System reliability analysis of soil slopes with general slip surfaces using multivariate adaptive regression splines

A data driven multivariate adaptive regression splines (MARS) based algorithm for system reliability analysis of earth slopes having random soil properties under the framework of limit equilibrium method of slices is considered. The theoretical formulation is developed based on Spencer method (valid for general slip surfaces) satisfying all conditions of static equilibrium coupled with a nonlinear programming technique of optimization. Simulated noise is used to take account of inevitable modeling inaccuracies and epistemic uncertainties. The proposed MARS based algorithm is capable of achieving high level of computational efficiency in the system reliability analysis without significantly compromising the accuracy of results.     

Efficient system reliability analysis of soil slopes using multivariate adaptive regression splines-based Monte Carlo simulation

System effects should be considered in the probabilistic analysis of a layered soil slope due to the potential existence of multiple failure modes. This paper presents a system reliability analysis approach for layered soil slopes based on multivariate adaptive regression splines (MARS) and Monte Carlo simulation (MCS). The proposed approach is achieved in a two-phase process. First, MARS is constructed based on a group of training samples that are generated by Latin hypercube sampling (LHS). MARS is validated by a specific number of testing samples which are randomly generated per the underlying distributions. Second, the established MARS is integrated with MCS to estimate the system failure probability of slopes. Two types of multi-layered soil slopes (cohesive slope and c–φ slope) are examined to assess the capability and validity of the proposed approach. Each type of slope includes two examples with different statistics and system failure probability levels. The proposed approach can provide an accurate estimation of the system failure probability of a soil slope. In addition, the proposed approach is more accurate than the quadratic response surface method (QRSM) and the second-order stochastic response surface method (SRSM) for slopes with highly nonlinear limit state functions (LSFs). The results show that the proposed MARS-based MCS is a favorable and useful tool for the system reliability analysis of soil slopes.     

Full probabilistic design of slopes in spatially variable soils using simplified reliability analysis method

ABSTRACT

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.     

考虑空间变异性条件下的边坡稳定可靠度高效敏感性分析

在有限数据条件下,可靠度敏感性分析是研究各种不确定性因素对边坡失稳概率影响规律的重要途径。基于直接蒙特卡洛模拟和概率密度加权分析方法提出了一种高效边坡稳定可靠度敏感性分析方法。所提出的方法通过随机场表征岩土体参数的空间变异性,并采用局部平均理论建立岩土体参数的缩维概率密度函数,用于概率密度加权分析中高效、准确地计算不同敏感性分析方案对应的边坡失稳概率。最后,通过一个工程案例--詹姆斯湾堤坝说明了所提出方法的有效性和准确性。结果表明：在敏感性分析过程中,所提出的方法只需要执行一次直接蒙特卡洛模拟,避免了针对不同敏感性分析方案重新产生随机样本和执行边坡稳定分析,节约了大量的计算时间和计算资源,显著提高了基于蒙特卡洛模拟的敏感性分析计算效率;在概率密度加权分析中采用岩土体参数的缩维概率密度函数能够准确地计算边坡失稳概率,避免了有偏估计,使概率密度加权分析方法适用于考虑空间变异性条件下的边坡稳定可靠度敏感性分析问题。     

System reliability analysis of slopes using multilayer perceptron and radial basis function networks

This paper presents a system reliability analysis method for soil slopes on the basis of artificial neural networks with computer experiments. Two types of artificial neural networks, multilayer perceptrop (MLP) and radial basis function networks (RBFNs), are tested on the studied problems. Computer experiments are adopted to generate samples for constructing the response surfaces. On the basis of the samples, MLP and RBFN are used for establishing the response surface to approximate the limit state function, and Monte Carlo simulation is performed via the MLP and RBFN response surfaces to estimate the system failure probability of slopes. Experimental results on 3 examples show the effectiveness of the proposed methodology.     

Reliability analysis of piles in spatially varying soils considering multiple failure modes

In most limit state design codes, the serviceability limit checks for drilled shafts still use deterministic approaches. Moreover, different limit states are usually considered separately. This paper develops a probabilistic framework to assess the serviceability performance with the consideration of soil spatial variability in reliability analysis. Specifically, the performance of a drilled shaft is defined in terms of the vertical settlement, lateral deflection, and angular distortion at the top of the shaft, corresponding to three limit states in the reliability analysis. Failure is defined as the event that the displacements exceed the corresponding tolerable displacements. The spatial variability of soil properties is considered using random field modeling. To illustrate the proposed framework, this study assesses the reliability of each limit state and the system reliability of a numerical example of a drilled shaft. The results show the system reliability should be considered for the serviceability performance. The importance measures of the random variables indicate that the external loads, the performance criteria, the model errors of load transfer curves and soil strength parameter are the most important factors in reliability analysis. Moreover, it is shown that the correlation length and coefficient of variation of soil strength can exert significant impacts on the calculated failure probability.     

Effect of 2D spatial variability on slope reliability: A simplified FORM analysis

To meet the high demand for reliability based design of slopes, we present in this paper a simplified HLRF(Hasofere Linde Rackwitze Fiessler) iterative algorithm for first-order reliability method(FORM). It is simply formulated in x-space and requires neither transformation of correlated random variables nor optimization tools. The solution can be easily improved by iteratively adjusting the step length. The algorithm is particularly useful to practicing engineers for geotechnical reliability analysis where standalone(deterministic) numerical packages are used. Based on the proposed algorithm and through direct perturbation analysis of random variables, we conducted a case study of earth slope reliability with complete consideration of soil uncertainty and spatial variability.     

Probabilistic analysis of strip footings resting on a spatially random soil using subset simulation approach

The failure probability of geotechnical structures with spatially varying soil properties is generally computed using Monte Carlo simulation (MCS) methodology. This approach is well known to be very time-consuming when dealing with small failure probabilities. One alternative to MCS is the subset simulation approach. This approach was mainly used in the literature in cases where the uncertain parameters are modelled by random variables. In this article, it is employed in the case where the uncertain parameters are modelled by random fields. This is illustrated through the probabilistic analysis at the serviceability limit state (SLS) of a strip footing resting on a soil with a spatially varying Young's modulus. The probabilistic numerical results have shown that the probability of exceeding a tolerable vertical displacement (P _e) calculated by subset simulation is very close to that computed by MCS methodology but with a significant reduction in the number of realisations. A parametric study to investigate the effect of the soil variability (coefficient of variation and the horizontal and vertical autocorrelation lengths of the Young's modulus) on P _e was presented and discussed. Finally, a reliability-based design of strip footings was presented. It allows one to obtain the probabilistic footing breadth for a given soil variability.     

Effect of spatial variability of shear strength on reliability of infinite slopes using analytical approach

This paper develops an analytical approach for reliability analysis of infinite slope stability in presence of spatially variable shear strength parameters. The analytical approach considers spatial autocorrelation of each parameter and cross-correlations between different parameters. It is robust, computational efficient and provides insight to the importance of spatial correlation scale on slope reliability analysis. This paper also explores the difference in continuous and discrete random fields and emphasizes the importance of fine discretization in relation to correlation scale. Finally, it shows that conditioning the stability analysis with information about trends and spatial data leads to reliability assessments with less uncertainty.     

Risk de-aggregation and system reliability analysis of slope stability using representative slip surfaces

This paper develops a risk de-aggregation and system reliability approach to evaluate the slope failure probability, p_f, using representative slip surfaces together with MCS. An efficient procedure is developed to strategically select the candidate representative slip surfaces, and a risk de-aggregation approach is proposed to quantify contribution of each candidate representative slip surface to the p_f, identify the representative slip surfaces, and determine how many representative slip surfaces are needed for estimating the p_f with reasonable accuracy. Risk de-aggregation is performed by collecting the failure samples generated in MCS and analyzing them statistically. The proposed methodology is illustrated through a cohesive soil slope example and validated against results from previous studies. When compared with the previous studies, the proposed approach substantially improves the computational efficiency in probabilistic slope stability analysis. The proposed approach is used to explore the effect of spatial variability on the p_f. It is found that, when spatial variability is ignored or perfect correlation assumed, the p_f of the whole slope system can be solely attributed to a single representative slip surface. In this case, it is theoretically appropriate to use only one slip surface in the reliability analysis. As the spatial variability becomes growingly significant, the number of representative slip surfaces increases, and all representative slip surfaces (i.e., failure modes) contribute more equally to the overall system risk. The variation of failure modes has substantial effect on the p_f, and all representative surfaces have to be incorporated properly in the reliability analysis. The risk de-aggregation and system reliability approach developed in this paper provides a practical and efficient means to incorporate such a variation of failure modes in probabilistic slope stability analysis.     

The critical scale of fluctuation for active lateral forces in spatially variable undrained clays

This study addresses the phenomenon of the critical scale of fluctuation (SOF) for active lateral force (P_a) in undrained clay when there is a spatial variability in the clay. The phenomenon is significant under shear strength (τ_f) random fields but is insignificant under unit weight (γ) random fields. It is found that the phenomenon of the critical SOF is connected to the nature of the spatial averaging, which is “line averaging” under τ_f random fields and is “area averaging” under γ random fields. The former averaging effect (line) is significantly weaker than the latter (area), so the tendency for the critical slip plane to seek for a favorable location is stronger for the τ_f random field than for the γ random field. Hence, the phenomenon of the critical SOF is more pronounced under τ_f random fields than under γ random fields. The underlying mechanisms for the phenomenon of the critical SOF will be explored in this paper.     

An intelligent response surface method for analyzing slope reliability based on Gaussian process regression

Problems in geotechnical engineering inevitably involve many uncertainties in the analysis. Reliability methods are important for evaluating slope stability and can take the uncertainties into consideration. In this paper, a novel intelligent response surface method is proposed in which a machine learning algorithm, namely Gaussian process regression, is used to approximate the high-dimensional and highly nonlinear response hypersurface. An iterative algorithm is also proposed for updating the response surface dynamically by adding the new training point nearest to the limit state surface to the initial training database at each step. The proposed Gaussian process response surface method is used to analyze three different case studies to assess its validity and efficiency. Direct Monte Carlo simulation is also carried out in each case to serve as the benchmark. Comparing with other methods confirms the accuracy and efficiency of the novel intelligent response surface method, which requires fewer performance function calls and avoids the need to normalize the correlative non-normal variables.     

SVM在地下工程可靠性分析中的应用

将支持向量机应用到地下工程可靠性分析中,通过将支持向量机分别与一阶二次矩和蒙特卡洛结合,提出了基于支持向量机的可靠性分析方法,利用数值模拟构造学习样本,通过支持向量机学习,建立变形与随机变量之间映射关系的支持向量机表达,进而实现隧道极限状态函数及其偏导数的显式表达,从而计算隧道的可靠性指标。该方法避免了传统可靠性分析的缺点。算例分析结果表明,该方法计算效率高、结果可靠,对含有大量随机变量的复杂岩土工程可靠性分析具有很大的潜力,具有广泛的应用前景和工程价值。     

基于随机响应面法的边坡系统可靠度分析

提出了一套基于随机响应面法的边坡系统可靠度分析方法。该方法首先从大量潜在滑动面中筛选出代表性滑动面。针对每条代表性滑动面,采用Hermite多项式展开建立其安全系数与土体参数间的非线性显式函数关系（即随机响应面）。然后,采用直接蒙特卡洛模拟计算边坡系统失效概率。在蒙特卡罗模拟中,采用所有代表性滑动面的随机响应面计算每一组样本所对应的边坡最小安全系数。最后,以两个典型多层边坡系统可靠度问题为例验证了该方法的有效性。结果表明：文中提出的边坡系统可靠度分析方法能够有效地识别边坡代表性滑动面,具有较高的计算精度和效率,并且确定代表性滑动面时无需计算滑动面间的相关系数。同时该方法可以有效地计算低失效概率水平的边坡系统可靠度,为含相关非正态参数的边坡系统可靠度问题提供了一条有效的分析途径。此外,多层边坡可能同时存在多条潜在滑动面,基于单一滑动面（如临界确定性滑动面）或者部分代表性滑动面进行边坡系统可靠度分析均会低估边坡失效概率。     

Generation of correlated properties in heterogeneous porous media

The spatial distribution of rock properties in porous media, such as permeability and porosity, often is strongly variable. Therefore, these properties usefully may be considered as a random field. However, this variability is correlated frequently on length scales comparable to geological lengths (for example, scales of sand bodies or facies). To solve various engineering problems (for example, in the oil recovery process) numerical models of a porous medium often are used. A need exists then to understand correlated random fields and to generate them over discretized numerical grids. The paper describes the general mathematical methods required to do this, with one particular method (the nearest neighbor model) described in detail. How parameters of the mathematical model may be related to rock property statistics for the nearest neighbor model is shown. The method is described in detail in one, two, and three dimensions. Examples are given of how model parameters may be determined from real data.     

Conditional random field reliability analysis of a cohesion-frictional slope

Discarding known data from cored samples in the reliability analysis of a slope in spatially variable soils is a waste of site investigation effort. The traditional unconditional random field simulation, which neglects these known data, may overestimate the simulation variance of the underlying random fields of the soil properties. This paper attempts to evaluate the reliability of a slope in spatially variable soils while considering the known data at particular locations. Conditional random fields are simulated based on the Kriging method and the Cholesky decomposition technique to match the known data at measured locations. Subset simulation (SS) is then performed to calculate the probability of slope failure. A hypothetical homogeneous cohesion-frictional slope is taken as an example to investigate its reliability conditioned on several virtual samples. Various parametric studies are performed to explore the effect of different layouts of the virtual samples on the factor of safety (FS), the spatial variation of the critical slip surface and the probability of slope failure. The results suggest that whether the conditional random fields can be accurately simulated depends highly on the ratio of the sample distance and the autocorrelation distance. Better simulation results are obtained with smaller ratios. Additionally, compared with unconditional random field simulations, conditional random field simulations can significantly reduce the simulation variance, which leads to a narrower variation range of the FS and its location and a much lower probability of failure. The results also highlight the great significance of the conditional random field simulation at relatively large autocorrelation distances.     

Effects of anisotropy in correlation structure on the stability of an undrained clay slope

Slopes are mainly naturally occurred deposits, so slope stability is highly affected by inherent uncertainty. In this paper, the influence of heterogeneity of undrained shear strength on the performance of a clay slope is investigated. A numerical procedure for a probabilistic slope stability analysis based on a Monte Carlo simulation that considers the spatial variability of the soil properties is presented to assess the influence of randomly distributed undrained shear strength and to compute reliability as a function of safety factor. In the proposed method, commercially available finite difference numerical code FLAC 5.0 is merged with random field theory. The results obtained in this study are useful to understand the effect of undrained shear strength variations in slope stability analysis under different slope conditions and material properties. Coefficient of variation and heterogeneity anisotropy of undrained shear strength were proven to have significant effect on the reliability of safety factor calculations. However, it is shown that anisotropy of the heterogeneity has a dual effect on reliability index depending on the level of safety factor adopted.     

Optimization of site exploration program for improved prediction of tunneling-induced ground settlement in clays

Excessive settlement caused by tunneling during construction often damages adjacent infrastructures and utilities. Such excessive settlement can also present a challenge in the maintenance of subways during their operation. Thus, it is important to be able to accurately predict tunneling-induced settlement. The uncertainties in geotechnical parameters, however, can lead to either an overestimation or an underestimation of the tunneling-induced settlement. Such uncertainties can arise from many sources such as spatial variability, measurement error, and model error; in this paper, the focus is on the geotechnical parameters characterization from site exploration. The goal here is to determine an optimal level of site exploration effort so that effective predictions of the tunneling-induced settlement in clays can be achieved. To this end, a Monte Carlo simulation-based numerical model of site exploration is first established to generate artificial test data. Then, a series of parametric analyses are performed to investigate the relationship between the level of site exploration effort and the accuracy of the tunneling-induced ground settlement prediction. Through the assumed different levels of site exploration effort, statistics of soil parameters are estimated using the maximum likelihood method and the tunneling-induced ground settlement is then analyzed using the probabilistic method, and finally the effect of site exploration effort is assessed. The knowledge generated from this series of analyses is then used to develop the proposed framework for selecting an optimal site exploration program for improved prediction of the tunneling-induced ground settlement in clays. Examples are presented to illustrate the proposed framework and demonstrate its effectiveness and significance.     

System reliability-based analysis of cantilever retaining walls embedded in granular soils

Reliability-based analysis of cantilever retaining walls requires consideration of different failure mechanisms. In this paper, the reliability of soil-wall system is assessed considering two failure modes: rotational and structural stability, and the system reliability is assumed as a series system. The methodology is based on Monte Carlo Simulation (MCS), and it deals with the variability of the design parameters in the limit equilibrium analysis of a wall embedded in granular soil. Results of the MCS indicate that the reliability of the failure components increases exponentially by increasing the variability of design parameters. The results of the system reliability indicate how the system reliability is different from the component reliabilities. The strength of the weakest component influences the reliability of the system. The system reliability index increases with the wall section gradually. However it remains constant for the rotational failure mode.