Slope Reliability Analysis Research Articles

Reliability Analysis of Earth Slopes Considering Spatial Variability

The paper presents a computational procedure for reliability analysis of earth slopes considering spatial variability of soils under the framework of the Limit Equilibrium Method. In the reliability analysis of earth slopes, the effect of spatial variability of soil properties is generally included indirectly by assuming that the probabilistic critical slip surface is the same as that determined without considering spatial variability. In contrast to this indirect approach, in the direct approach, the effect of spatial variability is included in the process of determination of the probabilistic critical surface itself. While the indirect approach requires much less computational effort, the direct approach is definitely more rigorous. In this context this paper attempts to investigate, with the help of numerical examples, how far away are the results obtained from the indirect approach from that obtained from the direct approach. In both the approaches, it is required to use a model of discretization of random fields into finite random variables. A few such models are available in the literature for one-dimensional (1D) as well as two-dimensional (2D) spatial variability. The developed computational scheme is based on the First Order Reliability Method (FORM) coupled with the Spencer Method of Slices valid for limit equilibrium analysis of general slip surfaces. The study includes bringing out the computational advantages and disadvantages of the three commonly used discretization models. The sensitivity of the reliability index to the magnitudes of the scales of fluctuation has also been studied.

Response surface methods for slope reliability analysis: Review and comparison

This paper reviews previous studies on developments and applications of response surface methods (RSMs) in different slope reliability problems. Based on the review, four types of soil slope reliability analysis problems are identified from the literature, including single-layered soil slope reliability problem ignoring spatial variability, single-layered soil slope reliability problem considering spatial variability, multiple-layered soil slope reliability problem ignoring spatial variability, and multiple-layered soil slope reliability problem considering spatial variability, which are referred to as “Type I–IV problems” in this study. Then, the computational efficiency and accuracy of four commonly-used RSMs (namely single quadratic polynomial-based response surface method (SQRSM), single stochastic response surface method (SSRSM), multiple quadratic polynomial-based response surface method (MQRSM), and multiple stochastic response surface method (MSRSM)) are systematically compared for cohesive and c–ϕ slopes, and their feasibility and validity in the four types of slope reliability problems are discussed. Based on the comparison, some suggestions for selecting relatively appropriate RSMs in slope reliability analysis are provided: (1) SQRSM is suggested as a suitable method for the single-layered soil slope reliability problem ignoring spatial variability (i.e., Type I problem); (2) MQRSM is applicable to the multiple-layered soil slope reliability problem ignoring spatial variability (i.e., Type III problem); and (3) MSRSM is suggested to solve slope reliability problems (including single-layered and multiple-layered slopes) considering spatial variability (i.e., Type II and IV problems).

Probabilistic Stability Analysis of the Bois Brule Levee Considering the Effect of Spatial Variability of Soil Properties Based on a New Discretization Model

The paper presents a computational procedure for reliability analysis of earth slopes for which the probabilistic critical slip surfaces have been determined considering spatial variability of soils. The effect of spatial variability has been taken into account based on a newly developed discretization model which is free from the shortcomings of the discretization models available in the literature and all these shortcomings are demonstrated numerically in this paper. The developed algorithm is based on the First Order Reliability Method coupled with the Spencer Method of Slices valid for limit equilibrium analysis of general slip surfaces. A case study of the Bois Brule Levee has been reanalyzed using the developed technique and several observations have been discussed. The result obtained in this study clearly shows that the proposed model is considerably more conservative than other available models.

Artificial Bee Colony Algorithm Optimized Support Vector Regression for System Reliability Analysis of Slopes

AbstractProbabilistic stability analysis is an effective way to take uncertainties into account in evaluating the stability of slopes. This paper presents an intelligent response surface method for system probabilistic stability evaluation of soil slopes. Artificial bee colony algorithm (ABC) optimized support vector regression (SVR) is used to establish the response surface to approximate the limit-state function. Then Monte Carlo simulation is performed via the ABC-SVR response surface to estimate system failure probability. The proposed methodology is verified in three case examples and is also compared with some well-known or recent algorithms for the problem. Results show that the new approach is promising in terms of accuracy and efficiency.

Efficient 3-D reliability analysis of the 530 m high abutment slope at Jinping I Hydropower Station during construction

Reliability-based slope stability analysis and design show promise, but few attempts have been made to study the 3-D reliability of high rock slopes. This paper aims to investigate the 3-D reliability of practical rock slopes and to explore the effects of reinforcement measures on slope reliability. A non-intrusive stochastic finite difference method is adopted to perform 3-D reliability analysis of high rock slopes, in which response surface functions are adopted to establish the explicit functions between the factors of safety for multiple slope failure modes and the significant random variables. The left abutment slope at Jinping I Hydropower Station in China is taken as a case study. A parametric sensitivity analysis is carried out to identify the significant uncertain parameters. The effects of two reinforcement measures, namely pre-stressed cables and shear-resistant concrete plugs, on the reliability of the rock slope during construction are highlighted. The results indicate that the 3-D reliability of complex rock slopes can be efficiently evaluated using a non-intrusive stochastic finite difference method. It provides an effective tool for solving 3-D reliability problems of complex rock slopes in practice. The combined support system with pre-stressed cables and three shear-resistant concrete plugs adopted in the Jinping I left abutment slope can effectively restrain the slope deformation and ensure the slope stability during construction. The disturbances induced by the excavation blasting during construction have a significant influence on the engineered slope reliability.

A probabilistic non-linear finite element analysis for slope stability problem

This paper deals with the slope stability problem by a finite element reliability analysis considering the spatial variation of soil strength parameters. In this work, the spatial variation is described using random field theory. To simplify the implementation of the proposed procedure for slope reliability analysis, elastic perfectly plastic constitutive model describing the soil behaviour using the Mohr coulomb yield criterion is adopted. The application of the proposed approach for the slope reliability analysis is performed using a performance function expressed in terms of stress fields mobilised along a circular failure surface. For this purpose, a computation of the stress gradient based on the incremental theory of plasticity is used. The results of the proposed procedure are checked by a probabilistic method based on the combination of the Bishop's model and the Monte Carlo simulation. The numerical examples have elucidated the efficiency and the validity of the present procedure to slope reliabi...

A Study of Probabilistic FEMs for a Slope Reliability Analysis Using the Stress Fields

In this paper, the applicability and the effectiveness of the probabilistic finite element methods (FEMs) such as the perturbation method, and the Spectral Stochastic Finite Element Method (SSFEM) applied to the reliability analysis of the slope stability have been studied. The results were checked by the Monte Carlo simulation and a direct coupling ap-proach combining the deterministic finite elements code and First Order Reliability Method (FORM) algorithm. These methods are presented considering the spatial variation of soil strength parameters and Young modulus. The random field is used to describe the spatial variation. Also, the reliability analysis is conducted using a performance function formulat-ed in terms of the stochastic stress mobilized along the sliding surface. The present study shows that the perturbation method and SSFEM can be considered as practical methods to conduct a second moment analysis of the slope stability taking into account the spatial variability of soil properties since good results are obtained with acceptable estimated rela-tive errors. Finally, the perturbation method is performed to delimit the location of the critical probabilistic sliding surfac-es and to evaluate the effect of the correlation length of soil strength parameters on the safety factor. In addition, the two methods are used to estimate the probability density and the cumulative distribution function of the factor of safety.

System reliability analysis of layered soil slopes using fully specified slip surfaces and genetic algorithms

This paper presents a new approach to identify the fully specified representative slip surfaces (RSSs) of layered soil slopes and to compute their system probability of failure, Pf,s. Spencer's method is used to compute the factors of safety of trial slip surfaces, and the first order reliability method (FORM) is employed to efficiently evaluate their reliability. A custom-designed genetic algorithm (GA) is developed to search all the RSSs in only one GA optimization. Taking advantage of the system aspects of the problem, such RSSs are then employed to estimate the reliability of the slope system, and a proposed linearization approach—based on first or second order (FORM or SORM) reliability information about the identified RSSs—is used to efficiently estimate Pf,s. Three typical benchmark-slopes with layered soils are adopted to demonstrate the efficiency, accuracy and robustness of the suggested procedure, and advantages of the proposed method with respect to alternative methods are discussed. Results show that the proposed approach provides reliability estimates that improve previously published results, emphasizing the importance of finding good RSSs—and, especially, good (probabilistic) critical slip surfaces that might be circular or non-circular—to obtain good estimations of the reliability of soil slope systems.

Slope reliability analysis using length-based representative slip surfaces

This paper develops an empirical approach to slope reliability considering 2D spatial variability of soil properties with length-based representative slip surfaces. The correlation coefficient between factors of safety for two slip surfaces is analogous to that between two slip surfaces which is quantified based on the lengths within different elements of the slip surface. A large number of potential slip surfaces are categorized into separate slip surface groups within each of which the correlation coefficient between any two slip surfaces is higher than the threshold value. A slip surface with the minimum factor of safety is considered to be the representative slip surface for each of slip surface groups. The factor of safety is minimized among all the representative slip surfaces, and the minimum one is regarded as the output for each set of random samples generated by Monte Carlo simulation (MCS). The methodology is validated through comparison with other methods. It is found that slope failure probability, p f, increases as the scale of fluctuation, λ y (in vertical direction), and λ x (in horizontal direction) increase. The effect of non-circular failure mechanism was studied at different λ x and λ y values. Significant larger failure probabilities are provided with non-circular failure mechanism than with circular failure mechanism. For 2D spatial variability case, two orders of magnitude larger failure probability is noticed.

Reliability analysis method for slope stability based on sample weight

The single safety factor criteria for slope stability evaluation, derived from the rigid limit equilibrium method or finite element method (FEM), may not include some important information, especially for steep slopes with complex geological conditions. This paper presents a new reliability method that uses sample weight analysis. Based on the distribution characteristics of random variables, the minimal sample size of every random variable is extracted according to a small sample t-distribution under a certain expected value, and the weight coefficient of each extracted sample is considered to be its contribution to the random variables. Then, the weight coefficients of the random sample combinations are determined using the Bayes formula, and different sample combinations are taken as the input for slope stability analysis. According to one-to-one mapping between the input sample combination and the output safety coefficient, the reliability index of slope stability can be obtained with the multiplication principle. Slope stability analysis of the left bank of the Baihetan Project is used as an example, and the analysis results show that the present method is reasonable and practicable for the reliability analysis of steep slopes with complex geological conditions.

Bootstrap method for characterizing the effect of uncertainty in shear strength parameters on slope reliability

This paper aims to propose a bootstrap method for characterizing the effect of uncertainty in shear strength parameters on slope reliability. The procedure for a traditional slope reliability analysis with fixed distributions of shear strength parameters is presented first. Then, the variations of the mean and standard deviation of shear strength parameters and the Akaike Information Criterion values associated with various distributions are studied to characterize the uncertainties in distribution parameters and types of shear strength parameters. The reliability of an infinite slope is presented to demonstrate the validity of the proposed method. The results indicate that the bootstrap method can effectively model the uncertain probability distributions of shear strength parameters. The uncertain distributions of shear strength parameters have a significant influence on slope reliability. With the bootstrap method, the slope reliability index is represented by a confidence interval instead of a single fixed index. The confidence interval increases with increasing factor of slope safety. Considering both the uncertainties in distribution parameters and distribution types of shear strength parameters leads to a higher variation and a wider confidence interval of reliability index.

Enhancement of random finite element method in reliability analysis and risk assessment of soil slopes using Subset Simulation

Random finite element method (RFEM) provides a rigorous tool to incorporate spatial variability of soil properties into reliability analysis and risk assessment of slope stability. However, it suffers from a common criticism of requiring extensive computational efforts and a lack of efficiency, particularly at small probability levels (e.g., slope failure probability P f < 0.001). To address this problem, this study integrates RFEM with an advanced Monte Carlo Simulation (MCS) method called “Subset Simulation (SS)” to develop an efficient RFEM (i.e., SS-based RFEM) for reliability analysis and risk assessment of soil slopes. The proposed SS-based RFEM expresses the overall risk of slope failure as a weighed aggregation of slope failure risk at different probability levels and quantifies the relative contributions of slope failure risk at different probability levels to the overall risk of slope failure. Equations are derived for integrating SS with RFEM to evaluate the probability (P f ) and risk (R) of slope failure. These equations are illustrated using a soil slope example. It is shown that the P f and R are evaluated properly using the proposed approach. Compared with the original RFEM with direct MCS, the SS-based RFEM improves, significantly, the computational efficiency of evaluating P f and R. This enhances the applications of RFEM in the reliability analysis and risk assessment of slope stability. With the aid of improved computational efficiency, a sensitivity study is also performed to explore effects of vertical spatial variability of soil properties on R. It is found that the vertical spatial variability affects the slope failure risk significantly.

System reliability analysis of slope with stochastic response surface method

System reliability analysis of slope with stochastic response surface method

An algorithm in generalized coordinate system and its application to reliability analysis of seismic slope stability of high rockfill dams

This paper introduced generalized coordinate system into reliability analysis involving correlated random variables. Correlation between two random variables is reflected by rotation of the corresponding coordinate axes and correlation coefficient between the two variables is identical to cosine of angle between the axes. The equivalent matrix formulation for the Hasofer–Lind reliability index represents the shortest distance from the failure surface to the origin in the generalized coordinate system of standardized normal variables. A simple optimization algorithm for reliability computing was established based on the perspective of reliability index and the gradient of limit state function in generalized coordinate system. Based on the algorithm and the Ordinary method of slices, a probabilistic approach for evaluating the seismic slope stability of high rockfill dams was developed and was applied to the slope reliability analysis of an actual high rockfill dam under seismic loading. The results of the slope reliability analysis are discussed and conclusions are presented in this paper.

A multiple response-surface method for slope reliability analysis considering spatial variability of soil properties

This paper proposes a multiple response-surface method for slope reliability analysis considering spatially variable soil properties. The scales of fluctuation of soil shear strength parameters are summarized. The effect of theoretical autocorrelation functions (ACFs) on slope reliability is highlighted since the theoretical ACFs are often used to characterize the spatial variability of soil properties due to a limited number of site observation data available. The differences in five theoretical ACFs, namely single exponential, squared exponential, second-order Markov, cosine exponential and binary noise ACFs, are examined. A homogeneous c–ϕ slope and a heterogeneous slope consisting of three soil layers (including a weak layer) are studied to demonstrate the validity of the proposed method and explore the effect of ACFs on the slope reliability. The results indicate that the proposed method provides a practical tool for evaluating the reliability of slopes in spatially variable soils. It can greatly improve the computational efficiency in relatively low-probability analysis and parametric sensitivity analysis. The extended Cholesky decomposition technique can effectively discretize the cross-correlated non-Gaussian random fields of spatially variable soil properties. Among the five selected ACFs, the squared exponential and second-order Markov ACFs might characterize the spatial correlation of soil properties more realistically. The probability of failure associated with the commonly-used single exponential ACF may be underestimated. In general, the difference in the probabilities of failure associated with the five ACFs is minimal.

Cumulative PSO-Kriging model for slope reliability analysis

The particle swarm optimization (PSO) algorithm is introduced in the Kriging modeling process to overcome the limits of pattern search method's single-point search scheme as well as its heavy dependence on the initial guess solution when obtaining the optimal correlation parameters. PSO-Kriging is proved to give better simulation by interpolating and extrapolating the unobserved points. In reliability analysis, cumulative formation of repeatedly using sampling points in previous iterations is introduced into PSO-Kriging and the classic response surface method (RSM). Cumulative formation can take full advantage of available sampling information and avoid reciprocating oscillation in the iterative process. One explicit nonlinear limit state function example demonstrated that cumulative scheme can make both PSO-Kriging and RSM much more effective, no matter latin hypercube sampling (LHS) or iteratively interpolating sampling (IIS) approach is utilized. Cumulative PSO-Kriging seems to be even more stable and efficient. Two slope reliability analysis examples including a practical nuclear plant breakwater's reliability analysis problem proved that the proposed cumulative PSO-Kriging is very suitable for the reliability analysis of real engineering structures.

Fuzzy Mathematical Method of Slope Reliability Analysis in Surface Mine under Seepage Conditions

With the development of the economy, the increasing height of slopes in surface mine will seriously threaten the security in Coal Mine Area. Therefore, how to accurately evaluate slope stability has become one important problem. Based on reliability theory and fuzzy theory, a new reliability analysis method-fuzzy point evaluation method is presented in this paper. It is more practical and efficient for slope reliability assessment because traditional evaluation method doesn’t consider the fuzziness of parameters. As a typical example, we applied the technique to a surface mine case under seepage conditions, and made comparison with the traditional reliability analysis approach. The results indicate that the conclusions from both methods are essentially consistent. However, because considers the fuzziness of parameters, fuzzy point evaluation method has more actual instructive significance comparing with the traditional mothod.

Sensitivity-based reliability analysis of earth slopes using finite element method

For slope stability analysis, an alternative to the classical limit equilibrium method (LEM) of slices is the shear strength reduction method (SRM), which can be integrated into finite element analysis or finite difference analysis. Recently, probabilistic analysis of earth slopes has been very attractive because it is capable to take the soil uncertainty into account. However, the SRM is less commonly extended to probabilistic framework compared to a variety of probabilistic LEM analysis of earth slopes. To overcome some limitations that hinder the development of probabilistic SRM stability analysis, a new procedure based on recursive algorithm FORM with sensitivity analysis in the space of original variables is proposed. It can be used to deal with correlated non-normal variables subjected to implicit limit state surface. Using the proposed approach, a probabilistic finite element analysis of the stability of an existing earth dam is carried out in this paper.

Multiple response surfaces for slope reliability analysis

SummaryThis paper develops a multiple response surfaces approach to approximate the limit state function for slope failure by second‐order polynomial functions, to incorporate the variation of the most probable slip surfaces, and to evaluate the slope failure probability pf. The proposed methodology was illustrated through a cohesive soil slope example. It is shown that the pf values estimated from multiple response surfaces agree well with those pf values that have been obtained by searching a large number of potential slip surfaces in each Monte Carlo simulation (MCS) sample.The variation of number of the most probable slip surfaces is studied at different scale of fluctuation (λ) values. It is found that when full correlation assumed for each of random fields (i.e., spatial variability is ignored), the number of the most probable slip surfaces is equal to the number of random fields (in this study, it is 3). When the spatial variability grows significantly, the number of the most probable slip surfaces or number of multiple response surfaces firstly increases evidently to a higher value and then varies slightly. In addition, the contribution of a specific most probable slip surface varies dramatically at different spatial variability level, and therefore, the variation of the most probable slip surfaces should be accounted for in the reliability analysis. The multiple response surfaces approach developed in this paper provides a limit equilibrium method and MCS‐based means to incorporate such a variation of the most probable slip surfaces in slope reliability analysis. Copyright © 2014 John Wiley &amp; Sons, Ltd.

Effect of Spatial Variability of Soil Properties on Slope Reliability Using Random Finite Element and First Order Second Moment Methods

Spatial variability of soil properties is an important factor that affects the probability of slope failure. Random finite element method (RFEM) is the most effective method to evaluate the probability of failure of slopes. In this study, the probability of slope failure is calculated using Monte Carlo simulation based on displacement based criteria. The probability of failure of an undrained cohesive slope is studied using RFEM, first order second moment (FOSM) method and FOSM combined with variance reduction theory, called modified FOSM (MFOSM). A comparison among FOSM, MFOSM and RFEM is shown for different values of undrained shear strength and scales of fluctuation. As FOSM does not consider the spatially varying nature of soil properties, it gives a constant probability of failure for varying scales of fluctuation, a parameter that represents the spatial variability of soil properties. In general, MFOSM has similar trend with RFEM, however, MFOSM cannot demonstrate the failure mechanism for spatially varying soils. The use of RFEM and determination of realistic scales of fluctuation, coefficient of variation of soil properties is recommended for slope reliability analysis.