elastic±plastic soil models for 2D FE analyses of tunnelling

7/28/2019 elasticplastic soil models for 2D FE analyses of tunnelling

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A comparison of elasticplastic soil models for

2D FE analyses of tunnelling

G. Oettl*, R. F. Stark, G. HofstetterInstitute for Strength of Materials, University of Innsbruck, Austria

Received in revised form 7 August 1998; accepted 10 August 1998

Abstract

Based on 2D FE analyses, simulating the excavation of a tunnel and subsequent lining with

shotcrete, the impact of the employed soil model on the predicted displacements and stresses

in the soil mass as well as on the predicted sectional forces in the shotcrete lining is investi-

gated. In particular, four dierent soil models are considered: linearelastic constitutive rela-

tions, the elasticplastic models according to the DruckerPrager and to the MohrCoulombcriterion as well as an elasticplastic cap model. The computed results are compared with

available eld data for the vertical strains. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

For the numerical analysis of the excavation of a tunnel by means of the nite

element method (FEM), it is generally accepted now that an elasticplastic material

model for the soil to model the non-linear behavior of the soil should be employed.There exists a large variety of models which have been proposed in recent years to

characterize the stressstrain and failure behavior of soil media. All these models

have their own advantages and limitations which depend to a large degree on the

particular application. The most severe drawback associated with rened and

sophisticated models is related to the larger number of required parameters, some of

them often cannot be determined from standard tests. Therefore, commonly, the

relatively simple material models with a yield surface according to the Drucker

Prager or to the MohrCoulomb criterion and an associated or a non-associated

ow rule are employed in practice. Although it is well known that these simple

models have certain inherent shortcomings, more rened soil models, such as the

Computers and Geotechnics 23 (1998) 1938

0266-352X/98/$see front matter # 1998 Elsevier Science Ltd. All rights reserved.

P I I : S 0 2 6 6 - 3 5 2 X( 9 8 ) 0 0 0 1 5 - 9

* Corresponding author.


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Cam-clay model or cap models are still not commonly used in geotechnical engi-

neering in many parts of the world.

In common engineering practice the choice of the soil model employed for a spe-

cic job mostly depends on both the expertise of the analyst and the economicaspects in terms of computation cost. However, application of a dierent soil model

will lead to dierent results. This might be one of the reasons for a considerable deal

of discomfort among practitioners related to the reliability of such computations.

Hence, it might be interesting to investigate the range of predicted response, result-

ing from the application of dierent soil models. To this end, a numerical study for a

2D FE analysis of the excavation of a tunnel, based on four soil models, is con-

ducted. Apart from linearelastic constitutive relations for the soil, the MohrCou-

lomb failure surface (Fig. 1), the DruckerPrager compressive cone and the

DruckerPrager tensile cone (Fig. 2), each of them treated as yield surfaces within

the framework of ideal plasticity with a non-associated ow rule, are employed. In

addition, a cap model [8] with a non-associated ow rule for the DruckerPrager

type failure envelope and with an associated ow rule for the strain hardening cap, is

used [Fig. 3(a) and (c)]. The main dierence between the DruckerPrager and Mohr-

Coulomb type models on the one hand and the cap model on the other hand is in the

prediction of deformations under predominantly compressive stress states. The for-

mer models have yield surfaces which are open in the direction of the hydrostatic

compressive axis (Figs. 1 and 2), i.e. they assume linearelastic soil response for

predominantly compressive stress states. This feature constitutes a severe physical

shortcoming of the models, since in reality, soil behavior under a hydrostatic stressstate is certainly non-linear. The cap model, however, avoids this model deciency

by using a closed yield surface with a strain hardening cap. The hardening of the cap

is dened by a non-linear relation between the volumetric plastic strain and the

hydrostatic pressure as described in Section 3 and shown in Fig. 3(b). For plastic soil

behavior the latter is represented by the point of intersection of the cap with the

hydrostatic axis.

In a FE analysis, when simulating the construction of a tunnel, results strongly

depend on the applied sequential scheme of excavating and shotcrete lining. More-

over, in a time-independent analysis the results for a particular step of excavation

and lining placement will substantially depend on the assumption of when the

Fig. 1. MohrCoulomb model.

20 G. Oettl et al./Computers and Geotechnics 23 (1998) 1938


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shotcrete lining is regarded to be active. This assumption is crucial, since it deter-

mines to which extent the deformations of the soil mass due to the current excava-

tion step have already developed before the lining is in place. Clearly, this inuences

not only the stress and deformation distribution in the soil, but gures as a sig-

nicant design parameter for the lining element.

The paper is organized as follows. Section 2 contains a brief description of the soil

models, employed for simplied 2D numerical simulations of excavating and lining

of a tunnel, presented in Section 3. Finally, in Section 4 a comparison of the results

for the dierent soil models is presented. In addition, the computed vertical strains

in the soil are compared with available eld data, taken from Ref. [1].

2. Material models

2.1. DruckerPrager model

From the mathematical point of view, the DruckerPrager criterion is the most

convenient choice because of its simplicity and its straightforward numerical imple-

mentation. In 3D principal stress space the failure surface associated with this cri-

terion is a right-circular cone as shown in Fig. 2 which can be expressed by thefollowing equation

F' 0I1

J2p

k 0Y I

where I1 and J2 are the rst and second invariants of the stress tensor and the stress

deviator tensor, respectively

I1 1

3'iiY J2

1

2sijsij P

and 0 and k are material constants. However, in the present study, as frequently

encountered in practice, these parameters are not directly available from experiments.

Fig. 2. DruckerPrager model.

G. Oettl et al./Computers and Geotechnics 23 (1998) 1938 21


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Fig. 3. Cap model: (a) yield surface; (b) hardening behavior of the cap; (c) ow potential.



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Rather the friction angle, 0, and the cohesion value, c, for the MohrCoulomb

model are given. Thus, 0 and k must be expressed in terms of0 and c. Matching the

DruckerPrager model with the external apices of the MohrCoulomb failure sur-

face we get

0 2 sin 0

3p

3 sin 0 Y k 6c os0

3p

3 sin 0 X Q

Besides this compressive meridian matching a tensile meridian matching yields the

following relations

0 2 sin 0

3p 3 sin 0Y k 6c os03p 3 sin 0

X R

There are further ways of matching the two criterions which, however, will not be

considered in this context.

The ow rule, dening the direction of the plastic ow is given by

4p l dG

d'Y S

where G represents a plastic potential and l is a positive scalar quantity dening the

amplitude of the plastic ow. For non-associative plasticity, i.e. for G T F, theplastic potential is selected so that its derivative with respect to the stress tensoryields

dG

d'ij 2 ij 1

2

J2p sij T

with ij and sij denoting the Kronecker delta and the stress deviator tensor, respec-

tively. 2 is dened by a given dilation angle 2 and relations analogous to Eqs. (3)

or (4). From a comparison of Eq. (6) with the derivative of F, given in Eq. (1), with

respect to 'ij it can be seen that the ow rule is associative with respect to deviatoricplastic ow and non-associative for the volumetric plastic component, as 2 T 0.In the numerical example described in Section 3, the Drucker-Prager model with

isochoric plastic ow, i.e. a ow potential with a vanishing dilatancy angle 2is used;

thus, 2 0.Regarding the hardening behavior, a DruckerPrager model with exclusively

elasticperfectly plastic material response is considered in this study.

2.2. MohrCoulomb model

In 3D principal stress space the failure surface associated with the classical MohrCoulomb criterion is an irregular hexagonal pyramid, its axis coinciding with the

hydrostatic axis (Fig. 1). The function of this failure surface can be formulated by



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means of the MohrCoulomb material parameters 0 and c and the stress invariants

I1Y J2 and as

F'Y 13

I1 sin 0

J2p

sin %3

J2

3

ros %

3

sin 0 c os0 0Y U

or alternatively in terms of the HaighWestergaard coordinates $Y&Y

F$Y&Y

2p

$sin 0

3p

& sin %3

& os %

3

sin 0

6

pc os 0 0 V

with 044 %3

being the deviatoric polar angle (Lode angle). $ and & are given in

terms of the hydrostatic and deviatoric stress invariants as $

I1a

3p

and &

2J2p

,

respectively. Instead of the formulation, given in Eqs. (7) and (8), a smooth, single-

surface approximation is adopted in the FE package [2] used for this study. The

approximation of the Mohr-Coulomb criterion is a particular case of a general

three-parameter criterion [3] given by

f$Y&Y Af &2 mf Bf & rf Y e Cf $ cf 0X W

For a specic choice of the parameters AfY Bf and Cf the general failure criterion Eq.

(9) is reduced to a particular one, e.g. the MohrCoulomb criterion. mf and cf arethe friction and cohesion parameter, respectively. rf describes the shape of the failure

surface in a deviatoric plane and is given by

rf Y e 41 e2 os2 2e 12

21 e2 os 2e 1

41 e2 os2 5e2 4ep X IH

The eccentricity parameter e depicts the ``out-of-roundness'' of the deviatoric trace.

When e is determined by

e 3 sin 03 sin 0 Y II

the generalized failure criterion Eq. (9) is calibrated to t exactly the MohrCou-

lomb surface on both extension and compression meridians, which leads to the

smooth failure surface. Hence, with the denition for e according to Eq. (11), for the

extension and the compression meridians, 0 and %3

, respectively, the elliptic

function Eq. (10) takes the values 1/e and 1, respectively. Expressing all other para-

meters of the generalized criterion Eq. (9) in terms of the MohrCoulomb friction

angle 0 and cohesion c, we get

Af 0Y mf 1Y cf 1Y IP



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Bf 3 sin 0

24

pc os0

Y Cf 1

3

pctn0X IQ

The ow potential adopted in [2] takes the form

g$Y & Ag&2 mgBg& rge Cg$X IR

It should be noticed that the radius rg is independent of the Lode angle, i.e.

rg rf 0Y e 1

eIS

resulting in a DruckerPrager type ow potential with rg being the radius for the

extension meridian. The other parameters AgY BgY CgY mg determining the ow

potential Eq. (14) are evaluated in an analogous way from Eqs. (12) and (13) but

using the dilatancy angle 2 instead of the friction angle 0. In this study a non-asso-

ciative ow rule, strictly speaking, isochoric plastic ow (2 0) is assumed for theMohrCoulomb model.

As in the case of the DruckerPrager model, no hardening is considered in the

example presented in Section 3, when this type of MohrCoulomb model is employed.

2.3. Cap model

The cap model adopted in this study is an extended DruckerPrager model. Itsyield surface consists of a DruckerPrager cone for shear-type failure and of an

elliptical cap for volumetric plastic compaction.

Using the invariants p, q and r given by

p 13

'iiY q

3

2sijsij

rY r 9

2sijsjkski

13

IT

allows the DruckerPrager failure surface to be written as

Fs

tYp

tEp tn

d

0Y

IU

where t is the deviatoric stress measure given by

t q2

1 1K

1 1K

r

q

34 5X IV

The formulation in Eq. (17) is consistent with the one used in [4] and is related to

Eq. (1) by the following expressions

tn 3

3p

0Y d

3p

kX IW

K is a material parameter that represents the ratio of the distance of stress points onthe tensile and compressive meridian from the hydrostatic axis in a specic devia-

toric plane of the yield surface. In this study no dependence on the third deviatoric



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invariant, r, is taken into account in this model, i.e. t q, requiring K 1 in Eq.(18). K 1 implies the classical yield surface according to DruckerPrager by whichthe soil strength in tension is likely to be overestimated.

As already mentioned previously, the MohrCoulomb material parameters haveto be converted to equivalent DruckerPrager parameters. Matching the parameters

to provide the same ow and failure response in plane strain [4] results in

sin 0 tn

39 tn2 2p

9 tn tn 2 Y c os0 d

39 tn2 2p9 tn tn 2X PH

Evaluating Eq. (20) for associated ow 2 we get

rtn

3p sin 01 1

3sin

2 0

qHfd IgeY d c 3p os01 1

3sin

2 0

q PI

and for nondilatant ow 2 0 we obtain

rtn

3p

sin 0Y d c

3p

os 0X PP

In this study and d were determined by means of Eq. (21). However, it is easily

veried, that there is only a small dierence between associated and nondilatant owwithin the range of typical values of the friction angle.

The cap yield surface has an elliptical shape [Fig. 3(a)] and is written as

FctYp p pa2 Rt1 a os

!2s Rd pa tn 0Y PQ

where R is a material parameter controlling the shape of the cap. is a small num-

ber used to dene a smooth transition surface between the DruckerPrager cone and

the cap dened as

FttYp p pa2 t 1

os

d pa tn

!2s d pa tn 0X PR

Whereas the material response is perfectly plastic for stress points located on the

DruckerPrager shear failure surface, for stress points on the cap, hardening is

taken into account. This hardening behavior is governed by the evolution parameter

pa, dened as

pa pb Rd1 R tn PS



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The tunnel is constructed in three working cycles, dealing sequentially with the

crown, the bench and the bottom of the tunnel as indicated in Fig. 6. In the analysiseach working cycle is modelled by two computation steps, simulating the excavation

of the soil material in the rst step and the placement of the shotcrete lining in the

Fig. 4. Set up of test example: (a) domain under consideration; (b) cross section of the tunnel.

Fig. 5. FE-model.



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second step. In this study it is assumed that the deformations along a newly gener-

ated free surface of the tunnel, resulting from the excavation in the current working

cycle, are already present when the shotcrete lining is installed in the current work-

ing cycle. Hence, the lining installed in a specic cycle is only stressed due to defor-

mations generated by the excavation in subsequent working cycles. It might be

argued that this is not a very realistic assumption, since it requires the soil to be

stable without support in the current excavation stage. Although this scenario is met

for the problem at hand, it might not hold for other types of soil or dierent con-struction conditions. However, the motivation for this assumption is to check whe-

ther the computed displacements provide an upper bound for the measured

deformations. On the other hand, if the computed displacements based on this

assumption are smaller than the measured ones, this could be regarded as an evi-

dence for shortcomings of the soil model. Moreover, it is not precisely known to

which extent the lining, placed in a specic working cycle, is actually loaded by the

excavation in that cycle. This depends on various parameters, like time-dependent

material response of the soil and the shotcrete, which are not taken into account by

the employed model.

Driving a tunnel is certainly a 3D problem. To account for the 3D-eects, com-monly, in 2D FE analyses, a partial initial stress relief is assumed for the modelled

cross section, when excavation and lining are simulated in the analysis. Dierent

stress relief methods have been proposed in the literature. Probably the two most

commonly used approaches are the load reduction method and the stiness reduc-

tion method. Both methods have been successfully applied in practice, although,

they may lead to signicantly dierent results when complex excavation stages and

non-linear material behavior have to be modelled [5]. Applying a stress relief method

in a 2D analysis yields some deformations in the soil before the lining is installed.

Thus, the actually 3D states of stress and deformation in the vicinity of the working

area, indicated by deformations in the soil mass ahead of the tunnel face, areaccounted for in an approximate manner. Consequently, this part of the deforma-

tions does not lead to stresses and strains in the shotcrete lining but of course, it is

Fig. 6. Construction sequences of the tunnel.



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accompanied by a change of the stress state in the soil. Certainly the results depend

on the specic stress relief method applied and therefore would veil the eect of

dierent non-linear soil models. Therefore it would not be appropriate to take par-

tial stress relief into account. Rather a complete stress relief is employed.The mentioned simplications with respect to the excavation and lining scheme

and the idealization of the material behavior of the shotcrete lining are justied,

since the primary concern of this paper is to study predictions of dierent soil

models, commonly used in the analysis for tunnel excavation.

In the present study three elasticplastic soil models described in the previous

section are used. Material properties for the soil and the shotcrete lining are taken

from [1] and are summarized in Table 1. With respect to the soil parameters and

their determination, some remarks contained in [1], are given subsequently. Young's

modulus E was determined from the results of oedometer tests by assuming Pois-

son's ratio to be 0.38. The parameters for the failure envelope followed from results

of conventional triaxial tests. Since both, the angle of internal friction, 0, and the

cohesion, c, depend on the strain rate, drained and undrained tests were performed.

For drained tests which were performed at a strain rate of 0.1 mm/m.day the para-

meters were found to be 0 25 and c 0. Higher strain rates, e.g. 210 mm/m.day,lead to 0 12 13 and c&100 kN/m2. Since the strain rates measured in situ weresomewhere between 0.5 and 8 mm/m.day, for the analysis the parameters were

assumed to be 0 20X4 and c=73 kN/m2. With respect to the application of thecap model, a shape parameter K 1 was assumed for the shear-type failure surface.The parameter which denes the size of the transition zone between DruckerPrager cone and cap was taken to be 0.001, i.e. virtually no transition zone was

assumed. The hardening law used in [1] is an exponential relationship between

hydrostatic pressure and volumetric plastic strain, whose governing parameters were

matched with the results of oedometer tests by means of a trial and error procedure.

In terms of the input data for the cap model implemented in ABAQUS, this expo-

nential function was resolved in a piecewise linear function relating the hydrostatic

compression yield stress, pb, and the corresponding volumetric plastic strain, 4plvol.

The parameters associated with the initial location of the cap at the onset of the

Table 1Material parameters for soil and shotcrete

Notations Units Soil Shotcrete

Specic weight kN/m3 19 25

Young's modulus E kPa 59,000 2.8.107

Poisson's ratio # 0.38 0.30

Cohesion c kPa 73

Friction angle 0 20.4 Shape parameter K 1.0

Shape parameter 0.001

Shape parameter R 0.64 Initial cap position pb j0 kPa 80 4

plbol j0 0.11



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analysis were given by pb j0 80 k and 4plvol j0% 0X11. Finally the factor R, con-trolling the shape of the cap, followed from an estimation by means of a constant

volume test performed in a shear box. In terms of ABAQUS parameters for Vien-

na's clayey silt R was found to be 0.64.The soil was modelled by means of 2D bilinear isoparametric continuum elements

with 4 nodes each, assuming plane strain conditions (Fig. 5). Using low-order ele-

ments, special attention has to be paid to the volume locking phenomenon. As

pointed out in [6], dilatant (2 b 0 or contractant 2 ` 0 plasticity imposes essen-tially the same kinematic constraint upon elements as in the case of isochoric

2 0 plastic ow. As described above, isochoric plastic ow was assumed in allcases but one. Hence, elements based on the so-called B-concept, which may safely

be used for isochoric plastic ow, were employed.

The shotcrete lining was also modelled with four noded plane strain isoparametric

continuum elements. To ensure improved bending behavior, elements enriched with

incompatible modes were used.

4. Comparison of the numerical results

The results presented in this section were computed using the FE-packages ABA-

QUS [4] and Z_SOIL [2]. For the analyses based on the DruckerPrager and the

MohrCoulomb soil models, Z_SOIL was used. Moreover, the FE-package

AFENA [7] was employed to check the results of the commercial nite elementprograms. Although AFENA does not use the smooth approximation of the Mohr

Coulomb yield surface, the results agreed quite well with those from Z_SOIL. The

results for the cap model were obtained from ABAQUS.

Figs. 7 and 8 contain the computed surface settlements and the computed dis-

placements of the soil at the boundary of the cross section of the tunnel after exca-

vation and lining of the tunnel have been completed. As one would expect, the

smallest surface settlements and the smallest displacements of the boundary of the

Fig. 7. Surface settlements.



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cross section of the tunnel are obtained using the linearelastic model (EL) for the

soil. The magnitude of the surface settlements predicted by the MohrCoulomb

model (MC) is between those, obtained for the DruckerPrager tensile cone (DP-EX) and the DruckerPrager compressive cone (DP-CO). Figs. 7 and 8 clearly

indicate that the surface settlements and the displacements of the soil at the bound-

ary of the cross section of the tunnel, based on the cap model (CA), are considerably

larger than the respective values, prognosticated by the commonly employed simple

elasticplastic models. As already pointed out in the introduction, the main dier-

ence between the latter and the cap model is given by the treatment of pre-

dominantly compressive stress states. As can be seen from Figs. 7 and 8 this

renement with respect to the representation of the soil behavior has a considerable

impact on the predicted deformations. The striking changes of the displacements at

the interfaces between dierent excavation stages (Fig. 8) is very much due to thespecic assumptions of the model, i.e. to simulate the excavation of the soil and the

installation of the liner in two consecutive steps. Again, this eect is by far more

pronounced for the cap model than for the elastic model or the other simple elastic

plastic models. Regarding the surface settlements both with respect to the maximum

settlement and the width of the settlement trough, the dierence between the cap

model and the simple models is even more distinct (Fig. 7). This is the consequence

of the elasticplastic material response, being no longer conned to a relatively small

region around the tunnel when employing a cap model.

Figs. 9 and 10 contain plots of the distribution of the vertical and the horizontal

normal stresses in the soil along three sections stretching nine meters away from theshotcrete lining for the nal stage. Fig. 5 shows the position and orientation of these

sections. For comparison, the primary stresses drawn in dashed lines, are also

Fig. 8. Soil displacements of the cross section of the tunnel.



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plots also contain some eld data (MEAS), given in [1], which were obtained from

tests with sliding micrometers. One might argue that considering the simplifying

assumptions made in the analysis, a comparison with in situ measurements is

somehow arbitrary. These assumptions, however, were made in order to get an

upper bound for the displacements. Consequently, we cannot expect perfect agree-

ment between experimental data and numerical results. However, what the plots do

show is a tendency to grasp the real behavior in a better way the more sophisticatedthe employed model is. In particular it can be seen that the vertical strains, predicted

by the linear elastic model are generally underestimated compared with eld data. A

Fig. 11. Vertical strains in the soil, relative to the primary state.



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further general tendency reected by the three plots of Fig. 11 is, that the results

based on the cap model constitute the other extreme, i.e. these results tend to over-

estimate the vertical strains, which might be expected as a consequence of the sim-

plifying assumptions for the analysis. For the vertical sections above and below thetunnel, for which the plots refer to regions being in a state of relaxation, the fan of

results for the dierent models is relatively narrow. In these regions of unloading

there is not much dierence between the response, predicted by the dierent models.

However, for the plot of the vertical strains along the horizontal section the picture

is quite dierent. For this region with compressive loading the cap model is the only

model which does not underestimate the vertical strains. Hence, the ndings seem to

conrm that by means of the cap model an upper bound for the displacements is

determined. Unfortunately, no eld data were available close to the tunnel.

All models but the cap model employed in this study do not take into account the

plastic compaction of the soil under predominantly hydrostatic pressure. This,

however, seems to be one of the key aspects and the surface settlement caused by

driving the tunnel may be governed by this eect to a considerable amount. Since

the vertical strains in these areas of compaction and, consequently, also the surface

settlement are overestimated by the cap model in this study, the results predicted by

this model are conservative. This might be crucial in urban areas.

In Fig. 12 distributions of the normal force and the bending moment in the shot-

crete lining are plotted. Since the shotcrete shell for the bottom of the cross section

of the tunnel was placed after the deformations due to excavating the bottom have

occurred, there are no stresses in the respective part of the shotcrete lining. Asexpected, the maximum normal force and the maximum bending moment in the

Fig. 12. Bending moment and normal force in the shotcrete lining.



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lining are predicted in the upper part of the lining, irrespective of the employed soil

model. However, there are considerable dierences in the predicted maximum values

of the normal force in the lining. Modelling the soil behavior using the cap model or

the DruckerPrager tensile cone yields maximum values for the normal force, whichare about 25% larger than the respective value, predicted on the basis of the Mohr

Coulomb model for the soil. As expected, smaller values of the normal force in the

lining are predicted employing the DruckerPrager compressive cone for the soil. By

contrast, except for the DruckerPrager compressive cone, which yields smaller

bending moments, there is almost no dierence for the predicted maximum value of

the bending moment between the dierent elasticplastic soil models.

5. Conclusions

In a numerical study of a 2D FE analysis dealing with multistage excavation and

lining of a tunnel, dierent material models for the representation of the soil beha-

vior have been employed. In particular, linearelastic constitutive relations, the

elasticplastic models according to the DruckerPrager and to the MohrCoulomb

criterion as well as an elasticplastic cap model were taken into account. Signicant

dierences of the predicted surface settlements have been found between the simple

elasticplastic models, based on a DruckerPrager or MohrCoulomb type yield

surface, and the cap model. Such signicant dierences are also observed for the

predicted deformations of the boundary of the cross section of the tunnel. Com-parison with eld data clearly shows, that an analysis based on the current simpli-

fying assumptions cannot describe the response of the soil with desired accuracy.

However, focussing on dierent soil models, this comparison reveals, that the cap

model seems to be superior to the DruckerPrager and MohrCoulomb model in

the sense that an upper bound for the displacements is obtained by accounting for

the nonlinear soil behavior under predominantly compressive stress states. At least

for the selected example this feature yields conservative results with respect to the

deformations in the soil due to the excavation. However, further investigations

would be necessary to improve the unloading/reloading behavior of the model.

Acknowledgements

The study described in this paper was partially funded by the Austrian Science

Foundation, FWF (Fonds zur Fo rderung der wissenschaftlichen Forschung), under

project number S08005-TEC. This support is gratefully acknowledged.

References

[1] Kropik Ch. Three-dimensional elasto-viscoplastic nite element analysis of deformations and stresses

resulting from the excavation of shallow tunnels. Ph.D. thesis, University of Technology of Vienna,

1994.



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Documents

elastic±plastic soil models for 2D FE analyses of tunnelling