base stability analysis of braced deep excavation in undrained anisotropic clay with upper bound...

SCIENCE CHINA Technological Sciences

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*Corresponding author (email: [email protected])

• Article • September 2014 Vol.57 No.9: 1865–1876

doi: 10.1007/s11431-014-5613-2

Base stability analysis of braced deep excavation in undrained anisotropic clay with upper bound theory

WANG LiZhong* & LONG Fan

College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China

Received November 13, 2013; accepted May 29, 2014; published online August 15, 2014

It is imperative to evaluate factor of safety against basal heave failure in the design of braced deep excavation in soft clay. Based on previously published field monitoring data and finite element analyses of ground settlements of deep excavation in soft clay, an assumed plastic deformation mechanism proposed here gives upper bound solutions for base stability of braced deep excavations. The proposed kinematic mechanism is optimized by the mobile depth (profile wavelength). The method takes into account the influence of strength anisotropy under plane strain conditions, the embedment of the retaining wall, and the locations of the struts. The current method is validated by comparison with published numerical study of braced excava-tions in Boston blue clay and two other cases of excavation failure in Taipei. The results show that the upper bound solutions obtained from this presented method is more accurate as compared with the conventional methods for basal heave failure anal-yses.

braced deep excavation, base stability, upper bound theory, soil anisotropy

Citation: Wang L Z, Long F. Base stability analysis of braced deep excavation in undrained anisotropic clay with upper bound theory. Sci China Tech Sci, 2014, 57: 18651876, doi: 10.1007/s11431-014-5613-2

1 Introduction

Basal heave failure of braced deep excavation in soft clay usually causes huge damage to buildings nearby, thus it becomes prerequisite to evaluate factor of safety against basal heave failure in the design of the same. Base stability analysis of deep excavation in undrained clay is generally analyzed with two types of conventional methods: the bear-ing capacity method and the slip failure method [1–7]. However, the factors of safety calculated with different conventional methods vary, and the mechanisms assumed by them have certain difference with the actual mechanism. Moreover, conventional methods ignore the embedment depth of the retaining wall and anisotropy of soil. Therefore, the conventional methods are controversial in practical en-

gineering. New analytical methods based on strict limit analysis theory are required for better evaluation of base stability with consideration of soil anisotropy.

The admissible kinematical velocity fields for upper bound limit analysis of basal heave failure can be catego-rized as Terzaghi mechanism (analogical mechanism used in bearing capacity method) and circular slip mechanism. Chang adopted the modified Terzaghi mechanism (similar to the mechanism proposed by Bjerrum and Eide as shown in Figure 1(a)) considering base roughness [2,8]. Huang et al. took the effects of wall embedment into consideration which is similar to the mechanism proposed by Eide et al. as shown in Figure 1(b) [3,9]. Liao and Su adopted the circular slip mechanism for upper bound analysis as shown in Fig-ure 1(c) [10]. With different mechanisms for upper bound analysis, the calculated upper bound solution is more accu-rate when the assumed mechanism is more practical [11].

1866 Wang L Z, et al. Sci China Tech Sci September (2014) Vol.57 No.9

Consquently, the mechanisms which fit the failure dis-placements induced by excavation better are more appealing to obtain the accurate solution. On the other hand, the finite element limit analysis was pioneered for base stability con-sidering the wall embedment, as done by Ukritchon et al. [12], with linear programming. Nevertheless, the finite ele-ment limit analysis has not been adequate for engineering design, and its numerical failure mechanisms are far from definite.

The mobilizable strength design (MSD) method was originally proposed to predict displacements induced by excavation [13,14]. The plastic deformation mechanism in MSD method (Figure 2) is based on the incremental lateral displacement of retaining wall subjected to excavation be-neath the lowest level of support, as proposed by O’Rourke [15], which is assumed to conform to a cosine function. It is also assumed that wall embedment does not alter the basal failure mechanism in the soil, but does contribute to the stability due to the elastic strain energy stored in flexure. Refinement of the deformation mechanism presented in MSD method could improve the base stability analysis, with consideration of fitting the kinematic failure mechanism to

field monitoring data and finite element analysis results of deep excavation deformations in soft clay.

It is widely accepted that the anisotropic behavior of the soft soil has a major effect on deep excavation stability [16]. The anisotropy of the undrained shear strength has two components: the inherent anisotropy and the stress-induced anisotropy. The inherent anisotropy results from the anisot-ropy of the soil skeletal structure such as the preferred sed-imentation orientation of soil particles, and the stress- in-duced anisotropy rises from the anisotropic consolidated stress state.

Different testing methods do have influences on the measured value of strength of soil, such as direct simple shear test and triaxial test. That is because different testing methods have different stress paths and failure surfaces. The measured value of undrained shear strength is quite close to the actual value provided the stress path and failure surface are constricted to the actual conditions. Trixial tests include two stages: the consolidation stage and the shear stage. The K0-consolidated undrained triaxial tests can simulate the in-situ K0 consolidation stress state precisely, and the stress path can be controlled to simulate the actual stress path in the

Figure 1 Conventional mechanism for basal stability analysis. (a) Terzaghi and Bjerrum mechanisms without wall embedment; (b) Terzaghi and Eide mechanisms with wall embedment; (c) slip failure mechanism.

Figure 2 The modified incremental displacement profile.

Wang L Z, et al. Sci China Tech Sci September (2014) Vol.57 No.9 1867

shear stage. Hence the undrained shear strength of undis-turbed soft soil obtained from the K0-consolidated undrained triaxial tests (CK0UE and CK0UC tests) is widely adopted to describe the anisotropy of soil by the researchers in ge-otechnical engineering [17]. Since most base stability prob-lems of deep excavations are analyzed under plane strain conditions, the undrained shear strength must be converted from under triaxial conditions to plane strain conditions.

The purpose of this work is to propose a reasonable and relatively simple method to analyze base stability of braced deep excavation in anisotropy undrained clay. For compar-ison with other conventional methods, this paper overlooks the contribution of wall flexure energy to the stability. Fol-lowing the MSD method, a new kinematic mechanism that fits better with the observed deformation induced by braced excavation is proposed to analyze basal stability of braced deep excavation in anisotropic undrained clay within the framework of plasticity; furthermore, this mechanism can be optimized by the mobile depth. The proposed method adopts the strength criterion of undrained anisotropic clay and its parameters are obtained from the K0-consolidated undrained strength theory presented by Su et al [16] and Wang et al [17]. The applicability of the proposed method is validated with three case studies.

2 Plastic deformation mechanisms

Osman and Bolton proposed the plastic deformation profile for a multipropped wall supporting excavation in soft clay based on the incremental displacement profile proposed by O’Rourke, as shown in Figure 2. The incremental lateral displacement of a retaining wall subjected to excavation beneath the lowest level of support is assumed to be

m 2= 1 cos ,2

w ywl

(1)

where w=incremental wall displacement at any distance y from the lowest level of support; wm=maximum incremen-tal wall displacement; and l=full wavelength of the defor-mation pattern. According to the plastic deformation mech-anism proposed by Osman and Bolton, the ground surface movement behind the retaining wall follows the defor-mation pattern of the retaining wall, which means the in-cremental ground displacements conform to the same cosine function of eq. (1), by replacing y with x [13]. The incre-mental lateral displacement profile, as shown in Figure 2, is modified based on the field observations of ground settle-ment curves induced by excavation in soft clay and numer-ical studies on ground movement [18–20]. The modified displacement profile is assumed to be

2m 24 1 8exp .

2

yw w yl l

(2)

The maximum incremental lateral displacement point of the modified profile is located a quarter of the wavelength l away from the lowest level of support, so the maximum incremental ground settlement also occurs in the quarter of the wavelength. Figure 3 shows that the assumed displace-ment profile is in better agreement with the field monitoring ground settlement data and results of FEM analysis [18–20]. The shear strain increments in the plastic deformation mechanism proposed in this paper concentrate to the areas closer to the retaining wall as compared with that proposed by Osman and Bolton.

Following the plastic deformation mechanisms proposed by Osman and Bolton (Figure 4(a)), the plastic deformation mechanisms associated with the incremental displacement profile proposed by the authors for a braced excavation in soft clay are shown in Figure 4(b). With the soil beneath the lowest level of support excavated, the retaining wall is as-sumed to be fixed incrementally in position and rotate at the lowest level of support. Similar to the plastic deformation mechanisms proposed by Osman and Bolton [13], the re-taining wall and soil are deforming compatibly and the soil deformation follows the function of eq. (2). The dashed lines with arrows (Figure 4) point the direction of plastic flow, along which the displacement is constant. The soil is assumed to deform continuously inside the plastic mecha-nisms zones, while the soil outside the plastic mechanisms zones is assumed to be rigid. Mitchell and Soga [21] de-scribed the typical stiffness degradation curve along with the typical strain levels developed in geotechnical construc-tions as shown in Figure 5. For retaining walls of excava-tion, the strain level is beyond the elastic zone, which has minute strain level within 0.01%. The strain induced by braced excavation is mostly pre-yield plastic strain, and hence the elastic zones are neglected in the deformation

Figure 3 Comparison of assumed incremental displacement profile with field monitoring ground settlement curves and results of FEM analysis.


Figure 4 The plastic deformation mechanisms. (a) Proposed by Osman and Bolton [13]; (b) proposed by the authors.

Figure 5 The stiffness degradation curve along with the typical strain levels developed in geotechnical constructions.

mechanism, similar to the assumption of Osman and Bolton [13].

Osman and Bolton assumed that the relationship between wavelength l and the length of the wall s beneath the lowest level of support should be

.l s (3)

Osman and Bolton selected value of considering the wall tip fixing conditions. For retaining walls embedded into a stiff layer beneath the soft clay such that the wall tip is fully fixed, the wavelength is equal to the wall length beneath the lowest support (=1). For short walls embedded in soft clay, the restriction on the wall tip depends on the ratio of embedded depth to excavation depth so that the wavelength is larger than the wall length (>1). While α in Osman and Bolton’ method is constant depending only on the wall tip fixing conditions, the kinematic mechanism in this paper is optimized through the parameter within the framework of minimum energy dissipation principle.

Four different deformation patterns are assumed from the plastic deformation mechanisms for braced excavation as shown in Figure 4(b). In rectangle zone ABDC, by taking the top of the retaining wall as the origin, the deformation pattern of zone ABDC is given in rectangular coordinates as follows

2m 24 1 8exp

2

xv v xl l

， (4a)

0. w (4b)

For plane strain conditions, the shear strain increment is obtained as follow

2 2m 2 24 16 1 81 exp .

2

w x x

l l l (4c)

In fan zone CDE, by taking the apex of the fan zone (D) as the origin, the deformation pattern of zone CDE is given in polar coordinates as follows

2m 24 1 8exp

2

ru v rl l

， (4d)

0, ru (4e)

where r=radial distance from the apex of the fan zone (D). The shear strain increment is obtained as follow

2 2m 3 264 1 8exp .

2

w r r

l l (4f)

Similarly, the shear strain increments of zone EFH and zone FIH are given as eqs. (4g) and (4h).

2 2m3 2

64 1 8exp ,2

w r h r hl l

(4g)

where r=radial distance from the apex of the fan zone (F);


and h= distance between the lowest level of support and the excavation level. And

2 2m 2 24 16 1 81 exp ,

2

w h r r h

l l l (4h)

where r=distance along FH between point F and the inter-section with a plastic flow line.

3 Anisotropic undrained shear strength under plane strain conditions

Conventional methods simplified the undrained strength with depth and assumed it as isotropic by taking the un-drained strength su as the value of su,DSS, which is roughly equal to the average value of su,PSA and su,PSP [8,13,15]. The subscript DSS, PSA and PSP identify the undrained strengths obtained from direct simple shear test, plane strain active test and plane strain passive test respectively. However, the ap-proximation will lead to errors as the strength anisotropy of soil has great influence on the results of base stability anal-ysis.

Considering K0 consolidation effect and material anisot-ropy, an undrained strength criterion under plane strain conditions for soft clay, which was presented by Su et al. and derived from the criterion proposed by Prevost [16,23].

Taking the anisotropy of soil and the inclination of shear plane under plane strain conditions into consideration, the undrained plain strain strength sups along the shear plane proposed by Su et al. [16] can be expressed as follows

ups uccos (2 ), s s g (5a)

2 2m 2 24 16 1 81 exp ,

2

w h r h r

l l l (5b)

where uer ;

uc

sA

s

r

r

2 1 2;

3 1

An

A

2r2 1;

3

A

K

r1 ; A sups=undrained shear strength along the shear

plane under plane strain conditions; suc=undrained shear strength obtained from the K0-consolidated undrained com-pression (CK0UC) tests; and sue=undrained shear strength obtained from the K0-consolidated undrained extension (CK0UE) tests. ϕ′ is the average effective friction angle of soil, and β is the angle between the vertical plane and the shear plane. Su and Liao proved that whether the value of ϕ′ (ϕc′ or ϕe′) has little influence on the results [22]. It is also noted that =0° and =90° correspond to axial compression and extension in plane strain conditions respectively.

When the engineers do not have access to K0-consolidated triaxial compression and extension tests, a relatively simple and reliable method for obtaining undrained shear strength has been provided in this paper. Considering K0

consolidation-induced anisotropy, a theoretical formula for predicting the undrained shear strength under triaxial stress condition based on critical state soil mechanics and the fact that the initial yield surface is a rotated, Wang et al. [17] derive distorted ellipse on the p′-q plane. This formula is coherent with the experimental data, and the undrained tri-xial compression and extension shear strength gain ratio of normally K0-consolidated soil can be given by

0u

0

2 2 2K0nc 0 K0nc 0 K0nc

220 K0nc

1 26

+ 2,

2

v

M Ks

M M

M M

(6)

where 0

K0nc0

3 1= ;1+2

KK

2 2K0nc K0nc

0K0nc

3= ;

3

M

s

c=1 ;

CC

in which Cs is swelling index, and Cc is com-

pression index, M is the slope of critical state line. Then the Ar in eq. (5) can be given as

0 K0ncr

0 K0nc.

MA

M (7)

A simple and reliable method for obtaining the undrained shear strength by substituting eq. (6) and eq. (7) into eq. (5) has been provided as an alternative for the determination of anisotropic undrained shear strength when the K0 consoli-dated undrained triaxial test values are unavailable.

While considering the effect of anisotropic undrained shear strength and direction of shear plane, the assumed mechanism is divided into four typical zones where the di-rection of major principal stresses of each zone are illus-trated in Figure 6. In Figure 6, is taken to be 45° for Zone ABDC, zone DEG and zone FHE. Zone CDG is taken to be plane strain axial compression (=0°) condition and zone FIH is taken to be plane strain axial extension condition (=90°). Hence the plane strain undrained shear strength sups of different zones can be calculated from eq. (5).

4 Methods for basal heave failure analysis

Generally, basal heave failure of deep excavation is ana-lyzed using two typical methods: the bearing capacity method and the slip failure method. The bearing capacity method presented by Su et al. is based on the upper bound bearing capacity theory proposed by Atkinston [4,16]. The assumed plastic deformation mechanism is shown in Figure 7(a), which consists of three zones: compression zone CDE, radial shear zone EDF and extension zone FDH. The bear-ing capacity, Qu, is obtained by the equilibrium of incre-


Figure 6 Direction of the major principal stresses.

mental dissipation and work done by external forces based on upper bound theory [16]. The vertical load acting on CD plane, Q, denoted the weight of soil in rigid zone ABDC after subtracting the shear resistance along plane AC. In addition, the factor of safety for this method is determined as

u .Q

FsQ

(8)

However, while the vertical load acting on CD plane (Q) is configured from the weight of soil in rigid zone ABDC by subtracting the shear resistance along plane AC, Su et al. ignored shear resistance along the soil-wall interface. It will lead to an underestimation of factor of safety, so the shear resistance along plane BD should be taken into account for comparing upper bound analytical methods.

The slip failure method is widely used in many Asian countries such as China and Japan [5,6]. The method pro-posed by Heish et al. assumes a circular failure surface cen-tered at the lowest level of strut as shown in Figure 7(b) [7].

The driving force is the weight of soil in zone ABFE and the resisting force is shear resistance along plane IHC. The factor of safety for this method is defined as ratio of the resisting moment (Mr) to the driving moment (Md) of the lowest strut (point D):

r

d.

MFsM

(9)

However, in the above equation, the shear resistance along plane AC has been ignored for calculating the resist-ing moment, which may underestimate the factor of safety. The resisting moment should take the undrained shear re-sistance along the whole failure plane IHCA into considera-tion for upper bound analytical methods comparison.

The two conventional methods assume the retained soil to be rigid and ignore its shear strength, which obviously disagrees with reality. In the plastic deformation mechanism, the shear strength of the whole plastic deformation zones is taken into account for analysis. The factor of safety for the proposed method is defined as the ratio of the dissipated energy to the work done by external forces (eq. (10)). For a given step of a braced excavation, the dissipated energy of the soil in the mechanism depends on the kinematical mechanism, where the value of (1) determines the wavelength of the kinematical mechanism, the minimum value of Fs is obtained through searching the critical value of .

ups

V

sV

dV,

dV

sFs

v (10)

where s=total unit weight of soil.

5 Verification with Case Histories

The proposed method is validated by the results of nonline-ar finite element analyses of braced excavation in Boston

Figure 7 Plastic deformation mechanism of: (a) the bearing capacity method (Su et al. [16]); (b) the failure surface method (Heish et al. [7]).


Blue Clay and two basal heave failure cases of braced deep excavation in Taipei [7,16,20].

5.1 Case history 1

The MIT-E3 constitutive model was used by Hashash and Whittle for the nonlinear finite element analysis of excava-tion in Boston blue clay [20]. The plane strain braced exca-vation is 40 m wide and is retained by 0.9 m thick walls. The braces are assumed to be rigid and spaced at equal ver-tical intervals, h=2.5 m. Assumed that the possible failure of the diaphragm wall is not considered, the failure excava-tion depths, Hf, of different wall lengths (L=12.5, 20.0, 40.0, 60.0 m) of normally consolidated clay as given by Hashash and Whittle are shown in Figure 9.

The parameters used for normally consolidated Boston Blue Clay from Hashash and Whittle are listed in Table 1. By substituting the parameters into eq. (6) and eq. (10), the relationship between factors of safety and excavation depths of different wall lengths (L=12.5, 20.0, 40.0, 60.0 m) can be calculated by the method proposed by the authors as sum-marized in Figure 9. At the same time, the results of FEM nonlinear analysis by Hashash and Whittle are compared in Figure 9.

For excavations with a certain wall length L and excava-tion depths H, the value of parameter is obtained through searching the minimum value of factors of safety. The rela-tionship between factor of safety Fs and parameter of different wall lengths L and excavation depths H are shown in Figure 8. For short wall lengths L=12.5 m and L=20.0 m, Fs first decrease to the minimal value then increase as increase. For long wall lengths L=40.0 m and L=60.0 m, Fs increase with α. It is found that the kinematical mechanism can be optimized with =2 for short wall lengths L=12.5 m and L=20.0 m, and α=1 for long wall length L=40.0 m and L=60.0 m by minimization of dissipated energy. Consider-ing the results of the proposed method shown in Figure 9, the increase of factor of safety (Fs) from point A to B is the result of installation of the lowest strut, and as the excava-tion depth increases, the Fs decreases from point B to C until the critical state is reached. In general, the failure depths (Fs=1) calculated by the proposed method are smaller than the nonlinear FE results. For wall length L=40.0 m and L=60.0 m, the failure depths predicted by proposed method is the same as FE analysis. While, for wall length L=12.5 m and L=20.0 m, the maximum difference of failure depth between results predicted by the proposed method and the nonlinear FE analysis is within 5%, which implies that the assumed kinematic mechanism is very close to the defor-mation mechanism of FE analysis.

To investigate the rationality of proposed plastic mecha-nism for basal stability analysis, safety factors with the plastic mechanism proposed by Osman and Bolton is com-pared, under the same presumption of isotropic soil strength

suDSS. The parameter is assumed to be 2 for short wall length L=12.5 m and L=20.0 m and 4/3 for long wall length L=40.0 m and L=60.0 m in accordance to the Osman and Bolton’ mechanism. In contrast, in the proposed method, the parameter is found by minimization of factor of safety. Moreover, it is assumed to be 2 for short wall length L=12.5 m and L=20.0 m and 1 for long wall length L=40.0 m and L=60.0 m. Figure 10 (a) shows the failure mechanisms of the proposed method and Osman and Bolton’ method (=2) for short wall length L=12.5 m and L=20.0 m. In fact, the same value of =2 is obtained by minimization of safety factor of the proposed method. Figure 10 (a) shows that the difference lies in the various incremental displacement pro-file. The cosine curve is the profile adopted by Osman and Bolton, while the bold line similar to the distorted Gaussian curve is that proposed by the present study. Figure 10 (b) shows the failure mechanisms of the proposed method and Osman and Bolton’ method of long wall length L=40.0 m and L=60.0 m. Osman and Bolton’ method assumed α as a constant (=4/3). For the proposed method, α is obtained as 1 through optimization.

In Table 2, the factors of safety calculated with the mechanism proposed in this paper (where anisotropy of soil is ignored) and the mechanism proposed by Osman and Bolton [13] are compared on the basis of neglecting anisot-ropy of soil. The difference lies in the mechanism, and the comparison of the two mechanisms is judged by the failure depths obtained from the FE analysis by Hashash and Whit-tle [20]. For failure depths of different wall lengths obtained

Table 1 The parameters for Boston Blue Clay

Parameter Value

Total unit weight of soil, s (kN/m3) 18.0

Average effective friction angle, ′ (o) 33.0

Undrained triaxial compression strength ratio, suc/v′ 0.330

Undrained triaxial extension strength ratio, sue/v′ 0.155 Undrained strength ratio of direct simple shear test,

suDSS/v′ 0.21

Vertical effective stresses, v′ (kPa) 8.19z+24.5

Figure 8 Relationship between factor of safety and parameter α of case history 1.


Figure 9 Relationship between factor of safety and excavation depth for excavation in Boston Blue Clay. (a) Wall depth, L=12.5 m; (b) wall depth, L=20.0 m; (c) wall depth, L=40.0 m; (d) wall depth, L=60.0 m.

Figure 10 The failure mechanisms of case history 1 with wall length. (a) L=12.5 m and L=20.0 m; (b) L=40.0 m and L=60.0 m.

from FE analysis by Hashash and Whittle [20], the factors of safety with the plastic mechanism proposed in this paper are reasonably marginally lesser than 1.0 for this work ne-glects the stored energy in bending wall , while those calcu-lated with mechanism proposed by Osman and Bolton are obviously overestimated. Since the soil is assumed to be isotropic in both the mechanisms compared here, it is proved that for base stability analysis the proposed mecha-nism is more suitable than the plastic deformation mecha-

nism of Osman and Bolton.

5.2 Case history 2

The applicability of the proposed method in practical exca-vations is confirmed by the published case history of a braced excavation failure in Taipei [16].

The second failure case history is about a deep rectangu-lar excavation site, with dimension of about 100 m in length


and 26 m in width, and 13.45 m in depth. The excavation was retained by a 0.7 m thick and 24.0 m deep diaphragm wall, which was braced at a horizontal spacing varying from 4.1 m to 5.8 m. The excavation site was located at a backfill area of a watercourse, and the subsoil profile of the excava-tion site is shown in Figure 11. The top 8.7 m is backfill placed between 1980 and 1982, and the level ranging from ground surface level (GL) 10.7 m to 44.7 m is deep, soft, silty clay (CL). The groundwater level is at about GL 2.8 m. Basal heave failure occurred about two and half hours after the completion of the final stage of excavation (GL 13.45 m), and the entire internal bracing system collapsed.

The geotechnical boring data show that the geological conditions above the soft silty clay (GL 10.7 m) are quite uniform. To reinvestigate the strength parameters for un-disturbed in-situ soil, Su et al. sampled the CL soil (between GL 10.7 m and GL 44.7 m) across the street of the exca-vation site and conducted CK0UC and CK0UE tests [16]. The test results of CL are presented in Table 3, in addition, the

Table 2 Comparison of Fs calculated with plastic mechanism proposed by Osman and Bolton and the authors

Wall length, L (m) 12.5 20.0 40.0 60.0 Failure depth, Hf (m) (Hashash and

Whittle [20]) 10.0 15.0 22.5 30.0

Factor of safety with mechanism by the authors

0.85 0.89 0.93 0.95

Figure 11 Subsoil profile of case history 2.

Table 3 The parameters for case history 2

Depth (m) Soil type s (kN/m3) ′ Undrained strength ratio

0.0–4.5 SM 20.29 33.0° ——

4.5–8.7 ML 19.50 —— ——

8.7–10.7 SM 19.70 32.0° ——

10.7–44.7 CL 18.82 c′=29.2° e′=37.3°

suc/v′=0.271 sue/v′=0.189

average effective friction of soil used in the analysis is taken as the average value of c′ and e′.

The model for base stability analysis is simplified by taking the mechanical properties of soil above GL -10.45 as a homogenous soil layer similar to CL. Three methods are used in this case study, including method M1—the method proposed in this paper, method M2—the bearing capacity method considering the shear resistance along the soil-wall interface as proposed by Su et al [16] and method M3—the slip failure method considering shear resistance of the re-tained soil as proposed by Hsieh et al. [7]. For method M1, the relationship between factor of safety Fs and parameter α is shown in Figure 12, where the optimized parameter α is found to be 2 by minimization of factor of safety, while, its failure mechanism is shown in Figure 13. By adopting the three methods for base stability analysis of case history 2, the relationship between factor of safety (Fs) and excavation depth is shown in Figure 14.

The increase of Fs from point A to B is due to the instal-lation of the lowest level of strut. The location and installa-tion of struts cannot be considered in method M2 and M3. The failure depth derived from method M1, M2 and M3 is 13.6 m, 14.7 m and 15.0 m, respectively. The failure depth


Figure 13 The failure mechanism of case history 2.


Figure 14 Relationship between factor of safety and excavation depth for case history 2.

obtained from M1 is quite close to the critical excavation depth Hf=13.45 m, while Fs derived from method M2 and M3 is overestimated. It can be seen from Figure 13 that method M1 is more sensitive to variation of excavation depth as compared with method M2 and M3, which means the presented method is more accurate for predicting failure depth and early warning of engineering accident.

5.3 Case history 3

The real-time application of the proposed method is vali-dated by another published case history of a braced excava-tion failure in Taipei [7].

The failure case history 3 has a rectangular shape, 45 m long and 12.3 m wide. A 0.5 m thick diaphragm wall, the depth of which ranges from 13.8 to 17.0 m, retains the ex-cavation. The average wall depth and the final excavation depth are 15.4 and 9.3 m respectively. The subsoil profile of the excavation site is shown in Figure 15. The level ranging from ground surface level 5.5 to 41.0 m is silty clay (CL). The groundwater level is at about GL 1.5 m. Basal heave failure occurred when the site was excavated down to the final excavation depth.

Soil conditions were reinvestigated at a site 500 meters from the excavation and a series of CK0UC and CK0UE tests were conducted [7]. The test results are presented in Table 4, and the average effective friction of soil adopted in the analysis is taken as the average of c′ and e′.

Three methods are used in case study 3, including meth-od M1—the method proposed in this paper, method M2—the bearing capacity method considering the shear resistance along the soil-wall interface by Su et al. [16] and method M3—the slip failure method considering shear re-sistance of the retained soil by Hsieh et al. [7]. The dia-phragm wall length ranges from 13.8 to 17.0 m (average wall length, L=15.4 m) and the wall length of 15.4 and 17.0 m are analyzed using these three methods. For method M1,

the relationship between factor of safety Fs and parameter α of wall lengths of L=15.4 m and L=17.0 m are shown in Figure 16, and it is found that the optimized parameter α=2 by minimization of factor of safety. The failure mechanisms of wall lengths of L=15.4 m and L=17.0 m are shown in Figure 17 (a) and (b). The relationship between factor of safety and excavation depth of different wall lengths of L=15.4 m and L=17.0 m are shown in Figure 18(a) and Fig-ure 18 (b) evaluated with three analytical methods.

For average wall length L=15.4 m, the critical excavation depths derived from method M1, M2 and M3 are 9.1, 9.5 and 9.7 m, respectively. For maximum wall length L=17.0 m, the critical excavation depths derived from method M1, M2 and M3 are 9.4, 9.9 and 10.3 m, respectively. The minimal critical excavation depth is obtained from method

Figure 15 Subsoil profile of case history 3.

Table 4 The parameters for case history 2

Depth (m) Soil type s (kN/m3) ′ Undrained strength ratio

0.0-1.5 Fill 18.0 —— —— 1.5-5.5 ML 18.0 —— ——

5.5-8.5 CL 17.7 c′=27.7° e′=31.8°

suc/v′=0.311 sue/v′=0.172

8.5-41.0 CL 18.0 c′=25.2° e′=34.9°

suc/v′=0.288 sue/v′=0.168



Figure 17 The failure mechanism of case history 3.

Figure 18 Relationship between factor of safety and excavation depth for case history 3. (a) Wall length L=15.4 m; and (b) wall length L=17.0 m.

M1. The Fs obtained at the final excavation depth of aver-age wall length (L=15.4 m) is smaller than 1.0, while that obtained at the final excavation depth of maximum wall length (L=17.0 m) is larger than 1.0 by the proposed method. As limited details about the accident is reported, it can be deduced that the basal heave failure occurs first at the side of minimal wall length and then propagates to the side of maximum wall length.

6 Conclusion

Based on previous published field monitoring data and finite element analysis of ground settlement induced by excavation in soft clay, an assumed plastic deformation mechanism with depth optimization is proposed for base stability analysis of braced excavations in undrained clay. In comparsion with Osman and Bolton’s mechanism, the maximum displace-ment of ground surface movement in the proposed mecha-nism occurs closer to the retaining wall, and fits the actual ground settlement curves better. The method adopts the anisotropic strength criterion under plane strain conditions. The parameters needed for the criterion including the un-drained shear strength and effective friction angle, which can be determined from the CK0UC and CK0UE tests. The ani-sotropic undrained shear strength can be obtained by this

relatively simple and reliable method that would be highly useful for the engineers who donot have access to K0 con-solidated undrained triaxial tests.

The possible application of the proposed method is vali-dated by three case studies. In general, the failure depths predicted by the proposed method are very close to the re-sults of finite element analysis for case history 1. It is also proved that the presented mechanism is much more accurate than the mechanism associated with the cosine function profile under the same presumption of isotropic soil strength. The proposed method is also validated by two basal heave failure cases in Taipei, which show that the proposed method in this paper is more precise for predicting failure depth as compared with the capacity method and the failure surface method. The factor of safety in the proposed method is obtained by optimizing failure depth. Furthermore, the case studies show that for excavation in soft clay with rela-tively short embedded wall length, α=2. However, for ex-cavation with embedded wall length long enough, α=1. And for mediate conditions, 1<α<2. All of which depend on the optimization of kinematic mechanism to minimize the fac-tor of safety.

This work was supported by the National Science Foundation for Distin-guished Young Scholars of China (Grant No. 51325901) and the State Key Program of National Natural Science of China (Grant No.51338009).


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