benchmarking in geotechnics-1 part-ii

PART II: REFERENCE SOLUTION AND PARAMETRIC STUDY ABSTRACT

The influence of different modelling assumptions on the results of a numerical analysis of the deep

excavation problem, which has been introduced in part I of this report, is discussed. Based on a

reference solution, which claims to represent closely the actual behaviour observed in situ, a

comprehensive parametric study is performed, identifying modelling assumptions which have a high

influence on the calculated displacement behaviour and the bending moments in the wall, and others

which do not significantly alter the results obtained. The parameters investigated include wall friction,

domain chosen for the analysis, discretisation, length and prestressing force of ground anchors,

constitutive models and variation of input parameters for stiffness and strength. It can be concluded

from this study that care must be taken when setting up a numerical model because the sum of

various assumptions, not considered to be of large importance when looked at it individually, may

significantly influence the outcome of the numerical calculation.

1 INTRODUCTION

In part I of this report it has been shown that results obtained from a numerical analysis of a practical

problem may scatter significantly depending on assumptions made in establishing the numerical

model. Not only constitutive model and material parameters, but also choice of element type for

modelling soil, structural elements and soil/structure interaction, finite element discretisation and

boundary conditions will influence results. Although the individual influence of each of these modelling

details on the deformation behaviour under working load conditions may not be very significant, the

sum of all these assumptions, which have to be made when establishing a computational model, may

finally lead to quite different results, entirely depending on the user and the software utilized. This has

been demonstrated in part I of this report where results from a benchmark exercise have been

discussed.

In order to study the influence of various modelling assumptions without introducing software specific

features, a systematic parametric study for the deep excavation problem described in part I is

performed with a particular code. Based on a reference solution a series of calculations has been

made varying only one parameter or modelling detail at once so that its influence could be separated.

It is emphasized that no exact solution exists for the this problem, but it is claimed that the model

discussed here is reasonably advanced, at least from a practical point of view, so that realistic results

can be expected.

C O M P UT A TI O N A LGEOTECHNICSG R O UP p a ge 26

The commercial finite element code PLAXIS V7.2 is used and the constitutive model employed is the

so-called Hardening Soil model. A brief description of the main features of this elastic-plastic

constitutive model is provided. However, it is emphasized that using this particular code and

constitutive model does not imply that it is a recommendation for using this software. It is argued that

the analyses shown in the following are representative for an advanced engineering analysis and the

findings of this study also hold for other programmes and other constitutive models with similar

features.

2 PROBLEM DEFINITION

Although the problem considered for this study corresponds to the one described in part I of this report

some details of the specification are repeated her because some minor changes have been made and

additional model parameters are defined for the reference analysis.

2.1 Geometry, basic assumptions and computational steps

The general layout has been given already in part I of this report but is shown here again for continuity

(Figure 1). The domain analysed has been chosen as follows: width = 150 m, depth = 100 m. The

mesh consists of approximately 1800 6-noded elements, which is refined in areas where high stress

gradients can be expected. The mesh was deliberately chosen to be relatively fine in order to minimize

the discretisation error (Figure 2).

23.3m

8.0m

8.0m

19.8m

27°

27°

0.8m

-32.00m = base of diaphragm wall

excavation step 3 = -14.35m

GW = -3.00m below surface

0.00m

excavation step 2 = - 9.30m

excavation step 1 = - 4.80m

30 m

Specification for anchors:prestressed anchor force: 1. row: 768KN 2. row: 945KN 3. row: 980KN

horizontal distance of anchors: 1. row: 2.30m 2. row: 1.35m 3. row: 1.35mcross section area: 15 cm2

Young's modulus E = 2.1 e8 kN/m2

top of hydraulic barrier = -30.00m

excavation step 4 = -16.80m-17.90m 23.8m 8.0m

27°

120 m

68

m

sand

γ'=γ'sand

x

z

Fig. 1 Geometry and excavation stages


Fig. 2 Finite element mesh for reference solution

The following assumptions have been postulated:

- plane strain

- influence of diaphragm wall construction is neglected, i.e. initial stresses without wall, then

wall "wished-in-place" (weight of wall, γb = 24 kN/m3, considered as difference to soil

weight)

- diaphragm wall modelling: beam elements (E b = 30e6 kPa, < b = 0.15, d = 0.8 m)

- interface elements between wall and soil

- horizontal hydraulic cut off at -30.00 m is not considered as structural support, the same

mechanical properties as for the surrounding soil are assumed

- hydrostatic water pressures corresponding to water levels inside and outside excavation

(groundwater lowering is performed in steps in advance to the respective excavation step)

- anchors are modelled as rods, the grouted body as membrane element (geotextile

element in PLAXIS terminology) which guarantee a continuous load transfer to the soil

- given anchor forces in Figure 1 are design loads

The following computational steps have been performed:

stage 0: initial stress state (given by σv = γz, σh = Koγz, Ko = 0.43)

stage 1: activation of diaphragm wall and groundwater lowering to -4.90 m

stage 2: excavation step 1 (to level -4.80 m)

stage 3: activation of anchor 1 at level -4.30 m and prestressing

stage 4: groundwater lowering to -9.40 m

stage 5: excavation step 2 (to level -9.30 m)


stage 6: activation of anchor 2 at level -8.80 m and prestressing

)

and prestressing

)

istance and prestressing loads for anchors follow from Figure 1.

2.2 Brief description of PLAXIS Hardening Soil model

The so-called "Hardening Soil model" is an elastic-plastic constitutive model which incorporates shear

The basic characteristics of the model can be summarizes as following:

stress for triaxial stress paths

Figure 3 includes the stress – strain relation for triaxial compression on which the "Hardening-Soil

ig. 3 Hyperbolic stress-strain relation of the Hardening Soil model for drained triaxial compression test


stage 8: excavation step 3 (to level -14.35 m

stage 9: activation of anchor 3 at level -13.85 m


stage 11: excavation step 4 (to level –16.80 m

D

hardening and volumetric hardening with an associated flow rule for the cap yield surface and a non-

associated flow rule for the deviatoric yield surface (references see part I).

- stress dependent stiffness according to a power law

- hyperbolic relationship between strain and deviatoric

- distinction between primary loading and unloading / reloading

- failure according to Mohr-Coulomb criterion

model" is based. Figure 4 schematically shows the deviatoric yield surfaces described by the

hardening parameter (function of plastic shear strains). In addition to deviatoric hardening, volumetric

hardening is considered and consequently plastic volumetric strains occurring for hydrostatic or Ko-

stress paths are taken into account. Figure 5 contains the complete yield surface in p-q space.

F


Fig. 4 Schematic representation of deviatoric hardening in the Hardening Soil model

Fig. 5 Schematic representation of th dening Soil model in p-q space

In addition to the Mohr Coulomb strength parameters the Hardening Soil model requires the following

E50 : secant modulus for primary loading at 50% of the maximum deviatoric stress in triaxial

Eur : modulus for unloading / reloading at a reference stress (= σ3´)

Eoedref : value of tangent modulus in one-dimensional compression at a reference stress (= Φ1´)

The stress dependence of the modulus is defined by an exponent m. pref denotes the reference stress.

E50 =

p'

q

Mohr-Coulomb'sc

he Bruchgerade

e Har

input parameters representing the stiffness of the soil (a complete list of input parameters is given in

section 2.3):

ref

compression test at a reference stress (= σ3´)

ref

⎟⎟⎠

⎞⎜⎜⎝

⎛

p + c’ - c E ref

3

m

ref50

ϕσϕ

cotcot


The stress dependence of the unloading modulus Eur is defined in analogy to the relation for E50. The

so-called “cap” used to describe the volumetric strain behaviour is defined by the stiffness modulus of

the oedometer test (Eoed) which is also stress dependent.

2.3 Material parameters for Hardening Soil model

The basic set of material parameters used to obtain the reference solution is based on data available

in the literature and also on published experimental data from triaxial and one-dimensional

compression test for Berlin sand. It is emphasized at this point that parameter identification for

numerical analysis for a given soil based on experimental data is not an easy task. For more

advanced constitutive models this procedure depends highly on the model employed and cannot be

generalized.

Although the Hardening Soil model takes into account the stress dependency of stiffness for primary

loading as well as unloading/reloading stress paths, three layers (see Table 1) are introduced in order

to increase this effect and to take into account the high stiffness at low strains, which will be prevailing

in most of the deeper layers of the domain analysed, at least in a very approximate way.

Rinter in Table 1 determines the reduction of strength parameters ν and c in the interface elements as

compared to the surrounding soil (tanνinter = Rinter tanν, cinter = Rinter c). The stiffness of the interface is

reduced as well. A value of 1 kPa is introduced for the cohesion which improves numerical stability,

this is however not strictly required.

depth of layer E50ref Eur

ref Eoedref ν Ρ c <ur pref m Rf Rinter

m kPa kPa kPa ° ° kPa - kPa - - -

0 - 20 45 000 180 000 45 000 35 5 1.0 0.2 100 0.55 0.9 0.8

20 - 40 75 000 300 000 75 000 38 6 1.0 0.2 100 0.55 0.9 0.8

> 40 105 000 315 000 105 000 38 6 1.0 0.2 100 0.55 0.9 -

Table 1 Material parameters for HS-Model - reference solution

2.4 Material parameters for structural elements

diaphragm wall EA = 2.4e7 kN/m EI = 1.28e6 kNm2/m < = 0.15 w = 7.5 kN/m/m


anchor row 1 EA = 2.87e5 kN

anchor rows 2 and 3 EA = 3.22e5 kN

membrane elements for modelling grout body (anchor row 1) EA = 4.92e5 kN/m

membrane elements for modelling grout body (anchor rows 2 and 3) EA = 8.38e5 kN/m

Anchor row 1 and rows 2 and 3 have been modelled with different cross sections respectively.

3 RESULTS FOR REFERENCE SOLUTION

In the following the most relevant results obtained for the reference solution are presented. Unlike

otherwise stated the last construction stage is considered. In addition a few results for the first

excavation step (no anchors installed) are shown.

Deformed MeshExtreme total displacement 46.55*10-3 m

(displacements scaled up 100.00 times)

Fig. 6 Deformed mesh (detail) - reference solution


In Figure 6 the deformed mesh is shown and in Figure 9 the surface settlements are plotted for the

first and final excavation stage. Settlements increase from approximately 5 mm for the first stage to

over 15 mm for the final stage, which can be considered to be a very plausible result. Figure 7 depicts

the lateral displacement of the wall together with the inclinometer measurements, again for the first

and final excavation step. The measurements for the final stage have been corrected for lateral

movement of the base of the wall which is not reflected in the inclinometer measurement but most

likely to occur (see also comments in part I of this report). Figure 8 shows calculated bending

moments.

horizontal displacement [mm]

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

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measurement (final stage)measurement correctedreference solution (final stage)measurement (1. excavation stage)reference solution (1. excavation stage)

bending moments [kNm/m]

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final stage1. excavation stage

Fig. 7 Wall deflection - reference solution Fig. 8 Bending moments - reference solution


distance from wall [m]

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verti

cal d

ispl

acem

ent o

f sur

face

[mm

]

-30

-25

-20

-15

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-5

0

5

10

final stage1. excavation stage

Fig. 9 Surface settlements - reference solution

4 INFLUENCE OF VARIOUS MODELLING ASSUMPTIONS

In this chapter modelling assumptions such as the dimensions of the domain analysed, modelling of

wall friction and grout body of anchors and variation in stiffness and strength parameters are

investigated. The influence of the constitutive model is addressed in chapter 5.

4.1 Influence of groundwater lowering

For the reference solution the groundwater lowering was performed in steps in advance to the

corresponding excavation steps. In order to study the influence of performing the groundwater

lowering in one step to the final level of drawdown (as specified in the benchmark exercise discussed

in part I of this report) an analysis has been made where the groundwater level was lowered in the

numerical model to the final level of -17.90 m below surface before the first excavation step. The

results of this analysis follow from Figures 10 to 12 and it can be seen that the maximum wall

deflection increases by approximately 10 mm, when the Hardening Soil model is used. However as

will be discussed in chapter 5 the influence of this modelling detail depends on the constitutive model

employed.



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verti

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ispl

acem

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f sur

face

[mm

]

-30

-25

-20

-15

-10

-5

0

5

10

reference solution(final stage)1 step GW-lowering(final stage)reference solution(1. excavation step)1 step GW-lowering(1. excavation step)

Fig. 10 Surface settlements - influence of groundwater lowering


-1000 -800 -600 -400 -200 0 200 400 600

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[m]

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reference solution(final stage)1 step GW-lowering(final stage)reference solution(1. excavation step)1 step GW-lowering(1. excavation step)


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

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reference solution(final stage)1 step GW-lowering(final stage)reference solution(1. excavation step)1 step GW-lowering (1. excavation step)

Fig. 11 Wall deflection - influence of Fig. 12 Bending moments - influence of

groundwater lowering groundwater lowering


4.2 Influence of wall friction

For the reference solution the interface elements implemented in the code PLAXIS have been used in

a standard way, i.e. the factor Rinter was used in order to reduce the strength properties of the interface

elements with respect to the surrounding soil (see section 2.3). The elastic stiffness of the interface

elements is governed by a so-called "virtual thickness" which is based on the average element size of

the mesh adjacent to the wall. For the reference solution Rinter = 0.8 has been assumed, a value which

is based on experience. In order to study the effect of this parameter an analysis has been performed

changing Rinter to 0.5. It follows from Figures 13 to 15 that this parameter has a significant influence on

the displacements. The horizontal displacement of the top of the wall increases by approx. 25 mm and

the settlement behind the wall by approx. 15 mm. Bending moments to not change significantly. In

order to evaluate the influence of the elastic properties of the interface elements a calculation was

performed with a reduced virtual thickness as compared to the default value set in PLAXIS, thus the

stiffness of the interface is increased. This results in a reduction of the maximum horizontal

displacement of the wall in the order of 5 mm (Figure 14).

It is obvious from these results that input parameters for modelling wall / soil interaction have to be

chosen very carefully, which is however a difficult task because the elastic stiffness of an interface is

not a well defined mechanical property. Although results presented here are related to the particular

interface element formulation implemented in PLAXIS it can be expected that other formulations will

show a similar sensitivity to input parameters.


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verti

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acem

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[mm

]

-40

-35

-30

-25

-20

-15

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-5

0

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10

reference solutionRinter = 0.5

Fig. 13 Surface settlements - influence of wall friction



-70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0

-70 -65 -60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0

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reference solution Rinter = 0.8 (final stage) Rinter = 0.5 (final stage)Rinter = 0.8 t_virt = 0.01 (final stage)reference solution(1. excavation stage)Rinter = 0.5 (1. excavation stage) Rinter = 0.8 t_virt = 0.01(1. excavation stage)


-1000 -800 -600 -400 -200 0 200 400 600

-1000 -800 -600 -400 -200 0 200 400 600

dept

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[m]

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reference solutionRinter = 0.5

Fig. 14 Wall deflection - influence of wall friction Fig. 15 Bending moments - influence of wall friction

4.3 Influence of discretisation

In this analysis a relatively coarse mesh was chosen, reducing the number of elements from approx.

1800 to 460 (Figure 16). However, zones with high stress gradients have been refined again and

therefore the differences in displacements are very small (Figure 17). Examining bending moments

one would argue that the mesh may be slightly to crude to obtain a smooth distribution and therefore

some differences as compared to the reference solution can be observed (Figure 18).


Fig. 16 Finite element mesh - coarse mesh


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reference solutioncoarse mesh


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[m]

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reference solutioncoarse mesh

Fig. 17 Wall deflection - influence of discretisation Fig. 18 Bending moments - influence of discretisation


4.4 Influence of domain analysed

For the reference solution the domain analysed was chosen as 150 x 100 m for width (W) and depth

(D) of the mesh respectively. In order to study the influence of the discretised domain chosen the

following analyses have been performed: W x D = 150 x 70, W x D = 100 x 100, W x D = 100 x 70 and

W x D = 200 x 150 m. It follows from Figures 19 to 21 that bending moments are hardly influenced but

displacements (horizontal displacements of wall and surface settlements) differ in the order of approx.

6 mm, the deeper meshes resulting in smaller surface settlements. This is a result of the vertical

upwards displacements caused by the elastic unloading due to excavation which increases with

deeper meshes. The width of the mesh has practically no influence on the surface settlements.


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

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reference solution D=100 W=150D=100 W=100 D=70 W=100D=70 W=150D=150 W=200reference solutionD=100 W=150D=100 W=100D=70 W=100D=70 W=150D=150 W=200

final stage

1. excavation stage


-1000 -800 -600 -400 -200 0 200 400 600

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reference solution D=100 W=150D=100 W=100 D=70 W=100 D=70 W=150 D=150 W=200

Fig. 19 Wall deflection - influence of Fig. 20 Bending moments - influence of domain analysed domain analysed



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[mm

]

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reference solution D=100 W=150D=100 W=100 D=70 W=100 D=70 W=150 D=150 W=200

Fig. 21 Surface settlements - influence of domain analysed

4.5 Influence of modelling ground anchors


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verti

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[mm

]

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reference solutionno membrane elementfor load transferno membrane elementfree anchor length increased

Fig. 22 Surface settlements - influence of modelling grout body

When using the code PLAXIS the load transfer from the free length of the ground anchors into the

ground can be conveniently modelled with membrane elements. These elements, which have no

bending stiffness but axial stiffness only, allow a continuous load transfer from the membrane element

to the ground along its entire length and avoid a concentrated point load at the end of the free anchor

length. Of course this modelling technique is only applicable for working load conditions because the

limiting pull out force cannot be taken into account correctly with this simple model. To emphasize the


importance of a continuous load transfer along the grout body two analyses without membrane

elements have been performed. In the first one the free anchor length has been kept the same as in

the reference solution and in the second one the free anchor length has been increased by half of the

length of the grout body in order to compensate for not modelling the load transfer in more detail.

Figures 22 to 24 clearly show that care must be taken when choosing the model representing the

ground anchor and grout body.


-140 -120 -100 -80 -60 -40 -20 0

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-1250 -1000 -750 -500 -250 0 250 500 750 1000

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Fig. 23 Wall deflection - influence of Fig. 24 Bending moments – influence of modelling grout body modelling grout body

4.6 Influence of free anchor length

The strong influence of the free anchor length could be seen from the previous section already but is

addressed again in this chapter. One analysis assumed an increase of free anchor length of 8 m as

compared to the reference solution, the other calculation a decrease of 4 m. The length of the grout

body (8 m) has not been changed. Figures 25 to 27 reveal a significant increase of displacements for


the short anchor (-4 m) and roughly the same amount of decrease in displacements for the long

anchor (+8 m), thus the relationship is, as expected, highly nonlinear. Of course the choice of the free

anchor length, and some other variations discussed in the following, are design assumptions and not

modelling assumptions at the discretion of the numerical analyst, but they are included here to

complete the parametric study.


-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0

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reference solutionanchor -4manchor +8m


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reference solutionanchor - 4 manchor + 8 m

Fig. 25 Wall deflection - influence of free Fig. 26 Bending moments - influence of free

anchor length anchor length



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[mm

]

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reference solutionanchor -4m anchor +8m

Fig. 27 Surface settlements - influence of free anchor length

4.7 Influence of anchor prestress force

A variation of +/- 25% of the anchor prestress force as compared to the reference solution has been

assumed and again a nonlinear relationship between calculated displacements and applied prestress

force in the anchors is observed (Figures 28 to 30), i.e. a reduction of anchor force of 25% results in

an increase of lateral displacements and settlements of approx. 40%. Maximum bending moments

increase roughly at the same amount.


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[mm

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reference solutionprestress force +25% prestress force -25%

Fig. 28 Surface settlements - influence of anchor prestress force



-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0

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reference solutionprestress force +25% prestress force -25%


-1000 -800 -600 -400 -200 0 200 400 600

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reference solutionprestress force +25%prestress force -25%

Fig. 29 Wall deflection - influence of Fig. 30 Bending moments - influence of anchor prestress force anchor prestress force

4.8 Influence of wall thickness

In the reference solution a diaphragm wall of thickness d = 0.8 m was assumed, and as follows from

Figure 31 an increase to d = 1.2 m does reduce the maximum horizontal displacement, but not

significantly. Surface settlements also do not change much. Of course bending moments increase

dramatically (Figure 32).



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reference solution(final stage)wall120 cm (final stage)reference solution(1. excavation stage)wall 120 cm(1. excavation stage)


-1250 -1000 -750 -500 -250 0 250 500

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reference solution(final stage)wall120 cm(final stage)reference solution(1. excavation stage)wall 120 cm(1. excavation stage)

Fig. 31 Wall deflection - influence of Fig. 32 Bending moments - influence of wall thickness wall thickness

4.9 Influence of variation in stiffness and strength parameters of soil

In part I of this report the difficulties in defining appropriate material parameters, in particular stiffness

properties, have been briefly addressed. In order to quantify the uncertainties in material properties on

the solution of the deep excavation problem considered here, the stiffness parameters have been

varied by +/-25% and the strength parameters by +/-10% respectively. In the latter case the friction

angle for all layers has been varied, but the dilatancy angle has been kept the same as in the

reference analysis. The variation in stiffness parameters was introduced by changing E50ref, Eoed

ref and

Eurref for all layers accordingly. The results follow from Figures 33 to 38 and the large differences in

calculated horizontal displacements and settlements emphasize again that careful considerations are

required in determining appropriate input parameters.



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reference solution(final stage)stiffness+25%(final stage)stiffness -25%(final stage)reference solution(1. excavation stage)stiffness +25%(1. excavation stage)stiffness -25%(1. excavation stage)


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reference solutionstiffness +25%stiffness -25%

Fig. 33 Wall deflection - influence of soil stiffness Fig. 34 Bending moments - influence of soil stiffness


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verti

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acem

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f sur

face

[mm

]

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-5

0

5

10

reference solutionstiffness +25%stiffness -25%

Fig. 35 Surface settlements - influence of soil stiffness



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reference solution(final stage)friction angle +10%(final stage)friction angle -10%(final stage)reference solution(1. excavation stage)friction angle +10%(1. excavation stage)friction angle -10%(1. excavation stage)


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reference solutionfriction angle +10%friction angle -10%

Fig. 36 Wall deflection - influence of soil strength Fig. 37 Bending moments - influence of soil strength


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verti

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acem

ent o

f sur

face

[mm

]

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-5

0

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reference solutionfriction angle +10%friction angle -10%

Fig. 38 Surface settlements - influence of soil strength


It has to be mentioned that the variation in friction angle has different consequences depending on the

constitutive model employed. A reduction of 10% increases the lateral displacement of the top of the

wall from 30 mm to approx. 40 mm when using the Hardening Soil model, whereas the same variation

leads to an increase from 20 mm to 40 mm when employing the Mohr-Coulomb model with parameter

set MC_1 (see section 5.2.3 and Figure 59)

4.10 Wall deflection and anchor forces for all construction stages

Wall deflection

In Figures 39 and 40 the development of the lateral displacements of the diaphragm wall are shown

for the reference solution and the solution where the groundwater lowering was simulated in one step

(see section 4.1). The significant increase of lateral displacement in excavation steps 3 and 4 is clearly

evident.


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GW-lowering 1excavation 1anchor 1GW-lowering 2excavation 2anchor 2GW-lowering 3excavation 3anchor 3GW-lowering 4final excavation


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GW-lowering tofinal levelexcavation 1anchor 1excavation 2anchor 2excavation 3anchor 3final excavation

Fig. 39 Wall deflection for all construction Fig. 40 Wall deflection for all construction steps - reference solution steps - groundwater lowering 1 step


Anchor forces

Figures 41 and 42 show the development of anchor forces for rows 1 and 2 with progressing

excavation. An increase in anchor forces is calculated in row 1 for construction step 5 and row 2 for

construction step 8 respectively. This holds for all analyses, but depending on the modelling

assumptions the amount of increase of force differs. In Figure 43 the change in anchor forces for rows

1 and 2 is depicted for the cases where length of anchors and prestress forces have been varied. It

follows that the anchor length does not have a significant influence on the change of anchor forces,

but - as expected - the applied prestress force has.

computational step

3 4 5 6 7 8 9 10 1

forc

e in

anc

hor [

kN/m

]

0

100

200

300

400

500

reference solutionRinter = 0.5 GW-lowering 1 stepno membrane elementwall 120 cmstiffness +25%stiffness -25%friction angle +10%friction angle -10%

1

Fig. 41 Development of anchor forces - upper row

computational step

6 7 8 9 10

forc

e in

anc

hor [

kN/m

]

0

100

200

300

400

500

600

700

800

900

1000

reference solutionRinter = 0.5GW-lowering 1 stepno membrane elementswall = 120 cmstiffness +25%stiffness -25%friction angle +10%friction angle -10%

11

Fig. 42 Development of anchor forces - middle row


computational step

3 4 5 6 7 8 9 10 1

forc

e in

anc

hor [

kN/m

]

0

100

200

300

400

500

600

700

800

900

1000 reference solutionfree length +8mfree length -4mprestress force +25%prestress force -25%reference solutionfree length +8mfree length -4mprestress force +25%prestress force -25%

upper row

middle row

1

Fig. 43 Development of anchor forces - upper and middle row - influence of anchor length and prestress

force

5 INFLUENCE OF CONSTITUTIVE MODEL

Because linear elastic - perfectly plastic constitutive models are still widely used in practice some

analyses are performed with a Mohr-Coulomb model, and for purposes of comparision linear elastic

analyses are also included and compared to the reference solution.

5.1 Elastic analysis

In this elastic analysis again three soil layers are introduced in order to take into account an increase

of stiffness with depth in an approximate way. A constant Young's modulus in each layer is assumed

(Table 2), based on the values given in the specification of the benchmark problem for Berlin sand

(see part I of this report).

depth of layer E < Rinter

m kPa - -

0 - 20 47 000 0.3 0.8

20 - 40 244 000 0.3 0.8

> 40 373 000 0.3 -

Table 2 Young's - moduli for elastic analysis

The results of this analysis clearly show the well known fact that calculations based on linear elastic

material behaviour do not produce a realistic deformation pattern, which is evident from Figures 44


and 45 indicating a heave of the surface behind the wall of approximately 20 mm. In Figures 45 to 47

results for an analysis where the groundwater lowering is performed in one step are also included and

it follows that - as expected - for elastic material behaviour both analysis do not differ much, in contrary

to the elastic-plastic analyses. In Figure 48 the development of wall deflection with progressing

excavation is plotted and the differences, qualitatively and quantitatively, to elastic-plastic analyses are

evident (compare to Figure 39).

Deformed MeshExtreme total displacement 56.66*10-3 m(displacements scaled up 200.00 times)

Fig. 44 Deformed mesh (detail) - elastic solution


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reference solutionelasticGW-lowering stepwiseelasticGW-lowering 1 step

Fig. 45 Surface settlements - elastic solution



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reference solution(final stage)elastic GW-lowering stepwise(final stage)elasticGW-lowering 1 step(final stage)reference solution(1. excavation stage)elasticGW-lowering stepwise(1. excavation stage)


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reference solutionelasticGW-lowering stepwiseelasticGW-lowering 1 step

Fig. 46 Wall deflection - elastic solution Fig. 47 Bending moments - elastic solution



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GW-lowering 1excavation 1anchor 1GW-lowering 2excavation 2anchor 2GW-lowering 3excavation 3anchor 3GW-lowering 4final construction stage

Fig. 48 Wall deflection - elastic solution

5.2 Elastic-perfectly plastic analyses (Mohr-Coulomb model)

5.2.1 MC-model 1 and 2

Analysis MC_1 corresponds to the elastic analysis discussed in the previous section as far as Young's

moduli are concerned but introduces the Mohr-Coulomb failure criterion.

depth of layer E ν Ρ c < Rinter

m kPa ° ° kPa - -

0 - 20 47 000 35 5 1.0 0.3 0.8

20 - 40 244 000 38 6 1.0 0.3 0.8

> 40 373 000 38 6 1.0 0.3 -

Table 3 Material parameters MC-model 1


Analysis MC_2 assumes a linear increase of stiffness with depth according to the specification of the

benchmark problem for Berlin sand (see part I), assuming a minimum value of E = 33 000 kPa. The

complete list of parameters used for these analyses is given in Tables 3 and 4.

depth of layer E ν Ρ c < Rinter

m kPa ° ° kPa - -

0 - 5 33 000 35 5 1.0 0.3 0.8

5 - 20 33 000 - 66 500 (linear) 35 5 1.0 0.3 0.8

20 - 40 200 000 - 282 000 (linear) 38 6 1.0 0.3 0.8

40 - 100 282 000 - 446 000 (linear) 38 6 1.0 0.3 -

Table 4 Material parameters MC-model 2


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elastic GW-lowering stepwiseelastic GW-lowering 1 stepMC_1 GW-lowering stepwiseMC_2 Gw-lowering stepwiseMC_1 GW-lowering 1 step MC_2 GW-lowering 1 step


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elasticGW-lowering stepwiseelasticGW-lowering 1 stepMC_1 GW-lowering stepwiseMC_2 GW-lowering stepwise MC_1 GW-lowering 1 step MC_2 GW-lowering 1 step

Fig. 49 Wall deflection - MC-models 1 and 2 Fig. 50 Bending moments - MC-models 1 and 2


Figures 49 to 51 compare the MC-analyses with the elastic solution for both cases of groundwater

lowering (stepwise and one step) and it is apparent that the groundwater lowering in one step

increases the horizontal displacement of the top of the wall by approximately 15 mm, which is slightly

more than in the reference solution (the reference solution is not included because the stiffness

parameters are not comparable in these cases). It is evident that MC-models tend to produce a

surface heave in a similar way as the elastic solutions. Bending moments do not differ significantly

with the exception of the lower end of the wall.


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elasticGW-lowering stepwiseelasticGW-lowering 1 stepMC_1 GW-lowering stepwiseMC_2 GW-lowering stepwise MC_1 GW-lowering 1 step MC_2 GW-lowering 1 step

Fig. 51 Surface settlements - MC-models 1 and 2

5.2.2 MC-model 3 and 4

Here the Young's moduli are back-calculated for each layer corresponding to the initial stiffness

introduced in the Hardening Soil model of the reference solution in the middle of each layer (Table 5).

MC_3 uses the loading stiffness (E50ref) and MC_4 the unloading stiffness (Eur

ref). It follows from the

results shown in Figures 52 to 54 that high differences are obtained for displacements but less for

bending moments. Again the deficiencies of elastic-perfectly plastic models become apparent, in

particular when looking at the displacements behind the surface in Figure 54.

depth of layer E (MC_3) E (MC_4) ν Ρ c < Rinter

m kPa kPa ° ° kPa - -

0 - 20 32 000 128 000 35 5 1.0 0.3 0.8

20 - 40 90 000 360 000 38 6 1.0 0.3 0.8

> 40 196 000 588 000 38 6 1.0 0.3 -

Table 5 Material parameters MC-model 3 and 4



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reference solutionMC_3 (E-loading) MC_4 (E-unloading)


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reference solutionMC_3 (E-loading)MC_4 (E-unloading)

Fig. 52 Wall deflection - MC-models 3 and 4 Fig. 53 Bending moments - MC-models 3 and 4


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reference solutionMC_3 (E-loading) MC_4 (E-unloading)

Fig. 54 Surface settlements - MC-models 3 and 4


5.2.3 Parameter variation with MC-model 1

In this section a limited parametric study, similar to section 4, is performed with the parameter set of

MC_1 as basic analysis. It is interesting to see that with certain, however not very realistic,

assumptions the Mohr-Coulomb model calculates a similar lateral deflection of the wall as the

Hardening Soil model (Figure 55). A match of surface settlements however cannot be achieved

(Figure 57). Bending moments are not influenced so much (Figure 56). Again the strong influence on

the results of the assumptions made for wall friction is obvious. It is worth noting that the Poisson's

ratio has a pronounced influence in the Mohr-Coulomb model, which is not the case in the Hardening

Soil model because in the elastic-plastic constitutive model Poisson's ratio does influence primarily

unloading / reloading stress paths. In Figure 58 the wall deflection is shown for all construction steps

and in Figure 59 the influence of a +/- 10% variation of the friction angle is shown. It can be seen that

the reduction of the friction angle leads to a much higher relative increase of displacements as

compared with the Hardening Soil model (Figure 36 in section 4.9).


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MC_1Poisson = 0.2Poisson = 0.4Rinter = 0.5reference solution(Hardening Soil)


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MC_1Poisson = 0.2Poisson = 0.4Rinter = 0.5reference solution (Hardening Soil)

Fig. 55 Wall deflection - MC-variations Fig. 56 Bending moments - MC-variations



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MC_1Poisson = 0.2Poisson = 0.4Rinter = 0.5reference solution(Hardening Soil)

Fig. 57 Surface settlements - MC-variations


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GW-lowering 1excavation 1anchor 1GW-lowering 2excavation 2anchor 2GW-lowering 3excavation 3anchor 3GW-lowering 4final construction stage


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MC_1(final stage)friction angle +10%(final stage)friction angle -10%(final stage)MC_1(1. excavation stage)friction angle +10%(1. excavation stage)friction angle -10%(1. excavation stage)

Fig. 58 Wall deflection - MC_1 Fig. 59 Wall deflection - influence of strength

in MC_1 model


6 SUMMARY AND CONCLUSION

A reference solution for a deep excavation problem utilizing a commercial finite element code and an

elastic-plastic constitutive model as been presented in part II of this report. Based on this solution a

comprehensive study was performed in order to evaluate quantitatively the influence of various

modelling assumptions on calculated displacements and bending moments. Some analyses assuming

elastic and elastic-perfectly plastic material behaviour confirmed the well known fact that these very

simple constitutive models are not suited for predicting realistic deformations for these types of

problems.

It is emphasized that the reference solution is not an exact solution which in fact does not exist for

such a problem. However it is claimed that the solution can be considered as a good approximation

because care has been taken in establishing the numerical model and in choosing input parameters.

Therefore very similar results can be expected if the problem is solved with other finite element codes

and constitutive models but a perfect match cannot be expected.

No emphasis was put on parameter identification in this study but it has become evident from the

benchmark study presented in part I of this report that the choice of appropriate input parameters is of

paramount importance for any numerical analysis and a sound knowledge of the effect of input

parameters on the results, which depend on the constitutive model employed, is required in order to

achieve sensible results. The analyses considering a realistic scatter in stiffness and strength

properties presented in part II of this report confirm this aspect.

Finally it is concluded that benchmark exercises and parametric studies as presented in this report are

necessary and very helpful in improving the validity and reliability of numerical predictions, and that

there is a strong need for formulating guidelines and recommendations for numerical analysis in

practice.
