Practical second‐order reliability analysis applied to foundation engineering

A practical and efficient approach of implementing second‐order reliability method (SORM) is presented and illustrated for cases related to foundation engineering involving explicit and implicit limit state functions. The proposed SORM procedure is based on an approximating paraboloid fitted to the limit state surface in the neighborhood of the design point and can be easily carried out in a spreadsheet. Complex mathematical operations are relegated to relatively simple user‐created functions. The failure probability is calculated automatically based on the reliability index and principal curvatures of the limit state surface using established closed‐form SORM formulas. Four common foundation engineering examples are analyzed using the proposed method and discussed: immediate settlement of a flexible rectangular foundation, bearing capacity of a shallow footing, axial capacity of a vertical single pile, and deflection of a pile under lateral load. Comparisons with Monte Carlo simulations are made. In the case of the laterally loaded pile, the friction angle of the soil is represented as a one‐dimensional random field, and pile deflections are computed based on finite element analysis on a stand‐alone computer package. The implicit limit state function is approximated via the response surface method using two quadratic models. Copyright © 2011 John Wiley & Sons, Ltd.     

Probabilistic analysis of underground rock excavations using response surface method and SORM

Probabilistic analysis of underground rock excavations is performed using response surface method and SORM, in which the quadratic polynomial with cross terms is used to approximate the implicit limit state surface at the design point. The response surface is found using an iterative algorithm and the probability of failure is evaluated using the first-order and the second-order reliability method (FORM/SORM). Independent standard normal variables in U-space are chosen as basic random variables and transformed into correlated non-normal variables in the original space of random variables for constructing the response surface. The proposed method is first illustrated for a circular tunnel with analytical solutions considering Mohr–Coulomb (M–C) and Hoek–Brown (H–B) yield criteria separately. The failure probability with respect to the plastic zone criterion and the tunnel convergence criterion are estimated from FORM/SORM and compared to those obtained from Monte Carlo Simulations. The results show that the support pressure has great influence on the failure probability of the two failure modes. For the M–C model, the hypothesis of uncorrelated friction angle and cohesion will generate higher non-performance probability in comparison to the case of negatively correlated shear strength parameters. Reliability analyses involving non-normal distributions are also investigated. Finally, an example of a horseshoe-shaped highway tunnel is presented to illustrate the feasibility and validity of the proposed method for practical applications where numerical procedures are needed to calculate the performance function values.     

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.     

特定场地下土工构筑物的几何可靠性分析

针对特定场地下土工构筑物的正常使用极限状态,采用近年发展的几何可靠性方法计算了多种构筑物的可靠度指标。考虑同一场地下的钻孔灌注桩、抗浮锚杆和CFG桩单桩加载变形测试曲线的离散性,各曲线回归参数呈现差异并可作为随机变量,进而探讨了各曲线回归参数间的相关性及联合分布特性。基于这些回归参数的联合发散概率密度等值线,即随机变量刚好达到极限承载能力状态,该几何可靠性算法可在随机变量的原始空间求得土工构筑物的可靠度指标。通过比对该几何可靠度指标与常规的一次可靠性算法成果,验证了该几何可靠性计算技术的可行性。计算表明,几何可靠性评价模型实施简便,易于被工程技术人员接受。     

Reliability analysis of serviceability performance for an underground cavern using a non-intrusive stochastic method

This paper proposes a non-intrusive stochastic analysis procedure for reliability analysis of the serviceability performance of an underground cavern with an implicit limit state function. This procedure is formulated on the basis of the stochastic response surface method (SRSM) and the deterministic finite element method. First, the SRSM is briefly introduced and implemented through a MATLAB code. Then, the software SIGMA/W is used to perform a deterministic finite element analysis. Next, a link between the MATLAB code and SIGMA/W is developed to automatically pass exchange data between the two platforms. Finally, two examples are presented to illustrate the capacity and validity of the proposed procedure. In the first example, a closed-form limit state function is adopted to validate the SRSM by comparing it with the results obtained from a direct Monte Carlo simulation. In the second example, the serviceability performance of an underground cavern is analyzed to illustrate the capacity of the proposed procedure to handle a reliability problem with an implicit limit state function. The proposed procedure does not require the user to modify the existing deterministic finite element code. The deterministic finite element analysis and the probabilistic analysis are decoupled. This is a major practical advantage because realistic probabilistic analyses are made possible. The SRSM can produce sufficiently accurate reliability results. Furthermore, the method is much more efficient than the direct Monte Carlo simulation. Sensitivity analyses show the effect of the variability of input random variables and the correlation between them on: (1) the probability density functions, (2) the first four order statistical moments, and (3) the probability of failure, which is investigated and discussed.     

Slope reliability analysis accounting for spatial variation

A practical and transparent procedure is described for implementing a generalized limit equilibrium method via cell-object-oriented constrained optimization in the spreadsheet platform. The formulation allows switching among the Spencer, Bishop simplified and wedge methods on the same template by specifying different side-force inclination and different constraints of optimization. Search for the critical circular or non-circular slip surface is possible. The deterministic procedure is extended probabilistically by implementing the first-order reliability method via constrained optimization of the equivalent dispersion ellipsoid in the original space of the random variables. This procedure is illustrated for an embankment on soft ground, and for a clay slope in southern Norway, both involving spatially correlated soil properties. The effects of autocorrelation distance on the results of reliability analysis are studied. Shear strength anisotropy is modelled via user-created simple function codes in the programming environment of the spreadsheet. The meaning of probability of failure is discussed.     

岩土工程可靠度分析的神经网络四阶矩法

针对岩土工程的功能函数强非线性且难以显式表达的特点,提出了基于人工神经网络的四阶矩法,充分利用了基本随机变量的统计信息。首先利用神经网络对结构的隐式功能函数进行拟合,求得基本随机变量在均值点处的功能函数值及其偏导数,然后利用泰勒级数展开的方法由基本随机变量的前四阶矩求得功能函数的前四阶矩,并借助于Pearson系统获得功能函数的更高阶矩。在此基础上,通过最大熵原理确定以功能函数各阶矩为约束的功能函数的概率密度函数,最后由一次积分得到结构的失效概率。通过数值算例和工程实例不同方法的对比分析,表明基于神经网络的结构可靠度分析四阶矩方法是可行的,有效的,能够满足岩土工程可靠度分析的要求。     

Relevance vector machine‐based response surface for slope reliability analysis

A new computing method is proposed for reliable analysis. The limit state function is implicit and nonlinear in reliability analysis of slopes and is difficult to apply by traditional reliability methods, especially in large‐scale project applications. Relevance vector machines (RVMs) are capable of approximating the limit state function without the need for additional assumptions regarding the function form, as opposed to traditional polynomial response surfaces. RVMs were adapted to obtain the limit state function. We propose an RVM‐based response surface method combined with the first‐order reliability method for slope reliability analysis and describe its step‐by‐step implementation. The reliability index obtained from the proposed method shows excellent agreement with traditional response surface method results. Copyright © 2017 John Wiley & Sons, Ltd.     

边坡稳定模糊随机可靠度分析

考虑边坡稳定评价中存在的两种不确定性--模糊性和随机性,以模糊随机变量为基本变量,建立模糊随机功能函数和模糊随机极限方程,得到模糊可靠指标和模糊可靠度。本文给出模糊随机可靠度的计算方法,给出实例并与经典随机可靠度比较,得到更合理的结果。     

Moving least squares method for reliability assessment of rock tunnel excavation considering ground-support interaction

A practical approach is proposed in this paper for the reliability assessment of rock tunnel excavations using the moving least squares method (MLSM) and the uniform design. The failure probability is computed by the first-order and the second-order reliability method (FORM/SORM), which is based on the generated MLSM response surface (MLSM-RS) via an iterative algorithm. The proposed approach is first implemented in the analysis of a circular tunnel that consists of three limit state functions to illustrate the efficiency and accuracy of the approach. Then, the method is applied to a non-circular tunnel to demonstrate the feasibility and validity of the method for practical problems, in which numerical procedures are commonly employed to solve the implicit limit state functions.     

Slope reliability analysis using a support vector machine

The first-order second-moment method (FOSM) reliability analysis is commonly used for slope stability analysis. It requires the values and partial derivatives of the performance function with respect to the random variables for the design. Such calculations can be cumbersome when the performance functions are implicit. Implicit performance functions are normally encountered when the slope is geologically complicated and the limit equilibrium method (LEM) is used for the stability analysis.

To address this issue, this paper presents a support vector machine (SVM)-based reliability analysis method which combines the SVM with the FOSM. This method employs the SVM method to approximate the implicit performance functions, thus arriving at SVM-based explicit performance functions. The SVM method uses a small set of the actual values of the performance functions obtained via the LEM for complicated slope engineering. Using the SVM model, a large number of values and partial derivatives of the performance functions can be obtained for conventional reliability analysis using the FOSM. Examples are given to illustrate the proposed SVM-based slope reliability analysis. The results show that the proposed approach is applicable to slope reliability analysis which involves implicit performance functions.     

钻孔灌注桩单桩承载力的可靠度研究

采用无量纲随机变量的极限状态方程,结合天津市钻孔灌注桩试桩资料进行可靠度分析。采用了JC法和Monte Carlo法两种方法进行可靠指标的计算,并对影响可靠指标的各种因素进行了分析。还计算了单桩承载力的抗力分项系数。     

带压开采煤层底板突水安全可靠性分析

为合理评价孙疃煤矿10煤层带压开采的安全可靠性,建立了10煤层隔水底板稳定性可靠度分析模型。考虑底板隔水层厚度、底板岩石容重、抗拉强度等参数为随机变量,采用斯列萨列夫安全隔水层厚度计算公式建立极限状态方程。在研究随机变量概型的基础上,采用JC法、最优化法以及基于最优化的蒙特卡罗法计算隔水底板的可靠度以及失效概率,并探讨了该矿10煤层安全开采的可靠指标,避免了单纯依据定值作为突水判据的某些不合理性。     

边坡可靠度分析的一种新的优化求解方法

介绍了Low & Tang提出的一种新的可靠度优化求解方法,并将之用于边坡可靠度分析中：该方法适用于任何概率分布的相关变量,不必计算当量正态均值和方差、相关变量独立变换,直接在变量的原始空间内搜索边坡的最小可靠指标和概率临界滑面,可采用任何合适的约束优化方法进行求解,方法清晰简洁。边坡可靠度分析常用的滑面有2个：最小安全系数（变量均值处）对应的确定性临界滑面和最小可靠指标对应的概率临界滑面,但这2个滑面在有些情况下差别较大,Hassan & Wolff提出了一种简化方法可以方便地获得概率临界滑面,但由于方法简单,受到质疑。通过一系列算例分析,优化求解方法得到的概率临界滑面和Hassan & Wolff的简化方法滑面非常接近,显示了简化方法的有效性,值得在工程实践中推广。     

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.     

Genetic algorithms in probabilistic finite element analysis of geotechnical problems

In application to numerical analysis of geotechnical problems, the limit-state surface is usually not known in any closed form. The probability of failure can be assessed via the so-called reliability index. A minimization problem can naturally be formed with an implicit equality constraint defined as the limit-state function and optimization methods can be used for such problems. In this paper, a genetic algorithm is proposed and incorporated into a displacement finite element method to find the Hasofer–Lind reliability index. The probabilistic finite element method is then used to analyse the reliability of classical geotechnical systems. The performance of the genetic algorithm (GA) is compared with simpler probability methods such as the first-order-second-moment Taylor series method. The comparison shows that the GA can produce the results fairly quickly and is applicable to evaluation of the failure performance of geotechnical problems involving a large number of decision variables.     

A rational method for development of limit state for liquefaction evaluation based on shear wave velocity measurements

This papers presents a new approach for developing a limit state for liquefaction evaluation based on field performance data. As an example to illustrate the new approach, a database that consists of, among many other features, in situ shear wave velocity measurements and field observations of liquefaction/non‐liquefaction in historic earthquakes is analysed. This database is first used to train a neural network to classify liquefaction/non‐liquefaction based on soil resistance parameters and load parameters. The successfully trained and tested neural network is then used to establish a limit state, a multiple dimension boundary that separates ‘zone’ of liquefaction from ‘zone’ of non‐liquefaction. The limit state yields cyclic resistance ratio for a given set of soil resistance parameters. Examination of all cases in the database show that the developed limit state has a high degree of accuracy in predicting the occurrence of liquefaction/non‐liquefaction. The developed neural network model can accurately predict the cyclic resistance ratio of soils. Copyright © 2000 John Wiley & Sons, Ltd.     

Efficient Probabilistic Algorithm Illustrated for a Rock Slope

Summary  A new spreadsheet-based algorithm for the first-order reliability method (FORM) is illustrated for a two-dimensional rock slope of Hong Kong. The new algorithm combines inverse distribution functions and a refined Newton method with the automatic constrained-optimization search of the design point in the original space of the random variables; it obviates the need for computations of equivalent normal means and equivalent normal standard deviations. In the rock slope analysis, the versatile 4-parameter beta distribution is used in lieu of a truncated normal distribution. Probabilities of failure inferred from reliability indices are compared with those from Monte Carlo simulations. The effects of parametric correlations on the required reinforcing force for a target reliability index value are studied. The intuitive perspective of an expanding equivalent dispersion ellipsoid in the original space of the basic random variables is also described as it is the basis from which the new approach evolved. Author’s address: Dr. Bak Kong Low, School of Civil and Environmental Engineering, Nanyang Technological University, Block N1 #1b-40, 50 Nanyang Avenue, 639798 Singapore     

Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables

This paper proposes a stochastic response surface method for reliability analysis involving correlated non-normal random variables, in which the Nataf transformation is adopted to effectively transform the correlated non-normal variables into independent standard normal variables. Transformations of random variables that are often used in reliability analyses in terms of standard normal variables are summarized. The closed-form expressions for fourth to sixth order Hermite polynomial chaos expansions involving any number of random variables are formulated. The proposed method will substantially extend the application of stochastic response surface method for reliability problems. An example of reliability analysis of rock slope stability with plane failure is presented to demonstrate the validity and capability of the proposed stochastic response surface method. The results indicate that the proposed stochastic response surface method can evaluate the reliability of rock slope stability involving correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that for the first-order reliability method, and it is much more efficient than direct Monte-Carlo simulation. The results also show that the number of collocation points selected should ensure that the Hermite polynomial matrix has a full rank so that different order SRSMs can produce a robust estimation of probability of failure for a specified performance function. Generally, the accuracy of SRSM increases as the order of SRSM increases.     

深埋隧道衬砌稳定可靠度的Kriging插值法与遗传算法协同优化求解法

首先利用遗传算法(GA)的全局优化能力,解决了Kriging插值预测算法中相关参数的精确求解问题。针对衬砌结构力学状态表征量与基本参数是高度非线性关系而无法明晰表达的难题,建立了基于Kriging插值预测法的状态表征量等代显式通用模型。以抗拉强度控制参量为例,提出了采用拉丁抽样等代显式通用模型的具体隧道工程衬砌结构力学状态方程构造程序。基于以状态方程为约束条件建立的可靠度指标计算模式,构建起Kriging与GA协同优化的隧道衬砌稳定可靠度求解方法。实例应用对比表明,该方法与精确解的相对误差为4.8%,计算工作量仅为其2.5%。与目前已有的同类方法相比,具有明显的优越性。