Slope Reliability Analysis Research Articles

Modeling multivariate cross-correlated geotechnical random fields using vine copulas for slope reliability analysis

Modeling multivariate cross-correlated random fields plays an important role in reliability analysis of geotechnical systems. The cross-correlation among random fields is commonly characterized by the Nataf transformation using a correlation coefficient matrix, which is equivalent to considering the Gaussian dependence structure from the perspective of the copula theory. In this study, a generic approach for modeling multivariate cross-correlated geotechnical random fields based on vine copulas is proposed. This approach can incorporate the diversity and non-Gaussianity of dependence structures in the cross-correlation characterization. A four-variate geotechnical dataset is used to verify the proposed approach and the results illustrate that the statistics of generated random field realizations are comparable with the predefined values. Furthermore, the application of the proposed approach to reliability analysis of soil slopes with spatially varying soil parameters (cohesion, friction angle, and unit weight) is presented. The results indicate that considerably different probabilities of slope failure are produced by different vine-copula models. The conventional Nataf transformation (i.e., vine-Gaussian copula model) may considerably underestimate the probability of slope failure. The difference of reliability results for different vine models is generally enlarged as the level of the failure probability becomes lower. The proposed approach provides a more flexible way for geotechnical practitioners to characterize the cross-correlation among geotechnical random fields compared with the Nataf transformation, and is applicable to reliability analysis of various geotechnical systems.

Non-intrusive reliability analysis of unsaturated embankment slopes accounting for spatial variabilities of soil hydraulic and shear strength parameters

Existing reliability analyses of embankment slopes usually considered the spatial variabilities of shear strength parameters and saturated hydraulic conductivity separately. Additionally, the variability in the soil–water characteristic curve (SWCC) was ignored. A non-intrusive approach is proposed for reliability analyses of unsaturated embankment slopes under four different cases on a steady-state seepage analysis. The spatial variabilities of hydraulic parameters (including fitting parameters, a and n, of SWCC and saturated hydraulic conductivity) and shear strength parameters are accounted for simultaneously. To illustrate the proposed approach, a hypothetical embankment under unsaturated seepage is investigated. The soil parameters of the embankment and foundation are modeled as lognormal random fields and lognormal random variables, respectively, to take into account the inherent uncertainties. Parametric sensitivity studies are then performed to account explicitly for the influence of the spatial variation of hydraulic parameters on the reliability of embankment slope stability. The proposed approach can act as a practical and rigorous tool for estimating the reliability of embankment slopes with small levels of probability of failure (i.e., 10−4–10−3). An interesting finding that the probability of failure of the unsaturated embankment slope decreases with the variability of the fitting parameter n of SWCC is observed and explained.

3D slope reliability analysis based on the intelligent response surface methodology

At present, there are two problems in 3D slope reliability analysis: (1) the slope stability analysis method; (2) how to effectively extend the reliability analysis method from 2D to 3D slopes. This paper combines the advantages of the finite element method and the limit equilibrium method, develops a 3D slope stability analysis finite element sliding surface stress method program, and introduces the radial basis function network (RBFN) intelligent response surface method, which has the characteristics of strong adaptability, high fault tolerance, greater flexibility, and strong nonlinearity. With the response surface function built for the general framework for an intelligent response surface methodology for the reliability analysis of a 3D slope system, the reliability analysis of a slope is extended from 2D to 3D. The process involves generating samples, creating an intelligent response surface, and calculating the failure probability of the 3D slope system. Through the reliability analysis of typical examples and a comparison with Monte Carlo simulation (MCS) method results, the accuracy, feasibility, and superiority of the proposed intelligent response surface methodology for application to the reliability analysis of a 3D slope system were verified. A comparison with the analysis results of the 3D slope shows that the 2D slope stability calculation is too conservative and seriously overestimates the probability of slope instability. Furthermore, 3D analysis can consider the true stress state of a slope and is able to specify a definite range for a sliding body, thereby providing a basis and reference for determining reinforcement regions.

Finite difference probabilistic slope stability analysis based on collocation-based stochastic response surface method (CSRSM)

Significant damage has been observed due to the landslide along the East–West motorway section, located near Ain Bouzian–El-Harrouch region, Northeast Algeria. In this paper, a probabilistic study was carried out to assess the stability of a slope, with a total height of 60 m and varying inclination angles. Two cases were considered with and without the presence of the groundwater table. To investigate the failure probability of the slope, the collocation-based stochastic response surface method was employed. The input random parameters were the Young modulus E, cohesion C, and friction angle φ, where probabilistic system response is the factor of safety. To identify the effective contribution of each random parameter in the variability of the system response, a global sensitivity analysis based on Sobol indices was conducted. Also, a parametric study was realized to inspect the effect of input geotechnical parameter variations on the reliability of slope stability. The result showed that the slope reliability analysis is strongly influenced by the inherent variability of friction angle and hydrogeological ground conditions.

Barton-Bandis criterion-based system reliability analysis of rock slopes

Several potential failure modes generally exist in rock slopes because of the existence of massive structural planes in rock masses. A system reliability analyses method for rock slopes with multiple failure modes based on nonlinear Barton-Bandis failure criterion is proposed. The factors of safety associated with the sliding and overturning failure modes are derived, respectively. The validity of this method is verified through a planar rock slope with an inclined slope top and tension crack. Several sensitivity analyses are adopted to study the influences of structural-plane parameters, geometric parameters, anchoring parameters and fracture morphology on the rock slopes system reliability.

Jackknifing for modeling sampling properties of soil statistics for geotechnical reliability analysis

This study introduces the jackknife method to model the statistical uncertainty in the joint distribution of geotechnical parameters derived from a small sample and its effect on geotechnical reliability. A numerical example using simulated data is adopted to validate the accuracy of the jackknife method. The following three real examples are studied to illustrate and demonstrate the jackknife method: (1) the reliability analysis of an infinite slope, (2) the serviceability limit state (SLS) reliability analysis of piles, and (3) the reliability analysis of a single-layered slope. The results indicate that sample statistics and resulting reliability index estimated from a small sample show visible statistical uncertainty. The jackknife method has a good accuracy and efficiency in modeling the sampling properties of sample statistics and resulting reliability index. The jackknife method overcomes the drawback of inefficiency associated with the bootstrap method, and can be applied to both the simple and complex geotechnical problems. By applying the jackknife method, an interval estimate of reliability index at a specified confidence level instead of a point estimate of reliability index is derived. The interval estimate of reliability index not only includes the point estimate of reliability index, but also quantifies the upper and lower bounds within which the point estimate of reliability index may vary. A larger sample size produces smaller statistical uncertainty in sample statistics and resulting reliability index, which provides an incentive for geotechnical engineers to draw more data of geotechnical parameters in a typical site investigation.

Importance sampling for system reliability analysis of soil slopes based on shear strength reduction

ABSTRACT Probabilistic methods are increasingly used to quantify the effect of uncertainties on slope stability assessment. The shear strength reduction method can be used for slope stability analysis without prior assumption on the shape of the failure surfaces, and hence is an attractive method for slope stability analysis. Since it is computationally more demanding than limit equilibrium methods, reliability analysis based on the shear strength reduction method is often considered as computationally challenging. In this paper, an importance sampling method is suggested to conduct system reliability analysis of soil slopes based on the shear strength reduction method. In the suggested method, the design points are first identified using a response surface method. Then, a sampling function is constructed centred at the design points. An efficient method is used to identify the failure samples based on the shear strength reduction method. Results from two illustrative examples show that the suggested importance sampling method is an effective and robust tool for system reliability analysis of soil slopes.

Reliability Analysis of Layered Soil Slopes Considering Different Spatial Autocorrelation Structures

It is widely recognized that different geological formations often vary differently in space. Therefore, soil properties from different layers should be modeled by different autocorrelation functions (ACFs) to reflect such soil heterogeneity. However, the same ACFs are frequently used for different soil layers in slope reliability analysis for simplicity purpose in the literature. The present work is a study on the effects of ACFs on the reliability analysis of layered soil slopes, where the soil properties of different layers are considered by different ACFs. Five commonly used classical ACFs and the non-classical Whittle–Matérn model were investigated in this study. Cholesky decomposition and Monte Carlo simulation were used to simulate the spatial variability of the soil properties and estimate the probability of failure (Pf) of slopes, respectively. Illustrative examples with various parametric studies show that when the soil properties from different layers are characterized by the same ACFs, the Pf of the studied slopes is comparable with that estimated using different ACFs for different soil layers. This indicates that the type of ACF has only a small impact on the slope reliability assessment. However, the Pf may be underestimated by the single exponential ACF and overestimated by the cosine exponential ACF. The scale of fluctuation of the soil properties influences the slope reliability more than the ACFs. In addition, the smoothness parameter in the non-classical model has a significant influence on the reliability of the slope, where Pf increases with the increase of the smoothness parameter.

Reliability analysis of slopes stabilised with piles using response surface method

Slopes stabilised with piles are seldom analysed considering uncertainties in the parameters of the pile-slope system. Reliability analysis of the pile-slope system quantifies the degree of uncertainties and evaluates the safety of the system. In the present study, the reliability analysis of a slope stabilised with piles is performed using the first-order reliability method (FORM) based on Hasofer-Lind approach. The implicit performance function associated with the factor of safety (FS) of the slope is approximated using the response surface method. The analyses are carried out considering the design matrices formulated based on both the 2k factorial design augmented with a centre run (2k fact-centred design) and face-centered cube design (FCD). The finite element method is used as the deterministic model to compute the FS of the pile-slope system. Results are compared with the results of the Monte Carlo simulation. It is observed that the optimum location of the row of piles is at the middle of the slope to achieve the maximum FS. The results show that the reliability of the system is not uniform for different pile configurations, even if the system deterministically satisfies the target factor of safety (FSt) criterion. The FSt should be selected judiciously as it is observed that the reliability of the system changes drastically with the FSt level. The results of the 2k fact-centred design and FCD are in good agreement with each other. The procedure of the FCD is computationally costly and hence the use of 2k fact-centred design is recommended, provided the response of the system is sufficiently linear over the factorial space.

Reply to the discussion on “Probabilistic characterization of the soil-water retention curve and hydraulic conductivity and its application to slope reliability analysis”

Reply to the discussion on “Probabilistic characterization of the soil-water retention curve and hydraulic conductivity and its application to slope reliability analysis”

Discussion on “Probabilistic characterization of the soil-water retention curve and hydraulic conductivity and its application to slope reliability analysis” by L. Wang, L. Tang, Z. Wang, H. Liu, and W. Zhang

The statistical characteristics of the soil-water retention curve parameters has an important role in unsaturated soil slope stability analysis under rainfall. However, such its application is questionable for reliability analysis of a hypothesis slope considering a steady state condition of seepage analysis.

Reliability analysis of slopes using the improved stochastic response surface methods with multicollinearity

It is known that uncertainties in geotechnical engineering are unavoidable. Slope reliability analysis is vital because the failure of a slope may cause great loss. In the reliability analysis of slopes, the stochastic response surface method (SRSM) provides an effective way to address the non-convergence of calculation results and has the advantage of high accuracy for highly nonlinear problems. However, multicollinearity, defined as the existence of exact correlations or highly correlated relationships among the explanatory variables, is not considered in the SRSM, leading to local solutions. To address the multicollinearity existing in slope reliability analysis, the least absolute shrinkage and selection operator (Lasso)-based SRSM, the ridge regression-based SRSM, the elastic net regression-based SRSM, and the stepwise regression-based SRSM are proposed in this paper. Monte Carlo simulations and the traditional SRSM are used to validate the accuracy of the proposed methods. It is found from the numerical results that the stepwise regression-based SRSM is the most competitive among the four proposed methods in addressing the uncertainties of slope stability due to its accuracy and efficiency.

Slope Reliability Analysis Considering Variability of Shear Strength Parameters

The traditional safety factor method cannot consider the uncertainty in the slope system, and the reliability theory based on probability and statistics can more objectively evaluate the degree of influence on the slope, which has become a trend of slope stability evaluation. In this paper, the spatial variability of soil strength parameters cohesion c, internal friction angle φ is considered to build a stochastic variable model, the Monte Carlo method is adopted to study the effect of reliability index β for two parameters. The results shows that, the soil parameters c, have a significant impact on the reliability index β, however, φ is more sensitive in the calculation process which have a greater impact on the calculation of the safety factor and reliability index. The steeper slope has less reliability index. For high variability of the steep slope, a reliable indicator of φ variability is no longer sensitive. In actual engineering, the correlation of soil parameters c and is often not considered, so the reliable index calculated in this case is smaller than the actual one.

Probabilistic characterization of the soil-water retention curve and hydraulic conductivity and its application to slope reliability analysis

Abstract Soil-water retention curve (SWRC) and hydraulic conductivity are two indispensable inputs in unsaturated seepage analysis for analyzing water flow in unsaturated soils, which can be obtained through direct measurements. Generally, the test data of the SWRC and unsaturated hydraulic conductivity obtained from direct measurements are limited because such measurements are time-consuming and costly. Thus, it is hard to determine an SWRC and hydraulic conductivity with certainty from a probabilistic perspective. How to quantify the uncertainty in them remains an unresolved question. This paper aims to propose a Bayesian approach to characterize the SWRC and hydraulic conductivity. The proposed approach is illustrated through a slope example. Results show that the proposed Bayesian approach is not only able to quantify the uncertainty in the estimated SWRC and hydraulic conductivity, but also capable of predicting the SWRC and hydraulic conductivity with reasonable accuracy. This facilitates the determination of both the SWRC and hydraulic conductivity in geotechnical applications for engineers by making use of the available test data. Moreover, the proposed approach offers a viable way to provide a robust estimate of slope failure probability by virtue of Bayesian model averaging, allowing SWRC parameter uncertainty and model selection uncertainty to be considered simultaneously in reliability analysis. This addresses the dilemma of over-reliance on a single SWRC model in geotechnical reliability analysis involving unsaturated soils. It is worthy to note that among the four candidate models, three of them belong to one category which may have similar abilities, other parametric models in different categories need to be considered in future research.

Probabilistic Assessment for Slope Using the Generalized Chebyshev Inequalities

AbstractThis paper aims to estimate the probability of failure (Pf) of slope based on Chebyshev inequalities. Traditionally, for a slope reliability analysis, the supposed distribution types of she...

Reliability analysis of slopes considering spatial variability of soil properties based on efficiently identified representative slip surfaces

Abstract Slope reliability analysis considering inherent spatial variability (ISV) of soil properties is time-consuming when response surface method (RSM) is used, because of the “curse of dimensionality”. This paper proposes an effective method for identification of representative slip surfaces (RSSs) of slopes with spatially varied soils within the framework of limit equilibrium method (LEM), which utilizes an adaptive K-means clustering approach. Then, an improved slope reliability analysis based on the RSSs and RSM considering soil spatial variability, in perspective of computation efficiency, is established. The detailed implementation procedure of the proposed method is well documented, and the ability of the method in identifying RSSs and estimating reliability is investigated via three slope examples. Results show that the proposed method can automatically identify the RSSs of slope with only one evaluation of the conventional deterministic slope stability model. The RSSs are invariant with the statistics of soil properties, which allows parametric studies that are often required in slope reliability analysis to be efficiently achieved with ease. It is also found that the proposed method provides comparable values of factor of safety (FS) and probability of failure (Pf) of slopes with those obtained from direct analysis and literature.

Copula-based approach coupling information diffusion distribution for slope reliability analysis

Uncertain probability distributions and interdependencies of geotechnical parameters have profound impacts on slope reliability analysis, especially under incomplete probabilistic information. In this paper, a copula-based approach coupling information diffusion (ID) distribution is proposed to reduce such uncertainties. Firstly, ID theory is introduced to establish marginal distribution of strength parameter. Copulas are subsequently performed to characterize the corresponding dependence structure and link the proposed ID margins. At last, equivalent samples are simulated and plugged into Monte Carlo simulation (MCS) to estimate slope reliability. The effect of the proposed method is validated by a slope case. Results show that the proposed ID algorithm holds remarkable superiority in estimating the marginal distribution with considering random volatility of geotechnical parameter. For slope reliability, both marginal distributions and interdependencies of geotechnical parameters barely affect the factor of safety (FS) but significantly impact failure probability (pf). Unreasonable estimation of probability distribution may lead to large deviation in calculation of slope failure probability. Comparably, the proposed copula-based approach coupling ID distribution can give a reasonable and robust reliability results with considering random volatility.

Accounting for 2D Spatial Variation in Slope Reliability Analysis

AbstractWithin the framework of the limit equilibrium method of slices, a slope reliability analysis accounting for two-dimensional (2D) spatial variation requires a model to discretize a random fi...

Slope reliability analysis using the geotechnical random field method

Inherent soil variability, measurement errors and transformation uncertainty are the three primary geotechnical uncertainties. Random field theory (RFT) is the most versatile and practical tool utilised in reliability analyses. RFT has conventionally been considered to model the inherent variability, with other geotechnical uncertainty sources usually ignored, although some degree of measurement error is incorporated in geotechnical test measurements. In addition, RFT induces a series of non-geotechnical uncertainties during simulation, including correlation function errors and uncertainty in statistical parameters. In this paper, a geotechnical random field algorithm is introduced for the evaluation of slope reliability considering different geotechnical uncertainty sources. Accordingly, a drained c–ϕ slope was considered for reliability analysis considering geotechnical uncertainties. It was shown that, by adding measurement error and transformation uncertainty to the inherent variability, the probability of failure is noticeably increased. Slope reliability analyses using the algorithm proved to give conservative results in comparison with approximate, stationary and non-stationary stochastic methods in the face of inherent soil variability.

Subset simulation for efficient slope reliability analysis involving copula-based cross-correlated random fields

This study proposes a subset simulation (SS)-based approach for efficient slope reliability analysis involving copula-based cross-correlated random fields of cohesion (c) and friction angle (ϕ) of soils. First, the copula theory for modeling the cross-correlation between c and ϕ is briefly introduced. The algorithms for generating the copula-based cross-correlated random fields of c and ϕ are detailed. Then, the SS for efficient slope reliability analysis involving copula-based cross-correlated random fields of c and ϕ is explained. Finally, two slope examples with the same geometry but different sources of probability information are presented to illustrate and demonstrate the proposed approach. The results indicate that the proposed approach has both good accuracy and efficiency in slope reliability analysis involving the copula-based cross-correlated random fields of c and ϕ at low failure probability levels. The copula theory for characterizing the cross-correlated random fields can consider both the Gaussian and non-Gaussian dependence structures between c and ϕ. The copula selection has a significant impact on slope reliability with spatially variable c and ϕ. The probabilities of slope failure produced by different copulas differ considerably. This difference increases with decreasing probability of slope failure. The commonly-used Gaussian copula may lead to a significant underestimate of the probability of slope failure. The reasonable identification of the best-fit copula for characterizing the cross-correlated random fields of c and ϕ based on the test data is highlighted in practical slope reliability analysis.