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Landslide is a uniquely dynamic large-deformation process that can present serious threat to human lives and infrastructures. The natural soil properties often exhibit inherent spatial variability, which affects the landslide behavior significantly. This paper focuses on combined Monte Carlo simulation and three-dimensional (3D) dynamic large-deformation finite-element (LDFE) analysis using the coupled Eulerian-Lagrangian method to investigate the whole runout process of landslide induced by the earthquake in spatially varying soil. The results from LDFE analysis show that the mean value of runout distance in spatially varying soil is significantly higher than that of the deterministic value obtained from a homogeneous slope due to the slope failure developed along the weakest path in soils. The mean runout distance increases and converges with increasing slope length in 3D-LDFE stochastic analysis. The advantages and necessities of 3D-LDFE analysis were illustrated by comparing it with two-dimensional (2D) LDFE analysis of landslide in spatially varying soil. The results show that the calculated mean runout distance using 3D-LDFE method is at least 16.1% higher than that calculated using 2D-LDFE analysis. Finally, a linear regression formula was established to estimate the mean runout distance of landslide due to horizontal inertia acceleration. Such a formula may facilitate the risk assessment of landslide in practical engineering.

Artificial ground freezing (AGF) has been widely used as a temporary soil stabilization and waterproofing technique in geotechnical practices (e.g., tunnel construction). Many sources of uncertainty exist during AGF. Firstly, groundwater seepage flow can adversely affect the freezing efficacy. Secondly, freeze pipe inclination inevitably occurs during installation, which is likely to yield an unfrozen path and elevate construction risk. Thirdly, as a key soil parameter, the spatial variability in thermal conductivity can also affect the freezing process. In this work, a unit cell model of freeze pipes is established by a coupled thermo-hydraulic finite element method to examine the effects of these sources of uncertainty. The pipe inclination is considered in the unit cell model by prescribing various values of freeze pipe spacing. The thermal conductivity of soil solid is simulated as a three-dimensional lognormal random field to account for the spatial variability of soil. Results are tabulated to evaluate the additional freezing time required in the AGF system due to the existence of these uncertainties. The findings are capable of determining a reasonable range of freeze pipe spacings and the corresponding critical seepage velocity, and can offer practitioners a rule of thumb for estimating freeze pipe spacing.

It is common to encounter a diversity of design scenarios with different values and statistics of loads and/or geotechnical parameters in design practice of engineering geology and geotechnical engineering. Maintaining a uniform level of reliability close to the target reliability under various design scenarios is a key goal of geotechnical reliability-based design (RBD), which is a fundamental challenge for semi-probabilistic RBD methods because a wide range of design scenarios can be involved in geotechnical design practice. This paper develops an efficient RBD updating approach under a full probabilistic design framework, which combines direct Monte Carlo simulation (MCS) with a sample reweighting technique to efficiently obtain final designs under different design scenarios by a single simulation run, avoiding repeatedly performing direct MCS for different design scenarios. This leads to considerable computational saving, particularly as the number of design scenarios concerned is large. The proposed approach is illustrated through a pad foundation design example and a rock slope design example. Results show that the proposed approach properly calculates failure probabilities of different possible designs under different design scenarios by a single run of direct MCS, provided that sufficient failure samples are generated in the direct MCS run to cover failure regions of different design scenarios, which are used to determine the updated final designs. Reliability of the final designs obtained from the proposed approach for various design scenarios is generally uniform and is close to the target reliability level prescribed for the design purpose. Moreover, the proposed approach is able to link site investigation efforts to design saving in an efficient and straightforward manner. It also provides insights into the sensitivity of the final design to uncertain parameters involved in design.

This paper presents a new copula-based earthquake early warning (EEW) decision-making strategy, aiming to characterize missed-alarm and false-alarm probabilities for an on-site EEW and determine an optimum threshold at which EEW should be set off with the lowest missed-alarm and false-alarm probabilities combined. On the basis of an existing PD3-PGV (PD3: peak ground displacement within the first three seconds after P-wave arrives; PGV: peak ground velocity) on-site EEW, the analysis shows that a copula model consisting of the Lognormal distribution, Weibull distribution, and Frank copula can satisfactorily model the PD3-PGV joint probability distribution. Accordingly, the optimum PD3 triggering thresholds for different PGV warning thresholds from 5 to 35 cm/s are presented for future references in the use of the PD3-PGV on-site EEW with maximum reliability.

This study performs a statistical analysis on shear strength parameters of rock mass for hydropower projects in China. First, a large database comprising 1139 groups of field direct shear tests on rock mass from 100 hydropower projects in China is compiled. Second, the best-fit probability distributions of shear strength parameters are identified using Akaike Information Criterion (AIC). Finally, new characteristic and design values of shear strength parameters of rock mass are suggested. The results indicate that shear strength parameters increase with increasing quality of rock mass. In particular, the shear strength parameters of discontinuities in rock mass are significantly smaller than those of rock mass itself. The discontinuities in rock mass should be paid more attention during geological investigations. There is a large variation in shear strength parameters, and the variability of cohesion is larger than that of tangent of friction angle. The derived statistics of shear strength parameters of rock mass in this study provide an update to the values suggested in Chinese codes, and serve as a reference for hydropower projects with few site-specific test data and similar geological conditions for partial-factor design of dams.

Determining the sensitivity of monitoring variables is essential to field monitoring design for effectively monitoring the safety and reliability levels of geotechnical structures in uncertain environment. Reliability sensitivity analysis of monitoring variables provides a rational approach for identifying sensitive monitoring variables and is capable of accounting for geotechnical uncertainties. It, however, can be computationally expensive, especially when sophisticated numerical models (e.g., finite difference model, FDM) are involved and repeated simulation runs are required. This paper proposes a reliability sensitivity analysis method that leverages on the robustness of direct Monte Carlo simulation (MCS) and the Bayesian Updating with Structural Reliability Methods. The proposed approach allows performing the reliability sensitivity analysis of a monitoring variable by a single run of direct MCS, avoiding repeated simulation runs for different possible observational values of a given monitoring variable. Illustrative examples demonstrate the capability of the proposed approach in identifying the most sensitive monitoring variables among candidates. It is possible to achieve a significant reduction in the number of evaluations of numerical models for reliability sensitivity analysis of monitoring variables using the proposed approach.

This paper develops a Bayesian approach for model comparison and characterization of the bivariate distribution of c′ and ϕ′ using limited site-specific data. The copula approach is presented to model the bivariate distribution of c′ and ϕ′. The Bayesian model comparison method is developed to select the most probable bivariate distribution model of c′ and ϕ′. The most probable model is used to characterize the joint probability density function (PDF) of c′ and ϕ′ under the Bayesian framework. The developed approach is illustrated and validated using real data of c′ and ϕ′ for clays from the core wall of Xiaolangdi rockfill dam in China.

This paper aims to investigate the impact of sample size on geotechnical probabilistic model identification. First, the copula approach is presented to model the bivariate distribution of geotechnical parameters. Thereafter, the AIC scores are adopted to identify the best-fit marginal distribution and copula. Second, the variation of AIC scores because of small sample size is investigated using simulated data. Finally, the impact of the variation of AIC scores on identification of the best-fit marginal distribution and copula is examined. The minimum sample sizes for geotechnical data are also suggested to obtain a correct identification of the probabilistic models. The results indicate that the AIC scores estimated from a small sample exhibit large variation. The variation of the AIC scores has a significant impact on probabilistic model identification. The marginal distributions and copulas have a low percentage of correct identification when sample size is small. The percentages of correct identification for the marginal distributions and copulas increase with increasing sample size. The correlation coefficient between geotechnical parameters has a much larger impact on probabilistic model identification than the COV of geotechnical parameters. The suggested minimum sample sizes for geotechnical data are useful for guiding practical geotechnical site investigation.

The random field theory is often utilized to characterize the inherent spatial variability of material properties . In order to incorporate sampled data from site investigations or experiments into simulations, a patching algorithm is developed to yield a conditional random field in this study. Comparison is conducted between the proposed algorithm and the conventional Kriging algorithm to underscore the former’s advantages in simulating material properties with limited sampled data. Unlike the Kriging algorithm that interpolates the entire spatial domain, the proposed algorithm restricts the influence domain of sampled data within a reasonable range, which is determined as a function of the scale of fluctuation. The simulated conditional random field via the proposed algorithm is stationary in mean and variance; thus, it would be preferable for situations with a few known data. Additionally, a tunnel excavation model is considered to exemplify the effectiveness of the proposed algorithm. By virtue of Monte-Carlo simulations, maximum tunnel convergence modeled by unconditional and conditional random fields is analyzed in a statistical manner. The results indicate that the proposed algorithm can effectively reduce the uncertainty of prediction in responses. Furthermore, the proposed algorithm is also applicable with a sparse sampling pattern.

Quantitative risk assessment of landslide-induced surges is often a prerequisite for formulating rational strategies to reduce the disaster severity degree of surrounding residents and infrastructures facilities. In this study, soils are simulated by a random field and the random finite element method is utilized to obtain several relevant parameters (e.g., volume of slide mass, slide velocity, inclination angle and length of slip surface) of reservoir bank slopes for evaluating the landslide-induced surges hazard. A modified risk index is proposed to evaluate the landslide-induced surges hazard, which is more straightforward than the initial surge height. Quantitative risk assessment is conducted based in the introduced risk index for the occurrence of reservoir bank landslides. Compared with a uniform soil slope, a slope considering spatially variable soils generally has a higher initial surge height, which implies a more disastrous surge hazard.

A simplified reliability analysis method is proposed for efficient full probabilistic design of soil slopes in spatially variable soils. The soil slope is viewed as a series system comprised of numerous potential slip surfaces and the spatial variability of soil properties is modelled by the spatial averaging technique along potential slip surfaces. The proposed approach not only provides sufficiently accurate reliability estimates of slope stability, but also significantly improves the computational efficiency of soil slope design in comparison with simulation-based full probabilistic design. It is found that the spatial variability has considerable effects on the optimal slope design.

An efficient analytical method for quantifying the correlation between performance functions of different slope failure modes in spatially variable soils is proposed, and its performance in slope system reliability analysis is investigated. First, a new correlation coefficient (NCC) is proposed to evaluate the correlation among slope failure modes considering spatial variability. For comparison and verification, the simulation-based correlation coefficient (SCC) is also presented. Second, appying these two types of correlation coefficients, the effects of soil spatial variability on the representative slip surfaces (RSSs) and the system probability of slope failure are investigated using different system reliability methods, including a probabilistic network evaluation technique, a risk aggregation approach, and a bimodal bounds method. A single-layered cohesive slope is investigated to illustrate the validity of the proposed NCC. The results indicate that the proposed NCC can efficiently and accurately quantify the correlation among slope failure modes considering soil spatial variability. The number of RSSs indicated by the NCC is in good agreement with the number obtained using the SCC. The system failure probabilities of slope stability obtained with the SCC and the NCC using a risk aggregation approach are generally comparable. Also, the system reliability bounds of slope stability obtained using the NCC are relatively close together and comparable to those obtained using the SCC. Thus, the NCC shows good performance when evaluating the correlation among slope failure modes, and was effectively applied to analyze a single-layered cohesive slope considering soil spatial variability.

Geological uncertainty appears in the form of one soil layer embedded in another or the inclusion of pockets of different soil type within a more uniform soil mass. An efficient coupled Markov chain (CMC) model has been proposed to simulate geological uncertainty in the past. This model, however, cannot be directly applied to geotechnical engineering. The primary problem lies in the estimation of horizontal transition probability matrix (HTPM), one key input of the CMC model. The HTPM is difficult to estimate due to the wide spacing between boreholes in the horizontal direction. Hence, a practical method for estimating the HTPM is proposed based on borehole data. The effectiveness of this method can be evaluated using the following approach. Several virtual borehole outcomes are simulated using artificial HTPM. The HTPM estimated from the virtual boreholes is compared with the artificial HTPM. Boreholes collected from Perth, Australian are adopted to illustrate the proposed method for HTPM estimation. The results show that the estimated HTPM agrees well with the artificial HTPM if the artificial (real) HTPM or vertical transition probability matrix (VTPM) is strongly diagonally dominant (diagonal element is larger than the sum of off-diagonal elements). The estimated HTPM is not sensitive to the borehole layout scheme as the diagonal dominancy of the HTPM is strong.

This paper aims to develop an efficient geotechnical reliability-based design (RBD) approach using Monte Carlo simulation (MCS). The proposed approach combines a recently developed MCS-based RBD approach, namely expanded RBD approach, with an advanced MCS method called “Subset Simulation (SS)” to improve the computation efficiency at small probability levels that are often concerned in geotechnical design practice. To facilitate the integration of SS and expanded RBD, a generalized surrogate response f is proposed to define the driving variable, which is a key parameter in SS, for expanded RBD of geotechnical structures (e.g., soil retaining structures and foundations). With the aid of the proposed surrogate response, failure probabilities of all the possible designs in a prescribed design space are calculated from a single run of SS. Equations are derived for integration of the surrogate response-aided SS and expanded RBD, and are illustrated using an embedded sheet pile wall design example and two drilled shaft design examples. Results show that the proposed approach provides reasonable estimates of failure probabilities of different designs using a single run of the surrogate response-aided SS, and significantly improves the computational efficiency at small probabilities levels in comparison with direct MCS-based expanded RBD. The surrogate response-aided SS is able to, simultaneously, approach the failure domains of all the possible designs in the design space by a single run of simulation and to generate more complete design information, which subsequently yields feasible designs with a wide range of combinations of design parameters. This is mainly attributed to the strong correlation between the surrogate response and target response (e.g., factor of safety) of different designs concerned in geotechnical RBD.

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).

Geological uncertainty appears in the form of one soil layer embedded in another or the inclusion of pockets of different soil type within a more uniform soil mass. Uncertainty in factor of safety (FS) and probability of failure (Pf) of slope induced by the geological uncertainty is not well studied in the past. This paper presents one approach to evaluate the uncertainty in FS and Pf of slope in the presence of geological uncertainty using borehole data. The geological uncertainty is simulated by an efficient coupled Markov chain (CMC) model. Slope stability analysis is then conducted based on the simulated heterogeneous soils. Effect of borehole layout schemes on uncertainty evaluation of FS and Pf is investigated. The results show that borehole within influence zone of the slope is essential for a precise evaluation of FS statistics and Pf. The mean of FS will converge to the correct answer as the borehole number increases.

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.

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.

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.

This study introduces the vine copula model to model the multivariate distribution of multiple soil parameters. First, the conventional bivariate and multivariate copulas are presented to model the joint probability distribution of soil parameters. Then, the procedure for modeling the multivariate distribution of soil parameters using the vine copula model is explained. Finally, two soil databases (CLAY/5/345 and CLAY/6/535) containing complete multivariate data of multiple soil parameters are studied to demonstrate the validity of the vine copula model. The results indicate that there exist different levels of correlation between all pairs of soil parameters. The dependence structure among multiple soil parameters shows obvious diversity and non-Gaussianity, which cannot be adequately characterized by the commonly-used multivariate normal distribution. The vine copula model performs well in modeling the multivariate distribution of multiple soil parameters. It can effectively consider the diversity and non-Gaussianity in the dependence structure among multiple soil parameters. In comparison with the original distributions, the coefficients of variation (COVs) for the conditional distributions of soil parameters may be significantly reduced. The incorporation of more information may produce smaller COVs for the conditional distributions. The reduced COVs for soil parameters can be eventually converted to cost savings in the reliability-based design of geotechnical structures, which provides an incentive for geotechnical engineers to collect more information of soil parameters from various sources.

It is well known that the shear strength parameters [cohesion (c) and friction angle (ϕ)] are important parameters for evaluating deformation and stability of geotechnical structures, such as slope stability, bearing capacity of foundations, and earth pressure of retaining walls. As far as the reliability analysis of these geotechnical structures is concerned, the shear strength parameters are typically treated as uncertain parameters (Griffiths et al. 2011; Cherubini 2000; Abd Alghaffar and Dymiotis-Wellington 2007). Furthermore, it is widely accepted that c and ϕ are negatively correlated in the literature (e.g., Low 2007; Li et al. 2011; Tang et al. 2012, 2013a). To evaluate the reliability of geotechnical structures exactly, the joint cumulative distribution function (CDF) or probability density function (PDF) of shear strength parameters should be known. It is concluded by the previous studies (e.g., Low 2007; Li et al. 2011; Tang et al. 2012, 2013a) that the negative correlation between cohesion and friction angle has a significant effect on geotechnical reliability and ignoring such a correlation would lead to an overestimate of the probability of failure.

This paper presents joint probability modeling for debris flows’ maximum impact pressure and total sediment discharge through the use of the copula theory that has been proven useful to joi...

This paper aims to propose a bootstrap method for characterizing the uncertainty in probabilistic models and its effect on geotechnical reliability. First, the copula theory is briefly introduced. Second, both the uncertainties in parameters and type of the best-fit marginal distributions and copulas are characterized by the bootstrap method. Finally, four load-test datasets of load-settlement curves of piles are used to illustrate the proposed method. The serviceability limit state reliability analysis of piles is presented to illustrate the practical application of the proposed method. The results indicate that the bootstrap method can effectively characterize the uncertainty in probabilistic models derived from a small sample. Through bootstrapping, the uncertainties in both the parameters and type of the specified probabilistic models are simultaneously incorporated into geotechnical reliability analyses. The probability of failure of piles is represented by a confidence interval at a specified confidence level instead of a single fixed probability, which enables the engineers to make a more informed decision.

The objective of this paper is to investigate the effect of copulas for constructing the bivariate distribution of shear strength parameters on system reliability of geotechnical structures. First, the bivariate distribution of shear strength parameters is constructed using copulas. Second, the implementation procedure of system reliability analysis using direct Monte Carlo simulation (MCS) is developed. Finally, the system reliability of a retaining wall and a rock wedge slope is presented to explore the effect of copula selection on geotechnical system reliability. The results indicate that the system reliability of geotechnical structures under incomplete probability information could not be determined uniquely because the bivariate distribution of cohesion and friction angle with given marginal distributions and correlation coefficient could not be determined uniquely. The copulas for modeling dependence structure between cohesion and friction angle have a significant influence on the system reliability of geotechnical structures. Such an influence includes two phases separately. The first phase is that the dependence structure between shear strength parameters characterized by copulas affects the reliability of single failure mode, depending on the marginal distributions, dependence structure between shear strength parameters, and reliability level of each failure mode. The second phase is that the reliability of each failure mode influences on system reliability, only depending on reliability level of each failure mode and correlations among various failure modes. It is important to distinguish between the effect of copula selection on reliability of each failure mode and that on geotechnical system reliability.

Slope reliability under incomplete probability information is a challenging problem. In this study, three copula-based approaches are proposed to evaluate slope reliability under incomplete probability information. The Nataf distribution and copula models for characterizing the bivariate distribution of shear strength parameters are briefly introduced. Then, both global and local dispersion factors are defined to characterize the dispersion in probability of slope failure. Two illustrative examples are presented to demonstrate the validity of the proposed approaches. The results indicate that the probabilities of slope failure associated with different copulas differ considerably. The commonly used Nataf distribution or Gaussian copula produces only one of the various possible solutions of probability of slope failure. The probability of slope failure under incomplete probability information exhibits large dispersion. Both global and local dispersion factors increase with decreasing probability of slope failure, especially for small coefficients of variation and strongly negative correlations underlying shear strength parameters. The proposed three copula-based approaches can effectively reduce the dispersion in probability of slope failure and significantly improve the estimate of probability of slope failure. In comparison with the Nataf distribution, the copula-based approaches result in a more reasonable estimate of slope reliability.

In order to conduct the numerical simulation of rock and soil slope strengthened by the micro-pile with reinforced bar, FEM strength reduction is combined with the program of FLAC which possesses the function of analyzing tensile and shear failure. The micro-pile is under plastic state due to its mechanic features. Solid element and ideal elastic-plastic constitutive model of Mohr-Coulomb are applied as the rock and soil mass and pile. The calculation model can work out the safety factor of slope, the dynamic changing process of pile deformation and failure, the failure mode of slope after strengthened by micro-pile. Then the structural element of pile is used to simulate piles, which can obtain the bending moment and shear force. The beam element is used to simulate the coupling beam. So the layout principle of inner force before the failure can be calculated.

Due to the special mesh structure, geo-grid material can avoid local subsidence of filling material, reduce uneven settlement of soil mass to the largest degree and improve the whole stability of soil mass, so the reinforced earth-retaining wall with geo-grid is widely used in engineering. Meanwhile, researches on its dynamic characters are not enough and it is hard to judge the whole stability of reinforced earth-retaining wall under seismic condition. When unstable failure happens, the location of failure surface can hardly be identified. These disadvantages have seriously limited the development of this supporting method and cause unsafe potentials for engineering. Based on the FEM strength reduction dynamic analysis and combined with practical engineering, the paper conducts stability analysis on the reinforced earth-retaining wall of geo-grid under seismic condition and the research achievements provide a new thinking for the seismic design of reinforced earth-retaining wall with geo-grid.

The traditional slope reliability methods may not efficiently carry out slope reliability considering inherent spatial variability (ISV) of soil properties. This paper aims to develop a two-stage dimension reduction method for efficient slope reliability analysis considering ISV of soil properties. First, the framework of slope reliability analysis based on surrogate model is provided. Second, the two-stage dimension reduction method is proposed systematically. Thereafter, the implementation procedure for slope reliability analysis using the proposed method is summarized. The validity of two-stage dimension method is illustrated with a multiple-layered soil slope. The results indicate that the proposed two-stage dimension method can reduce the number of original model evaluations significantly and improve the efficiency of slope reliability analysis considerably. The surrogate model based on the proposed method is proved to be unbiased. The loss of accuracy in modeling ISV has a slight influence on the minimum factor of safety, sliding path and sliding volumes of slope. The proposed method can yield sufficiently accurate reliability results, which is more efficient than direct Monte Carlo simulation. For the multi-layered soil slope reliability considering ISV of soil properties examined in this study, the proposed method performs well in terms of accuracy and efficiency.

We consider a generalized fourth-order nonlinear Schrödinger (NLS) equation. Based on the ansatz method, its bright, dark single-soliton is constructed under some constraint conditions. Furthermore, combining the Riccati equation extension approach, we also derive some exact singular solutions. With several parameters to play with, we display the dynamic behaviors of bright, dark single-soliton. Finally, the condition for the modulational instability (MI) of continuous wave solutions for the equation is generated. It is hoped that our results can help enrich the nonlinear dynamics of the NLS equations.

The paper adopts the interface element to simulate the joints so as to systematically and quantitatively study the deformation around tunnel, the mechanic state of lining and the stability under different inclining angles of joints. The result shows that the deformation around tunnel deteriorates mainly along the joint during the inner convergence effects of surrounding rock. When the inclining angle α=45°, the deformation around the tunnel is most serious, followed by that when α=90°, α=60°, α=30° and α=0°. At the same time, the influence of inclining angle on the distribution of the axial force of lining is comparatively small. But the distribution of bending moment and shear change obviously where the joints penetrate the tunnel. The tunnel stability of surrounding rock is the poorest when α=90° and the tunnel is most stable when α=0°. The stability of surrounding rock changes little when α is between 30° and 60°. The research result provides an effective calculation method for the forecast of deformation, the design of structure and the stability analysis of jointed tunnel. It is also helpful for the monitoring of construction and calculation of jointed tunnel in the future.

A copula-based method is presented to investigate the impact of copulas for modeling bivariate distributions on system reliability under incomplete probability information. First, the copula theory for modeling bivariate distributions as well as the tail dependence of copulas are briefly introduced. Then, a general parallel system reliability problem is formulated. Thereafter, the system reliability bounds of the parallel systems are generalized in the copula framework. Finally, an illustrative example is presented to demonstrate the proposed method. The results indicate that the system probability of failure of a parallel system under incomplete probability information cannot be determined uniquely. The system probabilities of failure produced by different copulas differ considerably. Such a relative difference in the system probabilities of failure associated with different copulas increases greatly with decreasing component probability of failure. The maximum ratio of the system probabilities of failure for the other copulas to those for the Gaussian copula can happen at an intermediate correlation. The tail dependence of copulas has a significant influence on parallel system reliability. The copula approach provides new insight into the system reliability bounds in a general way. The Gaussian copula, commonly used to describe the dependence structure among variables in practice, produces only one of the many possible solutions of the system reliability and the calculated probability of failure may be severely biased.

This paper aims to investigate the impact of copula selection on geotechnical reliability under incomplete probability information. The copula theory is introduced briefly. Thereafter, four copulas, namely Gaussian, Plackett, Frank, and No. 16 copulas, are selected to model the dependence structure between cohesion and friction angle. A copula-based approach is used to construct the joint probability density function of cohesion and friction angle with given marginal distributions and correlation coefficient. The reliability of an infinite slope and a retaining wall is presented to demonstrate the impact of copula selection on reliability. The results indicate that the probabilities of failure of geotechnical structures with given marginal distributions and correlation coefficient of shear strength parameters cannot be determined uniquely. The resulting probabilities of failure associated with different copulas can differ considerably. Such a difference increases with decreasing probability of failure. Significant difference in probabilities of failure could be observed for relatively small coefficients of variation of the shear strength parameters or a strong negative correlation between cohesion and friction angle. The Gaussian copula, often adopted out of expedience without proper validation, may not capture the dependence structure between cohesion and friction angle properly. Furthermore, the Gaussian copula may greatly underestimate the probability of failure for geotechnical structures.

The modeling of joint probability distributions of correlated variables and the evaluation of reliability under incomplete probability information remain a challenge that has not been studied extensively. This article aims to investigate the effect of copulas for modeling dependence structures between variables on reliability under incomplete probability information. First, a copula-based method is proposed to model the joint probability distributions of multiple correlated variables with given marginal distributions and correlation coefficients. Second, a reliability problem is formulated and a direct integration method for calculating probability of failure is presented. Finally, the reliability is investigated to demonstrate the effect of copulas on reliability. The joint probability distribution of multiple variables, with given marginal distributions and correlation coefficients, can be constructed using copulas in a general and flexible way. The probabilities of failure produced by different copulas can differ considerably. Such a difference increases with decreasing probability of failure. The reliability index defined by the mean and standard deviation of a performance function cannot capture the difference in the probabilities of failure produced by different copulas. In addition, the Gaussian copula, often adopted out of expedience without proper validation, produces only one of the various possible solutions of the probability of failure and such a probability of failure may be biased towards the non-conservative side. The tail dependence of copulas has a significant influence on reliability.

The stability of reservoir landslide would be influenced obviously by the fluctuation of water level, especially when the water descends, which is different from common landslide. Due to the unsteady seepage of underground water inside slope caused by the change of water level, the stability analysis of reservoir landslide through fluid-solid coupling is very complicated. At present, most people hold the view qualitatively that the less the permeability coefficient is and the faster the water level changes, the more unfavorable it is to the stability. This view lacks quantitative basis. Based on FEM strength reduction, the paper analyzes the influence of different sets of descending speed and permeability coefficient on the stability of reservoir landslide through fluid-solid coupling analysis. The paper also conducts the relevant analysis on the change of the characteristics of seepage field inside the slope, which provides basis for the study of the failure mechanism and the forecast of reservoir landslide.

The failure of slope is a gradual accumulation process. Under the effect of many interior and exterior factors, some parts in the slope reach yield with the increase of stress; sliding surface forms gradually till complete transfixion; with the plastic strain continuous increases, overall failure happens on the slope. Traditional analysis method cannot display the mechanic conditions and the whole process of deformation, transfixion of sliding surfaces and failure. Meanwhile, FEM strength reduction can quantitatively show the deformation features and the process of occurrence and development of sliding surface. Based on the previous researches, the paper classifies slopes according to the features of rock and soil and the slope structure. Through analyzing the graphs of deformation and the nephograms of plastic strain under different reduction factors or safety factors, the researchers can directly find the deformation tendency of slopes and the whole process of the extension, transfixion and failure of sliding surface with the reduction of safety factor. So, the failure mechanism of slope can be found intuitively, which can provide effective basis for the prevention and governance of slopes.

A knowledge-based clustered partitioning (KCP) approach is improved to determine the reliability index and probability of failure of a rock slope. The Nataf transformation is adopted to transform the correlated non-normal random variables involved in the KCP approach into independent standard normal variables. An improved KCP technique is proposed to search the design point and calculate the reliability index. Two illustrative examples are presented to demonstrate the capability and validity of the proposed approach. The results indicate that the improved KCP-based reliability method can be applied to evaluate the reliability of rock slopes involving multiple correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that of the traditional KCP using the bisection method, and it is much more efficient than Monte Carlo simulation. The improved KCP-based reliability method is especially suitable for dealing with an implicit performance function with a large number of random variables, which is often involved in slope reliability analysis.

Determining the joint probability distribution of correlated non-normal geotechnical parameters based on incomplete statistical data is a challenging problem. This paper proposes a Gaussian copula-based method for modelling the joint probability distribution of bivariate uncertain data. First, the concepts of Pearson and Kendall correlation coefficients are presented, and the copula theory is briefly introduced. Thereafter, a Pearson method and a Kendall method are developed to determine the copula parameter underlying Gaussian copula. Second, these two methods are compared in computational efficiency, applicability, and capability of fitting data. Finally, four load-test datasets of load-displacement curves of piles are used to illustrate the proposed method. The results indicate that the proposed Gaussian copula-based method can not only characterize the correlation between geotechnical parameters, but also construct the joint probability distribution function of correlated non-normal geotechnical parameters in a more general way. It can serve as a general tool to construct the joint probability distribution of correlated geotechnical parameters based on incomplete data. The Gaussian copula using the Kendall method is superior to that using the Pearson method, which should be recommended for modelling and simulating the joint probability distribution of correlated geotechnical parameters. There exists a strong negative correlation between the two parameters underlying load-displacement curves. Neglecting such correlation will not capture the scatter in the measured load-displacement curves. These results substantially extend the application of the copula theory to multivariate simulation in geotechnical engineering.

SUMMARY This paper aims to propose a procedure for modeling the joint probability distribution of bivariate uncertain data with a nonlinear dependence structure. First, the concept of dependence measures is briefly introduced. Then, both the Akaike Information Criterion and the Bayesian Information Criterion are adopted for identifying the best-fit copula. Thereafter, simulation of copulas and bivariate distributions based on Monte Carlo simulation are presented. Practical application for serviceability limit state reliability analysis of piles is conducted. Finally, four load–test datasets of load–displacement curves of piles are used to illustrate the proposed procedure. The results indicate that the proposed copula-based procedure can model and simulate the bivariate probability distribution of two curve-fitting parameters underlying the load–displacement models of piles in a more general way. The simulated load–displacement curves using the proposed procedure are found to be in good agreement with the measured results. In most cases, the Gaussian copula, often adopted out of expedience without proper validation, is not the best-fit copula for modeling the dependence structure underlying two curve-fitting parameters. The conditional probability density functions obtained from the Gaussian copula differ considerably from those obtained from the best-fit copula. The probabilities of failure associated with the Gaussian copula are significantly smaller than the reference solutions, which are very unconservative for pile safety assessment. If the strong negative correlation between the two curve-fitting parameters is ignored, the scatter in the measured load–displacement curves cannot be simulated properly, and the probabilities of failure will be highly overestimated. Copyright © 2011 John Wiley & Sons, Ltd.