3DF2 5 Validation 3D Plaxis

PLAXIS 3D FOUNDATION

Validation Manual Version 2

TABLE OF CONTENTS

i

TABLE OF CONTENTS

1 Introduction..................................................................................................1-1

2 Soil model problems with known theoretical solutions.............................2-1 2.1 Bi-axial test with linear elastic model....................................................2-1 2.2 Bi-axial shearing test with linear elastic model .....................................2-3 2.3 Bi-axial test with mohr-coulomb model ................................................2-5 2.4 Triaxial test with hardening soil model..................................................2-7 2.5 Phi-c reduction and comparison with Bishop’s method.........................2-9

3 Elasticity problems with known theoretical solutions ..............................3-1 3.1 Strip footing on elastic Gibson soil........................................................3-1 3.2 Flexible tank foundation on elastic saturated soil ..................................3-4

4 Plasticity problems with theoretical collapse loads...................................4-1 4.1 Bearing capacity of strip footing............................................................4-1 4.2 Bearing capacity of a circular footing....................................................4-4

5 Consolidation................................................................................................5-1 5.1 One-dimensional consolidation..............................................................5-1

6 Structural element problems ......................................................................6-1 6.1 Bending of floor elements......................................................................6-1 6.2 Bending of wall elements.......................................................................6-3 6.3 Bending of shell elements ......................................................................6-5 6.4 Bearing capacity of ground anchors.......................................................6-8 6.5 Performance of springs ........................................................................6-11 6.6 Performance of interface elements in soil with constant cohesion.......6-12 6.7 Performance of interface elements in soil with varying cohesion........6-15

7 Single pile and pile group in overconsolidated clay ..................................7-1 7.1 Introduction............................................................................................7-1 7.2 Numerical simulation of the single pile behaviour (pile load test) ........7-1

7.2.1 Geometry of the model ..............................................................7-3 7.2.2 Material properties .....................................................................7-5 7.2.3 Modelling the single pile............................................................7-6

7.3 Numerical simulation of the pile group action.......................................7-8 7.3.1 Effect of initial stresses ............................................................7-13

7.4 Conclusions..........................................................................................7-14

8 Validation of embedded piles – the Alzey bridge pile load test................8-1 8.1 General...................................................................................................8-1 8.2 Finite element model..............................................................................8-2 8.3 Material properties .................................................................................8-3

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ii PLAXIS 3D FOUNDATION

8.4 Results ................................................................................................... 8-5 8.5 Discussion and conclusions ................................................................... 8-7

9 Piled raft foundation in Frankfurter clay..................................................9-1 9.1 Introduction ........................................................................................... 9-1 9.2 Frankfurt subground and methodology to develop the piled raft........... 9-1 9.3 Example of a high-rise building on Frankfurt subsoil ........................... 9-2 9.4 Geometry ............................................................................................... 9-2 9.5 Loads ..................................................................................................... 9-3 9.6 Numerical model ................................................................................... 9-5

9.6.1 Soil Parameters ..........................................................................9-5 9.6.2 3D Finite element model............................................................9-6 9.6.3 Calculations ...............................................................................9-7

9.7 Inspect output ........................................................................................ 9-8 9.8 Conclusion and Outlook ...................................................................... 9-10

10 Application of the ground anchor facility................................................10-1 10.1 Introduction ......................................................................................... 10-1 10.2 Deep excavation with prestressed ground anchors .............................. 10-3 10.3 Results ................................................................................................. 10-5 10.4 Comparison of 3D results with 2D reference solution......................... 10-7 10.5 Conclusions ......................................................................................... 10-9

11 References ..................................................................................................11-1

INTRODUCTION

1-1

1 INTRODUCTION

The performance and accuracy of PLAXIS 3D FOUNDATION has been carefully tested by carrying out analyses of problems with known theoretical solutions. A selection of these benchmark analyses is described in Chapters 2 to 6. PLAXIS 3D FOUNDATION has also been used to carry out predictions and back-analysis calculations of the performance of full-scale structures as additional checks on performance and accuracy.

Soil model problems: A selection of soil model problems with known theoretical solutions is presented in Chapter 2.

Elastic benchmark problems: A large number of elasticity problems with known exact solutions is available for use as benchmark problems. A selection of elastic calculations is described in Chapter 3; these particular analyses have been selected because they resemble the calculations that PLAXIS might be used for in practice.

Plastic benchmark problems: A series of benchmark calculations involving plastic material behaviour is described in Chapter 4. This series includes the calculation of collapse loads for two different footings. As for the elastic benchmarks, only problems with known exact solutions are considered.

Consolidation: The performance of consolidation has been verified with the classical one-dimensional problem in Chapter 5.

Structural element problems: In Chapter 6 the performance of structural elements has been verified with known theoretical solutions.

Case studies: PLAXIS has been used extensively for the prediction and back-analysis of full-scale projects. This type of calculations may be used as a further check on the performance of PLAXIS provided that good quality soil data and measurements of structural performance are available. Some such projects are published in the PLAXIS Bulletin, on the internet site: http://www.plaxis.nl and are available at PLAXIS. Four validation examples can be found in the last chapters in this manual.

VALIDATION MANUAL

1-2 PLAXIS 3D FOUNDATION

SOIL MODEL PROBLEMS WITH KNOWN THEORETICAL SOLUTIONS

2-1

2 SOIL MODEL PROBLEMS WITH KNOWN THEORETICAL SOLUTIONS

A series of calculations is described in this chapter. In each case the analytical solutions may be found in many of the various textbooks on elasticity solutions, for example Giroud (1972) and Poulos & Davis (1974).

2.1 BI-AXIAL TEST WITH LINEAR ELASTIC MODEL

Input: A bi-axial test is conducted on a volume of 1x1x1 m as shown in Figure 2.1. The bottom-left is fixed in all direction and the front, left and rear planes are fixed horizontally.

σ1

σ1

σ2

Figure 2.1 Bi-axial test geometry

The lateral pressure σ2 is represented by a distributed load on the right plane. The axial load σ1 is represented by a distributed load on the top and bottom plane. The density γ is set to zero, the remaining properties of the soil are:

E = 1000 kN/m2 ν = 0.25

The sample is subjected to the following loading tests: lateral loading of σ2 = -1 kN/m2, axial loading of σ1 = -1 kN/m2 and bi-axial loading of σ1 = σ2 = -1 kN/m2.

VALIDATION MANUAL


Output: The displacement of the upper right corner for the three loading tests is:

Phase 1: ux = 0.9375 mm, uy = -0.3125 mm, uz = 0 mm

Phase 2: ux = 0.3125 mm, uy = -0.9375 mm, uz = 0 mm

Phase 3: ux = uy = -0.625 mm, uz = 0 mm

Since a block of unit length is considered, the values of these displacement components are equal to the strains in corresponding directions.

Verification: The theoretical solution of the amount of strain is:

( )( )E

zzyyxxxx

σσνσε

+−=

( )( )E

zzxxyyyy

σσνσε

+−=

( )( ) ( )yyxxzzyyxxzz

zz Eσσνσ

σσνσε +=→=

+−= 0

The theoretical strain is the following in each phase:

Test 1: σxx = -1 kN/m2 σyy = 0 kN/m2 σzz = -0.25 kN/m2

εxx = -0.9375⋅10-3 εyy = 0.3125⋅10-3 εzz = 0 Test 2:

σxx = 0 kN/m2 σyy = -1 kN/m2 σzz = -0.25 kN/m2

εxx = 0.3125⋅10-3 εyy = -0.9375⋅10-3 εzz = 0

Test 3: σxx = -1 kN/m2 σyy = -1 kN/m2 σzz = -0.5 kN/m2

εxx = -0.625⋅10-3 εyy = -0.625⋅10-3 εzz = 0

Theoretical and calculated values are in agreement with each other.


2-3

2.2 BI-AXIAL SHEARING TEST WITH LINEAR ELASTIC MODEL

Input: A bi-axial shearing test is conducted on a volume with the same properties as given in Section 2.1. The sample is subjected to a shear loading of 1 kN/m2 as shown in Figure 2.2. Additionally, the line (1, 0) – (1, 1) in the y = -1 m plane is fixed in y- and z-directions.

Figure 2.2 Bi-axial shearing test initial geometry (top) and result (bottom)

Output: The resulting deformations are shown in Figure 2.2, the shear strain is 2.5⋅10-3.

VALIDATION MANUAL


Verification: The shear modulus is equal to:

( )2kN/m400

5.21000

12==

+=

υEG

and the shear strain is:

31 2.5 10400

xyxy G

σγ −= = = ⋅

The computational results are in agreement with the theoretical solution.


2-5

2.3 BI-AXIAL TEST WITH MOHR-COULOMB MODEL

Input: A bi-axial test is conducted on a volume identical to the one presented in section 2.1. The material behaviour is now modelled by means of the Mohr-Coulomb model. The confining pressure σ2 is represented by vertical distributed load on the right side plane. The axial load σ1 is represented by distributed loads on top and bottom planes. The density γ is set to zero, the remaining model parameters are:

E = 1000 kN/m2 ν = 0.25

c = 1 kN/m2 ϕ = 30°

The sample is subjected to the following loading scheme: bi-axial loading of σ1 = σ2 = -1 kPa, axial loading of σ1 = -2 kPa and further axial loading to σ1 = -10 kPa. A tolerated error of 0.001 is used.

Output: The soil fails at an axial stress σ1 = -6.48 kN/m2 as shown in Figure 2.3.

2 4 6 8 O

-1

-2

-3

-4

-5

-6

-7

eps-y [‰]

sig'-y [kN/m2]

Figure 2.3 Results of the bi-axial loading test with the Mohr-Coulomb model, axial stress versus axial strain

VALIDATION MANUAL


Verification: The theoretical solution to the failure of the sample is given by the Mohr–Coulomb criterion:

0cossin22

2121 =⋅−⋅+

+−

= ϕϕσσσσ

cf

Failure occurs in compression at:

46.6sin1

cos2sin1sin1

21 −=−

⋅−−+

⋅=ϕ

ϕϕϕσσ c kN/m2

The error in the numerical solution is therefore 0.3 %.


2-7

2.4 TRIAXIAL TEST WITH HARDENING SOIL MODEL

Input: A triaxial test is conducted on a volume of 1x1x1 m as shown in Figure 2.4. The soil behaviour is modelled by means of the Hardening Soil model. The bottom-left is fixed in all directions and the left and rear planes are fixed horizontally. The pressure σ2 is represented by a distributed load on the right plane and σ3 is represented by a distributed load on the front plane. The axial load σ1 is represented by a distributed load on the top and bottom planes. The density γ and ν are set to zero, the remaining model parameters are:

450 100.2 ⋅=refE kN/m2 4100.2 ⋅=oedE kN/m2 4100.6 ⋅=ref

urE kN/m2

1' =refc kN/m2 °= 35'ϕ °= 5'ψ

σ 1

σ 1

σ 2σ 3

Figure 2.4 Triaxial test geometry

The sample is subjected to the following loading: isotropic loading to -100 kN/m2, (after which displacements are reset to zero), axial compression until failure and axial extension until failure.

Output: The triaxial sample fails at σ1 = -373.3 kN/m2 in compression and at σ1 = -26.4 kN/m2 in extension as can be seen in Figure 2.5.

VALIDATION MANUAL


-60 -30 0 30 60 90 0

-100

-200

-300

-400

eps-y [‰]

sig'-y [kN/m2]

Extension Compression

-26 kPa

-373 kPa

Figure 2.5 Compression and extension results of triaxial test with the Hardening Soil model, axial stress versus axial strain

Verification: The theoretical solution to the failure of the sample is given by the Mohr-Coulomb criterion:

0cossin22

3131 ≤⋅−⋅+

+−

= ϕϕσσσσcf

so that failure occurs in compression at:

9.372sin1

cos2sin1sin1

31 −=−

⋅−−+

⋅=ϕ

ϕϕϕσσ c kN/m2

and failure occurs in extension at:

1.26sin1

cos2sin1sin1

31 −=+

⋅++−

⋅=ϕ

ϕϕϕσσ c kN/m2

The calculated and theoretical values are in good agreement with each other.


2-9

2.5 PHI-C REDUCTION AND COMPARISON WITH BISHOP’S METHOD

Figure 2.6 Geometry of the embankment

Input: In this chapter the stability of an embankment is calculated by means of phi-c reduction. The situation is compared with a 2D calculation and with Bishop’s slip circle method (see for example Verruijt, 1983). The embankment has a slope of 1:2, a height of 4.5 m and a width of 9.0 m. A load is applied to an area of 3.0x1.0 m on top of the embankment (Figure 2.6). The Mohr-Coulomb model is used and the unit weight γ is set to 16 kN/m3. The remaining properties of the soil are:

E = 2600 kN/m2 υ = 0.3

c = 5 kN/m2 φ = 20° The initial stresses are generated using gravity loading. Then the embankment is subjected to the following analyses:

• Phi-c reduction without additional loading • Phi-c reduction after external loading of 30 kPa • Applying an external load of 100 kPa to simulate failure

Output: The initial safety factor without external loading is 1.55, the safety factor with external loading to 30 kPa is found to be 1.23 (Figure 2.7). The embankment fails at an external load of 59 kPa.

VALIDATION MANUAL


Figure 2.7 Load-displacement curve

Verification: The results of PLAXIS 2D and 3D calculations are very similar. The safety factors are compared with Bishop’s slip circle method (Figure 2.8). From the Bishop’s slip circle method a safety factor of 1.56 is obtained for the initial situation. This value agrees with the PLAXIS calculation.

Smallest Safety FactorBishop F= 1.564

Figure 2.8 Bishop’s slip circle method result

Input: In addition safety factors are calculated for different situations where the load is only applied partially in order to see the influence of 3D effects. The following areas have been subsequently loaded to 30 kPa: 3x3 m, 3x6 m, 3x12 m and 3x18 m (Figure 2.9).


2-11

Figure 2.9 Incremental displacements after Phi-c reduction for the different loading areas

Figure 2.10 Load-displacement curve

The safety factor decreases with increasing load as expected (see Figure 2.10). The situation in which an area of 3x18 m is loaded is comparable to the situations as considered in the first part of this section.

VALIDATION MANUAL


ELASTICITY PROBLEMS WITH KNOWN THEORETICAL SOLUTIONS

3-1

3 ELASTICITY PROBLEMS WITH KNOWN THEORETICAL SOLUTIONS

A series of elastic benchmark calculations is described in this Chapter. In each case the analytical solutions may be found in many of the various textbooks on elasticity solutions, for example Giroud (1972) and Poulos & Davis (1974).

3.1 STRIP FOOTING ON ELASTIC GIBSON SOIL

Input: Figure 3.1 shows the 3D mesh and the soil data for a ‘plane strain’ calculation of the settlement of a strip load on Gibson soil. (Gibson soil is an elastic layer in which the shear modulus increases linearly with depth). Using z to denote depth, the shear modulus, G, used in the calculation is given by: G = 100 · z. With a Poisson’s ratio of 0.495, the Young’s modulus varies by: E = 299 · z. In order to prescribe this variation of Young’s modulus in the material properties window the reference value of Young’s modulus, Eref, is taken very small and the Advanced option is selected from the Parameters tab sheet. The reference level yref is entered as 0.0 m, being the top of the geometry.

1.0 m

7.0 m

Gibson soil

υ = 0.495

G = 100 z kN/m2

4.0 m

1.0 m 1/2 B

p = 10 kN/m2

z

Figure 3.1 Problem geometry

Output: An exact solution to this problem is only available for the case of a Poisson’s ratio of 0.5; in the PLAXIS calculation a value of 0.495 is used for the Poisson’s ratio in order to approximate this incompressibility condition. The numerical results show an almost uniform settlement of the soil surface underneath the strip load as can be seen from the displacement shadings plot in Figure 3.2. Figure 3.3 shows the shadings of the total stresses. The computed settlement is 46.4 mm at the centre of the strip load.

VALIDATION MANUAL


Figure 3.2 Vertical displacement shadings

Figure 3.3 Total stresses in soil


3-3

Verification: The analytic solution is exact only for an infinite half-space, whereas the PLAXIS solution is obtained for a layer of finite depth. However, the effect of a shear modulus that increases linearly with depth is to localise the deformations near the surface; it would therefore be expected that the finite thickness of the layer has only a small effect on the results. The exact solution for this particular problem, as given by Gibson (1967), gives a uniform settlement beneath the load of magnitude:

Settlement = α2p

with α = 100 for this case. The exact solution for this case gives a settlement of 50 mm. The numerical solution is 7% lower than the exact solution, which is partly due to the finite depth. If, for instance, the thickness of the soil layer is increased to 100 m, the settlement calculated by PLAXIS becomes 49 mm and the error is only 2%.

VALIDATION MANUAL


3.2 FLEXIBLE TANK FOUNDATION ON ELASTIC SATURATED SOIL

Problem: In this case a flexible tank on elastic saturated soil is tested. The test includes the verification of the settlement of the centre of the tank for the condition of homogeneous, isotropic soil of finite depth.

Input: The dimensions of the tank used in the test calculation are shown in Figure 3.4. The tank is founded on a homogeneous, isotropic soil of infinite depth. The tank will impose a pressure difference in the soil of ∆q = 263.3 kN/m2. The remaining soil properties are:

E = 95.8 MN/m2 ν = 0.499

Figure 3.4 Flexible tank foundation

Output: The vertical settlement of the surface at the centre of the tank, calculated by PLAXIS, is 73.6 mm. A coarse mesh has been used for this calculation. The vertical displacements and the deformed mesh are shown in the Figure 3.5 and Figure 3.6.


3-5

Figure 3.5 Vertical displacements

Figure 3.6 Deformed mesh

VALIDATION MANUAL


Verification: The settlement at the centre of the tank is given by:

EqRI

u py

Δ=

Where Ip is the influence coefficient, which can be determined with Figure 3.7.

Figure 3.7 Influence coefficients for settlement under uniform load over circular area

The settlement at the centre of the tank is therefore:

074.010008.95

15.135.233.263=

⋅⋅⋅

=yu m

This is in good correspondence with the numerical value from PLAXIS 3D FOUNDATION.

PLASTICITY PROBLEMS WITH THEORETICAL COLLAPSE LOADS

4-1

4 PLASTICITY PROBLEMS WITH THEORETICAL COLLAPSE LOADS

Two footing collapse problems involving plastic material behaviour are described in this chapter. The first involves a strip footing on a cohesive soil with strength increasing linearly with depth and the second involves a smooth square footing on a frictional soil.

4.1 BEARING CAPACITY OF STRIP FOOTING

Problem: In practice it is often found that clay type soils have a strength that increases with depth. This type of strength variation is particularly important for foundations with large physical dimensions. A series of plastic collapse solutions for rigid plane strain footings on soil with strength increasing linearly with depth, has been derived by Davis and Booker (1973). These solutions may be used to verify the performance of PLAXIS for this class of problems.

Input: The dimensions and material properties used in the test calculation are shown in Figure 4.1. In fact, only half of the symmetric problem is modelled. The cohesion at the soil surface, cref, is taken to be 1 kN/m2 and the value of the cohesion gradient in the advanced settings, cincrement, is 2 kN/m2/m, using a reference level, yref = 0 m (= top of the layer). The stiffness at the top is given by Eref = 299 kN/m2 and the increase of stiffness with the depth is defined by Eincrement = 598 kN/m2/m. Calculations are carried out for the case of a rough (x- and z-direction are fixed) and a smooth footing (x- and z-direction are free).

1.0 m

4.0 m

No tension cut-off

υ = 0.495

G = 100 c kN/m2

2.0 m

1.0 m 1/2 B

p

φ = 0 °

clayer

c 5 kN/m2

1 kN/m2


Output: The calculated maximum average vertical stress under the smooth footing is 7.82 kN/m2, giving a bearing capacity of 15.6 kN. For the rough footing this is 9.28 kN/m2, giving a bearing capacity of 18.6 kN. The computed load-displacement curves are shown in Figure 4.2. The deformed mesh for the smooth footing is shown in Figure 4.3.

VALIDATION MANUAL


10 20 30 40 50 O

2

4

6

8

10

Uy [mm]

sig'-y [kN/m2]

Figure 4.2 Stress-displacement curves

Verification: The analytical solution derived by Davis & Booker (1973) for the mean ultimate vertical stress beneath the footing, pmax, is:

( ) ⎥⎦

⎤⎢⎣

⎡++==

42max

depthlayer

Bcc

BFp πβ

Where B is the footing width and β is a factor that depends on the footing roughness and the rate of increase of clay strength with depth. The appropriate values of β in this case are 1.27 for the smooth footing and 1.48 for the rough footing. The analytical solution therefore gives average vertical stresses at collapse of 7.8 kN/m2 for smooth footing and 9.1 kN/m2 for the rough footing. These results indicate that the errors in the PLAXIS solution are 0.3% and 2% respectively.

Directional dependence: In addition the infinite long strip is modelled along the x-axis with the same parameters. The deformed mesh is shown in Figure 4.4. The results are exactly the same as obtained from the above calculation with the strip modelled along the z-axis. There is no directional dependency.


4-3

Figure 4.3 Deformed mesh (smooth) when the strip is modelled along the z-axis.

Figure 4.4 Deformed mesh (smooth) when the strip is modelled along the x-axis.

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4.2 BEARING CAPACITY OF A CIRCULAR FOOTING

Input: Figure 4.5 shows the mesh and material data for a smooth rigid circular footing with a radius of 1 m on a frictional soil. The thickness of the soil layer is taken to be 4 metres and the material behaviour is represented by the elasto-plastic Mohr-Coulomb model. The footing is represented by a distributed load on a plate with high flexural rigidity, but low normal stiffness. Around the footing an interface has been modelled, extending 0.5 metres below the footing. A virtual thickness of 0.3 metres has been assigned to this interface. During the ultimate level 3D plastic staged construction calculation the load is increased until failure.

footing

load

1.0 m

4.0 m E = 2400 kN/m2

c = 1.6 kN/m2

υ = 0.20

φ = 30°

x

y

5.0 m

5.0 m

z

x z

γ= 16 kN/m3


Output: The load-displacement curve for the footing is shown in Figure 4.6. The final vertical load at failure is 227 kN/m2. During the calculation a higher vertical load of 242 kN/m2 is obtained and the final, lower, collapse load is only obtained if sufficient additional calculation steps are permitted. For this calculation a total of 1000 calculation steps have been used. Figure 4.7 shows the absolute displacement shadings at failure.


4-5

0 2 4 6 8 10 12 0

50

100

150

200

250

|U| [m]

Load [kN/m2]

Figure 4.6 Load –displacement curve

Figure 4.7 Absolute displacement shadings at failure

Verification: The exact solution for this collapse load problem for a circular footing is derived by Cox (1962). For γ⋅R/c = 10 and ϕ = 30°. Cox presents the exact solution:

Pmax = 141 ⋅ c = 141 ⋅ 1.6 = 225.6 kN/m2

The relative error of the end result calculated with PLAXIS is less than 1%.

VALIDATION MANUAL


CONSOLIDATION

5-1

5 CONSOLIDATION

In this Chapter, the results of a one-dimensional consolidation analysis in PLAXIS 3D FOUNDATION are compared to an analytical solution.

5.1 ONE-DIMENSIONAL CONSOLIDATION

Input: Figure 5.1 shows the finite element mesh for the one-dimensional consolidation problem. The thickness of the layer is 1.0 m. The layer surface (upper side) is allowed to drain while the other sides are kept undrained by imposing closed consolidation boundary conditions. These are the standard boundary conditions in PLAXIS 3D FOUNDATION. An excess pore pressure, p0, is generated by using undrained material behaviour and applying an external load p0 in the first (plastic) calculation phase. In addition, ten consolidation analyses are performed to ultimate times of 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10, 20, 50 and 100 days respectively.

H

p0

Figure 5.1 Problem geometry and finite element mesh

Output: Figure 5.2 shows the calculated relative excess pore pressure versus the relative vertical position as marked. Each of the above consolidation times is plotted. Figure 5.3 presents the development of the relative excess pore pressure at the (closed) bottom.

E = 1000 kN/m2 ν = 0.0 k = 0.001 m/day γw = 10 kN/m3 H = 1.0 m

VALIDATION MANUAL


0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Relative excess pore pressure p / p0

Rel

ativ

e ve

rtic

al p

ositi

on y

/ H 0,01

0,02

0,05

0,1

0,2

0,5

1

2

Figure 5.2 Development of excess pore pressure as a function of the sample height

1e-2 0.1 1 10 100 0

0.2

0.4

0.6

0.8

1

Time [day]

Relative Excess Pore Pressure p/p0

Figure 5.3 Development of excess pore pressure at the bottom of the sample as a function of time

HtcT v2=

γ w

oedv

kEc =

( )( )( )ν-+ν

E-νEoed 2111

=

T=

CONSOLIDATION

5-3

Verification: The problem of one-dimensional consolidation can be described by the following differential equation for the excess pore pressure p:

2

2

z pc

t p

v ∂∂=

∂∂

where:

( )( )( ) yHz EE

kEc oed

w

oedv −=

−+−

==νν

νγ 211

1

The analytical solution of this equation, i.e. the relative excess pore pressure, p / p0 as a function of time and position is presented by Verruijt (1983):

( ) ( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛

−−

=−∞

∑ Htc

π j- Hy π j-

j

πtz

pp v

j

j=2

22

1

10 412exp

212cos

1214,

This solution is presented by the continuous lines in Figure 5.2. It can be seen that the numerical solution is close to the analytical solution, but has two distinct points of difference. First, the excess pore pressure initially calculated is 0.98 p0, instead of 1.0 p0. This is due to the fact that the pore water in PLAXIS is not completely incompressible. See Undrained behaviour in Section 3.5 of the Reference Manual for more information. Secondly, the consolidation rate is slightly lower than the theoretical consolidation rate. This is caused by the implicit time integration scheme used.

VALIDATION MANUAL


STRUCTURAL ELEMENT PROBLEMS

6-1

6 STRUCTURAL ELEMENT PROBLEMS

A series of structural element/elastic benchmark calculations is described in this chapter. In each case the analytical solutions may be found in many of the various textbooks on elasticity solutions, for example Giroud (1972) and Poulos & Davis (1974).

6.1 BENDING OF FLOOR ELEMENTS

Input: For the verification of a floor element two problems are considered. These problems involve a single line load and a uniformly distributed load on a plate respectively, as indicated in Figure 6.1. For these problems a plate of 1 m length and 1m width has been selected. The properties, dimensions and the loads of the plate are:

E = 1·106 kN/m2 G = 5·105 kN/m2 ν = 0.0

d = 0.1 m F = 100 kN/m q = 200 kN/m2

Plates cannot be used individually. A single cluster may be used to create the geometry. The two plates are added to the top work plane with a spacing in between. Use line fixities on the end points of the plate. A coarse mesh is sufficient to model the situation. In the Initial conditions mode the soil cluster can be deactivated so that only the plates remain.

Figure 6.1 Loading scheme for testing plates

Output: The results of the two calculations are plotted in Figure 6.2, Figure 6.3 and Figure 6.4. For the extreme moments and displacements we find:

Line load: Mmax = 25.22 kNm/m umax = 25.5 mm

Distributed load: Mmax = 25.58 kNm/m umax = 31.8 mm

VALIDATION MANUAL


Figure 6.2 Computed distribution of moments

Figure 6.3 Computed shear forces

Figure 6.4 Computed displacements

Verification: As a first verification, it is observed from Figure 6.2 that PLAXIS yields the correct distribution of moments. For further verification we consider the well-known formulas as listed below. These formulas give approximately the values as obtained from the PLAXIS analysis.

Point load: kNm2541

max == FlM 25mm481 3

max ==EIFlu

Distributed load: kNm2581 2

max == qlM mm25.31384

5 4

max ==EI

l qu

The error of the results of PLAXIS is less than 2.5 %.


6-3

6.2 BENDING OF WALL ELEMENTS

Input: For the verification of a wall element the same two problems are considered as in the last section. These problems involve a single line load and a uniformly distributed load on a plate respectively, as indicated in Figure 6.5. For these problems a plate of 1 m length and 1m width has been selected. The properties, dimensions and the loads of the plate are:

E = 1·106 kN/m2 G = 5·105 kN/m2 ν = 0.0

d = 0.1 m F = 100 kN/m q = 200 kN/m2

Plates cannot be used individually. A single cluster may be used to create the geometry. The two plates are added to the geometry, taking care that there is a gap between the plates and the boundaries of the problem. Use line fixities on the top and bottom sides of the plates. A coarse mesh is sufficient to model the situation. In the Initial conditions mode the soil cluster can be deactivated so that only the plates remain.

Figure 6.5 Loading scheme for testing walls

Output: The results of the two calculations are plotted in Figure 6.6, Figure 6.7 and Figure 6.8. For the extreme moments and displacements we find:

Line load: Mmax = 25.00 kNm/m umax = 25.6 mm

Distributed load: Mmax = 25.46 kNm/m umax = 31.8 mm

The error of the results of PLAXIS is less than 2.5 %.

VALIDATION MANUAL


Figure 6.6 Computed distribution of moments

Figure 6.7 Computed shear forces

Figure 6.8 Computed displacements


6-5

6.3 BENDING OF SHELL ELEMENTS

The wall of a circular pile can be modelled in PLAXIS using curved shell elements. By using this element, 3 types of deformations are taken into account: shear deformation, compression due to normal forces and obviously bending.

Input: A ring with a radius of R = 1 m and a width of 1 m is considered. The Young's modulus and the Poisson's ratio of the material are taken respectively as E = 1·106 kN/m2 and ν = 0. For the thickness of the ring cross section, H, several different values are taken so that we have rings ranging from very thin to very thick. To model such a ring one point of the ring is fixed with respect to translation. The other side is allowed to move freely and a load F = 1.0 kN/m is applied at that side. Geometric non-linearity is not taken into account.

Output: The calculated vertical deflections at the top point are presented in Figure 6.9. The deformed shape of the ring is shown in Figure 6.10. The calculated normal force at the belly of the ring is 0.50 kN for all different values of ring thickness. The calculated bending moment at the belly is 0.182 kNm for all different values of ring thickness. Typical graphs of the bending moment and normal force are shown in Figure 6.11.

100

101

102

103

104

105

106

107

0 0.1 0.2 0.3 0.4 0.5 0.6

AnalyticalPLAXIS

Euy/F

H/R

Figure 6.9 Calculated deflections compared with analytical solutions

VALIDATION MANUAL


Figure 6.10 Deformed and original ring

a Normal forces b Bending moments

Figure 6.11

Verification: The analytical solution for the deflection of the ring is given by Blake (1959), and the analytical solution for the bending moment and the normal force can be found from Roark (1965). The vertical displacement at the top of the ring is given by the following formula:

⎥⎦⎤

⎢⎣⎡

+−+=

λ...

EFu y 2

2

121637009137881 λλ

with λ =HR

< F/2 > < >

0.181 FR


6-7

The solid curve in Figure 6.9 is plotted according to this formula. It can be seen that the deflections calculated by PLAXIS fit the theoretical solutions very well. Only for a very thick ring some errors are observed, which is about 4 percent for H/R = 0.5. But for thin rings the error is nearly zero. The analytical solution for the bending moment and normal force at the belly is 0.181 kNm and 0.5 kN respectively. Thus even for very thick rings the error in the bending moment and normal force is almost zero.

VALIDATION MANUAL


6.4 BEARING CAPACITY OF GROUND ANCHORS

To verify the bearing capacity of the grout body of a ground anchor the anchors are pre-stressed up to pulling out of the grout body. The test is performed with a ground anchor in loose sand as well as in dense sand. The skin friction along the grout body is considered to be either constant or linear for both soil material models.

Input: The ground anchor is attached to a concrete block. This block is modelled by a non-porous linear elastic material model:

γ = 25 kN/m3 E = 2.80·107 kN/m2 G = 1.217·107 kN/m2 ν = 0.15

The loose sand as well as the dense sand are modelled by a Mohr-Coulomb material model. The properties of the loose sand are:

γunsat = 17 kN/m3 γsat = 20 kN/m3 E = 4.5·104 kN/m2 ν = 0.3

c = 1 kN/m2 φ = 35° ψ = 5°

The properties of the dense sand are:

γunsat = 17 kN/m3 γsat = 20 kN/m3 E = 10.5·104 kN/m2 ν = 0.3

c = 1 kN/m2 φ = 35° ψ = 5°



6-9

The grout body of the ground anchor has a length of 4 m. The remaining properties of the grout body are:

E = 2·107 kN/m2 Diameter = 0.125 m

The anchor is elastic with a stiffness EA = 4.095·105 kN.

The pull out force of the ground anchor can be estimated as 200 kN in case of loose sand and as 752 kN in case of dense sand (after Ostermayer & Barley, 2003). The skin friction properties of the ground anchor can now be determined as:

Loose sand: Constant skin friction (CS): Ttop,max = 50 kN/m2 Tbot,max = 50 kN/m2 Linear skin friction (LS): Ttop,max = 100 kN/m2 Tbot,max = 0 kN/m2

Dense sand: Constant skin friction (CS): Ttop,max = 188 kN/m2 Tbot,max = 188 kN/m2 Linear skin friction (LS): Ttop,max = 376 kN/m2 Tbot,max = 0 kN/m2

The mesh density is considered to be medium. After the initial conditions, only one phase is added in which the anchor is pre-stressed up to pulling out of the grout body.

Figure 6.13 Curve of the axial force in the anchor against the displacements

VALIDATION MANUAL


Output: In Figure 6.13 the axial force in the anchor is plotted against the displacements. In loose sand, the maximum axial force in the anchor is 202 kN both for a constant skin friction distribution and a linear skin friction distribution. In dense sand, the maximum axial force in the anchor is 760 kN for both skin friction distributions.

Verification: The results in loose sand as well as in dense sand indicate an error of about 1%.

In both soil materials, the ground anchor shows a less stiff behaviour in case of a trapezoidal skin friction distribution. This is caused by early slip at the bottom of the grout body, when the prestress is still rather low, resulting in larger displacements. In case of a constant skin friction distribution, the whole grout body remains elastic, and thus smaller displacements, until a higher value of the pre-stress force is reached.


6-11

6.5 PERFORMANCE OF SPRINGS

Springs are used to transport forces to the outside world. Springs are fully fixed on one side and are connected to the geometry on the other side. They only transport forces parallel to their direction and have no stiffness perpendicular to their direction. In the following example the performance of springs connected to floors and walls is verified.

Input: Two floors of 2 x 2 m are modelled (Figure 6.14). Each floor is loaded by a distributed load of 100 kN/m2, acting downwards. One floor is directly supported by 4 vertical springs on the corners. The second floor is supported by two walls. The walls in turn are supported by vertical springs on their lower corner points. For stability two horizontal springs acting in x-direction are added at the bottom center of the walls, and two horizontal springs acting in z-direction are added to the floor.

All springs have a spring stiffness EA/L = 103 kN/m. All walls and floors have a Young’s modulus E = 108 kN/m2, Poisson ratio ν = 0 and a thickness d = 0.1 m.

Verification: The resulting force in all springs is equal to –100.00 kN. The vertical displacement of the corners of the floor directly supported by springs is –100.34 mm, which is a relative error of 0.3 %. For the second case, with the floors supported by walls, the vertical displacement of the bottom corner points of the walls is equal to -100.00 mm exactly.

Figure 6.14 Geometry of floors and walls supported by springs

VALIDATION MANUAL


6.6 PERFORMANCE OF INTERFACE ELEMENTS IN SOIL WITH CONSTANT COHESION

The soil-structure interaction effects can be modelled suitably in PLAXIS 3D FOUNDATION using interface elements. It enables the user to consider a more flexible interface between the structure and the soil, thereby capturing properly a realistic distribution of stresses, strains and deformations in the system. To verify the use of interface elements in PLAXIS 3D FOUNDATION, the problem of a vertically sliding block in a soil with constant cohesion has been analysed and the failure load has been obtained. A very good resemblance between the PLAXIS computed result and the theoretical result has been observed.

Input: The system consists of two blocks with a wall in between (Figure 6.15). The left block has been kept fixed and the right block was allowed to move freely. A vertical distributed load of 100 kN/m2 was applied on the top surface of the right hand block. The interface property between the wall and the right hand block has been considered by assigning a certain value to the parameter Rinter to the right hand block. However, the value of Rinter between the wall and the left hand block was fixed at 1.0, thereby forcing the left block-wall combination to behave rigidly. The properties of the blocks and the wall are narrated below. The block on the left side has a depth of 5 m (Figure 6.15b) whereas the block on the right side has a depth of 4 m. The wall has also a height of 5 m. The left block-wall system is acting rigidly and is fixed. The finite element model of the system is shown in Figure 6.16.

a. Plan view b. Vertical cross section

Figure 6.15 Block-wall system

The material behaviour of both blocks is represented by the Mohr-Coulomb model. The unit weight as well as the friction and dilatancy angles are set to zero. The other undrained properties of the left hand block are as follows:

Left block - fixed

Right block – sliding vertically

Wall in between 2 blocks

4 m 4 m 0.1m

4 m 5 m

4 m 4 m

1 m


6-13

6103 ⋅=E kN/m2, 25=c kN/m2, 3.0=ν and Rinter 0.1= .

The other undrained properties of the right hand block are as follows:

7103 ⋅=E kN/m2, 25=c kN/m2, 3.0=ν and Rinter 2.0= .

The properties of the wall are:

9105 ⋅=E kN/m2, 0=γ kN/m3, 15.0=ν and 1.0=d m.

Figure 6.16 Finite element model of the problem

Output: The numerical analysis has been performed in PLAXIS 3D FOUNDATION to obtain a failure load qf of 5 kN/m2 (ΣMstage = 0.05), giving an equivalent vertical force Ff = 5·4·4 = 80 kN. Figure 6.17 demonstrates the load-displacement curve.

VALIDATION MANUAL


Figure 6.17 Load-displacement curve at a point in the right block near the interface

Verification: As the right block is considered weightless and the friction angle is taken as zero, the ultimate load applied on the block will only depend on the cohesion of the interface. The uniform vertical load on the top horizontal surface of the right hand block would intend to move the block and ultimately it would be on the verge of sliding when the applied external load would be exactly equal to the resisting vertical cohesion along the right block-wall interface.

Thus, the final force Ff causing the onset of sliding could be computed as follows:

Ff = cRinterhw = 25·0.2·4·4 = 80 kN

This value coincides exactly with the value of the failure load as obtained by PLAXIS 3D FOUNDATION.


6-15

6.7 PERFORMANCE OF INTERFACE ELEMENTS IN SOIL WITH VARYING COHESION

In addition to the previous verification of interface elements, another problem of a vertically sliding block in a soil with varying cohesion has been analysed and the failure load has been obtained. A very good resemblance between the PLAXIS computed result and the theoretical result has been observed.

Input: The model is simular to the previous example. Only, in this case the cohesive strength in the right block-wall interface is increasing with depth. This variation is shown in Figure 6.18 where the cohesion at the top of the soil is considered as 20 kN/m2 and is assumed to vary at a rate of 5 kN/m2/m with depth.

Figure 6.18 Variation of cohesion with depth of the soil layer

The remaining properties of both soil blocks and the wall are similar to the previous verification.

Output: The value of the failure load qf at the moment of onset of sliding obtained from the numerical analysis through PLAXIS 3DF is 6 kN/m2 (ΣMstage = 0.06), giving an equivalent vertical force Ff = 6·4·4 = 96 kN. Figure 6.19 shows the load-displacement curve.

c = 20 kN/m2

c = 45 kN/m2

c = 40 kN/m2

cincr = 5 kN/m2/m

1 m

4 m

Top surface

VALIDATION MANUAL


Figure 6.19 Load-displacement curve at a point in the right block near the interface for varying cohesion

Verification: As the right block is considered weightless and the friction angle is taken as zero, the ultimate load on the block will only depend on the cohesion of the interface. The uniform vertical load on the top horizontal surface of the right hand block would intend to move the block and ultimately it would be on the verge of sliding when the applied external load would be exactly equal to the resisting vertical cohesion along the right block-wall interface. In the case of varying cohesion, the value of the failure force Ff may be computed as:

( ) 96442.02

4020=⋅⋅⋅

+== ∫

Ainterf dAcRF kN

where A denotes the surface of the interface. This value coincides exactly with the value of the failure load as obtained by PLAXIS 3D FOUNDATION.

SINGLE PILE AND PILE GROUP IN OVERCONSOLIDATED CLAY

7-1

7 SINGLE PILE AND PILE GROUP IN OVERCONSOLIDATED CLAY

(by Y. El-Mossallamy, Ain Shams University)

In order to validate the program, a pile load test in Germany has been analysed. The load test investigated both the load-settlement behaviour of a single pile and that of a pile group. The behaviour of the single pile has been analysed using both PLAXIS 3D FOUNDATION as well as PLAXIS V8. Subsequently, the behaviour of the pile group has been analysed using PLAXIS 3D FOUNDATION.

7.1 INTRODUCTION

The load settlement behaviour of the piles in a pile group is totally different from the behaviour of the corresponding single pile. The group action represents the behaviour of the pile group compared to that of the single pile. Pile group action plays an important role for the behaviour of piled foundation both under vertical tension and compression loads and under horizontal loads. The group action of pile groups under vertical compression loads will be dealt with in this example.

As no possibility exists to take into account -in an adequate manner- the soil disturbance caused due to pile installation by theoretical means (El-Mossallamy, 1999), pile load tests on single piles are frequently carried out to determine the load-settlement behaviour of a single pile. On the other hand it is costly and time consuming to carry out load tests on pile groups. Therefore, the pile group action is considered either by adapting simple correlations, or by comparing the pile group to simplified foundation shapes, or by applying advanced numerical analyses. The application of three dimensional finite element analyses to determine the pile group action will be demonstrated in this example.

7.2 NUMERICAL SIMULATION OF THE SINGLE PILE BEHAVIOUR (PILE LOAD TEST)

An extensive research program related to bored piles in overconsolidated clay was conducted by Sommer & Hambach (1974) to optimise the foundation design of a highway bridge in Germany. Load cells were installed at the pile base to measure the loads carried directly by pile base. Figure 7.1 gives the layout of the pile load test arrangement. The measured load-settlement curves and the distribution of loads between base resistance and skin friction are shown in Figure 7.2. The upper 4.5 m subsoil consist of silt (loam) followed by tertiary sediments down to great depths. These tertiary sediments are stiff plastic clay similar to the so-cal1ed Frankfurt clay, with a varying degree of overconsolidation. A pile load test is often used to verify the numerical modelling of pile behaviour in Frankfurt overconsolidated clay (El-Mossallamy 2004). The groundwater table is about 3.5 m below the ground surface. The considered pile has

VALIDATION MANUAL


a diameter of 1.3 m and a length of 9.5 m. It is located completely in the overconsolidated clay. The loading system consists of two hydraulic jacks working against a reaction beam. This reaction beam is supported by 16 anchors. These anchors were installed vertically at a depth between 15 and 20 m below the ground surface at a distance of about 4 m from the tested pile, in order to minimize the effect of the mutual interaction between the tested pile and the reaction system (Figure 7.1.a). Vertical and horizontal loading tests were carried out. The loads were applied in increments and maintained constant until the settlement rate was negligible. Both the applied loads and the corresponding displacements at the tested pile head were measured. Additionally the soil displacements near the pile at different depths were measured using deep settlement points (Figure 7.1b).

Figure 7.1 Layout of the pile load test and the measurement points

Figure 7.2 Measured load-settlement curves and distribution of loads between base

resistance and skin friction


7-3

7.2.1 GEOMETRY OF THE MODEL

In order to analyse the behaviour of the single pile, at first a model has been made in PLAXIS V8 using an axisymmetric model for a completely homogeneous soil. A mesh of 15 m width and 16 m depth has been used. At the axis of symmetry the pile has been modelled with a length of 9.5 m and a diameter of 1.3 m. The soil is modelled as a single layer of overconsolidated stiff plastic clay, with properties as given in Table 7.1. The groundwater table is located at 3.5 m below the soil surface. Along the length of the pile an interface has been modelled. This interface extends to 0.5 m below the pile, in order to allow for sufficient flexibility around the pile tip. The resulting mesh composed of high order 15 node elements is shown in Figure 7.3.

Figure 7.3 The resulting 2D axisymmetric mesh

Figure 7.4 The dimensions of the 3D Foundation model

VALIDATION MANUAL


Secondly, a model has been made using PLAXIS 3D FOUNDATION. A working area 50 m x 50 m has been used. The pile is modelled as a solid pile using volume elements in the centre of the mesh. Interfaces are modelled along the pile. The soil consists of a single layer of overconsolidated stiff plastic clay, with properties as given in Table 7.1. The load is modelled as a distributed load at the pile top. 6 different meshes with different levels of refinement were applied to check the sensitivity of the mesh refinement on the results. Table 7.2 summarizes the main properties of the 6 tested meshes. This table also lists the number of elements used to model the pile in vertical direction. Figure 7.5 shows the different finite element meshes composed of 15 node volume elements.

Figure 7.5 The finite element meshes used for the 3D analyses.


7-5

7.2.2 MATERIAL PROPERTIES

The required soil parameters were determined based on the conducted laboratory and in-situ tests as well as on experience gained in similar soil conditions, see Table 7.1. The concrete pile is modelled as a non-porous linear elastic material with Young’s modulus E = 3·107 kN/m2, Poisson ratio ν = 0.2 and unit weight γ = 24 kN/m3. For the overconsolidated clay layer, two different material models have been considered. Table 7.1 Model parameters for different soil data sets Parameter Name Overcons.

Clay 1 Overcons. Clay 2

Silt (Loam)

Unit

Material model Model Mohr-Coulomb

Hardening Soil

Mohr-Coulomb

-

Type of material behaviour Type Drained Drained Drained - Unsaturated soil weigth γunsat 20 20 19 kN/m3 Saturated soil weigth γsat 20 20 19 kN/m3 Young’s modulus E 6·104 - 1·104 kN/m2 Poisson ratio ν 0.3 - 0.3 - Secant stiffness refE50 - 4.5·104 - kN/m2

Oedometer stiffness refoedE - 4.5·104 - kN/m2

Unloading-reloading stiffness refurE - 9·104 - kN/m2

Power m - 0.5 - - Unloading-reloading Poisson ratio

νur - 0.2 - -

Cohesion c 20 20 5 kN/m2 Friction angle ϕ 22.5 22.5 27.5 ° Dilatancy angle ψ 0 0 0 ° Lateral earth pressure coeft. K0 0.8 0.8 0.5 -

Table 7.2 Applied meshes for the three dimensional analyses (15 node wedge elements) Model name No. of elements / nodes

in top work plane Total no. of elements / nodes for the whole 3D mesh

No. of layers in pile

Variety - 01 106 / 237 742 / 2238 4 Variety - 02 292 / 609 2044 / 5865 4 Variety - 03 350 / 741 2450 / 7060 4 Variety - 04 350 / 741 3150 / 8862 5 Variety - 05 350 / 741 3850 / 10664 7 Variety - 06 350 / 741 5250 / 14268 10

VALIDATION MANUAL


7.2.3 MODELLING THE SINGLE PILE

Initial stresses were generated using the K0-procedure in the 2D axisymmetric case and using gravity loading in the 3D analyses. In both cases the initial K0 value in the overconsolidated clay was taken 0.8. Pore pressures were generated based on a phreatic level. The actual load test was simulated by applying a distributed load at the top of the pile.

Figure 7.6 shows the load-settlement curves for the different 3D analyses. The vertical displacement of the top of the pile has been plotted. The results are similar up to 2000 kN, almost equal to the working load. At higher load levels, the results of meshes 3, 4, 5 and 6 show little differences. These results demonstrate the stability of the program. Nevertheless, it is recommended to check the sensitivity of the mesh refinement on the results for each individual case.

Figure 7.6 Results of different finite element meshes.

Figure 7.7 shows a comparison between the different numerical models. There is a good agreement between the results of different numerical models and those of the pile load test up to a working load of about 2000 kN. Nevertheless, the three dimensional analysis shows a relatively stiff behaviour at higher load level in comparison with the axisymmetric results for the same initial conditions. The effect of the initial stresses on the load settlement behaviour of a single pile as well as on a pile group will be discussed in more details in Section 7.3.1.


7-7

Figure 7.7 Comparison between the results of different numerical models and

measured results.

Figure 7.8 Deformation results using PLAXIS 3D FOUNDATION.

VALIDATION MANUAL


Figure 7.8 demonstrates some deformation results of PLAXIS 3D FOUNDATION for Variety 6 (see Table 7.2). At higher load levels, plastic deformation of the soil controls the settlement behaviour of the pile. These plastic deformations are concentrated in a narrow zone around the pile shaft. Outside this plastic narrow zone the soil behaviour remains mainly elastic. Therefore, the settlement trough under working loads (of 1500 kN (Figure 7.8a)) is wider than that under loads near the ultimate load level (of 4000 kN (Figure 7.8b)).

7.3 NUMERICAL SIMULATION OF THE PILE GROUP ACTION

From the pile load test of the single pile, it was determined that the ultimate skin friction was about 60 kN/m2. Subsequently, an allowable skin friction of 30 kN/m2 was selected for the foundation design, as at the corresponding load level, the settlement of the tested pile was measured to be in the order of 3 mm. A settlement of 3 mm was deemed to be acceptable for the bridge design. The bridge piers consists of 2 pillars, each founded on a separate pile group. The foundation piles have a diameter of 1.5 m and a length of 24.5 m with 6 piles under each pillar. The pile arrangement is shown in Figure 7.9a. The settlement of the entire foundation should be about 3 mm if there were no group action. The load-settlement behaviour of the whole foundation was monitored during and after the construction to obtain information on the group action. The load settlement relationship of one of the monitored pillars (Sommer/Hambach, 1974) is shown in Figure 7.9b.

Figure 7.9 Foundation layout and load settlement behaviour


7-9

The average measured settlement of the pillar was about 9.0 mm. The difference between the expected settlement and the measured value demonstrates the importance of considering the pile group action to predict a reliable settlement of the whole foundation. A three dimensional finite element analysis is applied to investigate its reliability determining the pile group action. The results of the boundary element method (El-Mossallamy, 1999) will be used to compare with the results of the 3D finite element analyses.

The load settlement behaviour of a single foundation pile (pile length 24.5 m and pile diameter 1.5 m) was calculated using both the 3D-FEM as well as the BEM (El-Mossallamy, 1999). In both cases the same soil parameters were used for the clay layers as in the verification analysis of the single pile in homogeneous soil conditions, see Table 7.1. In this analysis the top layer of silt is also taken into account. Figure 7.10 shows the 3D finite element mesh used to simulate the behaviour of a single foundation pile.

Figure 7.10 3D finite element mesh to simulate the behaviour of a single foundation pile.

Figure 7.11 shows a comparison between the different conducted analyses. The load settlement relationship up to a working load of about 3 MN is mainly linear. Furthermore, the different models behave very similar up to a load of about 7 MN (about twice the working load).

For the analysis of the pile group, three different mesh refinements were used, see Figure 7.12. Table 7.3 summarizes the main properties of the 3 different meshes.

VALIDATION MANUAL


Figure 7.11 Load settlement behaviour of a single foundation pile.

Figure 7.12 3D finite element meshes to simulate the foundation behaviour.


7-11

Table 7.3 Main properties of the 3 meshes used for the analysis of the pile group. Variety No. of elements / nodes

in top work plane Total no. of elements / nodes for the whole 3D mesh

No. of pile subdivisions

Variety - 01 164 / 417 1804 / 5249 7 Variety - 02 161 / 412 2093 / 6038 8 Variety - 03 429 / 956 8151 / 22120 14

The calculated results of the load settlement behaviour of the whole pile group are shown in Figure 7.13. The different meshes give almost the same result up to 32 MN (twice the working load). Mesh variations 2 and 3 yield a good agreement at higher loads. The calculated settlement at the working load of 16 MN is about 10 mm and agrees well with the measurements.

Figure 7.13 Load-settlement behaviour of the whole foundation

Shadings of equal settlement at the ground surface are shown in Figure 7.14a to demonstrate the 3D results. The settlement of the foundation alone is shown in Figure 7.14b. It can be recognized that the mutual interaction between the two pillars leads to some tilting of both pillars. The calculated tilting reaches about 1:3500. These results show the ability of PLAXIS 3D FOUNDATION to predict the load settlement behaviour of pile groups under working conditions in order to check the serviceability requirements.

VALIDATION MANUAL


Figure 7.15 compares the behaviour of the single pile with the average behaviour of the pile group under the same average load. The calculated pile group action, resulting from the 3D finite element analyses as well as from the boundary element analyses (El-Mossallamy, 1999) can be determined to be in the order of 3.0. This value agrees well with the results of the conducted measurements. These results demonstrate the ability of PLAXIS 3D FOUNDATION to predict the pile group action.

a) b) Figure 7.14 Deformation results of the bridge pillar using PLAXIS 3D FOUNDATION.

a) Settlement at the ground surface. b) Settlement of the foundation plate

Figure 7.15 Pile group action


7-13

7.3.1 EFFECT OF INITIAL STRESSES

The previous analyses show that the load-settlement behaviour of a foundation pile can be accurately modelled under working load conditions. For instance Figure 7.13 shows that the correct settlement under working load conditions can be predicted for a pile group, and that this predicted settlement is not strongly influenced by the mesh refinements. On the other hand, the ultimate bearing capacity of the pile is strongly influenced by several factors, amongst which mesh refinements and the initial stress state. Figure 7.7, Figure 7.11 and Figure 7.15 show a comparison between results with different initial stresses. From each of these figures it can be seen that an accurate prediction of the settlement under working loads can be obtained. However, the ultimate bearing capacities obtained from these analyses depend strongly on the modelling scheme followed. Figure 7.15, for example, shows that the difference in ultimate bearing capacity of a single pile obtained using the boundary element method (El-Mossallamy, 1999) and PLAXIS 3D FOUNDATION amounts to approximately 3 MPa.

2

3

Figure 7.16 Effect of initial stresses on the calculation results.

VALIDATION MANUAL


Figure 7.16 summarizes the results of the comparison for the behaviour of the pile load test, the single foundation pile and the whole pillar foundation, in order to demonstrate the effect of the initial stresses in more detail. Once again, this figure shows significant differences in predicted ultimate bearing capacity for different models, but also for different initial stress conditions. For example for the single pile, the deformation under working load conditions is hardly influenced by the initial value of K0, but the ultimate bearing capacity may change as much as 3 MPa. The same trend is seen for the pile group.

7.4 CONCLUSIONS

The load settlement behaviour of the piles in overconsolidated clay is almost linear up to the working load. Therefore, the initial stresses have almost no effect on the results up to the working loads. On the other hand, the initial stresses have a dominant effect on the pile behaviour under higher load levels. The calculated ultimate bearing capacity depends strongly on the initial stresses. The results of Figure 7.16 show that the PLAXIS 3D FOUNDATION analyses have a good agreement with the results of the PLAXIS V8 (axisymmetric modelling with 15-node elements) under the working load. At higher load level, the PLAXIS 3D FOUNDATION analyses show stiffer behaviour than the axisymmetric analyses and predict a higher ultimate bearing capacity. Therefore, the ultimate bearing capacity should be checked using independent conventional methods. Nevertheless, it can be concluded that the calculated deformation under working conditions (serviceability limit analyses) can be adequately determined using PLAXIS 3D FOUNDATION.

VALIDATION OF EMBEDDED PILES – THE ALZEY BRIDGE PILE LOAD TEST

8-1

8 VALIDATION OF EMBEDDED PILES – THE ALZEY BRIDGE PILE LOAD TEST

(by H. K. Engin, Middle East Technical University)

In order to validate the embedded piles option, the Alzey Bridge pile load test of a single pile has been analysed (Engin, 2007). This load test has already been used to validate the behaviour of a volume pile in Chapter 7. Constant as well as trapezoidal skin friction distribution of the embedded pile is considered. A comparison between the results of both embedded piles and the volume pile will be made.

8.1 GENERAL

An extensive research program related to bored piles in overconsolidated clay was conducted by Sommer & Hambach (1974) to optimise the foundation design of a highway bridge in Germany.

Load cells were installed at the pile base to measure the loads carried directly by pile base. Figure 8.1a and Figure 8.1b give the layout of the pile load test arrangement. The measured load-settlement curves and the distribution of loads between base resistance and skin friction are shown in Figure 8.1c.

The total pile capacity is about 3230 kN. The upper 4.5 m subsoil consist of silt (loam) followed by tertiary sediments down to great depths. These tertiary sediments are stiff plastic clay similar to the so-cal1ed Frankfurt clay, with a varying degree of overconsolidation. A pile load test is often used to verify the numerical modelling of pile behaviour in Frankfurt overconsolidated clay (El-Mossallamy, 2004). The groundwater table is about 3.5 m below the ground surface.

The considered pile has a diameter of 1.3 m and a length of 9.5 m. The test pile is a bored pile; hence the soil disturbance around the pile is expected to be very low. It is located completely in the overconsolidated clay.

The loads were applied in increments and maintained constant until the settlement rate was negligible. Both the applied loads and the corresponding displacements at the tested pile head were measured. These test results are compared with the results obtained with embedded piles using PLAXIS 3D FOUNDATION.

VALIDATION MANUAL


Figure 8.1 Layout of the pile load test and the measured points (After El-Mossallamy,

1999)

8.2 FINITE ELEMENT MODEL

The pile load test is modelled by PLAXIS 3D FOUNDATION. A working area of 20 m x 20 m has been used. The pile is modelled by an embedded pile element in the centre of the mesh. The load is modelled by a point load on top of the embedded pile. In case of a volume pile, the load is modelled by a distributed load on top of the volume pile. The soil is modelled by a single layer overconsolidated stiff plastic clay. The mesh size can be said to be medium to fine for both embedded piles as well as the volume pile. The mesh of the model is given in Figure 8.2.

a. 2D mesh b. 3D mesh

Figure 8.2 Finite Element Model


8-3

8.3 MATERIAL PROPERTIES

The soil consists of a single layer of overconsolidated stiff plastic clay modelled by the Hardening Soil material model, with properties as given in Table 8.1. The pre-consolidation stress is defined by means of a POP of 50 kN/m2.

Table 8.1 Summary of soil parameters used in analysis

The material properties of the embedded pile are defined in Table 8.2. A distinction has been made between an embedded pile with a constant skin friction (CS) and an embedded pile with a trapezoidal skin friction (TS) (Figure 8.3). The strength of the embedded pile is defined according to the field test results that give the total and base resistances mobilized.

The volume pile is defined by a linear elastic non-porous material model with the same properties as given in Table 8.2.

Parameter Name OC Clay Unit

Material model Model HS - Type of material behaviour Drained Drained - Unsaturated soil weight γunsat 20 kN/m3 Saturated soil weight γsat 20 kN/m3

Secant stiffness refE50 4.5·104 kN/m2

Oedometer stiffness refoedE 2.715·104 kN/m2

Unloading-reloading stiffness refurE 9·104 kN/m2

Power m 1.0 - Unloading-reloading Poisson ratio νur 0.2 - Cohesion c’ 20 kN/m2 Friction angle ϕ’ 20 o

Dilatancy angle ψ 0 o Lateral earth pressure coefficient for normal consolidation K nc

0 0.658 -

Lateral earth pressure coefficient K 0 0.8 -

Over-consolidation ratio OCR 1 - Pre-overburden pressure POP 50 kN/m2

Interface reduction factor Rinter 1.0 -

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Table 8.2 Material properties of the embedded pile

a. Constant skin friction b. Trapezoidal skin friction

Figure 8.3 Pile capacity models defined for Alzey Brigde test pile

Parameter Name Constant skin friction

Trapezoidal skin friction

Unit

Young’s modulus E 3·107 3·107 kN/m2 Weight γ 5 5 kN/m3

Properties type Type Massive circular pile

Massive circular pile -

Diameter Ø 1.3 1.3 m Cross section area Α 1.327 1.327 m2 Moment of inertia against bending around the third axis I3 0.140 0.140 m4

Moment of inertia against bending around the third axis I2 0.140 0.140 m4

Moment of inertia against oblique bending I23 0 0 m4

Skin friction distribution Type Linear Linear - Maximum traction allowed at the top of the embedded pile Ttop,max 201.368 19.18 kN/m

Maximum traction allowed at the bottom of the embedded pile Tbot,max 201.368 383.560 kN/m

Base resistance Fmax 1320 1320 kN

10000 kN

1320 kN

383.56 kN/ m

19.18 kN/ m

10000 kN

1320 kN

201.368 kN/m

201.368 kN/m


8-5

8.4 RESULTS

The capacity of the embedded pile is defined by constant as well as trapezoidal skin friction distribution. Intermediate steps are also checked in order to obtain the base resistance curves. Skin friction curves are obtained by subtracting the base resistance from the total load – displacement curve. In Figure 8.4 it can be seen that the embedded pile model by a constant friction distribution is quite in agreement with the pile load test results. Mobilization of skin friction and base resistance could almost catch the real behaviour.

Alzey Brigde Single Pile Load Test

PILE CAPACITY

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50

Settlement (mm)

Load

(kN

)

Total Load

Skin Friction

Base Resistance

PILE CAPACITY

HS-CS

HS-CS-Base Res.

HS-CS-Ave. Skin Friction

Figure 8.4 Pile test curves and Plaxis 3D Foundation embedded pile results in case of a constant skin friction.

The results of the embedded pile model with a trapezoidal skin friction distribution are given in Figure 8.5. It can be seen that the results of the embedded pile model with a trapezoidal friction distribution is also in agreement with the pile load test results.

The total load displacement curve as well as the skin friction and base resistance curves of the volume pile are plotted with the pile load test and embedded pile model test curves (Figure 8.6). It can be seen that the volume pile gives an overestimated behaviour due to an overestimated behaviour of the base resistance. However, the skin friction curve is in good agreement with the real behaviour.

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PILE CAPACITY

0

500

1000

1500

2000

2500

3000

3500

0 5 10 15 20 25 30 35 40 45 50

Settlement (mm)

Load

(kN

)

Total Load

Skin Friction

Base Resistance

PILE CAPACITY

HS-TS

HS-CS

HS-TS-Base Res.

HS-TS-Ave. SkinFriction

Figure 8.5 Pile test curves and Plaxis 3D Foundation embedded pile results in case of a

trapezoidal skin friction.


PILE CAPACITY

0

500

1000

1500

2000

2500

3000

3500

4000

0 5 10 15 20 25 30 35 40 45 50

Settlement (mm)

Load

(kN

)

Total Load

Skin Friction

Base Resistance

PILE CAPACITY

HS-CS

Volpile-Base Res.

VolPile-Ave. Skin Friction

VolPile

Figure 8.6 Pile Test, Plaxis 3D Foundation embedded pile with a constant skin friction

and volume pile results


8-7

8.5 DISCUSSION AND CONCLUSIONS

Modelling of piles is a difficult job since there are many parameters affecting the pile behaviour. Even if the soil is perfectly modelled, deviations from actual behaviour occur due to pile installation. Also the coarseness of the mesh influences to some extend the load – displacement behaviour.

In this pile load test the installation effect is negligible. Therefore, it is a good starting point to validate the embedded pile. The pile test is modelled by an embedded pile with a constant or trapezoidal skin friction distribution as well as by means of a volume pile. The results show that it is convenient to use embedded piles to model piles, especially bored piles. It should be noted that pile load test data should be available in order to define the embedded pile capacity properly. The volume pile seems to overestimate the pile bearing capacity. However, when decreasing the strength of the soil below the volume pile, a better approximation can be found.

It is clearly observed in this study that use of embedded piles has a great potential in modelling pile foundations easily and effectively, since the major criteria is the satisfaction of ultimate load capacity. Although this study had shown the efficiency of embedded piles, the adequateness in pile groups can be a question, since the group effect as well as installation effect are not considered.

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PILED RAFT FOUNDATION IN FRANKFURTER CLAY

9-1

9 PILED RAFT FOUNDATION IN FRANKFURTER CLAY

(by Y. El-Mossallamy, Ain Shams University)

9.1 INTRODUCTION

The piled raft foundation has shown its validity in the last two decades as a very economic geotechnical foundation type, where the structural loads are carried partly by the piles and partly by the raft contact stresses. The structural serviceability requirements regarding the settlements and tilting of buildings can be fulfilled with relatively fewer piles in comparison with a pure piled foundation. This foundation system was successfully applied in stiff as well as soft subsoil. An innovative application of the piled raft is its special adjustment to cases of foundations with large load eccentricities or very different loaded parts of buildings to avoid the need of complex settlement joints especially below ground water table. Extensive measurements of the load transfer mechanism of piled raft foundations during and after the construction were performed to verify the design concept and to prove the serviceability requirements.

Calculation procedures to model the behavior of such complex three-dimensional problems have been developed since the 1970s (e.g. Butterfield and Banerjee, 1971; Poulos and Davis, 1980; Randolph, 1983). But some important requirements concerning the raft stiffness, the nonlinear behavior of the pile support and the slip developing along the pile shafts even under working loads were not sufficiently considered in these analyses. For these reasons improved numerical models based on three dimensional finite element method are applied taking into account all above mentioned effects. A traditional 3D finite element technique with the appropriate soil constitutive laws presents a powerful tool to model this complex soil-structure interaction problem. Nevertheless, the main disadvantage applying the 3D FE analyses is the need of a huge number of volume elements which can exceed the available computer capacities. To cover this problem, a new technique combined the so called embedded pile model with the 3D finite element model was developed by PLAXIS B.V. under the name PLAXIS 3D FOUNDATION Version 2. The following chapters present an example demonstrating the ability of this program to deal with a complex piled raft. A case history in Frankfurt will be resolved applying this program.

9.2 FRANKFURT SUBGROUND AND METHODOLOGY TO DEVELOP THE PILED RAFT

Most of the high-rise buildings in Frankfurt are founded on the so-called Frankfurter clay, which developed 2 to 10 million years ago as a result of the sedimentation in the Tertiary sea in the Mainz basin. In the town center, the clay layer measures up to 100 meters and includes limestone banks, lignite coal lenses and layers of calcareous sand. The groundwater level is mostly just above the clay surface and circulates in the fissured

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limestone banks and sand lenses resulting in different confined aquifer pressures. The clay is geologically overconsolidated through older, already eroded sediments and volcanic rock.

9.3 EXAMPLE OF A HIGH-RISE BUILDING ON FRANKFURT SUBSOIL

The 120 m building with a 4-storey underground basement has an L shape (Figure 9.1) with a load eccentricity of about 7.0 m. By applying the concept of piled raft foundation it was possible to construct the foundation without settlement joints between the tower and the adjacent 4-storey underground garage. The piles were placed eccentrically below the tower to balance the load eccentricity.

Foundation and subsoil conditions

General information Height (m) 114 Foundation area (m²) 1930 Raft thickness (m) 3.5 - 1.0 Foundation depth (m) -15.75 Groundwater - 6.0 Slenderness ratio 3.5 No of piles 25 Pile length (m) 22 Pile diameter (m) 1.3

Figure 9.1 General layout

9.4 GEOMETRY

The foundation of the building has a total area of about 1930 m². Only 25 large diameter bored piles were constructed beneath the raft as a piled raft foundation. The pile arrangements are shown in Figure 9.2. The rafts are 3.5 meters thick in the middle and 1.0 m at the edges. The raft base lies at a depth of 15.75 meters below the soil surface. The piles where designed with a diameter of 1.3 m and a length of 22 m. The total working loads reach about 900 MN.


9-3

Figure 9.2. Foundation dimensions and pile arrangement

9.5 LOADS

The applied loads are given in Figure 9.3, Figure 9.4 and Figure 9.5.

Figure 9.3 Point loads in MN

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Figure 9.4 Line loads in kN/m

Figure 9.5 Distributed loads in kN/m²


9-5

9.6 NUMERICAL MODEL

9.6.1 SOIL PARAMETERS

The soil stress-strain relationship was modeled by means of the Hardening Soil model. The main advantage of this constitutive law is its ability to consider the stress path and its effect on the soil stiffness and its behavior. The parameters of this model are summarized in Table 9.1.

Table 9.1 Model parameters of the different soil layers Parameter Name Filling Quaternary

Sand/Gravel Overconsolidated Clay

Unit

Material model Model Hardening Soil

Hardening Soil

Hardening Soil -

Unsaturated soil weigth

γunsat 8 11 10 kN/m3

Saturated soil weigth

γsat 18 19 20 kN/m3

Permeability coefficient in horizontal direction

kx, kz 10-3 10-3 2.5·10-5 m/sec

Permeability coefficient in vertical direction

ky 10-3 10-3 0.01 kx m/sec

Secant stiffness refE50 20 30 35 MN/m2

Unloading-reloading stiffness

refurE 50 75 105 MN/m2

Power m 0.5 0.5 1.0 -

Cohesion c - - 20 kN/m2 Friction angle ϕ 30 35 20 °

Unloading-reloading Poisson ratio

νur 0.2 0.2 0.2 -

Failure ratio Rf 0.9 0.9 0.9 -

Lateral earth pressure coeft.

K0 0.5 0.43 0.8 -

For the concrete piles and raft, a linear elastic material set was applied using the concrete weight and its stiffness. The elastic modulus of the concrete was chosen equal to 30000 MPa with a Poisson’s ratio of 0.2. The skin friction of the pile is assumed to

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start with 60 kPa at the pile head and increased with depth to reach 120 kPa at the pile tip. The pile base resistance was taken equal to 2.5 Mpa.

9.6.2 3D FINITE ELEMENT MODEL

Work planes are defined at each level where a discontinuity in the geometry or the loading occurs in the initial situation or in the construction process. Figure 9.6 shows the applied three dimensional finite element mesh. The main model geometries are given in Figure 9.7 and Figure 9.8.

Figure 9.6 3D FE-Model

Figure 9.7 Applied loads


9-7

Figure 9.8 Modeling the raft and the embedded piles

9.6.3 CALCULATIONS

The initial conditions should be generated using the K0-procedure. A value of K0 = 0.8 is applied to consider the effect of overconsolidation.

The aim of the calculation is to determine the average settlement of the rafts under working load (serviceability limit state).

It is for the user to determine the necessary calculation phases, but effects that may be taken into account are: • Installation of shoring system. • Modelling the excavation phases • Installation of the piles and foundation. • Application of the load from the superstructure (working loads)

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9.7 INSPECT OUTPUT

Figure 9.9 demonstrates the raft settlements under working loads. Figure 9.10, Figure 9.11, Figure 9.12 and Figure 9.13 show the load distribution among the individual piles within the pile group.

Figure 9.9 Foundation settlement under working loads

Figure 9.10 Results of normal force distribution along all piles - Outer piles


9-9

Figure 9.11 Results of normal force distribution along all piles - Middle piles

Figure 9.12 Results of normal force and skin friction distribution along an edge pile

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Figure 9.13 Results of normal force and skin friction distribution along a middle pile

It can be recognized that the contribution of the edge piles by carrying the loads is very small. This is due to the presence of the outer wall that works also as shoring system, which is modeled as fully connected with the foundation raft. The effect of the outer walls can be investigated applying a new model in which the outer walls are not modeled.

9.8 CONCLUSION AND OUTLOOK

The piled raft foundation can be modeled using the embedded piles. The maximum measured settlement is about 6 cm, which shows a good agreement with the calculated values. The calculated values are smaller than the measured value due to the modelling of the shoring system completely fixed with the raft. The results should be further compared with cases where the piles are modeled using volume elements. There is still need of a horizontal interface element to investigate the raft contact stresses under the floor in a direct manner. The embedded piles help to reduce the required number of elements needed to model the complex three dimensional feature of piled rafts. The experience with this model type should be gathered with time and shared among the PLAXIS users. The effect of the shoring system on the behavior of piled raft needs further investigation.

APPLICATION OF THE GROUND ANCHOR FACILITY

10-1

10 APPLICATION OF THE GROUND ANCHOR FACILITY

(by F. Tschuchnigg and H. Schweiger, Graz University of Technology)

10.1 INTRODUCTION

The ground anchor in PLAXIS 3D FOUNDATION consists of two different parts. The first part represents the free anchor length and the second part the grout body. The free length is modelled as a node-to-node anchor, which represents the connection between the grout body and e.g. a diaphragm wall, and the grout body consists of embedded beam elements, which are line elements with a special interface to model the grout-soil interaction. For the definition of the ground anchor eight input values are required (Figure 10.1 and Table 10.1).

Figure 10.1 System layout

Table 10.1 Input values for a ground anchor GEOMETRY INPUT GROUT BODY PROPERTIES A starting point of the ground anchor E stiffness of the grout body

[kN/m2] B ending point of the ground anchor ∅ diameter of the grout body [m] Ltotal total anchor length Lgrout grout body length α inclination angle [°] Anchor properties Skin resistance EA axial stiffness of the anchor rod

[kN] Ttop,max maximum skin traction at the top

of the ground anchor [kN/m] Fmax maximum force of the anchor rod

(for elastoplastic behaviour) [kN] Tbot,max maximum skin traction at the

bottom of the ground anchor [kN/m]

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The soil-interaction is defined with the two separate values for skin resistance along the grout body. Thus it is possible to define a constant, linear or trapezoidal distribution of skin resistance. The maximum interaction force between the soil and the grout body is directly applied in the “interface” of the embedded beam.

It is pointed out that this represents the skin resistance at failure (i.e. when the pull out force is reached) and that the skin traction distribution below full mobilisation is influenced by the specified limiting distribution. In reality mobilisation will start at the top of the grout body and only close to the pull out force (failure) the skin traction at the bottom should be mobilised. In the embedded pile the mobilisation follows the predefined shape from the beginning (also at the bottom). However, tests have shown that this does not have a noticeable influence on the global behaviour of an anchored structure under working load conditions.

Another important point is, that for forces close to the theoretical pull out force numerical failure may occur due to plasticity in the soil adjacent to the grout body. Although this is of course possible in reality, in the model it may be artificial and caused by the fact that the grout body is a line element. To overcome this problem in ultimate limit state conditions it is necessary to work with an enlarged diameter of the grout body. This virtual diameter of the grout body is defined as follows:

Dvirtual=f · Dreal

In this equation f is the factor for the enlargement, and a value of f in the range of 2 – 4 is suggested. This does not affect the pull out force (this is an input due the input of the limiting skin resistance and the length of the grout body) and has minor effect on the behaviour under working load conditions.

It follows, and the user must be aware of this, that when using this option in PLAXIS 3D FOUNDATION the maximum pull out force is an INPUT and cannot be OBTAINED from the analysis.


10-3

10.2 DEEP EXCAVATION WITH PRESTRESSED GROUND ANCHORS

In order to demonstrate the application of the ground anchors in PLAXIS 3D FOUNDATION, some results from a practical example, namely a deep excavation in Berlin sand, are presented. This example was chosen for testing the ground anchor facility under working load conditions because a 2D reference solution was available. The model dimensions and material sets for the soil layers have been taken from the 2D reference solution (Figure 10.2).

Figure 10.2: Geometry and subsoil conditions

The diaphragm wall has been modelled as a continuum element (Figure 10.3), with linear elastic material behaviour and a stiffness Eref=3.0·E7 kN/m2. The hydraulic cut off does not act as a structural element, the properties are the same as for the soil (sand 20 – 40m).

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Figure 10.3 3D view of the model (20944 elements)

To obtain the current porewater distribution inside the excavation the porewater pressure was defined after each groundwater lowering (with user defined pore pressure distribution).

The ground anchors have different spacing and prestress forces in the different layers and therefore the anchor rods have different properties. The properties of the grout body are the same in all rows (Table 10.2).

Table 10.2 Ground anchor properties Properties of the node-to-node anchor material type EA prestress force spacing anchor row1 elastic 2.87·E5 kN 768 kN 2.30m anchor row2 elastic 3.20·E5 kN 945 kN 1.35m anchor row3 Elastic 3.20· E5 kN 980 kN 1.35m Properties of the grout body grout body 2·E7 kN/m2 diameter 0.125m Aim of the test was to see if the embedded pile model (employed for the grout body) works well in working load conditions and therefore the skin resistance in the grout body has been defined about two times the expected axial load in the node-to-node anchor. In the different calculations the material model, the shape of the limiting skin


10-5

resistance and the enlargement of the grout body have been varied (Table 10.5). In the following tables the soil properties for the MC and the HS-Model are summarized.

Table 10.3: Soil parameters for the HS-model γunsat γsat k E50,ref Eoed,ref Eur,ref m

soil layer [kN/m3] [kN/m3] [m/day] [kN/m2] [kN/m2] [kN/m2] [-] sand 0-20m 17.0 20.0 - 45000 45000 180000 0.55

sand 20-40m 17.0 20.0 - 75000 750000 300000 0.55 sand >40m 17.0 20.0 - 105000 105000 315000 0.55

vur pref K0 nc cref Φ ψ Rinter soil layer [-] [kN/m2] [-] [kN/m2] [°] [°] [-]

sand 0-20m 0.2 100 0.426 1.0 35.0 5.0 0.8 sand 20-40m 0.2 100 0.384 1.0 38.0 6.0 0.8

sand >40m 0.2 100 0.384 1.0 38.0 6.0 -

Table 10.4: Soil parameters for the MC-model γunsat γsat Eref v cref φ ψ

soil layer [kN/m3] [kN/m3] [kN/m2] [-] [kN/m2] [°] [°] sand 0-20m 17.0 20.0 47000 0.3 1.0 35.0 5.0

sand 20-40m 17.0 20.0 244000 0.3 1.0 38.0 6.0 sand >40m 17.0 20.0 373000 0.3 1.0 38.0 6.0

Table 10.5 Variations in the different calculations material model shape of the skin friction distribution f-factor

calculation 1 HS constant 1 calculation 2 HS linear 1 calculation 3 HS linear 2 calculation 4 HS linear 4 calculation 5 HS linear 4 calculation 6 MC linear 1 In calculation 5 the stiffness of the grout body has been changed according to the ratio of the real diameter (0.125m) to the fictitious enlarged diameter (0.125·f=0.5m).

10.3 RESULTS

It follows from Figure 10.4 that neither the variation of the predefined limiting skin resistance of the grout body nor the f-factor for the enlargement of the grout diameter have a significant influence on the axial forces predicted under working load conditions. However the distribution of the mobilised skin traction along the grout body is not what one would expect in reality (Figure 10.5). If the Mohr Coulomb model is employed the results are slightly different (Figure 10.6).

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700,00

720,00

740,00

760,00

780,00

800,00

820,00

840,00

anchor row 1 GW -9,3m excavation 2 anchor row 2 GW -14,35m excavation 3 anchor row 3 GW -17,90m excavation 4

calculation phase

axia

l for

ce[k

N]

calculation 1calculation 2calculation 3calculation 4calculation 5

Figure 10.4 Axial forces in the first anchor row (calculation 1, 2, 3, 4, 5)

Figure 10.5 Mobilised skin friction and axial force – first anchor row (after excavation 4, calculation 1)

With respect to the horizontal displacements there is a trend that wall deflection with a linear predefined shape of the skin friction is slightly higher than the one with constant skin traction distribution. It is also notable that by increasing the f-factors for the virtual grout body diameter displacements in horizontal direction become smaller. The differences are in the order of 10%. With the MC-Model the highest deformations in horizontal direction are located around the grout body (Figure 10.7), whereas with the HS-Model this is not the case. This effect also occurs with the assignment of a high f-factor.


10-7

The settlements behind the diaphragm wall are in the range of 11mm (almost the same for the different variations) with the HS model, but with the MC model there is a heave of more than 14mm, an effect which is well known.

660,00

680,00

700,00

720,00

740,00

760,00

780,00

800,00

820,00

840,00

860,00

anchor row 1 GW -9,3m excavation 2 anchor row 2 GW -14,35m excavation 3 anchor row 3 GW -17,90m excavation 4

calculation phase

axia

l for

ce [k

N]

calculation 2calculation 6

Figure 10.6 Axial forces in the first anchor row – calculation 2 vs calculation 6

calculation 2 (HS-model) calculation 6 ( MC-model)

Figure 10.7 Horizontal displacements

10.4 COMPARISON OF 3D RESULTS WITH 2D REFERENCE SOLUTION

In Figure 10.8 axial forces in the first anchor row from calculation 2 (HS-model and f-factor=1) are compared with the axial forces from the 2D reference solution. In PLAXIS V8 the grout body of a ground anchor is modelled with geogrid elements. These elements have an axial stiffness but no bending stiffness. The axial forces from PLAXIS V8 are in the dimension [kN/m] and to compare these results with the 3D analysis it is necessary to divide the axial forces from PLAXIS 3D FOUNDATION by the anchor spacing of the different rows. One can see that the axial forces from the 3D calculations are in a

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very good agreement to the reference solution. The deviation of the forces in the node-to-node anchor between both calculations is less than 4%. Also the vertical displacements behind the diaphragm wall from the 3D calculation (Figure 10.9) are very similar to the ones obtained from the 2D solution (both maximum value and distribution).

300

310

320

330

340

350

360

370

380

anchor 1 GW -9,3m excavation 2 anchor 2 GW -14,35m excavation 3 anchor 3 GW -17,90m excavation 4calculation phase

axia

l for

ce [k

N/m

]

first anchor row/calculation 2first anchor row/2D reference solution

Figure 10.8 Axial forces calculation 2 vs. 2D reference solution

Figure 10.9: Vertical displacements behind the diaphragm wall – comparison reference

solution with calculation 2


10-9

10.5 CONCLUSIONS

A deep excavation supported by a diaphragm wall and three rows of anchors has been analysed utilizing PLAXIS 3D FOUNDATION with the ground anchor option.

The results from the 3D calculation with the HS-Model compare well to the 2D reference solution (both with respect to anchor forces and displacements) and as a consequence from the parametric study it can be concluded that it is not necessary to artificially increase the diameter of the grout body for working load conditions.

Concerning the distribution of the skin friction along the grout body, it is obvious that the mobilisation is not realistic. The reason is, that also at working load conditions the distribution of the skin friction is strongly influenced by the distribution in the failure state, which is an input. Due to the fact that the limiting skin friction is an input the grout body length has no or minor influence on the result and therefore the length cannot be determined from the analysis.

Compared to the HS-Model the MC-Model predicts significantly larger deformations around the grout body. The virtual enlargement of the grout body diameter (f-factor) does not change the results significantly for working load conditions.

However for ultimate limit state calculations the f-factor becomes important, because in these calculations a premature failure (i.e. a failure below the theoretical pull out force) may occur when f=1.0. To overcome this problem it is essential to work with a virtual grout body enlargement.

It follows from this study that the ground anchor concept in PLAXIS 3D FOUNDATION is efficient for working load conditions, but for ultimate limit state analysis assumptions such as the f-factor, mesh coarseness and stiffness parameters of the soil (adjacent to the grout body) may have a significant influence on the result. It is emphasized again the maximum pull out force is an INPUT to the analysis and not a RESULT.

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REFERENCES

11-1

11 REFERENCES

[1] Bakker K.J., (2000), Soil Retaining Structures; development of models for structural analysis. Dissertation (Delft University of Technology). Balkema, Rotterdam.

[2] Blake, A., (1959), Deflection of a Thick Ring in Diametral Compression, Am. Soc. Mech. Eng., J. Appl. Mech., Vol. 26, No. 2.

[3] Butterfield, R., and Banerjee, P.K., (1971). The elastic analysis of compressible piles and pile groups. Géotechnique, Vol. 21, No. 1, pp. 43-60.

[4] Butterfield, R., and Banerjee, P.K., (1971). The problem of pile group-pile cap interaction. Géotechnique, Vol. 21, No. 2, pp. 135-142.

[5] Cox, A.D., (1962), Axially-symmetric plastic deformations - Indentation of ponderable soils. Int. Journal Mech. Science, Vol. 4, 341-380.

[6] Davis, E.H. and Booker J.R., (1973), The effect of increasing strength with depth on the bearing capacity of clays. Geotechnique, Vol. 23, No. 4, 551-563.

[7] Engin, H.K., Septanika, E.G., and Brinkgreve, R.B.J., (2007). Improved Embedded Beam Elements for the Modelling of Piles. Numog X.

[8] Gibson, R.E., (1967), Some results concerning displacements and stresses in a non-homogeneous elastic half-space, Geotechnique, Vol. 17, 58-64.

[9] Giroud, J.P., (1972), Tables pour le calcul des foundations. Vol.1, Dunod, Paris. [10] Van Langen, H., (1991). Numerical Analysis of Soil-Structure Interaction. PhD

thesis Delft University of Technology. PLAXIS users can request copies. [11] Mattiasson, K., (1981), Numerical results from large deflection beam and frame

problems analyzed by means of elliptic integrals. Int. J. Numer. Methods Eng., 17, 145-153.

[12] McMeeking, R.M., and Rice, J.R., (1975). Finite-element formulations for problems of large elastic-plastic deformation. Int. J. Solids Struct., 11, pp. 606-616.

[13] El-Mossallamy, Y., (1999). Load-settlement behaviour of large diameter bored piles in over-consolidated clay. Proceeding of the 7th. International Symposium on Numerical Models in Geotechnical Engineering, Graz, Austria, September 1999, pp. 443-450.

[14] El-Mossallamy, Y., (2004). The Interactive Process between Field Monitoring and Numerical Analyses by the Development of Piled Raft Foundation. Geotechnical innovation, International symposium, University of Stuttgart, Germany, 25 June 2004, pp. 455-474.

[15] El-Mossallamy, Y., Lutz, B., and Richter, Th., (2006). Innovative application and design of piled raft foundation. 10th International Conference on Piling and Deep Foundations, Amsterdam, Netherlands.

[16] Ostermayer, H. and Barley, T., (2003). Ground Anchors. Geotechnical Engineering Handbook, Vol. 2, pp. 169-219.

[17] Poulos, H.G. and Davis, E.H., (1974), Elastic solutions for soil and rock mechanics. John Wiley & Sons Inc., New York.

[18] Randolph, M.F. and Wroth, C.P., (1978). Analysis of deformation of vertically loaded piles. ASCE, Vol. 104, No. GT12, pp. 1485-1488.

[19] Roark, R. J., (1965), Formulas for Stress and Strain, McGraw-Hill Book Company.

VALIDATION MANUAL


[20] Sagaseta, C., (1984), Personal communication. [21] Sommer, H. and Hambach, P., (1974). Großpfahlversuche im Ton für die

Gründung der Talbrücke Alzey. Der Bauingenieur, Vol. 49, pp. 310-317 [22] Verruijt, A., (1983), Grondmechanica (Geomechanics syllabus). Delft University of

Technology.

3DF2 5 Validation 3D Plaxis

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