Reliability Analysis of Heterogeneous Slope Considering Effect of Distribution Types

Reliability Analysis of Heterogeneous Slope

Considering Effect of Distribution Types

Shuihua JIANG a , Dianqing LI b and Bowen WEI a a

School of Civil Engineering and Architecture, Nanchang University, Nanchang, Jiangxi 330031, China

b

State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China

Abstract. The effect of probability distributions on the slope reliability is highlighted in this study since the lognormal

distribution is often used to characterize the variability of soil properties to avoid negative values. A multiple response-surface based MCS method for slope reliability analysis in spatially variable soils is proposed. A heterogeneous slope consisting of three soil layers (including a weak layer) is studied to demonstrate the validity of the proposed method and explore the effect of probability distributions for soil shear strength parameters on the reliability of the heterogeneous slope. The results indicate that the proposed method provides a practical tool for evaluating the reliability of heterogeneous slopes in spatially variable soils. The difference in the probabilities of failure of the heterogeneous slope associated with the lognormal, normal, gamma and beta distributions is significant when the spatial variability of multiple soil properties is incorporated. In general, the probability of failure may be underestimated as the lognormal distribution is used for characterization of the random fields of soil properties.

Keywords. heterogeneous slope, weak layer, spatial variability, reliability analysis, probability distribution

1. Introduction

The marginal probability distribution of soil properties is one of the most important parameters in probabilistic analysis and design of geotechnical structures (Lumb, 1970; Popescu et al., 2005). For simplicity, it is often assumed that the soil properties follow lognormal distribution to avoid negative values due to limited site observation data (Griffiths and Fenton, 2004; Huang et al., 2010; Wang et al., 2011; Cho, 2012; Jiang et al., 2014, 2015; Li et al., 2014, 2015; Li and Chu, 2015). In the literature (Zhou et al., 1999; Jimenez and Sitar, 2009; Jiang et al., 2011; Zeng et al., 2014), the effect of the type of probability distributions on geotechnical responses and reliability has been preliminarily investigated to clarify the rationale behind the commonly-used lognormal distribution. Zhou et al. (1999) found that the type of probability distributions for the coefficient of consolidation has a significant effect on the consolidation results. Jimenez and Sitar (2009) indicated the probability distributions for Young’s modulus have a significant impact on settlement results. Jiang et al. (2011) concluded that the failure

probability of a rock slope is sensitive to the probability distribution of the shear strength parameters. However, few attempts have been made to investigate the effect of probability distributions on the slope reliability when the spatial variability of shear strength parameters is incorporated. Additionally, to explore the effect of different types of probability distributions (i.e., lognormal, normal, gamma and beta) on slope reliability, an effective and general approach is to perform a series of parametric sensitivity studies. Theoretically, direct Monte-Carlo simulation (MCS) can be used for such a purpose. However, it could be time-consuming for parametric sensitivity analyses because numerous similar slope stability analyses are carried out repeatedly. The objective of this study is to propose a multiple response-surface based MCS method for slope reliability analysis and parametric sensitivity analysis in spatially variable soils. First, a multiple response-surface based MCS method is presented. Then, a heterogeneous slope consisting of three soil layers (including a weak layer) is investigated to illustrate the proposed method. Finally, a parametric sensitivity study is carried out to explore the

© 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:10.3233/978-1-61499-580-7-730

effect of probability distributions on the slope reliability.

2. Multiple Response-Surface Based MCS Method

To facilitate the calculation of the minimum factor of safety (FSmin) in MCS-based reliability analysis of slope, this study applies a quadratic polynomial chaos expansion (PCE) without cross terms to construct a response surface function (RSF) for each potential slip surface. To correctly identify the critical slip surface underlying a slope, Ns different potential slip surfaces are randomly generated to cover the entire failure domain of the slope, in which the value of Ns is frequently on the order of magnitude of 103 ~ 104 (e.g., Zhang et al., 2011; Li and Chu, 2015). The quadratic RSF between the factor of safety for the jth potential slip surface and input original random variables (such as shear strength parameters) involved in slope stability analysis is expressed as follows (Li et al., 2015): 2 , , 1, , , 1 1 1 ( ) ( ) c N n n j i j i j j i j i i j i i i i FS X

¦

¦

¦

1

= X , ,Xi, ,Xn

X   is the vector of input

random variables in the physical space, in which n is the number of input random variables;

1, 1, , 1, ,

= , , , , , ,

j a j b j bn j c j cn j

a   is the vector

of unknown coefficients with a size of Nc = 2n+1; <_{i j,} ( )˜ is a general PCE.

A sample design method using (2n+1) combinations is adopted to determine the unknown coefficients in Eq. (1) (e.g., Zhang et al., 2011; Li et al., 2015). After that, Ns quadratic RSFs are constructed to explicitly express the relations between the factors of safety and the original random variables. Thereafter, an extended Cholesky decomposition technique (Fenton, 1997; Li et al., 2015) is employed in this study to generate realizations of cross-correlated non-Gaussian random fields in the physical space. The kth realization of

cross-correlated non-Gaussian random fields can be obtained conveniently based on the kth random sample, which is substituted into Eqs. (1) and (2) to calculate the FSmin,

( ) 1,2, ,min s ( , ) where 1, 2, , k k min j MCS j N FS FS ª_¬X x yº_¼ k N   (2) where k( , ) x y

X is the kth realization of cross-correlated non-Gaussian random fields, (x, y) is the coordinate of an arbitrary location in a 2-D space. The probability of slope failure is then evaluated using direct MCS with a total of NMCS random samples as follows:

( ) f 1 1 1.0 MCS N k min k MCS p I FS N

¦

where I[.] is an indicator function. For a given random sample, I[.] is taken as the value of 1 for

1.0 min

FS  . Otherwise, it is set to zero. Note that the evaluation of the factors of safety during the MCS does not require deterministic slope stability analyses again, but only involves the evaluation of the algebraic expressions in Eq. (1). Hence, the computational cost for the proposed method is reduced substantially. Note that the RSFs remain unchanged for different statistics (e.g., coefficient of variation, COV, cross-correlation coefficient, marginal distribution and scale of fluctuation, SOF). In other words, there is no need to re-calibrate the RSFs during parametric sensitivity studies. Therefore, the proposed method provides an efficient way to perform the sensitivity analyses and to explore the influences of statistics on slope reliability.

3. Illustrative Example

3.1. Application to a Heterogeneous Slope with a Weak Layer

A heterogeneous slope example (see Figure 1) that consists of three soil layers (including a weak layer with a thickness of 0.5 m and a centroid Y-coordinate of 3.75 m) is investigated to demonstrate the validity of the proposed method. The groundwater with hydraulic heads of 11 m at the left side and 5 m at the right side is

also considered. The slope has a height of H = 10.0 m and an inclination of D = 45°. A typical random field element model for the considered slope is shown in Figure 2. The random field mesh consists of 4-noded quadrilateral elements, which are degenerated into 3-noded triangular elements at the sloping mesh boundary. The random fields are discretized into 1210 elements and 1281 nodes. A total of Ns = 2992 potential slip surfaces is randomly generated to cover the entire failure domain of the slope.

Figure 1. Profile of a hypothetic slope with a weak layer and

groundwater.

Figure 2. Random field element model and slope stability

analysis results (FS = 1.317).

The statistical properties of the soil parameters for the heterogeneous slope are summarized in Table 1. Cohesion c1     angle I1            undrained shear strength cu of the clay in the foundation and friction angle I2    layer are characterized by four lognormal random fields. The single exponential autocorrelation function with the horizontal SOF,

Gh = 40 m and vertical SOF, Gv = 4.0 m is used. The cross-correlation coefficient between c1 and

I1 is considered as

1,1

c I

U c c= -0.5. The minimum

factor of safety based on the mean values of the shear strength parameters (c1 I1cu, I2 is

obtained as 1.317 using Bishop’s simplified method. The corresponding critical failure surface passes through the bottom of the weak layer (see Figure 2).

Table 1. Statistical properties of soil parameters

Soil type Parameters Mean COV Distribution Sandy soil c1 5 0.225 Lognormal I1 46 0.15 Lognormal J1 (kN/m 3 ) 20 ü ü

Clay cu (kPa) 50 0.24 Lognormal J2 (kN/m3) 20 ü ü Weak layer c2 0 ü ü I2 10 0.225 Lognormal J3 (kN/m3) 20 ü ü 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1:1 line FS mi n d ete rm in ed f ro m s im plif ie d B is ho p m eth od FS

min determined fromstochastic response surfaces

Figure 3. Validation of multiple response surfaces in the

heterogeneous slope example.

The multiple quadratic RSFs between the factors of safety associated with 2992 potential slip surfaces and 1820 original random variables (i.e., 610 c1, 610 I1, 540 cu and 60 I2) are first constructed, which requires a total of 3641 deterministic slope stability analyses. To verify the effectiveness of multiple response surfaces in calculating FSmin of slope stability, Figure 3 compares the FSmin values of slope stability obtained from the 2992 response surfaces and those obtained from the deterministic analysis (e.g., Bishop’s simplified method) of slope stability using 100 sets of random samples. The FSmin values obtained from the two approaches agree well with each other. This indicates that the multiple response surfaces are good enough to obtain the FSmin in this example. The method reported in Lu and Zhang (2007) and Cho (2012) is then adopted here to simulate globally non-stationary random fields of three-layered soil Distance (m) 0 5 10 15 20 25 30 El ev at ion ( m ) 0 5 10 15 Embankment J1= 20 kN/m3 c1kPa I1!"e Foundation J2= 20 kN/m3 cu= 50 kPa Weak layer J3= 20 kN/m3 c2#I2#e Y Distance (m) 0 5 10 15 20 25 30 El evat ion ( m ) 0 5 10 15

Critical slip surface (FSmin= 1.317)

shear strength parameters. The extended Cholesky decomposition technique is used to generate 200,000 realizations of cross-correlated non-Gaussian random fields of c1I1cu, I2. Based on these, 200,000 FSmin values are obtained using Eq. (2) and the probability of slope failure is calculated as 2.68×10-2, which agrees fairly well with 2.74×10-2 obtained from the Latin Hypercube Sampling (LHS) with 10,000 random samples and evaluations of FSmin using the Bishop’s simplified method.

Generally, the location of the weak layer in the heterogeneous slope can be uncertain due to limited site data in geotechnical practice (Cao et al., 2014). In order to account for the uncertainty of the location of the weak layer properly in slope reliability analysis, a discrete random variable, Y, is used to characterize the vertical location of the weak layer. The probability mass function of Y is given by

where Y denotes the centroid Y-coordinate of the weak layer as shown in Figure 1. The probability of slope failure obtained from the proposed method is 3.03×10-2, which is well consistent with that (3.19×10-2) obtained from the LHS method with 10,000 random samples. The proposed method also accurately estimates the probability of slope failure incorporating the uncertainty of underground stratigraphy. It should be pointed out that the total required number of deterministic slope stability analyses increases dramatically to 18,205 when the uncertainty in the location of the weak layer is considered. This is because the RSFs need to be re-calibrated for each different location of the weak layer.

The effect of probability distributions on the slope reliability incorporating spatial variability of shear strength parameters is investigated using the proposed method. For illustrative purpose, only the reliability results associated with the fixed weak layer (Y = 3.75 m, see Figure 1) are presented. Figures 4 (a) and (b) present the

10 20 30 40 50 60 1E-3 0.01 0.1 Gv = 4.0 m, U c1I1 = -0.5 Lognormal Normal Gamma Beta P ro ba bility o f f ailu re

Horizontal scale of fluctuationGh (m)

(a) Horizontal scale of fluctuation

1.0 2.0 3.0 4.0 5.0 6.0 1E-3 0.01 0.1 Gh = 40 m, U c1I1 = -0.5 Lognormal Normal Gamma Beta P ro ba bility o f f ailu re

Vertical scale of fluctuationG_v (m) (b) Vertical scale of fluctuation

Figure 4. Effect of horizontal and vertical scales of

fluctuation on the probability of failure

probabilities of slope failure associated with four probability distributions (lognormal, normal, gamma and beta) for various horizontal and vertical SOFs, respectively. The difference in the probabilities of slope failure for different distributions exceeds one order of magnitude, and increases as the horizontal and vertical SOFs decrease, respectively. The probabilities of failure associated with the commonly-used lognormal distribution are significantly smaller than those associated with the other three distributions. This indicates that applying the commonly-used lognormal distribution to characterize the variability of shear strength parameters may lead to relatively conservative designs. In addition, the results underlying the normal and beta distributions almost remain the same, which are larger than those associated with the gamma distribution. The multiple quadratic RSFs between the factors of slope safety and the

original random variables do not rely on the statistics (e.g., marginal distributions, SOFs) of the soil properties. Hence, only a total of 3641 runs of slope stability analyses are performed as the marginal distributions and SOFs change.

4. Conclusions

A multiple response-surface based MCS method is proposed for slope reliability analysis in spatially variable soils. A heterogeneous slope which consists of three soil layers (including a weak layer) is studied to illustrate the proposed method and explore the effect of probability distributions on the slope reliability. Several conclusions can be drawn from this study:

1. The multiple response-surface based MCS method can effectively evaluate slope reliability when the spatial variability of multiple soil properties is incorporated. It achieves very high computational efficiency in the parametric sensitivity analysis and in exploring the influences of statistics (e.g., marginal distributions, SOFs) on the slope reliability.

2. The type of probability distributions used for characterization of the random fields of soil shear strength parameters has significant effect on the reliability of a heterogeneous slope. The probability of slope failure associated with the commonly-used lognormal distribution may be underestimated.

3. The potential slip surfaces are assumed as circular failure mechanisms in this study which may be not appropriate when the compound failure mechanism dominates. Thus, a deterministic slope stability analysis method that allows compound slip surfaces needs to be included in the proposed approach.

Acknowledgments

This work was supported by the National Science Fund for Distinguished Young Scholars (Project No. 51225903), the National Basic Research Program of China (973 Program)

(Project No. 2011CB013506), the National Natural Science Foundation of China (Project No. 51409139).

References

Cao, Z., Wang, Y. (2014). Bayesian model comparison and selection of spatial correlation functions for soil parameters. Structural Safety 49, 10-17.

Cho, S.E. (2012). Probabilistic analysis of seepage that considers the spatial variability of permeability for an embankment on soil foundation. Engineering Geology

133-134, 30-39.

Fenton, G.A. (1997). Probabilistic methods in geotechnical engineering. In: Workshop presented at ASCE

GeoLogan’97 conference, Logan, Utah.

Griffiths, D.V., Fenton, G.A., (2004). Probabilistic slope        $   % Journal of

Geotechnical and Geoenvironmental Engineering

130(5), 507-518.

Huang, J., Griffiths, D.V., Fenton, G.A. (2010). System reliability of slopes by RFEM. Soils and Foundations

50(3), 343-353.

Jiang, S.H., Li, D.Q., Zhang, L.M., Zhou, C.B. (2014). Slope reliability analysis considering spatially variable shear strength parameters using a non-intrusive stochastic finite element method. Engineering Geology 168, 120-128.

Jiang, S.H., Li, D.Q., Cao, Z.J., Zhou, C.B., Phoon, K.K. (2015). Efficient system reliability analysis of slope stability in spatially variable soils using Monte Carlo simulation. Journal of Geotechnical and Geoenvironmental Engineering, 141(2), 04014096.

Jiang, T., Liu, J., Yuan, B., Wang, S. (2011). Influence of probability distribution of shear strength parameters on reliability-based rock slope analysis. Slope Stability and

Earth Retaining Walls 65-72. http://dx.doi.org/10.

1061/47627 (406)9.

Jimenez, R., Sitar, N. The importance of distribution types on finite element analyses of foundation settlement.

Computers and Geotechnics 2009, 36(3), 474-483.

Li, D.Q., Qi, X.H., Zhou, C.B., Phoon, K.K. (2014). Effect of spatially variable shear strength parameters with linearly increasing mean trend on reliability of infinite slopes. Structural Safety 49, 45-55.

Li, D.Q., Jiang, S.H., Cao, Z.J., Zhou, W., Zhou, C.B., Zhang, L.M. (2015). A multiple response-surface method for slope reliability analysis considering spatial variability of soil properties. Engineering Geology 187, 60-72. Li, L., Chu, X. (2015). Multiple response surfaces for slope

reliability analysis. International Journal for Numerical

and Analytical Methods in Geomechanics, 39(2),

175-192.

Lu, Z., Zhang, D. (2007). Stochastic simulations for flow in nonstationary randomly heterogeneous porous media using a KL-based moment-equation approach.

Multiscale Modeling and Simulation SIAM 6(1),

228-245.

Lumb, P. (1970). Safety factors and the probability distribution of soil strength. Canadian Geotechnical

Popescu, R., Deodatis, G., Nobahar, A. (2005). Effects of random heterogeneity of soil properties on bearing capacity. Probabilistic Engineering Mechanics 20(4), 324-341.

Wang, Y., Cao, Z.J., Au, S.K. (2011). Practical reliability analysis of slope stability by advanced Monte Carlo simulations in a spreadsheet. Canadian Geotechnical

Journal 48(1), 162-172.

Zhang, J., Zhang, L M, Tang, W.H. (2011). New methods for system reliability analysis of soil slopes. Canadian

Geotechnical Journal 48(7), 1138-1148.

Zeng, P., Senent, S., Jimenez, R. (2014). Reliability analysis of circular tunnel face stability obeying Hoek–Brown failure criterion considering different distribution types and correlation structures. Journal of Computing in

Civil Engineering, http://ascelibrary.org/doi/abs/10.

1061/(ASCE)CP.1943-5487.0000464.

Zhou, W., Hong, H.P., Shang, J.Q. (1999). Probabilistic design method of prefabricated vertical drains for soil improvement. Journal of Geotechnical and Geoenvironmental Engineering 125(8), 659-664.