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ALiteratureReviewonBehaviorofShallowFoundationsonJointedRockMass

RESEARCH·MAY2015

DOI:10.13140/RG.2.1.4017.2968

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124

Availablefrom:DipalokeMajumder

Retrievedon:16October2015

BEHAVIOUR OF SHALLOW FOUNDATION ON

JOINTED ROCK MASS

A Seminar Report

Submitted in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in

Civil Engineering

(with specialization in Geotechnical Engineering)

by

DIPALOKE MAJUMDER

(Enrolment No.: 14910011)

Under the Guidance of

Dr. M. N. VILADKAR

DEPARTMENT OF CIVIL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY ROORKEE

ROORKEE - 247667, UTTARAKHAND, INDIA

MAY, 2015

II

CANDIDATE’S DECLARATION

It is certified that the work which is presented in the seminar report entitled “Behaviour of

shallow foundation on jointed rock mass” has been carried out in Department of Civil Engineering

at Indian Institute of Technology Roorkee under the supervision of Dr. M. N. Viladkar, Professor of

Geotechnical Engineering Group, Department of Civil Engineering, Indian Institute of Technology

Roorkee, Roorkee, India.

I further declare that the matter embodied in this seminar report, has not been submitted by me

for the award of any other degree.

Date: Dipaloke Majumder

Place: IIT Roorkee Enrolment No.: 14910011

CERTIFICATE

This is to certify that the above statement made by the candidate is correct to the best of my

knowledge.

Dr. M. N. Viladkar

Professor

Department of Civil Engineering

Indian Institute of Technology, Roorkee

Roorkee-247667, Uttarakhand, India

III

ACKNOWLEDGEMENT

With immense pleasure, I would like to express my deep sense of gratitude to my supervisor,

Dr. M. N. Viladkar, Professor, Geotechnical Engineering Group, Department of Civil Engineering,

Indian Institute Technology Roorkee, for being the source of inspiration and for providing valuable

advice, resourceful guidance in all respect throughout this work. It is due to his continuous

encouragements for which this report could be brought to the current shape.

My sincere gratitude goes to my Parents for supporting me in my every success and failure in

my life. I would also acknowledge my gratefulness to my friends and seniors for providing their

supports, thoughts and suggestions.

Date:

Place: IIT Roorkee Dipaloke Majumder

IV

ABSTRACT

In last few decades, immense growth in infrastructural development makes the

appropriate available locations scarce for foundation construction of heavy structures like high

rise buildings, bridges, transmission line towers etc. As rocks are inherently stronger and stable

to withstand heavy loads compared to soil, the foundation engineers always prefer rock or rock

mass as a foundation material. However, behaviour of jointed rocks is very complex due to its

non-homogeneity and anisotropy. In order to design foundations on jointed rocks, two criteria

must be fulfilled. One criterion is the shear failure criterion of the ground and other is the

settlement criterion under resultant load.

In the present study, literature on bearing capacity and settlement characteristics of

different types of shallow foundations resting on horizontal or sloping, isotropic or anisotropic

jointed rock mass subjected to different types of loading, has been reviewed. Some accessible

provisions of Indian standard code of practice have been stated in brief. In addition, some

existing analytical (Limit equilibrium, limit analysis and characteristic line method), numerical

(FEM, DEM and ANN), experimental and empirical studies relevant to the problem, have been

discussed. Finally, the report concludes with few useful suggestions.

Exploration of available studies revealed that most of the studies deal with the problem

of bearing capacity of strip footing on horizontal surface of jointed rock mass. However, very

limited number of studies have focussed on bearing capacity of other types of footings

(rectangular, circular, combined or raft) on slopping jointed rock mass and subjected to

eccentric-inclined loading. Moreover, adequate attention has not been paid so far on the

formulation of pressure-settlement and pressure-tilt characteristics of footings on rock mass.

Therefore on a conclusive note, the study finds these as grey areas and further research can be

conducted these areas.

V

CONTENTS

Title Page No.

Candidate’s Declaration II

Acknowledgement III

Abstract IV

Contents V

List Of Figures VI

List Of Tables X

Notations IX

1.0 Introduction 1

1.1 General 1

1.2 Objective 1

1.3 Outline Of Report 2

2.0 Literature At A Glance 4

3.0 Codal Provisions 4

4.0 Limit Equilibrium Method 6

5.0 Limit Analysis Method 8

6.0 Characteristics Line Method 28

7.0 Numerical Methods 34

8.0 Experimental Methods 47

9.0 Empirical Methods 50

10.0 Contact Pressure Distribution on Rock Mass 51

11.0 Critical Comments 59

12.0 Conclusions 60

References 61

Appendix – I 64

VI

LIST OF FIGURES

Figure

No. Title

Page

No.

1 Various types of foundations (Wyllie, 2005) 2

2 Analysis of bearing capacity of blocks, (A) Failure of small block and (B) Failure

of large block (Meyerhof 1953)

7

3 Tangential line to modified Hoek-Brown failure criterion (Yang and Yin 2005) 9

4 Symmetrical failure mechanism for bearing capacity (Yang and Yin 2005) 9

5 Variation of bearing capacity with surcharge load (Yang and Yin 2005) 10

6 Failure mechanism for seismic bearing capacity of a footing on a slope (Yang 2009) 11

7 Problem definition (Merifield et al. 2006) 12

8 Bearing capacity factor for weightless rock (Merifield et al. 2006) 13

9 Average finite element limit analysis values for bearing capacity factor N

(Merifield et al. 2006)

14

10 Failure mechanisms, (a) Prandtl-type mechanism, M1 and (b) Multi-wedge

translation mechanism, M2 (Saada et al. 2007)

15

11 Variation of ultimate bearing capacity uq with surcharge load

0q for different

failure mechanisms (Sadda et al. 2007)

16

12 Variation of ultimate bearing capacity uq with unit weight of rock and surcharge

load 0q (Sadda et al. 2007)

16

13 Elements used in lower bound analysis: (a) Four node rectangular extension

element, (b) Three node triangular extension element and (c) Three node triangular

element (Sutcliffe et al. 2003)

17

14 Linearized Mohr-Coulomb yield function ( 6p )(Sutcliffe et al. 2003) 18

15 Variation of bearing capacity with joint orientation – one joint set (Sutcliffe et al.

2003)

19

16 Variation of bearing capacity with joint orientation – two joint set (Sutcliffe et al.

2003)

19

17 Bearing capacity against joint set orientation for : (a) one joint set and (b) two joint

set (Sutcliffe et al. 2003)

19

VII

Figure

No. Title

Page

No.

18 Lower bound capacity model (Bell, 1915) for rock mass with one joint set (Prakoso

and Kulhawy 2004)

20

19 Lower bound bearing capacity of strip footings on jointed rock masses with – (a)

One joint set, (b) Two joint sets (Prakoso and Kulhawy, 2004)

22

20 Comparison of csN of the proposed lower bound model with other models (Prakoso

and Kulhawy, 2004)

22

21 Bell’s approach for bearing capacity estimation (Singh and Rao, 2005) 24

22 Parabolic strength criterion (Singh and Rao, 2005) 24

23 Chart for computing (a) joint factor fJ and (b) cr (Singh and Rao, 2005) 25

24 Charts for computing (a) average strength enhancement and (b) lower bound

strength enhancement (Singh and Rao, 2005)

26

25 Failure mechanisms and corresponding hodographs: (a) TS1 and (b) OS1 (Imani et

al., 2012)

27

26 Comparison of results among the present upper bound solution, methods of Ausilio

and Conte (2005) and Hansen et al. (1987) (Imani et al. 2012)

28

27 Variation of Quw/Qu with Dw/B for (A) Фi = Фj=35o and Ci = 5 MPa and (B) Фi =

35o, Ci = 0.1 Mpa and Cj = 0 (Imani et al. 2012)

28

28 Shallow footing on horizontal surface subjected to vertical load (Serrano et al.,

1994)

29

29 Characteristic network under the foundation (Serrano et al., 1994) 30

30 Variation of Riemann’s invariant with instantaneous friction angle (Serrano and

Olalla 1994)

31

31 Variation of load coefficient ( )N with normalized external load *

01 on

boundary 1 and inclination of load on boundary 2 (Horizontal surface, 0 )

(Serrano and Olalla, 1994, 1996)

32

32 Variation of Load Factor N with Normalized External Load on Boundary 1,

with Horizontal Ground and Vertical External Loads (Serrano et al. 2000)

33

33 Geometry and boundary conditions of the calculation domain (Clausen 2012) 34

34 Variation of final load up with no. of degree of freedom dofn (Clausen 2012) 34

VIII

Figure

No. Title

Page

No.

35 Variation of bearing capacity and near-failure displacement for 10GSI (Clausen

2012)

36

36 Dimensions, velocity field and boundary conditions of distinct element model in the

homogeneous medium (Salari-rad et al. 2012)

37

37 Possible anisotropic failure mechanisms depending on dip of joints: MI, M1, M2,

MC, MS and MD (Serrano and Olalla 1998)

38

38 Variation of bearing capacity with dip angle of joint set (Salari-rad et al. 2012) 38

39 Comparison of bearing capacity obtained by DEM and lower bound limit analysis

method by Sutcliffe et al. 2004 (Salari-rad et al. 2012)

39

40 Rock mass specimens (a) Type A and (b) Type B (Bindlish et al. 2013) 40

41 For type-A specimen results obtained from UDEC analysis (a) Load versus

settlement curve and (b) failure mode and major stress contours (ϴ = 45°, s = 0)

(Bindlish et al. 2013)

40

42 Comparison of ultimate bearing capacity predicted by UDEC model with

experimental values for (a) type-A specimens and (b) type-B specimens (Bindlish

et al. 2013)

40

43 Finite element model for elasto-plastic analysis of surface strip footing

(Shekhawat and Viladkar 2014)

41

44 Load intensity vs settlement characteristics with respect to load (a)

eccentricity and (b) inclination (Shekhawat and Viladkar 2014)

41

45 Bearing capacity factor0N model validation curve (Shekhawat and

Viladkar 2014)

42

46 Schematic illustration of proposed ANN (MLP) network (Ziaee et al. 2014) 43

47 Comparison of measured and predicted ultq values using ANN model: (a)

training (learning and validation) data and (b) testing data (Ziaee et al. 2014)

45

48 Comparison of ultq values among ANN model predicted, experimental and

Goodman (1989) values (Ziaee et al. 2014)

46

49 The percentage relative importance histogram of each input variable for

predicting ultq based on the Garson’s algorithm (1991)

46

IX

Figure

No. Title

Page

No.

50 Rock mass specimens with 0 90 @15 are: (a) Type A, (b) Type B, (c)

Type C (Bindlish et al. 2012)

48

51 (a) Rock mass specimen with variation in joint set angle and side slope

and (b) Bearing capacity test apparatus for J0090-SL45-ED00 (Shukla et al.

2014)

49

52 Pressure Bulb Resulting From Loading of An Elastic Half Plane By (a) Vertical

Line Load, (b) Shear Line Load and (c) Inclined Line Load (Goodman 1989)

53

53 Two Dimensional Joint Model with (a) Horizontal Major Joints ( 0 ) and (b)

Inclined Major Joints with An Inclination Angle (Gaziev and Erlikhman 1971)

51

54 Stress Concentration Patterns Induced By Partial Surface Loading on 2D Joint

Models with Different Values of (Gaziev and Erlikhman 1971)

54

55 Comparison of Pressure Bulbs Predicted By Anisotropic Continuum Model and

Discrete Joint Model (Singh 1973b)

55

56 Pressure Bulb Under Line Loads in Jointed Rocks Calculated By Bray (1977)

(Goodman 1989)

56

57 Maximum Shear Stress Contours Due To Partial Surface Loading For R = 10

(Oda et al. 1993)

57

58 Stress Distribution Beneath Loaded Area, Estimated By Jointed Rock Model in

Semi-Infinite Model with One Joint Set (Agharazi et al. 2012)

58

X

LIST OF TABLES

Table

No. Title

Page

No.

1 Net allowable bearing pressureaq based on rock material as per IS 12070:1987 5

2 Net allowable bearing pressureaq based on RMR as per IS 12070:1987 5

3 Maximum and differential settlements of buildings on rock mass (clause 5.2.4; IS

13063:1991)

5

4 Ultimate bearing capacity uq of strip footing (Yang and Yin 2005) 10

5 Comparison of ultimate bearing capacity for various quality of weightless rock

mass (Merifield et al., 2006)

14

6 Comparison of bearing capacity factor N for weightless rock, 0D and

10im (Sadda et al. 2007)

16

7 Joint inclination parameter fn , (Ramamurthy 1993) 23

8 Coefficient a for cj estimation (Singh et al. 2002) 23

9 Benchmark values (Clausen 2013) 36

10 Weight and bias values between input and hidden layer (Ziaee et al. 2014) 44

11 Weight and bias values between hidden layer and output layer (Ziaee et al. 2014) 45

XI

NOTATIONS

B ,0B = Width of footing or column

D = Disturbance factor

,i jE E = Modulus of elasticity of intact rock and jointed rock respectively

mE = Modulus of deformation

,p mF F = Partial safety factors

GSI = Geological strength index of rock masses

H = Thickness of rock block

1H = Height of wall from foundation footing

I = Moment of inertia 3 12bd

I1 = Riemann’s invariant

fJ = Joint factor

nJ = Joint frequency in loading direction

K = Coefficient of buckling or effective length

L = Lower limit of normalized variables

bL = Buckling length or effective length of column

1L = Length of deflected part of wall/raft or centre to centre distance between columns

0, , , , ,c q csN N N N N N = Bearing capacity coefficients

N = Load factor

crP = Buckling load of column

RMR = Rock mass rating

S

B = Ratio of joint spacing to footing width

mS = Maximum settlement of strip footing subjected to eccentric inclined load

mlS = Joint spacing of i th family

perS = Permissible settlement

0S = Settlement of strip footing subjected to central vertical load

XII

U = Upper limit of normalized variables

kV = Hidden layer weights

kW = Input layer weights

X = Unknown stresses or component of applied load in parallel to joint planes

nX = Normalized value of variable X

max min,X X = Maximum and minimum value of variables

a = Empirical coefficient

kbias and hbias = Input and hidden layer biases respectively

0,c ,rc S = Cohesion of rock-material

', 'c = Equivalent Mohr-Coulomb parameters

jc and j = Cohesion and friction angle of joint sets respectively

e = Load eccentricity

i = Load inclination

i1 = Inclination of load at boundary 1

hk = Horizontal seismic coefficient

n = Number of discontinuity families

bm = Reduced value of the material constant im

,im s = Hoek-Brown parameters of intact rock

fn = Modified joint inclination parameter for shallow foundation

0p = Normal pressure on the equivalent free surface

tp = Maximum bending tensile stress at point of wedge

q = Overburden pressure

0q = Equivalent surcharge load

aq = Net allowable bearing pressure

, ,, , ,ult u u h ult ANNq q p P q = Ultimate bearing capacity of rock mass

r = Joint shear strength parameter / Radius of circular footing

XIII

cs , qs = Shape factors

t = Tilt of strip footing

= Inclination of slope / Semi wedge angle 45 2

= Angle between joint and vertical axis

i , i = Angle of wedge

, = Geo-mechanical parameters

= Unit weight of geo-material

' = Submerged unit weight of rock mass

i = Frequency of i th discontinuity family (m)

= Poison’s ratio.

= Angle of internal friction of geo-material

t = Tangent angle

r = Friction angle rock material

1 = Instantaneous friction angle

*

1o = Normalized major principal stress

1 = Major effective principal stress

3 = Minor effective principal stress

'

3max = Upper limit of confining stress

c = Uniaxial compressive strength of rock at failure

cj ,ci = Uniaxial compressive strength of jointed rock mass and intact rock respectively

n = Normal stress on plane

r = Stress at a distance r from point of application of load

t = Tensile strength of rock

1 = Inclination of major principal stress

= Shear stress on the plane

= Joint set orientation with horizontal plane

1

1.0 INTRODUCTION

1.1 GENERAL

A foundation engineer frequently comes across the problem of foundations of heavily loaded

structures like high rise buildings, bridges, transmission line towers etc. As rocks are inherently stronger

and stable to withstand heavy loads, compared to soil, the foundation engineers always prefer rocks or

rock mass as a foundation material.

Shallow foundation is a type of foundation whose depth to size ratio is less than 1.0. The

shearing resistance of soil in the sides of the foundation is generally neglected (IS 6403: 1981).

Rock mass is a non-homogeneous, anisotropic and discontinuous medium. It consists of intact

rocks separated by geological discontinuities such as joints, faults and bedding planes. Generally, the

behaviour of rock mass is governed by the interaction of intact blocks with these discontinuities in the

presence of an applied force. The compressive strength of intact rock is in the range of 1 MPa to 200

MPa. But due to the presence of these weak planes, joints and other discontinuities in the rock mass,

compressive strength and modulus of the mass are significantly lower and the correct assessment of

bearing capacity of foundations on jointed rock mass is a complex problem.

The ability of rock to sustain substantial shear and tensile forces facilitates the engineers to

construct many type of structures on rock mass. Examples of such structures are

Buildings, dams, bridges etc. which produce vertical and inclined loads on the foundation.

Machine foundations which produce vibrating loads on the foundation.

Heavily loaded industrial structures, thermal power plants etc.

The anchorages for suspension bridges, transmission line towers and other tie down anchors

which develop uplift forces.

Rock socketed piers which sustain substantial loads in both compressive and uplift situations.

Depending upon the magnitude, direction of loading and the geotechnical conditions of bearing

area, rock foundations are classified as - spread footings, socketed piers and tension foundations, shown

in Fig. 1 (Duncan C. Wyllie 2005).

1.2 OBJECTIVE

The objective of present study is to review of available literatures on bearing capacity and

settlement characteristics of different types of shallow foundations resting on horizontal or slopping,

isotropic or anisotropic jointed rock mass subjected to different types of loading.

2

Fig. 1 Various Types of Rock Foundations: (a) Spread Footing Located at Crest of Steep Slope;

(b) Dam Foundation with Resultant Inclined Loading (Spread Footing); (c) Socketed Pier to

Transfer Structural Load to Elevation Below Base of Adjacent Excavation and (d) Tie-Down

Anchors, with Staggered Lengths, to Prevent Uplift of Submerged Structure

(Duncan C. Wyllie 2005).

1.3 OUTLINE OF REPORT

A brief introduction of the problem, necessity and objective of the present study have been

discussed in section 1.

Section 2 contains broad classification of methodologies and appendix – I in which all the

literatures are given in tabular form.

Different IS code provisions related to shallow foundation on rock mass have been mentioned

in section 3.

Available literatures corresponding to different analytical and numerical methodologies on

bearing capacity of shallow foundation have been discussed briefly in section 4, 5, 6 and 7.

3

In section 8 and 9, various experimental and empirical studies on bearing capacity of shallow

foundation on rock mass have been discussed respectively.

Few studies on distribution of contact pressure in rock mass under shallow foundation are

summarised in section 10.

Some critical comments on the reviewed literatures have been made in section 11.

Finally, the report has been concluded in section 11 where few useful suggestions and

conclusions have been discussed.

4

2.0 LITERATURE AT A GLANCE

The well-established methodologies, used in the solution of ultimate bearing capacity of

shallow foundation resting on jointed rock mass, can be classified into main four groups, namely,

i. Analytical methods

a. Limit equilibrium analysis

b. Limit analysis

Lower bound

Upper bound

c. Characteristic line method

ii. Numerical methods

iii. Experimental methods

iv. Empirical methods

After discussing the salient features of each technique, detailed review of available literatures

with special reference to foundation on jointed rock masse has been presented briefly in Appendix – I.

3.0 CODAL PROVISIONS FOR SHALLOW FOUNDATIONS ON ROCK MASS

According to IS 6403:1981, ultimate bearing capacity is the intensity of loading at the base of

foundation which would cause shear failure of the sub-soil.

Some numerical formulae are provided in this code for calculating the ultimate net bearing

capacity of strip footings. It also provides guidelines to take into account the shape of footing,

inclination of loading, depth of embedment and effect of water table and guidelines about the mode of

failure of footing.

The recommended methods to determine ultimate net bearing capacity for cohesion less soil

0c , are established on (a) relative density, (b standard penetration resistance value and (c) static

cone penetration test.

Also, provisions for determination of ultimate net bearing capacity of cohesive soil 0

have been recommended for the three types of soil conditions, viz., homogeneous soil, two layered soil

and desiccated soil.

According to IS 12070:1987, the recommended values of allowable bearing capacity for

various rocks are shown in Table 1 and Table 2. These are conservative values to be adopted when no

field tests data is available for the estimation of allowable bearing pressure and the corresponding

settlement.

5

A detail guidelines for site investigation and laboratory testing of rock masses for obtaining the

geological and geotechnical data is provided in IS 13063:1991.

The most critical combination of all type of loads acting on the foundations shall be considered

as per the field condition (IS 875: 1987).

Table 1 Net Allowable Bearing Pressureaq Based on Rock Material (IS12070: 1987)

Rock Material aq (MPa)

Massive crystalline bed rock including granite, diorite, gneiss, trap, hard lime stone

and dolomite 10.0

Foliated rocks such as schist or slate in sound condition 4.0

Bedded limestone in sound condition 4.0

Sedimentary rock, including hard shales and sandstones 2.5

Soft or broken bedrock (excluding shale) and soft limestone 1.0

Soft shale 0.4

Table 2 Net Allowable Bearing Pressureaq Based on RMR (IS12070: 1987)

Classification no. I II III IV V

Description of rock Very good Good Fair Poor Very Poor

RMR 100-81 80-61 60-41 40-21 20-0

aq 0.6-4.5 4.5-2.9 2.9-1.5 1.5-0.6 0.6-0.4

Table 3 Maximum and Differential Settlements of Buildings on Rock Mass (IS 13063: 1991)

Sl.

No. Type of structure

Maximum

settlement,

mm

Differential

isolated

footing

Settlement

of raft

foundation,

mm

Angular

isolated

footing

Distortion

raft

foundation

1 For steel structure 12 10.0033L 10.0033L 1 300 1 300

2 For reinforced

concrete structures 12 10.0015L

10.002L 1 666 1 500

3 For plain bricks

block walls in multi

storeyed buildings

(a) For 1 2 3L H

(b) For 1 2 3L H

12

12

10.00025L

10.00033L

---

---

1 400

1 300

---

---

4 For water towers and

silos 12 --- 10.0025L

---

1 400

1L = Length of deflected part of wall/raft or centre to centre distance between columns 1H = Height of wall from foundation footing

6

The allowable bearing pressure,aq on the rock due to foundation should be less than or equal

to the safe bearing capacity of rock foundation. The effect of eccentricity and foundation interface

should to be considered for aq estimation. The total settlement S of the foundation should be less than

or equal to permissible settlement,perS i.e. perS S . Various permissible limits of total and differential

settlements are given in this code which are shown in Table 3.

Several guide lines for constructing foundations of horizontal and sloping rock/rock mass have

been provided in this code. It also gives some guide lines for the treatment of defects of rock masses

which lies below the foundations.

4.0 LIMIT EQUILIBRIUM METHOD

The limit equilibrium method is a widely known technique for obtaining approximate solution

for stability problems which include bearing capacity of footing, lateral earth pressure of retaining walls

and stability of slopes. Some common assumptions are made in limit equilibrium method, as follows:

i. The soil/rock mass obeys the Mohr-Coulomb failure criterion,

tanc (3.a)

ii. A failure surface of simple shape, viz., planar, circular, log spiral or combination of

these is assumed.

iii. The distribution of stress along the failure surface is also assumed.

iv. The general shape of different regions in the failure zone remains unchanged (straight slip line

remains straight) irrespective of the consideration of the weight.

v. Principle of super position holds good.

With the above assumptions, each stability problem is reduced to determining the most critical

location of failure surface of the chosen shape. Although, having a simple formulation, one limitation

of the limit equilibrium method has been the neglect of the stress-strain response of rock mass. As this

method considers the equilibrium conditions only, so the solutions obtained are mostly approximate.

Many researcher (Terzaghi 1943, Meyerhof 1953, Bisnoi 1968, Kulhawy and Goodman 2005, Zhu et

al. 2001, (Silvestri 2003), (Sahu 2009)) have developed bearing capacity solutions using this

methodology.

Meyerhof (1953) proposed a theory for the solution of the problem of bearing capacity of

shallow footing on horizontal rock and concrete blocks using limit equilibrium method. According to

this theory, due to applied vertical loading on rock blocks, a wedge is formed immediately below the

footing at the time of failure. If the block thickness is less and if the applied load exceeds the tensile

strength of rock material, then a tensile crack initiates progressively downwards and splits the block as

7

Fig. 2 Analysis of Bearing Capacity of Blocks, (A) Failure of Small Block and (B)

Failure of Large Block (Meyerhof 1953)

shown in Fig. 2 (A). If the block is large, compared to footing, shearing along rupture surface occurs as

shown in Fig. 2 (B). For the case of splitting failure, Eq. (2), (3) and (4) have been proposed for bearing

capacity calculation of strip footing (Meyerhof 1953).

2

2cot cot

2 cot8

cot

t

u

Hp

Bq c

H

B

(2)

6 cot

12 cot 2 cot

t h

H Bp p

H B H B

(3)

8

2tan 2 tanhp q c (4)

hp = resultant horizontal splitting pressure acting at a depth of 0.25 cotB

In shearing failure mode, Eq. (5) and (6) have been proposed for bearing capacity estimation of

strip and circular footing, respectively.

u cq cN (5)

0u cq cN p N (6)

5.0 LIMIT ANALYSIS METHOD

In compare to limit equilibrium method, limit analysis method considers the stress-strain

response of rock mass in an idealized approach. This idealisation, known as normality principle or

plastic flow rule, establishes two limit theorems which forms the foundation of limit analysis. The

plastic limit theorems of Druker, Prager and Greenberg (1952) are conveniently utilized to obtain the

lower and upper bounds of the collapse load. For a body or assemblage of bodies of elastic-perfectly

plastic material, the two theorems have been stated as

(i) Lower Bound Theorem: If an equilibrium distribution of stress can be found which

balances the applied load and nowhere violates the failure criterion, the rock mass will not

fail, or will be just on the verge of failure.

(ii) Upper Bound Theorem: The rock mass will collapse if there is any compatible pattern

of plastic deformation for which the rate of work of the external loads exceeds the internal

energy dissipation.

According to the above theorems, a stress field which follows the lower bound theorem will

produce a lower bound solution. A compatible failure mechanism (velocity field or flow pattern)

satisfying all conditions of upper bound theorem will produce an upper bound solution. If the upper and

lower bounds provided by the velocity field or stress field coincide, the exact value of collapse or limit

load is determined (Chen and R 1968).

Many researchers (Sutcliffe et al. 2003, Prakoso and Kulhawy 2004, Singh and Rao 2005, Yang

and Yin 2005, Merifield et al. 2006, Sadda et al. 2007, Yang 2009, Imani et al. 2012) utilized the limit

analysis method to obtain the solutions of bearing capacity of shallow foundation on rock masses.

Yang and Yin (2005) studied the upper bound solution of ultimate bearing capacity of a strip

footing using modified Hoek-Brown failure criterion and generalised tangential technique, shown in

Fig. 3.

The rock mass was assumed to be isotropic, homogeneous and idealized as a perfectly plastic

material which followed the associated flow rule. The footing was subjected to central vertical load

9

under plain strain condition. A symmetrical translation failure mechanism composed of rigid triangular

blocks was used in this analysis (shown in Fig. 4).

Fig. 3 Tangential Line to Modified Hoek-Brown Failure Criterion (Yang and Yin 2005)

Fig. 4 Symmetrical Failure Mechanism For Bearing Capacity (Yang and Yin 2005)

Equating the work rates of external loads to the total internal energy dissipation rates the general

equation (7) for the ultimate bearing capacityuq was obtained.

2

00 4 5 6 1 2 0 0 32

2u t

Bq c B f f f f f q B f

(7)

where

11 sin 1 sincos tan sin

1 tan2 2sin 2sin

n

nt tt t t

t c t

t t

mn mn sc

m n m

(8)

1 2 3 4 5, , , ,f f f f f = non-dimensional functions = , ,i i tf

i , i = angle of wedge in Fig. 2.5 ( i =1,…, n )

t = tangent angle in Fig. 4

10

In this upper bound analysis, the critical value of the ultimate bearing capacity was obtained by

optimizing the above expression (7) with respect to ,i i and t . Extending the work of Collins et al.

(1988), bearing capacity factor N was calculated by Eqn. (9) and compared with solution of Eqn. (7)

0.5

c t cN N c s (9)

where 2tan 45 / 2 exp tan 1 cotc t t tN

100

exp9 3

GSIs

D

From the results, it was perceived that the maximum difference between two values were less

than 0.5% which was indicated the efficiency of generalized tangential technique for determining

bearing capacity of a strip footing. Effects of surcharge load and self-weight of rock on bearing capacity

were also investigated, shown in Table 4 and Fig 5. It was concluded, the contribution related to c

can be separated from the uq whereas the contribution related to

0q and cannot be separated from

theuq . Numerical results for five types of rocks were presented for practical use in tabular form.

Table 4 Ultimate Bearing Capacity uq of Strip Footing (Yang and Yin 2005)

Surcharge 0q (kPa)

Unit weight (kN/m2)

20 21 22 23 24

10 14.352 14.367 14.383 14.399 14.413

20 14.540 14.553 14.568 14.582 14.597

30 14.717 14.731 14.745 14.759 14.772

40 14.888 14.901 14.914 14.927 14.940

when, 017, 0, 10 , 30, 1.0 .i cm D MPa GSI B m Unit of

uq : MPa

Fig. 5 Variation of Bearing Capacity with Surcharge Load for

10, 0, 10 ,GSI 30i cm D Mpa and 0.0 (Yang and Yin 2005)

11

Yang (2009) investigated the effect of horizontal seismic force on the bearing capacity of a

strip footing on rock slopes using the method of Yang and Yin (2005). An earthquake has two possible

effects on the bearing capacity. One is to increase the driving forces and the other is to decrease the

shearing resistance of the rock mass. In this analysis, only the driving forces during the earthquake was

considered and shearing strength was assumed to be unaffected. The expression for bearing capacity

uq has been presented by Eq. (10).

01 2 3 4 0 5 6

1 1

1

sin cos 2u h h t

t h t

Bq g k g g k g q g g c

k

(10)

where

1 2 6, ,...,g g g = non-dimensional functions given by Soubra (1999) = 0 , , ,t i tf B

hk = horizontal seismic coefficient

Fig. 6 Failure Mechanism for Seismic Bearing Capacity of A Footing on Slope (Yang 2009)

The bearing capacity for static case has been obtained from Eq. (10) by putting 0hk and by

setting 0hk in Eq. (10) the seismic bearing capacity has been obtained. Considering the influence of

slope angle and neglecting the effect of surcharge load and self-weight of rock mass, the expression

for N was derived by Eq. (11) compared with solution of Eqn. (10) for the validation of proposed

method. The maximum difference between two values (<0.5%) indicated the effectiveness of

“generalized tangential” technique for determining bearing capacity of a strip footing resting on slope

for both static and seismic cases also.

0.5

c t cN N c s (11)

where 2exp 2 tan tan 45 1 cot2

tc t tN

12

From the results of analysis, it was found that the failure surfaces become shallower as the hk

increases. However, failure surface depth increases with increase in inclination angle. For five types of

rocks and 0 ,5 ,..25 ,30 , the values of N were calculated and presented in tabular form for

practical use.

In this work, it was assumed that the rock mass will follow a specific failure mechanism. In

reality, at time of yielding rock masses do not follow any single failure mechanism. Rocks below footing

was assumed to be isotropic and homogeneous. But in practical, rock masses were highly anisotropic

and non-homogeneous in nature.

Fig. 7 Problem Definition (Merifield et al. 2006)

Merifield et al. (2006) have been applied limit analysis method to evaluate the ultimate bearing

capacity of surface strip footing resting on rock mass (Fig. 7). The generalised Hoek-Brown criterion

(2002) was used to calculate the strength of rock mass. The ultimate bearing capacity solution was

obtained by employing finite elements coupled with the upper and lower bound limit theorems of

classical plasticity.

In this analysis, the rock mass was idealized as a homogeneous and isotropic continuum. The

problem was treated as a plain strain problem. Influence of unit weight, UCS, GSI of rock mass, Hoek-

Brown parameter have been studied.

The results have been presented in terms of a bearing capacity factor N in graphical form.

From the results of upper and lower bound analysis, it was found that the true collapse load was

bracketed to within ±2.5% for both weightless and ponderable rock foundations. The effect of ignoring

rock weight can lead to a very conservative estimate of the bearing capacity particularly for the case of

13

poor quality rocks (GSI < 30). Impact of Hoek-Brown parameter im and weight of rock mass on the

bearing capacity factor N were shown in Fig. 8 and Fig. 9 respectively.

Fig. 8 Bearing Capacity Factor for Weightless Rock (Merifield et al. 2006)

The bearing capacity of rock mass also has been calculated using equivalent Mohr-Coulomb

parameters ', 'c (Hoek et al. 2002) from Eq. (12) and Eq. (13) and compared it with the results of

limit analysis. It was noticed, the former method overestimated the bearing capacity up to 157% for

good quality rocks, shown in Table 5. The results of this finite element limit analysis also has been

compared to the results of Serano et al. (2000) and a close agreement between two results was found.

One of the limitation of this method is that unrealistic values of input parameters were considered in

this analysis.

1

3 3

1

3

1 2 1 ' ''

6 '1 2 1

1 2

a

ci b n b n

a

b b n

a s a m s mc

am s ma a

a a

(12)

1

31

1

3

6 '' sin

2 1 2 6 '

a

b b n

a

b b n

am s m

a a am s m

(13)

14

Fig. 9 Average Finite Element Limit Analysis Values for Bearing Capacity Factor N

(Merifield et al. 2006)

Table 5 Comparison of Ultimate Bearing Capacity for Various Quality of Weightless Rock

Mass (Merifield et al. 2006)

Rock quality uq (MPa) Hoek-

Brown

30 0.25 ci 30 0.75 ci

uq (MPa)

Serrano et al.

(2000)

uq (Mpa) Mohr-

Coulomb

uq (Mpa) Mohr-

Coulomb

Very poor 6.7 12.0 (+46%) 15.3 (+87%) 6.5 (-3%)

Average 98.2 156.4 (+59%) 161.0 (+63%) 94.4 (-4%)

Very good 886.0 2279.4 (+157%) 1614.6 (+82%) 870.4 (-1%)

Saada et al. (2007) have been investigated the problem of assessing the ultimate bearing

capacity of shallow foundations resting on the rock mass using the kinematical approach of upper bound

limit analysis theory. Closed form solutions of rock failure criterion have been derived and applied for

estimating the bearing capacity.

It has been assumed that the rock mass is isotropic and homogeneous. The strength properties

of rock mass has been defined by the modified Hoek-Brown Criterion (Hoek et al. 1988, 2002). Two

kinds of failure mechanisms of rock mass were considered: (a) generalised Prandlt-type failure

mechanism (M1) with Mohr-Coulomb failure criterion, applicable for soil and rock material and (b)

Multi-wedge translation failure mechanism (M2), presented originally by Soubra (1999) for bearing

capacity calculation of foundations resting on a Mohr-Coulomb soil (Fig. 2.10).

15

Fig. 10 Failure Mechanisms, (a) Prandtl-Type Mechanism, M1 and (b) Multi-Wedge

Translation Mechanism, M2 (Saada et al. 2007)

Influence of HB parameters, disturbance factor, GSI, unit weight, surcharge load have been

considered in this study. The results of this analysis have been compared to the results of Yang and Yin

(2005) and Merifield et al. (2006) and the proposed method was found to be efficient reasonably for

bearing capacity estimation (shown in Fig 11 and Table 6 respectively). The effects of loading

parameters (surcharge load and unit weight of rock mass) on the ultimate bearing capacity have been

shown in Fig. 12. From the results, a linear dependency of bearing capacity uq with unit weight of

rocks has been observed. However, linear dependency of uq with

0q has not been maintained always.

One of the major limitations of this analysis is that the rocks with few discontinuities can’t be analysed

with this approach.

16

Fig. 11 Variation of Ultimate Bearing Capacity uq with Surcharge Load

0q for Different

Failure Mechanisms (0 1B m, 0D , 10im ,GSI = 30, 10c Mpa and 0 ),

(Saada et al. 2007)

Table 6 Comparison of Bearing Capacity Factor Nσ for Weightless Rock when D = 0 and

mi = 10 (Saada et al. 2007)

GSI N (Saada Et Al., 2008) N (Merifield Et Al.,2006) Relative Difference (%)

10 11.561 11.427 -1.1

20 17.484 17.796 -0.3

30 19.513 19.396 -0.6

40 18.582 18.472 -0.6

50 16.746 16.678 -0.4

60 14.784 14.376 -0.3

70 12.977 12.939 -0.3

80 11.402 11.376 -0.2

Fig. 12 Variation of Ultimate Bearing Capacity uq with Unit Weight of Rock and Surcharge

Load 0q (

0 1B m, 0D , 17im ,GSI = 30 and 10c Mpa), (Saada et al. 2007)

17

Sutcliffe et al. (2003) have been carried out rigorous lower bound limit analysis of surface strip

footings to evaluate the bearing capacity of jointed rock mass. In this analysis, linearized Mohr-

Coulomb yield criterion has been used to generate a statically admissible stress field in conjunction

with finite element programming.

The problem has been formulated assuming plain strain conditions. The jointed rock mass has

been treated as homogeneous, anisotropic and perfect plastic material. Only small deformations were

considered in this analysis under limit load. The numerical procedure which has been presented in this

paper is discussed in following six steps.

i) The problem was defined as a surface strip footing of width B resting on a layer of jointed rock

and bearing capacity ( , , , , )u i i iq f c c where i joint set number.

ii) Failure criterion of jointed rock material was defined by the Mohr-Coulomb criterion,

expressed by Eqns. (14) and (15).

2 2 2( ) (2 ) (2 cos ( )sin ) 0r x y xy x yF c (14)

2 21sin 2 ( ) 2cos 2 (sin cos sin 2 ) tan 0

2x y xy i x y xy iFi c (15)

iii) The lower bound theorem was formulated using three types of linear elements (shown in Fig.

13). Each node was associated with three stresses, ,x y and xy .

iv) The equilibrium equations of stresses was satisfied throughout each element and at every point

along joints.

v) At every nodal points, boundary conditions were imposed in following form:

n q constant and t constant.

Fig. 13 Elements Used in Lower Bound Analysis: (a) Four Node Rectangular

Extension Element, (b) Three Node Triangular Extension Element and

(c) Three Node Triangular Element (Sutcliffe et al. 2003)

18

Fig. 14 Linearized Mohr-Coulomb Yield Function ( 6p )(Sutcliffe et al. 2003)

vi) The Mohr-Coulomb yield criterion i.e. Eq.(14) and Eq.(15) were formulated as linearized yield

function expressed by Eq. (16) and (17).

,k x k y k xyA B C E 1,2,..., ,k p (16)

,k x k y k xy iA B C c 2 1, 2 ,k p i p i (17)

where , , , ( , , , , )i iA B C E f k p c

vii) The collapse load uq was obtained by integrating the normal stresses along boundary.

viii) The solution of unknown stresses X was obtained from eq. (18) which defines a

statically admissible stress field and the corresponding uq defines a rigorous lower bound

solution on the true collapse load.

Minimise ,TC X

Subject to 1 1

2 2

A X B

A X B

(18)

The results of this analysis were represented in graphical form in terms of normalised bearing

capacity q c against joint orientation , shown in Fig, 15, 16 and 17. From the analysis it was

found that presence of one or two joint sets in rock mass can reduce the bearing capacity by up to 60%

or 87% respectively. The inclusion of another third joint, vertically oriented, results in a further loss in

ultimate bearing capacity up to 40% as compared to the results for a rock mass with two joint sets.

However, the overall reduction in strength is significantly affected by the variation of cohesive and

frictional strength of these joints, orientation of joints with horizontal and relative angle between joints,

shown in Fig. 15. When compared, bearing capacity values obtained in this paper, were found to be

19

lower than both the displacement FEM result of Alehossein et al. (1992)(shown in Fig. 17) and the slip-

line results of Davis (1980). The major limitation of this analysis was the assumption of the linearized

MC yield criterion. Because in practical jointed rocks yields non-linearly.

Fig. 15 Variation Of Bearing Capacity With Dip Angle – One Joint Set (Sutcliffe et al. 2003)

Fig. 16 Variation of Bearing Capacity with Dip Angle – Two Joint Set (Sutcliffe et al. 2003)

Fig. 17 Bearing Capacity Against Joint Set Orientation For : (a) One Joint Set And (b) Two

Joint Sets (Sutcliffe et al. 2003)

20

Prakoso and Kulhawy (2004) have been presented the bearing capacity solutions for strip

footings on jointed rock masses considering one and two sets of discontinuities. The solutions have

been obtained by using a lower bound bearing capacity model (Bell’s model, 1915) in conjunction with

a simple discontinuity strength model. A parametric study has been carried out to evaluate the impacts

of strength of rock material and joints, number and orientation of joint sets on the bearing capacity of

rock masses.

It has been assumed that both rock material and joints will follow Mohr-Coulomb failure

criterion and failure will occur along the joints only. For the evaluation of bearing capacity the authors

have been proposed the following Eq. (19) where csN is the bearing capacity factor.

ult r csq c N (19)

Fig. 18 Lower Bound Capacity Model (Bell, 1915) For Rock Mass with One Joint Set

(Prakoso and Kulhawy 2004)

The Bell model (Bell, 1915) which has been used in this study, is shown schematically in Fig.

18. The procedure for bearing capacity calculation which has been presented in this paper, illustrated

below.

Step I: Strength calculation of zone I

i) The rock mass strength1r is calculated using Eq. (20) by considering

3 0 .

1 2 tan 452

rr rc

(20)

ii) Discontinuity strength 1 j is calculated with the help of Eq. (20) for 90j . When

90 90j , the strength was calculated by Eq. (21).

21

1

2

1 tan tan 90 sin 2 90

j

j

j

c

(21)

This calculation procedure was repeated for n number of discontinuities.

iii) The strength 1 I

of zone I was calculated by Eq. (22).

1 1 1 1 1 2 1min , , ,...,I r j j jn (22)

Step 2: Strength calculation of zone II

i) The confining stress 3 II

was established as

3 1II I (23)

ii) The rock material strength1r was calculated using Eq.(24).

2

1 3 tan 45 2 tan 452 2

r rr II rc

(24)

iii) Discontinuity strength 1 j was calculated with the help of (24) for j . When

90j the strength was calculated using Eq. (25).

3

1 3

2 2 tan

1 tan tan sin 2

j II j

j II

j

c

(25)

This calculation procedure was repeated for n number of discontinuities.

iv) The strength 1 II

(or ultq ) of zone I was calculated by Eq. (26).

1 1 1 1 1 2 1min , , ,...,ult II r j j jnq (26)

The results of the proposed model has been presented in terms of bearing capacity factor csN

in graphical form as shown in Fig. 19 and Fig. 20. It has been observed that strength and geometric

parameters have significant influence on bearing capacity of rock mass. The results of proposed model

have been compared with the results of other researchers (Davis 1980, Booker 1991, Alehossein et al.

1992 and Yu and Sloan 1994), shown in Fig. 20. The major limitations of this method are that the effects

of the rock mass weight, embedment and joint set spacing were not considered in bearing capacity

solutions.

22

Fig. 19 Lower Bound Bearing Capacity of Strip Footings on Jointed Rock Masses with – (a)

One Joint Set, (b) Two Joint Sets (Prakoso and Kulhawy, 2004)

Fig. 20 Comparison Of csN of Proposed Lower Bound Model with Other Models

(Prakoso and Kulhawy, 2004)

Singh and Rao (2005) have been suggested a methodology to evaluate the ultimate bearing

capacity of anisotropic non-Hoek-Brown jointed rock masses on horizontal surface. A Hoek-Brown

rock mass is an isotropic material whereas the non-Hoek-Brown rock mass is an anisotropic material.

It has small joint spacing compared to footing width with few joint sets and strength of all joint sets is

equal.

In this study, the foundation has been assumed to be smooth and shallow. For the applicability

of this method at least two regular and continuous joint sets are necessary and the block size is in the

order of one-fifth or less of foundation width. It has been considered that the bearing capacity is

sensitive to the properties of weakest joint set and is estimated using Bell’s approach (Jumikis, 1965

and Wyllie, 1992) (Fig. 21). The input parameters have been required for this approach are the number

23

of joint sets, the joint spacing, the friction angle along the joint planes, UCS of the intact rock, depth of

footing and similar moisture condition as in the field.

The concept of joint factor fJ (Ramamurthy and Arora 1994), presented by Eq. (27), has

been modified by Singh and Rao (2005) in order to establish a parabolic strength criterion of rocks for

bearing capacity evaluation (Fig. 22).

nf

f

JJ

n r (27)

exp( )cj ci fa J (28)

2

1 3 31 2cj j ci jA A for 3 ci (29)

0.77

, 1.23j avg ciA

(30)

0.72

,lower_bound 0.43j ciA

(31)

where

1 3, = Major and minor principal effective stresses at failure

nJ = Joint frequency in loading direction

fn = Modified joint inclination parameter for shallow foundation, obtained from Table 7

r = Joint shear strength parameter tan j

cj ,ci = Uniaxial compressive strength of jointed rock mass and intact rock respectively

a = an empirical coefficient, depends on failure mode of rock, obtained from Table 8

Table 7 Joint Inclination Parameter fn (Ramamurthy 1993)

Joint orientation 0 10 20 30 40 50 60 70 80 90

Inclination parameter

fn 1.0 0.814 0.634 0.465 0.306 0.071 0.046 0.105 0.460 0.810

Table 8 Coefficient a for cj Estimation (Singh et al. 2002)

Failure mode Splitting / shearing Sliding Rotation

Coefficient a -0.0123 -0.0180 -0.0250

24

Fig. 21 Bell’s Approach for Bearing Capacity Estimation (Singh and Rao, 2005)

Fig. 22 Parabolic Strength Criterion (Singh and Rao 2005)

A step-by-step procedure has been explained for ultimate bearing capacity assessment in this

study, illustrated below.

(i) Element I of fig. 2.22 is considered at first and over-burden pressure is computed using

Bell’s approach.

(ii) For both joint sets, joint factor fJ and cj are calculated using Eq. (27) and (28) in

horizontal direction. The cj of element I is the minimum of cj values of two joint sets.

25

(iii) Assuming 3 equal to over burden pressure,

1 of element one at failure is calculated

using strength criterion Eq. (29).

(iv) Next, considering element II of Fig. 2.22, fJ and

cj are calculated accordingly in vertical

direction for both joint sets.

(v) Assuming 3, 1,II I , 1,II of element II in vertical direction is calculated using Eq. (29).

Finally, the ultimate bearing capacity ultq of rock mass is 1,II of element II.

Also some graphs have been presented for the calculation of bearing capacity by the proposed method,

shown in Fig. 23 and Fig. 24. The major limitations of this method are as follows

(i) It is not applicable to rock blocks with columnar geometry.

(ii) It is not applicable if the block size is large or there is only one joint set.

(iii) The bearing capacity results, obtained by the proposed methodology, is not validated with any

experimental or field data or existing literature.

Fig 23 Chart for Computing (a) Joint Factor fJ and (b) cr (Singh and Rao, 2005)

26

Fig. 24 Charts for Computing (a) Average Strength Enhancement and (b) Lower Bound

Strength Enhancement (Singh and Rao, 2005)

Imani et al. (2012) investigated the effect of ground water and joint spacing on the bearing

capacity of submerged jointed rock under strip foundations using upper bound theorem of limit analysis.

In this analysis, rock mass containing two orthogonal joint set was considered. Mohr-Coulomb criterion

was used for both intact rocks and joints. Orientation of joints equal to 15°, 30° and 45° were taken into

consideration. The concept of ‘spacing ratio’ (SR) (Serrano and Olalla 1996) was used to account the

effect of joint spacing.

Shape of four assumed failure mechanisms were obtained by numerical analysis using UDEC.

It was seen that the TS1 and OS1 mechanisms, shown in Fig. 25, were produced the optimum bearing

capacity values. Equating the total external work to the total energy dissipation, the general expression

for the ultimate bearing capacity of submerged jointed rocks (uwq ) was obtained by Eq.(32). Where the

values of Bearing capacity coefficients ( ,cj ciN N , qN ) are depends on angles , , i and j and the

submerged bearing capacity coefficient subN value is obtained from Eq. (33).

1

2

sub

uw j cj i ci qq c N c N qN BN (32)

' '

(1 )sub ww

dN N N

B

(33)

The proposed upper bound solution is able to take into account different depths of water

table beneath the footing. From results it was revealed that submergence of rock below footing

reduces the contribution of the rock weight in bearing capacity. The maximum reduction in

27

bearing capacity occurred when dip of joint 15 and the minimum occurred when 45 .

It was also observed that the effect of submergence of rock mass on the bearing capacity

increases with increasingj (Fig. 27). However, this effect decreases with increasing

j ic c

ratio. The results of this analysis were compared to the results of Hansen et al. (1987) and

Ausilio & Conte (2005), and good agreements were observed among them, shown in Fig 26.

In this study only two continuous and orthogonal joint sets were considered with

specific orientations. In reality, rock mass containing any number of joint sets with random

orientation may exists in the field.

Fig. 25 Failure Mechanisms and Corresponding Hodographs: (a) TS1 and (b) OS1

(Imani et al., 2012)

28

Fig. 26 Comparison of Results Among The Present Upper Bound Solution, Methods of Ausilio

and Conte (2005) and Hansen et al. (1987) (Imani et al. 2012)

Fig. 27 Variation of Quw/Qu with Dw/B for (a) Фi = Фj=35o and Ci = 5 MPa and (b) Фi = 35o, Ci =

0.1 Mpa and Cj = 0 (Imani et al. 2012)

29

6.0 CHARACTERISTICS LINE METHOD

Serrano and Olalla (1994) have been proposed a methodology for bearing capacity

quantification based on characteristics method (Sokolovskii 1960, 1965) coupled with Hoek and Brown

failure criterion (Hoek and Brown 1980). The rock mass was considered as an ideal homogeneous,

isotropic, continuous, plastic and weightless material. The strip footing was subjected to central vertical

and inclined load on horizontal or on sloping ground. The necessary input parameters were the type of

rock, UCS of intact rock, Bieniawski classification (RMR parameter) and specific unit weight of rock.

Fig. 28 Shallow Footing on Horizontal Surface Subjected to Vertical Load

(Serrano and Olalla 1994)

A step by step procedure for determining ultimate bearing capacity has been described in this

paper for six different cases. Among those cases, case-1 where foundation surface is horizontal (Fig.28)

and normal loads are acting on the two boundaries 1 and 2, is discussed below.

(i) Hoek-Brown parameters are determined using equations (34) and (35).

0

100exp

14.45

RMRm m

(34)

100

exp60.3

RMRs

(35)

(ii) Geo-mechanical parameters ( , ) are calculated using equation (36) and (37).

0 100exp

8 14.45

cm RMR

(36)

2

0

8 100exp

60.3

RMR

m

(37)

(iii) Boundary 1 conditions: The values of major principal stress (1 H ), normalized major

principal stress ( *

1o H ), inclination (i1) of load at boundary 1 are determined.

(iv) Calculation of data at boundary 1: Inclination (1 ) of the major principal stress is obtained

from Eq. (38). Instantaneous friction angle (1 ) and Riemann’s invariant (I1) are calculated

using expressions (39) and (40) respectively.

30

2

i (38)

1

1*

1

1sin

1 2 o

(39)

1

cot ln cot2 2

I

(40)

(v) Calculation of data in boundary 2: Inclination 2 of major principal stress and invariants (I2)

is obtained using slope angle ( ) and inclination (i2) of load at boundary 2 from Eq. (38).

Instantaneous friction angle (2 ) is calculated from both Fig. 30 and Eq. (39) and the minimum

value of 2 is considered.

Fig. 29 Characteristic Network Under the Foundation (Serrano and Olalla 1994)

(vi) Calculation of ultimate bearing capacity (hP ): The ultimate bearing capacity (

hP ) of rock mass

is obtained using Eq. (41) where the value of N can be obtained from Eq. (42).

( )hP N (41)

where

22

2 2

cot 1 sin sincos cos 1

2 sin 2 tan tan

iN i i

(42)

31

Some nomograms have been presented to obtain the value of ultimate bearing capacity factor

N for a strip footing. The major limitation of this method is that the effect of self-weight of rock mass

is not incorporated in solution which is actually affects the bearing capacity.

Fig. 30 Variation of Riemann’s Invariant with Instantaneous Friction Angle

(Serrano and Olalla 1994)

Serrano and Olalla (1996) have been extended their theory published in 1994 to consider the

effects of spacing ratio of foundation (SR) and scale effect on ultimate bearing capacity. The SR was

defined by expression (43).

1 1

1n n

i

i iml

SR B BS

(43)

where B = foundation width (m); mlS = joint spacing of i th family;

i =frequency of i th discontinuity

family (m); n = no. of discontinuity families.

A statistical analysis was carried out to evaluate the partial safety factor (Fp) related to geo-

mechanical parameter variations. Some values of partial safety factor (Fm) was also suggested to

32

consider the uncertainty of brittle failure of rock mass. Finally, the values of proposed allowable bearing

pressures were compared with the values from existing code of practice and a reasonable agreement

was found, depending on some specific situations (Serrano and Olalla 1996).

Fig. 31 Variation of Load Coefficient ( )N with Normalized External Load *

01 on

Boundary 1 and Inclination of Load on Boundary 2 (Horizontal Surface, 0 )

(Serrano and Olalla, 1996)

Serrano, Olalla and Gonzalez (2000) have been modified the method of Serrano and Olalla

(1994, 1996), by using the modified Hoek-Brown criterion (1992) which leads to a better assessment

of behaviour of highly fractured rock masses (RMR ≤ 25). This method is valid under the assumptions

of plain strain, homogeneity, isotropy and weightless rock media. The bearing capacity of rock mass

has been calculated using the theory of Serrano and Olalla (1994, 1996), but with an exception of using

modified Hoek-Brown criterion (1992) instead of original Hoek-Brown criterion (1980). Expressions

for ultimate bearing capacity ( )hP using this approach were given below,

( )h n nP N (44)

where

33

2

1

2

2 202 02 02

2 22

11 sin

1 sin 1 sincos cos 1 sin

1 sin sinsin

n

n n

nN i n i n i

n

n

(45)

1

1

1

2

n

n

n c n

m n

(46)

1

1

1

2

n n

n

n

s

m nm

(47)

and n Modified Hoek-Brown parameter ranges from 0.50 to 0.65 (Hoek et al. 2002)

A nomogram was presented for obtaining N factor directly, for the simple case of vertical

loads on horizontal ground, shown in Fig. 32. Self-weight of rock masses was not considered in this

formulation also.

Fig. 32 Variation of Load Factor N with Normalized External Load on Boundary 1, with

Horizontal Ground and Vertical External Loads (Serrano et al. 2000)

34

7.0 NUMERICAL METHODS

Clausen (2012) has been investigated the problem of bearing capacity of a circular surface

footing resting on horizontal rock mass. The standard displacement finite element method coupled with

a convergence extrapolation scheme have been implemented to develop numerical codes.

The problem has been treated as an axis symmetric. The behaviour of rock mass have been

described by the generalised Hoek-Brown failure criterion (Hoek et al. 2002, 2006) in conjunction with

linear elasticity and perfect plasticity.

The problem geometry (width b and height h) and boundary conditions have been shown in

Fig. 33. The footing nodes have been fixed in the horizontal direction. Triangular six-node linear strain

elements with two degree of freedom in each node has been considered in this analysis. The mesh sizes

have been selected after a detailed sensitivity analysis, shown in Fig. 34.

Fig. 33 Geometry and Boundary Conditions of the Calculation Domain (Clausen 2012)

Fig. 34 Variation of Final Load up with No. of Degree of Freedom dofn (Clausen 2012)

35

The bearing capacity coefficient N and normalised near failure displacement 95( )U

corresponding to 0.95 up are calculated using Eq. (49) and (50).

u ciN p (48)

95

0.95 u rm

ci

p EU

r (49)

where up = ultimate bearing capacity of rock mass

r = radius of circular footing

1 2100

1 exp 75 25 11rm

DE

D GSI

GPa (Hoek et al. 2006)

or if intact rock modulus is known

1 2

0.021 exp 60 15 11

rm i

DE E

D GSI

GPa (Hoek et al. 2006)

D = disturbance factor which was assumed to be = 0

= poison’s ratio which was assumed to be = 0.3

The values of , iGSI m and 2ci r ratio have been varied to study their influence on N and

95U . The results have been found to lie within the 1% of the exact solutions. The results have been

presented in the form of charts, for different values of GSI ranges from 10 to 100, to facilitate their use

in practical design. A bearing capacity chart for 30GSI has been shown in Fig. 35.

It have been found that for poor quality rock 30GSI the rock mass weight has a significant

impact on the bearing capacity and the near displacement failure. However, the self-weight has almost

no effect for higher quality rocks. A comparison has been made between the results of this analysis and

the results obtained using equivalent Mohr-Coulomb parameters (Hoek et al., 2002) and a very poor

agreement has been observed. However, the bearing capacity solutions of Serrano and Olalla (2002)

has been in good agreement with the proposed solutions. The authors have represented some results in

tabular form of bearing capacity of circular footing as benchmark for other researchers for comparison,

shown in Table 9. One of the limitation is that unrealistic value of disturbance factor (D = 0) is assumed

in this analysis.

36

Fig. 35 Variation of Bearing Capacity and Near-Failure Displacement For 30GSI

(Clausen 2012)

Table 9 Benchmark Values of Different Variables (Clausen 2013)

GSI 2ci r im N

90U 95U b r h r

10 125 7.5 0.176 0.38 0.45 10 8

20 2000 22.5 0.851 1.74 2.03 22 12

30 22.5 1.381 2.83 3.30 38 35

40 250 35.0 3.544 7.16 4.36 20 11

50 5000 10.0 1.678 3.42 4.03 15 12

60 35.0 7.053 14.4 16.8 35 32

80 500 10.0 5.818 11.9 13.9 12 10

100 250 22.5 23.91 48.1 56.3 12 11

Salari-rad et al. (2012) have been investigated the problem of bearing capacity of shallow

foundation rested on anisotropic discontinuous rock mass using a distinct element based software

UDEC. A strip footing of width 5 m has been modelled on rock mass containing single joint set (Fig.

36). The dimension (width 70 m × depth 30 m) of the model has been selected by sensitivity analysis.

The rock material has been assumed to be isotropic and homogeneous. Therefore, anisotropy

occurs only due to presence of single joint set in rock mass. The numerical analysis was carried out by

assuming Mohr-Coulomb and Hoek-Brown failure criterion for joints and rock material respectively.

Six failure mechanisms introduced by Serrano and Olalla (1998) were assumed as basic possible failure

37

mechanisms, shown in Fig. 37. Variation of failure mode and bearing capacity with dip angle and shear

strength of joint sets was investigated in this study.

The outcomes of this analysis indicated that the ultimate bearing capacity of rock mass

containing one joint set varies between 27% and 86% of intact rock. MC is the most common failure

mechanism whereas minimum bearing capacity values are obtained in MS mechanism. Variation of

bearing capacity and failure mechanisms of rock mass with joint dip angle is shown in Fig. 38. It was

also observed, the bearing capacity of rock mass decreases with decreasing shear strength of plane of

weakness. The obtained results were compared with the results of Sutcliffe et al. (2004) (Fig. 39). It

was seen that the difference between the two results increases when the footing is prepared to fail along

joint planes. One of the limitation of this analysis is that rock mass containing only single joint set has

been considered which is rarely exists in nature.

Fig. 36 Dimensions, Velocity Field and Boundary Conditions of Distinct Element Model in

Homogeneous Medium (Salari-Rad et al. 2012)

38

Fig. 37 Possible Anisotropic Failure Mechanisms Depending on Dip of Joints: Homogeneous

and Isotropic (MI), Conditioned by Boundary 1 (M1), Conditioned by Boundary 2 (M2), with A

Central Wedge (MC), Simple Mechanism (MS), and Double Mechanism (MD)

(Serrano and Olalla 1998)

Fig. 38 Variation of Bearing Capacity with Dip Angle of Joint Set (Salari-Rad et al. 2012)

39

Fig. 39 Comparison of Bearing Capacity Obtained by DEM and Lower Bound Limit Analysis

Method by Sutcliffe et al. 2004 (Salari-Rad et al. 2012)

Bindlish et al. (2013) simulated the problem of ultimate bearing capacity of a strip footing on

jointed rock mass subjected to central vertical load using distinct element based software UDEC and

compared the results of analysis with the experimental results of Bindlish et al. (2012). A series of

numerical tests were performed in plain strain condition. The behaviour of the foundation, blocks and

joints of rock masses were defined by the elastic model, Mohr-Coulomb model and Coulomb-Slip

model respectively. Two types of rock mass specimens (shown in Fig. 40) were considered in

simulation, viz.,

(i) Type A: Rock mass contains two perpendicular continuous joint sets with 90 .

(ii) Type B: Rock mass specimen having two perpendicular joint set, one continuous and

another stepped with 90 .

The load intensity versus settlement graphs were analysed, obtained from the UDEC model, to

evaluate the bearing capacity. Pressure bulb area and ultimate bearing capacity of strip footing for type-

A specimen were predicted by the proposed model reasonably well where the failure mechanism was

governed by the pre-existing discontinuities. A typical load intensity versus settlement graph and a

pressure bulb has shown in Fig. 41. Numerical simulation result were appeared to be very similar to the

experimental data, shown in Fig. 42(a). However, this model had failed to anticipate the ultimate bearing

capacity of Type-B specimen where failure had occurred through intact material (Fig. 42(b)) as this

model had failed to simulate the initiation of new fractures. Furthermore, it was perceived that the

governing failure modes are shearing and splitting.

40

Fig. 40 Rock Mass Specimens (a) Type A and (b) Type B (Bindlish et al. 2013)

Fig. 41 For Type-A Specimen Results Obtained From UDEC Analysis (a) Load Versus

Settlement Curve and (b) Failure Mode and Major Stress Contours (ϴ = 45° and s = 0)

(Bindlish et al. 2013)

Fig. 42 Comparison of Ultimate Bearing Capacity Predicted by UDEC Model with

Experimental Values For (a) Type-A Specimens and (b) Type-B Specimens

(Bindlish et al. 2013)

41

Shekhawat and Viladkar (2014) studied the behaviour of shallow strip footing resting on

jointed rock mass using FEM coupled with modified Hoek-Brown failure criterion (2002). The double

tangent method (Lutenegger and Adams 1998) was employed for evaluating the ultimate bearing

capacity. In this analysis, the footing was subjected to eccentric-inclined loading. The jointed rock mass

was treated as isotropic, homogeneous, elasto-plastic continuum. The numerical modelling of the

problem was done using software package PLAXIS-3D. Ten node tetrahedral elements and six node

plate elements were used to discretization the numerical model as shown in Fig. 43.

Fig. 43 Finite Element Model For Elasto-Plastic Analysis of Surface Strip Footing

(Shekhawat and Viladkar 2014)

The effect of eccentricity e , inclination i of load and geological strength index GSI of rock

mass were studied in this analysis. A rigorous parametric analysis was carried out to study the pressure-

settlement and pressure-tilt ( )t characteristics of strip footing. Variation of pressure-settlement

behaviour with eccentricity e and inclination angle i parameters were shown in Fig. 44.

Fig. 44 Load Intensity vs Settlement Characteristics with Respect To (A) Load Eccentricity

and (B) Load Inclination (Shekhawat and Viladkar 2014)

42

From the results of numerical simulations non-dimensional correlations were produced to

envisage the ultimate bearing capacity, settlement and tilt of footing. For different values of GSI

different sets of correlations were developed. A set of correlations for GSI = 40 are shown below.

2

2 20.002 0.044 0.181 0.013 0.12 7 05 0.105u

ci

q e ei i i E i

B B

(50)

2

2 2

0

0.027 0.282 5.408 0.011 0.291 1.163 0.061 1.044eS e ei i i i i

S B B

(51)

2

2 2

0

0.005 0.963 21.81 0.001 0.122 3.605 0.045 1.108mS e ei i i i i

S B B

(52)

and 1sin

2

m eS St

B e

(53)

For validation of this numerical model, bearing capacity factor 0 u ciN q was calculated

and compared with the values of 0N obtained from Kulhawy and Carter (1992) and Serrano (2000).

As shown in Fig. 45, very close agreement was observed among those results.

Fig. 45 Bearing Capacity Factor0N Model Validation Curve

(Shekhawat and Viladkar 2014)

43

The major limitations found in this method of analysis were:

(i) Weight of rock mass were not considered in calculation which results in conservative design

(Merifield at al., 2006).

(ii) Jointed rock mass was assumed to be isotropic and homogeneous which rarely exists in nature.

(iii) Unrealistic initial values of various input parameters have been considered in this FEM

modelling like disturbance factor D is equal to zero.

A new model has been proposed by Ziaee et al. (2014) for the prediction of bearing capacity

of shallow foundation on rock mass applying artificial neural network (ANN). Conventional procedure,

based on the fixed connection weights and bias factor of an ANN structure (Multilayer perception

networks), has been carried out for computational purposes. An artificial neural network is a

computational simulated system that follows the neural networks of human brain. The ANN model (Fig.

46) which has been presented in this paper for the formulation ofultq , consists of

One invariant input layer, with 4 (n = 4) arguments

One hidden layer having 5 nodes (m = 5)

One invariant output layer with 1 node providing the value of ultq

Fig. 46 Schematic Illustration of Proposed ANN (MLP) Network (Ziaee 2014)

In this model, rock mass has been treated as an equivalent continuum medium. Ultimate bearing

capacity ( ,ult ANNq ) has been assumed to be a function of RMR , UCS (uq ), ratio of joint spacing to

footing width S B and internal friction angle ( ) of rock mass as shown in Eq. (55).

44

, , , ,ult ANN u

Sq f RMR q

B

(54)

A comprehensive data base (49 socket tests, 40 plate load tests and 13 large scale footing load

test results) have been considered for this model development. The database primarily comprises of

results on circular and square footings of various sizes tested on various types of rock masses. All input

( , , , )u uRMR q S B q values were normalized ,( , , , )n u n nnRMR q S B using Eq. (56) such that they

lie in a range of 0.05 - 0.95.

min

max min

n

X XX U L U

X X

(55)

where

nX = Normalized value of variable X

max min,X X = Maximum and minimum value of variables

U = Upper limit of normalized variables, 0.95U

L = Lower limit of normalized variables, 0.05L

An ANN based formulation of ultq (MPa) has been presented in this paper using Eq. (57).

0.04788 5.22 1748.59 0.95ultq A (56)

where 5

1 1 j

khF

k

VA bias

e

(57)

1 , 2 3 4j n k u n k k k k

n

SF RMR W q W W W bias

B

(58)

In Eq. (58), k has been represented number of hidden layer neurons. Various values of input layer

weights kW , input layer biases kbias , hidden layer weights kV and hidden layer biases hbias

have been presented in Table 10 and Table 11 respectively.

Table 10 Weight and Bias Values Between Input and Hidden Layer (Ziaee et al. 2014)

Weights Number of hidden neurons k

1 2 3 4 5

1kW -4.3660 1.2050 1.0146 -2.3774 1.0715

2kW -5.9392 2.6041 -3.3564 -0.1235 3.7980

3kW 4.4797 1.5636 3.7741 -5.3471 0.3445

4kW -0.8778 2.2540 0.9362 3.3533 3.6613

kbias -2.8214 -2.5686 1.1696 0.8654 -4.7052

45

Table 11 Weight and Bias Values between Hidden Layer and Output Layer (Ziaee et al. 2014)

Weights Number of hidden neurons k

1 2 3 4 5 hbias

kV 6.3774 9.4936 -2.8366 -0.8674 -1.3602 -4.3811

The methodology for bearing capacity calculation, proposed by the authors, are illustrated below.

Step 1: Normalization of input data ( , ,uRMR q S B anduq ): The normalized input neurons

,, , &n u n nnRMR q S B are calculated with the help of input data base by Eq. (55).

Step 2: Calculation of hidden layer: Using Eq. (58), the input values jF of five neurons in the hidden

layer were determined.

Step 3: Prediction of ultq : Finally, the ultimate bearing capacity is predicted using Eq. (56).

The proposed model has been predicted the ultq values with an acceptable degree of accuracy

(Fig. 47). The obtained results have been compared with results of Goodman model (Goodman R. E.,

1989) and very close agreement has been observed between them (Fig. 48). A parametric study has

been performed to evaluate the sensitivity of this ANN model to the variation of each independent

variable based on Garson’s algorithm (Garson D., 1991). The sensitivity analysis indicated that ultq is

more sensitive to uq and S B compared to RMR and , as shown in Fig 49. The major limitation of

this method is the negligence of anisotropic behaviour of foundation rock mass in ultimate bearing

capacity formulation as it assumes the rock mass as an equivalent continuum.

Fig. 47 Comparison of Measured and Predicted ultq Values Using ANN Model: (a) Training

(Learning and Validation) Data and (b) Testing Data (Ziaee et al. 2014)

46

Fig. 48 Comparison of ultq Values among ANN Model Predicted, Experimental and

Goodman (1989) Values (Ziaee et al. 2014)

Fig. 49 The Percentage Relative Importance Histogram of Each Input Variable For Predicting

ultq Based on Garson’s Algorithm (1991) (Ziaee 2014)

47

8.0 EXPERIMENTAL METHODS

Few investigators have carried out experimental studies of shallow foundations resting

horizontal or sloping jointed rock mass and subjected to central vertical load.

Dunham et al. (2005) have been performed four series of centrifuge tests on using model rock

specimens made from a mixture of sand, bentonite, cement, and water. Tests have been conducted to

determine the effect of horizontal and vertical discontinuities in the bedrock on the load settlement

response of a rigid, shallow footing. A prototype square footing, measuring 1 m × 1 m, has been

simulated in the study. Horizontal and vertical joints, in-filled with compressible material, have been

simulated by incorporating thin seams of Styrofoam.

A parametric study was carried out to observe their influence on bearing capacity of rock mass.

Results of this study have been compared to the commonly available methods used to predict the

ultimate bearing capacity of footings on jointed rock mass. It has been concluded that the existing

methods to predict the bearing capacity are conservative for the range of design parameters studied in

this research (Dunham et al. 2005).

Bindlish et al. (2012) has been conducted an experimental study on a rigid footing placed on

top surface (semi-infinite) of confined jointed rock mass and loaded up to failure consequently. The

experiments were carried out under plain strain conditions keeping the size of rock elements one-fifth

times that of the footing width. The footing was assumed to be smooth and shallow. Effect of moisture

was not considered in this test. The influences of intact rock properties, characteristics of joint intensity,

joint orientations, interlocking conditions and type of structure on the bearing capacity of jointed rock

mass were studied.

A series of model tests were performed on synthesized rock mass (Plaster of Paris mixed with

medium sand) specimens of size 750 750 150 mm3. The size of elemental blocks and elemental

plates, used to construct the specimen, were 25 25 75 mm3 and 750 150 25 mm3 respectively.

Three types of jointed blocky specimens (shown in Fig. 50) were prepared, viz., Type – A (two sets of

orthogonally intersecting continuous joints), Type – B (two orthogonal joint sets, one continuous and

the other stepped) and Type – C (one continuous joint set). Size of footing plate was 150 150 mm2.

The tests have been conducted in a specially designed and fabricated bearing capacity test apparatus of

1000 kN capacity.

Based on the experimental results, a methodology (Bindlish et al. 2012) has been suggested to

determine the ultimate bearing capacity of shallow foundation placed on anisotropic rock mass for both

continuous as well as discontinuous joints. This method is basically a modification over the Singh and

Rao (2005) approach which over-estimates the anisotropy of rock mass. The results obtained from this

methodology matched closely with the experimental values. The results of tests indicated that the

48

impact of anisotropy in case of shallow foundations is much less than envisaged in the study of Singh

et al. (2002). Also, stepping of joints significantly enhances the bearing capacity if the dip angle of

continuous joints are less than 45° due to interlocking. Splitting and shearing are the governing failure

mode of the rock mass beneath the footing.

Fig. 50 Rock Mass Specimens with = 0° to 90° are: (a) Type – A, (b) Type – B and

(c) Type – C (Bindlish et al. 2012)

Shukla et al. (2014, 2014) studied the problem of bearing capacity of shallow foundation on

slopping anisotropic rock mass with continuous joints experimentally as well as analytically. The tests

were performed on in plain strain condition. The rock mass was remained unconfined at slopping side.

A series of experiments were conducted on rock mass model of size 750 750 150 mm3,

shown in Fig. 51(a). The rock mass models have constructed using sand stone element of size

25 25 75 mm3. The joint angle and slope angle of rock specimens with horizontal were

varied from 15 to 90 and 30 to90 , respectively, at an interval of15 . The footing was placed at the

edge of slope and at a distance of 15 cm from slope edge. Tests were conducted in a specially designed

and fabricated bearing capacity test apparatus of 200 ton capacity, shown in Fig. 51(b).

49

Fig. 51 (a) Rock Mass Specimen with Variation in Joint Set Angle and Side Slope and (B)

Bearing Capacity Test Apparatus For J0090-SL45-ED00 (Shukla et al. 2014a; b)

The bearing capacity of shallow footing were also estimated analytically using Euler’s buckling

theory (Caver’s 1981). In this analysis, the buckling loadcrP was calculated by Eq. (60) where the value

of modulus of elasticity jE of jointed rock was obtained from Eq. (61) (Ramamurthy and Arora 1994).

2

2

jcr

b

K E IP

B BL

(59)

2exp 1.15 10r j i fE E E J (60)

where crP = Buckling load of column

bL = Buckling length or effective length of column

I = Moment of inertia 3 12bd

B = Width of column

K = Coefficient of buckling or effective length

jE and iE = Modulus of elasticity of jointed and intact rock respectively

It was observed that joint angle, distance of footing from edge and mode of failure are the

governing parameters in bearing capacity estimation on slope apart from rock mass properties. Buckling

and sliding have been found to be dominant failure mode. For the case of footing on the slope edge,

bearing capacity of jointed rock mass is half of the total buckling load capacity. It was also observed

from experiments, when joint angle 0 ,15 ,30 (buckling failure), average settlement of footings

are less but bearing capacity is higher than the cases when 45 ,60 (combination of sliding and

buckling failure). Buckling resistance has been found to be always greater than sliding resistance for

50

continuous jointed rocks. This method is applicable only for the rock mass with two continuous joint

sets. It is not applicable when the rock mass is heavily fractured or when the joints are not continuous.

9.0 EMPIRICAL METHODS

Several empirical expressions are available in literature for the estimation of bearing capacity

of shallow foundations resting on jointed rock masses.

The following expression (62) was developed by Pauker (1889) for bearing capacity estimation

of shallow strip footing on jointed rock mass. The major limitation of this method is that for 0fD ,

Eq. (62) gives 0ultq which is unrealistic (Ramamurthy 2011).

4tan 45 2ult fq D (61)

A rigorous bearing capacity expression (63) was presented by Terzaghi (1943) assuming

general shear failure of jointed rock mass where ,c qN N and N are bearing capacity factors. Shape

factors are represented bycs and qs .

0.5ult c c f q qq cN s D N BN s (62)

Bishnoi (1968) has been proposed Eq. (64) for bearing capacity evaluation of shallow

foundation considering the influence of footing size (B) and with respect to the joints spacing (s). It was

found that Eq. (64) overestimated the bearing capacity as it did not consider the rotation and sliding of

rock blocks within the zone of influence.

1

11

1

N

N

ult ci

sq N

N B

(63)

where 2tan 452

N

(64)

Considering failure along two planes, Coates (1970) developed expression (66) for calculation

of bearing capacity of a strip footing on rock surface.

0.5ult c qq cN qN BN (65)

where 45tan 452cN (66)

51

6tan 452qN (67)

and 1qN N (68)

Ladanyi and Roy (1971) studied the problem of bearing capacity of shallow strip footing on

rock mass containing two inclined joint sets. Assuming 1 and

2 as the dip angle of steep and shallow

joint sets, expression (70) was proposed for bearing capacity estimation(Ladanyi and Roy 1971).

11 1

1

1tan

ult

cq pN N

(69)

where 21 2 2

2

2 tan 1tan

cp B N N

(70)

Bowels (1988) suggested expression (72) for bearing capacity evaluation based on RQD

value and unconfined compressive strength of rock (rq ) (Bowles 1997).

2

ult rq q RQD (71)

Goodman (1989) developed the following expression (2.75) for estimation of bearing capacity

of strip footing on heavily fractured rock mass ignoring the cohesion of rock mass (Goodman 1989).

2

1 sin

1

ult ci

ci

q

N

(72)

10.0 CONTACT PRESSURE DISTRIBUTION IN ROCK MASS

Distribution of stresses at any point beneath shallow foundation in intact or heavily fractured

rock masses under the loads (concentrated, line or distributed) are analogous to the stress distribution

in soil beneath the shallow footing. This can be evaluated using the method of Boussinesque or

Westergaard by idealizing the intact or heavily fractured rock mass as semi-infinite, homogeneous,

elastic and isotropic medium. On the other hand, distribution of stresses in anisotropic jointed rock mass

is quite different from intact rock due to presence of joints.

Many researcher (Gaziev and Erlikhman 1971, Sauma 1971, Singh 1972, Bray 1977, Oda et al.

1993, Agharazi et al. 2012, Bindlish et al. 2013 and so on) have been studied the distribution of stresses

in rock mass below shallow foundations. For typical loading, the various approaches are discussed in

this section.

52

For a line load acting normal to the surface of an intact or heavily fractured rock mass, the radial

stress at any point is obtained by Eq. (74) and the tangential and shear stresses are zero i.e. 0 and

0r .

2 cos

r

P

r

(73)

where P = Load per unit length

, r = polar co-ordinate of the point under consideration

For a constant load, the graphical representation of Eq.(73), results into a circle tangent to the point of

application of load P and centred at depthrP , known as pressure bulb (Fig. 52.a).

Similarly, for a shearing load Q the stress distribution is entirely radial and it is obtained from

Eq.(75). The Eq. (75) is graphically presented in Fig. 52.b where the left circle represents tensile stress

and the right circle represents compressive stresses.

2 sin

r

Q

r

(74)

When the line load R is inclined with vertical axis the radial stress is given by Eq. (76). and

locus of r for constant value of R is shown in Fig. 52.c.

12 sinr

R

r

(75)

Gaziev and Erlikhman (1971) have been conducted a series of model tests on synthesized

rectangular rock specimens to study the distribution of stresses in jointed rock mass under shallow

foundation. The specimens were consisted two orthogonal joint sets – one was continuous (set I),

inclined at angle of 0°, 30°, 45°, 60° and 90° with horizontal while the other joint set was stepped

(set II) as shown in Fig. 53. The test specimens were loaded through a flexible footing of width B. The

stress distribution under footing was measured by strain gauges embedded in the plaster blocks in the

specimen. The results have been presented in Fig. 54. From the results it was observed that stress

concentration occurs along parallel and perpendicular to the continuous joints especially for the case of

45 .

53

Fig. 52 Pressure Bulb Resulting From Loading of An Elastic Half Plane By (a) Vertical Line

Load, (b) Shear Line Load and (c) Inclined Line Load (Goodman 1989)

54

Fig. 53 Two Dimensional Joint Model with (a) Horizontal Major Joints ( 0 ) and (b)

Inclined Major Joints with An Inclination Angle (Gaziev and Erlikhman 1971)

Fig. 54 Stress Concentration Patterns Induced By Partial Surface Loading on 2D Joint Models

with Different Values of (Gaziev and Erlikhman 1971)

55

Singh (1973) has been investigated the stress field in jointed rock mass under shallow

foundation using anisotropic continuum model (Singh 1973a) coupled with FEM and compared it with

the results obtained from the discrete joint model as shown in Fig 55. This computations have been

revealed excellent agreement between the FEM predictions of the two models except the regions of

high stress gradients near loaded area. The principal stresses in jointed rock mass were observed to be

distributed to a considerable depth along joints and to some extent across joints. This phenomenon is

more pronounced in a rock mass of joints with low shear stiffness.

Fig. 55 Comparison of Pressure Bulbs Predicted By Anisotropic Continuum Model and

Discrete Joint Model (Singh 1973b)

Bray (1977) has been studied the stress contours beneath a shallow footing for a line load. The

layered rock mass was idealized as a transversely isotropic medium. Expressions (77), (78) and (79)

have been presented for the determination of stress field by assuming 0 and 0r .

2

2 2 2 2 2

cos sin

cos sin sin cosr

h X Yg

r g h

(76)

2

11 n

Eg

k S

(77)

2

2 1 12

1 1s

Eh g

E k S

(78)

56

where X and Y are the components of load P in the parallel and perpendicular to the joint planes

respectively. The constants g and h are non-dimensional quantities describing the properties of rock

mass. S is the average spacing of between joints. kn and ks are the normal and shear stiffness of joints.

The pressure bulbs obtained though Eqn. (76) for a constant load have been presented in Fig. 56. The

results of this approach were compared with the experimental results of Gaziev and Erlikhman (1971)

and a close aggrement between two results were found.

Fig. 56 Pressure Bulb Under Line Loads in Jointed Rocks Calculated By Bray (1977)

(Goodman 1989)

Oda et al. (1993) have been formulated an elastic stress-strain relation in terms of crack tensor.

Using this stress-strain relation coupled with finite element analysis, the distribution of stress fields

below shallow footing have been investigated. The problem rock mass has been modelled using a finite

mesh of four-node quadrilateral isoperimetric elements. A vertical stress has been realised uniformly

on the rock surface thorough a footing of B = 10 cm. The stress contours obtained applying the proposed

57

method are shown in Fig. 57 in which each contour represents an equivalent line of maximum shear

stressm normalized by the unit surface pressure. Hence, the study found that the stress concentration

pattern in rock mass is governed by the ratio of normal stiffness to shear stiffness (R). The stress

concentration was occurred in two directions; parallel and perpendicular directions to the major joints

for high value of R. However, the stress concentration is restricted to the direction parallel to the major

joints only when R equal to unity.

Fig. 57 Maximum Shear Stress Contours Due To Partial Surface Loading For R = 10

(Oda et al. 1993)

58

Agharazi et al. (2012) formulated a three dimensional constitutive model for stress and

deformation analysis of jointed rock mass containing up to three joint sets with arbitrary spatial

configurations. The model was developed based on superposition of deformations of the representative

elemental volume components. The Mohr-Coulomb failure criterion coupled with this constitutive

model was implemented in FLAC3D to study the stress and deformation behaviour of a plate load test

on an intensely jointed rock mass.

Fig. 58 Stress Distribution Beneath Loaded Area, Estimated by Jointed Rock Model in Semi-

Infinite Model with One Joint Set (Agharazi et al. 2012)

59

In FLAC3D a semi-infinite body of rock with one joint set was modelled. The dimensions of the

rock model were 26×26×15 m (W×L×D). A distributed normal stress of P = 10 MPa was realized on

a circular area of diameter d = 1.0 m in diameter, on the top centre of the model. The stress distribution

beneath the loaded area in the model with one joint set is shown in Fig. 58. Although, no discrete

discontinuities was existed the effectiveness of this model in capturing the influence of the joints in

stress distribution is observed in Fig. 58.

11.0 CRITICAL COMMENTS

From the above reviewed literatures following gaps in the study have been observed.

i. Several methods have been proposed for solution of this problem and almost all the methods come

with a set of conventions such as infinite strip footing, plain strain condition, no inertial force,

weightless rock mass, undisturbed rock mass etc. makes the situation impractical or unrealistic.

ii. Almost, all the research works (Serrano and Olalla 1994, Sutcliffe at al. 2003, Merifield et al. 2006,

Ziaee et al. 2014) have been conducted on strip foundations. However, very few researchers have

worked on other type of foundations like square footing (Ziaee et al. 2014, Bindlish et al. 2012,

2013), circular footing (Clausen 2012), and raft footing (Justo et al. 2013).

iii. It has been observed that adequate amount of work (Prakoso and Kulhawy 2004, Saada et al. 2009

and so on) has been carried out on shallow footings on horizontal ground surface. On the other hand,

bearing capacity of footing on slopping rock mass have been investigated by a few investigators

(Serrano and Olalla 1994, Yang 2009, Shukla 2014).

iv. The research works (Meyerhof 1953, Merifield et al. 2006, Imani et al 2012, and so on) have been

mostly conducted on footings subjected to central vertical load. Conversely, very few scientists have

worked on footings subjected to inclined (Serrano and Olalla, 1994) or eccentric-inclined loading

(Shekhawat and Viladkar 2014).

v. The studies are concentrated mainly on the bearing strength of rock mass. Hardly, any (Clausen

2012; Shekhawat and Viladkar 2014) attention has been paid regarding the formulation of pressure-

settlement and pressure- tilt characteristics of footing on rock mass.

vi. A very little amount of investigation (Imani et al. 2012) has been conducted on the variation of

bearing capacity due to presence of water table.

vii. The tests, referred in the literature, are mostly laboratory tests so far. Hence, the validation of

analytical or experimental results with field data is necessary before the proposed methods are used

in field.

60

12.0 CONCLUSION

Assessment of behaviour of shallow foundation on jointed rock mass is one of the most classical

problem in the field of geotechnical engineering. Numerous research work have been conducted in this

field which are broadly classified in four categories: (i) Analytical methods, (ii) Numerical methods,

(iii) Experimental methods and (iv) Empirical methods. Few available literatures, relevant to the

problem of ultimate bearing capacity of jointed rock mass and distribution of contact pressure beneath

shallow foundations have been reviewed in this report.

Several IS codeal provisions regarding the construction of shallow foundations on various types

of rocks, permissible settlements and different bearing capacity estimation processes also have been

discussed.

The stress concentration under shallow footing occurs in two directions, parallel and normal to

the major joints. Orientation of joints and joint stiffness governs the stress distribution in rock mass.

The important parameters which governs the behaviour of shallow foundation on jointed rock

mass are internal frictional angle and cohesion of intact rock and joints, quality of rock mass, number

of joint sets, orientation of joints, uniaxial compressive strength of intact rock, joint surface

characteristics, failure mode of rock mass, Hoek-Brown parameters, unit weight of rock, surcharge load

and water table depth.

Therefore, on a conclusive note, the present study finds some fields, mentioned in the previous

section, as grey areas and further research can be carried out in these areas.

61

REFERENCES

1. Agharazi, A., Martin, C. D., and Tannant, D. D. (2012). “A three-dimensional equivalent

continuum constitutive model for jointed rock masses containing up to three random joint sets.”

Geomechanics and Geoengineering, 7(4), 227 – 238.

2. Bindlish, A., Singh, M., and Samadhiya, N. K. (2012). “Ultimate Bearing Capacity of Shallow

Foundations on Jointed Rock Mass.” Indian Geotechnical Journal, 42(September), 169–178.

3. Bindlish, A., Singh, M., and Samadhiya, N. K. (2013). “Modelling of Ultimate Bearing

Capacity of Shallow Foundations resting on Jointed Rock Mass.” Indian Geotechnical Journal,

42(3), 169–178.

4. Bowles, J. E. (1997). Foundation Analysis and Design Fifth Edition. Engineering Geology, The

McGraw-Hill Companies, Inc.

5. Chen, W. F., and R, S. C. (1968). Limit Analysis and Limit Equi Librium. Bethlehem.

6. Clausen, J. (2012). “Bearing capacity of circular footings on a Hoek-Brown material.”

International Journal of Rock Mechanics and Mining Sciences, Elsevier, 57, 34–41.

7. Duncan C. Wyllie. (2005). Foundations on Rock. Taylor & Francis Group.

8. Dunham, L., Valsangkar, A. J., and Schriver, A. B. (2005). “Centrifuge modeling of rigid

square footings on weak jointed rock.” Geotechnical Testing Journal, 28(2), 133–143.

9. Gaziev, E., and Erlikhman, S. (1971). “Stress and strains in anisotropic foundations Gaziev et

al.” Symp. on Rock Fracture, ISRM (Nancy, paper II-1).

10. Goodman, R. E. (1989). Introduction to Rock Mechanics, 2nd Edition.pdf. Willey and Sons.

11. Hoek, E., and Brown, E. T. (1980). “Empirical Strength Criterion for Rock Masses.” Journal

of the Geotechnical Engineering Division.

12. Hoek, E., Carranza-torres, C., and Corkum, B. (2002). “Hoek-brown failure criterion – 2002

edition.” Narms-Tac, 267–273.

13. Hoek, E., Marinos, P., and Marinos, V. (2006). “Variability of the engineering properties of

rock masses quantufied by geological strength index; the case of ophiolities with special

emphasis on tunneling.” Bulletin of Engineering Geology and the Environment, 65(2), 129–

142.

14. Imani, M., Fahimifar, A., and Sharifzadeh, M. (2012). “Upper bound solution for the bearing

capacity of submerged jointed rock foundations.” Rock Mechanics and Rock Engineering,

45(4), 639–646.

15. IS 13063: (1991). Structural Safety of Buildings on Shallow Foundations on Rocks. Bureau of

Indian Standards.

16. IS 6403: (1981). Code of Practice for shallow Foundations. Bureau of Indian Standards, New

Delhi.

17. IS 875: (1987). Code of Practice for Design Loads ( Other Than Earthquake ) for Buildings

and Structures Code of Practice.

62

18. IS12070: (1987). Code of Practice for Design and Construction of Shallow Foundations on

Rocks. Bureau of Indian Standards.

19. Ladanyi, B., and Roy, A. (1971). “Some aspects of bearing capacity of rock mass.” Proceedings

of the Seventh Canadian Symposium on Rock Mechanics, 161 – 190.

20. Lutenegger, A. J., and Adams, M. T. (1998). “Bearing Capacity of Footings on Compacted

Sand.” Fourth International Conference on Case Histories in Getechnical Engineering, 1216

– 1224.

21. Merifield, R. S., Lyamin, a. V., and Sloan, S. W. (2006). “Limit analysis solutions for the

bearing capacity of rock masses using the generalised Hoek-Brown criterion.” International

Journal of Rock Mechanics and Mining Sciences, 43(6), 920–937.

22. Meyerhof, G. G. (1953). “The bearing capacity of concrete and rock*.” Magazine of Concrete

Research, 4(12), 107–116.

23. Oda, M., Yamabe, T., and Ishizuka, Y. (1993). “Elastic stress and strain in jointed rock masses

by means of crack tensor analysis.” Rock Mechanics and Rock Engineering, 26(2), 89 – 112.

24. Prakoso, W. a., and Kulhawy, F. H. (2004). “Bearing Capacity of Strip Footings on Jointed

Rock Masses.” Journal of Geotechnical and Geoenvironmental Engineering, 130(December),

1347–1349.

25. Ramamurthy, T. (2011). Engineering in Rocks for Slopes, Foundations and Tunnels. PHI

Learning Private Limited, New Delhi.

26. Ramamurthy, T., and Arora, V. K. (1994). “Strength predictions for jointed rocks in confined

and unconfined states.” International Journal of Rock Mechanics and Mining Sciences &amp;

Geomechanics Abstracts, 31(I), 9–22.

27. Saada, Z., Maghous, S., and Garnier, D. (2007). “Bearing capacity of shallow foundations on

rocks obeying a modified Hoek-Brown failure criterion.” Computers and Geotechnics,

35(July), 144–154.

28. Sahu, T. (2009). “Analysis of Footings on Upper Surface of Slopes Under Eccentric Inclined

Loads.” IIT Roorkee.

29. Salari-Rad, H., Mohitazar, M., and Rahimi Dizadji, M. (2012). “Distinct element simulation of

ultimate bearing capacity in jointed rock foundations.” Arabian Journal of Geosciences, 6(11),

4427–4434.

30. Serrano, A., and Olalla, C. (1994). “Ultimate Bearing Capacity of Rock Masses.” International

Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 31(2), 93–106.

31. Serrano, A., and Olalla, C. (1996). “Allowable Bearing Capacity of Rock Foundations Using a

Non-linear Failure Criterium.” International Journal of Rock Mechanics and Mining Sciences

& Geomechanics Abstracts, 33(4), 327–345.

32. Serrano, A., Olalla, C., and Gonza, J. (2000). “Ultimate bearing capacity of rock masses based

on the modified Hoek - Brown criterion.” International Journal of Rock Mechanics and Mining

Sciences, 37(6), 1013–1018.

33. Shekhawat, P., and Viladkar, M. N. (2014). “Behaviour of Shallow Foundations on Jointed

Rock Mass.” IIT Roorkee.

63

34. Shukla, D. K., Singh, M., and Jain, K. K. (2014a). “Bearing Capacity Of Footing On Slopping

Anisotropic Rock Mass.” IMPACT: International Journal of Research in Engineering &

Technology (IMPACT: IJRET), 2(6(April)), 217–232.

35. Shukla, D. K., Singh, M., and Jain, K. K. (2014b). “Variation In Bearing Capacity Of Footing

On Slopping Anisotropic Rock Mass.” 2(6 (June)), 85–98.

36. Silvestri, V. (2003). “A limit equilibrium solution for bearing capacity of strip foundations on

sand.” Canadian Geotechnical Journal, 40(Bowles 1988), 351–361.

37. Singh, B. (1973a). “Continuum characterization of jointed rock masses Part I- The Constituve

Equations.” International Journal of Rock Mechanics and Mining Sciences & Geomechanics

Abstracts, 10(October), 311–335.

38. Singh, B. (1973b). “Continuum Characterization of Jointed Rock Masses, Part II- Significance

of Low Shear Modulus.” International Journal of Rock Mechanics and Mining Sciences &

Geomechanics Abstracts, 10(4), 337–349.

39. Singh, M., and Rao, K. S. (2005). “Bearing Capacity of Shallow Foundations in Anisotropic

Non-Hoek–Brown Rock Masses.” Journal of Geotechnical and Geoenvironmental

Engineering, 131(August), 1014–1023.

40. Singh, M., Rao, K. S., and Ramamurthy, T. (2002). “Strength and deformational behaviour of

a jointed rock mass.” Rock Mechanics and Rock Engineering, 35(1), 45–64.

41. Sutcliffe, D. J., Yu, H. S., and Sloan, S. W. (2003). “Lower bound solutions for bearing capacity

of jointed rock.” Computers and Geotechnics, 31(1), 23–36.

42. Yang, X. L., and Yin, J. H. (2005). “Upper bound solution for ultimate bearing capacity with a

modified Hoek-Brown failure criterion.” International Journal of Rock Mechanics and Mining

Sciences, 42(4), 550–560.

43. Yang, X.-L. (2009). “Seismic bearing capacity of a strip footing on rock slopes.” Canadian

Geotechnical Journal, 46(1975), 943–954.

44. Ziaee, A. S. (2014). “Explicit formulation of bearing capacity of shallow foundations on rock

masses using artificial neural networks : application and supplementary studies.”

Environmental Earth Sciences, Volume 73(Issue 7), pp 3417–3431.

64

APPENDIX – I

Table I.1 Literature Review at a Glance

S.

No.

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

1 Pauker

(1889)

BC of shallow strip

footing at certain

depth on heavily

fractured rock mass

Empirical method; failure

planes were assumed under

footing

Footing depth,

density of rock mass,

internal friction angle

Simple expression

developed for bearing

capacity estimation

Failed to predict

bearing capacity at

surface ( 0fD )

2 Terzaghi

(1943)

BC of shallow strip

footing at certain

depth on rock mass

Limit equilibrium method Cohesion, angle of

internal friction, unit

weight of rock, depth

and width of footing,

shape factor

Simple expression

proposed for bearing

capacity evaluation

Stress-strain response

of rock mass is

neglected

3 Meyerhof

(1953)

BC of surface strip

footing on concrete

and rock blocks

Limit equilibrium method;

lab experiments on concrete

and rock blocks

Footing width, block

thickness, cohesion,

friction angle

Equations presented for BC

estimation of strip and

circular footing; splitting

and shearing is dominant

failure mode

Not applicable for

anisotropic rock mass

4 Bishnoi

(1968)

BC of strip footing

on heavily

fractured rock mass

with open vertical

joint set

Empirical method; footing

size with respect to joint

spacing is considered

Friction angle, UCS,

ratio of footing width

to joint spacing

Expression proposed for

bearing capacity evaluation

considering footing size to

joint spacing ratio

Not considered

surcharge loading

65

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year Problem Statement Methodology

Parameters

considered Major Contributions Limitations

5 Coates

(1970)

BC of strip footing on

rock surface

Failure along two planes

was considered

Cohesion, friction

angle, surcharge,

footing width, rock

unit weight

Expression proposed for

BC evaluation

Provide reasonable

results only for friction

angle 0 45

6 Gaziev and

Erlikhman

(1971)

Contact pressure

distribution in jointed

rock mass below

shallow footing

Series of experiments

conducted on rock mass

specimen built with

synthesized plaster blocks

Joint orientation Produce stress

concentration pattern due to

partial loading for different

joint orientations

Only two orthogonal

joint sets were

considered.

7 Ladanyi and

Roy (1971)

BC of strip footing on

rock mass with

inclined joint sets

Assuming the development

of active and passive wedge

below strip footing

Cohesion and friction

angle of joints,

footing width, unit

weight of rock

Expression proposed for

BC evaluation considering

joint dip angle

Considered only two

joint sets; effect of

surcharge load needs

to be considered

separately

8 Singh (1973) Analysis of

deformation and stress

in anisotropic jointed

rock mass

Anisotropic continuum

characterization and

discrete joint model

coupled FEM

Joint stress

concentration factor,

normal and shear

stiffness of joints,

joint spacing, joint

orientation

Validated the continuum

model (Singh 1973b); three

case study were conducted;

deformation and stress

contour in jointed rock

mass below strip footing

Failed to predict

stresses in high stress

gradient regions near

loaded area.

66

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year Problem Statement Methodology Parameters considered Major Contributions Limitations

9 Bray

(1977)

Stress distribution in

jointed rock mass under

surface point load

Assuming rock mass as

equivalent isotropic

medium

Joint orientation, normal

and shear stiffness,

distance of load

application point, E, ν

Provide equations for

stresses and pressure bulbs

in rock mass containing

one joint set

Not considered no. of

joint set, joint spacing,

rock mass quality

10 Bowles

(1988)

BC of strip footing

resting on jointed rock

mass

Based of experimental

data obtained from

various types of rocks

UCS and RQD Expression proposed for

BC evaluation considering

RQD

Approximate results

11 Goodman

(1989)

BC of strip footing on

heavily fractured rock

mass

Assuming crushing of

rock below footing with

confining pressure ci

UCS (ci ) and friction

angle of intact rock

Expression proposed for

BC evaluation

Cohesion and weight

of rock is neglected

12 Oda et al.

(1992)

Analysis of stress fields

below strip footing in

jointed rock mass

Elastic stress strain

relation using crack

tensor; finite element

method

Ratio of normal to shear

stiffness, joint

orientation,

Provide stress strain

relation using crack tensor;

stress distribution pattern

in rock mass below footing

Consider only two

joint sets; Not consider

stiffness ratio of

natural joints

13 Serrano

and Olalla

(1994)

Ultimate BC of spread

foundation on rock

masses subjected to

central vertical load

BC quantification by HB

criterion (1980) and

Characteristics method for

resolving differential

equation system

Type of rock, UCS of

intact rock, RMR

parameter, unit weight

of rock, HB parameters

A new methodology

proposed for Ultimate BC

estimation. Nomograms

presented for BC factor

N calculation

Considered weight less

rock mass and plane

strain condition; Not

validated with

experimental data.

67

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year Problem Statement Methodology Parameters considered Major Contributions Limitations

14 Serrano

and Olalla

(1996)

Allowable BC of spread

Foundation on rock

masses subjected to

central vertical/inclined

load Using A Non-

Linear Failure Criterion

Application of modified

HB criterion (1988).

Statistical analysis of exp.

data. Comparison of

present results with code

values & previous

literature

Type of rock, UCS of

intact rock, RMR

parameter, unit weight

of rock, Spacing ratio of

foundation, scale effect,

slope angle, Safety

factor, HB parameters

Introduced spacing ratio

and scale effect parameter;

Nomograms prepared to

cal. BC for various

situation; Safety factor

values are proposed.

Considered weight less

rock mass and plain

strain condition;

Safety factor values

given for limited

values of RMR and

UCS

15 Serrano

and Olalla

(2000)

Ultimate BC of spread

foundation on heavily

fractured rock mass

subjected to central

vertical/inclined load

Modified HB criterion

(1992) and Characteristics

line method

Type of rock, UCS of

intact rock, RMR

parameter, unit weight,

Spacing ratio, scale

effect, slope angle,

Safety factor, HB

parameters

A modified methodology

proposed for ultimate BC

estimation considering

heavily fractured rock;

Nomograms presented for

BC factor N calculation

Plain strain condition,

weightless rock mass,

absence of initial force

16 Sutcliffe,

Yu, Sloan

(2003)

Plane strain BC of

surface strip footing on

jointed rock mass with

one, two and three joint

sets subjected to central

vertical load

FEM formulation of

lower bound limit

theorem; Mohr-Coulomb

yield criterion

cohesive and frictional

strength of joint sets,

orientation of joints and

relative angle between

joints

BC solution of rock mass

with one, two and three

joint sets; effect of

different parameter on BC

variation; comparison with

Alehossein et al. (1992)

and Davis (1980)

Linearized yield

criterion, not

applicable for heavily

jointed or intact

isotropic rock mass.

68

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

17 Prakoso,

Kulhawy

(2004)

BC of strip footing

on jointed rock

mass with one and

two joint sets

subjected to central

vertical load

Lower bound BC model

(Bell’s model 1915) with

Mohr-Coulomb yield

criterion

Cohesion and friction

angle of both rock

material and joints;

no. and orientation of

joint sets

A new methodology

proposed for BC

calculation; effect of

strength parameters on BC

presented; solutions

compared with Alehossein

et al. (1992) and Yu &

Sloan (1994)

Rock mass weight,

embedment and joint

set spacing were not

considered; not

applicable for heavily

jointed or intact rock

mass.

18 Singh and

Rao (2005)

Bearing capacity

(BC) of shallow

strip footing on

Anisotropic Non-

Hoek-Brown (HB)

rock mass subjected

to vertical load

Lower bound limit analysis

using Bell’s approach.

Parabolic strength criterion

with modified fJ concept

No of joint sets, joint

spacing, friction

angle along joint

planes, UCS of intact

rock, footing depth

A bearing capacity (BC)

calculation procedure was

proposed. Charts presented

to simplify BC

computation.

Not applicable for

columnar blocks or

blocks with one joint

set. Block size < 0.2 of

footing. Not validated

with experimental data

19 Yang and

Yin (2005)

BC of shallow strip

footing on

horizontal HB rock

mass subjected to

central vertical load

Upper bound limit analysis

with generalized tangential

technique; HB criterion;

multi wedge translational

failure mechanism

HB parameter, GSI,

surcharge load,

disturbance factor,

unit weight

BC solution; parametric

study; tables for BC factor

estimation of five rock

types; solutions compared

with Collins et al (1988)

Plain strain solution;

not applicable for

anisotropic rock mass

69

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

20 Merifield,

Lyamin and

Sloan

(2006)

BC of surface strip

footings on jointed

rock mass

subjected to central

vertical load

Numerical limit analysis

with FEM; generalized HB

criterion(2002)

Unit weight, UCS,

GSI, footing width,

HB parameters

Solution are within 2.5% of

collapse load; Charts

presented BC factor N

estimation; solutions

compared with Kulhawy

and Karter (1992) and

Serrano et al. (2000)

Unrealistic values

considered for

different parameters in

yield criterion

approximation; not

applicable for

anisotropic rock mass

21 Hoek and

Marinos

(2007)

Non-linear yield

criterion for

isotropic (intact or

heavily fractured)

rock mass

Empirical strength

envelope originated by

linear regression analysis of

large number of

experimental data

Hoek-Brown

parameters ,bm s and

exponent a , GSI,

disturbance factor

Simple non-linear strength

or failure criterion

presented the strength

behaviour of isotropic rock

mass

Effect of intermediate

principal stress is

neglected; Not

applicable for

anisotropic rock mass

22 Saada,

Maghous,

Garnier

(2007)

BG of shallow strip

foundation on

Hoek-Brown rock

mass subjected to

central vertical load

Kinematical approach of

upper bound limit analysis

with generalized HB

criterion; two failure

mechanisms: generalized

Prandlt type and multi

wedge translation failure

Hoek-Brown

parameters, GSI,

Disturbance factor,

surcharge load, unit

weight

A BC estimation method

proposed; effect of

surcharge and unit weight

on BC investigated;

solutions compared with

Yang and Yin (2005)

Merifield et al. (2006)

Not applicable for

anisotropic rock mass

with few joint sets

70

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

23 Saada,

Maghous,

Garnier

(2007)

BC of strip footing

on jointed rock

mass near slopes

under vertical and

horizontal seismic

load

Kinematical approach of

upper bound limit analysis;

HB criterion; pseudo-static

approach for seismic

analysis; four different

failure mechanisms used

HB parameters, GSI,

surcharge load, unit

weight, horizontal

seismic load, slope

angle, distance from

slope edge

Seismic BC solution in

terms of BC factors; Tables

for BC factor estimation;

effect of geometric,

strength and load

parameters studied;

Variation of seismic

force and direction is

not considered; plain

strain solution; not

applicable for

anisotropic rock mass

24 Yang

(2009)

BC of strip footing

on Hoek-Brown

rock mass at slope

edge under vertical

load and horizontal

seismic load

Upper bound limit analysis

with generalized tangential

technique; HB criterion;

multi wedge translational

failure mechanism; pseudo-

static seismic analysis

HB parameter, GSI,

surcharge load, unit

weight, horizontal

seismic coefficient,

slope angle, BC

factor for static and

seismic conditions

Tables presented for

Seismic BC factor

estimation for five rock

types; effect of seismic

coefficient and slope angle

on critical slip surfaces

investigated

Plain strain solution;

not applicable for

anisotropic rock mass;

Variation shearing

strength due to earth

quake is not

considered

25 Labuz and

Zang

(2012)

Mohr-Coulomb

yield criterion for

isotropic rock mass

Empirical linear strength

envelope developed from

Mohr (1900) and Coulomb

(1776) failure envelopes

Cohesion c and

angle of internal

friction

Simple linear strength or

failure criterion; clear

physical significance of

parameters c and Ф

Intermediate principal

stress is neglected; Not

applicable for

anisotropic rock mass

71

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

26 Imani et al.

(2012)

BC of submerged

jointed rock

foundations

subjected to central

vertical load

Upper bound limit analysis;

DEM analysis of failure

analysis; spacing ratio

Joint spacing, c &

of rock material and

joints, dry and

submerged unit

weight, water table

depth, joint dip angle

Submerged BC equation

presented; parametric study

conducted; results

compared with Hansen et

al. (1987) and Ausilio and

Conte (2005)

Not applicable for

anisotropic or heavily

fractured rock mass;

27 Bindlish,

Singh and

Samadhiya

(2012)

Ultimate BC of

surface square

footing on non-HB

rock mass subjected

to vertical load

No. of experiments

conducted on synthetic rock

mass. Results compared

with the method of Singh

and Rao (2005)

Rock properties,

joint intensity, type

of structure,

Joint orientations,

interlocking

conditions.

A modified BC calculation

procedure was proposed for

rock mass with continuous

& discontinuous joint sets

Not applicable for

columnar blocks or

blocks with one joint

set. Block size < 0.2 of

footing. Moisture

effect of is not

accounted

28 Salari-Rad

et al. (2012)

Modelling of

Ultimate BC of

Shallow Footing on

jointed rock mass

containing single

joint set subjected

to vertical load

DEM modelling by UDEC

in plane strain condition.

Rock material and joint

modelled by HB and MC

criterion

failure mechanisms,

Dip angle, shear

strength of joint sets

and other properties

of jointed rock mass

Due to single joint set BC

varies upto 27 - 86 % of

intact rock.

BC of rock mass reduces

with decreasing joint shear

strength

2-D modelling

method. Only single

joint set considered.

Not consider joint

frequency. Not

validated with

experimental data

72

Table – I.1 Literature Review at a Glance (contd.)

Sl.

No

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

29 Clausen

(2012)

BC of circular

surface footings on

jointed rocks

subjected to central

vertical load

Numerical coding by

displacement FEM with

convergence extrapolation

scheme; HB failure

criterion (2006)

HB parameters, GSI,

rock mass unit

weight, UCS, radius

of footing, BC

factors

Sensitivity analysis; BC

equations presented;

parametric study was done;

nomograms presented;

solutions compared with

Hoek et al. (2002)

Serrano and Olalla (2002)

Not applicable for

anisotropic jointed

rock mass; results not

compared with any

field data

30 Agharazi et

al. (2012)

Stress and

deformation

analysis of rock

mass containing up

to three joint sets

MC criterion, 3D

constitutive model based on

superposition of REV

components deformation;

FLAC3D

Joint orientation,

stiffness, cohesion,

friction angle, shear

modulus

Provide a 3D constitutive

model for rock mass up to

three joint sets; stress

distribution in jointed rock

in plate load test

This constitutive

model is not applicable

when footing size is

small relative to joint

spacing

31 Bindlish,

Singh and

Samadhiya

(2013)

Modelling of

Ultimate BC of

Shallow Footing on

non-HB rock mass

subjected to vertical

load

DEM modelling using

UDEC, programming with

FISH. UDEC model results

compared with the

experimental results of

Bindlish et al. (2012)

Rock properties, joint

intensity, type of

structure,

Joint orientations,

interlocking

conditions

UDEC model predicted BC

reasonably when failure

through pre-existing

fractures or continuous

joints

Failed to predict BC

when failure through

intact rock due to

initiation of new

fractures

73

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

32 Justo

(2013)

Case study of raft

foundation of high

rise building; BM

and deformation

calculation of the

raft foundation

FEM modelling using

PLAXIS 3D; analytical

calculation using rock

classification systems and

three material models:

elastic, MC and hardening

model

Rock classification

systems, elasticity

modulus, UCS, unit

weight, MC

parameters

FEM modelling of raft

footing; sensitivity analysis

for mess fineness;

hardening model is most

effective in settlement

estimation; variation of BM

with elasticity modulus

observed; results compared

with field data

---

33 Shukla,

Singh and

Jain (2014,

2014)

BC of strip footing

at edge and at a

distance from edge

of slopping

anisotropic rock

mass subjected to

vertical load

No. of experiments

conducted on synthetic rock

mass. Results compared

with the BC obtained by

Euler’s buckling theory

with modified fJ concept

Joint angle with

principal axis, joint

frequency, joint

strength, UCS,

elasticity modulus

and failure mode of

rock masses

A new method proposed for

BC estimation. Joint angle

edge distance and failure

mode is dominating

parameter. Resistance of

buckling > sliding for

continuous jointed rock

mass

Not applicable for

heavily jointed rock

mass or rock mass

with discontinuous

joints

74

Table I.1 Literature Review at a Glance (contd.)

S.

No.

Author /

Year

Problem

Statement Methodology

Parameters

considered Major Contributions Limitations

34 Ziaee et al.

(2014)

BC of surface strip

foundation on rock

mass with vertical

joint set subjected

to central vertical

load

Statistical analysis of large

no. of experimental data by

ANN based on fixed

connection weights and

bias factors; parametric

study by Garson’s

algorithm

RMR, UCS, ratio of

joint to footing

width, internal

frictional angle

A new BC equation

presented; sensitivity and

parametric analysis

conducted; BC is more

sensitive to UCS and ratio

of joint to footing width;

results compared

experimental data and

Goodman (1989)

Effect of shape of

footing and anisotropic

behaviour of

foundation rock mass

was not considered

35 Shekhawat

and

Viladkar

(2014)

BC, settlement and

tilt behaviour of

surface strip

footing on HB rock

mass subjected to

eccentric and

inclined load

Numerical FEM analysis

with modified HB criterion;

double tangent method

(Luteneger and Adams

1989)

HB parameters, GSI,

UCS, Disturbance

factor, load

eccentricity and

inclination, UBC,

Settlement and tilt of

footing

Pressure-settlement

characteristics of surface

strip footing studied; Non-

dimensional correlations

developed for BC,

settlement and tilt

estimation; results validated

with results of Kulhawy &

Carter (1992) and Serrano

(2000)

Not applicable for

anisotropic rock mass;

weight of rock mass is

not considered;

unrealistic initial

values of various

parameters are

considered in FEM

modelling

75