Robust Design of Rock Slopes with Multiple Failure Modes Modeling Uncertainty of Estimated Parameter Statistics with Fuzzy Number

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1 Robust Design of Rock Slopes with Multiple Failure Modes Modeling Uncertainty of Estimated Parameter Statistics with Fuzzy Number Changjie Xu a, Lei Wang b*, Yong Ming Tien c, Jian-Min Chen c, C. Hsein Juang b a Center of Coastal and Urban Geotechnical Engineering, Zhejiang University, Hangzhou , China b Glenn Department of Civil Engineering, Clemson University, Clemson, SC 29634, USA c Department of Civil Engineering, National Central University, Jhongli City, Taoyuan County 32001, Taiwan * Corresponding author ( lwang6@g.clemson.edu) Abstract: The variability of shear characteristics of rock discontinuities is often difficult to ascertain. Thus, even with the reliability-based design (RBD) approach, which allows for consideration of the uncertainty of input parameters, the design of a rock slope system may be either cost-inefficient (overdesign) or unsafe (under-design), depending on whether the variation of input parameters is overestimated or underestimated. The uncertainty about the variation of input parameters is a critical issue in a RBD. This paper presents a feasible approach to addressing this problem using robust design concept. First, the uncertainty of the estimated statistics of input parameters (such as rock properties) is represented by fuzzy sets, which requires only the knowledge of lower and upper bounds of the estimated statistics. Then, the robust design concept is implemented to ensure that the final design is insensitive to, or robust against, the uncertainty of the estimated statistics of input parameters. The design methodology is demonstrated with an application to the design of a rock slope system with multiple failure modes. This design methodology, termed Robust Geotechnical Design (RGD), aims to achieve a certain level of design robustness, in addition to meeting safety and cost requirements. In this paper, the RGD framework is realized through a multi-objective optimization, as it involves three requirements, safety, cost, and robustness. The significance of the design methodology is demonstrated with an example of rock slope design. Key words: Uncertainty; Reliability; Rock slope; Failure probability; Robust design; Fuzzy sets; Optimization. 1

2 Introduction The design of rock slopes is often carried out in the face of uncertainty especially regarding rock properties. Earlier design approaches using the deterministic methods rely on factor of safety (FS) that is calibrated to engineering experience to compensate for the uncertainty. More recently, various probabilistic approaches, including reliability-based design methods, have been adopted to aid in the design of rock slopes (Hoek 2006; Jimenez-Rodriguez et al. 2006; Jimenez-Rodriguez and Sitar 2007; Low 2007; Low 2008; Penalba et al. 2009; Li et al. 2011; Lee et al. 2012; Park et al. 2012a&b; Tang et al. 2012). However, the results of the reliability-based design of rock slopes are often affected by the accuracy of the statistical characterization of the rock properties. However, the fully accurate statistical characterization of rock properties along the rock discontinuities often requires a high cost in site investigation and laboratory testing, which may not be feasible for each design project. In this paper, we mainly focus on how to make rational decision with only very limited data, which is a common scenario due to budget constraint in practice. The variation of shear resistance properties of rock discontinuities may be affected by the inherent variability of its strength and roughness, limited availability of quality samples and testing, uncertainty in the adopted transformation models, and so on. With only very limited data, the variation of rock properties, in terms of the coefficient of variation (COV), often can only be estimated and expressed in a range based on published literature and engineering judgment (Duncan 2000; Feng and Hudson 2004; Hoek 2006). In a typical reliability analysis of the stability of rock slopes, a fixed value is often arbitrarily assumed for the COV of each strength parameter within the perceived range (Lee et al. 2012). However, the outcome of the reliability-based design is often very sensitive to the 2

3 assumed COVs, and as a result, the design can be either cost-inefficient or unsafe depending on whether these COVs are under- or over-estimated. A possible solution to this dilemma is to adopt the robust design concept with which the design robustness against the variation of the estimated COVs is achieved. In other words, the effect of the variation of the estimated COVs is minimized through robust design (Juang and Wang 2013; Wang et al. 2013). It should be noted that the design robustness may be measured in various ways (Taguchi 1986; Chen et al. 1996; Lagaros et al. 2010), and in this paper, it is measured in terms of the variation of the system response (in terms of failure probability of the system). Thus, the design is considered more robust if the variation of the system response is getting smaller. Historically, the robust design concept was originated from Industry Engineering for quality control (Taguchi 1986; Chen et al. 1996). This concept has since been applied to many design fields, including structural design and mechanical design (Marano and Quaranta 2008; Lagaros et al. 2010; Lee et al. 2010). The writers have also adapted this concept to formulate a robust geotechnical design (RGD) methodology for geotechnical engineering problems (Juang and Wang 2013; Wang et al. 2013). The RGD methodology seeks to achieve a certain level of design robustness, in addition to meeting safety and cost requirements. In this paper, we adapted this methodology to design rock slope systems with multiple failure modes. Herein, a rock slope design is considered robust if the variation in the failure probability is insensitive to the variation of noise factors (mainly uncertain COVs of rock properties). To evaluate the variation in the failure probability of the designed rock slope, which is used as a means of measuring the design robustness, the uncertain COVs of rock properties are modeled using fuzzy numbers (see Section 2.1). These uncertainties in the estimated COVs, modeled as fuzzy numbers, are propagated through the entire analysis process, rendering a measure of the 3

4 variation of the response of the rock slope system. In this paper, the robust design of a rock slope is to find an optimal design that is insensitive to the variation of noise factors by carefully adjusting design parameters. The noise factors refer to the parameters that cannot be controlled by the designer. For the rock slope problem, the noise factors mainly refer to the uncertain rock properties. The design parameters are the parameters that can be determined and controlled by the designer, such as the geometry of the design for the rock slope problem. Here, the RGD framework is realized through multi-objective optimization, in which the safety requirement is satisfied by the reliability constraint and the design is optimized with respect to cost and robustness. For the rock slope design, the objectives of cost and robustness are usually conflicting with each other, and the multi-objective optimization generally yields a Pareto Front (Cheng and Li 1997), which is a collection of multiple non-dominated optimal designs, as opposed to a single best design. The Pareto Front essentially describes a trade-off relationship between robustness and cost for all non-dominated designs that satisfy the safety requirement, thus it enables selection of the best design according to a desired cost range or a target robustness level. A design example of a rock slope system is used to demonstrate the significance of the proposed RGD approach. The example is a hypothetical rock slope with two potential unstable blocks, which involves multiple failure modes Framework for Reliability-Based RGD Using Fuzzy Sets 2.1 Modeling parameter statistics using fuzzy numbers The variation of the failure probability of the rock slope system is naturally affected by the variation of the unknown COVs of input parameters (such as rock strength parameters). 4

5 However, it generally difficult to characterize accurately the variation of these COVs given limited quality data and spatial variability. Nevertheless, the engineer can usually characterize these COVs as a range, as often reported in the literature. Thus, a reasonable compromise is to construct a fuzzy number (Zadeh 1965; Zadeh 1978; Juang et al. 1992; Juang et al. 1998; Luo et al. 2011) using the estimated range. For example, if the COV is estimated with a range of [a, b], where a is the lowest conceivable value (or low bound) and b is the highest conceivable value (or upper bound), then this COV (denoted as x in Figure 1) may be represented by a fuzzy set (or fuzzy number) as shown in Figure 1. A fuzzy set (Zadeh 1965) is mathematically defined as a set of ordered pairs, [ x, ( x)]. The membership grade ( x), ranging from 0 to 1, is used to characterize the degree of belief that a member x belongs to this set. A fuzzy number is a fuzzy set that is normal and convex, in which the shape of the membership function is single-humped and has at least one value with a membership grade (or degree of belief) of 1. A triangular fuzzy number is often used in the geotechnical applications, which is characterized with three values: a low bound, an upper bound, and a mode. The mode has a membership grade of 1, the highest possibility, to represent COV of rock properties (parameter x in Figure 1). As the value of the parameter departs from the mode, the degree of belief for this value to represent the parameter x decreases, and when the value reaches the lower bound (or the upper bound), the degree of belief is reduced to zero (Juang et al. 1992; Juang et al. 1998; Luo et al. 2011). Figure 1(a) shows a fuzzy number, in which the highest membership grade (degree of belief) occurs at x equal to the mode m. In this fuzzy number, the lowest membership grade (degree of belief) occurs when x is equal to either lower bound a or upper bound b. Note that if m = (a+b)/2, the fuzzy number will be symmetric. On the other hand, if the mode (m) is not 5

6 equal to (a+b)/2, the fuzzy number will be non-symmetric. A symmetric fuzzy number is the most preferred choice to represent an uncertain variable when only knowledge of lower and upper bounds is known (Juang et al. 1992). In this paper, the uncertainty in the estimated COV of an input parameter is represented by a symmetric fuzzy number. It should be noted that the above approach of modeling the uncertainty of the estimated COV is not a limitation of the proposed robust design methodology but rather a choice that works well with the stability problem of rock slopes with discontinuities. 2.2 Reliability-Based RGD Approach Using Fuzzy Sets The reliability-based robust geotechnical design (RGD) methodology developed by the authors (Juang and Wang 2013; Wang et al. 2013) is adapted for rock slope design considering multiple failure modes in this paper. In particular, the uncertainty of the estimated COVs of rock properties is modeled with fuzzy numbers as available rock properties data are generally very limited, which makes it difficult to characterize the uncertainty with a probability distribution function, and the formulation for the entire analysis process is modified accordingly. The modified reliability-based RGD framework is summarized in six steps as follows (in reference to Figure 2): (1) Establish the deterministic model for stability analysis of rock slope system. For a given slope, a proper deterministic model for rock slope analysis can be selected based on the sliding mechanism and the number of removable (unstable) blocks. The removable blocks are referred to as the rock masses geometrically isolated by discontinuity planes (Giani 1992). In this paper, for the design case of a rock slope composed of two blocks separated by a vertical tension crack as shown in Figure 3, the deterministic model proposed by Jimenez-Rodriguez et al. (2006) is employed, which is summarized in 6

7 Appendix (Appendix A, ESM only). (2) Classify the input parameters and specify the design domain for the rock slope. In the context of Robust Geotechnical Design (RGD), the input parameters are classified into two categories: design parameters and noise factors. The design parameters refer to the parameters that can be easily and accurately controlled by the designer and can be treated as fixed values, while the noise factors mainly refer to inherent properties of geo-materials that are difficult to control by the designer. In this paper, noise factors mainly refer to uncertain rock properties (e.g., shear properties of discontinuity). For the design of rock slope, there are generally two categories for design and remedial measures. One is to reduce the slope height and slope face angle; the other is to reinforce the slope by rock bolts and anchors. As noted by Hoek (2006), the design of rock slope by reducing the height and face angle of slope is generally more cost-efficient than that by reinforcing by rock bolts and anchors. Furthermore, the rock bolts may endure significant strength reduction due to long-term corrosion, creep and deterioration effects, which are quite difficult for long-term maintenance (Wang et al. 2013). In this study, for illustration purpose, the design of rock slope is realized by manipulating the two design parameters, slope height H and slope angle. The feasible ranges for these design parameters must be chosen and these parameters are usually modeled as discrete values in the design space. (3) Model the uncertainty in the estimated statistics of noise factors using fuzzy numbers. In routine geotechnical exploration, the mean values of geotechnical parameters can usually be adequately estimated even with a small sample of data (Wu et al. 1989). However, the parameter statistics such as the coefficient of variation (COV) and correlation coefficient (ρ) of rock properties are quite difficult to ascertain with a small sample of data; the COV 7

8 and correlation coefficient of rock properties are usually estimated based on the typical ranges reported in the literature and with engineering judgment (Hoek 2006). Under these circumstances, the statistics (e.g., COV and ρ) of rock properties (these are noise factors in the context of robust design) should be characterized logically with a range. For an uncertain parameter that is characterized with only a lower bound and an upper bound, a fuzzy set approach can be employed (Juang et al. 1992; Juang et al. 1998; Sonmez et al. 2004). In the proposed RGD framework, the uncertainty in the estimated COVs and correlation coefficients of rock parameters is modeled as a symmetric fuzzy number, which requires only the knowledge of the estimated range [a,b], as depicted in Figure 1. (4) Process the fuzzy data to obtain the variation of failure probability using fuzzy point estimate method for robustness evaluation. As noted previously, the robustness of a rock slope design is measured in this paper by the variation (in terms of standard deviation) of the failure probability caused by variations of the statistics of noise factors, and the smaller variation in the failure probability indicates the greater robustness. If the statistics of rock strength properties (noise factors) can be ascertained, the computed failure probability will be a fixed value. However, when the statistics of noise factors cannot be ascertained (for example, can only be described with a range and modeled with fuzzy number), the computed failure probability will be uncertain. In this step, the variation of the failure probability for each of the totally Y possible designs in the design space will be computed, as depicted with the 3 rd loop in Figure 2. In this paper, the system reliability algorithm (Jimenez-Rodriguez et al. 2006; Jimenez-Rodriguez and Sitar 2007) is used to compute the failure probability of the rock slope given a fixed set of statistics of noise factors. The uncertainty in the computed probability of failure caused by the 8

9 uncertainty of input statistics of noise factors (expressed as fuzzy numbers) is then evaluated using the fuzzy set method. Considering a performance function y = g (x 1, x 2,..., x M ), in which M is the number of input uncertain statistics of noise factors (M equals seven in the case study presented later) and y is the outcome of an algorithm for evaluation of system failure probability. For each uncertain statistical parameter with only the knowledge of a range [a,b], a fuzzy number is assigned as shown in Figure 1 (a). After the fuzzy numbers are defined, the vertex method (Dong and Wong 1987; Juang et al. 1998) is used for propagation of the set of fuzzy numbers through the adopted system reliability algorithm. The vertex method is based on the α-cut concept, in which a fuzzy number is discretized into a group of α-cut intervals. By drawing a horizontal line at a selected membership value ( x) = k, an interval of two points can be obtained as shown in Figure (b). For example, when is set as 0.2 for α ranged from 0 to 1, six different α-cut levels can be obtained (α = 0, 0.2, 0.4, 0.6, 0.8 and 1.0). The step size of adequate for the fuzzy set-based reliability analysis (Luo et al. 2011). = 0.2 is generally At each α level, the intervals for each input variable are obtained, and the combinations of vertexes (i.e., the lower bound and the upper bound for each input variable at a given α level) are determined. For an analysis with M input variables, the number of combination for vertexes (denoted as Z in Figure 2) is 2 M. Each combination of input parameters is entered into the adopted system reliability algorithm (also referred to herein as the performance function), which yields a solution (in this case, a failure probability). Repeating this process for all combinations, a set of 2 M solutions is obtained. According to the vertex method (Dong and Wong 1987), the maximum value and the minimum value of 9

10 obtained 2 M solutions are taken as the resulting interval of the performance function y at the specific α level (see the 1 st loop of Figure 2). Repeating the above analysis for each of the six α levels (N = 6), six corresponding intervals for y are obtained. These six α levels and the corresponding intervals define the final fuzzy number that represents the output or the system response (see the 2 nd loop of Figure 2). With the resulting fuzzy number that represents the system response, the fuzzy-based point estimate method (Fuzzy-based PEM) proposed by Dodagoudar and Venkatachalam (2000) can be used to evaluate the mean and standard deviation of this system response. First, 223 denote the lower bound and the upper bound of the resulting fuzzy number as g and k g, k k = 1, 2,, N, where N is the number of α cut levels (N = 6 in this case). The sum of the function values at each α level considering the correlation effect between variables is calculated as (Dodagoudar and Venkatachalam 2000): 227 w p g p g (1) r r r k k k 228 where p and p are weighting factors, given by the following expression (Dodagoudar 229 and Venkatachalam 2000): 230 p 1 M 1 M i1 ji1 M i1 ij (1) 1 2 i 2 (2) 231 where M is the number of fuzzy input variables; ij is the correlation coefficient between 232 fuzzy variables x i and x j ; and (1) i is the skewness coefficient of the fuzzy variable x i. 233 The r th moment of the function is calculated as: 10

11 N r k w k r k1 (3) EW [ ] N With the first two moments of the output fuzzy number (the system response, or in this case, the failure probability) being obtained by Eq. (3), the mean and standard deviation of the failure probability can be determined. Specifically, the mean and standard deviation of the 238 failure probability (denoted as p and p ) are calculated as follows: EW [ ] (4) p [ ] { [ ]} (5) p E W E W The standard deviation of the failure probability computed by Eq. (5) is used herein as a measure of design robustness. Greater design robustness is achieved with a smaller standard deviation of the evaluated failure probability. It should be noted that, although the traditional sensitivity analysis using upper and lower bounds of the input parameters can give us some insights of the rock slope problem, however, such sensitivity analysis using upper and lower bounds of the input parameters cannot tell us whether a design is robust, which requires the knowledge of the variation of the system response. Processing fuzzy number data through the analytical model allows us to evaluate the variation of the system response. The process involves rigorous mathematics but in a straightforward and systematic way. (5) Establish the Pareto Front optimized with respect to both robustness and cost. Geotechnical designs often involve multiple criteria, including the requirements for safety, cost and robustness. In this paper, a multi-objective optimization is performed to obtain the optimal designs. The mean failure probability obtained by Eq. (4) is first compared with a target (acceptable) failure probability to screen out the unsatisfactory designs. Once the safety 11

12 constraint is satisfied, all acceptable designs are optimized with the other two objectives on cost and robustness. In this paper, the cost for a given design is simplified as the volume of rock mass to be excavated, and the robustness is measured with the standard deviation of the failure probability. The volume of rock mass to be excavated is determined by the difference between initial volume of rock slope and volume of rock slope after remediation (Wang et al. 2013). The multi-objective optimization is carried out using Non-dominant sorting genetic algorithm (NSGA-II), developed by Deb et al. (2002). No single best design can be obtained if the objectives are conflicting with each other. Thus, the multi-objective optimization often leads to a group of non-dominated designs that are optimal to all objectives, which collectively defines a Pareto Front (Cheng and Li 1997; Deb et al. 2002). As will be shown later, the obtained Pareto Front offers a trade-off relationship between cost and robustness. This trade-off relationship enables to reach a more informed decision especially when a desired cost or robustness level is selected. (6) Determine feasibility robustness to aid in the decision making. In addition to Pareto Front, an index called feasibility robustness (Parkinson et al. 1993) may be used as a decision-making aid. For the rock slope design, the feasibility robustness may be defined as the confidence probability that the failure probability of slope, as designed, meets the constraint of a target failure probability. Symbolically, the feasibility robustness is expressed as follows (Wang et al. 2013): P[( p p ) 0] P (6) f T where p f is the computed failure probability of the slope system, which is not a fixed value 12

13 278 because of the uncertainty in the statistics of noise factors, as reflected in step 4; pt is the 279 target failure probability (in this paper, pt , which corresponds to a target reliability 280 index T 2.5, is adopted as per Low 2008); and P 0 is an acceptable confidence probability (say, 85%) specified by the designer. To compute the feasibility robustness, P[( p p ) 0], an equivalent counterpart in form of P[( T ) 0], where is the reliability index that corresponds to the failure f T 284 probability p f and is the target reliability index ( = 2.5), may be evaluated (Wang et T T al. 2013). In fact, Eq. (6) can be re-written as: P[( T ) 0] ( ) P0 (7) 287 where Φ is the cumulative standard normal distribution function, and is defined as: where T 2.5 (8) and are the mean and the standard deviation of computed by the Fuzzy-based PEM described and formulated previously. Thus, instead of using the confidence probability, the index, referred to herein as the feasibility robustness index, is used to measure the design robustness. A feasibility robustness index of 0 corresponds to a confidence probability level of 50% that the reliability 294 (or failure probability) target will be satisfied. A feasibility robustness index of = corresponds to a confidence probability level of 84.13% that the reliability (or failure 296 probability) target will be satisfied, and = 2 will raise the confidence probability to %. As will be shown in the illustrative example later, the feasibility robustness index 13

14 298 ( ) provides an easy-to-use measure of robustness to aid in the decision-making Example: Robust Design of Rock Slope with Multiple Failure Modes 3.1 Rock slope with multiple failure modes Rock slopes are often composed of several potentially unstable blocks. In such cases, the rock slope should be modeled as a rock slope system, as it may involve multiple failure modes. For illustration purposes, a simple rock slope composed of two blocks separated by a vertical tension crack (Jimenez-Rodriguez et al. 2006) is used as an example to illustrate the rock slope with multiple failure modes. The position of tension crack separating two blocks is considered random, which may be located at the slope top or slope face (see Figure 3). The rock slope is considered safe if the factor of safety of block A is grater than unity using the stability model by Hoek and Bray (1981) with a slight modification to account for the interaction between blocks. Based on the condition of interaction between the two blocks, two distinct scenarios are possible. In scenario 1, block B is stable by itself and there is no interaction between the two blocks. In scenario 2, block B is unstable and tends to slide such that an interaction force will be imposed on block A (Jimenez-Rodriguez et al. 2006). Detailed formulation for factor of safety for both blocks is summarized in Appendix (Appendix A, ESM only). The rock slope composed of two removable blocks depicted in Figure 3 is used to demonstrate the reliability-based robust geotechnical design. The initial geometry of the slope is defined by a slope height of H = 25 m and a slope face angle of = 50. The location of the slip surface is assumed to be certain with a dip angle = 32, and the unit weight of rock 3 ( 25 kn/m ) is considered a fixed value. 14

15 As noted by Jimenez-Rodriguez et al. (2006), the parameters describing rock properties along the slip surface, as well as the position of tension crack and water depth should be considered as random variables in the reliability analysis of rock slope with two removable blocks. Thus, totally seven random parameters, listed in Table 1, are considered in the design analysis. Specifically, cohesion along the slip surface of block A and block B (c A and c B ), friction angle along the slip surface of block A and block B ( A and B ), as well as friction angle along the contact surface between two blocks ( AB ) are assumed as truncated normal random variables (Hoek 2006). The ratio of the tension crack depth filled with water z w is assumed to follow the exponential distribution with a mean of 0.25 and truncated to the 330 range [0, 0.5] (Low 2007). The location of tension crack X B (see Figure 3) is modeled as a non-symmetric beta distribution, in order to represent the common observations that tension cracks are more commonly presented at the top of the slope (Hoek and Bray 1981; Jimenez-Rodriguez et al. 2006). The probability density function of this beta distribution bounded by [a, b] with two shape parameters q and r is given by (Ang and Tang 2007): 335 q1 r1 ( x a) ( b x) f ( x) a x b 1 q1 r1 qr1 x (1 x) dx( b a) 0 (9) 336 Furthermore, cohesion and friction angle are assumed to be negatively correlated with 337 correlation coefficients c, c, A A B B 0.35 (Lee et al. 2012). On the other hand, shear strength parameters between the two blocks are assumed positively correlated with correlation coefficients c, c,,, = 0.3 (Jimenez-Rodriguez et al. 2006). All other A B A AB AB B A B random variables are assumed independent. The correlation structure of these variables is shown in Table 2. 15

16 System reliability approach for rock slope with multiple failure modes For the reliability analysis of a rock slope system with multiple failure modes, a disjoint cut-set formulation (Jimenez-Rodriguez et al. 2006; Jimenez-Rodriguez and Sitar 2007) may be employed. The slope system is modeled as a series of sub-systems each with parallel components (see Figure 4). Each sub-system is a cut-set that addresses a particular failure mode, and within a given cut-set, there are three parallel components each defined by a 348 limit state function g i (see Table 3). Note that cut-sets are disjointed (meaning that there is 349 no intersection in any two cut-sets). Symbolically, Ck Cl, for k l and 350 k, l 1, 2, NC, where C k and C l are two given cut-sets, and N C is the number of 351 cut-sets Based on the cut-set formulation, the failure probability of the complete rock slope system can be obtained by taking the summation of individual failure probabilities of these cut-set as follows (Jimenez-Rodriguez and Sitar 2007): N C P( E) P E i ick (10) k1 where E represents the failure event of the entire rock slope system; P is the probability of a given event (for example, PE ( ) is the probability of event E); E i represents the failure event of i th component of k th cut-set C k ; the notation components of a cut-set C k. E i represents the intersection of the three i Ck For a given cut-set (failure mode) C k, the probability of failure can be calculated as follows (Ditlevsen and Madsen 1996; Jimenez-Rodriguez 2004): P ick E i ( βc, R ) (11) k 16

17 363 where is the cumulative standard multi-normal distribution function; βck is the vector composed of reliability index ( ) of each component i in the cut-set C k ; R is the correlation i matrix between components within the cut-set C k. The reader is referred to Ditlevsen and Madsen (1996) for further details of this formulation. The probability of failure for each of the four cut-sets (failure modes) can be computed using Eq. (11), then the failure probability of the rock slope system can be obtained using Eq. (10). As demonstrated by Jimenez-Rodriguez (2004), this approach is computationally effective and provides a good approximation to the exact probability of failure given by Monte Carlo simulation. 3.3 Traditional reliability-based design of rock slope system With the defined statistics and correlation structure of input parameters listed in Table 1 and Table 2, the hypothetical rock slope system that is composed of two removable blocks (Figure 3) can be analyzed with the traditional reliability-based design method. The design space for design parameters should first be specified for the reliability-based design. For the two design parameters of the hypothetical rock slope (see Figure 3 and Tables 1 and 2), slope height H may be in the range of 20 m to 25 m, and slope angle may be in the range of 40 to 50. The upper bounds of design parameters are chosen based on the initial slope condition and the lower bounds of design parameters are chosen based on the geometry requirement of the deterministic model. For convenience in the rock slope construction, H is rounded to nearest 0.2 m and is rounded to nearest 0.2. Thus, H can take from 26 discrete values and can take from 51 discrete values, with totally 1326 possible designs in the design space. 17

18 If the statistics of the rock properties can be accurately characterized (so that they can be described with fixed values as listed in Tables 1 and 2), the probability of failure of the rock slope system for each possible design in the design space can be computed using the system reliability approach (Jimenez-Rodriguez and Sitar 2007) discussed previously. The results of these system reliability analyses are shown in Figure 5. As expected, the failure probability increases with the increase of slope height and slope angle. The design with the least cost while satisfying the reliability requirement (i.e., the computed failure probability is less than a threshold value) is usually selected as the final design (Zhang et al. 2011). For illustration purpose, the target failure probability p T (as per Low 2008) is used as the reliability constraint in this paper. The final design following the traditional reliability-based design method yields a design with design parameters H = 25 m, = 45.6 and a design cost of 43.8 units [note: 1 unit = unit cost ($) of excavated volume of rock mass in m 3 /m]. However, the final design H = 25 m, = 45.6 is based on the assumption of fixed parameter statistics (i.e., COVs and correlation coefficients) listed in Table 1. In practice, the statistics of rock properties are usually difficult to ascertain. However, with the aid of published COVs of rock properties and engineering judgment, these statistics may be estimated as a range. For example, the coefficient of variation (COV) of cohesion c, denoted as COV[] c, typically ranges from 10% to 30% (Lee et al. 2012); the COV of friction angle, denoted as COV[ ], typically ranges from 5% to 10% (Alejano et al. 2012). In addition, the correlation coefficient between c and, denoted as, c, typically ranges from -0.1 to -0.6 (Lee et al. 2012). These published numbers can be used as a guide along with engineering judgment to characterize the uncertainty of the statistics of these rock properties. 18

19 To examine the possible effect of the uncertainty in the estimated statistics of rock properties on the final design of the hypothetical rock slope system, c, is assumed as a fixed value of -0.35, and COV[] c and COV[ ] are assumed to vary within their typical ranges. Table 4 lists the final designs for various assumed COV levels obtained using the traditional reliability-based design approach. It is found that the final design obtained with the traditional reliability-based design approach is very sensitive to the levels of COVs of rock properties. Under the assumption of the lowest variability level ( COV[] c = 0.1 and COV[ ] = 0.05), all designs in the design space satisfy the reliability requirement. In this scenario, the design that represents the initial slope system, which of course requires no excavation, is the least cost design that satisfies the reliability requirement (i.e., the evaluated failure probability 418 is less than a threshold value of pt ). However, under the assumption of the highest variability level ( COV[] c = 0.3 and COV[ ] = 0.1), all designs in the design space, including the one with the highest cost (H = 20 m, = 40 with design cost of units), cannot satisfy the reliability requirement. The implication from the results shown in Table 4 is that the final design through traditional reliability-based design approach is dependent on assumed variability levels of rock properties. If the variation of rock properties is overestimated, the resulting final design will not be cost-efficient. On the other hand, if the variation of rock properties is underestimated (for example, when actual variation of rock properties is higher than the assumed variation), the initially acceptable design (e.g., H = 25 m, = 45.6 based on the assumed statistics listed in Tables 1 and 2) will no longer meet the reliability requirement. As 19

20 shown in Table 5, under the actual variation of rock properties ( COV[] c = 0.3 and COV[ ] = 0.1), the computed failure probability for the initially acceptable design (H = 25 m, = 45.6 ) is , much higher than the target probability of failure ( pt ). 3.4 Reliability-based robust design of rock slope with multiple failure modes For the hypothetical rock slope composing of two removable blocks (Figure 3), the design parameters are the slope height H and slope angle. As discussed previously, the design space consists of 1326 possible designs with different combination of H and. For this problem, the noise factors are mainly associated with uncertain rock properties, including c A, c B, A, B, and AB. Here, seven statistical parameters, namely COV[ c A], COV[ c B], COV[ A], COV[ B], COV[ AB], c,, and c, A A B B, may be treated as fuzzy numbers. For demonstration purpose, the typical ranges for COV[] c from 10% to 30%, COV[ ] from 5% to 10%, and c, from -0.1 to -0.6, as reported in the literature, are used as the basis for establishing fuzzy numbers that characterize these uncertain statistical parameters. Of course, local experience and engineering judgment can play a significant role in selecting a proper range to best characterize the uncertainty of these statistics. As an example, for the statistical parameters COV[ c A] and COV[ c B], the fuzzy number is constructed with only the knowledge of lower bound (a = 0.1) and upper bound (b = 0.3), with the implied mode m = 0.2, as shown in Figure 1. Following the RGD procedure outlined previously (in reference to Figure 2), the mean and standard deviation of the failure probability for the hypothetical rock slope system, 449 denoted as p and p, can be obtained using Fuzzy-based PEM for each of the designs in the design space. For illustration purpose, the resulting mean and standard 20

21 deviation of selected designs with H = 20 m, 21 m,..., 25 m are shown in Figure 6 and Figure 7, respectively (note: the initial geometry of the slope is H = 25 m and = 50 ). Then, the multi-objective optimization is set up as follows: Find d = [H, ] Subjected to: H {20m, 20.2m, 20.4m,, 25m } and {40, 40.2, 40.6,, 50 } p pt Objectives: Minimizing the standard deviation of failure probability ( p ) Minimizing the cost for rock slope design. Using the NSGA-II algorithm, 64 designs out of the 1326 designs are found satisfactory with respect to all constraints and most optimal with respect to both objectives of 464 robustness (measured with p ) and cost. These 64 designs are more optimal than other designs in the design space, but within the set of 64 designs, none of them is superior to any others in all objectives. These 64 designs collectively form a Pareto Front, as shown in Figure 8. Recall that no design that belongs to the Pareto Front can be improved with respect to one design objective without weakening the performance in the other objective (Deb et al. 2002). Thus, the Pareto Front in this case offers a trade-off relationship between cost and robustness (in terms of the standard deviation of the failure probability). Figure 9 further depicts the relationship between cost and robustness for all designs on the established Pareto Front (Figure 8), where the robustness is now measured by the feasibility robustness index. Recall that every point on the Pareto Front is a non-dominated design that satisfies the safety requirement (i.e., the mean failure probability must be smaller than a target failure probability). If the least cost design is desired by the engineer, the final 21

22 476 design of H = 25 m and = 45.2 (the lowest point shown in Figure 9) will be selected, which 477 costs 48.1 units. This design has a feasibility robustness index of = 0.27, which is equivalent to saying that there is a 60.57% chance (or confidence probability) that the design will meet the safety requirement in face of the variation of the parameter statistics. If the uncertainty of the estimated parameter statistics is unavoidable and an estimate of this uncertainty (variation) represents the best knowledge we have, then it may be desirable to select the final design based on a trade-off consideration using the established Pareto Front or the cost- relationship derived from it. For example, an increase in the feasibility 484 robustness to = 1, which increases the chance of satisfying the safety requirement to %, will result in a final design of H = 25 m and = 42.6 (note: this data is not revealed in Figure 9) that costs 77.6 units. Thus, the final design may be selected based on a prescribed feasibility robustness 488 level (for example, = 1 or 2, and so on). In this case, we are confident that the final design will have a certain percentage of satisfying the safety requirement (for example, 84.13% with = 1; 97.72% with = 2, see Table 6) even with the existence of uncertainty in the estimated parameter statistics. Alternatively, the designer can select the final design as the one that gives the highest feasibility robustness under a certain cost. Either way, the Pareto Front established using the reliability-based RGD methodology, or the cost- relationship derived from it, provides a useful decision making tool that can aid in selecting the final design in a more rational and easily-communicated way. To help with selection of the final design using the developed Pareto Front (see Figure 8 or Figure 9), a further step can be undertaken to identify the knee point on the Pareto Front. 22

23 Based on the definition given by Deb and Gupta (2011), the knee point is the most preferred design, since it requires a large sacrifice in one objective to gain a minor improvement in the other objective. The normal boundary intersection method (Deb and Gupta 2011) may be used to identify the knee point on the Pareto Front. A boundary line is first constructed by connecting the two extreme points on the Pareto Front, and then the point on the Pareto front that has the maximum distance to this boundary line is identified as the knee point (see annotation in Figure 8 and Figure 9). Based on the normal boundary intersection method, the same knee point is identified for Figure 8 and Figure 9, which is the design with H = 25 m and θ = 40 that costs units. Below this cost level, it requires a large sacrifice on robustness to achieve a minor gain in cost-efficiency (reduction in cost). On the other hand, above this cost level, it requires a large sacrifice in cost-efficiency to achieve a minor gain in robustness improvement (i.e., increase of feasibility robustness or reduction of variation of failure probability). This knee point concept provides an additional tool for selecting the most preferred design on the Pareto Front Concluding Remarks In this paper, a fuzzy set approach is incorporated into the reliability-based Robust Geotechnical Design (RGD) framework to deal with the uncertainty in the estimated statistics of rock properties for design of a rock slope system. Use of a fuzzy number to describe or model the uncertainty in the estimated statistics (such as COV) is deemed appropriate, as the amount of quality data is generally very limited. Construction of a fuzzy number for such situation requires only the knowledge of a highest conceivable value and a lowest conceivable value, which enables the engineer to quantify uncertainty based on the reported ranges from 23

24 literature and augmented with local experience and engineering judgment. It should be noted, however, that use of the fuzzy set (or fuzzy number) approach within the RGD framework is only for its practicality, and should not be viewed as a limitation of the reliability-based RGD methodology. The proposed design approach has been demonstrated with an application to the design of rock slope system considering multiple failure modes, and its effectiveness has been demonstrated with the results presented in this paper. In typical geotechnical practice, statistical parameters such as the coefficient of variation and the correlation coefficient between noise factors (e.g., uncertain rock properties in this paper), which are required in a reliability-based design, are difficult to ascertain. If these statistics of noise factors are overestimated or underestimated, the final design obtained from the traditional reliability-based design will be either cost-inefficient or unsafe. By considering explicitly the robustness against the uncertainty in the estimated statistics of rock properties, the RGD approach reduces the adverse effect of such uncertainty. In fact, by considering three objectives, safety, robustness, and cost, in the design using a multi-objective optimization within the RGD framework, a set of optimal, non-dominated designs, which form a Pareto Front collectively, can be obtained. The established Pareto Front is shown to be a useful decision-making tool, which has been demonstrated in the rock slope design example presented in this paper. It should be noted that the proposed RGD approach is mainly for improving decision making under scarce and incomplete information. The RGD approach is not a methodology to compete with reliability-based design, but is a complementary to reliability-based design under scarce and incomplete information. If sufficient information is available for a comprehensive assessment of all uncertainty involved in rock slope design (e.g., the 24

25 probability distributions of all input parameters can be fully and accurately characterized), the traditional reliability-based design approach will suffice for rock slope design. Under the scenario of scarce and incomplete information, the RGD approach often can reduce the adverse effect of the uncertainty associated with the estimated statistical parameters ACKNOWLEDGMENTS The study on which this paper is based was supported in part by National Science Foundation through Grant CMMI The results and opinions expressed in this paper do not necessarily reflect the view and policies of the National Science Foundation. 25

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