two-dimensional base stability of excavations in soft soils using fem

Two-dimensional base stability of excavations insoft soils using FEM

Hamdy Faheema,*, Fei Caia, Keizo Ugaia, Toshiyuki Hagiwarab

aDepartment of Civil Engineering, Gunma University, Kiryu, Gunma 376-8515, JapanbNishimatsu Construction Co. Ltd., Japan

Received 15 April 2002; received in revised form 16 August 2002; accepted 30 September 2002

Abstract

The 2D base stability of excavations is evaluated using FEM with reduced shear strength.The numerical results indicate that the base stability of excavations is significantly influencedby the ratio of the depth to the width of the excavations, the thickness of the soft soil layerbetween the excavation base and hard stratum, the depth of the walls inserted below the

excavation base, and the stiffness of the walls around the excavation. Closed form equationsconsidering embedment depth under the base of braced excavation has been proposed basedon the upper bound analysis and compared with the FE results.

# 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Finite-element analysis; Soft clay; Base stability; Upper bound analysis

1. Introduction

Excavations with large scale and depth are increasingly used for undergroundspace developments. Evaluation of the safety factor against base instability isimportant in the design of excavations in soft soils. The undrained condition usuallygoverns the stability of an excavation immediately after construction, and a totalstress analysis is pertinent. Several limit equilibrium analysis methods are availablefor evaluating the base stability of excavations. The calculated safety factors varywith different methods. Up to the present, the safety factor of excavations has beenusually predicted using the methods proposed by Terzaghi [1] or Bjerrum and Eide[2]. However, Terzaghi’s method is only used for shallow or wide excavation with

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H=B4 1:0, where B and H are the width and the depth of the excavation, respec-tively, as shown in Fig. 1. For Bjerrum and Eide’s method, some modification to theNc-values is necessary when a hard stratum is present near the excavation base. Thisis because the failure surface will not extend fully across the base of the excavation,and a deep foundation failure mode is no longer realistic.The two above-mentioned methods were developed before the introduction of

stiffer retaining wall systems such as diaphragm walls and secant piles. Hence, thesafety factors given by these methods do not consider the stiffness (EI) of the wall,which has significant influence on the base stability in soft soils. The depth of thewall penetration below the excavation base, D, is also not considered, in these twomethods.In this study, the finite element method with reduced shear strength is used to

evaluate the 2D base stability of excavations. The influences of H=B, the claythickness under the base of excavation (T), the wall stiffness (EI), the inserted depthof the wall (D), and the net distance between the end of the wall and the hard stra-tum (Tn) have been considered on the base stability. Closed form equations con-sidering the embedment depth under the base of braced excavation have beenproposed. The effects of strain softening and anisotropy have not been considered inthe finite element method as in the limit equilibrium method.

2. FEM with reduced shear strength

The FEM with reduced shear strength has been applied to analyze the slope stabi-lity in two-dimensional situations by Zienkiewicz et al. [3], Ugai [4], Matsui et al. [5]and Griffiths et al. [6], and three-dimensional situations by Ugai et al. [7] and Cai et al.[8]. The safety factor is evaluated by the gradual reduction of the shear strengthparameters (c, �) of the soil inducing the divergence (failure) of the nonlinear analysis.

Fig. 1. Geometry of braced excavation.

142 H. Faheem et al. / Computers and Geotechnics 30 (2003) 141–163

The reduced shear strength parameters c0f and �0f will replace the corresponding

values of c0 and �0 in the shear strength equation �f ¼ c0 þ �0tan�0 to c0F ¼ c

0=F and�0F ¼ tan�1ðtanð�0Þ=FÞ: So the equation with reduced shear strength can be written as

�fF ¼ c0F þ �0tan�0

F:The initial value of F is assumed to be sufficiently small so as to produce a nearly

elastic problem. Then the value of F is increased step by step until finally a globalfailure develops [7]. The load step is controlled by the increment of the safety factor.If convergence cannot be reached after, for example, 1000 iterations, the F value justbefore this load step is taken to be the unique safety factor Fs. This method is calledthe shear strength reduction FEM (SSRFEM).The details of the FEM with reduced shear strength can be seen elsewhere Cai et

al. [8]. The numerical comparisons have shown that the FEM with reduced shearstrength is a reliable and robust method for assessing the safety factor of slope andcorresponding critical slip surface. One of the main advantages of the FEM withreduced shear strength is that the safety factor emerges naturally from the analysiswithout the user having to commit to any particular form of the mechanism a priori[6]. Furthermore, it is possible for the FEM with reduced shear strength to evaluatethe slope stability under a general framework even for the case of base stability ofcircular excavations [9].A 2D finite element program is used for the present analysis, where the return-

mapping algorithm is used for stress integration, and the secant Newton method isused to accelerate the convergence of the modified Newton–Raphson scheme [9].The isoparametric elements with eight nodes are used to model the soil and wallelements. In order to obtain the precise results it is found that at least six rows ofelements must be put below the bottom of the wall, where the soil failure shouldtake place. The horizontal distance from the wall to the outer boundary of theexcavation should be not less than at least 2H. The vertical distance in the meshunder the base of excavation must be fine as possible which depends on the thicknessof the clay under the base of excavation; for example, the vertical distance is 0.25 min case of T ¼ 3:0 m, and 0.5 m in the case of T ¼ 4:50 m.The reduced integration is used to avoid the numerical difficulties, known widely

as ‘locking’, arisen with modeling incompressible soft soils [9]. The lower ordertechniques with reduced integration have been used with success in a wide range ofgeotechnical collapse problems including slope stability and earth pressures [10]. Thesoil is modeled as linear elastic-perfectly plastic with the Mohr–Coulomb yield sur-face and the wall is assumed to behave linear elastically. The shear strength of thethin interface element was given as 0.01, 0.1, 0.5 times the shear strength of the clayand the obtained results were almost the same. So, the shear strength of the clay-wall interface is found to give slight influence on the base stability. The excavateddepth Hð Þ of all excavations analyzed is 9 m, and the other geometrical parametersare changed case by case. The mechanical properties of the soft clay are shown inTable 1.To show that the effect of the wall roughness is minor, the nodal displacement

vector for the case of H/B=1.0 and D=0.0 has been plotted as shown in Fig. 2.From that figure it can be seen that the relative nodal displacement between the wall

H. Faheem et al. / Computers and Geotechnics 30 (2003) 141–163 143

and the thin interface element is very small which means that the wall and the thinelement displaced together in the downward direction. This agrees with theassumption of the boundary condition of the mesh, whereas the wall is fixed in thehorizontal direction and free in the vertical direction. For this reason the shearstrength of the clay-wall interface has almost no influence on the Nc values.

3. Upper bound analysis

The closed form equations considering embedment depth under the base of bracedexcavation have been proposed based on the upper bound analysis. The loads aredetermined by equating the external work to the internal rate of dissipation inassumed deformation modes (or velocity fields) that satisfy velocity boundary con-ditions and strain and velocity compatibility conditions. The dissipation of energy in

Table 1

Mechanical properties of the soft clay

Parameter Value

Undrained shear strength, su (kPa) 35.0

Young’s modulus, Eu (kPa) Eu=250 suPoisson’s ratio, � 0.49

Unit weight, � (kN/m3) 20.0

Fig. 2. Velocity field for H/B=1.0, D=0.0, T ¼ Tc, and the shear strength of thin element=0.1 su.


plastic flow associated with such a field can be computed from the idealized stress/strain rate relation (or the so-called flow rule). If the soil layer of the thickness Dbelow the excavation base acts as a load (�D), and the wall is rigid, the followingequations could be derived from the energy dissipation assumption. There are threecases that could be considered; (1) where there is no embedded depth of the wall andT4Tc, (2) where there is embedded depth of the wall and Tn � Tc, (3) where thereis embedded depth of the wall and Tn � Tc: Tc is called the critical depth and equalto B=

ffiffiffi2

p. The assumptions and failure mechanisms are shown in Figs. 3–10, where V

is the velocity of plastic flow. The value of �H=su in the upper bound analysis isequivalent to the Nc-value in the FE analysis.

3.1. Case (1): D=0.0 and T4Tc

Terzaghi-type (I):

NC ¼�H

Su¼ 5:71þ

H

Tð1Þ

Prandtl-type (I):

Nc ¼ 6:14þ1ffiffiffi2

pH

Tð2Þ

Prandtl-type (II):

Nc ¼ 2 1þ3

4�� 1 � 3 þ tan1

� �þ tan3 þ

1ffiffiffi2

pH

Tð3Þ

Fig. 3. Failure mechanism and the assumptions based on Terzaghi-type (I) mechanism.


where 3 is a variable and must be chosen to minimize Nc-value.

cos1 ¼2T

Bcos3; Vv ¼

ffiffiffi2

pVcos1cos3

Prandtl-type (III):

Nc ¼ 2 1þ3

4�� 1 þ tan1

� �þ1ffiffiffi2

pH

Tð4Þ

Fig. 4. Failure mechanism and the assumptions based on Prandtl-type (I) mechanism.

Fig. 5. Failure mechanism and the assumptions based on Prandtl-type (II) mechanism.


1 ¼ cos�12T

B

� �

3.2. Case (2): D50.0 and Tn�Tc

Terzaghi type (II):

N ¼ 5:71þffiffiffi2

p HþD

Bð5Þ

Fig. 6. Failure mechanism and the assumptions based on Prandtl-type (III) mechanism.

Fig. 7. Failure mechanism and the assumptions based on Terzaghi-type (II) mechanism.


Prandtl-type (IV):

Nc ¼ 6:14þHþD

Bð6Þ

3.3. Case (3): D�0.0 and Tn�Tc

Terzaghi type (II):

Fig. 8. Failure mechanism and the assumptions based on Prandtl-type (IV) mechanism.

Fig. 9. Failure mechanism and the assumptions based on Prandtl-type (V) mechanism.


Nc ¼ 5:71þHþD

Tnð7Þ

Prandtl-type (V):

Nc ¼ 6:14þHþDffiffiffi2

pTn

ð8Þ

Prandtl-type (VI):

Nc ¼ 2 1þ3

4�� 1 � 3 þ tan1

� �þ tan3 þ

1ffiffiffi2

pH

Tð9Þ

where 3 is a variable and must be chosen to minimize Nc-value.

cos1 ¼2T

Bcos3;Vv ¼

ffiffiffi2

pVcos1cos3

4. Results and discussions

Parameter values of the wall and excavation geometry, which are used in the FEcalculations, are shown in Table 2.The base instability can be indicated using the normalized maximum rebounding

displacement on the excavation base, �,

Fig. 10. Failure mechanism and the assumptions based on Prandtl-type (VI) mechanism.


� ¼�maxEusu Hþ Tð Þ

ð10Þ

where �max is the maximum nodal displacement in the base of excavation. Thelocation of that point is not fixed, but it depends on the geometry of the excavation.�max can be found at the middle of excavation if T5Tc, else, it is located near thewall. The normalized displacement � increases with the reduction factor of the shearstrength F, and dramatically develops with F trending to the safety factor of theexcavation bases, as indicated in Fig. 11. This is similar to the results for two-dimensional excavations, reported by Smith et al. [11] and Goh [12].The safety factor can be determined from the ��F curve, where the nodal

displacement indicates a rapid increase in the deformation as shown in Fig. 11.However, for larger H/B ratios or in case of increasing the embedded wall depthwith a few exceptions, the nodal displacement curve did not indicate a well-defined asymptotic failure value. In these cases the Nc values can be determinedas follows:

Table 2

Wall and excavation geometry and their values used in the FE calculations

Excavation depth in all cases, H (m) 9.0

Effect of H/B

T=Tc B/ffiffiffi2

p

H=B 0.5, 0.75, 1.0, 1.5, 2.0, 3.0, 4.0

D 0.0

Effect of EI

H/B 0.75, 1.50

Wall stiffness, EI (MNm2) 22.5, 45, 90, 180, 360

D/T 0.0, 0.278, 0.50, 0.722

T/Tc 1.0

Effect of T

T/Tc 0.236, 0.471, 0.707, 1.0, 1.5, 2.0

H/B 0.5, 0.75, 1.0, 1.5, 2.0

D 0.0

Effect of D

(1) T=TcH/B 0.5, 0.75, 1.0, 1.5

D/T 0.25, 0.50, 0.75

(2) T<TcH/B 0.50

T/Tc 0.471, 0.70

D/T 0.25, 0.50, 0.75

(3) T>TcH/B 0.50, 0.75, 1.0, 1.50

D/H 0.25, 0.50, 0.75

T/Tc 1.25, 1.50, 2.0


(a) For larger H/B ratios, the reason why the nodal displacement curve did notindicate a well-defined asymptotic failure value is that the used mesh iscoarse. Therefore, to obtain a well-defined asymptotic failure value, it isnecessary to use finer mesh in the FE analysis than used before, in that casethe Nc value can be determined easily as shown in Fig. 11.

(b) When the embedded wall depth increases as shown in Figs. 17 and 18 thenodal displacement curve did not indicate a well-defined asymptotic failurevalue. In that case the maximum normalized displacement � for a well-defined asymptotic failure value (i.e., case of D=0.0) in the same figures ischosen as a reference for all different embedded depth cases, since theobtained displacement in case of D=0.0 is enough to cause failure.

4.1. Effect of H/B (D=0, T5Tc)

The results of the FEM with reduced shear strength indicate that the location ofhard stratum has influence on the Nc-value, when the thickness of the soft soil layerbelow the excavation base is less than the critical depth, i.e. T4Tc ¼ B=

ffiffiffi2

p. When

T5Tc, and D ¼ 0, the influence of H=B on the base stability is indicated in Fig. 12,where Nc is given by

Nc ¼ Fs�H=su ð11Þ

where Fs is the safety factor of the base of excavation. The factor of safety Fs can beobtained from Fig. 11 based on the FE analysis. The factor of safety is taken atwhich nodal displacements indicate a rapid increase in the deformation. By

Fig. 11. Normalized displacement versus shear strength reduction factor, F (D ¼ 0:0).


substituting the value of factor of safety obtained from Fig. 11 into Eq. (11) Nc canbe obtained.Fig. 12 indicates that the Nc-value, predicted using the FEM with reduced shear

strength, is close to Eq. (1) for the shallow or wide excavation with H=B4 1:0.When H=B � 1:0 the Nc-value of the FEM increases with the ratio of H=B and arehigher than Bjerrum & Eide’s method. Also the Nc-value, predicted with the FEM isless than Eq. (2) when H=B � 1:0. For Eqs. (1) and (2) T must be chosen equal to Tcin the case of T � Tc, and T is exactly its value if T4Tc. Nc is a dimensionlesscoefficient depending on the geometry of the excavation.Also, from Fig. 12 it can be seen that in case of T=Tc ¼ 2:0 there is a little decrease

in safety factor which was not considered in both Terzagi and Prandtl-typemechanisms. From Fig. 12 the FEM solutions give less safety factors than thoseobtained from the upper bound analysis.The failure mechanism by the FEM can be indicated using nodal displacement

vectors induced by the shear strength reduction just before the failure takes place.The failure mechanism for H=B=1.0 with T ¼ Tc is shown in Fig. 13 which issimilar to that of the upper bound analysis shown in Fig. 8 with D ¼ 0:0.The failure mechanism for H=B=4.0 with T ¼ Tc is shown in Fig. 14, which is a

deep excavation case, and also similar to Fig. 8. From this figure it can be concludedthat, even if the excavation is deep compared to the width, the nodal vector dis-placement starts from the ground surface, which means that the total failure isoccurring, although Bjerrum and Eide’s method assumes that for deep excavationthe failure will be local.

Fig. 12. Effect of H=B on Nc-value (D=0 and T5Tc).


4.2. Effect of T/Tc (D=0)

In this section the effect of clay thickness under the excavation base is considered.The clay thickness under the excavation base has influence on the stability of thebase as shown in Fig. 15. It can be concluded that the safety factor increases withdecreasing the thickness of clay under the base of excavation.When T=Tc � 1:0, the increase of clay depth more than the critical depth will

slightly reduce the Nc-value. However the reduction of the Nc-value is smaller than5% as shown in Fig. 15. From that figure it can be seen that the presence of the hardstratum close to the excavation base increases the Nc-value when T=Tc � 1:0. This isbecause the size of the yielding zone is influenced, and the displacement of the soilbeneath and around the excavation is restrained. Fig. 15 also shows that the Nc-values of Eq. (1) and Eq. (2) are larger than that of the FEM with reduced shearstrength.When H=B ¼ 1:0, and T=Tc ¼ 0:24, the failure mechanism is shown in Fig. 16. It

is obvious that the failure zone is restrained, and similar to that assumed in theTerzaghi-type (I) and Prandtl-type (I) (Figs. 3 and 4) for the shallow or wide exca-vations with limited thickness of the soft soil layer in the two-dimensional situations.This cannot be considered by Bjerrum and Eide’s method. When T=Tc � 0:5 thedifference of the Nc-values between the FEM with reduced shear strength, Eqs. (1)and (2) are small as shown in Fig. 15.

Fig. 13. Nodal displacement vectors for H=B ¼ 1:0, D ¼ 0:0; and T ¼ Tc.


Fig. 14. Nodal displacement vectors for H=B ¼ 4:0, D ¼ 0:0; and T ¼ Tc.

Fig. 15. Influence of T=Tc on Nc-value (D=0).


When T=Tc ¼ 1:0 the difference of the Nc-values between the FEM with reducedshear strength, Eqs. (1) and (2) is smaller than 5 as shown in Table 3.

4.3. Effect of T/Tc (D�0)

The results of the FEM with reduced shear strength indicate that when T=Tc4 1:0the thickness of the clay layer has influence on the Nc-values. The effect of theembedded depth is shown in Fig. 17. This figure shows that, the safety factorincrease with increasing D=Tc, in case of T4Tc, that is because the wall resists themovement of soil towards the base of excavation, which increases the safety factor.When T=Tc � 1:0 the influence of D on Fs values is indicated in Fig. 18. For clay

extending to a considerable depth below the base of excavation (T � Tc) the effect ofincreasing the embedded depth of the wall will leads to a marginal increase in thesafety factor. Figs. 17 and 18 show that, for clay that has bigger depth (T ¼ 2Tc)below the base of excavation, with increasing the embedded wall depth, the increase

Fig. 16. Nodal displacement vectors for D ¼ 0:0, H=B ¼ 1:0; and T=Tc ¼ 0:24.

Table 3

The Nc-values under T=Tc=1.0

H=B FEM Eq. (1) Eq. (2)

0.50 6.33 6.4 6.64

0.75 6.64 6.76 6.89

1.00 6.94 7.11 7.14


of safety factor will be smaller than that in the case of T4Tc. For D=T ¼ 0:60 theincrease of safety factor in case of T4Tc is about more than 50%, while in case ofT ¼ 2Tc the increase of embedded depth of the wall until D=T ¼ 0:60 has increasedthe safety factor with about 20% only.

Fig. 17. Effect of embedded depth (T ¼ Tc).

Fig. 18. Effect of embedded depth on safety factor (T ¼ 2Tc).


With increasing the wall embedded depth, the safety factor increases, whereas theincrease in the safety factor is smaller in the case of large clay thickness under thebase of excavation than in the case of small clay thickness. The calculated safetyfactor is 1.58 in case of H=B ¼ 1:50, D=T ¼ 0:40; and T ¼ 2Tc, and 1.75 in case ofH=B ¼ 1:50, D=T ¼ 0:70; and T ¼ 2Tc. Local views at the end of the wall for thesetwo cases are shown in Figs. 19 and 20. From these figures it can be seen that, theshorter embedded depth the higher resistance to the soil movement. Fig. 19 showsthat the nodal displacement of the wall at its end is significantly smaller than thenodal displacement of the adjacent soil. This means that the wall resistance to thesoil movement was high. While in the case of large embedded depth the nodal dis-placement of the wall at its end is almost the same as the adjacent soil, this meansthat the resistance of the wall to the soil movement was small. Fig. 21 is the globalview of the failure mechanism for the case of which local view is shown in Fig. 19.Fig. 21 shows that the failure mechanism is compatible with the assumption in Fig. 8for the closed form equation by Prandtl-type (IV).

4.4. Effect of D and wall stiffness

The analyses of the effect of the wall stiffness on the safety factor shows that, thereare relations between the embedded depth and the wall stiffness as shown in Fig. 22.The safety factor increases with increasing the stiffness of the wall, which depends onthe embedded depth of the wall. The increase in Fs as D increases is likely becausethe wall stiffness reduces the tendency of the clay behind the wall and adjacent to thebase of excavation to be displaced toward the excavation. The relative nodal dis-

Fig. 19. Nodal displacement vectors when H=B ¼ 1:50, D=T ¼ 0:40; and T ¼ 2Tc (local view at the end

of the wall).


Fig. 20. Nodal displacement vectors when H=B ¼ 1:50, D=T ¼ 0:70; and T ¼ 2Tc (local view at the end

of the wall).

Fig. 21. Nodal displacement vectors when H=B ¼ 1:50, D=T ¼ 0:40; and T ¼ 2Tc.


placement for low rigidity wall and rigid wall are shown in Figs. 23 and 24. Thecalculated safety factors for these two cases are 1.53 and 1.74, respectively.This study employs the 8-noded quadrilateral elements with reduced integration

for the displacement finite element analysis. Whilst this element may sometimes givereasonable estimates of the limit load, it frequently gives deformation modes that

Fig. 22. Effect of embedded depth and wall stiffness on safety factor when H=B ¼ 0:75.

Fig. 23. Nodal displacement vectors when H=B ¼ 1:50, D=T ¼ 0:60; T=Tc ¼ 1:0, and EI=45 MNm2

(local view at the end of the wall).


contain ‘near mechanisms’ Sloan et al. [10]. For that reason the deformed mesh hasbeen drawn as shown in Fig. 25. It can be seen that, the deformed mesh containsalmost no barrel elements, which does not affect the result of collapse load. Thedeformed mesh does not appear particularly serious.

Fig. 24. Nodal displacement vectors when H=B ¼ 1:50, D=T ¼ 0:60;and T=Tc ¼ 1:0, and EI=360 MNm2

(local view at the end of the wall).

Fig. 25. Deformed mesh in case of H/B=1.0, T=Tc ¼ 1:0, and 8-noded quadrilateral element results withreduced (2*2) integration.


4.5. Closed form equations considering embedded depth

The closed form equations for the Nc-value, considering the embedded depthunder the base of braced excavations, have been derived in Section 3. The derivedformula for the Nc-value in Section 3 can be applied only in the case of rigid walls,which are modeled by fixing the horizontal displacement of walls in the FE analysis.Also, the minimum of the Nc-value must be chosen among the upper bound equa-tions. The results of the calculation are shown in Fig. 26 for rigid walls.Fig. 26 shows that there are good agreements between the FE analysis and the

closed form formula for rigid walls. Fig. 27 shows the results in case of T4Tc andthat the part of the wall (EI=360 MNm2) embedded under the base of excavationsis free for displacement. It is indicated that the closed form equations can be appliedand give good results until a critical value of the wall stiffness �s ¼ EI=�D

4 for thebase stability, where the critical wall stiffness is about 55 for the case of H=B ¼ 0:5,D=T ¼ 0:5, and EI=360 MNm2. The definition of the wall stiffness for the basestability is similar to the system stiffness proposed by Clough et al. [13]. When thewall stiffness for the base stability is smaller than the critical value, the Nc-valueobtained from the FE analysis becomes smaller than that obtained from the closedform equations as shown in Fig. 27. This means that these closed form equationscannot be used for the base stability of walls with the wall stiffness smaller than thecritical value about 55. This is because with increasing the wall embedded depthunder the base of excavations the wall stiffness decreases and the wall displaces withthe surrounding soil.

Fig. 26. Nc-values from the closed form equations and FE analysis in the case of rigid wall.


5. Conclusions

The Nc-values of excavations are predicted using the FEM with reduced shearstrength and compared with the ones from the closed form equations derived by theupper bound theorem. The influences of H=B, T, EI, Tn, and D on the Nc-values ofexcavations are evaluated. As the results of the FEM with reduced shear strength,the following conclusions are obtained:

1. The Nc-values of excavations increase almost linearly with H=B whenH=B4 1:0, and slightly increase withH=BwhenH=B > 1:0; in case ofD ¼ 0:0.

2. The presence of the hard stratum close to the base of excavation (T � TC)increases significantly the Nc-value of excavations when D ¼ 0:0.

3. The influence of D on the Nc-values of excavations depends on the wallstiffness.

4. To obtain good results using FEM for the 2D base stability the followingfactors must be considered: (i) The horizontal distance from the wall to theouter border of the excavation must be at least 2H; (ii) The vertical distancein the mesh under the base of excavation should be as fine as possible; (iii)The minimum number of rows in the mesh under the wall is not less than six.

5. The failure mechanism and the Nc-values obtained from the upper boundanalysis have shown the tendency similar to the ones by the FEM when thewall is rigid enough.

Fig. 27. Nc-values from the closed form equations and FE analysis in the case of wall under the base of

excavation (EI=360 MNm2).


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two-dimensional base stability of excavations in soft soils using fem

Documents

base of excavation t

base of braced excavation

theexcavation base

base instability isimportant

design of excavations

safety factor of excavations

wall d

wide excavation withcomputers