System Failure Probability of c-φ Soil Slope Stability using Vertical Random Fields

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(1)Geotechnical Safety and Risk V T. Schweckendiek et al. (Eds.) © 2015 The authors and IOS Press. This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License. doi:10.3233/978-1-61499-580-7-209. 209. System Failure Probability of c-Soil Slope Stability using Vertical Random Fields Liang LI, Xuesong CHU School of Civil Engineering, Qingdao Technological University, China. Abstract. The need for system failure probability, pf , of soil slope stability that takes into account the spatial variation of soil properties has been acknowledged by the geotechnical profession. The spatial variations of cohesion and friction angle for a c- soil slope were modeled by one dimensional vertical random field respectively. The cross correlation of these two random fields was modeled by a simple decaying function. The effect of random field discretization (i.e., local average size) on pf was studied. The results show that the influence of local

(2) f cannot be ignored (relative error is greater than 10%) if.

(3) is greater than 0.2. The influence of local averag

(4) f is negligible (relative error is smaller than 10%) if variance

(5) less than 1.0. It is highly recommended to take into account the variance reduction of local averages and correlation between local averages using random fields for the calculation of pf

(6) 1.0 if no more than 10% relative error is expected. Keywords. System failure probability, random fields; Monte Carlo simulation. 1. Introduction The system failure probability of soil slope stability is well known to the geotechnical profession and it could be evaluated by probabilistic approach. However, traditionally, the system failure probability, pf of soil slope stability is evaluated on one (critical slip surface) or a limited number of slip surfaces (Hassan and Wolff, 1999; Xue and Gavin, 2007; Low et al., 2007; Li et al., 2009; Zhang et al., 2011; Ji and Low, 2012; Ji et al., 2012; Zhang et al., 2013). In recent years, pf is assessed by Finite Element Method (FEM) (Griffiths and Fenton, 2004; Xu and Low, 2006; Griffiths et al., 2009; Jiang et al., 2014) or Limit Equilibrium Method (LEM) incorporating Monte Carlo Simulation (MCS) and direct search for the critical slip surface (Wang et al., 2011;; Li et al., 2013a, 2014a, 2014b). The spatial variation of soil properties is one of sources of uncertainty (Phoon, 1999) and it should be included in the evaluation of pf. When random field is adopted to model the spatial variation of soil properties, the variance reduction of local averages and correlation between local averages have not been given. sufficient attention. Furthermore, the pf of cohesive soil slope stability is reported in literatures with little focus on the c- stability. This paper focuses on the pf of a c- soil slope stability. The spatial variations of cohesion (c) and friction angle () are modeled by vertical random fields. The variance reduction of local averages and correlation between local averages are considered. The influence of local average size on the pf is investigated. Finally, the recommendation is given.. 2. System Failure Probability 2.1. Summary of System Failure Probability by MCS In slope stability analysis, the pf is usually evaluated along a large but finite number of slip surfaces. Let N denote the number of slip surfaces considered in a slope, and Si denote the ith slip surface, i=1, 2, . . ., N. The slope can be treated as a series system, i.e., system failure occurs when failure occurs along any slip surface (Chowdhury and Xu, 1995). Let X=(x1,x2,…,xm).

(7) L. Li and X. Chu / System Failure Probability of c-ϕ Soil Slope Stability Using Vertical Random Fields. 210. denote all random variables involved in the slope reliability analysis, where m is the number of random variables, FSmin(X) denote the minimum Factor of Safety (FS) among N slip surfaces for a given X value. Then, the limit state function or performance function for system reliability of slope stability is expressed as FSmin(X)-1=0. The pf can be estimated using MCS as: pf |. 1 n ¦ I >FS min X i

(8) 1@ ni 1. (1). where n is the number of samples in MCS and Xi is the ith sample of X. I>FS min X

(9) 1@ is a function indicating the failure domain and defined as: 1 FS min X

(10) 1 I>FS min X

(11) 1@ ® ¯0 FS min X

(12) t 1. (2). with HS algorithm and particle swarm optimization algorithm for efficient location of the deterministic critical slip surface. The inhouse software is further developed in this study to allow reliability analysis of slope stability.. 3. Case Study of c-Slope 3.1. Geometry As shown in Figure 1, The c- soil slope has a height H=10m and slope angle of 45q. The c- soil slope is underlain by a hard stratum at 14 m below top of the slope. The shear strength of the c- unit weight of soil is . The stability of the c- soil slope is assessed using Bishop Method (Bishop, 1955). y(m). 1 1. …. 2.2. Selecting Representative Slip Surfaces 10m.

(13) y.

(14) L01i.

(15) L01. (i) (i+1). c (p). (p). 4m o.

(16) L0i.

(17) L1

(18) L0. c (i) c (i+1). …. Although the pf is evaluated based on N slip surfaces, the contribution of different slip surfaces to the pf is likely to be different (Zhang et al. 2011; Li et al., 2013a, 2014b). A finite number of representative slip surfaces are selected from N slip surfaces by using ‘Hassan and Wolff’ procedure. Its main idea is to locate the critical slip surfaces corresponding to Xj= ( , …, , , ,…, ), where

(19) and

(20) are the mean value and standard deviation of random variable xk. k=1, 2, …, m, j=1, 2, …, m. Finally, m representative slip surfaces are selected to determine FSmin(X) instead of using N slip surfaces thereby reducing computational effort.. (1) (2). c (1) c (2).

(21) Li. x(m) Hard Stratum. Figure 1. c- soil slope example.. Table 1. Values and distributions of input variables. Variable c. Distribution Normal (a vector with a length of p) Normal (a vector with a length of p) Deterministic * :Cov =Coefficient of variation. Statistics Mean=15kPa Cov*=0.30 Mean=23° Cov*=0.10 19kN/m3. 3.2. Input Variables 2.3. Software Package for Deterministic Stability Analysis Deterministic slope stability analysis must be performed to locate m representative slip surfaces. An in-house Fortran-based software package for deterministic slope stability analysis has been developed and used successfully in previous studies (Li et al., 2010, 2013a, 2013b, 2014b; Li and Chu, 2011, 2015a, 2015b; Chu et al., 2015). The software package is equipped. As summarized in Table 1, the is taken as deterministic value of 19 kN/m3, and each of c and are modeled by a one-dimensional random field spatially varying along the vertical direction. The spatial variability of c or with depth is modeled by a homogeneous normal random field with an exponentially decaying correlation structure. Consider the one-dimensional random field of for example. Let (yi)be the value of.

(22) L. Li and X. Chu / System Failure Probability of c-ϕ Soil Slope Stability Using Vertical Random Fields. Uc(1),M (i ). friction angle at depth yi. The correlation ij between (yi) and (yj) is given by: (3) K ij exp 2 yi y j / OM.

(23). Where is the effective scale of fluctuation of . As shown in Figure.1, the 14-m-thick cohesive soil layer is divided into p

(24) -m-thick sub-layers, and c and at each sub-layer is represented by an entry in a c and vector with a length of p(for example (1), (2),…, (p)).

(25) is the local average size. As listed in Table 1, the mean and standard deviation of c are equal to 15kPa and 4.5kPa (i.e., 30% coefficient of variation), respectively. The mean and standard deviation of ° and 2.3° (i.e., 30% coefficient of variation). 3.3. Variance Reduction of Local Average Let u and denote the mean value and standard deviation of , and u!

(26) ) and "!

(27) ) denote the mean value and standard deviation of for each

(28) -m-thick sub-layers. The variance reduction factor, #$ : 2. (4). correlation between different local averages such (1) (i) is calculated as: (Vanmarcke, 1977). UM 1

(29) ,M i

(30) . In. study,. '+c'+. 0.1 0.01 Hassan and Wolff. 0.001 @

(31) '<. 0.0001. 0.00001 1. 10 100 . 1000. (a)

(32) . (5). 2'L1'Li * 'L1

(33) * 'Li

(34). this. Consider three different values of

(35) $

(36) =2.0m, 1.0m and 0.5m. Then, 30 runs are + 8< 1000 m (starting from 1.0 m to 11.0 at equal interval of 0.5 m and 15.0 m, 20.0 m, 30 m, 40 m, ><$8<<$8><$<<$8<<<&?

(37) + specified, the # by Eq.(4), and then u!

(38) ), "!

(39) ) and uc(

(40) ), "c(

(41) ) are determined respectively. Secondly, m representative slip surfaces are selected by ‘Hassan and Wolff’ procedure (indicated by Hassan and Wolff in the following figures) to determine FSmin(X). For comparison, FSmin(X) is also minimized among a large number (N) of slip surfaces (indicated by direct search in the following figures). n=1,000,000 sets of random samples are generated in Monte Carlo Simulation (MCS).. 1. 'L01 p 2* 'L01i

(42) 'L02 2* 'L0i

(43). Uc(1),c(i ). 3.4. Case Studies. 1. Therefore, V M ( 'y ) = V M * 'y

(44) and u!

(45) &=u. V c('y ) can be determined likewise. The. 'L0 2* 'L0

(46) 'L012* 'L01

(47) 2'L1'Li * 'L1

(48) * 'Li

(49). (6). where c- is the cross correlation coefficient between c (-0.5is assumed in this study).. Failure probability. § V M ( 'y ) · ¸ * 'y

(50) ¨ ¨ V ¸ © M ¹ 2 'y 2 ·º § 0.5OM · ª §¨ 2'y O ¸¸ «2 ¨¨ 1 e M ¸» ¸» © 'y ¹ «¬ ¨© OM ¹¼. Uc M UM (1),M (i ). and. UM (1),M (i ) . Regarding the cross. $ assumed (Ji and Low, 2012), that is:. Failure probability. Uc M Uc(1),c(i ). 211. 0.1 0.01 Hassan and Wolff. 0.001. @

(51) '8<. 0.0001. 0.00001 1. 10 100 Scale of fluctuation . (b) 1

(52) . 1000.

(53) L. Li and X. Chu / System Failure Probability of c-ϕ Soil Slope Stability Using Vertical Random Fields. 212. 1 Failure probability. Failure probability. 1 0.1 0.01 ^ K_

(54) '<>. 0.001. 0.01

(55) '<

(56) '<

(57) '8<

(58) '8<

(59) '<>

(60) '<> . 0.001. 0.0001. @

(61) '<>. 0.0001. 0.1. 0.00001 1. 0.00001 1. 10 100 Scale of fluctuation . Figure 2. Comparison of pf from Hassan and Wolff procedure and direct search.. Figure 2 (a), (b) and (c) show the pf versus +

(62) =2 m, 1 m, and 0.5 m, respectively. It is noticed that the pf obtained from direct search agrees well with that by Hassan and Wolff procedure. This indicates that the representative slip surfaces located by Hassan and Wolff procedure can dominate the slope failure. . ^ K_

(63) '8< @ _

(64) '8< @ _

(65) '<> @ _

(66) '< ^ K_

(67) '< ^ K_

(68) '<>. 10000 1000 100 10 1 1. 10. 100. 1000. Figure 4. Comparison of pf with variance reduction and no variance reduction at different

(69) . (c)

(70) m. 100000. 10 100 . 1000. 1000. . Figure 3. Comparison of computational time from different procedures. Figure 3 shows the computational effort for each of 30 runs from direct search and Hassan K

(71) ' $

(72) '8

(73) '<>O h method takes more computational time than ‘Hassan and Wolff’ procedure takes. For example, When

(74) '8< $ Q 8> $ and Hassan and Wolff procedure takes 10 minutes. Figure.3 shows that the Hassan and Wolff procedure can efficiently evaluate the FSmin(X) with reasonable accuracy.. Figure 4 shows the pf calculated by Hassan and Wolff procedure with variance reduction (i.e., determining u!

(75) & $"!

(76) & and u!

(77) &$"!

(78) & using Eq.(4) and the correlation between local averages using Eqs.(5)~(6) and those without considering variance reduction (i.e., using u$", uc$"c as the approximation of u!

(79) &$"!

(80) & and u!

(81) &$"!

(82) & and correlation between different local averages evaluated at midpoints of each sub-layer). It is noticed from Figure.4 that the pf + value with variance reduction differentiates with that without variance reduction. This difference becomes larger as the l

(83) . Y

(84) '< +'8< $ example, the pf with variance reduction is 0.00005, and the value without variance reduction is 0.008. The pf with variance reduction is 0.00008 and it is 0.0003 without variance reduction

(85) '<>+'8< Let p1 and p2 denote the pf with variance reduction and without variance reduction, respectively. The relative difference is defined as: =. | | . × 100%. (7). Figure > \

(86) / + K

(87) ] + <$ \ smaller than 10%..

(88) L. Li and X. Chu / System Failure Probability of c-ϕ Soil Slope Stability Using Vertical Random Fields. &'()*. 10000 1000 100 10.

(89) '<

(90) '8<. 1.

(91) '<>. 0.1 0.01 0. 0.2. 0.4. 0.6. 0.8. 1. 1.2. %. !

(92) ` \

(93) / + 10. &'(+*.

(94) '<. 213. !&

(95) system failure probability pf is negligible (relative error is smaller than 10%) if variance reduction and correlation between local averages are considered for the cases that the ratio

(96) to scale of fluctuation is less than 1.0. (3) It is highly recommended to take into account the variance reduction of local averages and correlation between local averages using random fields for the system failure probability of soil slope stability. The ratio of local average.

(97) 8< if no more than 10% relative error is expected..

(98) '8<. Acknowledgements. 0 0. 0.2. 0.4. 0.6. 0.8. 1. 1.2. The present work was supported by National Natural Science Foundation of China (Grant Nos. 51274126) and China Scholarship Council (CSC). The financial supports are gratefully acknowledged.. %. Fig. 6 ` {

(99) +. The pf values with variance reduction at

(100) =0.5m are treated as the reference value for comparative study on the influence of local

(101) | p3 denote the pf with variance reduction at

(102) '<>, p4 denote the pf with variance reduction at larger local average size values. The relative difference Ƀ is defined as: | | (8) = × 100% . Figure 6 shows the Ƀ value versus

(103) ] +Ƀ is lower than 10% w

(104) ]+ 1.0.. 4. Conclusions The followings may be noted: !8&

(105) system failure probability pf cannot be ignored (relative error is greater than 10%) if ignoring variance reduction and correlation between local

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