94 Han et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2013 14(2):94-100

Journal of Zhejiang University-SCIENCE A (Applied Physics & Engineering) ISSN 1673-565X (Print); ISSN 1862-1775 (Online) www.zju.edu.cn/jzus; www.springerlink.com E-mail: [email protected]

2D and 3D stability analysis of slurry trench in frictional/cohesive soil*

Chang-yu HAN, Jin-jian CHEN, Jian-hua WANG†‡, Xiao-he XIA (Department of Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China) †E-mail: [email protected] Received Oct. 6, 2012; Revision accepted Jan. 7, 2013; Crosschecked Jan. 23, 2013

Abstract: A 2D and 3D kinematically admissible rotational failure mechanism is presented for homogeneous slurry trenches in frictional/cohesive soils. Analytical approaches are derived to obtain the upper bounds on slurry trench stability in the strict framework of limit analysis. It is shown that the factor of safety from a 3D analysis will be greater than that from a 2D analysis. Compared with the limit equilibrium method, the limit analysis method yields an unconservative estimate on the safety factors. A set of examples are presented in a wide range of parameters for 2D and 3D homogeneous slurry trenches. The factor of safety increases with increasing slurry and soil bulk density ratio, cohesion, friction angle, and with decreasing slurry level depth and trench depth ratio, trench width and depth ratio. It is convenient to assess the safety for the homogeneous slurry trenches in prac- tical applications.

Key words: Limit analysis, Stability, Slurry trench, Diaphragm wall doi:10.1631/jzus.A1200257 Document code: A CLC number: TU46

1 Introduction extrusion in a slurry supported trench. Oblozinsky et al. (2001) proposed suggestions for the practical de- Slurry trenches are used in the construction of sign on the stability of the slurry trench based on the groundwater cutoff walls and subsurface structural elasto-plastic finite-element method (FEM) analysis. diaphragm walls. They are also the first stage of the Filz et al. (2004) presented a method for analyzing building process of barettes (they were used as deep global stability, and pointed out that increasing the foundations of the Petronas Towers in Kuala Lum- bentonite concentration of the slurry had beneficial pur). The stability of slurry trench has attracted great impacts on stability. Fox (2004) employed the limit attention among geotechnical researchers and indus- equilibrium method to estimate the stability factors try. The safety factor is an important index in the for general 2D and 3D stability of a slurry-supported design of slurry trench. Theoretical analysis, nu- trench. Based on the diaphragm wall construction of merical analysis and experiments have been under- two open-cut metro stations, Xu et al. (2011) studied taken to investigate slurry trench stability (Ng and the influence factors of stability of slurry trench sides Lings, 1995; Ng et al., 1995; Tsai and Chang, 1996; during diaphragm wall building in soft soil by ana- Ng and Yan, 1998; 1999; Tsai et al., 1998; 2000). lyzing the fullness coefficient of slurry trenches cor- Tsai (1997) presented an analytical method to responding to different controlling parameters. Based evaluate the stability of weak sublayers against lateral on the upper bound limit analysis theorem, Han et al.

(2012a) developed a 2D and 3D analysis of slurry trench for local and overall stability for cohesive soil. ‡ Corresponding author This paper is aimed at developing a 2D and 3D * Project (Nos. 41002095, 41172251 and 41272317) supported by the National Natural Science Foundation of China analysis of slurry trench stability for frictional/ © Zhejiang University and Springer-Verlag Berlin Heidelberg 2013 cohesive soil. Formulas for the slurry trench stability Han et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2013 14(2):94-100 95

analysis are obtained through theoretical derivation discontinuity. In Fig.1, radii of r0, r and rh, angles of with the basis of limit analysis theory, and then rota- θ0, θB, θsr and θh, and distances of h and hsr are pre- tional mechanisms are presented for slurry trench sented; ω is the angular velocity of the region ABC for stability. Examples are provided to illustrate the rotation; and φ is the internal friction angle. safety factor influence by the trench width and depth ratio (B/h), friction angle (φ), cohesion (c), slurry and 0 B sr  soil bulk density ratio (γsr/γ), and distance from slurry O  h Log-spiral surface level to trench top and trench depth ratio (hsr/h). ro r L Slurry level B A 2 Limit analysis theorems hsr  D v0 Guide wall rh

The limit analysis method is based on the ex- h tensions of the maximum work principle derived by Trench Hill (1948), and was given in the form of theorems by h'  C Drucker et al. (1952). Limit analysis aims at evalu- vh ating bounds on the limit load inducing or resisting failure in structures built of perfectly plastic materi- als. An upper bound can be obtained from the kine- matic method, in which the kinematically admissible Fig. 1 2D rotational mechanism for frictional/cohesive velocity field defines the possible mechanism of soil failure. The strain rates resulting from the velocity field must satisfy the flow rule that is associated with Slurry trench can be considered as a special the yield condition of the material, and the velocities situation of the slope, when the slope angle is 90°. satisfy the boundary conditions (Chen, 1975). Expressions of the rate of internal energy dissipation The most common yield condition used for soils (D) and the rate of work done by soil weight (Ws) for is the Mohr-Coulomb function, which contains two the 2D failure mechanism can be found in (Chen, material constants: the internal friction angle φ and 1975). the cohesion c. The perfectly plastic idealization of The work rate of the soil weight (Ws) can be Coulomb yield criterion with associated flow rule written as implies that any plastic deformation must be accom- panied by an increase in volume (for details please 3 Wrfffs 0[], 11 12 13 (1) refer to (Michalowski, 1995)). where γ is the unit weight of soil, ω is the angular

velocity of the region ABC for rotation, and r0 are 3 2D analysis of slurry trench stability shown in Fig. 1. The functions f1, f12 and f13 will be

explained later in Eqs. (11)–(13). In this study, a 2D rotational mechanism with a The internal dissipation of energy occurs along log-spiral surface AC is shown in Fig. 1, in which the the discontinuity surface AC. The total internal dis- failure surface is assumed to pass through the toe of sipation rate of energy (D) is found by integration the slurry trench. The log-spiral slip surface is used over the whole surface. for problems of slope stability (Chen et al., 2003;

Loukidis et al., 2003; Zhu et al., 2003; Jiang et al., cr 2 2009) and foundation stability (Kumar and Ghosh, D  0 [e2(h  0 )tan  1]. (2) 2007; Han et al., 2012b). The region ABC rotates as a 2tan rigid body around the center of rotation O with the materials below the logarithmic surface AC remaining The slurry velocity (vsr) during rotation about at rest. Thus, the surface AC is a surface of velocity axis O is 96 Han et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2013 14(2):94-100

vr  sin . (3) 1 sr x2 f11 2 [(3tan coshh sin ) 3(1 9 tan ) (11)

The slurry pressure is 3(h  0 ) tan  e (3tan cos00  sin  )],

1 prh  sec , (4) f12ff 02(2cos 0 02 ) , (12) sr sr x2 6

1 ()tanh  0  where γ is the unit weight of the slurry in the trench. ff13e[sin()sin]hh 0 02 sr 6 (13) The functions r , r and h″ are defined by ( ) tan  x1 x2 h 0 [cos002f cosh e ], 3( )tan  32 f esecsin()h 0   cos 14 sr sr h (14) rr h e,()tanh 0 (5) x1 0 [sin(hhsr ) 3sin(  sr  )], cossr f [e2(hh00 )tan  1][sin e (  )tan   sin ], (15) cos 15h 0 rr h e,()tanh 0 (6) x2 cos 0   f01cotB sin 0 sr  arccos   , (16) hr sin r sin . (7) 2 xx21w    1sin(2cscsin)ff01 0 01B 0 

 sin0  The rate of external work (Wsr) due to the slurry   arctan , (17) B  ()tanh  0  pressure on CD is cosh e 

 Wrh h 2  tan  d  where sr sr  x2 sr 1 3323( )tan  h 1 r eh 0 sec sin (  ) (8) sr 0 sr sr h f01  , (18) 12 h sin e()tanh  0  sin sr h 0 [sin(srhh ) 3sin(  sr )]. sin(hh 0 ) cos  f02  sin sin (19) For a slurry trench of given geometry, it is pos- hh [sin e()tanh  0  sin ], sible to evaluate the safety factor, defined as h 0

where h is the distance from slurry level to trench D  Wsr sr F  . (9) top. The upper-bound theorem of limit analysis gives Ws an upper bound for the value of the safety factor, F.

The function F has a minimum value when θ and θ The same expression of safety factor (F) for 0 h satisfy the conditions: cohesive soil can be found in (Han et al., 2012a).

The safety factor (F) in Eq. (9) can then be ob- FF tained by substituting Eqs. (1), (2) and (8) into Eq. (9)  0, 0 . (20)  as follows: 0 h

1 c f15 F  4 3D analysis of slurry trench stability 2tan(hfff ) 11 12 13 (10) 1  f  sr 14 . A 3D rotational mechanism is shown in Fig. 2 12  fff11 12 13 for frictional/cohesive soil with a logarithmic helicoid surface, in which the failure surface is assumed to In Eqs. (1) and (10), f11, f12, f13, f14, f15, θsr and θB pass through the top and the toe of the slurry trench. are functions of soil strength parameters and geome- The same shape of this mechanism is considered by try of the slip surface, which can be defined as Michalowski and Drescher (2009), Han et al. (2013), follows: and Xia et al. (2012) for the evaluating safety factor Han et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2013 14(2):94-100 97

of slope. Soil over the failure surface rotates about the The slurry velocity during rotation about axis O center of rotation O. Failure surface AC is the velocity is discontinuous surface. Assume the mechanism can be vr  sin . (23) specified completely by three variables. For conven- sr x2 ience, we select the angles θ0, θh and rr00 / .

 The slurry pressure is 0 B sr  O   E h 0A= r 0 prhRd2sec,  22   (24) 0A = r0 sr sr x 2 A r  Slurry level B A x hsr where γsr is the unit weight of the slurry in the trench. D R r  v0 The rate of external work (W ) due to the slurry Guide wall sr C  y pressure is h

a h h' WrhRd2tand222 Trench sr w  x2  C w vh  r 33cos  e3(h  0 )tan  (25) sr 0 h

h 2tan(tantan)x    2srd.  2 sr cos  Fig. 2 3D rotational mechanism for frictional/cohesive soil The safety factor in Eq. (9) can then be obtained With the slope angle β=90°, expressions of the by substituting Eqs. (21), (22), (25) into Eq. (9) as rate of internal energy dissipation (D) and the work follows: rate of soil weight (Ws) for the 3D failure mechanism can be found in (Michalowski and Drescher, 2009; crffcot( ) 2  Han et al., 2013; Xia et al., 2012). F  022sr21. (26) 2( ff23 24 ) The work rate of the soil weight (Ws) can be written as In Eqs. (21), (22), (25) and (26), rm, R, x1, x2, y1,  xy  B 11 2 a, d, f21, f22, f23 and f24 are functions of soil strength Wryyxs 2 (m )cosddd 0 0 a (21) parameters and geometry of the slip surface which  xy h 21(ry )cosddd2  yx . can be defined as follows:  0 d m  B  rr  rr  The internal dissipation of energy (D) can be rRm ,, (27) computed: 22 22 2 2 22 x12Rax,,,  Rdy 1  Rx (28) 22 2( )tan  Dcr 2cotcoseh 0 sin cos 0  h 0 h ()tanh  0   arrdr00mm,e,  r (29) sin cos  h sin 22 Rdd (22) 333(h  0 )tan  3 fr21 0 ecosh B cos   (30)  h 2tan(tantan)x2sr  222B cos   d, sin Ra d ,  2 0  3  sr cos  0 sin   h sin 2(h  0 )tan  2 fx22 ecosdh 3 2  B cos  where the functions rm, R, x1, x2, y1, a, d will be ex- (31)  plained in Eqs. (27)–(29). The two integrals relate to 2 B cos  sin013 x d , the work rate in the upper portion of the slope, in the 0 sin   xy range (θ0, θB), and in the remaining part of the slope B 11 2 fryyx23 (m )cosddd  , (32) (θB, θh). 0 0 a 98 Han et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2013 14(2):94-100

 xy fryyxh 21()cosddd,2   (33) Figs. 6 and 7 also show the comparisons between 24  0 d m B the safety factor calculated from this study and from the limit equilibrium method by Fox (2004). It can be where seen that the trends of the upper bound results are similar to those of Fox (2004), but the factors of ()tan 00 ()tan  safety obtained from this study are higher than those rr00e,e. r r (34) presented by Fox (2004). It means that, the factors of safety derived from the limit analysis method are The function F has a minimum value when θ0, θh higher than those calculated form the limit equilib- and rr / satisfy the conditions: 00 rium method. The limit equilibrium analyses lead to a

FF  F ,3D,B/h=1, this study 0, 0,  0. (35) 1.6  ,3D,B/h=5, this study 000h (/)rr ,2D, this study 1.4 , 3D,B/h=1, Han et al. (2012a) ,3D,B/h=5, Han et al. (2012a) ,2D, Han et al. (2012a) 1.2 5 Results and discussion F 1.0

The estimates of safety factor F were obtained 0.8 using a procedure for the given slurry and soil bulk 0.6 density ratio, slurry level depth and trench depth ratio, 0.50 0.55 0.60 0.65 0.70 cohesion, friction angle, trench width and depth ratio γsr /γ

(in 3D), soil bulk density and trench depth. Inde- Fig. 3 Comparison of results with different γsr/γ for h /h=0.05, c=20 kPa, γ=18 kN/m3, and h=20 m pendent variables in the procedure were: angles θ0 sr and θh, and ratio rr / (in 3D). These parameters 00 1.2 were varied by a small increment in computational loops, and the process was repeated until the mini- 1.0 mum of F was reached, with the increments of 0.1° 0.8 used for angles θ0 and θh, and 0.01 for ratio rr00 / . F 0.6 In this section, the solutions obtained by Han et ,3D,B/h=1, this study ,3D,B/h=5, this study al. (2012a) and Fox (2004) will be compared for 0.4 ,2D, this study , 3D,B/h=1, Han et al. (2012a) various parameters, including γ /γ, h /h, c, φ and B/h.    sr sr 0.2 , 3D, B/h=5, Han et al. (2012a) The results of these computations are graphically ,2D, Han et al. (2012a) 0.1 0.2 0.3 0.4 0.5 represented in Figs. 3–7. The differences in 3D and h /h sr 2D factors of safety of slurry trenches can be meas- Fig. 4 Comparison of results with different hsr/h for ured by the vertical distance between the respective γ /γ=0.6, c=20 kPa, γ=18 kN/m3, and h=20 m sr lines in these figures. It can be found that the factors 1.4 of safety increase with increasing slurry and soil bulk ,3D,B/h=1, Han et al. (2012a) 1.3 ,3D,B/h=5, Han et al. (2012a) density ratio, cohesion, friction angle, and with de- ,2D, Han et al. (2012a) 1.2 creasing slurry level depth and trench depth ratio, ,3D,B/h=1, this study 1.1 ,3D,B/h=5, this study trench width and depth ratio. ,2D, this study 1.0 In Figs. 3–5, and 7 the data points corresponding F to φ=0° (cohesive soils), represent the solutions ob- 0.9 tained by Han et al. (2012a). The analytical upper 0.8 bound solutions of this study are higher than those 0.7 presented by Han et al. (2012a). The friction angle 0.6 5 10152025 φ>0° is the reason why the factors of safety obtained c (kPa) from this study are greater than those presented by Fig. 5 Comparison of results with different c for γsr/γ=0.6, 3 Han et al. (2012a). hsr/h=0.05, γ=18 kN/m , and h=20 m Han et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2013 14(2):94-100 99

analysis gradually when the ratio of B/h increases. 5 3D, B/h=0.5, Fox (2004) Using 2D solutions to evaluate the stability of 3D 3D, B/h=1, Fox (2004) slurry trenches may underestimate factors of safety. 4 2D, Fox (2004) 3D, B/h=0.5, this study Compared with the limit equilibrium method, the 3D, B/h=1, this study 3 2D, this study limit equilibrium method yields a conservative esti- F mate on the safety factors. 2 References 1 Chen, J., Yin, J.H., Lee, C.F., 2003. Upper bound limit analysis 5 1015202530 of slope stability using rigid finite elements and nonlinear φ (°) programming. Canadian Geotechnical Journal, 40(4): Fig. 6 Comparison of results with different φ for c=20 kPa, 742-752. [doi:10.1139/t03-032] 3 γsr/γ=0.6, hsr/h=0.05, γ=18 kN/m , and h=20 m Chen, W.F., 1975. Limit Analysis and Soil Plasticity. Elsevier Scientific Publication Co., Amsterdam and New York. 2.0 ,3D, Fox (2004) Drucker, D., Prager, W., Greenberg, H., 1952. Extended limit 1.8 ,3D, Fox (2004) design theorems for continuous media. Quarterly of Ap- ,2D, Fox (2004) plied Mathmatics, 9(4):381-389. 1.6 ,2D, Fox (2004) ,3D, this study Filz, G.M., Adams, T., Davidson, R.R., 2004. Stability of long 1.4 ,3D,Han et al. (2012a) trenches in sand supported by bentonite-water slurry. ,2D, this study F 1.2 , 2D, Han et al. (2012a) Journal of Geotechnical and Geoenvironmental Engi- 1.0 neering, 130(9):915-921. [doi:10.1061/(ASCE)1090-0241 0.8 (2004)130:9(915)] Fox, P.J., 2004. Analytical solutions for stability of slurry 0.6 trench. Journal of Geotechnical and Geoenvironmental

0.4 Engineering, 130(7):749-758. [doi:10.1016/(ASCE)1090- 246810 B/h 0241(2004)130:7(749)] Han, C.Y., Wang, J.H., Xia, X.H., Chen, J.J., 2012a. Limit Fig. 7 Variation of safety factor F with B/h for c=20 kPa, 3 analysis for local and overall stability of slurry trench in γsr/γ=0.6, hsr/h=0.05, γ=18 kN/m , and h=20 m cohesive soil. International Journal of Geomechanics, in press. [doi:10.1061/(ASCE)GM.1943-5622.0000268] conservative estimate on the safety of slurry trenches. Han, C.Y., Xia, X.H., Wang, J.H., 2012b. Upper bound solu- Using limit analysis can not only provide a useful way tions of ultimate bearing capacity of curved footing. Chi- of analyzing the stability of slurry trenches, but can nese Journal of Geotechnical Engineering, 34(2):230-236 also avoid the shortcomings of the arbitrary assump- (in Chinese). tions of the limit equilibrium method (Yu et al., Han, C.Y., Xia, X.H., Wang, J.H., 2013. Analytical Solutions 1998). for Three-Dimensional Stability of Coastal Slope. Huang, Y., Wu, F.Q., Shi, Z.M., Ye, B. (Eds.), New Frontiers in Engineering Geology and the Environment. Springer Berlin Heidelberg, 9:181-185. 6 Conclusions Hill, R., 1948. A variational principle of maximum plastic work in classical plasticity. The Quarterly Journal of Mechanics and Applied Mathematics, 1(1):18-28. The upper bound method for the 2D and 3D Jiang, P.M., Sheng, H., Lu, C.F., Mei, G.X., 2009. Rigorous stability analysis of slurry trenches in frictional/ solution of the slope ultimate bearing capacity. Chinese cohesive soil are presented in this paper. It is inves- Journal of Engineering Mechanics, 26(Suppl. 1):77-80 (in tigated via examples for the effects of slurry and soil Chinese). bulk density ratio, slurry level depth and trench depth Kumar, J., Ghosh, P., 2007. Ultimate bearing capacity of two interfering rough strip footings. International Journal of ratio, cohesion, friction angle, trench width and depth Geomechanics, 7(1):53-62. [doi:10.1061/(ASCE)1532- ratio on slurry trench stability. The factor of safety 3641(2007)7:1(53)] increases with increasing slurry and soil bulk density Loukidis, D., Bandini, P., Salgado, R., 2003. Stability of seis- ratio, cohesion, friction angle, and with decreasing mically loaded slopes using limit analysis. Geotechnique, slurry level depth and trench depth ratio, trench width 53(5):463-479. [doi:10.1680/geot.53.5.463.37509] Michalowski, R.L., 1995. Slope stability analysis: A kinema- and depth ratio. The factor of safety from a 3D anal- tical approach. Geotechnique, 45(2):283-293. [doi:10. ysis will approach the factor of safety from a 2D 1680/geot.1995.45.2.283] 100 Han et al. / J Zhejiang Univ-Sci A (Appl Phys & Eng) 2013 14(2):94-100

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