Ductile structural system reliability analysis using adaptive importance sampling

This paper presents a hybrid technique for efficient system reliability estimation of large ductile framed structures. The proposed procedure starts with a simple enumeration scheme, but quickly changes to an adaptive importance sampling scheme to make the process more efficient and easier to implement. The method solves the problem of including the effect of multiple failure sequences in an importance sampling scheme, for the system reliability estimation of large structures. The enumeration method is used to identify the first complete failure sequence. This failure sequence defines the initial failure domain for starting the adaptive sampling process. A weighted multi-modal sampling density is used to account for the contribution of different regions in the sampling domain to the system failure probability. As the simulation progresses, the failure domain is gradually modified to include the effect of other significant failure sequences and arrive at an accurate estimate of the system failure probability.     

A benchmark study on importance sampling techniques in structural reliability

Several widely used importance sampling methods for the estimation of failure probabilities are compared. The methods are briefly reviewed, and a set of evaluation criteria for the comparison of the methods is chosen. In order to perform a fair comparison the developers of the schemes were asked to solve a number of problems selected in view of the evaluation criteria. Their solutions are presented and discussed. Conclusions about the performances of the schemes under different circumstances are given.     

Search-based importance sampling

Importance sampling as a special technique in Monte Carlo probability integration has been shown to be a highly efficient and rather unrestricted method. Non-Gaussian and dependent random variables and nonlinear limit functions can be treated relatively easily and with reasonable rates of convergence. A major draw-back, however, is the need to identify so-called “interesting” or “important” regions for integration. Reference to first-order second-moment (FOSM) methods may help, as well as numerical maximization routines applied. Each involves certain difficulties. An alternative procedure, based on directing and correcting the importance sampling function as sampling is carried out, is presented herein. In particular it is possible to have a multi-modal sampling function.     

Multinormal integrals by importance sampling for series system reliability

A structural system with multi-failure modes can be modeled as a series system if it fails whenever any of the failure mode occurs. Applying FORM, failure probability of a series system can be expressed using a complementary standard multinormal integral. However, the integral is increasingly more difficult as the dimension increases. Importance sampling method can be used to deal with such multi-fold integration. Considering the fact that the optimal importance sampling function can be determined for a linear limit state function in a uncorrelated standard normal space, this paper proposes an importance sampling function for multinormal integral as a linear combination of such optimal sampling functions. The accuracy and applicability of the method are investigated using numerical examples.     

Asymptotic importance sampling

An importance sampling technique is described which is based on theoretical considerations about the structure of multivariate integrands in domains having small probability content. The method is formulated in the original variable space. Sampling densities are derived for a variety of practical conditions: a single point of maximum loglikelihood; several points; points located at the intersect of several failure surfaces; and, bounded variables. Sampling in the safe domain is avoided and extensive use is made of noncartesian as well as surface coordinates. The parameters of the importance sampling densities are taylored in such a way as to yield asymptotic minimum variance unbiased estimators. The quality and the efficiency of the method improves as the failure probability decreases. Parameter sensitivies are easily computed owing to the use of local surface coordinates. Several examples are provided.     

A fast and efficient response surface approach for structural reliability problems

The reliability analysis of large and complex structural requires approximate techniques in order to reduce computational efforts to an acceptable level. Since it is, from an engineering point of view, desirable to make approximative assumptions at the level of the mechanical rather than the probabilistic modeling, simplifications should be carried out in the space of physically meaningful system- or loading variables.Within the context of this paper, a new adaptive interpolation scheme is suggested which enables fast and accurate representation of the system behavior by a response surface (RS). This response surface approach utilizes elementary statistical information on the basic variables (mean values and standard deviations) to increase the efficiency and accuracy. Thus the RS obtained is independent of the type of distribution or correlations among the basic variables which enables sensitivity studies with respect to these parameters without much computational effort.Subsequently, the response surface is utilized in conjunction with advanced Monte Carlo simulation techniques (importance sampling) to obtain the desired reliability estimates.Numerical examples are carried out in order to show the applicability of the suggested approach to structural systems reliability problems. The proposed method is shown to be superior both in efficiency and accuracy to existing approximate methods, i.e., the first order reliability methods.     

Time-dependent reliability of ageing structures: an approximate approach

Concrete structures may deteriorate over time due to aggressive service environments, leading to a reduction in their strengths, stiffnesses and reliabilities. In general, the assessment of time-dependent reliability of ageing structures must consider uncertainties in structural deterioration as well as non-stationarities in the structural load processes. This paper develops an approximate method for assessing the impact of structural deterioration and non-stationary live loads on structures, which requires only low-dimensional integration and reduces the cost of assessing time-dependent reliability over a service life extending to 50 years significantly. This approximate method is demonstrated through several examples. The importance of non-stationarities in the resistance and load processes on time-dependent reliability is illustrated and the accuracy of the method is confirmed in several cases utilising Monte Carlo simulation.     

Structural analysis methods for system reliability

The appropriateness of the use of traditional methods of structural analysis in reliability analyses is examined using two examples: partially restrained steel beams and the stability of earth slopes. It is found that new and innovative methods of structural analysis may cause simplifications in the system reliability analysis.     

Adaptive radial-based importance sampling method for structural reliability

In this paper an adaptive radial-based importance sampling (ARBIS) method is presented. The radial-based importance sampling (RBIS) method, excluding a β-sphere from the sampling domain, is extended with an efficient adaptive scheme to determine the optimal radius β of the sphere. The adaptive scheme is based on directional simulation. The underlying basic methods are presented briefly. Several numerical examples demonstrate the efficiency, accuracy and robustness of the scheme. As such, the ARBIS method can be applied as a black-box and is of particular interest in applications with a low probability of failure, for example in structural reliability, in combination with a small number of stochastic variables.     

Evaluation of accuracy and efficiency of some simulation and sampling methods in structural reliability analysis

Numerous simulation and sampling methods can be used to estimate reliability index or failure probability. Some point sampling methods require only a fraction of the computational effort of direct simulation methods. For many of these methods, however, it is not clear what trade-offs in terms of accuracy, precision, and computational effort can be expected, nor for which types of functions they are most suited. This study uses nine procedures to estimate failure probability and reliability index of approximately 200 limit state functions with characteristics common in structural reliability problems. The effects of function linearity, type of random variable distribution, variance, number of random variables, and target reliability index are investigated. It was found that some methods have the potential to save tremendous computational effort for certain types of limit state functions. Recommendations are made regarding the suitability of particular methods to evaluate particular types of problems.     

An assessment of high-order bounds for structural reliability

Second-order bounds are often used to estimate the reliability of series structural systems. However, the quality of second-order bounds is poor when the failure modes are highly correlated. This paper examines the merit of third and higher-order bounds. First, the general form of the nth-order upper and lower bounds is derived. Second, an efficient algorithm for computing the n-dimensional multinormal integral is presented. Finally, the performance of the third and fourth-order bounds is examined in three numerical examples. These bounds can provide significant improvement over the second-order bounds, particularly when the number of significant failure modes is small (less than 20) or when the correlations between modes are somewhat localized.     

Application of line sampling simulation method to reliability benchmark problems

A procedure denoted as Line Sampling (LS) has been developed for estimating the reliability of static and dynamical systems. The efficiency and accuracy of the method is shown by application to the subset of the entire spectrum of the posed benchmark problems [Schuëller GI, Pradlwarter HJ, Koutsourelakis PS. Benchmark study on reliability estimation in higher dimensions of structural systems. In URL: http://www.uibk.ac.at/mechanik/Publications/benchmark.html. Institute of Engineering Mechanics, Leopold-Franzens University, Innsbruck, Austria, 2004], i.e. in particular linear systems with random properties. The notion of design point excitation for non-linear systems is discussed and its use extended for reliability estimations of conservative non-linear MDOF systems considering critical conditional excitation.For solving the hysteretic MDOF system with uncertain structural parameters subjected to general Gaussian excitation, however, the general applicable subset procedure [Au SK, Beck JL. Estimation of small failure probabilities in high dimensions by subset simulation. Probab Eng Mech 2001;16:263–277] has been used combined with Importance Sampling.     

钢框架结构抗震可靠度的概率重要性分析

概率重要度是一类特殊的参数灵敏度,对于结构可靠度设计、优化和评定等具有重要的价值。本文引进四类重要性测度,即重要性向量α、γ、δ和η。前两个测度分别描述标准正态空间内和原始空间内随机变量的本质特征和相对重要性程度以及可靠指标对设计点变化的灵敏性;后两个测度分别描述可靠指标对随机变量均值和标准差变化的灵敏性。采用基于FORM的有限元可靠度方法对钢框架结构进行抗震可靠度分析和概率重要性分析,以一个实际工程结构为例,分析了承载能力和变形能力极限状态抗震可靠度的概率重要性,结果表明:四类重要性测度的变化规律基本一致,将概率重要度很小的随机变量作为确定性变量处理,可以显著地提高大型复杂结构可靠度分析的计算效率。     

非线性Mohr-Coulomb破坏准则下边坡可靠度上限

传统边坡可靠度分析往往在岩土参数服从线性Mohr-Coulomb(简称线性M-C)破坏准则的假设条件下进行,并且常常采用极限平衡法或有限元法计算安全系数。然而,岩土介质破坏准则具有一定的非线性。为能更加实际地描述岩土破坏机理和得到严格精确的解,基于非线性MohrCoulomb(简称非线性M-C)破坏准则,结合极限分析上限法和蒙特卡洛法,进行边坡可靠度上限分析。当非线性参数m=1时,与等效的线性M-C破坏准则进行对比计算,验证了方法的可行性。同时,将初始粘聚力、内摩擦角arctan(c0/σt)和非线性参数作为随机变量且服从截断正态分布,进行了参数变异性和敏感性影响分析。研究表明:非线性M-C破坏准则下,边坡可靠度随初始粘聚力、内摩擦角arctan(c0/σt)和非线性参数变异性的增大而减小;边坡可靠度随初始粘聚力和内摩擦角arctan(c0/σt)的增大而增大,随非线性参数的增大而减小。     

A Monte Carlo method based on first- and second-order reliability approximations

A Monte Carlo sampling method based on both first- and second-order approximations to the failure region is proposed for the calculation of structural failure probabilities. The method applies to smooth failure surfaes and involves consideration of the hyperplane tangential to the failure surface at the design point and a hyperparabolic approximation to the failure surface. The efficiency of the method is illustrated with two examples and comparisons made with some other Monte Carlo methods.     

System reliability analysis by enhanced Monte Carlo simulation

The main focus of this paper is on the development of a Monte Carlo based method for estimating the reliability of structural systems. The use of Monte Carlo methods for system reliability analysis has several attractive features, the most important being that the failure criterion is usually relatively easy to check almost irrespective of the complexity of the system. The flip side of such methods is the amount of computational efforts that may be involved. However, by reformulating the reliability problem to depend on a parameter and exploiting the regularity of the failure probability as a function of this parameter, it is shown that a substantial reduction of the computational efforts involved can be obtained.     

基于概率密度演化的锈蚀混凝土梁时变可靠性分析

建立了锈蚀钢筋混凝土梁的极限状态函数，其中引入了锈蚀不均匀系数考虑钢筋锈蚀不均匀性和随机性。根据概率守恒原理引出锈蚀钢筋混凝土梁极限状态函数的广义概率密度演化方程，并介绍了其TVD(total variation diminishing)差分格式的有限差分方法。运用“吸收边界条件”提出了基于概率密度演化理论的锈蚀钢筋混凝土梁时变可靠度计算方法。以三个锈蚀钢筋混凝土梁为例，展示广义概率密度演化方程以及时变可靠度的计算结果。通过100万次Monte Carlo模拟方法以及2范数误差指标考察了概率密度演化及时变可靠度计算结果的精准度。二阶矩信息和可靠度的对比结果显示：概率密度演化法能够以较小的计算代价，较为精确地捕捉锈蚀钢筋混凝土梁在服役期间的概率密度演化信息，较为准确地预测其时变可靠度。     

Variance reduction by truncated multimodal importance sampling

This paper presents an importance sampling method for system reliability analyses of structures with failure domains defined by linear or appropriately linearized surfaces. Truncated simulation as a new technique is suggested. It provides an advantage of locating all samples in the failure domain and thus increasing computation efficiency. The variance of the estimator is evaluated by an analytically derived upper bound. It is compared with that of the conventional Monte Carlo method by a variance change factor for conservative estimation of the increase in accuracy and efficiency. The upper bound of variance can be used for a priori determination of the required sample size, given an acceptable maximum error associated with a confidence level. Various application examples of both series and parallel systems are included for illustration.     

An effective approximation to evaluate multinormal integrals

In structural system reliability theory, the evaluation of multivariate normal distributions is an important problem. Numerical integration of multinormal distributions with high accuracy and efficiency is known to be impractical when the number of distribution dimensions is large, typically greater than five. The paper presents a practical and effective approach to approximate a multinormal integral by a product of one-dimensional normal integrals, which are easy to evaluate. Examples considered in the paper illustrate a remarkable accuracy of the approximation in comparison with exact integration. Unlike a first-order multinormal approximation widely used in the literature, this method does not involve any iterative linearization, minimization or integration. Computational simplicity with high accuracy is the major advantage of the proposed method, which also highlights its potential for estimating reliability of structural systems.