System Reliability Analysis of Rock Slope Stability Involving Correlated Failure Modes

Transcription

1 KSCE Journal of Civil Engineering (20) 5(8): DOI 0.007/s Geotechnical Engineering System Reliability Analysis of Rock Slope Stability Involving Correlated Failure Modes Dian-Qing Li*, Shui-Hua Jiang**, Yi-Feng Chen***, and Chuang-Bing Zhou**** Received June 8, 200/Accepted February 23, 20 Abstract This paper aims to propose a systematic quantitative method for system reliability evaluation of rock slope with plane failure involving multiple correlated failure modes. A probabilistic fault tree approach is presented to model system reliability of rock slope. An n-dimensional equivalent reliability method is employed to perform the system reliability analysis of the slope involving multiple correlated failure modes. Reliability sensitivity analyses at three different levels, namely, the single limit state function level, single failure mode level, and system reliability level, are carried out to study the effect of variables on reliability. An example is presented to demonstrate the validity and capability of the proposed approach. The results indicate that the system reliability of rock slope involving multiple correlated failure modes can be evaluated efficiently using the proposed approach. The system probability of failure is overestimated if the correlations between different failure modes are ignored. The relative importance of different failure modes to the system reliability can differ considerably. The sensitivity coefficients of basic random variables strongly depend on the selected sensitivity analysis level. The system reliability is sensitive to the location of the tension crack and the percentage of the tension crack filled with water. Keywords: rock slope, system reliability, n-dimensional equivalent method, probabilistic fault tree, correlated failure modes. Introduction It is widely recognized that there are often many uncertainties in the analysis of rock slope stability owing to inadequate information for site characterization and inherent variability and measurement errors in geological and geotechnical parameters (Phoon, 2008; Phoon and Kulhawy, 999a; Phoon and Kulhawy, 999b). Therefore, reliability-based approaches that allow the systematic and quantitative treatment of these uncertainties have become a topic of increasing interest in rock slope engineering. There has been extensive reliability analysis of rock slope stability (Low, 2008; Duzgun and Bhasin, 2009; Li et al., 2009; Li et al., 20). Generally, reliability analysis of rock slope stability can be divided into two categories. One is reliability analysis for a single failure mode of rock slope stability. The other is system reliability analysis for multiple failure modes of rock slope stability. The reliability of a rock slope with a single failure mode has been investigated extensively in the literature (Tamimi et al., 989; Genske and Walz, 99; Pathak and Nilsen, 2004; Low, 2007; Low, 2008; Duzgun and Bhasin, 2009). For example, Tamimi et al. (989) investigated the reliability of a rock slope with plane failure through Monte Carlo simulation. Duzgun and Bhasin (2009) carried out a probabilistic analysis of the Oppstadhornet rock slope with plane failure using First-Order Reliability Method (FORM). However, a rock slope may involve several potential failure modes, in which case the overall reliability depends on individual failure modes as well as the correlations between failure modes. Therefore, the system reliability of rock slopes should be investigated. Regarding the system reliability of a rock slope with multiple failure modes, Low (997) investigated the system reliability of a rock wedge with four failure modes employing Cornell s bound method (Cornell, 967). Jimenez-Rodriguez et al. (2006) explored a disjoint cut-set formulation to model the system reliability of a rock slope with plane failure in which each cut set corresponds to a failure mode of the rock slope. Similarly, such an approach was further applied to the system reliability analysis of a rock wedge with four failure modes (Jimenez-Rodriguez *Professor, State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, Wuhan, , P. R. China (Corresponding Author, dianqing@whu.edu.cn) **Ph.D Candidate, State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, Wuhan , P. R. China ( jiangshuihua-2008@63.com) ***Professor, State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, Wuhan , P. R. China ( csyfchen@whu.edu.cn) ****Professor, State Key Laboratory of Water Resources and Hydropower Engineering Science, Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering, Ministry of Education, Wuhan University, Wuhan , P. R. China ( cbzhou@whu.edu.cn) 349

2 Dian-Qing Li, Shui-Hua Jiang, Yi-Feng Chen, and Chuang-Bing Zhou and Sitar, 2007). Li et al. (2009) studied the system reliability of a rock wedge employing an n-dimensional equivalent reliability approach. Fadlelmula et al. (2008) compared the system probability of the failure of a wedge, obtained using Cornell s bound method, for the Coulomb linear shear failure criterion with that for the Barton-Bandis non-linear shear failure criterion. However, the range between the upper and lower bounds of the system probability of failure determined using Cornell s bound method may be too wide to be meaningful (Grimmelt and Schueller, 982). Moreover, the correlation between pairs of potential failure modes of the rock slope cannot be taken into account by Cornell s bound method. The disjoint cut-set formulation (Jimenez-Rodriguez et al., 2006) cannot account for correlations between different failure modes of the rock slope, which further results in overestimation of the system probability of failure of the rock slope. Although such correlations may be taken into consideration by Monte Carlo simulations, the approach is too time-consuming to be of practical interest to engineers. In addition, sensitivity analyses at the system reliability level have not been investigated sufficiently. For the system reliability analysis of rock slopes, the lognormal distribution is often used to describe cohesion (Jimenez- Rodriguez et al., 2006). The lognormal distribution can exclude negative values. However, the lognormal distribution is within the range of (0, + ), which is not suitable for cohesion because the cohesion of the rock mass is bounded. Compared with the lognormal distribution, the four-parameter beta distribution is more versatile as demonstrated by Low (2007, 2008), and it can be used to model the cohesion associated with the system reliability of rock slopes. Therefore, it is necessary to develop a method for system reliability analysis considering the correlations between different failure modes of the rock slope. The objective of this paper is to propose a system reliability approach for evaluating the stability of rock slopes involving multiple correlated failure modes. Firstly, limit state functions for a rock slope with plane failure are formulated. The system aspects of the rock slope stability analysis are represented by a probabilistic fault tree (Thacker et al., 2006). The versatile four-parameter beta distribution is then used in lieu of a normal distribution or a lognormal distribution to describe the location of the tension crack and the cohesion and friction angle along the failure surface. A truncated normal distribution and truncated exponential distribution are used to describe the reinforcing force and the percentage of the tension crack filled with water, respectively. Thereafter, an n-dimensional equivalent method is employed to analyze the reliability of the rock slope. Sensitivity analyses of reliability at three different levels, namely, the single limit state function level, single failure mode level, and system reliability level, are carried out to evaluate the effect of changes in variables on the slope stability. Finally, an example is presented to illustrate the proposed method. 2. Formulation of the Limit State Function for a Rock Slope with Plane Failure For illustrative purposes, a simple sliding mass composed of two blocks separated by a vertical tension crack is considered, as shown in Fig. (Jimenez-Rodriguez et al., 2006). To account for the effect of rock reinforcement, a passive force with uncertain magnitude is applied at the toe of the slope. In addition, the location of the tension crack is assumed to be random. Generally, for plane failure of the considered rock slope to occur, four geometrical conditions must be satisfied (Hoek and Bray, 98): (a) a continuous plane on which sliding occurs must strike parallel or nearly parallel (within approximately ±20 o ) to the slope face, (b) the dip of the failure plane must be less than the dip of the slope face, (c) the dip of the failure plane must be greater than the angle of friction of this plane, and (d) surfaces of separation that provide negligible resistance to sliding must be present in the rock mass to define the lateral boundaries of the failing block. In the stability analysis of a rock slope with plane failure, the limit equilibrium method is usually employed. The factor of safety for the considered rock slope with plane failure is calculated by resolving all forces acting on the slope into components parallel and normal to the sliding plane. The vector sum of the shear forces acting down the plane is termed the driving force. The product of the total normal forces and the tangent of the friction angle, plus the cohesive force, is termed the resisting force. The factor of safety of the sliding block is the ratio of the resisting forces to the driving forces. Fig.. Geometrical Definition of the Stability Model of A Rock Slope (Modified from Jimenez-Rodriguez et al., 2006): (a) Tension Crack at Slope Top, (b) Tension Crack at Slope Face 350 KSCE Journal of Civil Engineering

3 System Reliability Analysis of Rock Slope Stability Involving Correlated Failure Modes To conduct reliability analysis of the considered rock slope, a clear distinction between success and failure events should be established. Therefore, it is assumed that the slope performance is satisfactory when the factor of safety of block A is greater than that determined using the stability model proposed by Hoek and Bray (98) with proper modifications to account for the interaction between two blocks. Following Jimenez-Rodriguez et al. (2006), two different cases may be identified in the analysis, depending on the interaction between blocks A and B as follows. In case, block B is stable by itself; i.e., there is no interaction between blocks. In case 2, block B is unstable; i.e., block B will tend to slide, which will impose an interaction force, I F, on block A. 2. Case : No Interaction between Two Blocks In case, the factor of safety against sliding for block A is expressed as: c FS A A A + ( Tcosθ W A cosψ p U A Vsinψ p )tanφ A = A () W A sinψ p + Vcosψ p Tsinθ where c A is the cohesion along the failure surface for block A; φ A is the friction angle; A A is the area of contact with the failure surface; W A is the weight of block A; U A and V are the forces induced by water pressure; T is the reinforcing force; θ is the inclination of the reinforcing force, which is equal to the angle between the normal direction of the failure surface and the reinforcing force; and ψ p is the dip of the slope failure plane. A A, U A, and V are given by: A A = ( H z)cscψ p U A = --γ 2 w z w ( H z)cscψ p V 2 --γ 2 = w z w where H is the height of the slope; z is the distance between the slope top and the bottom of the vertical tension crack; z w is the vertical height of water in the tension crack; and γ w is the unit weight of water. The weight of block A depends on the location of the tension crack. For the tension crack located at the top of the slope as illustrated in Fig. (a), W A is given by: W A = --γ (5) 2 rock H 2 [( ( zh ) 2 )cotψ p cotψ f ] where γ rock is the unit weight of rock and ψ f is the slope face angle. For the tension crack located at the slope face as illustrated in Fig. (b), W A is given by: W A = --γ (6) 2 rock H 2 [( z H) 2 cotψ p ( cotψ p tnaψ f ) ] Similarly, the factor of safety against sliding for block B is expressed as: c FS B A B + ( W B cosψ p U B + Vsinψ p )tanφ B B = W B sinψ p Vcosψ p (2) (3) (4) (7) where c B is the cohesion of block B along the failure surface; φ B is the friction angle; A B is the area of contact with the failure surface; W B is the weight of block B; and U B is the force induced by water pressure. A B and U B are given by: A B = zcscψ p (8) U B = --γ (9) 2 w z w2 cscψ p Again, the expression for the weight of block B is dependent on the location of the tension crack. For the tension crack located at the top of the slope, one obtains: W B = --γ 2 rock z 2 cotψ p For the tension crack located at the slope face, one obtains (0) W B = --γ 2 rock H 2 [ cotψ p ( ( zh ) 2 ( cotψ p tanψ f ) ) cotψ f ] () There is transition from one condition to another when the tension crack coincides with the slope crest, that is when zh = ( cotψ f tanψ p ) (2) 2.2 Case 2: Interaction between Two Blocks When block B is unstable (i.e., the factor of safety for block B is less than ), the block tends to slide. In this case, there are two possible outcomes. One is that block A is stable under the extra load due to block B, and thus, the slope is considered stable. The other is that block A is unstable, and consequently, the slope fails. The factors of safety corresponding to blocks A and B are similar to those presented previously. We denote the interaction force between the two blocks by I F, which has a direction inclined at an angle φ AB with respect to the surface of the tension crack as shown in Fig.. Applying the same approach proposed by Hoek and Bray (98), the factors of safety for blocks A and B are expressed as: FS A c A A A + [ Tcosθ + W A cosψ p U A Vsinψ p I F sin( ψ p γ AB )]tanφ = A W A sinψ p + Vcosψ p + I F cos( ψ p φ AB ) Tsinθ (3) c FS A A B + [ W B cosψ p U B + Vsinψ p + I F sin( φ p γ AB )]tanφ B = B W B sinψ p Vcosψ p I F cos( ψ p φ AB ) (4) All the symbols in Eqs. (3) and (4) are as defined previously. Because of the unknown quantity I F in both Eqs. (3) and (4), it is assumed that the factor of safety for block B in Eq. (4) is equal to. I F is thus obtained and can be substituted back into Eq. (3) to solve the factor of safety for block A. The slope is considered to be stable if FS A > and unstable otherwise. Equations () to (4) were initially derived by Hoek and Bray (98), Hoek (2000), and Jimenez-Rodriguez et al. (2006), and Vol. 5, No. 8 / November 20 35

4 Dian-Qing Li, Shui-Hua Jiang, Yi-Feng Chen, and Chuang-Bing Zhou are reproduced in this study because they are used in the subsequent system reliability analysis of the rock slope. It should be noted that, unlike in the work of Jimenez-Rodriguez et al. (2006), the reinforcing force is not assumed to be normal to the failure surface in this study; thus, the reinforcing force appears in Eqs. () and (3). 3. System Reliability To conduct the system reliability analysis of a rock slope, the first step is to identify the relevant failure modes from information of the stability model of the rock slope and forces acting on the rock slope. For the considered stability model of a rock slope shown in Fig., according to the location of the tension crack and the interaction between two blocks, four failure modes are identified as follows (Jimenez-Rodriguez et al., 2006). In failure mode, the tension crack is located at the top of the slope, and there is no interaction between blocks. In failure mode 2, the tension crack is located at the top of the slope, and there is interaction between blocks. In failure mode 3, the tension crack is located at the face of the slope, and there is no interaction between blocks. In failure mode 4, the tension crack is located at the face of the slope, and there is interaction between blocks. These failure modes provide a basis for the formulation of limit state functions associated with the considered stability problem of rock slopes. The reliability of a general structural system is evaluated taking the following steps (Ang and Tang, 984). First, the probability of failure of each parallel system is evaluated. Second, the correlations between the parallel systems due to common variables or correlated variables are evaluated. Finally, the probability of failure for the series system is evaluated on the basis of the results obtained from the first two steps. Evaluation of the correlation between a pair of parallel systems can be easily carried out if the safety margins for the parallel systems are linear. However, this is not the case in general. Therefore, an alternative is to investigate the possibility of introducing an equivalent linear safety margin for each parallel system, which is discussed later. Jimenez-Rodriguez et al. (2006) used a disjoint subset formulation in which the performance of the system is modeled as a series assembly of disjointed parallel sub-systems. The total probability of failure of the system is obtained as the sum of the individual failure modes. That is, the correlations between different failure modes are not considered, which leads to overestimation of the system probability of failure for the rock slope. To evaluate the correlations between different failure modes efficiently, a probabilistic fault tree is developed to model the general system problem and an n-dimensional equivalent reliability method initially proposed by Li et al. (2009) is again adopted, both of which are described below. 3. Probabilistic Fault Tree With respect to system reliability evaluation, fault tree analysis provides an organized means for identifying sources of structural system failure and their interactions that may lead to one or more failure paths. We focus here on how these failure paths are modeled using a probabilistic fault tree (Thackeret al., 2006), which provides a systematic way to manage multiple failure modes. A probabilistic fault tree (Thacker et al., 2006) has three major characteristics: bottom events, combination gates, and the connectivity between the bottom events and gates. Only AND and OR gates are currently included in the probabilistic fault tree approach. The AND gate is used to model a parallel system while the OR gate is used to model a series system. The limit state functions are defined in the bottom events so that the correlations between different failure modes represented by the limit state functions can be considered. In a conventional fault tree approach, however, probability values are first assigned to all bottom events, which are assumed to be independent. Next, the probabilities are propagated through the logic gates of the fault tree to calculate the probability of failure. Through a probabilistic fault tree, all the failure modes can be defined, and the correlations between different failure modes can be taken into consideration in a more rational way. Note that a failure mode can involve one or more limit states. By combining all failure modes and corresponding limit states, a system limit state surface can be constructed piece by piece. On the basis of the four aforementioned failure modes of the considered rock slope, the system reliability of the rock slope can be modeled by a probabilistic fault tree as shown in Fig. 2. Note that the performance of the system is modeled as a series of allparallel systems (failure modes). That is, the overall system (rock slope) fails when any failure mode occurs. For each failure mode involving several components, failure occurs only when all components in the corresponding parallel subsystem fail. The performance of each component is defined by a limit state function. Table gives the physical interpretations of the limit state functions and the definitions of the limit state functions corresponding to each failure mode. 3.2 n-dimensional Equivalent Reliability Method Consider first the case where a series system is composed of n components. A useful first step is to transform all safety margins associated with the n components to their standardized forms using FORM: Fig. 2. Probabilistic Fault Tree for the System Reliability of a Rock Slope 352 KSCE Journal of Civil Engineering

5 System Reliability Analysis of Rock Slope Stability Involving Correlated Failure Modes Table. Physical Interpretation of Limit State Functions in the System Reliability of the Rock Slope Stability (After Jimenez-Rodriguez et al., 2006) Limit state function Physical interpretation Eqs. g = z H( cotψ f tanψ p ) 0 Tension crack at top of slope (2) g 2 = { FS B g 0} 0 Block B is unstable (without interaction from A), given tension crack located at top of slope (7), (0) g 3 = { FS B ( g > 0) } 0 Block B is unstable (without interaction from A), given tension crack located at face of slope (7), () g 4 = { FS A g 0, g 2 > 0} 0 Block A is unstable, given tension crack at top of slope and block B stable (no interaction occurs) (), (5) g 5 = { FS A ( g 0, g 2 0) } 0 Block A is unstable, given tension crack at top of slope and block B not stable (interaction occurs) (3), (5) g 6 = { FS A ( g > 0, g 3 > 0) } 0 Block A is unstable, given tension crack at face of slope and block B stable (no interaction occurs) (), (6) g 7 = { FS A ( g > 0, g 3 0) } 0 Block A is unstable, given tension crack at face of slope and block B not stable (interaction occurs) (3), (6) Z i = α T i Y + β i i =, 2,, n (5) where α i = [ α, α 2,..., αn ] T is a unit vector and Y is a vector of standard normal variables. β i is the ith reliability index of the ith component. Generally, the probability of failure for such a series system is defined by P f = Φ n ( β; [ ρ] ) (6) where β = ( β, β 2,..., βn ) is a vector in which the components are the reliability indices of the failure elements; [ρ] is the correlation matrix for the linear and normally distributed safety margins of the failure elements; and Φ n ( ) is the n-dimensional standard multinormal integral with correlation coefficient matrix [ρ]. The corresponding reliability index β s is β S = Φ [ Φ n ( β; [ ρ] )] (7) To develop the n-dimensional equivalent method, the equivalent component concept proposed by Gollwitzer and Rackwitz (983) is used herein. Gollwitzer and Rackwitz (983) supposed that there is an equivalent linear safety margin representing series or parallel systems. The standard safety margin Z e can be expressed as Z e = α et Y + β e (8) where β e is the equivalent reliability index for the considered series or parallel systems, and is a unit vector. The equivalent linear safety margin should meet two conditions. Firstly, the equivalent reliability index β e is equal to the nominal reliability index β S ; i.e., α e = [ α e, αe... 2,, αn e] T (9) Secondly, the sensitivities of β e and β S with respect to changes in the basic random variables remain the same; i.e., β e ( Y) Y k Y = 0 -- β S( Y) = l Y k Y = 0 (20) in which β e ( Y) and β S ( Y) respectively represent the equivalent reliability index and nominal reliability index for the vector Y with a small increase vector Y ; and l is a proportionality coefficient. The vector α e must be evaluated numerically. In other words, the detailed equations to calculate α e were not given by Gollwitzer and Rackwitz (983). The vector α e can be determined using the proposed approach as follows. Let the vector Y of basic variables increase by an increment represented by vector Y. The corresponding probability of failure for the series system is then: P fs n ( Y) = P [ α T i ( Y+ Y) + β i 0] i = = Φ n ( β [ α] T Y; [ ρ] ) (2) where [α] T is a matrix consisting of the vector of direction cosines for each failure surface. On the basis of Eq. (2), the corresponding nominal reliability index increment for Y with an increment vector Y is given by: β S ( Y) = Φ [ p fs ( Y) ] = Φ [ Φ n ( β [ α] T Y; [ ρ] )] (22) The safety margin of the equivalent failure surface for Y with an increase of Y is: Z e ( Y) = α et Y α et Y + β e and the corresponding equivalent reliability index is: β e ( Y) = α e Y α e 2 Y 2 α e m Y m + β e (23) (24) where α e k is equal to the derivative of β e ( Y) with respect to Y k, expressed as: e β e ( Y) α k β S( Y) = = k = 2,, Y k l Y, m k Y = 0 Y = 0 (25) Here β S ( Y) is the nominal reliability index obtained from Eq. (22). The proportionality coefficient, l, is used to ensure that is a unit vector; i.e., α e l = m β S ( Y) Y k = k Y = 0 2 (26) The equivalent linear safety margin representing a parallel system can be constructed using a similar method. In general, the probability of failure for a parallel series system is: P f = Φ n ( β; [ ρ] ) (27) The equivalent linear safety margin for a parallel system is Vol. 5, No. 8 / November

6 Dian-Qing Li, Shui-Hua Jiang, Yi-Feng Chen, and Chuang-Bing Zhou determined by replacing Eqs. (7), (2) and (22) with: β S = Φ [ Φ n ( β; [ ρ] )] P f β S (28) n ( Y) = P [ α T i ( Y+ Y) + β i 0] = Φ n ( β + [ α] T Y; [ ρ] ) i = ( Y) = Φ [ Φ n ( β + [ α] T Y; [ ρ] )] (29) (30) The remaining derivation steps are the same as those for a series system. 4. An Illustrative Example 4. Case Geometry and Material Properties As an example, the rock slope stability model shown in Fig. is considered. The deterministic parameters adopted in the analyses are (Jimenez-Rodriguez et al., 2006) ψ p =32 o, ψ f =60 o, θ =0 o, γ rock = 25 kn/m 3, and γ w =9.8 kn/m 3. The effect of the height of the rock slope, H, ranging from 0 to 40 m, on the system reliability of the rock slope is considered. Table 2 presents the statistical distributions and parameters for the input variables (Jimenez-Rodriguez et al., 2006; Low, 2008). It should be noted that, instead of the lognormal distributions for cohesion used by Jimenez-Rodriguez et al. (2006), the distribution of cohesion is modeled by four-parameter (q, r, a, b) beta distributions, in which the first two parameters are shape parameters and the other two parameters define the lower and upper limits of the range. The reason for taking this approach is that the lognormal distribution is within the range of [0, + ], which is unsuitable because the cohesion in the rock slope stability model is bounded. Compared with the lognormal distributions, the four-parameter beta distribution is more versatile as demonstrated by Low (2008), and it can be symmetrical if q = r or asymmetrical if q r. With regard to the distribution of the location of the tension crack, since a tension crack is more commonly located at the top of the slope than at the slope face, an asymmetric beta distribution is used to model the distribution of the location of the tension crack. In addition, the vertical height of water in the tension crack should be less than the height of the tension crack, which is a bounded variable. Jimenez-Rodriguez et al. (2006) assumed that the drainage system of the slope prevents water levels from exceeding 50% of the height of the crack. A uniform distribution within [0, 0.5] is adopted in modeling the distribution of ξ zw. According to engineering practice, the level of water in a tension crack is usually low. Thus, a uniform distribution may not be suitable for the distribution of ξ zw. For this reason, a truncated exponential distribution is used for the distribution of ξ zw. The original exponential distribution has a mean of 0.25, and is truncated within the interval [0, 0.5]. The corresponding probability density function (PDF) and cumulative distribution function (CDF) can be given by: and f( x) Fx ( ) 4 = e 4x 0 x 0.5 e 2 e 4x = x 0.5 e 2 (3) (32) To avoid a negative reinforcing force, a truncated normal distribution is adopted to describe the variability of the reinforcing force T. Its PDF can be expressed as: f( x T ) = F ( x u ) F 0 ( x ) f ( x ) x 0 T x T x u (33) where x l and x u are the lower and upper bounds of x T, respectively, and F 0 (x l ) and F 0 (x u ) are the cumulative distribution functions for x l and x u, respectively. f 0 (x T ) is the original PDF of the normal distribution (Ang and Tang, 2007): f 0 ( x T ) exp 2πσ 2 -- xt µ 2 = x σ T + (34) where µ and σ are the mean and standard deviation of the normal distribution. For the considered example, µ = 50 kn and σ = 3 kn are used for the original normal distribution. It is well accepted that the cohesion and friction angle are negatively correlated. Therefore, correlation coefficients of ρ ca,φa = ρ cb,φb = -0.5 are used to model common shear test results in which the cohesion generally decreases as the friction angle increases and vice versa (Low, 2008; Hoek, 2000). Again, Variable Distribution Mean Table 2. Summary Statistics of Basic Random Variables in the Slope Stability Model Standard deviation Lower bound Upper bound References c A (kpa) Beta (Jimenez-Rodriguez et al., 2006; Low, 2007) c B (kpa) Beta φ A (deg) Beta (Jimenez-Rodriguez et al., 2006) φ AB (deg) Beta φ B (deg) Beta T (kn) Truncated normal ξ XB Beta ξ zw Truncated exponential (Jimenez-Rodriguez et al., 2006; Low, 2007) Note: The truncated normal distribution is adopted for T in this study but the mean and standard deviation of T are taken from Jimenez-Rodriguez et al. (2006). 354 KSCE Journal of Civil Engineering

7 System Reliability Analysis of Rock Slope Stability Involving Correlated Failure Modes Table 3. Correlation Structure of Random Variables Underlying the Slope Stability Model c A c B φ A φ AB φ B T ξ XB ξ zw References c A.0 c B Symmetric φ A φ AB (Jimenez-Rodriguez et al., 2006; φ B Low, 2007) T ξ XB ξ zw following Jimenez-Rodriguez et al. (2006), positive correlation coefficients of ρ φa,φb = r φa,φab = ρ φb,φab = ρ ca,cb = 0.3 are used. All other random variables in the rock slope stability model are assumed to be independent of each other. The corresponding correlation structure of random variables is shown in Table Analysis Results for System Reliability In this study, a C#-language-based computer program WHUREL (WuHan University Reliability computer program for rock slopes) was developed to calculate the reliability index β and the most probable failure points X*. WHUREL calculates the reliability of a component using FORM and the system reliability of a series or parallel system employing the n-dimensional equivalent method (Li et al., 2009). A major advantage of WHUREL is that the limit state functions, defined by the bottom events in the probabilistic fault tree as shown in Fig. 2, are expressed as a set of user-defined subroutines. To reflect the contribution of each failure mode to the system probability of failure for the rock slope, the probabilities of failure on a logarithmic scale for each failure mode of the rock slope are shown in Fig. 3 on the basis of data in Tables 2 and 3. Fig. 3 shows that the relative influences of the four failure modes on the system reliability can differ considerably. Failure mode contributes most to the system probability of failure, while failure mode 4 is the least significant. For instance, for a rock slope with height of 25 m, the probabilities of failure are and for failure modes and 4, respectively. In addition, the probabilities of failure for failure modes and 3 are significantly higher than those for failure modes 2 and 4. This indicates that the case in which there is no interaction between two blocks has a significantly higher probability of failure than the case in which there is interaction between two blocks. Therefore, the stability of block A should be a primary concern in the design of the rock slope. Measures can be taken to improve the stability of block A. For example, one can increase the passive reinforcing force at the toe of the slope. In this case, however, failure modes with interaction between two blocks may significantly contribute to the system probability of failure. The probability of failure for failure mode is highly sensitive to the height of the rock slope. For instance, when the height of the rock slope increases from 0 to 40 m, the probability of failure for failure mode increases from to To investigate the relative importance of each limit state function, namely g to g 7 as shown in Table, the probabilities of failure for these limit state functions should be determined. Take the rock slope with a height of 20 m as an example. The probabilities of failure calculated by FORM for g to g 7 are 8.7 0, ,.7 0 3, , ,.5 0, and , respectively. Note that the probability of failure for g is significant larger than those for the other six limit state functions, which means that the tension crack is most likely located at the top of the slope. Accordingly, failure modes and 2 are significantly more likely. The probability of failure for g 4 is , which is about nine times that for g 5. Such results indicate that failure mode is significantly more likely than failure mode 2 to occur, which is also consistent with the results shown in Fig. 3. The limit state function g 6 also produces a slightly high probability of failure, which means that block A is most likely to be unstable given that the tension crack is located at the face of the slope and block B is stable. In addition, the limit state functions g 2 and g 3 lead to small probabilities of failure, which indicates Fig. 3. Comparison among Probabilities of Failure for Different Failure Modes Fig. 4. Comparison among System Probabilities of Failure Obtained Using Different Methods Vol. 5, No. 8 / November

8 Dian-Qing Li, Shui-Hua Jiang, Yi-Feng Chen, and Chuang-Bing Zhou that block B is most likely to be stable regardless of the location of the tension crack. Figure 4 compares the system probabilities of failure of the rock slope using Cornell s bound method (Cornell, 967), the method of Jimenez-Rodriguez et al. (2006), and the n-dimensional equivalent method. The system probabilities of failure obtained using the n-dimensional equivalent method are fully within the range computed using Cornell s bound method, which indicates that the n-dimensional equivalent method is valid. On the other hand, the system probability of failure obtained using the method of Jimenez-Rodriguez et al. is higher than the Cornell upper bound because Jimenez-Rodriguez et al. (2006) assumed that the system probability of failure for the rock slope with multiple correlated failure modes can be taken as the sum of probabilities of failure associated with the considered four failure modes. Accordingly, the system probability of failure of the rock slope will be overestimated, especially for larger slope heights. For the rock slope with height of 40 m, the system probabilities of failure obtained using the n-dimensional equivalent method and the method of Jimenez-Rodriguez et al. are and , respectively. The negative correlation between cohesion and the friction angle along the failure surface is often ignored for the simplification of computations in traditional practice. To account for the effect of correlations between input variables on the system probability of failure of the rock slope, Fig. 5 compares the system probabilities of failure with and without consideration of correlations between variables. The red line in Fig. 5 corresponds to the system probability of failure for the case that all random variables in the rock slope stability model are assumed to be independent of each other. Note that if the correlations between input variables for rock slope stability analysis are not taken into account, the system probability of failure of the rock slope will be overestimated, especially for greater slope heights. Taking the rock slope with a height of 40 m as an example, the system probabilities of failure with and without consideration of correlations between variables are and , respectively. Fig. 6. System Reliability Indexes for Different Truncated Exponential Distributions of the Percentage of the Tension Crack Filled with Water The distribution of ξ zw is assumed to be a truncated exponential distribution within the interval [0, 0.5] as shown in Table 2. To account for the case that the tension crack is fully filled with water, Low (2008) used a truncated exponential distribution within the interval [0,.0] to model the distribution of ξ zw. Fig. 6 compares the system reliability indexes for the truncated exponential distribution within the interval [0, 0.5] with those for the truncated exponential distribution within the interval [0,.0]. For comparison, the original exponential distributions have the same mean of In Fig. 6, the reliability indexes for ξ zw truncated within the interval [0, 0.5] are significantly higher than those for ξ zw truncated within the interval [0,.0], especially for small slope heights. For the rock slope with height of 0 m, the system reliability index for ξ zw truncated within the interval [0, 0.5] is 3.5, which is significantly higher than 2.23 for ξ zw truncated within the interval [0,.0]. Therefore, if a good drainage system for the rock slope is established, the truncated exponential distribution within the interval [0, 0.5] is recommended. If a drainage system for the rock slope has average performance, the Fig. 5. Comparison between System Probabilities of Failure with and without Consideration of Correlations between Variables Fig. 7. Effect of the Inclination of the Reinforcing Force on the System Reliability Index of the Slope 356 KSCE Journal of Civil Engineering

9 System Reliability Analysis of Rock Slope Stability Involving Correlated Failure Modes truncated exponential distribution within the interval [0,.0] is recommended. To take into account the effect of the inclination of the reinforcing force on the system reliability index of the rock slope, Fig. 7 shows system reliability indexes for the inclination of the reinforcing force varying from 0 o to 90 o. The figure shows that the system reliability index is a maximum when the inclination of the reinforcing force is about 55 o, which is the optimal inclination of the reinforcing force. In addition, when the reinforcing force is normal to the plane of failure (i.e., the inclination of the reinforcing force is zero in Fig. ), the system reliability index is a minimum. Therefore, the direction of the reinforcing force should not be normal to the plane of failure for the considered slope stability model. 4.3 Sensitivity Analysis Sensitivity analyses are carried out using WHUREL at three different levels; namely, the single limit state function level, single failure mode level, and system reliability level. At the single limit state function level, the sensitivity coefficients of the underlying random variables only reflect their relative significance to the probability of failure of the single limit state function. At the single failure mode level, the sensitivity coefficients of the same random variables as for the single limit state function represent the relative significance to the probability of failure of the single failure mode that involves several limit state functions. Since the system reliability of the rock slope involves several failure modes, the resulting sensitivity coefficients of the same random variables indicate the sensitivities of the calculated system reliability to changes in the basic random variables. It should be noted that when the basic random variables are not independent, the sensitivity coefficients defined in this study are not informative in relation to the basic random variables owing to the transformation to independent standard normal space. For this reason, all the random variables in Table 2 are assumed to be independent, even though the statistics of variables in Table 2 are used again. Table 4 shows the FORM results for the case of single limit state functions associated with the rock slope stability. The results include the sensitivity coefficient α * denoting the sensitivity of the computed reliability results to the changes in the random variables as defined in FORM, together with the computed design points X* corresponding to the most likely failure point transformed back to the original space. At the single limit state function level, the reliability associated with g 0 in failure modes and 2 is only sensitive to the change in ξ XB. For g 2 0 in failure mode 2, c B, φ B, and ξ XB are significant random variables with high sensitivity coefficients. For g 4 0 in failure mode, φ A, ξ XB, and ξ zw are significant random variables with high sensitivity coefficients. Similarly, the reliability associated with g 6 0 in failure mode 3 is quite sensitive to φ A, ξ XB, and ξ zw, while it is insensitive to c B, φ AB, and φ B. These results show that ξ XB significantly affects the computed reliability associated with the aforementioned four limit state functions, and it is the most significant random variable. Thus, determination of the location of the tension crack with sufficient accuracy is paramount for adequate assessment of rock slope stability. To carry out sensitivity analyses at the single failure mode level, Fig. 8 shows the sensitivity coefficients of basic random variables for four failure modes in the case of H = 25 m. Note that the sensitivity coefficients of random variables in different failure modes can differ considerably. Take the sensitivity coefficient of φ B as an example. φ B is the most significant variable in failure mode 2. It is not, however, a significant variable in failure modes and 3. In general, the reliability for the considered four Fig. 8. Comparison among Sensitivity Coefficients of Random Variables for Different Failure Modes Table 4. Sensitivity Coefficients and Design Points of Basic Random Variables Variable g 0 g 2 0 g 4 0 g 6 0 X* α* X* α* X* α* X* α* c A (kpa) 2.00E E E-05.90E+0 -.4E-0.96E+0 -.E-0 c B (kpa).80e E E-0.80E+0.89E-06.80E+0.72E-05 φ A (deg) 3.60E E E E E E E-0 φ AB (deg) 3.00E E E E+0.89E E+0.72E-05 φ B (deg) 3.20E E E E+0.92E E+0.72E-05 T (kn) 5.00E E E E E E E-03 ξ XB 6.39E E E E E E E-0 ξ zw.42e E E E E E-0 8.8E-0 Note: α* and X* represent the sensitivity coefficients and the design points, respectively. Vol. 5, No. 8 / November

10 Dian-Qing Li, Shui-Hua Jiang, Yi-Feng Chen, and Chuang-Bing Zhou failure modes is very sensitive to ξ zw and ξ XB, while it is insensitive to φ AB and T. Figure 9 shows the sensitivity coefficients of the system reliability for various slope heights. It is seen that ξ zw is the most significant variable with the highest sensitivity coefficient regardless of the slope height. That is, the change in ξ zw significantly affects the system reliability of the rock slope. Therefore, a good drainage system for the slope can improve the slope stability efficiently. This conclusion agrees well with that drawn from the results for the single failure mode level. Additionally, the system reliability of the slope is quite sensitive to c A, φ A, and ξ XB, which indicates that the shear strength of the failure surface and the location of the tension crack are key factors in slope stability analysis, hence emphasizing the importance of a thorough geological investigation of discontinuities in the rock mass. The sensitivity coefficients of c B, φ AB, φ B, and T are almost equal to zero, which indicates that the changes in these four variables have no influence on the system reliability of the slope. From the sensitivity results for the three aforementioned levels, it is seen that the sensitivities of the reliability results with respect to basic random variables at different levels can differ considerably, and they highly depend on the selected sensitivity analysis level. 4.4 Reliability-based Design of Reinforcing Force T Having determined the system reliability of the considered rock slope, the reliability-based design of the rock slope can be performed readily for a desired probability of failure or reliability index. Since the reliability-based design typically aims at a target reliability index greater than 2.5, the corresponding probability of failure is less than (Low, 2007). For the considered rock slope stability model, a target system reliability index β T = 2.5 is used. For illustration, take the rock slope with a height of 25 m as an example. Fig. 0 shows the variation in the required T force with the inclination of the reinforcing force θ as defined in Fig.. Since the reinforcing force T required to achieve a target system reliability index of 2.5 is initially unknown, one would need to try different T values for each trial θ value. Furthermore, for different means of the reinforcing forces, the coefficient of variation of 0.06 of T remains the same. Fig. 0 shows that there is an optimal inclination of the reinforcing force corresponding to the smallest T value. The optimal inclination is about 55 o and the reinforcing direction for the specified target system reliability index is very close to horizontal. The resulting T force required is about 266 kn. In addition, the required T value can be determined easily for the target system reliability index. For instance, if a reinforcing direction of θ = 0 o, as assumed by Jimenez- Rodriguez et al. (2006), is adopted, the reinforcing force T required is 470 kn, which indicates that the reinforcing force being normal to the failure plane is not an effective measure for improving the stability of the rock slope. 5. Conclusions Fig. 9. Comparison of System Reliability Sensitivity Coefficients of Random Variables Fig. 0.Required Reinforcing Force T Values versus Inclination of the Reinforcing Force for a Target System Reliability Index of 2.5 This paper proposed a methodology for the evaluation of the system reliability of rock slope stability with consideration of multiple correlated failure modes. A probabilistic fault tree is used to model the system reliability of the rock slope stability. The n-dimensional equivalent method is employed for the reliability calculation. An example is presented to illustrate the proposed methodology. The system reliability of a rock slope with multiple correlated failure modes can be modeled using the probabilistic fault tree in an intuitive way. The correlations between different failure modes can be taken into consideration properly using the n-dimensional equivalent method. If such correlations are neglected, the system probability of failure of the rock slope will be overestimated. The relative importance of different failure modes to the system reliability differs greatly. In the example, failure modes and 3 are significantly more likely than failure modes 2 and 4, which indicates that the case in which there is no interaction between two blocks is significantly more likely than the case in which there is interaction between the two blocks. In addition, the reliability indexes for the percentage of the tension crack filled with water truncated within the interval [0, 0.5] are significantly higher than those for the percentage of the tension 358 KSCE Journal of Civil Engineering

11 System Reliability Analysis of Rock Slope Stability Involving Correlated Failure Modes crack filled with water truncated within the interval [0,.0], especially in the case of small slope heights. Accordingly, a good drainage system for the particular slope should be designed to improve the slope stability efficiently. In addition, there is an optimal inclination of the reinforcing force acting on the slope, which gives the maximum value of the system reliability index for the slope. Thus, the reinforcing measures should be designed carefully for the particular slope. Sensitivity analysis of random variables should be conducted at three different levels, namely, the single limit state function level, single failure mode level, and system reliability level. The sensitivity results highly depend on the selected sensitivity analysis level. In the example, at the system reliability sensitivity level, the shear strength parameters of the failure surface underlying block A, the location of the tension crack, and the percentage of the tension crack filled with water are significant variables with high sensitivity coefficients. Therefore, to improve the rock slope stability effectively, a good drainage system should be established and an adequate geological characterization of the rock mass should be conducted. Acknowledgements This work is supported by the National Natural Science Foundation of China (Project Nos , and ) and the Program for New Century Excellent Talents in University, Ministry of Education of China (Project No. NCET ). References Ang, H. S. and Tang, W. H. (984). Probability concepts in engineering planning and design, decision, risk and reliability, Vol. 2, New York. Ang, H. S. and Tang, W. H. (2007). 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