联合分布函数蒙特卡罗模拟及结构可靠度分析

针对复杂极限状态方程可靠度计算问题,提出了基于理论联合分布函数以及2 种近似联合分布函数的结构失效概率蒙特卡罗模拟方法,并给出了计算流程图.采用2 个算例证明了所提方法的有效性.结果表明:所提的失效概率模拟方法的计算精度很高,尤其适用于复杂极限状态方程的可靠度计算问题.2 种联合分布函数近似构造方法得到的失效概率精度相当,近似方法与精确方法结果的差异随失效概率的减小而增大,而且随着变量间相关性的增加而增加.当失效概率小于10-3时,近似方法的失效概率误差较大.     

First excursion probabilities for linear systems by very efficient importance sampling

An analytical study of the failure region of the first excursion reliability problem for linear dynamical systems subjected to Gaussian white noise excitation is carried out with a view to constructing a suitable importance sampling density for computing the first excursion failure probability. Central to the study are ‘elementary failure regions’, which are defined as the failure region in the load space corresponding to the failure of a particular output response at a particular instant. Each elementary failure region is completely characterized by its design point, which can be computed readily using impulse response functions of the system. It is noted that the complexity of the first excursion problem stems from the structure of the union of the elementary failure regions. One important consequence of this union structure is that, in addition to the global design point, a large number of neighboring design points are important in accounting for the failure probability. Using information from the analytical study, an importance sampling density is proposed. Numerical examples are presented, which demonstrate that the efficiency of using the proposed importance sampling density to calculate system reliability is remarkable.     

基于概率密度演化理论的结构抗震可靠性分析

运用随机过程的正交展开方法,将地震动加速度过程表示为由10个左右的独立随机变量所调制的确定性函数的线性组合形式。结合概率密度演化方法和等价极值事件的基本思想,研究了非线性结构的抗震可靠度分析问题。以具有滞回特性的非线性结构为例,对某一多自由度的剪切型框架结构进行了抗震可靠性分析。结果表明：按照复杂失效准则计算的结构抗震可靠度较之结构各层抗震可靠度均低。这一研究为基于概率密度函数的、精细化的抗震可靠度计算提供了新的途径。     

A load space formulation for probabilistic finite element analysis of structural reliability

A load space formulation for calculating the failure probability of complex structures for which the limit state functions are implicit is described in this paper. This formulation is used in conjunction with probabilistic finite element (PEE) analysis and employs a directional simulation to calculate the structural reliability. Apart from the advantage that a lower order space is used, the main advantage of the load space formulation proposed in this paper is that the number of inversions of the structural stiffness matrix and/or its gradients with respect to the material property random variables is reduced dramatically when compared with the usual Monte Carlo simulation (MCS) method. When used in a finite element reliability analysis, this procedure can save significant amounts of CPU time. Numerical examples are presented to show the efficiency and accuracy of the proposed approach.     

Mechanical/structural reliability evaluation through Esscher''s method of approximation

A simple and effective computational procedure is presented for computing probabilities of failure of mechanical and structural systems. The method provides very close approximate solutions for well behaved linear limit state functions with independent normal or nonnormal, continuous or discrete random variables. But the random variables should have moment generating functions. This method is also applicable when the limit state function is a hyper sphere. Numerical examples are given including one for hyper sphere limit state.     

考虑状态模糊性时广义失效概率计算的矩方法

针对失效状态和安全状态具有模糊性的广义可靠性分析问题,提出了一种广义失效概率计算的矩方法。所提方法首先将广义失效概率的积分区域依据功能函数的取值离散化,在离散化的积分区域中,极限状态函数对模糊失效域的隶属函数近似保持为常数,从而将模糊可靠性问题转化为一般的随机可靠性问题,进而可以利用近似的矩方法求得广义失效概率。该文给出了所提方法的实现步骤和原理,算例结果表明所提方法对于中低维问题具有很高的精度和效率。     

Reliability analysis of nonlinear laminated composite plate structures

A procedure for the reliability analysis of laminated composite plate structures subjected to large deflections under random static loads is presented. The nonlinear analysis of laminated composite plate structures is achieved via a corotational total Lagrangian finite element formulation which is based on the von Karman assumption and first order shear deformation theory. This formulation is applicable for the nonlinear analysis of plate structures with large rotations but moderate deformation and thus accurate enough to predict the behavior of the structures at the point of failure. The reliability assessment of laminated composite plate structures with random strength subjected to random loads is approached by the determination of limit state surfaces in load space. The limit space surfaces are obtained by performing a series of first ply failure analyses following different load paths in load space using the proposed nonlinear structural analysis technique and an appropriate failure criterion. A numerical technique is then proposed to evaluate the reliability of the plate structures. Examples of the reliability analyses of laminated plates with different layer orientations subject to random loads are given for illustration.     

结构动力可靠度计算的修正条件反应法

基于反应时程分解的子集模拟法是计算受随机激励作用的结构小失效概率的高效算法,该法将失效域划分为子集序列,通过分解反应时程来生成子集计算所需的样本。条件反应法计算动力可靠度时,通过线性反应来估计非线性反应,或通过单自由度结构的反应来估计多自由度结构的反应。该文将基于反应时程分解的子集模拟法用于条件反应法中,提出修正的条件...     

Set theoretic formulation of performance reliability of multiple response time‐variant systems due to degradations in system components

This paper presents a design stage method for assessing performance reliability of systems with multiple time‐variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time‐dependent limit‐state functions. System failure is defined as the non‐conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second‐order bounds. Further, for numerical efficiency probabilities are evaluated by first‐order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

结构动力可靠度计算的区域分解法和重要抽样法

Lambros Katafygiotis提出的区域分解法和S.K.Au提出的重要抽样法,被用来计算受高斯随机过程作用的线性动力体系的失效概率.在以上两种方法的基础上,给出了非平稳随机地震作用下,单自由度线性体系动力可靠度的区域分解法和重要抽样法的计算步骤,对以上两种方法补充给出了选取失效域下标的措施;其中区域分解法将结构反应用离散脉冲响应函数表示,补充给出了极限状态函数系数的具体确定方法.通过算例对比分析了区域分解法和重要抽样法的计算效率,结果表明当计算小失效概率时,区域分解法的计算效率同重要抽样法一样高效,而对比Monte-Carlo法有显著提高,且两种方法的计算精度接近.     

Towards error bounds of the failure probability of elastic structures using reduced basis models

Structural reliability methods aim at computing the probability of failure of systems with respect to prescribed limit state functions. A common practice to evaluate these limit state functions is using Monte Carlo simulations. The main drawback of this approach is the computational cost, because it requires computing a large number of deterministic finite element solutions. Surrogate models, which are built from a limited number of runs of the original model, have been developed, as substitute of the original model, to reduce the computational cost. However, these surrogate models, while decreasing drastically the computational cost, may fail in computing an accurate failure probability. In this paper, we focus on the control of the error introduced by a reduced basis surrogate model on the computation of the failure probability obtained by a Monte Carlo simulation. We propose a technique to determine bounds of this failure probability, as well as a strategy of enrichment of the reduced basis, based on limiting the bounds of the error of the failure probability for a multi‐material elastic structure. Copyright © 2017 John Wiley & Sons, Ltd.     

Fatigue strength functions in shear loading of fabric conveyor belts

The fatigue behaviour of three types of fabric conveyor belt subjected to shear loading is studied. The fatigue phenomenon of belts is explained by the decrease of shear strength, which is determined as an ultimate angle initiating the lamination failure of the central rubber layers. Tests with precycling have shown that the permanent shear strength is diminished during precycling and can be modelled by power functions of the number of cycles, called fatigue strength functions (FSFs). Two main hypotheses of FSFs' existence and state equivalence are tested. A comparison between the proposed method and a linear damage accumulation rule is also carried out for two-level loading.     

Evaluating small failure probabilities of multiple limit states by parallel subset simulation

A novel subset simulation algorithm, called the parallel subset simulation, is proposed to estimate small failure probabilities of multiple limit states with only a single subset simulation analysis. As well known, crude Monte Carlo simulation is inefficient in estimating small probabilities but is applicable to multiple limit states, while the ordinary subset simulation is efficient in estimating small probabilities but can only handle a single limit state. The proposed novel stochastic simulation approach combines the advantages of the two simulation methods: it is not only efficient in estimating small probabilities but also applicable to multiple limit states. The key idea is to introduce a “principal variable” which is correlated with all performance functions. The failure probabilities of all limit states therefore could be evaluated simultaneously when subset simulation algorithm generates the principal variable samples. The statistical properties of the failure probability estimators are also derived. Two examples are presented to demonstrate the effectiveness of the new approach and to compare with crude Monte Carlo and ordinary subset simulation methods.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

基于加权线性响应面法的支持向量机可靠性分析方法

针对估算非线性隐式极限状态函数的失效概率问题,提出了一种基于加权线性响应面法的支持向量机可靠性分析方法。首先采用加权线性响应面确定设计点,在线性响应面迭代的同时获得一定数量的样本,然后在这些样本和设计点附近补充抽取样本的基础上,采用具有良好小样本学习能力的支持向量机方法来训练样本,保证了在设计点周围获得更好的非线性极限状态函数的替代。这种方法既保证了对设计点的精确近似,又保证了对设计点附近非线性极限状态函数的良好近似,大大提高了失效概率的计算精度,为非线性隐式极限状态的可靠性分析提供了一种合理可行的方法。     

Approaches for predicting the probability of failure of bridges subjected to high-speed trains

The scope of this paper is to evaluate different approaches for the prediction of the probability of failure of uncertain railway bridges subjected to high-speed trains. The peak acceleration of a bridge, which is commonly the governing response quantity for dynamic bridge design and failure, depends strongly on the type of train and the train speed. Since in many cases the critical speeds related to response maximums are below the design speed and failure, and during operation the speed varies up to the design speed, the assessment of the probability of failure is not straightforward. In this contribution, several more sophisticated measures of the probability of failure of the bridge-train interaction problem are proposed, considering the peak acceleration as a function of the speed in a certain interval and the distribution of the actual train speed. These measures are tested on two random test bridges, taking into account the main sources of uncertainty, i.e. damping, track irregularities, and the environmental impact. The mechanical model used for the prediction of the dynamic bridge response is composed of a beam bridge crossed by a planar mass-spring-damper model of the train. In this simplest approach that considers explicitly dynamic bridge-train interaction, random irregularity profiles capture the effect of track irregularities. It is shown that in certain speed intervals the predicted probability of failure strongly depends on the underlying measure of the probability of failure. In the first example bridge, whose response is governed by a pronounced resonance peak, exceedance of the serviceability limit state is predicted by all measures at virtually the same speed. The second example problem, where track irregularities lead to considerable response amplifications, only some of the measures predict failure. The results of this study may serve as an impulse for a more in-depth discussion on the appropriate prediction of the probability of failure of bridge-train interaction.     

Stochastic finite element analysis of layered composite beams with spatially varying non-Gaussian inhomogeneities

This study focuses on the development of a stochastic finite element-based methodology for failure assessment of composite beams with spatially varying non-Gaussian distributed inhomogeneities. The material properties in the individual laminae are modeled as non-Gaussian random fields, whose probability density functions and the correlations are estimated from the test data. The non-Gaussian random fields are discretized into a vector of correlated non-Gaussian random variables using the optimal linear expansion scheme that preserves the second-order non-Gaussian characteristics of the fields. Subsequently, the estimates of the failure probability are obtained from Monte Carlo simulations carried out on the vector of correlated random variables. Issues related to the computational efficiency of the proposed framework and the variabilities in the material properties are discussed. Numerical examples are presented, which highlight the salient features of the proposed method.     

Hybrid perturbation-Galerkin methods for structural reliability analysis

A major challenge in the reliability analysis of complex structures is to determine the response function or surface from which the limit state function can be established and hence the failure probability of the structure can be predicted. The intention of this paper is to propose a new class of hybrid perturbation-Galerkin methods to determine the response function (surface) of the structure. The proposed methods, mainly including single-variable and double-variable approximations, are based on a combination of the perturbation technique and Bubnov–Galerkin projection, where various orders of summation terms of power polynomial expansions are used as the Galerkin trial functions or basis vectors. A series of examples is provided to demonstrate the accuracy and efficiency of the proposed method by comparison with various existing methods. It is found from the numerical examples that with a significantly lower computational effort with respect to the existing reliability methods, very accurate results can be obtained by the proposed methods.     

Reliability of a Multi‐State Computer Network Through k Minimal Paths Within Tolerable Error Rate and Time Threshold

A computer system is usually modeled as a network topology where each branch denotes a transmission medium and each vertex represents a station of servers. Each branch has multiple capacities/states due to failure, partial failure, and maintenance. Such a network is named a multi‐state computer network (MSCN). From the viewpoint of quality management, transmission error rate and transmission time are both critical performance indicators to assess Internet quality for system managers and customers. Within both tolerable error rate and time threshold, the addressed problem is concentrated on an MSCN for computing the probability that d units of data can be sent through multiple minimal paths simultaneously. Such a probability is named system reliability. A solution procedure including an efficient algorithm based on MPs is proposed to derive the lower boundary vectors (LBVs) meeting the requirements. Then system reliability, which is represented as the probability of union of subsets, can be subsequently evaluated by the LBVs. Copyright © 2015 John Wiley & Sons, Ltd.     

A new artificial neural network-based response surface method for structural reliability analysis

This paper presents a new artificial neural network-(ANN)based response surface method in conjunction with the uniform design method for predicting failure probability of structures. The method involves the selection of training datasets for establishing an ANN model by the uniform design method, approximation of the limit state function by the trained ANN model and estimation of the failure probability using first-order reliability method (FORM). In the proposed method, the use of the uniform design method can improve the quality of the selected training datasets, leading to a better performance of the ANN model. As a result, the ANN dramatically reduces the number of required trained datasets, and shows a good ability to approximate the limit state function and then provides a less rigorous formulation in the context of FORM. Results of three numerical examples involving both structural and non-structural problems indicate that the proposed method provides accurate and computationally efficient estimates of the probability of failure. Compared with the conventional ANN-based response surface method, the proposed method is much more economical to achieve reasonable accuracy when dealing with problems where closed-form failure functions are not available or the estimated failure probability is extremely small. Finally, several important parameters in the proposed method are discussed.