem Over Unlined Waste Area. Seattle: King CountyDepartment of Natural Resources Solid Waste Division,Washington State.

Kavazanjian, E. Jr., Matasovic, N., Bonaparte, R. andSchmertmann, G. R. 1995. Evaluation of MSW proper-ties for Seismic Analysis. Geoenvironment 2000, ASCEGeotech. Spec. Publ. No. 46, 2, 1126–1141.

Matasovic, N., Kavazanjian, E. Jr., and Anderson, R. 1998.Performance of solid waste landfills in earthquakes,Earthquake Spectra, Issue #2, Vol. 14, p. 319–334.

Youngs, R. R., Chiou, S.-J., Silva, W. J. and Humphrey, J. R.1997. Strong Ground Motion Attenuation Relationshipsfor Subduction Zone Earthquakes. Seismological ResearchLetters. Vol. 68, No. 1, Jan./Feb. 58–73.

USGS Probabilistic 1996. Seismic Hazard Deaggregationwebsite (http://eqint1.cr.usgs.gov/eq/html/deaggint.shtml).

285

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287

The usability analyses of HDPE leachate collection pipes in a solid waste landfill

F. MaWashington State Department of Ecology Solid Waste Program, Eastern Regional Office, Washington, USA

ABSTRACT: The objective of the FLAC modeling effort was to analyze the stress–strain behavior of HighDensity Polyethylene (HDPE) leachate collection pipes proposed for an eastern Washington State solid wastelandfill. Case one of the FLAC modeling assessed whether a 102 mm (4�) diameter Standard Dimension Ratio(SDR) 9 perforated HDPE leachate collection pipe could withstand the load of up to 64 m (210�) high columnof solid waste. Similarly case two of the FLAC modeling predicted how a 305 mm (12�) diameter perforatedSDR 11 HDPE pipe would perform under a solid waste load of as much as about 26 m (85�). The FLAC analy-ses allowed simulating the development of stresses and deformations in the HDPE leachate pipes as the solidwaste column grows. The model predictions were compared with results from a methodology included in theGuidelines for HDPE Pipes in Deep Fills (Petroff 1998) used by CH2MHILL (2002) in their design. Using theindustry standards of (1) ring compressive stress, (2) pipe deflection and (3) wall buckling, the FLAC resultswere very similar to the values in the CH2MHILL (2002) study. The main conclusion of the FLAC modeling isthat the proposed HDPE leachate collection pipes will be adequate to withstand the loadings associated with theproposed solid waste column heights.

1 INTRODUCTION

Finite difference analyses using the computer codeFLAC (Fast Lagrangian Analysis of Continua) wereconducted of a solid waste landfill expansion projectat the Graham Road Landfill, Spokane County,Washington State. The analyses were performed toconfirm the adequacy of modeling done by the proj-ect engineer, CH2MHILL. Specifically, the analysesfocused on predicting the stress–strain responses of:

i. a 102 mm (4�) diameter SDR 9 perforated HDPEleachate collection pipe under solid waste load upto 64 m (210�), and

ii. a 305 mm (12�) perforated SDR 11 HDPE pipeunder a solid waste load of about 26 m (85�) whenthe landfill is under final closure, respectively.

2 SITE CONDITIONS

The landfill accepts solid wastes from industries andother sources, but it does not accept municipal solidwastes. The leachate collection and recovery system(LCRS) consists of a 0.3 m (12�) thick granulardrainage layer with embedded, perforated HDPE pipesto collect and remove leachate. This LCRS directly

overlies the bottom composite liner of the landfill(CH2MHILL 2002).

3 MODEL CONFIGRATIONS

3.1 Consultant analyses

CH2MHILL (2002) evaluated the adequacy of the102 mm (4�) diameter SDR 9 LCRS header pipe and305 mm (12�) SDR 11 sump/pump access pipe by theprocedures included in “Guidelines for HDPE Pipes inDeep Fills” (Petroff 1998). The methodology assesses(1) ring compressive stress, (2) pipe deflection, and(3) wall buckling. The evaluation confirmed that the102 mm (4�) SDR 9 and the 305 mm (12�) SDR 11were adequate for the anticipated 64 m and 26 m(210� and 85�) of overlying fill, respectively. It shouldbe noted that the analysis assumes the bedding encap-sulating the pipe will be compacted to a minimum of90 percent of the maximum density as determined byASTM Procedure D698. This compacted zone mustextend immediately above the pipe and for 5 feet oneither side of the pipe. The specifications prepared byCH2MHILL (2002) accordingly required the com-paction of the drain rock around the pipe to the abovecited minimums.

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3.2 This study’s current modeling

The empirical equations provided in the Guidelinesfor HDPE Pipes in Deep Fills (Petroff 1998) predictthe maximum ring compressive stress and the maxi-mum pipe deflection. The FLAC modeling describedherein went on to predict the distribution of the axialstress, shear stress and bending momentum in thepipes as well as the stress–strain or deformation rela-tionship under plane strain conditions.

The following simplifications or assumptions weremade in the FLAC modeling.

i. The HDPE pipe is modeled with isotropic-elasticbeam segments; its material properties per unitlength are area A � t, moment of inertia I � t3/12,elastic modulus under plane strain e � ey /(1 � �2), where t is the pipe wall thickness, ey isthe Young’s modulus and � the Poisson ratio. Thematerial properties of the 102 mm (4�) SDR 9 and305 mm (12�) SDR 11 pipes are listed in Table 1.The effect of perforations in the HDPE pipes wasmodeled by reducing the wall thickness by onetwelfth as typical perforations account for thatmuch of the pipe mass.

ii. The gravel drainage layer is simulated as a perfectplastic Mohr-Coulomb material. The materialproperties include bulk modulus K, shear modu-lus Gs, friction angle and density. The average val-ues of density and friction angle of gravel are2.16 g/cm3 (135 lb/ft3) and 40 degree, respec-tively. As adapted and extended from McGrath

(1994) by Petroff (1998), the typical design valuesof one-dimensional constrained modulus Ms of soilincrease linearly with the increase of the soil over-burden pressure. This linear relationship (Petroff1998) was used in the FLAC modeling and was related to bulk modulus K and shear modulusGs by (K � Ms (1 � �)/(3 � (1 � �)) andGs � Ms(1 � 2�)/ (2(1 � �), respectively. Theelastic modulus Es is related to the constrainedmodulus Ms of the soil by Es � Ms(1 � �)(1 � 2�)/(1 � �)). The material properties of thegravel layer are listed in Table 2.

iii. The waste behavior is simulated by the elasto-plastic Modified Cam-Clay model in FLAC. Thebehavior of the HDPE pipes is the focus of themodeling effort here. The stress-strain response ofthe waste was of little interest. Thus, the wasteproperties approximated with values typical ofsoft clay at a density of 1441 kg/m3 (90 lb/ft3) wereused as the input parameters of the wastes. Themodel parameters are elastic shear modulusG � 5.12 � 104kPa (7430.6 psi), maximum elastic bulk modulus Kmax � 1.53 � 105kPa(22222.2 psi), density � a variable with depth(Kavazanjian et al. 1995), slope of elastic swellingliner � � 0.03, slope of normal consolidation line� � 0.13, frictional constant M � 1.418, precon-solidation pressure pc � 71.8 kPa (10.4 psi), refer-ence pressure p1 � 71.8 kPa (10.4 psi), andspecific volume at reference pressure on normalconsolidation lien �� 1.75.

288

Table 1. Material properties of HDPE pipes.

d t A I eSetting kg/m3 mm mm2 mm4 kPa

4�� SDR 9* 955.2 12.7 322.6 4.34E � 3 2.14E � 512�� SDR 11* 955.2 29.4 746.8 5.38E � 4 2.14E � 5

* Note: the values are obtained or deduced and deducted from Driscoppipe data sheetsand are values per 25.4 mm (1 inch) of the pipes.

Table 2. Material properties of gravel layer.*

Overburden Ms K GskPa (psi) kPa (psi) kPa (psi) kPa (Psi)

68.9 (10) 10,342.1 (1500) 6405.2 (929) 2957.9 (429)137.9 (20) 11,721.1 (1700) 7253.3 (1052) 3350.9 (486)275.8 (40) 14,479.0 (2100) 8963.2 (1300) 4136.9 (600)413.7 (60) 17,236.9 (2500) 10,673.1 (1548) 4922.9 (714)551.6 (80) 19,994.8 (2900) 12,376.1 (1795) 5715.8 (829)689.5 (100) 22,063.2 (3200) 13,658.5 (1981) 6301.8 (914)

*Note: the values are obtained or deduced and deducted from Petroff (1998) withPoisson ratio � � 0.3.

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iv. The HDPE pipe is connected to the surroundinggravel layer using the FLAC interface. The inter-face allows the relative slip between the HDPEpipe and the gravel. The interface input parame-ters per unit length are normal stiffnesskn � 411.6 N/mm (2341 lb/in), shear stiffnessks � 107.8 N/mm (619 lb/in), and friction angle� � 30°. Due to limited data availability on theinterface properties, more studies will be done insome future researches.

The details of the model grids for the two scenariosare shown in Figures 1 & 2. The horizontal dimensionof the grid was chosen such that a further increase inwidth had no material impact on the modeling results

of the pipe response. When executing the models, thewaste was added layer by layer to simulate a landfilloperation. As the waste pile grew, increasing the verti-cal overburden pressure, the modulus of the gravel layerincreased accordingly (Petroff 1998). Since the rela-tively small sizes of the HDPE pipes, no further refine-ments of the mesh around the pipe openings were done.

4 MODELING RESULTS

4.1 305 mm (12) SDR 11 HDPE LCRS headerpipe under solid waste load up to 26 m (85)

The axial compression, shear and moment distribu-tion for the 305 mm (12�) SDR 11 HDPE LCRSsump/pump pipe under solid waste of 26 m (85�) areshown in Figure 3. Since the results for pipes with orwithout perforations are very similar, only the latterare shown here. Numerical values of the data showngraphically in Figure 3 are presented in Table 3.

289

Figure 1. Cross section of the 305 mm (12�) SDR 11HDPE pipe in the landfill leachate collection layer under the26 m (85�) of solid wastes.

Figure 2. Cross section of the 102 mm (4�) SDR 9 HDPEpipe in the landfill leachate collection layer under the 64 m(210�) of solid wastes.

Figure 3. Predicted distributions of axial compression,shear and moment of the 305 mm (12�) SDR 11 HDPE pipewithout perforation under a simulated solid waste loading of26 m (85�).

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Figure 3a shows that the maximum hoop compres-sive force occurs near the pipe springline and isapproximately 50% larger than that at the pipe crownand invert. The maximum shear force occurs atapproximately 45 degrees below the pipe crown as inFigure 3b. Figure 3c shows that the moment near thepipe crown and springline are approximately 25%larger than that at pipe invert.

4.2 102 mm (4) SDR 9 HDPE LCRS header pipeunder solid waste load up to 64 m (210)

The axial compression, shear and moment distributionsfor the 102 mm (4�) SDR 9 HDPE LCRS header pipepredicted for 64 m (210�) of solid waste are shown inFigure 4. As in the earlier case, the results for pipes withor without perforations are very similar. Thus, only thelatter are shown here. The numerical results underlyingthe graphical data in Figure 4 are cited in Table 4.

Similar trends are evident in the loadings predictedin Figure 4 to those predicted for the 12� SDR 11 pipecase. However, the differences between the values ofhoop compression and moments at the pipe crown,springline and invert are much smaller, see Table 4

and Figures 4a&c. As in the case of the 305 mm (12�)SDR 11 pipe, the maximum shear force is predictedto occur at approximately 45 degrees below the pipecrown, see Figure 4b.

4.3 Pipe crown deflections

The predicted crown deflection of a buried pipe is oneof the key parameters in assessing the structural ade-quacy of a pipe in traditional pipe analyses. The pre-dicted displacement histories of the pipe crown andinvert are shown in Figures 5 & 6.

4.4 Comparing FLAC predictions to those oftraditional empirical formulas (Petroff 1998)

According to general thin beam theory, the normalstress in the pipe wall is the combination of normalstresses from the hoop force and the bendingmoments. It is expressed as follows:

(1)

290

Table 3. Numerical results corresponding to Figure 3.

Elem F-Shear F-axial Mom-1 Mom-2ID* Nod1* Nod2* N N N-m N-m

24 24 25 27.6 841.8 �4.60 5.1823 23 24 22.2 855.5 �4.15 4.6122 22 23 15.6 892.8 �3.82 4.1521 21 22 39.5 949.6 �2.98 3.8220 20 21 39.5 989.8 �2.16 2.9819 19 20 48.1 1068.2 �1.15 2.1618 18 19 74.0 1166.2 0.39 1.1517 17 18 78.0 1254.4 2.03 �0.3916 16 17 64.1 1332.8 3.36 �2.0315 15 16 86.7 1411.2 5.17 �3.3614 14 15 47.4 1479.8 6.16 �5.1713 13 14 �10.3 1489.6 5.95 �6.1612 12 13 8.8 1499.4 6.13 �5.9511 11 12 �8.1 1506.3 5.96 �6.1610 10 11 �7.45 1460.2 4.44 �5.969 9 10 �73.0 1381.8 3.20 �4.448 8 9 �49.9 1303.4 2.17 �3.207 7 8 �93.8 1244.6 0.21 �2.176 6 7 �75.3 1136.8 �1.36 0.215 5 6 �69.2 1048.6 �2.81 1.364 4 5 �84.1 989.8 �4.57 2.813 3 4 �38.6 912.4 �5.37 4.572 2 3 �35.6 860.4 �6.13 5.371 1 2 �35.7 837.9 �6.87 6.13

*Note: Elem ID 1 corresponds to the pipe segment at pipe crown with nods 1 and 2. Elem ID24 to the pipe segment at pipe invert. F-shear and F-axial are the shear and axial forces of theeach pipe segment, respectively. Mom-1 and Mom-2 are the moments of both ends of eachpipe segment.

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where � � Normal stress, kPa (psi); N � compres-sive hoop force, N (lb); A � cross-sectional area ofpipe wall, m2 (in2); M � bending moment, N-m (lb-in); I � inertia of wall cross-section, m4 (in4); andt � pipe wall thickness, m (in).

For a pipe under plane strain conditions, only a unitlength of pipe need to be considered in Equation 1.Using the modeling results of Tables 3 & 4 and thematerial properties and geometries of the pipes (Table1), the normal stress distributions from inner to outerfibers in the pipe wall are shown in Figures 7 & 8using Equation 1. Figures 7 & 8 clearly showed thatthere are only four locations where the pipe wall isunder uniform compression. For the rest of the pipesection, larger compressive stresses exist either in the

outer or inner fibers of the pipe wall. The higher com-pressive stresses from both the hoop and bendingeffects are predicted to occur at the crown, invert orspringline of the pipe. Furthermore, the maximumhoop and total compressive stresses occur at or nearthe springline of the pipe.

The predicted maximum hoop and compressivestresses and the crown deflection of the pipe from themodeling results are summarized in Table 5 along withthe results of CH2MHILL (2002). A comparison ofdata shows the CH2MHILL results based on Petroff(1998) are in relatively good agreement with theFLAC results. Note that the predicted displacements

291

Figure 4. Predicted distributions of axial compression,shear and moment per 25.4 mm (1�) of the 102 mm (4�) SDR9 HDPE pipe without perforation under solid waste loadingof 64 m (210�).

Table 4. Numerical results corresponding to Figure 4.

Elem F-Shear F-axial Mom-1 Mom-2ID* Nod1* Nod2* N N N-m N-m

12 12 13 33.0 661.5 �2.07 2.5611 11 12 65.9 698.7 �1.11 2.0710 10 11 80.7 751.7 0.07 1.119 9 10 54.1 792.8 0.87 �0.078 8 9 70.2 852.6 1.89 �0.877 7 8 58.2 941.8 2.74 �1.896 6 7 �40.1 933.0 2.16 �2.745 5 6 �38.5 895.7 1.59 �2.164 4 5 �95.1 861.4 0.20 �1.593 3 4 �100.4 789.9 �1.26 �0.202 2 3 �70.7 708.5 �2.30 1.261 1 2 �46.1 672.3 �2.97 2.30

*Note: Elem ID 1 corresponds to the pipe segment at pipe crownwith nods 1 and 2. F-shear and F-axial are the shear and axial forcesof the each pipe segment, respectively. Mom-1 and Mom-2 are themoments of both ends of each pipe segment.

Figure 5. Displacement and crown deflection of the buriedHDPE 305 mm (12�) SDR 11 Pipe.

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of CH2MHILL’s (2002) modeling are similar to thoseof the solid pipe in the FLAC analyses. However,CH2MHILL’s prediction of crown deflections arelower than those of FLAC when modeling a perforatedpipe. Overall the results of both methods are within orare very close to the allowable ranges of industry stan-dards of 6.89 � 103kPa ( 1000 psi) hoop compressivestress and 7.5% crown deflection (Wilson-Fahmy andKoerner 1994).

4.5 Buckling

Ideally, a more sophisticated pipe model would beemployed to account for the visco-plastic behavior ofHDPE pipes. This would allow incorporating creepand pipe buckling effects into the FLAC modelingresults. This was not done. Instead, the factor of safetyagainst buckling was assessed by simply computingthe ratio of computed stresses to the critical bucklingstresses as per Wilson-Fahmy and Koerner (1994).According to their study, the critical buckling stressesfor pipes of SDR 11 and 9 are 1.03 � 104 and1.17 � 104kPa (1500 and 1700 psi), respectively.

292

Table 5. Maximum stresses from FLAC and empirical formulas.

PipeMax-hoop Max-comp crown

Pipes Model stress stress deflectionSDR/D type kPa kPa %

11/12 FLAC1 2.01 � 103 3.68 � 103 3.6FLAC2 2.16 � 103 3.72 � 103 4.0CH2MHILL3 1.73 � 103 3.5

9/4 FLAC1 2.94 � 103 6.93 � 103 6.7FLAC2 3.10 � 103 6.99 � 103 7.7CH2MHILL3 3.78 � 103 6.8

1Note: Modeling results ignoring pipe perforations;2Note: Modeling results considering pipe perforations; and3Note: See CH2MHILL (2002) study for details.

-4 .00E+03

-2 .00E+03

0.00E+00

2.00E+03

4.00E+03

6.00E+03

8.00E+031

23

4

5

6

7

8

9

10

1112

1314

15

16

17

18

19

20

21

22

23

24

Inner fiber Outer fiber Hoop Comp.

-1.00E+03

0.00E+00

1.00E+03

2.00E+03

3.00E+034.00E+03

12 3

45

67

89

10

11

12

13

14

15

16

17

1819

2021

222324252627

2829

3031

32

33

34

35

36

37

38

39

40

41

4243

4445

4647 48

Inner fiber Outer fiber Hoop comp.

Figure 7. Hoop compressive stress and normal stressesalong the inner, outer fibers per 25.4 mm (1�) of the 102 mm(4�) SDR 9 HDPE pipe without perforation (unit in kPa).

Figure 8. Hoop compressive stress and normal stressesalong the inner and outer fibers per 25.4 mm (1�) of the305 mm (12�) SDR 11 HDPE pipe without perforationincluded (unit in kPa).

Figure 6. Displacement and crown deflection of the buriedHDPE 102 mm (4�) SDR 9 Pipe.

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Dividing the critical buckling stresses by the pipe wallhoop stresses in Table 5, yields factors of safety of4.79 and 3.79, respectively, for 12� SDR 11 and 4� SDR9 pipes with perforation. These values exceed industrypractice of a minimum factor of safety of 2.

5 CONCLUSIONS

First, the results presented in Figures 7 & 8 demon-strate graphically the predicted stress distributionalong the pipe wall. Based on these results, a stressinduced failure of the pipe would be judged accept-ably remote. If one where dealing with appreciablygreater waste depths one would expect pipe over-stressing to initiate near the springline of the pipe.

Second, the results based on the “Guidelines forHDPE Pipes in Deep Fills” (Petroff 1998) are rela-tively close to the FLAC modeling results. Therefore,the more sophisticated FLAC modeling is likely unnec-essary. It would seem warranted only when conven-tional empirical methods indicate the pipe stresses andcrown displacements approach the minimum acceptedfactor of safety on a critical project.

ACKNOWLEDGMENTS

The author wishes to thank Waste Management (WM)Northwest for its permission to publish the paper. Spe-cial thanks to the review and comment of Mr. RodgerNorth of WM.

Special thanks also to Mr. Jerald LaVassor ofWashington State Department of Ecology WaterResources Program Dam Safety Section for his invalu-able technical and editorial revisions of the paper.

REFERENCES

CH2MHILL. 2002. Cells 4 and 5 Design Report, GrahamRoad Recycling and Disposal Facility. Spokane,Washington: Waste Management, Inc.

McGrath, T. 1994. Analysis of Burns & Richard Solution forThrust in Buried Pipe. Simpson Gumpertz & Heger, Inc,Cambridge, Massachusetts.

Petroff, L. 1998. Guidelines for HDPE Pipes in Deep Fills,(written under the employment of PLEXCO).

Wilson-Fahmy, R.F. & Koerner, R.M. 1994. Finite ElementAnalysis of Plastic Pipe Behavior in Leachate Collectionand Removal Systems. Geosynthetic Research Institute,Drexel University.

293

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295

FLAC numerical simulations of the behavior of a spray-on liner for rock support

C.P. O’Connor & R.K. BrummerItasca Consulting Canada Inc, Sudbury, Ontario, Canada

G. SwanFalconbridge Ltd, Sudbury Operations, Sudbury, Ontario, Canada

G. Doyle3M Canada Co., Mining Division, London, Ontario, Canada

ABSTRACT: The current practice of bolting and screening of underground excavations is time consuming andlabor intensive and requires extensive materials handling. In pursuit of alternative rock support systems,Falconbridge Ltd. has experimented with several different spray-on liners. In co-operation with 3M Canada, a thinspray-on liner was developed with the intention of replacing screen and reducing the cycle time in rapid develop-ment mining. FLAC was used on this project to provide an efficient method of investigating alternative materialproperties without the expense that is typically incurred in full scale testing.

1 INTRODUCTION

The implementation of high speed development tech-niques necessitates the use of rapidly installed supportin order to meet the desired cycle time. Current prac-tices of bolting and screening are labor intensive andadd significant time to the development cycle. The useof spray-on-liners to act as membrane support in placeof screen is seen as the next step in the evolution ofrapid development.

The physical characteristics of membrane supportneed to be extensively tested. Full scale physical testingis expensive and time consuming and the number ofsuch tests needs to be limited. In order to fill the gapbetween laboratory measurements of liner propertiesand full scale trials of the material, a numerical modelof the testing apparatus was constructed. This providesa method by which a large number of potential linerformulations can be investigated without the expense offull scale testing.

Using the cable element capability of FLAC, a modelof the “baggage loading” testing apparatus was gener-ated to allow for a large number of simulations to becarried out to cover the wide range of propertiesobserved in the liner material.

2 LINER SUPPORT SYSTEM

The spray on liner support system developed by 3MCanada is a polymeric compound that contains approx-imately 40% water when initially applied. As the linerdries out (the rate is dependant on the ambient temper-ature, humidity, and air flow) the strength builds untilafter 24 to 72 hours it approaches its ultimate tensileand adhesive strength. The time dependence of thestrength of the liner is a critical aspect and one that isvery difficult to define in full scale testing apparatussince there are so many variables involved.

Shotcrete is often sprayed in excess of 4�, this linersystem only requires 2 to 3 mm final dried thickness toperform. When dealing with such a thin application,minor thickness variations become important.

Another important characteristic of this liner com-pound is the elongation potential. Depending on the formulation in use, strains ranging from 100 to 600%are possible. Having such a large capacity to deformshould help prevent violent failures due to stress build up. A stress–strain curve for some early liner for-mulations is shown in Figure 1.

One final challenge in understanding this material isthat it does not yield in a linear fashion. Instead, there

09069-35.qxd 08/11/2003 20:40 PM Page 295


is a well-defined initial yield point located near astrain of 30% after which the stiffness of the entiresystem reduces until the ultimate tensile strength isreached.

All of these different parameters (thickness, timedependent strength, complex yield curve, and exten-sive deformations) must be accounted for and

modeled. There are currently two tests in use for theliner material. The first is the “dog bone” test, per-formed on small samples of material and used to gen-erate the stress–stain curves. The second is the“baggage load” test in which a 1 m2 metal frame isfilled with rock and sprayed with a coating of linerand tested to failure in a press (see Fig. 2).

3 FLAC MODEL

A FLAC model was built to replicate the baggage loadtesting apparatus. This involved a complex interactionbetween several different types of materials and cableelements. The actual grid used in the model is shown inFigure 4. The liner is represented by a string of cableelements across the bottom boundary of the slabbygranitic material.

A key challenge in the modeling was obtaining thecorrect response of the liner as the stiffness changes inresponse to plastic deformation. To do this, a FISH func-tion was written in which the modulus of the materialwas dynamically adjusted within specific strain inter-vals defined in a table. This allows for automatic adjust-ments of the material properties during cycling ensuringan accurate response (see Fig. 5).

The stiffness of the material depends on the amountof drying of the liner, and therefore the modulus of thematerial changes over time (see Fig. 7). In order to prop-erly assess the effect of drying time on the performanceof the liner, a sensitivity analysis was required in orderto determine how the liner responds under differentconditions of drying time and thickness.

There is only a limited database of full scale testingon which to calibrate the model. Using this limited dataand the material properties that were expected to be pro-duced by the test (i.e. 4 hour drying time, 2 mm thick-ness), the model was calibrated as best as possible to thetest results available (see Fig. 6).

296

0

Str

ess

(MP

a)

Stress–strain curve of System I and II

System II with strong adhesion

Percent strain

System I

800700600500400300200100

Figure 1. Stress–strain curve for early revisions of the 3Mspray on liner product.

Figure 2. Baggage load testing frame with a 2 to 3 mmsprayed thickness of liner product.

2 to 3 mm liner support

Coarse Gravel

Loading Platen

Thin slabby granitic material

Steel Loading Frame

Figure 3. Geometry of the baggage loading test frame.

Figure 4. Grid used in the spray on liner modeling.

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297

1

2

3

4

5

6

7

8

9

10

Ten

sile

Str

ess

(MP

a)

FLAC Simulations of 3M Liner Systems in CANMET Testvs. Averaged Response Curves From Tensile Strength Testing

(Liner Thickness = 2mm)

0.00 0.10 0.20 0.30Strain

0.40 0.50 0.60 0.70

3M System 1 - FLAC

3M System 2 - FLAC

3M System 1 Averaged

3M System 2 Averaged

0

Figure 5. A stress–strain plot created with the FLAC FISH function compared to the actual strengths reported in materialtensile testing.

Str

ess

(MP

a)

Strain (mm/mm)

Figure 6. Full scale baggage loading test stress–strain curve.

1

10

100

1000

1 10

3M Mining Liner - 23˚C / 70% RHFor <50fpm and 300fpm Air Flow

STRENGTH/STIFFNESS, MPa; ELONGATION,%

Tensile Yield, <50fpm

Tensile Ultimate, 300fpm

Secant Modulus Yield, <50fpm

Adhesion, <50fpm

Elongation Yield, <50fpm

Tensile Yield, 300fpm

Tensile Ultimate, <50fpm

Elongation Ultimate, <50fpm

Adhesion, 300fpm

DR

YIN

G T

IME

, ho

urs

0.01 0.1 100 1000

Figure 7. Plot of adhesive and yield strength over time.

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4 RESULTS

Several model simulations were performed in order tocalibrate the model against the observed and measuredresponse of the liner.

One of the first full scale tests carried out was anapplication of the liner after 4 hours with no adhesion tothe rock (a silicone material prevented liner adhesion).Immediately it became apparent that the adhesion ofthe material was playing a much bigger role in the sta-bility of the baggage loading test than would be anti-cipated. Figures 8 & 9 show the actual results and theFLAC modeling results respectively for a liner that hasno adhesion to the rock. In both cases the liner quicklydeforms due to the low strength observed at this earlytime after spraying.

The second test in the series involved the baggageloading test with the material left to dry over a 24 hourperiod. In this case, adhesion was allowed to takeplace by lightly wetting the surface of the rock priorto application.

With this second test, the liner was stable undergravity loading from the dead weight of the rock in thetesting frame. The total displacement under this staticloading was 24 mm. This test was also the source of thestress–strain data generated in Figure 6. This test pro-vided valuable data for the calibration but also demon-strated that there was another effect at work that was notbeing accounted for in the modeling. The model pre-dicted a total displacement for this particular test at90 mm rather than the 24 mm observed in Figure 10.

In order to make up for the difference in displace-ments observed between the baggage loading tests andthe FLAC models, an investigation took place in whichthe stiffness of the material was increased until match-ing results were observed.

Modifying the model properties resulted in a curi-ous result. In order to match the performance of thebaggage loading test, the stiffness of the material hadto be increased to near 20 times its original value.When these modified properties were used, the resultsmatched quite well with the observed testing (Fig. 12).

298

Figure 8. Baggage loading test with no adhesion.

JOB TITLE: Falconbridge-3M Baggage Load Testing Model: 3M System 2

FLAC (Version 4.00)

LEGEND

27-Mar-02 14:56step 19190-3 860E-01 <x< 1.786E+00-1.300E+00 <y< 8.715E-01Grid plot

0 5E-1

Axial Force onStructure# 1 (Cable)

Max. value-1.100E+02

Itasca consulting Group, Inc.Minneapolis, Minnesota USA

0.000 0.400 0.800 1.200 1.500

0.500

0.200

-0.200

-0.500

-1.000

Figure 9. FLAC modeling results for the baggage loadingtests with no adhesion.

Figure 10. Baggage load test with adhesion – total staticdisplacement of 24 mm.

JOB TITLE: Falconbridge-3M beggage Load Testing Model: 3M System 2

FLAC (Version 4.00)

LEGEND

21-Mar-02 12:08step 21000-2.222E-02 <x< 1.422E+00-4.222E-01 <y< 1.022E+00

Grid plot

0 2E-1Axial Force onStructure# 1 (Cable)

Max. Value-4.934E+03

Itasca Consulting Group, Inc.Minneapolis, Minnesota USA

0.100 0.300 0.500 0.700 0.900 1.100 1.300

0.100

0.300

0.500

0.700

0.900

-0.100

-0.300

Figure 11. FLAC results for the modeling of the baggageloading test with adhesion and modified material properties.

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Based upon the results observed in the full scaletesting, it was obvious that some mechanism must beat work in order to account for the discrepancy in themodeling. There are three possible sources of uncer-tainty that would tend to artificially increase the stiff-ness of the material, the adhesion, and the impact ofblocks interlocking with each other and the frame.

Firstly, the FLAC model is a 2D model, while theactual baggage load test is three-dimensional.However, the actual discrepancy would appear to betoo large to be accounted for simply by this difference.

The adhesion certainly plays an important role inthe strength of the material and the model is unable tofully account for the impact of adhesion because of themethod by which the cable elements are used. Becausethe liner adheres to the slabs over most of its area, andonly deforms over a relatively small “gauge length”,this would appear to be the largest difference betweenthe actual liner and the model. This difference wouldhave the effect of making the liner much stiffer than itappears in a “dog bone” test.

The interlocking of the slabby blocks of material iseven harder to quantify. Each frame of material isloaded by hand and as a result there is a complexinteraction between the blocks supporting each otherand being supported by the frame rails. It is difficult totry and quantify this behavior without additional testingbeing done to determine the sensitivity of the system tothe loading of the frame.

Despite the relatively small dataset used in the cali-bration, the model itself has proven quite useful formodeling the baggage loading tests. The next step in theevolution of this testing will be to use this informationto develop a drift modeler that incorporates the infor-mation gathered from these trails.

5 CONCLUSIONS

The FLAC model has provided a valuable tool forreducing the cost of full scale testing. With this model,it is possible to anticipate the response of different formulations of the liner at different thicknesses andtime frames. Given the high cost of full scale tests, itcan be used to narrow down the testing regime to themost promising combinations of thickness, time, andliner properties in order to maximize the data collectedduring full scale tests. Further calibration against full-scale tests will help to reduce some of the uncertaintyinvolved with the model.

ACKNOWLEDGMENTS

The authors would like to thank all of those involvedin the preparation of the baggage load tests including3M Canada, Falconbridge Ltd, and CANMET.

299

Figure 12. Calibration used on the FLAC model to matchthe observed baggage loading test results.

JOB TITLE: Falconbridge-3M Baggage Load Testing Model: 3M System 2

FLAC (Version 4.00)

LEGEND

19-Mar-02 14:04step 21000-2.222E-02 <x< 1.422E+00-4.222E-01 <y< 1.022E+00

Grid plot

0 2E-1

Axial Force onStructure# 1 (Cable)

Max. Value-1.824E+03

Itasca Consulting Group, Inc.Minneapolis, Minnesota USA

0.100 0.300 0.500 0.700 0.900 1.100 1.300

-0.300

-0.100

0.100

0.300

0.500

0.700

0.900

Figure 13. FLAC output for baggage load testing withadhesion. Anticipated displacement was 90 mm instead ofthe actual 24 mm measured.

09069-35.qxd 08/11/2003 20:40 PM Page 299



301

A numerical study of the influence of piles in the passive zone ofembedded retaining walls

T.Y. Yap & C. PoundMott MacDonald Ltd, Croydon, Surrey, United Kingdom

ABSTRACT: Piles are often placed in the base of deep excavations to carry future structure loads or to reducebase heave. Where these piles are located close to retaining walls, they can provide additional resistance to the movement of the embedded length of the retaining wall. This paper discusses a series two-dimensional and three-dimensional analyses carried out using the finite difference programs FLAC and FLAC3D to investigate the increasein the passive resistance in front of the embedded retaining wall due to the presence of these piles. Two passive fail-ure mechanisms were identified; the first involved squeezing of the ground upward between the wall and the pilesand the second involved squeezing of the ground between the piles. The influence of pile and wall roughness, pilespacing and pile to wall separation was investigated in order to define which of the two passive failure mechanismswould govern and under what circumstances. Based on the results of the two-dimensional analyses a methodologywas developed to determine the limiting passive resistance allowing for the presence of the piles. Three-dimensionalanalyses were carried out which showed a close agreement with the results of the two-dimensional analyses.

1 INTRODUCTION

Excavations for building basements or transportation orutility tunnels are often carried out within retained cuts.Piles are often placed below the base of these excava-tions either to carry future structure loads or to reduceshort-term or long-term ground heave. Often to easepile construction or to reduce the overall constructionprogram, these piles are installed from the ground sur-face. If these piles are located close to the retainingwall, they can reduce wall deflection over the embeddedlength, which can be beneficial when considering theeffect of the construction works on adjacent structures.

Normally design of embedded retaining walls iscarried out using two-dimensional plane strain analy-ses. In such analyses piles would be represented as awall with smeared structural properties. In reality,depending on the pile spacing, diameter and proxim-ity of the wall, ground could be squeezed between thepiles and the conventional analyses could signifi-cantly overestimate the benefit of the piles. Thispaper presents a two-dimensional numerical studycarried out using the finite difference program FLACto investigate the earth pressures developed betweenrough, partially rough and smooth walls and piles in close proximity. A three-dimensional analysis wasalso used to investigate the validity of adopting asmeared representation of the piles when the piles are installed in soft clay.

2 TWO-DIMENSIONAL ANALYSES

2.1 Vertical section

Figure 1 illustrates the geometry of the problem wherea row of piles was located at a distance, d, from theembedded portion of a retaining wall of embeddedlength, H. The top of the model was taken to be the finalexcavation level for the construction. The boundaryconditions were such that no displacement was allowedon the base of the model and no horizontal or verti-cal displacement was allowed of the piles. In order to

H

d

Embeddedretaining wall

Piles

1.5H

Figure 1. Passive earth pressure problem (d � 20 m).

09069-36.qxd 08/11/2003 20:41 PM Page 301


reduce boundary effects, the bottom of the soil masswas located at a depth of 1.5 H. Beam elements wereused to model the embedded retaining wall and piles,with interfaces connecting the beam elements to thesoil. The interface strength properties were varied torepresent different values of wall roughness.

A linear elastic perfectly plastic soil model was usedthroughout these analyses, with the soil stresses limitedby the adoption of a Mohr-Coulomb failure criterion.The soil properties adopted for both undrained anddrained materials are listed in Table 1. The undrainedmaterial properties are typical of a soft clay, whereas thedrained properties are typical of a medium dense sand.

For both materials the initial horizontal stress wasgenerated using a coefficient of earth pressure at rest,ko, of 1.0, although the results obtained are not believedto be sensitive to the value of this parameter.

To determine the passive pressure, the wall wasforced towards the soil at a constant velocity and thereaction of the soil on the wall measured. Analyses werecarried out for a range of values of d between 0.5 m and20 m and for each analysis the limiting horizontal pas-sive resistance was determined. The wall and pile fric-tion was also varied between smooth and rough, withequal values of friction being used on both the wall andthe piles in all cases. The limiting horizontal passiveresistance forces, Pph, were used to back-calculate themobilized passive earth pressure coefficients, forundrained and drained soil materials, from the follow-ing equations:

(1)

(2)

where Kpc and Kp are the passive earth pressure coef-ficients associated with undrained and drained soilmaterials respectively.

2.1.1 Undrained materialFigures 2 & 3 show the limiting horizontal passiveresistance and the limiting passive earth pressure coef-ficients respectively for a 10 m deep wall embedded in

the undrained material. The limiting passive earth pres-sure coefficients are also given in Table 2. When thewall and piles were placed 20 m apart, the computedvalues of Kpc are very close to the theoretical valuesgiven in CP2 and reproduced in Table 2. BS8002 sug-gests that the passive resistance in a cohesive soil canbe approximated by the following equation:

(3)

The passive earth pressure coefficients predictedby this equation are also given in Table 2. The passive

302

Table 1. Material properties.

Setting Undrained Drained

Unit weight � (kN/m3) 15.5 18.0Young’s modulus E (MPa) 6.0 25.0Poisson’s ratio v 0.49 0.25Cohesion c (kPa) 20.0 0.0Friction angle � (degree) 0.0 30.0Dilation angle � (degree) 0.0 0.0, 30.0

1000

2000

3000

4000

5000

6000

1 10 100Distance between wall and piles (m)

Ho

rizo

nta

l Fo

rce

(kN

)

0.1

cw = 0cw = c/3cw = c/2cw = 2c/3cw = c

0

5

10

15

20

25

0.1 1 10 100

Kpc

Distance between wall and piles (m)

cw = 0cw = c/3cw = c/2cw = 2c/3cw = c

Figure 2. Influence of the distance of the walls on the mobi-lized ultimate load (undrained material).

Figure 3. Influence of the distance of the walls on the mobi-lized ultimate passive earth pressure coefficient (undrainedmaterial).

09069-36.qxd 08/11/2003 20:41 PM Page 302


earth pressure coefficient predicted by this equationare somewhat higher than the values predicted byFLAC or quoted in CP2.

As the distance between the wall and the pilesdecreases, Pph and Kpc computed for the rough andpartially rough walls increase. This is due to the fric-tional restraint developed on the piles. The higher thewall cohesion, cw, the larger the vertical restraintdeveloped and hence the higher the values of Pph andKpc. Conversely, for smooth walls Php and Kpc remainconstant even though the distance between the wallsdecreases to as little as 0.5 m.

Figure 4 shows a contour plot of shear strain incre-ment for the analysis with a 20 m separation between a rough wall and piles. The failure surface is clearlyshown comprising an arc adjacent to the wall and astraight portion up to the ground surface. There is a fanof intense shearing above the failure surface and adja-cent to the wall, with the ground more distant from thewall comprising a passive block with little or no internalshearing.

Figure 5 shows the influence of the distance d onthe computed horizontal stress acting directly on the

perfectly rough wall. The apparent localized reductionin the force at ground surface is due to the force actingover one half of the area represented by the otherforces. When the piles are located 10 m or 20 m fromthe wall the horizontal force profiles are nearly identi-cal although there is a slight divergence below 8 m.Inspection of Figure 4 shows that for the 20 m case the failure surface reaches the ground surface approx-imately 14 m from the wall. For the 10 m case thiswould not be possible and therefore the slightly higherhorizontal forces below 8 m are indicative of the inter-action of the failure surface with the piles. As the dis-tance d reduces there is a progressive increase inhorizontal force acting on the lower part of the wallalthough the forces at the top of the wall down to adepth of about 0.7 d remain unaffected by the presenceof the piles. Inspection of other analyses indicates thatthe depth over which the forces on the wall remainunaffected by the presence of the piles is dependent onthe wall roughness such that for a smooth wall andpiles this depth is approximately equal to d.

Further analyses were carried out and these analysesshowed that the limiting passive earth pressure coeffi-cients given in Table 2 were correct for different walllengths provided the wall to pile separation, d, was nor-malized by wall embedment depth, H.

The significant increase in horizontal force actingon the lower part of the embedded retaining wall ismatched by an increase in the force acting on the pilesat the same level. This suggests that the soil may besqueezed between the piles rather than forced upwards.On the other hand, the ground stresses over the top partof the wall and piles are limited by conventional passive failure.

303

Table 2. Passive earth pressure coefficients for anundrained material.

Wall and pile friction, cw

d(m) 0 c/3 c/2 2c/3 c

20 2.04 2.31 2.42 2.50 2.6110 2.03 2.34 2.47 2.58 2.745 2.03 2.67 2.96 3.23 3.683 2.03 3.11 3.62 4.10 4.992 2.03 3.66 4.45 5.21 6.641 2.01 5.33 6.95 8.54 11.70.5 2.01 8.68 12.0 15.2 21.6BS8002 2.00 2.31 2.45 2.58 2.83CP2 2.0 – 2.4 – 2.6

-10

-8

-6

-4

-2

00 100 200 300 400

0.5m1m2m3m5m10m20m

Dep

th b

elow

top

of w

all

(m).

Horizontal Force (kN)

Figure 5. Variation of earth pressure acting on a rough wallfor different wall to piles spacings.

Figure 4. Shear strain contours for a rough pile.

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2.1.2 Drained materialFigure 6 shows equivalent results for a drained mate-rial, with dilation angle, �, of 0°. When the embeddedretaining wall and the piles are placed far apart, Kp aresomewhat below the values given by CP2, except forthe smooth wall case. For a smooth wall and piles, thevalues of Kp remains constant at the theoretical valueregardless of the distance d. However, for rough andpartially rough walls, the values of Kp increase rapidlyas d decreases. Note that in Figure 6 the lines are cut offat a point where back-calculated values of Kp are largerthan 50. As for the results for an undrained material, an increase in the wall friction �w, leads to significantincrease in Kp. For high values of Kp it became increas-ingly difficult to obtain a reliable limit pressure sincethe compressive stresses developed in front of the wallbecame a significant proportion of the Young’s modulus.The effect of assuming an associated rather than a non-associated flow rule is shown in Figure 7. The drained

analyses with a 20 m separation between the walls andthe piles were rerun assuming a dilation angle of 30°for the material. Note that the general trend of Kp issimilar to that in Figure 5. The passive earth pressurecoefficient values for the material with the associatedflow rule are now equal to the values quoted in CP2.

2.1.3 Friction on pile surfaceThe failure process describes above will lead to shear-ing along a vertical surface which passes around andbetween the piles. Around the piles the friction mobi-lized on this surface will be given by the piles shaftfriction value whereas on the failure surface betweenthe piles, the strength mobilized is given by the soilstrength. The effective strength mobilized, c�, on anequivalent planar surface is given by the followingformula:

(4)

where D and S are the pile diameter and spacingrespectively. The parameter k� can be approximated bythe following relationship:

(5)

For most normal situations, the value of c�/c is closeto unity suggesting that the piled wall can normally beconsidered as rough. Assuming that the embedded wallis not rough, it is suggested, though not proven, that thevalue of Kp can be obtained by averaging the Kp valuefor the wall and that for the piles.

304

0

10

20

30

40

50

0.1 1 10 100Distance between wall and piles (m)

Kp

phiw = 0phiw = phi/3phiw = phi/2phiw = 2phi/3phiw = phi

Figure 6. Variation of wall to piles spacing on the passiveearth pressure coefficient for drained material.

0

1

2

3

4

5

6

0 20 40 60 80 100

Wall friction (phi %)

Kp

AssociatedNon-associated

Figure 7. Comparison of passive earth pressure coefficientfor associated and non-associated materials.

Table 3. Limiting passive earth pressure coefficients for adrained material with angle of friction of 30° and a dilationangle of zero.

Wall and pile friction, �w

d(m) 0 �/3 �/2 2�/3 �

20 2.94 3.96 4.44 4.57 4.7010 2.95 4.50 5.53 6.45 7.095 2.98 6.99 11.2 17.6 36.23 3.00 13.3 35.5 95.5 3542 3.01 34.3 141 639 –1 3.02 731 – – –0.5 3.01 – – – –CP2 3.0 4.0 – 4.9 5.8

09069-36.qxd 08/11/2003 20:41 PM Page 304


2.2 Horizontal section

The analyses above show that the presence of a piledwall in close proximity to the embedded length of a retaining wall can have a significant effect on themobilized passive resistance. However, piles are rarelyinstalled as a continuous wall; more normally they areinstalled at a spacing of between two and five timestheir diameter. When the piles are not in contact it ispossible for the ground located in front of the wall to beforced between the piles. This process cannot be mod-eled in a two-dimensional plane strain analysis of avertical section and could therefore limit the applica-bility of this type of analysis.

To investigate the probability of this form of behav-ior, a further set of two-dimensional analyses wasundertaken. The analyses considered a horizontal slicethrough the piles. A diagram showing the configurationof the model is shown in Figure 8. Symmetry throughthe middle of the pile and the mid-point between thepiles was assumed. The pile was prevented from mov-ing in all directions. The upper boundary of the modelhad an applied pressure equal to the initial in situ stress. The lower boundary of the model was dis-placed at a constant rate towards the pile. The pile wasconnected to the ground through an interface whichallowed both shear displacement of the ground aroundthe pile and separation of the ground from the pile onthe “back” side of the pile. Only undrained analyseswere undertaken using the soil properties given inTable 1. The resistance provided by the pile was mon-itored in two ways; firstly by determining the reac-tions on the grid points around the pile and secondly bydetermining the reaction on the lower boundary. The

difference in these two reactions at the limiting stateafter allowing for the magnitude of the in situ stresswas no more than 1% of the measured value.

Analyses were carried out for pile spacings rangingbetween 1.3 D and 10 D where D is the pile diameter,for different initial in situ stress conditions and witheither smooth or rough pile interface properties. Theforce exerted on the pile, P, was converted to a bearingcapacity factor, Ncp value as follows:

(6)

Figure 9 shows the effect on Ncp of varying pile spac-ing for two different in situ stress states of 0 kPa and200 kPa and for a rough or smooth pile interface. Forthe 0 kPa analyses with a smooth pile interface the valueof Ncp reaches a minimum for a pile spacing of about1.6 D with rapidly increasing values of Ncp for smallerpile spacings and gradually increasing values for largerpile spacings. The minimum value of Ncp is about 4.20.The Ncp values are consistently higher for the rough pileinterface analyses with a minimum Ncp value of about5.6 occurring at a pile spacing of about 2.5 D.

For the analyses with an in situ stress of 200 kPa theNcp values are higher than for the corresponding analy-sis with an in situ stress of 0 kPa. The analyses with asmooth pile interface shows a minimum Ncp of 8.2 at apile spacing of 2.0 D whereas the analyses with a roughinterface show a minimum Ncp of 10.9 at a pile spacingof 2.5 D. Both sets of analyses indicate that at higherpile spacings the Ncp value becomes constant. For the

305

D

S/2

Figure 8. Horizontal slice model.

0

2

4

6

8

10

12

14

16

1 3 5 7 9 10

S/DN

cp

smooth, 0 kParough, 0 kPasmooth, 200 kParough, 200 kPa

2 4 6 8

Figure 9. Comparison of passive resistance coefficientdeveloped on smooth and rough piles for different pile diam-eter to pile spacing ratios.

09069-36.qxd 08/11/2003 20:41 PM Page 305


smooth and rough pile interface analyses the limitingNcp values are 9.3 and 12.0 respectively. These are com-parable to the results given by Chen & Martin (2002).

Figure 10 shows the effect of varying the in situstress state for a constant pile spacing of 3.0 D. For thesmooth pile interface the Ncp value increases from4.63 to 9.1 as the in situ stress is increased from 0 kPato 70 kPa and is then constant for higher in situ stressstates. For the rough pile interface the response is sim-ilar with the Ncp value increasing from 5.67 to 11.1 atabout 70 kPa with the value constant at higher in situstresses. The reason for the change from a graduallyincreasing Ncp values at low stress to a constant valueat higher stresses can be resolved by inspection of thedeformation pattern around the piles. At stresses lessthan 70 kPa the ground is not in contact with the backof the pile whereas above this stress the ground is intouch with the pile over the whole pile circumference.As the stress is gradually increased from 0 kPa to70 kPa the length over which the ground is not in con-tact with the pile gradually reduces.

These analyses would appear to suggest that the pas-sive resistance provided by the pile would vary withdepth down the pile. Near the surface the restraint pro-vided by the pile would be least and the ground move-ment would lead to a gap developing on the side of thepile furthest from the wall.

2.3 Combined effect

The two dimensional horizontal section analyses haveshown that under high stress ground can be forcedbetween the piles. However, the ground forced between

the piles is resisted by a passive wedge behind the piles.It is suggested that, except at very close pile spacings,the wedge mobilized behind the piles is identical to thatwhich would have been mobilized if the piles had notbeen present. The restraint provided by the piles istherefore generally additive to the normal passivewedge. The piles provide resistance only in that portionof the passive wedge through which it passes. For asmooth wall the failure surface underlying the passivewedge rises at 45° from the toe of the wall. For a wallwith friction the failure surface rises at a shallowerangle (see Fig. 4). Conservatively it can be assumedthat the pile penetrates through the passive wedge to adepth D–h. The total passive resistance, Ptotal, per meterrun provided by this failure mode can therefore beexpressed as follows:

(7)

To decide whether failure will occur by squeezingof ground between the piles or by failure of ground infront of the piles the mode with the lower failure loadmust govern.

As an example the force required to develop the twofailure mechanisms has been calculated for the soil con-ditions described above with a 10 m deep wall with1.5 m diameter piles at 4.5 m centers located 2 m infront of the wall. The force for the combined failuremode is 1825 kN/m whereas for the failure mode infront of the piles, the force is 2103 kN/m. In this casefailure by squeezing of the ground between the piles ismore likely than failure by squeezing in front of thepiles. It is illustrative to note that passive failure wouldhave occurred at a force of 1297 kN/m if the piles hadnot been present. This illustrates the significant increasein the passive resistance caused by installing piles inthis location.

3 THREE-DIMENSIONAL ANALYSES

The analyses discussed above provide a basic under-standing of the mechanisms involved with piles in thepassive zone of embedded retaining walls, but theactual behavior is almost certainly more complex thanthe two dimensional analyses can show. It is conceiv-able that failure of a deeply embedded wall would occurby a combination of both mechanisms. It is also diffi-cult to judge the apparent horizontal stress acting in thehorizontal plane when assessing the resistance providedby the piles.

To provide more guidance on the equivalence of dis-crete passive-zone piles compared to an equivalentcontinuous wall a pair of three-dimensional analyseshave been carried out using FLAC3D. Both analysesconsidered clay with the properties given in Table 1.

306

0

2

4

6

8

10

12

0 50 100 150 200In situ stress (kPa)

Nc

Smooth pileRough pile

Figure 10. Comparison of passive resistance coefficientdeveloped on smooth and rough piles for different in situstress states.

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The clay layer was taken to be 30 m thick overlying ahard stratum. To reduce the size of the model only thatpart of the construction below final excavation levelwas considered. The overall width of the model was15 m which was deemed to be sufficiently wide to pre-vent interaction of the boundary with the wall or thepile. As in the two-dimensional analysis of the horizon-tal slice, symmetry was adopted on vertical planes per-pendicular to the wall through the center of the pilesand also through a point midway between the piles.

The wall was embedded 10 m into the clay and wasmodeled using liner elements which comprise triangu-lar plate elements connected to the ground through aninterface with normal and shear elastic and plasticproperties. Many propped embedded retaining wallsundergo maximum horizontal displacement at aroundfinal excavation (formation) level and therefore thewall was moved towards the soil by applying a hori-zontal force to the wall at formation level while alsopreventing rotation about a horizontal axis at thispoint. In one of the analyses, 1.5 m diameter pilesspaced at 4.5 m centers were modeled with the pile axeslocated 2.75 m in front of the wall, resulting in 2.0 mof soil between the wall and the nearest edge of thepiles. A close-up of this model is shown in Figure 11.In the second analysis the pile was substituted by acontinuous wall with equivalent smeared properties tothat of the discrete piles. The centerline of the equiva-lent wall was also located 2.75 m from the embeddedwall. Both the discrete piles and the equivalent wallwere modeled using solid brick elements rather thanstructural elements.

Both the discrete piles and the equivalent wallextended the full depth of the model and both wereassumed to be rigidly fixed in a hard stratum at thebase of the model. Rough interface properties wereconsidered between the embedded wall, the pile, theequivalent wall and the ground.

The wall and pile properties are given in Table 4.The equivalent wall properties were derived using thefollowing formulae, which ensured that the equiva-lent wall had the same bending and axial stiffness asthe discrete piles.

(8)

Where Ep and Es are the Young’s moduli of the pile andthe equivalent wall respectively, and t is the equivalentsmeared pile wall thickness.

The model was initially brought to equilibrium underthe in situ stress conditions and by fixing the horizontalmovement of the embedded wall. The reactions devel-oped on the embedded wall during this stage were thenapplied as a series of nodal forces acting on the wall forthe remainder of the analysis. The analysis was contin-ued by increasing the magnitude of the horizontal forceat the top of the wall in increments. After each increasein the force the model was allowed to reach equilibrium.

Figure 12 shows the results of the two analyses. Thesolid symbols show the displacement of the top of thewall versus applied force for the two analyses. In bothcases the displacements are initially small as the force isincreased. Up to an applied force of 1500 kN/m the wallmovement is very similar in the two analyses. Howeveras the force is increased above 1500 kN/m, the wallmovement in the analysis with discrete piles increasesrapidly and appears to become unlimited at an appliedforce of about 1850 kN/m. In the analysis with the pilesrepresented by an equivalent wall, the wall movementdoes not increase as rapidly and only becomes unlim-ited as the applied force approaches 2000 kN/m. Theselimiting values are very similar to the theoretical valuescalculated in section 2.3 above.

Also shown in Figure 12 as open symbols is the pilehead or equivalent wall top movement versus appliedforce. The pile movement is very similar to theembedded wall movement up to an applied force of1500 kN/m. In the analysis with discrete piles the pilestarts to move less than the wall as the applied force isincreased above 1500 kN/m. In the analysis with anequivalent wall, the equivalent wall only starts to moveless than the embedded wall when the applied load

307

Figure 11. General view of the FLAC3D pile model.

Table 4. Structural properties.

EquivalentSetting Wall Pile wall

Thickness/diameter (m) 1.0 1.5 1.35Young’s modulus E (GPa) 28.0 28.0 8.46Poisson’s ratio v 0.2 0.2 0.2

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exceeds 1850 kN/m. This effect is shown more clearlyin Figure 13 which plots the differential movementbetween the top of the pile or equivalent wall and thetop of the embedded wall for the two analyses. Positiverelative movements indicate movement of the walltowards the piles. The differential movement is small inboth analyses up to 1500 kN/m. In the analysis with dis-crete piles at higher applied forces the gap between thewall and the piles begins to close rapidly as the groundstarts to squeeze between the piles.

In the analysis with the equivalent wall, at appliedforces greater than 1500 kN/m, the gap between the

embedded wall and the equivalent wall increases up toan applied force of about 1750 kN/m. This is believedto be due to high ground stresses in front of the toe ofthe embedded wall causing the equivalent wall to rotateforward more rapidly at formation level. At even higherapplied forces the gap between the two walls reduces asground starts to be squeezed upwards between the twowalls. It is considered that the occurrence of significantdifferential movement between the embedded wall andthe pile is indicative of the onset of passive failure ofthe ground in front of the wall.

Figure 14 shows the horizontal displacement of theembedded wall and the piles at an applied force of1815 kN/m for the analysis with the discrete piles. Thehorizontal wall displacement far exceeds the pilemovement suggesting that there is failure of the groundpast the pile. Also shown is the horizontal deflection ofthe ground mid-way between the piles at the same distance from the wall as the pile axis. The pattern ofground displacement is complex with the section nearthe ground surface moving significantly less than thewall and only slightly more than the pile. This isbecause a passive wedge develops near the ground sur-face which daylights in front of or between the piles.Between 3 m and 10 m below ground level the groundmovement far exceeds the pile movement and is closerto the movement of the embedded wall. This clearlyshows that the ground is being squeezed between thepiles. Because of the restricted gap between the pilesand the incompressible nature of the undrained mate-rial, under certain situations the ground displacementbetween the piles could actually exceed the embeddedwall displacement.

In two-dimensional analyses piles or walls are oftenrepresented using structural elements. These structural

308

-50

0

50

100

150

200

250

300

350

500 1000 1500 2000Applied Load (kN/m)

Rel

ativ

e di

spla

cem

ent (

mm

)

Discrete Piles

Equivalent Wall

Figure 13. Relative wall and pile displacement in the three-dimensional analyses.

-30

-25

-20

-15

-10

-5

0

-200 0 200 400 600 800Horizontal displacement (mm)

Dep

th b

elow

form

atio

n le

vel (

m).

Embedded wall

Pile

Ground

Figure 14. Wall, pile and ground displacement profiles foran applied force of 1815 kN/m.

0

200

400

600

800

1000

1200

500 1000 1500 2000

Applied Force (kN/m)

Dis

plac

emen

t (m

m)

Discrete PilesEquivalent WallDiscrete PilesEquivalent WallStructural Elements

Figure 12. Wall and pile displacements in the three-dimensional analyses.

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elements have no physical thickness in the modelalthough their axial and bending stiffness is mod-eled correctly by assigning appropriate values of areaand moment of inertia. The structural elements aregenerally placed along the centerline of the piles orwall that they are intended to represent. To investigatethe effect that modeling the piles as structural elementshas on the predicted passive resistance the three-dimensional model was rerun with the piles repre-sented by shell elements with properties identical tothose of the equivalent wall given in Table 4. The shellelements were rigidly connected to the mesh and there-fore represented a rough wall. To be comparable to theother two analyses, the toe of the equivalent wall wasfixed against horizontal movements and was preventedfrom rotating around a horizontal axis. Because the structural elements have no physical thickness inthe model, 2.75 m of clay is now present between theembedded wall and the equivalent wall.

The results of the analysis are shown in Figure 12 ascrosses. The greater thickness of soil between theembedded wall and the equivalent wall gives a softerwall displacement response compared to the analysiswith the equivalent wall modeled using solid elements.The limiting passive pressure is also lower because ofthe greater separation between the two walls. FromTable 2 it can be seen that increasing the wall separa-tion from 2 m to 2.75 m for a 10 m deep wall has theeffect of reducing the passive earth pressure coefficientfrom 6.4 to about 5.3. This results in a reduction in thelimiting passive resistance from 2100 kN/m to about1835 kN/m which is very similar to the passive resist-ance predicted by this three-dimensional analysis.

4 DISCUSSION

There appears to be a good match between the passiveresistance obtained in the three-dimensional analysisand the predicted passive resistance made from theresults of the two-dimensional analyses despite the obvi-ous limitations of these analyses. A sensitivity studycarried out using the results of the two-dimensionalanalyses for the undrained material shows that squeez-ing of the ground between the piles is more likely when:

1. The embedded retaining wall and piles are roughrather than smooth.

2. The piles are spaced more widely.3. The piles are nearer to the wall.

Using the two-dimensional analyses it is possible toidentify the critical pile spacing defining the change in passive failure mechanism from squeezing of theground between the piles to squeezing upwards in frontof the piles. For a 10 m long embedded rough wall and

piles the critical pile spacing appears to vary from 2 pile diameters when the piles are located 2 m in front ofthe wall to 4 diameters when the piles are located 5 m in front of the wall.

5 CONCLUSIONS

The analyses have shown that piles installed in thepassive zone of embedded retaining walls can signifi-cantly increase the passive resistance mobilized infront of the retaining walls. The passive resistance issensitive to the distance of the piles to the wall andwhether the piles and wall are rough or smooth. Forgranular deposits the passive earth pressure coeffi-cient increases dramatically as the spacing betweenthe walls and the piles reduces and it is suggested thatpassive failure is unlikely to occur in this materialunless the piles are widely spaced or the wall has onlya shallow embedment.

Three-dimensional analyses showed a very similarlimiting passive resistance to a calculation based ontwo-dimensional analyses. However, the deflections tomobilize this passive resistance are large and may implyunacceptable movement of the retaining structure. Thepiles undergo significant lateral deflection, localizedbending and axial tension due to the movement of theembedded wall and therefore it is important that thesepiles are designed for these additional forces.

Where the passive failure mechanism does not com-prise squeezing of the ground between the piles, two-dimensional plane-strain analyses in which the piles arerepresented by a wall with equivalent smeared proper-ties will provide a safe estimate of the passive resist-ance. The analyses suggest that where piles are spacedat 3 diameters or less in a cohesive deposit, squeezing ofthe ground between the piles is unlikely to occur unlessthe piles are located nearer to the wall than 30% of theembedded length. Where the piles are represented by astructural element, the additional soil present in themodel between the embedded wall and the structuralmember will lead to a conservative estimate of the passive resistance.

REFERENCES

BS8002: 1994. Code of Practice for Earth RetainingStructures. British Standards Institution.

Chen C-Y & Martin, G.R. 2002. Soil-structure interactionfor landslide stabilizing piles. Computers andGeotechnics 29: 363–386.

Civil Engineering Code of Practice No. 2 1951. EarthRetaining Structures, Institution of Structural Engineers,London.

309

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Dynamic and thermal analysis

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313

A practice orientated modified linear elastic constitutive model for fireloads and its application in tunnel construction

E. Abazovic & A. AmonGeoconsult ZT GmbH, Salzburg, Austria

ABSTRACT: A spate of fires in tunnels over the past decade, causing serious loss of life and significantstructural damage, has led to new safety concepts in tunnel construction. Nowadays these concepts are alreadybeing considered during the design. In this context numerical methods represent a powerful tool for assessingthe structural forces in the lining and the change of material properties caused by thermal effects.

This article deals with the simulation of a fire within a tunnel by means of the program FLAC. The tunnel lin-ing is modeled by four-node continuum elements for simulating non-linear and time dependent temperaturevariation within the lining. The thermal effect is applied according to the fire load curve of the BEG-project, a future major railway section between Italy and Austria passing the Alps, at the inside of the lining. The coeffi-cient of thermal transmission between the thermal source and the lining is chosen such that the temperature-field within the lining corresponds to experimental data. Non-linear material behavior due to thermal loading isimplemented by varying the coefficient of thermal expansion.

1 INTRODUCTION

Thermo mechanical processes are very complex andthey are characterized by non-linear material behaviorand transient heat transfer mechanisms. Mechanicalprocesses are depicted through induced stresses as aresult of mechanical loads. Alterations of the elasticproperties, spalling and material failure are caused byfire loads.

In a case of tunnel fire the heat between the heatsource and the tunnel inner lining is transmitted byradiation and convection. These exchange mechanismsare dependent on various factors like brightness, airflow velocity, temperature difference and materialconductivity. Convection is time dependent due totransient conditions of the temperature gradient.

Heat interchange through radiation is characterizedby the difference of the fourth power of the temperaturequotient of the heat source and heat recipient. Heatinterchange through radiation is time dependent too.

The required calculation constants are difficult todetermine because of the above mentioned reasons,and should therefore be determined by experiment. Inaddition in absence of a material law describing thecomplex thermo mechanical processes, numerical sim-ulations are even more difficult. For this reason, thisarticle is intended to describe thermo mechanical

processes numerically through a modified linear elasticmaterial model by means of a user-defined function(FISH-function).

2 ASSUMPTIONS

It can be assumed that the major principal stress withina concrete tunnel lining tends to act in circumferentialdirection and the minor principal stress (radial direc-tion) can be neglected. In this case a uniaxial state ofstress within the tunnel lining prevails and the devia-toric stresses are negligibly small. The material behavioris determined by only the spherical tensor. Since tem-perature loads are also only influenced by the sphericaltensor and the temperature load is linearly propor-tional to temperature increase it is possible to obtain astress change by variation of the thermal coefficientof expansion.

The thermal load is applied in terms of a temperatureload at the inside of the tunnel lining. The appliedtemperature is equivalent to the fire load curve of the Brenner Eisenbahn Gesellschaft (BEG) project.The temperature increases linearly within seven min-utes from the initial temperature to the maximumtemperature of 1200°C which can be seen in Figure 1(Gresslehner 2001).

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The coefficient of heat transfer and the factor ofthermal conductivity are chosen until the temperaturegradients meet the experimental data, published by theUniversity of Innsbruck (Kusterle & Waubke 2001)(Fig. 2).

3 NON-LINEAR THERMAL CONSTITUTIVEMODEL

3.1 General

Due to mechanical loads the tunnel lining is undercompression (�M, Fig. 3) and normal stress (�M). Thisstate is the initial condition for the thermal calcula-tions. In reality the normal stress will further increasewithout change in strain under temperature load andsubsequently decrease due to the loss of the stiffness

parameters, until no further stresses can be taken, i.e.material destruction. The presentation in form of ausual stress–strain diagram is insufficient and inap-propriate as the process is still controlled by tempera-ture. To be able to define a material-law dependent ontemperature, the thermal process is depicted by analogyto the stress–strain behavior (dashed axes).

Due to material warming, the elementary volumeexpands linearly proportional to the temperatureincrease (�el) and the temperature expansion coeffi-cient, whereby the material behavior is temperatureindependent (Fig. 3, �0 � � � �1 � 70°C). Due torestrained thermal expansion in the closed ring struc-ture of a tunnel lining, the initial stress increase is cal-culated as:

(1)

(2)

where ��ElT � elastic stress increase due to temperature

increase; K � compression modulus; �0 � coeffi-cient of linear thermal expansion; �� temperatureincrease.

Within a temperature range between 70 and 700°Cthe elastic modulus decreases from 100% to 10% witha sudden drop to 0% thereafter (material destruction,spalling of concrete). The generated constraint andtemperature stresses which are caused by the loss ofthe elastic modulus drop afterwards to zero.

3.2 Governing equations

Based on Hook’s law in the tensor form the followingshall apply:

(3)

(4)

(5)

314

0

200

400

600

800

0Time [minutes]

Tem

pera

ture

[°C

]

30 60 90 120 150 180

1200

1000

Fire load curve BEG-1

Figure 1. Fire load curve.

00 5 10 15 20 25 30 35 40 45 50 55 60

Depth of temperature penetration [cm]

Tem

pera

ture

of c

oncr

ete

[°C

]

Temperature of concrete during BEG-1 fire

100

200

300

400

500

600

700

800

900

1000

1100

1200

180,ξ

360,ξ

540,ξ

1800,ξ

3600,ξ

5400,ξ

7200,ξ

10080,ξ

ξ

Figure 2. Depth of temperature penetration.

s1 s1

�ϑ

�M

+�

ϑ1 ϑ2ϑ0

−�

sq = −3K . a

q . ∆q = −3K . �q

sM = E . �1 = 3K . (1 − 2v) . �1

ϑ

el

Figure 3. Material behavior.

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where

[�Dij] � deviatoric part of stress tensor;

[�0ij] � volumetric part of stress tensor;

[�ij] � total stress tensor;[�D

ij] � deviatoric part of strain tensor;[�0

ij] � volumetric part of strain tensor;

G, K � shear and compression modulus, respectively(Skolska Knjiga 1996).

Based on the assumption that the deviatoric part ofthe stress tensor is negligibly small in a closed tunnellining the total stress then equals the volumetric partof the stresses:

(6)

(7)

Stress increase due to temperature increase has aninfluence only on the volumetric part of the stress ten-sor. The volumetric strains are linearly proportional tothe temperature increase and the thermal expansioncoefficient:

(8)

where [�ij�] � thermal strain tensor; �0 � coefficient

of linear thermal expansion; �1, �0 � temperature attime (1) and initial temperature; [ij] � Kronecker -tensor, respectively.

If the volumetric displacements are restrained thenstresses will be induced into the elementary volumeas follows:

(9)

(10)

The stresses within the elementary volume can becalculated as the sum of the stresses of the volumetrictensor (Fig. 4) and the stress increase as a result of thetemperature load (Eq. 10).

(11)

(12)

Within a temperature range between �1 � 70°C and�2 � 700°C the elastic modulus decreases approxi-mately linearly from 100 to 10% so that the followingrelation can be written:

(13)

where E�t � Young’s elastic modulus at time t for

temperature �; E0 � Young’s elastic modulus at timet � 0; �t � actual temperature at time t; �2, �1 �temperature at material failure and temperature atbeginning of material softening respectively.

The tensor form of Hook’s law written in incre-mental form for temperature loads gives:

(14)

The stress decrease observed in the temperature rangebetween �1 and �2 can not be achieved by a reductionof the bulk modulus because a zero or negative incre-ment characterized by a negative compression modu-lus is not possible. As the coefficient of temperatureexpansion (�0) is a linear part of the stress incrementit is possible to calculate a direct derivative of theequivalent thermal expansion coefficient which givespositive and negative increments. Analogous to Equa-tion 13 we can write:

(15)

Until a relaxation occurs at a temperature greater than�1 � 70°C the state of stress is determined bymechanical and thermal stresses and therefore a neg-ative equivalent coefficient of temperature expansionhas to be recalculated.

(16)

where the term [�Rij] � the volumetric deformation at

relaxation.

315

Figure 4. Cross section with surface load.

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At a temperature �2 � 700°C the stress in the ele-mentary volume has to be zero.

(17)

so that we can write:

(18)

(19)

(20)

where �I�1 � major principal stress at a temperature

�1 � 70°C and �1 � equivalent coefficient of thermalexpansion for temperature between 70°C and 700°C.

From Equation 20 we can easily obtain an equiva-lent coefficient of thermal expansion for the relax-ation area:

(21)

If the temperature in the elementary volume is greaterthan 700°C this effect is called as physical materialdestruction and the elastic modulus decreases to zero.As a zero value of the elastic constants within a numer-ical model is not possible a further correction of thecoefficient of expansion is necessary in order to achievea compensation of the increasing stresses, which wouldbe caused by static loads. The external load would causea negative extension (εM, Fig. 3) and therefore gener-ate a compressive stress (�M, Fig. 3) which should becompensated by thermal expansion. The maximumtemperature in the element can theoretically reach thevalue of the temperature source (�3 � 1200°C, fire loadcurve). Again, we can derive a temperature depend-ence of the coefficient of thermal expansion:

(21)

(22)

4 PRACTICAL APPLICATION OF A FIRELOAD WITHIN A TUNNEL

The investigated example is an NATM tunnel with anoverburden of 12 m and a uniform surface load of100 kN/m2. All calculations are performed with FLAC(Itasca 2000), a program for two-dimensional numer-ical calculations.

The discrete model consists of a matrix of 70 110four node continuum elements for the soil (Fig. 5)whereby the tunnel lining is also modeled by contin-uum elements to be able to implement a modified

316

Table 1. Constitutive constants.

Unit Friction Young’s Poisson’s Weight � Angle � Modulus E Ratio

Material [kN/m3] [Degree] [MPa] [–]

Soil 22.0 38.0 65.0 0.30Concrete 25.0 – 30,000 0.20

0.100

0.300

JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 120 minutes

0

LEGEND

19-May-03 8:03

step 45768

Thermal Time 7.2000E+03

-3.000E+01 <x< 1.300E+02

-1.200E+02 <y< 4.000E+01

Grid plot

2E 1

FLAC (Version 4.00)

Geoconsult ZT GmbH

Salzburg - Austria -0.200 0.000 0.200 0.400 0.600 0.800 1.000 1.200(*10^2)

(*10^2)

-0.100

-0.300

-0.500

-0.700

-0.900

-1.100

Figure 5. Discretisized model.

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linear-elastic material model. The lining and the adja-cent soil are linked with so called “Interface” ele-ments. For the tunnel lining a Young’s modulus ofE � 30 GPa and a Poisson’s ratio of � 0.2 has beenchosen. The soil is modeled according to an elastic–plastic material model, the Mohr–Coulomb failurecriterion. To keep the calculation time within reason-able time limits only one half of the system is mod-eled, introducing symmetry boundary conditions.

The horizontal displacements at the symmetry axisat the right boundary of the mesh are fixed as well asthe vertical displacements at the bottom boundary ofthe model. At the surface a constant uniformly distrib-uted load of 100 kN/m2 is applied. The initial stressstate is defined applying a lateral earth pressure coef-ficient of 40% of the vertical pressure. The imple-mentation of the tunnel lining is performed without anyrelaxation immediately after excavation.

After the static analysis, i.e. equilibrium within thesystem prevails, a temperature load as a function oftime is applied at the inside of the tunnel lining (Fig. 6).The coefficient of heat transfer �0 � 160 W/m2K andthe coefficient of thermal conductivity � � 1.6 W/mKwere varied such that the temperature gradient andthe velocity of thermal penetration corresponded toexperimental data. The values for the specific heatCV � 1000 Ws/kgK and the coefficient for thermalexpansion �0 � 1 � 10�51/K were chosen accordingto the literature.

The calculation was performed as a coupled time-dependent mechanical analysis where mechanical andthermal time steps, which were calculated in real time,changed cyclically.

5 VERIFICATION OF MATERIAL-MODELAND CALCULATION RESULTS

The verification of the temperature fields at differenttime steps is done by comparison of the experimentaldata with the temperature pattern within the tunnellining. As it can be seen in Figure 7 the numericalresults match well with the experimental values.

For verification of the material law a beam (1 mwide, 45 cm high) was modeled by using a 1 cm by1 cm zone size. A linear-elastic material model wasused with a Young’s modulus of 30 GPa and a Poisson’sratio of 0.25. Normal pressure of 3.0 MPa was appliedon the vertical boundaries and after static calculationa thermal load according to the fire load curve (Fig. 1)was applied on the bottom of the model. The “while-stepping” loop was used for calculating the equivalent

317

FLAC (Version 4.00)

5 10 15 20 25 30 35 40

0.200

0.400

0.600

0.800

1.000


LEGEND

27-May-03 11:51step 33549


Table Plot9 Minutes30 Minutes60 Minutes90 Minutes120 Minutes

GEOCONSULT ZT GmbHSalzburg - Austria

(10+03

)

(10-02

)

Figure 7. Depth of temperature penetration in FLAC.

FLAC (Version 4.00)

0 2E 0

Applied Heat Sources

0.100

0.300

0.500

0.700

0.900

0.900

LEGEND

19-May-03 8:03

step 45768


-4.259E+00 <x< 1.019E+01

-4.270E+00 <y< 1.018E+01

Grid plot

O Max Value = 1.302E+03


Geoconsult ZT GmbH

Salzburg - Austria

(*10^1)-0.300 -0.100 0.100 0.300 0.500 0.700

(*10^1)

-0.100

-0.300

Figure 6. Detail of tunnel lining with applied temperature load.

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thermal coefficient of expansion during the thermalanalysis. Figure 8 shows the evolution of the reversemajor principal stress of a zone depending on thetemperature within the center of the zone. It is apparentthat at the beginning the principal stress increases to a

temperature of 70°C, and afterwards decreases linearlyto zero at the temperature of 700°C.

The same effect can be seen in Figure 9 where theevolution of the major principal stress in one of the ele-ments of the tunnel lining boundary, depend on thetemperature at the center of the element, is depicted.

The effects of stress changes for the internal forces(axial force and bending moment) due to temperature

318

1 2 3 4 5 6 7 8 9

JOB TITLE : 90 minutes

FLAC (Version 4.00)

LEGEND

X-axis :ztemp2 (FISH)

Y-axis :Rev_Prin. stress 1( 50, 2)

HISTORY PLOT

Thermal Time 5.2000E+03step 102002-Jul-03 15:38

GEOCONSULT ZT GmbHSalzburg - Austria

(10+02

)

(10+07

)

2.400

2.000

1.600

1.200

0.800

0.400

0.000

Figure 8. Reverse principal stress vs. zone temperature.

FLAC (Version 4.00)

10 20 30 40 50 60 70


LEGEND

X-axis :ztemp6 (FISH)

Y-axis :Prin. stress 1( 130, 116)

HISTORY PLOT

Thermal Time 7.2000E+03step 4576820-May-03 9:34

Geoconsult ZT GmbHSalzburg - Austria

(10+01

)

-0.500

-1.000

-1.500

-2.000

-2.500

-3.000

(10+01

)

Figure 9. Major principal stress in dependence on temper-ature in element.

FLAC (Version 4.00)

39 40 41 42 43 44 45

-5.000

-4.500

-4.000

-3.500

-3.000

-2.500

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JOB TITLE : G4126 BEG-Stans/Terfens, RQ 9a - km 53+500, Fire Load - 12z0 minutes

LEGEND


Y-axis :m_his_130 (FISH)

HISTORY PLOT



(10+03

)

(10-01

)

Figure 10. Time progression of bending moment in shoulder.

FLAC (Version 4.00)

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1.000

1.500

2.000


LEGEND


Y-axis :m_his_239 (FISH)

HISTORY PLOT



(10+03

)

(10-01

)

Figure 11. Time progression of bending moment in roof.

FLAC (Version 4.00)

39 40 41 42 43 44 45

-3.800

-3.600

-3.400

-3.200

-3.000

-2.800

-2.600

-2.400

-2.200

-2.000


LEGEND

(10+03

)Geoconsult ZT GmbH


Y-axis :n_his_130 (FISH)

HISTORY PLOT


Salzburg - Austria

Figure 12. Time progression of normal force in shoulder.

FLAC (Version 4.00)

39 40 41 42 43 44 45

-3.200

-3.000

-2.800

-2.600

-2.400

-2.200

-2.000

-1.800

-1.600

-1.400


LEGEND

20-May-03 9:34step 45768


HISTORY PLOTY-axis :n_his_239 (FISH)X-axis :Number of steps

Geoconsult ZT GmbH

Salzburg - Austria(10

+03)

Figure 13. Time progression of normal force in roof.

09069-37.qxd 08/11/2003 20:48 PM Page 318


loading are presented for sections in the roof and shoul-der area. Until application of a fire load a negativemoment predominates (tension soil-sided) whereas inthe roof a positive moment (tension cavity-sided)predominates.

If the boundary elements do not reach the so-called“Creep-Temperature” it is apparent that at the begin-ning the negative moment in the shoulder increasessignificantly, but immediately afterwards the momentdecreases because of the loss of stress reception.

In the roof, constraint stresses develop from thetemperature. On account of this reason the tensilestresses at the inside of the tunnel lining are reducedor turn into compressive stresses. As a result of fireloads the positive moment reduces as well until theelements achieve the relaxation temperature. Afterwardsthe moments change their direction in dependence onthe temperature in the next element rows.

The normal force in the lining is calculated overthe projection of the stresses normal to the cross-sectionthrough integration over the thickness. From Figures 12& 13, it is obvious that the normal force increasessuddenly and decreases after failure of the materialwithin a part of the lining elements. Afterwards thenormal force levels around the initial value and staysmore or less constant.

6 CONCLUSIONS

From the calculation results it is apparent that anumerical simulation of thermo-mechanical processesis possible. The quality of the results is dependent onthe quality of the implemented material-law and onthe amount of experimental data on which the materialmodel and the thermal process can be calibrated.

The temperature pattern within the first centimeterof the section has the steepest temperature gradientsand is highly non-linear. In comparison to experimen-tal data the temperature interpolation in the center ofthe element (FLAC) of a discretisized model is linear.Because of the accuracy of the calculated bendingmoments and normal forces the relation of the height

to the length of the element as well as the number ofthe Gauss integration points is a very important factorfor the relation of the internal forces from the stressesof continuum elements. Therefore an optimum numberof elements as well as geometry of the elementsshould be achieved in order not to falsify the calcula-tion results.

Local phenomena like spalling and local loss ofthermal protection, which lead to irregular temperaturevariation within the cross-section of a concrete lining,are factors which influence the results considerably.These effects are not considered in this paper.

The internal forces, especially the moments areconsiderably influenced by the rate of temperaturespread. All calculations show that the moments arepredominately effected within the first minutes of afire case as soon the concrete has the full stiffness andstress reception capability. The negative moments(tensile stresses soil-sided) increase rapidly and after-wards decrease slowly because of the reduction of thestiffness and loss of stress reception capability withinthe heated zones of the lining. The positive moments(tensile stresses cavity-sided) show the same trendsand can change their sign in dependence of the initialstress state. These extreme values can provide impor-tant information for dimensioning the lining.

Within the first minutes of a fire load the normalforces in the tunnel lining show the same trend as thebending moments. As a result of fire loads a signifi-cant stress increase at the inside of the tunnel liningcan speed up the spalling of concrete.

REFERENCES

Itasca Consulting Group, Inc. 2000. FLAC – Fast LagrangianAnalysis of Continua, Version 4.0 User’s Manual.Minneapolis: Itasca.

Skolska Knjiga, 1996. Inzenjerski Prirucnik, Zagreb:Strucno-Znanstvena Redakcija Biblioteke.

Kusterle W., Waubke N.V., 2001. Baulicher Brandschutz –Betontechnologie, Innsbruck: Institut für Baustoffe undBauphysik der Universität Innsbruck.

Gresslehner K.H., 2001. Festlegung der BEG-1 Kurve.

319

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321

Seismic liquefaction: centrifuge and numerical modeling

P.M. Byrne & S.S. ParkDepartment of Civil Engineering, University of British Columbia, BC, Canada

M. BeatySenior Engineer, Calif. Dept. of Water Resources, Sacramento, CA, USA

ABSTRACT: A fully coupled effective stress dynamic analysis procedure for modeling seismic liquefactionis presented. An elastic plastic formulation is used for the constitutive model UBCSAND in which the yield lociare radial lines of constant stress ratio and the flow rule is non-associated. This is incorporated into the 2D versionof FLAC by modifying the existing Mohr-Coulomb model. This numerical procedure is used to simulate centrifugetest data from the Rensselaer Polytechnic Institute (RPI). UBCSAND is first calibrated to cyclic simple shear testsperformed on Nevada sand. Both pre- and post-liquefaction behavior is captured. The centrifuge tests are thenmodeled and the predicted accelerations, excess porewater pressures, and displacements are compared with themeasurements. The results are shown to be in general agreement when stress densification and saturation effectsare taken into account. The procedure is currently being used in the design of liquefaction remediation measuresfor a number of dam, bridge, tunnel, and pipeline projects in Western Canada.

1 INTRODUCTION

Displacements arising from seismic liquefaction can bevery large and are a major concern for earth structureslocated in regions of moderate to high seismicity. Liq-uefaction is caused by high porewater pressures result-ing from the tendency for granular soils to compactwhen subjected to cyclic loading. Remedial measurestypically involve attempts to prevent or curtail lique-faction so that displacements are reduced to tolerablelevels. Modifications can also be made to the structureso that larger displacements can be tolerated. In eithercase, the rational design for remediation requires areliable prediction of soil-structure response during thedesign earthquake.

State-of-practice procedures for evaluating liquefac-tion typically use separate analyses for liquefactiontriggering (e.g. Youd et al. 2001), flow slide (limitequilibrium with residual strength), and displacements(Newmark sliding block). While the results of the trig-gering evaluation are used as input into the flow slideand displacement evaluations, the analyses are other-wise independent. While this practice often provides a good screening level tool, these simplified totalstress analyses cannot reliably predict excess porewa-ter pressures, accelerations, or displacement patterns.

State-of-art procedures involve dynamic finite ele-ment or finite difference analyses using effective stress

procedures coupled with fluid flow predictions. Theseanalyses can estimate the displacements, accelerationsand porewater pressures caused by a specified inputmotion. Triggering of liquefaction, displacements andflow slide potential are addressed in a single analysis.Such analyses involve capturing the liquefaction behav-ior of a soil element as observed in laboratory tests,and then modeling the soil-structure as a collection ofsuch elements subjected to the design earthquake basemotion.

It is vital that these sophisticated procedures be ver-ified before they are used in practice. Instrumentedcentrifuge model tests can be used for verification andhave some advantages over observed field behavior.Centrifuge tests allow the measurement of displace-ments, input and induced accelerations, and porewaterpressures under field stress conditions. These tests cantherefore provide a useful database for verification ofnumerical modeling. This approach is used below.

2 LIQUEFACTION

Liquefaction is caused by the tendency of granular soilto contract when subjected to monotonic or cyclic shearloading. When this contraction is prevented or curtailedby the presence of water in the pores, normal stress istransferred from the soil skeleton to the water. This

09069-38.qxd 8/11/03 9:20 PM Page 321


can cause high excess pore pressures resulting in a verylarge reduction in shear stiffness. Large shear strainsmay occur, and the soil will dilate with these strainsunless the soil is very loose. This dilation causes theporewater pressure to drop and the stiffness to increase,which can limit the strains, induced by a load cycle.This behavior is illustrated in Figure 1 for monotonicloading.

It is this tendency of the soil skeleton to contract anddilate that controls its liquefaction response. Once theskeleton behavior is modeled, the response underdrained, undrained or coupled stress-flow conditionscan be computed by incorporating the bulk stiffness andflow of the pore fluid.

3 CONSTITUTIVE MODEL: UBCSAND

The simplest realistic model for soil is the classic Mohr-Coulomb elastic–plastic model as depicted in Figure 2.Soils are modeled as elastic below the strength enve-lope and plastic on the strength envelope with plasticshear and volumetric strains increments related by the

dilation angle, �. This model is really too simple forsoils since plastic strains also occur for stress statesbelow the strength envelope. The UBCSAND stress–strain model described herein modifies the Mohr-Coulomb model incorporated in FLAC to capture theplastic strains that occur at all stages of loading. Yieldloci are assumed to be radial line of constant stressratio as shown in Figure 3. Unloading is assumed to beelastic. Reloading induces plastic response but with astiffened plastic shear modulus.

The plastic shear modulus relates the shear stress andthe plastic shear strain and is assumed to be hyper-bolic with stress ratio as shown in Figure 4. Movingthe yield locus from A to B in Figure 3 requires a plas-tic shear strain increment, �

P�, as shown in Figure 4,

and is controlled by the plastic shear modulus, GP. Theassociated plastic volumetric strain increment, d�P

v, isobtained from the dilation angle �:

(1)

The dilation angle is based on laboratory data andenergy considerations and is approximated by

(2)

322

Shear Strain, γ

Shea

r St

ress

, τ

u

σv0

Shear Strain, γPore

Pre

ssur

e, u

Shear Strain, γ

Eff

ectiv

e st

ress

, σv

(a)

(b)

(c)

Contraction Dilation

σ'v0

τ

�

Figure 1. Undrained response of loose sand in simple shear:(a) stress–strain, (b) pore pressure, and (c) effective stressresponse.

Normal Effective Stress, s'Plastic Volumetric Strain Increment, dεv

Shea

r St

ress

, τPl

astic

She

ar S

trai

n In

crem

ent,

dγp

Elastic

Strength EnvelopePlastic StrainIncrement Vector f

p

Figure 2. Classic Mohr-Coulomb model.

s', dεv

A

B

Yield Locus

τ ,

dγp

φd

Plastic StrainIncrement Vector

f

p

Figure 3. UBCSAND model.

09069-38.qxd 8/11/03 9:20 PM Page 322


where �cv is the phase transformation or constant vol-ume friction angle and �d describes the current yieldlocus. A negative value of � corresponds to contraction.Contraction occurs for stress states below �cv and dila-tion above as shown in Figure 5. Additional informationon earlier but similar forms of UBCSAND is presentedby Puebla et al. (1997) and Beaty & Byrne (1998).

Elastic and plastic properties for the model aredefined as follows.

3.1 Elastic properties

The elastic bulk modulus, B, and shear modulus, Ge,are assumed to be isotropic and stress level dependent.They are described by the following relations where kBand kG are modulus numbers, PA is atmospheric pres-sure, and ��m is the mean effective stress:

(3)

(4)

3.2 Plastic properties

The plastic properties used by the model are the peakfriction angle �P the constant volume friction angle �cv,and plastic shear modulus GP, where

(5)

GiP Ge and depends on relative density, � is the

current shear stress, �f is the projected shear stress atfailure, and Rf is the failure ratio used to truncate thehyperbolic relationship.

The position of the yield locus �d is known for eachelement at the start of each time step. If the stress ratioincreases and plastic strain is predicted, then the yieldlocus for that element is pushed up by an amount ��das given by Equation 6. Unloading of stress ratio is con-sidered to be elastic. Upon reloading, the yield locus isset to the stress ratio corresponding to the stress reversalpoint.

(6)

The elastic and plastic parameters are highly depend-ent on relative density, which must be considered in anymodel calibration. These parameters can be selected by calibration to laboratory test data. The response ofthe model can also be compared to a considerable data-base for triggering of liquefaction under earthquakeloading in the field. This database exists in terms ofpenetration resistance, typically from cone penetration(CPT) or standard penetration (SPT) tests. A commonrelationship between (N1)60 values from the SPT andthe cyclic stress ratio that triggers liquefaction for a magnitude 7.5 earthquake is given by Youd et al.(2001). Comparing laboratory data based on relativedensity to field data based on penetration resistancerelies upon an approximate conversion, such as thatproposed by Skempton (1986):

(7)

Model parameters based on penetration resistanceand field observation may be useful for field conditionswhere it is very difficult to retrieve and test a represen-tative sample. However, this indirect method is notappropriate for simulation of centrifuge models. Cali-brations for this case should be based on direct labo-ratory testing of samples that are prepared in the samemanner as the centrifuge model.

323

Stre

ss R

atio

, h (

= τ

/s’)

A B

G p/ s'

Plastic Shear Strain, g p

Dgp

Figure 4. Hyperbolic stress–strain relationship.

Normal Effective Stress,

Shea

r St

ress

,

Dilation

Contraction

Figure 5. Zones of shear-induced contraction and dilation.

09069-38.qxd 8/11/03 9:20 PM Page 323


4 SIMULATION OF CYCLIC ELEMENT TEST DATA

A number of cyclic simple shear tests have been con-ducted on Fraser River sand at the University of BritishColumbia. The samples were prepared by air pluvia-tion with a target relative density Dr of 40% and testedat an initial vertical effective stress, ��v0, of 100 kPa.Samples were also tested at ��v0 of 200 kPa with a Dr of44%. Samples were subjected to cyclic shear underconstant volume conditions that simulate undrainedresponse at a range of cyclic stress ratios. Typicalexamples of measured response are shown in Figures 6& 7. From Figure 6a it may be seen that as the shearstress is cycled, the effective stresses decrease as thepore pressure ratio ru increases. This ratio ru is given by(u – u0)/��v0, where u0 and u are the initial and currentpore pressures. ru approaches unity after 5 cycles,which corresponds to a state of zero effective stress.Application of further cycles produce very large shearstrains in the range of 10 to 15% or more as shown inFigure 6b. However, the strain per cycle is limited asthe pore pressures drop with strain due to dilation.

Figures 6 & 7 also show the response predictedusing UBCSAND. The elastic and plastic parametersselected by the calibration were the same for bothtests. The model gives a reasonable representation ofthe observed response, although the final predictedstrains are less than measured for Figure 6. A summary

of the test results and the UBCSAND calibration areshown in Figure 8. The predicted and measured lique-faction response for ��v0 of 100 and 200 kPa is in closeagreement.

5 CENTRIFUGE TESTS

A simulation using UBCSAND was made of 2 cen-trifuge tests carried out at RPI as described in Table 1.In the centrifuge test, a small model is used that issubjected to a high acceleration field during the test.

324

-20-15-10-505

101520

0 20 40 60 80 100

Vertical Effective Stress (kPa)

Shea

r St

ress

(kP

a)

Drc=40% CSR=0.1(a)

-20-15-10

-50

5101520

-20 -15 -10 -5 0 5 10 15 20

Shear Strain (%)

Shea

r St

ress

(kP

a)

Test Calibration

Test Calibration

Drc=40% CSR=0.1(b)

-20-15-10-505

101520

0 20 40 60 80 100

Vertical Effective Stress (kPa)

Shea

r St

ress

(kP

a)

Test Calibration

Drc=40% CSR=0.15(a)

-20-15-10-505

101520

-20 -15 -10 -5 0 5 10 15 20

Shear Strain (%)

Shea

r St

ress

(kP

a)Test Calibration

Drc=40% CSR=0.15(b)

Figure 7. Stress path and stress–strain relationship(CSR 0.15).

01 10 100

0.05

0.1

0.15

0.2

No. of Cycles to Liquefaction

Cyc

lic S

tres

s R

atio

(CSR

)

Test: Dr=40%Test: Dr=44%

UBCSANDDr=44%Dr=40%

Figure 8. Predicted and measured liquefaction response ofFraser River sand.

Figure 6. Stress path and stress–strain relationship(CSR 0.1).

09069-38.qxd 8/11/03 9:20 PM Page 324


This has the effect of increasing its stresses by theratio of the induced acceleration divided by the accel-eration of gravity. This ratio or factor is 120 forModel 1 and 60 for Model 2 as indicated by Table 1.The centrifuge model under the increased accelerationfield can also be thought of as representing a prototype

that is 120 (Model 1) or 60 (Model 2) times largerthan the actual model. Results from the centrifuge testcan be presented at either the model or prototype scale.The prototype scale is used for this paper.

While in flight, a motion simulating an earthquaketime history is applied to the base of the model. Fordynamic similitude at the model scale, the earthquaketime scale must be decreased by a factor of 120 (Model1) or 60 (Model 2), and the earthquake accelerationincreased by the same factor. The engineering coeffi-cient of permeability k will also increase by this samefactor due to the increased unit weight of the fluid. kshould be decreased for hydraulic similitude, althoughit is not necessary to model a specific k. It is commonto use a fluid in the test that is 30 to 60 times more vis-cous than water to prevent rapid rates of dissipationthat might unduly curtail liquefaction effects.

Nevada sand was used for these centrifuge testsand its liquefaction and permeability (at 1 g usingwater as pore fluid) properties were obtained fromlaboratory tests (Arulmoli et al. 1992, Kammerer et al. 2000, Taboada-Urtuzuastegui et al. 2002). Itsmeasured liquefaction resistance together with theUBCSAND prediction is shown in Figure 9.

5.1 Model 1

Model 1 comprises a uniform horizontal sand layerhaving a thickness of 37 m (prototype scale) and aplacement density Dr of 55% as shown in Figure 10(Gonzalez et al. 2002). After application of the 120 gacceleration field, Dr was estimated to increase to 63%near the base due to the increase in stresses. The amountof densification was estimated from one-dimensionalcompression tests. The applied base motion is shown

325

Table 1. Centrifuge model tests.

RPI Model 1 RPI Model 2

Test condition Level SlopeDr 55% 40%Centrifuge acc. 120 g 60 gMax. ��v 380 kPa 100 kPaSoil depth 38 m 10 mFluid viscosity 60 �w 60 �w

01 10 100

0.1

0.2

0.3

0.4

0.5

No. of Cycles to Liquefaction

Cyc

lic S

tres

s R

atio

(C

SR) Test: Dr=43-46%

Test: Dr=60-63%Test: Dr=86-89%

UBCSANDDr=44%

Dr=62% Dr=88%

Figure 9. Liquefaction resistance of Nevada sand.

(a) Centrifuge Model 1 (b) FLAC Model 1

Z = 0.0 m

Z = 6.3 m

Z = 13.1 m

Z = 24.8 m

Z = 30.8 mZ = 37.0 m

Z = 38.1 m Ac8

Input Motion: 50 cycles, 0.2g, 1.5Hz

Pore Pressure Transducer Accelerometer

Measurements

120 g

Z = 1.3 mAc6

Ac5

Ac7

Ac4

Ac3

Ac2

Ac1

P7

P5

P1

P8

P6

P4

P3

P2

Navada sand(Dr=55%)

Figure 10. Centrifuge Model 1 and FLAC Model 1.

09069-38.qxd 8/11/03 9:20 PM Page 325


in Figure 11 and consisted of 50 cycles with an ampli-tude of 0.2 g and a frequency of 1.5 Hz. The key inputsincluding water bulk stiffness (Bf) for different layersin the numerical model are listed in Table 2.

The container for Model 1 consisted of slip “rings”that allowed differential horizontal displacements inthe vertical direction but not in the horizontal. Thiswas simulated in the FLAC model by “attaching” thevertical sides, Figure 10. The initial horizontal effec-tive stresses were set to 0.5 times the vertical effectivestresses.

The measured and predicted excess pore pressuresand accelerations for various depths are shown inFigure 12. The predicted accelerations are initiallyabout the same at all depths and approximately equal tothe base input value of 0.2 g. The accelerations decrease

326

0 10 20 30 40

Time (Sec)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Acc

eler

atio

n (g

)

Centrifuge Input Numerical Input

Figure 11. Based input motions of Model 1.

Table 2. Key input for Model 1 numerical analysis.

Bf afterspinup Permeability

Layer kG kB (kPa) (m/sec)

Top 952 2856 0.44 0.2 E5 5 E-5Middle 1020 3060 0.61 0.6 E5 5 E-5Bottom 1042 3126 0.67 1.2 E5 5 E-5

-0.4

-0.2

0

0.2

0.4

Acc

(g)

P(depth=1.3m)

-0.4

-0.2

0

0.2

0.4

Acc

(g)

M (measurement) P (prediction)

M(depth=1.3m)

M(depth=6.3m) P(depth=6.3m)

-0.4

-0.2

0

0.2

0.4

Acc

(g)


-0.4

-0.2

0

0.2

0.4

Acc

(g)


-0.4

-0.2

0

0.2

0.4

Acc

(g)


0 10 20 30 40Time (sec)

-0.4

-0.2

0

0.2

0.4

Acc

(g)

10 20 30 40Time (sec)


Figure 12a. Measured (left) and predicted (right) accelera-tions of Model 1.

0 10 20 300

20406080

100120140

EPP

(kP

a)

Measurement Prediction

0 10 20 30 400

50

100

150

200

250

300

EPP

(kP

a)

0 10 20 30 400

50100150200250300350

EPP

(kP

a)

0 10 20 30

Time (sec)

0

100

200

300

400

EPP

(kP

a)

Depth =13.1m

--

Depth = 24.8m

--

Depth = 30.8m

--

Depth = 37.0m

-- σ' vo

σ' vo

σ' vo

σ' vo

40

40

Figure 12b. Measured and predicted excess pore pressuresof Model 1.

09069-38.qxd 8/11/03 9:21 PM Page 326


over much of the model as the shaking continues. Thedecay of acceleration is most rapid in the upper layersand can be explained in terms of the excess porewaterpressures shown in Figure 12b. A large drop in acceler-ation response occurs when the measured excess porepressure reaches the initial vertical effective stress ��v0,which corresponds to a liquefied state. Measurementsshow that liquefaction occurs first near the surface andthen progresses downward. The accelerations andexcess pore pressures predicted using UBCSAND arein generally good agreement with the measurements.

The analysis described above incorporates the effectof densification due to the increased acceleration field.If this effect is not considered, and a uniform Dr of55% is used in the analysis, then liquefaction is pre-dicted to occur first at the base of the model ratherthan at the surface. The higher Dr at the base reversesthis trend and indicates the importance of stress den-sification in centrifuge tests.

Full saturation of the pores is difficult to achieve in acentrifuge test. The best fit with the data was obtainedassuming an initial placement saturation, or Sr, of 98%at atmospheric pressure. The pore pressure will increaseas the centrifuge acceleration is applied, and the result-ing increase in Sr is modeled using the gas laws.

In summary,

(a) UBCSAND provides a reasonable agreement tothe test results,

(b) ru 1.0 and liquefaction can occur at depths of40 m in medium dense sand strata,

(c) a large reduction in the accelerations can occurupon liquefaction,

(d) the effect of stress densification should beincluded, and

(e) the degree of saturation, Sr, must be considered.

5.2 Model 2

The cross section for Model 2 is shown in Figure 13and comprises a steep 1.5:1 slope in loose fine sandwith Dr 40% (Taboada-Urtuzuastegui et al. 2002).The base motion consists of 20 cycles of 0.2 g at a fre-quency of 1 Hz. The container for model 2 was rigidand this was simulated in the FLAC model by apply-ing the input motion to the vertical sides as well as thebase. The key inputs for Model 2 are listed in Table 3.Pore pressures and accelerations were measured awayfrom the face of the slope, approximating free fieldconditions, as well as adjacent to the slope.

The predicted and observed accelerations and porepressures in the free field are shown in Figures 14 & 15.As expected, similar trends are seen as for the levelground test of Model 1, i.e. ru of 100% and reducedaccelerations.

The accelerations and pore pressures near theslope are shown in Figures 16 & 17. It may be seen inFigure 16 that there is little or no reduction in theaccelerations. Instead, large upslope acceleration spikesoccur. Excess pore pressures are shown in Figure 17.Large negative excess pore pressure spikes occur thatcoincide in time with the upslope acceleration spikes.The slope is steep and the upslope acceleration of thebase tends to induce failure of the slope and relative

327

Figure 13. Cross section of Model 2 (Taboada-Urtuzuastegui et al. 2002).

Table 3. Key input for Model 2 numerical analysis.

Bf after spinup Permeability

Layer kG kB (kPa) (m/sec)

Free field 867 2601 0.22 1.0 E5 2.1 E-5

09069-38.qxd 8/11/03 9:21 PM Page 327


downslope movement. The soil dilates as it shears inthe downslope direction, producing negative porepressures which stiffen the shear modulus. Enoughstrength is mobilized through this dilation to arrest the

downslope movement and gives rise to the accelera-tion spike (Taboada-Urtuzuastegui et al. 2002).

UBCSAND provides a reasonable prediction ofthe accelerations and pore pressure response for the

328

Time (sec)

Acc

eler

atio

n (g

)

-0.4

0

0.4 AH1

-0.8

-0.4

0

0.4

0.8 AH5

5 10 15 2050 10 15 20

Time (sec)

-0.8-0.4

00.40.8 AH6

-0.2

0

0.2

-0.4

0.4

-0.8

-0.4

0.4

0.8

-0.8-0.4

0.40.8

-0.20.0

0.0

0.0

0.0

0.2Input

AH5

AH6

(a)

(a)

(b)

(c)

AH1

Input

(Taboada-Urtuzuastegui et al. 2002)

Figure 14. Measured (left) and predicted (right) accelerations at free field.

0

40

80

PP1

0

20

40

PP5

5 10 15 205-5

10 15 20Time (sec)Time (sec)

0

10

20

0

40

80

0

20

40

0

0

10

20

PP6PP6

PP5

PP1 (a)

(b)

(c)

ru = 1.0


Exc

ess

pore

pre

ssur

e (k

Pa)

Figure 15. Measured (left) and predicted (right) excess pore pressures at free field.

09069-38.qxd 8/11/03 9:21 PM Page 328


329

-0.2

0

0.2 Input

-0.4

0

0.4 AH2

-0.6-0.4-0.2

00.2

AH4

5 10 15 20

Time (sec)

-0.8

-0.4

0

0.4AH7

50 10 15 20Time (sec)

Acc

eler

atio

n (g

)

-0.2

0.0

0.0

0.0

0.0

0.2

-0.4

0.4

-0.6-0.4-0.2

0.2

-0.8

-0.4

0.4 (Taboada-Urtuzuastegui et al. 2002)

AH4

AH7

AH2

Input

(c)

(d)

(b)

(a)

Figure 16. Measured (left) and predicted (right) accelerations near the slope.

-10

0

10

20 PP2

0

10

20

30

PP3

-20

0

20 PP4

5 10 15 20Time (sec)

-30-20-10

010

PP7

50 10 15 20

Time (sec)

Exc

ess

pore

pre

ssur

e (k

Pa)

0

10

20

0

10

20

30

-20

0

20

-30-20-10

0

10


PP7

PP4

PP3

PP2(a)

(b)

(c)

(d)

ru = 1.0

Figure 17. Measured (left) and predicted (right) excess pore pressures near the slope.

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free field. More significant differences are observedfor locations near the slope. Some of these differencesare due to UBCSAND under predicting the dilativespikes. This requires further investigation. The meas-ured and predicted displacements after shaking areshown in Figures 18 & 19. It may be seen that both themagnitude and pattern of displacements are in gen-eral agreement.

In summary,

(a) UBCSAND provides reasonable agreement withthis centrifuge test, although further study isneeded for locations close to the sloping face,

(b) a decrease in accelerations after liquefaction wasnot observed near the slope,

(c) a large upslope acceleration spikes occurred nearthe slope,

(d) a decrease in pore pressure due to dilation corre-sponded with these upslope acceleration spikes, and

(e) the dilative spikes prevented very large displace-ments from occurring in this homogeneous finesand model.

6 SUMMARY

A fully coupled effective stress dynamic analysis procedure has been presented. The procedure is first calibrated by comparison with laboratory element test data and then verified by comparison with twocentrifuge model tests.

Model 1 represented a deep sand layer with a levelground condition. This model showed that high excessporewater pressure and liquefaction can occur todepths of 40 m in medium dense sands. Liquefactionfirst occurred at the surface and progressed down-ward under continued shaking. Accelerations above

330

Test No.1Test Conditions: 60g,60 times viscosity of waterMax. Acceleration: 0.25g

Displacement unit : m Magnitude

10

1.91+1.74 to1.911.58 to1.741.41 to1.581.24 to1.411.08 to1.240.91 to1.080.75 to 0.910.58 to 0.750.41 to 0.580.25 to 0.410.08 to 0.25

8

6

4

2

0

10

8

6

4

2

00510152025303535404550

Figure 18. Measured displacements for Model 2 from centrifuge test (Taboada-Urtuzuastegui et al. 2002).

Maximum Displacement = 2.6 m

Figure 19. Predicted displacements for Model 2 using UBCSAND.

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the depth of liquefaction showed a significantdecrease. The numerical model results were in goodagreement with the measurement when stress densifi-cation and saturation effects were included.

Model 2 represented a steep slope condition inhomogeneous loose fine sand. The results showed thatlarge upslope acceleration spikes occurred near the faceof the slope after liquefaction. These accelerationspikes corresponded with large negative excess porepressure spikes associated with dilation. It is theincrease in effective stress associated with these nega-tive pore pressure spikes that curtails the displacementsand makes the slope more stable than might be expectedunder cyclic loading. The overall pattern of predictedresponse is in reasonable agreement with the measure-ments, although both the acceleration and pore pressurespikes are under predicted by the UBCSAND analysis.

A new series of centrifuge tests are planned atCCORE (Centre for Cold Ocean Research), MemorialUniversity, Newfoundland, which will permit furtherverification and refinement of the numerical model.

REFERENCES

Arulmoli, K., Muraleetharan, K.K., Hossain, M.M. & Fruth,L.S. 1992. VELACS laboratory testing program, soil datareport. The Earth Technology Corporation, Irvine,California, Report to the National Science Foundation,Washington D.C., March.

Beaty, M. & Byrne, P. 1998. An effective stress model forpredicting liquefaction behaviour of sand. ASCE Geot.Special Pub. No. 75: 766–777.

Gonzalez, L., Abdoun, T. & Sharp, M.K. 2002. Modeling ofseismically induced liquefaction under high confiningstress.

Kammerer, A., Wu, J., Pestana, J., Riemer, M. & Seed, R.2000. Cyclic simple shear testing of Nevada sand forPEER Center project 2051999. Geotechnical EngineeringResearch Report No. UCB/GT/00-01, University ofCalifornia, Berkeley, January.

Puebla, H., Byrne, P.M. & Phillips, R. 1997. Analysis ofCANLEX liquefaction embankments: prototype andcentrifuge models. Can. Geotech. Journal, Vol. 34, No. 5:641–657.

Skempton, A.W. 1986. Standard penetration test proceduresand the effects in sands of overburden pressure, relativedensity, particle size, ageing and overconsolidation,Geotechnique 36, No. 3: 425–447.

Taboada-Urtuzuastegui, V.M., Martinez-Ramirez, G. &Abdoun, T. 2002. Centrifuge modeling of seismic behav-ior of a slope in liquefiable soil, Soil Dynamic andEarthquake Engineering, Vol. 22: 1043–1049.

Youd, T.L., Idriss, I. M., Andrus, R.D., Arango, I., Castro, G.,Christian, J.T., Dobry, R., Finn, W.D.L., Harder Jr., L.F.,Hynes, M.E., Ishihara, K., Koester, J.P., Liao, S.,Marcuson III, W.F., Martin, G.R., Mitchell, J.K.,Moriwaki, Y., Power, M.S., Robertson, P.K., Seed, R.B. &Stokoe, K.H. 2001. Liquefaction Resistance of Soils:Summary Report from the 1996 NCEER and 1998NCEER/NSF Workshops on Evaluation of LiquefactionResistance of Soils. ASCE J. of Geot. and Geoenv. Eng.,Vol. 127, No. 10: 817–833.

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333

Modeling the dynamic response of cantilever earth-retaining walls using FLAC

R.A. GreenDepartment of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI, USA

R.M. EbelingInformation Technology Laboratory, US Army Engineer Research and Development Center, Vicksburg, MS, USA

ABSTRACT: A research investigation was undertaken to determine the dynamically induced lateral earth pres-sures on the stem portion of a concrete, cantilever, earth-retaining wall. In total, the wall-soil column system was68.6 m in height, with the upper 6.1 m being composed of the cantilever wall retaining compacted backfill. A seriesof numerical analyses were performed using FLAC. The analyses consisted of the incremental construction of thewall and placement of the backfill, followed by dynamic response analyses, wherein the soil was modeled as elasto-plastic. This paper outlines the details of the numerical model used in the analyses. Particular attention is given tohow the ground motion was specified, determination of the wall and soil model parameters, and the modeling of thewall-soil interface. To benchmark the FLAC results, comparisons are presented between the FLAC results and theresults from simplified techniques for computing dynamic earth pressures and permanent wall displacement.

1 INTRODUCTION

1.1 Scope

A research investigation using FLAC was undertaken todetermine the dynamically induced lateral earth pres-sures on the stem portion of a concrete, cantilever,earth-retaining wall. The analyses consisted of theincremental construction of the wall and placement ofthe backfill, followed by dynamic response analyses,wherein the soil was modeled as elasto-plastic with aMohr-Coulomb failure criterion. The focus of thispaper is to outline the details of the numerical modelused in the analyses. Particular attention is given to howthe ground motions were specified, the wall and soilmodel parameters were determined, and the wall-soilinterface was modeled. To assess the validity of the pro-posed FLAC model, comparisons of the FLAC resultsare made with results from simplified analysis tech-niques for determining dynamic earth pressures (i.e.,Mononobe-Okabe approach) and for determining per-manent displacement of the wall (i.e. Newmark slidingblock approach).

1.2 Description of wall-soil system

The retaining wall analyzed was approximately 6.1 min height, retaining medium-dense, cohesionless, com-pacted fill (total unit weight: �t � 19.6 kN/m3; effective

angle of internal friction: �� 35°). Underlying thewall/backfill was approximately 62.5 m of naturallydeposited dense cohesionless soil (�t � 19.6 kN/m3;�� 40°). The groundwater table was well below thebase of the wall and was not considered in the analyses.

The geometry and structural detailing of the wallwere determined following the US Army Corps ofEngineers static design procedures (Headquarters, USArmy Corps of Engineers 1989, 1992), with the dimen-sions of the structural wedge (i.e. wall and containedbackfill) depicted in Figure 1. The properties of the con-crete and reinforcing steel used in the wall design are asfollows: unit weight of concrete: �c � 23.6 kN/m3; com-pressive strength of concrete: f�c � 27.6 MPa; and yieldstrength of reinforcement: fy � 413.4 MPa. Additionaldetails about the wall design and soil profile are givenin Green & Ebeling (2002).

2 NUMERICAL MODEL

2.1 Overview of FLAC model

The FLAC numerical model consisted of the upper9.1 m of the wall-soil system, comprising the wall/back-fill and approximately 3 m of the underlying naturaldeposit (foundation soil). Laterally, the FLAC modelwas approximately 22.9 m, to include approximately7.6 m of the foundation soil in front of the wall and

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approximately 15.3 m of the backfill/foundation soilbehind the wall (Fig. 2).

An elasto-plastic constitutive model, in conjunctionwith Mohr-Coulomb failure criterion, was used tomodel the soil. Elastic beam elements were used tomodel the concrete retaining wall, with the wall/back-fill being “numerically constructed” in FLAC similarto the way an actual wall would be constructed. Thebackfill was placed in 0.61 m lifts, for a total of tenlifts, with the model being brought to static equilib-rium after the placement of each lift. Such placementallowed realistic earth pressures to develop as the walldeformed and moved because of the placement of eachlift. The constructed retaining wall-soil model is shownin Figure 2.

The model consists of four sub-grids, labeled onethrough four in Figure 2. The separation of the founda-tion soil and backfill into sub-grids one and two wasrequired because a portion of the base of the retainingwall was inserted into the soil. Sub-grid three wasincluded so that free-field boundary conditions could

be specified along the lateral edges of the model (free-field boundary conditions cannot be specified acrossthe interface of two sub-grids). Sub grid four wasincluded for symmetry, but its inclusion was not neces-sary. The sub-grids were “attached” at the soil-to-soilinterfaces, as depicted by white lines in Figure 2, andinterface elements were used at the wall-soil interfaces.

The following sub-sections outline how the groundmotions were specified and the procedures used todetermine the various soil and wall model parameters.

2.2 Specification of input motions

Dynamic analyses can be performed with FLAC,wherein user-specified acceleration, velocity, stress, orforce time-histories can be input as exterior boundaryconditions or as interior excitations. A parametric studywas performed to determine the best way to specifythe ground motions in FLAC for earthquake analyses.The parametric study involved performing a series ofone-dimensional (1-D) site response analyses usingconsistently generated acceleration, velocity, and stresstime-histories. Generally, earthquake ground motionsare not defined in terms of force time-histories andtherefore were not considered in the parametric study.The use of stress time-histories in FLAC has the benefitof allowing the time-history to be specified at “quietboundaries,” thus simulating radiation damping.

Using a free-field acceleration time-history recordedat the surface of a USGS site class B profile duringthe 1989 Loma Prieta earthquake, a 1-D site responseanalysis was performed using a modified version ofSHAKE91 (Idriss & Sun 1992). The analysis was performed on a 68.6 m, 5% damped, non-degrading profile, wherein the acceleration time-history wasspecified as an outcrop motion. Interlayer accelera-tion and stress time-histories were computed at theprofile surface and at depths of 7.6 m, 10.7, 15.2, and 68.6 m (i.e. bedrock). Interlayer velocity time-histories were computed by integrating the interlayeracceleration time-histories using the trapezoidal rule.The interlayer acceleration, velocity, and stress time-histories were used as base motions in a series ofFLAC analyses, in which the acceleration time-histories at the surface of the FLAC profiles werecomputed. The profiles used in the FLAC analyseswere comparable to the SHAKE profiles down to thedepths corresponding to the interlayer motions. Anelastic constitutive relation, with 5% Rayleigh damp-ing, was used to model the soil layers in the FLACprofiles. The central frequency of the damping rela-tionship was set to the fundamental frequencies of therespective FLAC profiles.

Fourier amplitude spectra (FAS) and 5% damped,pseudo acceleration response spectra (PSA) werecomputed from the acceleration time-histories of thesurface motions of the SHAKE and FLAC profiles.

334

3

m

1

2 3

4

7.6m 15.3m

9.1

m

22.9m

6.1

m

Figure 2. Annotated FLAC model of the wall-soil system.

0.6m

0.2m

0.9m

HeelToe

Stem

Base

Backfill

0.5m

4m

6.1

m

2.4m

Figure 1. Dimensions of the structural wedge of the wall-soilsystem analyzed, wherein the term “structural wedge” refers toall that is shown above.

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Error analyses were performed on the spectra corre-sponding to the different profiles and different typesof specified input motions. In the error analyses, thespectra for the SHAKE motions were used as the“correct” motions. The word “correct” does not implythat SHAKE precisely models the behavior of an actualsoil profile subjected to earthquake motions. Rather,SHAKE gives the analytically correct motion for avisco-elastic profile with constant damping applied toall frequencies of motion. On the other hand, the FLACmodels used in this study give numerical approxima-tions of the correct analytical solution. The errors inthe FLAC spectral values were computed at a spec-trum of frequencies using the following expressions.

(1a)

(1b)

From the results of the parametric study, it wasdetermined that the specification of the input motionin FLAC in terms of stress time-histories gives theleast accurate results, wherein the stress time-histories were applied at a “quiet boundary” along thebase of the FLAC model. The errors corresponding tospecifying the motions in terms of acceleration andvelocity time-histories were essentially identical andconsiderably less than those associated with the stresstime-histories.

2.3 Development of input motions for wall analyses

As stated previously, the FLAC model of the soil-wallsystem consisted of only the upper 9.1 m of a 68.6 mprofile. To account for the influence of the soil profilebelow 9.1 m on the ground motions, the entire 68.6 mprofile, without the retaining wall, was modeled usinga modified version of SHAKE91. The interlayer motionat the depth corresponding to the base of the FLACmodel (i.e. 9.1 m) was computed. The input groundmotion used in the SHAKE analysis was the sameLoma Prieta motion used in the parametric study dis-cussed above. The motion was specified as a rock out-crop motion at the base of the 68.6 m soil column.

The small strain fundamental frequency of theretaining wall-soil system in the FLAC model wasestimated to be approximately 6 Hz. At larger strains,the fundamental frequency of the system will be lessthan the small strain value. To ensure proper excita-tion of the wall, the cutoff frequency in the SHAKEanalysis was set at 15 Hz. This value was selectedconsidering both the fundamental frequency of thewall-soil system and the fact that the input motion

had little energy at higher frequencies. The interlayermotion (at 9.1 m depth) computed using SHAKE wasspecified as an acceleration time-history along thebase of the FLAC model.

2.4 Model parameters for soil

The stress-strain behavior of the soil was modeled usingthe Mohr-Coulomb constitutive model. Four parame-ters are required for the Mohr-Coulomb model: effec-tive internal friction angle (��); mass density (�); shearmodulus (G); and bulk modulus (K). The first twoparameters (i.e., �� and �) are familiar to geotechni-cal engineers, where mass density is the total unitweight of the soil (�t) divided by the acceleration dueto gravity (g), i.e. � � �t/g. As stated previously, ��for the foundation soil was 40° and 35° for the back-fill. These values are consistent with dense naturaldeposits and medium-dense compacted fill. G and Kmay be less familiar to geotechnical engineers; there-fore, their determination is outlined below.

Several correlations exist that relate G to other soil parameters. However, the most direct relation isbetween G and shear wave velocity (vs):

(2)

�s may be determined by various types of site charac-terization techniques, such as cross hole or spectralanalysis of surface waves (SASW) studies.

Values for K can be determined from G andPoisson’s ratio (v) using the following relation:

(3)

v may be estimated using the following expression:

(4)

which was derived from the theory of elasticity (e.g.Terzaghi 1943), in conjunction with the correlationrelating Ko and �� proposed by Jaky (1944), i.e.Ko � 1 � sin(��). Using the above expression, v wasdetermined to be 0.26 and 0.3 for the foundation soiland backfill, respectively.

2.5 Model parameters for wall

The concrete wall was divided into five segmentshaving constant parameters, as illustrated in Figure 3,with each segment consisting of several 0.3 m elasticbeam elements. Four parameters were required todefine the mechanical properties of the elastic beam

335

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elements: cross sectional area (Ag); mass density (�);elastic modulus (Ec); and second moment of area (I ),commonly referred to as moment of inertia.

The basis for subdividing the wall into five seg-ments was the variation of the mechanical propertiesin the wall. A wall having a greater taper or largelyvarying steel reinforcement along the length of thestem or base would likely require more segments.

For each of the segments, Ag and � were readilydetermined from the wall geometry and the unit weightof the concrete (i.e. 23.6 kN/m3). Ec was computedusing the following expression (e.g. MacGregor 1992):

(5)

In this expression, f �c is the compressive strength ofthe concrete (e.g. 4000 psi for the wall being modeled),and both Ec and f �c are in psi. Because the structure iscontinuous in the direction perpendicular to the analy-sis plane, Ec computed using Equation 5 needed to bemodified to account for plane-strain conditions. Thismodification was done using the following expression(Itasca 2000, FLAC Structural Elements Manual).

(6)

where 0.2 was assumed for Poisson’s ratio for concrete.I is a function of the geometry of the segments, the

amount and location of the reinforcing steel, and theamount of cracking in the concrete, where the latter inturn depends on the static and dynamic load imposed

on the member. In dynamic analyses, it is difficult tostate a priori whether the use of sectional propertiescorresponding to uncracked, fully cracked, or someintermediate level of cracking will result in the largestdemand on the structure. However, I � 0.4 Iuncrackedwas used as a reasonable estimate for the sectionalproperties (Paulay & Priestley 1992).

2.6 Model parameters for wall-soil interface

Interface elements were used to model the interactionbetween the concrete retaining wall and the soil.However, FLAC does not allow interface elements tobe used at the intersection of branching structures(e.g. the intersection of the stem and base of the can-tilever wall). Several approaches were attempted bythe authors to circumvent this limitation in FLAC, withthe simplest and best approach, as found by the authors,illustrated in Figure 4. As shown in this figure, threevery short beam elements, oriented in the direction ofthe stem, toe side of the base, and heel side of thebase, were used to model the base-stem intersection.No interface elements were used on these three shortbeam elements. However, interface elements were usedalong the other contact surfaces between the soil andwall, as depicted by the hatched areas in Figure 4.

A schematic of the FLAC interface element is pre-sented in Figure 5. As may be observed from this figure,the interface element has four parameters: S � sliderrepresenting shear strength; T � tensile strength;kn � normal stiffness; and ks � shear stiffness. Theelement allows permanent separation and slip of thesoil and the structure, as controlled by the parameters

336

BeamElements

1

2

3

4

5

2.4m1.5m

6.1

m

4m

1.5m

1.5m

1.5m

1.5m

Figure 3. Numerical model of retaining wall using elasticbeam elements.

InterfaceElements

1.5m 2.4m

No InterfaceElements

BeamElements

4 m

6.1

m

Figure 4. Location of interface elements in the FLAC model.

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T and S, respectively. For the cohesionless soil beingmodeled, T � 0, while S was specified as a functionof the interface friction angle (). For medium-densesand against concrete, � 31° (Gomez et al. 2000b).

As a rule-of-thumb, the FLAC manual (Itasca 2000,Theory and Background Manual) recommends that knbe set to ten times the equivalent stiffness of thestiffest neighboring zone, i.e.:

(7)

In Equation 7, K and G are the bulk and shearmoduli, respectively, and �zmin is the smallest widthof a zone in the normal direction of the interfacingsurface. The max[ ] notation indicates that the maxi-mum value over all zones adjacent to the interface beused. The FLAC manual warns against using arbitrar-ily large values for kn, as is commonly done in finiteelement analyses, as this results in an unnecessarilysmall time step, and therefore unnecessarily longcomputational times.

The determination of the ks required considerablymore effort than the determination of the other interfaceelement parameters. In shear, the interface element inFLAC essentially is an elasto-plastic model, with anelastic stiffness of ks and yield strength S. ks valueswere selected such that the resulting elasto-plasticmodel gave an approximate fit of the hyperbolic-typeinterface model proposed by Gomez et al. (2000a,b).A comparison of the two models for initial loading(i.e. construction of the wall) is shown in Figure 6.

The procedure used to determine ks values for ini-tial loading is outlined below. The reader is referred toGomez et al. (2000a,b) for more details concerningtheir proposed hyperbolic-type model.

1. Compute �r using the following expression.

(8a)

where,

(8b)

(8c)

Ksi � dimensionless interface initial shear stiffnessof the interface; �n � normal stress acting on theinterface (determined iteratively in FLAC by firstassuming a small value for ks and then constructingthe wall); � interface friction angle � 31°; Rfj �failure ratio � 0.84; KI � dimensionless interfacestiffness number for initial loading � 21000;nj � dimensionless stiffness exponent � 0.8; �w �unit weight of water in consistent units as �r; andPa � atmospheric pressure in the same units as �n.The values for Rfj, KI, nj, and were obtained fromGomez et al. (2000a).

2. ks was computed using the following expression:

(9)

The above computed ks values were used only forthe initial construction of the wall. The ks valueswere changed after the construction of the wall andprior to the application of the earthquake loading tovalues consistent with the Gomez-Filz-EbelingVersion I load/unload/reload extended hyperbolicinterface model (Gomez et al. 2000b). The pro-cedure used to compute ks for the cyclic loading isoutlined below. Again, the reader is referred to thecited report for more details concerning this model.

337

T S ks

kn

zone

grid point grid point

Side A of Interface

zone zone

Side B of Interface

Figure 5. Schematic of the FLAC interface element (adaptedfrom Itasca 2000).

0.000 0.002 0.004 0.006 0.008 0.010

100

200

300

400

500

600

700

hyperbolicmodel

FLACmodel

�r

�s (ft)

τult

τf

τ (p

sf)

Ksi ks

Figure 6. Calibration of the FLAC interface model to thehyperbolic-type model proposed by Gomez et al. (2000a,b).

09069-39.qxd 08/11/2003 20:43 PM Page 337


(10a)

where,

(10b)

(10c)

Kurj � unload-reload stiffness number for interfaces;and Ck � interface stiffness ratio.

Using the above expressions, the interface stiffnesseswere computed for the interface elements identifiedin Figure 4. While the ks for unload-reload were higherthan the corresponding values for initial loading (i.e.,Equation 10a versus Equation 9), the values for kn werethe same for both initial loading and unload-reload.

2.7 Dimensions of finite difference zones

Proper dimensioning of the finite difference zones isrequired to avoid numerical distortion of propagatingground motions, in addition to accurate computationof model response. The FLAC manual (Itasca 2000,Optional Features Manual) recommends that the lengthof the element (�l) be smaller than one-tenth to one-eighth of the wavelength (�) associated with the highestfrequency (fmax) component of the input motion. Thebasis for this recommendation is a study by Kuhle-meyer & Lysmer (1973). Interestingly, the FLUSHmanual (Lysmer et al. 1975) recommends �l be smallerthan one-fifth the � associated with fmax, also refer-encing Kuhlemeyer & Lysmer (1973) as the basis forthe recommendation, i.e.:

(11a)

(11b)

� is related to the shear wave velocity of the soil (vs)and the frequency (f) of the propagating wave by thefollowing relation.

(12)

In a FLUSH analysis, it is important to note thatthe vs used in this computation is not that for small(shear) strains, such as measured in the field usingcross-hole shear wave test. Rather, in FLUSH, the vsused to dimension the elements should be consistent

with the earthquake-induced shear strains, frequentlyreferred to as the “reduced” vs by FLUSH users.Assuming that the response of the retaining wall willbe dominated by shear waves, substituting Equation12 into Equation 11a gives:

(13a)

or

(13b)

As may be observed from these expressions, thefinite difference zone with the lowest vs, for a given�l will limit the highest frequency that can passthrough the zone without numerical distortion. For theFLAC analyses performed in this investigation, 0.3 mby 0.3 m zones were used in sub-grids one and two;(refer to Figure 2). The top layer of the backfill hasthe lowest vs (i.e. 160 m/sec). Using Equations 13 and� � 0.3 m, the finite difference grid used in theFLAC analyses should adequately propagate shearwaves having frequencies up to approximately 53 Hz.This value is well above the 15 Hz cutoff frequencyused in the SHAKE analysis to compute the inputmotion for the FLAC analysis and well above the esti-mated fundamental frequency of the retaining wall-soil system being modeled (i.e. ≈6 Hz).

2.8 Damping

As stated previously, an elasto-plastic constitutivemodel, in conjunction with the Mohr-Coulomb failurecriterion was used to model the soil. Inherent to thismodel, once the induced dynamic shear stresses exceedthe shear strength of the soil, the plastic deformationof the soil introduces considerable hysteretic damping.However, for dynamic shear stresses less than the shearstrength of the soil, the soil behaves elastically (i.e. nodamping), unless additional mechanical damping isspecified. FLAC allows mass proportional, stiffnessproportional, and Rayleigh damping to be specified,where the latter provides a relatively constant level ofdamping over a restricted range of frequencies.

For the analyses performed, Rayleigh damping wasused, which required the specification of a dampingratio and corresponding central frequency. One- to two-percent damping ratio is commonly used as a lowerbound for non-linear dynamic analyses to reduce high-frequency spurious noise (e.g. Finn 1988). However,it was found by the authors that considerable high-frequency noise may still exist even when one- totwo-percent Rayleigh damping was specified; this isthought to be a numerical artifact of the explicit solu-tion algorithm used in FLAC. The damping levels in

338

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the last iteration of the SHAKE analysis used to com-pute the FLAC input motion may be used as an upperbound of the values for Rayleigh damping. Judgmentis required in selecting the damping ratio between thelower and upper bounds; three-percent Rayleigh damp-ing was used for most of the retaining wall analysesperformed by the authors. The central frequency cor-responding to the specified damping ratio is typicallyset to either the fundamental period (small strain) ofthe system being modeled (an inherent property ofthe wall-soil system) or predominant period of thesystem response (an inherent property of the wall-soilsystem and the ground motion). For the FLAC analysesperformed, the central frequency was set equal to thesmall strain fundamental frequency of the retainingwall-soil system (i.e. ≈6 Hz).

3 DISCUSSION

Several analyses were performed using the model ofthe wall-soil system described above, scaling the inputmotion to different peak ground acceleration values.To assess the adequacy of the model, the results fromthe FLAC analyses were compared with the results fromsimplified techniques for estimating the permanentwall displacement and the dynamic earth pressures.The purpose of the comparisons was only to provide areality check of the FLAC results, while true validationof the FLAC model would require a comparison withactual field observations. Comparisons of the resultsare discussed in the following sub-sections. However,the reader is referred to Ebeling & Morrison (1992)and Green and Ebeling (2002) for more detailed dis-cussions about the simplified techniques used.

3.1 Permanent wall displacement

Comparisons of the permanent relative displacements(dr) of the wall computed from the FLAC results and computed by Newmark sliding block analyses(Newmark 1965) of the structural wedge (Fig. 1) areshown in Figure 7a,b. dr was not computed directly byFLAC, but rather was computed by subtracting the totaldisplacement of the structural node at the intersectionof the stem and base of the wall from the total dis-placement of the grid point at the free-field boundaryat the same depth. As may be observed from Figure 7,dr computed from the FLAC results is about 0.33 m.

Newmark sliding block analyses of the structuralwedge (Fig. 1) were performed using the accelerationtime-history shown in Figure 8. This time-history wascomputed by FLAC at the free-field boundary at adepth corresponding to approximately mid-height ofthe structural wedge. In order to perform a Newmarksliding block analysis, a maximum transmissible accel-eration (N* g) has to be specified, which is the value

of acceleration imparted to the block resulting in afactor of safety against sliding equal to 1.0. Using theinterface friction angle between the concrete wall andfoundation soil (i.e. � 31°) in conjunction with theweight of the structural wedge, N* g was determinedto be approximately 0.22 g. The sliding block analysisresulted in dr � 0.55 m, as shown in Figure 7a, whichis considerably larger than that from the FLAC analysis.

One possible reason for the difference in the dr val-ues may be that the sliding block analysis did notaccount for additional sliding resistance resulting fromthe “plowing action” that occurs at the toe of the wall.

339

0 5 10 15 20 25 30 35 40-0.10.00.10.20.30.40.50.60.7

Time (seconds)

Perm

anen

t rel

ativ

edi

spla

cem

ent (

m)

Newmark

FLAC

Figure 7a. Comparison of the permanent relative displace-ments computed from the FLAC results and a Newmark slid-ing block analysis with N*�g � 0.22 g.

Newmark

0 5 10 15 20 25 30 35 40-0.10.00.10.20.30.40.50.60.7

Time (seconds)

Perm

anen

t rel

ativ

edi

spla

cem

ent (

m)

FLAC

Figure 7b. Comparison of the permanent relative displace-ments computed from the FLAC results and a Newmark slid-ing block analysis with N* g � 0.27 g.

15 20 25 30 35 405

-1.0

-0.5

0.0

0.5

1.0

Time (seconds)Acc

eler

atio

n (g

)

100

N*.g = 0.27g N*.g = 0.22g

Figure 8. Acceleration time-history used in the Newmarksliding block analysis of the structural wedge.

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Although the wall is not embedded in the foundationsoil in its initial, undeformed shape, the wall tends torotate around the toe as it translates away from thebackfill. As a result, the toe of the wall penetrates andplows through the foundation soil. Such a mechanismwas observed in the deformed FLAC mesh. To accountfor this additional resistance to sliding, N* g wasrecomputed assuming a friction angle of 35°, which isbetween the interface friction angle (i.e. � 31°) andthe �� of the foundation soil (i.e. 40°), with the revisedvalue of N* g � 0.27 g. A comparison of the perma-nent relative displacements computed from FLAC andthe sliding block analyses using the revised value ofN* g is shown in Figure 7b. As may be observed fromthis figure, the predicted displacements are in veryclose agreement, thus giving credence to the validityof the proposed FLAC model.

3.2 Dynamic earth pressures

The dynamically induced lateral earth pressures act-ing on the stem of the wall were computed by FLAC.The corresponding lateral earth pressure coefficients(KFLAC) were computed from these stresses using thefollowing expression (Green et al. 2003):

(14)

where PFLAC � the resultant of the FLAC computedstresses acting on the stem of the wall; �t � the totalunit weight of the backfill; H � the height of thewall; and kv � vertical inertial coefficient (assumedto be zero). Equation 14 was used to compute KFLACvalues at times corresponding to the peaks in thetime-history of the horizontal inertial coefficient (kh)acting away from the backfill (i.e. active-type condi-tions). A plot of the computed KFLAC values versus khis shown in Figure 9. Also shown in this figure are thelateral dynamic earth pressure coefficients (active:KAE; Passive: KPE) computed using the Mononobe-Okabe expressions for the wall-soil system (Okabe1924; Mononobe & Matsuo 1929). The reader isreferred to Green et al. (2003) and Ebeling & Morrison(1992) for details regarding the Mononobe-Okabedynamic earth pressure coefficients.

Two items are of particular note in Figure 9. First,in general, the KFLAC values are higher than the KAEfor values of kh less than about 0.4. This phenomenonis discussed in detail in Green et al. (2003) and is dueto the failure wedge in the backfill being composed ofseveral failure wedges rather than a single rigid wedge,as assumed in the Mononobe-Okabe expressions. Inshort, the difference in the KFLAC and KAE values isattributed to a shortcoming in the Mononobe-Okabeexpressions, rather than a shortcoming in the FLACmodel.

The second item of particular note in Figure 9 isthat FLAC predicts kh values as high as 0.5, while theupper bound value should be the maximum transmis-sible acceleration used in the sliding block analyses(i.e. N* � 0.27). A plot of the kh time-history com-puted by FLAC at the approximate center of gravity ofthe structural wedge is shown in Figure 10. It can beobserved from this figure that the kh values greaterthan 0.27 are associated with high-frequency motions(that contain little energy).

There are two possible reasons for kh � N*. First,this could simply be a numerical artifact of theexplicit algorithm used in FLAC, rather than a physi-cal phenomenon. However, the criterion that kh � N*is based on the premise that the structural wedge isperfectly rigid. For a flexible structural wedge, suchas the one modeled, higher modes of vibration couldbe excited in the structural wedge. This could result inhigh local kh values, while the global kh value for thestructural wedge (i.e. that which contributes to baseshear) is less than N*. Adding credence to the latterexplanation is that Wartman et al. (2003) observedkh � N* values in physical model tests of deformableblocks on an inclined plane. Additional analyses areunderway to determine exactly the cause of the highkh values.

340

0.0 0.4 0.5 0.6 0.7 0.8 0.9 1.00.1 0.2 0.30.0

1.0

0.5

1.5

2.0

2.5

3.0

3.5

4.0

Lat

eral

ear

th p

ress

ure

coef

fici

ent (

K)

KPE

KAE

kh

Figure 9. Comparison of FLAC and Mononobe-Okabedynamic lateral earth pressure coefficients.

15 20 25 30 35 400.0

1.0

Time (seconds)5 100

-1.0

-0.5

kh

0.5 N* = 0.27

Figure 10. kh at middle of structural wedge.

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The authors outline the details of a numerical modeland its calibration for use in computing the dynamicresponse of a cantilever retaining wall. The proposedmodel employs an elasto-plastic constitutive modelfor the soil in conjunction with the Mohr-Coulombfailure criterion. The wall is modeled with elastic beamelements using a cracked second moment of area(Icracked) equal to 0.4 Iuncracked. Interface elementsare used to model the wall-soil interface, wherein theinterface element parameters are those that give a bestfit of the Gomez et al. (2000a,b) hyperbolic interfacemodel. Based on comparisons with simplified tech-niques for dynamic lateral earth pressure and perma-nent relative displacement, the proposed wall modelis believed to yield valid results.

ACKNOWLEDGEMENTS

A portion of this study was funded by the Headquarters,US Army Corps of Engineers (HQUSACE) Civil WorksEarthquake Engineering Research Program (EQEN).Permission was granted by the Chief of the US ArmyCorps of Engineers to publish this information.

During the course of this research investigation,the authors had numerous discussions with other FLACusers. Of particular note were lengthy conversationswith Mr. C. Guney Olgun, Virginia Polytechnic andState University, Blacksburg, VA. Others who pro-vided valuable insight into FLAC modeling were Mr. Nason McCullough and Dr. Stephen Dickenson,Oregon State University, Cornvallis, OR; Drs. N. Dengand Farhang Ostadan, Bechtel Corporation, SanFrancisco, CA; Mr. Michael R. Lewis, BechtelSavannah River, Inc., Aiken, SC; Drs. Peter Byrneand Michael Beaty, University of British Columbia,Vancouver; and Dr. Marte Gutierrez, Virginia Poly-technic and State Uni-versity, Blacksburg, VA.

Review comments by Dr. William F. Marcuson, III,Emeritus Director, Geotechnical Laboratory, US ArmyEngineers Waterways Experiment Station, were alsogreatly appreciated.

REFERENCES

Ebeling, R.M. & Morrison, E.E. 1992. The Seismic Designof Waterfront Retain Structures. US Army TechnicalReport ITL-92-11, US Navy Technical Report NCELTR-939, US Army Engineer Waterways ExperimentStation, Vicksburg, MS. (http://itl.erdc.usace.army.mil/pdf/itl9211.pdf).

Finn, W.D.L. 1988. Dynamic Analyses in GeotechnicalEngineering. In J.L. Von Thun (ed), Earthquake Engi-neering and Soil Dynamics II – Recent Advances in

Ground-Motion Evaluation, Geotechnical Special Publi-cation 20, ASCE, 523–591.

Gomez, J.E., Filz, G.M., & Ebeling, R.M. 2000a. Developmentof an Improved Numerical Model for Concrete-to-SoilInterfaces in Soil-Structure Interaction Analyses, Report2, Final Study. ERDC/ITL TR-99-1, US Army Corps ofEngineers, Engineer Research and Development Center.(http://libweb.wes.army.mil/uhtbin/hyperion/ITL-TR-99-1.pdf).

Gomez, J.E., Filz, G.M., & Ebeling, R.M. 2000b. ExtendedLoad/Unload/Reload Hyperbolic Model for Interfaces:ParameterValues and Model Performance for the ContactBetween Concrete and Coarse Sand. ERDC/ITL TR-00-7, US Army Corps of Engineers, Engineer Research andDevelopment Center. (http://libweb.wes.army.mil/uhtbin/hyperion/ITL-TR-00-7.pdf).

Green, R.A., Olgun, C.G., Ebeling, R.M., & Cameron, W.I.2003. Seismically Induced Lateral Earth Pressures on aCantilever Retaining Wall. Proceedings: The Sixth USConference and Workshop on Lifeline Earthquake Engi-neering (TCLEE2003), ASCE, August 10–13, 2003,Long Beach, CA.

Green, R.A. & Ebeling, R.M. 2002. Seismic Analysis ofCantilever Retaining Walls, Phase 1. ERDC/ITL TR-02-3, US Army Corps of Engineers, Engineer Research andDevelopment Center. (http://libweb.wes.army.mil/uhtbin/hyperion/ITL-TR-02-3.pdf).

Headquarters, US Army Corps of Engineers. 1989. Retainingand Flood Walls. EM 1110-2-2502, Washington, DC.(http://www.usace.army.mil/inet/usace-docs/eng-manuals/em1110-2-2502/toc.htm)

Headquarters, US Army Corps of Engineers. 1992. StrengthDesign for Reinforced-Concrete Hydraulic Structures.EM 1110-2-2104, Washington, DC. (http://www.usace.army.mil/inet/usace-docs/eng-manuals/em1110-2-2104/toc.htm).

Idriss, I.M. & Sun, J.I. 1992. User’s Manual for SHAKE91:A Computer Program for Conducting Equivalent LinearSeismic Response Analyses of Horizontally Layered SoilDeposits. Center for Geotechnical Modeling, Departmentof Civil and Environmental Engineering, University ofCalifornia, Davis, CA.

Itasca. 2000. FLAC (Fast Largrangian Analysis of Continua)User’s Manuals. Minneapolis: Itasca Consulting Group,Inc.

Jaky, J. 1944. The Coefficient of Earth Pressure at Rest.Magyar Menok es Epitesz Kozloi (Journal of the Societyof Hungarian Architects and Engineers).

Kuhlemeyer, R.L. & Lysmer, J. 1973. Finite Element MethodAccuracy for Wave Propagation Problems. Journal of theSoil Mechanics and Foundations Division 99(SM5):421–427.

Lysmer, J., Udaka, T., Tsai, C.-F., & Seed, H.B. 1975.FLUSH: A Computer Program for Approximate 3-DAnalysis of Soil-Structure Interaction Problems. EERCReport No. EERC-75-30, Earthquake EngineeringResearch Center, University of California, Berkeley, CA.

MacGregor, J.G. 1992. Reinforced Concrete Mechanics andDesign. Englewood Cliffs: Prentice Hall.

Mononobe, N. & Matsuo, H. 1929. On the Determination ofEarth Pressures During Earthquakes. Proceedings: WorldEngineering Congress 9: 177–185.

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Newmark, N.M. 1965. Effects of Earthquakes on Dams andEmbankments. Geotechnique 15(2): 139–160.

Okabe, S. 1924. General Theory of Earth Pressures. JournalJapan Society of Civil Engineering 10(6): 1277–1323,plus figures.

Paulay, T. & Priestley, M.J.N. 1992. Seismic Design ofReinforced Concrete and Masonry Buildings. New York:John Wiley and Sons, Inc.

Terzaghi, K. 1943. Theoretical Soil Mechanics. New York:John Wiley and Sons, Inc.

Wartman, J., Bray, J.D., & Seed, R.B. 2003. Inclined PlaneStudies of the Newmark Sliding Block Procedure.Journal of Geotechnical and Geoenvironmental Engi-neering 129(8): in press.

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