Structural Analysis with Fuzzy Variables

Abstract: The analysis of existing structures requires engineers to model two types of uncertainty, cognitive and non-cognitive. The objective of this paper is to reexamine structural analysis methods by considering the cognitive type of uncertainty. Two analytical approaches are proposed for this purpose: (1) combining the displacement method with fuzzy arithmetic and (2) considering all possible permutations of extreme values of any uncertain variables in a structure using the displacement method. The first approach, which is based on fuzzy arithmetic, requires less computing time as compared with the permutations method but only obtains approximate solutions. However, the second approach produces the exact solution. For the purpose of illustration, the modulus of elasticity E is assumed to be an uncertain variable and is modeled as a triangular fuzzy number. The structural behavior was investigated due to this cognitive uncertainty in E. The results based on the second approach show that if E is a triangular fuzzy number, the member forces can be either fuzzy numbers or crisp values, depending on the structural type. In addition, modified definitions for fuzzy division and fuzzy subtraction are proposed in this paper. Applications of these modified definitions and proposed methods are also presented.     

基于支持向量机的边坡可靠性分析

将支持向量机与一阶二次矩方法结合,提出了边坡可靠性分析的支持向量机方法。利用极限平衡分析构造学习样本,通过支持向量机学习,建立安全系数与随机变量之间映射关系的支持向量机表达,进而实现边坡极限状态函数及其偏导数的显式表达,从而计算边坡的可靠性指标。该方法避免了传统可靠性分析的缺点。利用一个算例进行了分析,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。     

Multicut‐High Dimensional Model Representation for Structural Reliability Bounds Estimation Under Mixed Uncertainties

Abstract: In reliability analysis of structural systems involving both aleatory and epistemic uncertainties, in conjunction with multiple design points, every configuration of the interval variables is to be explored to determine the bounds on reliability. To reduce the computational cost involved, this article presents a novel uncertain analysis method for estimating the bounds on reliability of structural systems involving multiple design points in the presence of mixed uncertain (both random and fuzzy) variables. The proposed method involves Multicut‐High Dimensional Model Representation (MHDMR) technique for the limit state/performance function approximation, the transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. The limit state function approximation is obtained by linear and quadratic approximations of the first‐order HDMR component functions at the most probable point. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by the sample points, as compared to full‐scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through four examples involving explicit/implicit performance functions.     

既有建筑结构体系可靠性评估实用方法

回顾和总结了结构可靠度理论的发展历史以及既有建筑结构评估的研究现状,指出现有结构体系可靠性评估方法的不足.针对存在的问题,提出了基于子结构的既有建筑结构体系可靠性评估的简化方法,建议将子结构取为自然层,先对子结构进行可靠性评估,再根据子结构评定结果,通过结构自然层间的相关关系对结构体系可靠性进行评估.根据层次分析法,提出了一种适合工程应用的子结构中构件权重系数实用计算方法.工程实例分析结果表明,所提方法得出的评定结果能更好地符合实际情况.     

Lifetime cost optimization of structures by a combined condition–reliability approach

Cost optimization for the determination of the most effective maintenance strategy of deteriorating structures has recently been used by researchers and accepted by experts. The main reasons for this lie in strict requirements of budgetary efficiency and in reliability requirements for structures. Accessible data directly related to structural performance include the condition state, obtained from generally stipulated visual inspections, and the reliability state, derived from the inspection or monitoring programs combined with numerical computations. In this paper, a novel approach to cost models for condition and reliability profiles is described. Global cost optimization can be achieved by means of multi-objective optimization or by means of a Cost-Optimized Condition-Reliability Profile (COCRP) approach. Since the assessment of structures is under uncertainty, it is essential to embed the COCRP concept in a full probabilistic framework. Probabilistic COCRP computations can be performed, even for complex structures, within a reasonable time by using advanced Monte Carlo Methods such as Latin Hypercube Sampling. The proposed COCRP approach allows realistic and efficient treatment of structures that involve uncertainties and detects the parameters having the most significant effects on lifetime cost.     

Reliability analysis with nondestructive inspection

Nondestructive inspection tools are commonly used to inspect structures or structural components with resistance deterioration due to defect size growth. The quality of the tools is mainly defined by the rate of detecting a defect with defect size s, (s), and the accuracy in sizing a detected defect. The uncertainty of sizing a detected defect can be incorporated in limit state functions that include defect size, and a reliability evaluation can be carried out with the efficient first-order reliability method (FORM). The rate of detecting a defect can also be incorporated in the reliability evaluation of an inspected structure or structural component. This is done, in this paper, by introducing a standard normally distributed variate, Z, and defining a limit state function as a function of and (s). Advantages of using this limit state function, rather than using a limit state function based on the actual defect size and the critical defect size distributed according to the rate of detection curve, are discussed. It is shown that one only needs to use the mean rate of defect detection curve to consider the uncertainty in the rate of detection. The incorporation of the uncertainty in rate of detection for reliability updating analysis with inspection results, and for reliability-based selection of optimal inspection and maintenance schedule for resistance deteriorating structures are also presented. The proposed approach is illustrated by two examples in evaluating reliability with inspection information and in selecting an optimal inspection and maintenance schedule by minimizing the probability of time to failure before inspection and before the time at the end of remaining service life.     

Structural reliability analysis for implicit performance functions using artificial neural network

The Monte-Carlo simulation (MCS), the first-order reliability methods (FORM) and the second-order reliability methods (SORM), are three reliability analysis methods that are commonly used for structural safety evaluation. The MCS requires the calculations of hundreds and thousands of performance function values. The FORM and SORM demand the values and partial derivatives of the performance function with respect to the design random variables. Such calculations could be time-consuming or cumbersome when the performance functions are implicit. Such implicit performance functions are normally encountered when the structural systems are complicated and numerical analysis such as finite element methods has to be adopted for the prediction.To address this issue, this paper presents three artificial neural network (ANN)-based reliability analysis methods, i.e. ANN-based MCS, ANN-based FORM, and ANN-based SORM. These methods employ multi-layer feedforward ANN technique to approximate the implicit performance functions. The ANN technique uses a small set of the actual values of the implicit performance functions. Such a small set of actual data is obtained via normal numerical analysis such as finite element methods for the complicated structural system. They are used to develop a trained ANN generalization algorithm. Then a large number of the values and partial derivatives of the implicit performance functions can be obtained for conventional reliability analysis using MCS, FORM or SORM. Examples are given in the paper to illustrate why and how the proposed ANN-based structural reliability analysis can be carried out. The results have shown the proposed approach is applicable to structural reliability analysis involving implicit performance functions. The present results are compared well with those obtained by the conventional reliability methods such as the direct Monte-Carlo simulation, the response surface method and the FORM method 2.     

模糊随机可靠度计算的验算点法

本文通过引入模糊随机变量的确定率和广义可靠性指标,将工程结构的模糊随机可靠度与模糊随机信息的数字特征联系起来,提出了计算模糊随机可靠度的简化实用方法——验算点法,其计算工作量与普通可靠度计算的验算点大致相当。     

Seismic system reliability analysis of bridges using the multiplicative dimensional reduction method

Abstract

A combined method of finite element reliability analysis and multiplicative dimensional reduction method (M-DRM) is proposed for systems reliability analysis of practical bridge structures. The probability distribution function of a structural response is derived based on the maximum entropy principle. To illustrate the accuracy and efficiency of the proposed approach, a simply supported bridge structure is adopted and the failure probability obtained are compared with the Monte Carlo simulation method. The validated method is then applied for the system reliability analysis for a practical high-pier rigid frame railway bridge located at the seismic-prone region. The finite element model of the bridge is developed using OpenSees and the M-DRM method is used to analyse the structural system reliability under earthquake loading.     

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.     

基于区间截断法的地下结构模糊能度可靠性模型研究

根据地下结构稳定性影响因素的随机性和模糊性特点,首先,引进三角模糊数理论,建立地下结构物理力学参数的可能性分布函数,然后,在地下结构稳定性评价之模糊可靠性功能函数确定方法研究基础上,引进区间截断法和区间数运算规则,建立了功能函数值域区间的确定方法,并避免了功能函数值域的扩展问题,最后,通过研究地下结构模糊能度可靠性度量方法,建立了地下结构模糊可靠性指标的可能性分布曲线及其失效可能度的确定方法,从而建立了地下结构模糊能度可靠性分析模型与方法。工程实例分析与计算结果表明了本文模型和方法的合理性与可行性。     

Lifetime-oriented multi-objective optimization of structural maintenance considering system reliability, redundancy and life-cycle cost using GA

The need to design and construct structural systems with adequate levels of reliability and redundancy is widely acknowledged. It is as crucial that these desired levels are maintained above target levels throughout the life of the structure. Optimization has served well in providing safer and more economical maintenance strategies. Lifetime maintenance optimization based on system reliability has already been proposed. It is still needed, however, to incorporate redundancy in the lifetime maintenance optimization process. Treating both system reliability and redundancy as criteria in the lifetime optimization process can be highly rewarding. The complexity of the process, however, requires the automation of solving the optimization problem. Genetic algorithms (GAs) are used in this study to obtain solutions to the multi-objective optimization problems considering system reliability, redundancy and life-cycle cost (LCC). An approach to provide the optimization program the ability to optimally select what maintenance actions are applied, when they are applied, and to which structural components they are applied is presented. Two different strategies are proposed. The first strategy has the ability to optimally select mixed maintenance types to apply to different parts of the structure at the same time. This strategy can be used in cases where any combination of different maintenance options can be practically applied to any part of the structure. The application of this strategy on truss structures is shown in a numerical example. The second strategy can be used when a limited number of possibilities of practical maintenance options are available. The application of this strategy to bridge structures is shown in a numerical example. The greatest advantage of the proposed approach (both strategies) is its ability to avoid the application of maintenance interventions to structural components that are not critical.     

钢筋混凝土结构基于耐久性劣化度的可靠性分析

结构耐久性的不足将引发其性能的劣化,并导致结构可靠度的降低。根据已建立的两种劣化度模型及极限状态法,以碳化深度和锈胀开裂两种不同的耐久性劣化度为指标,提出了混凝土结构使用寿命全过程可靠度的计算方法。结合工程实例,以荷载作用下的混凝土结构在碳化环境中的劣化问题为例,构建极限状态函数,基于两种劣化度模型,对其运用可靠度理论进行分析,得到了基于耐久性劣化度的结构可靠度求算方法。研究结果表明：可靠度与结构的混凝土保护层厚度及碳化速率的统计数值或钢筋锈蚀量密切相关。计算方法可为分析在役工程的耐久性提供参考。参14     

Consistent distribution parameter estimation for reliability analysis

The estimation of parameters for structural reliability analysis is subject to a consistency principle that may be stated as follows. The distribution function for any random variable should be a monotonically non-increasing function of all sample values of the variables. This principle is violated by the conventional fitting of normal or lognormal distributions using sample mean and sample variance. An alternative method is suggested and shown not to violate the principle.In reliability work the quality of the fit in a tail region of a distribution is particularly important. Conventional methods of parameter estimation are unsuited for this purpose. A new approach to produce a fitting that is optimal for structural reliability applications is presented. This approach minimizes the weighted deviation from a set of target reliabilities over a specified set of loadings. Simple versions of this method are shown to satisfy the consistency principle. The method is illustrated by examples.     

抗震结构的模糊震害预测与损失估计

本文在我们提出的模糊随机振动和模糊动力可靠性分析的基础上,提出了一种抗震结构震害预测与损失估计的实际可行的方法。此方法考虑了预测地震烈度的模糊随机性、地震地面运动的随机过程性,结构震害等级的模糊性和结构在强震作用下的弹塑性。用此方法可以较好地预测抗震结构在使用期限内的失效概率和震害损失,从而为结构以可靠性为主要约束或以“结构造价和损失期望之和最小”为目标的结构优化设计提供了基础。     

Developing a Smart Structure Using Integrated Subspace‐Based Damage Detection and Semi‐Active Control

Damage detection in large building structures has always faced challenges due to analyzing the large amount of measured data. In this article, a new damage detection approach based on subspace method is proposed to identify damages using limited output data. Also, a new scheme is presented to develop a smart structure by integrating structural health monitoring with semi‐active control strategy. If damage occurs in such a structure under severe excitations, the proposed scheme has the capability to exert necessary control forces in order to compensate for damage and reduce simultaneously the dynamic response of the structure. The reliability and feasibility of the proposed method are demonstrated by implementing the technique to two shear building structures with semi‐active control devices. Results show that the damage could be identified accurately with saving time and cost due to less computation even under noise existence; and dynamic response is significantly reduced in the smart structure.     

An improvement of the response surface method

The coupling of the Monte Carlo method and the finite element method for the reliability analysis of structures leads often to a prohibitive computational cost. The response surface method is a powerful reliability method that approximates the limit state function with a polynomial expression using the values of the function at specific points. This type of analytical function replaces the exact limit state function in the Monte Carlo simulation. Therefore, the computational effort required for the assessment of the reliability of structural systems can be reduced significantly. The position of the sample points and the type of polynomial response surface have been investigated by several authors and the performance of the response surface method is still under discussion. In this paper an improvement of the response surface method is proposed. An iterative strategy is used to determine a response surface that is able to fit the limit state function in the neighbourhood of the design point. The locations of the sample points used to evaluate the free parameters of the response surface are chosen according to the importance sensitivity of each random variable. Several analytical and structural examples are considered to demonstrate the advantages of the proposed improvement.     

Structural dynamic optimal design based on dynamic reliability

In this work, dimension and shape optimization of structures under stochastic process excitation is addressed in the context of element or system dynamic reliability constraints, where the structural gross mass is taken to be the objective function. Firstly, based on the dynamic response analysis of truss structures under stochastic process loads, the dynamic reliability constraints are developed and simplified, and the normalization of design variables is discussed to avoid some variables being drowned by others during optimization due to their different dimensions and orders of magnitude. The optimal models of dimension and shape with element or system dynamic reliability constraints are then presented. Two numerical examples are finally used to illustrate the results of different optimal designs, which demonstrate that the efficiency to solve the structural optimization with dynamic reliability constraints can be significantly improved if the design variables and their initial values are selected properly.     

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。     

Probability analysis method using Fast Fourier transform and its application

This paper presents an efficient probability analysis method using Fast Fourier transform taking into account that (1) a probability density function and its characteristic function are a pair of Fourier transforms, and (2) the characteristic function of the sum of statistically independent random variables is given by the product of the characteristic functions of each random variable. The efficient probability analysis method is extended for the sum of correlated non-normal random variables with correlation coefficients expressed by the power of a common value. Expressing the correlated basic variables in terms of two sets of uncorrelated random variables, the characteristic function of their sum can be estimated approximately by the characteristic functions of the correlated basic variables. The probability density function of the sum is obtained using Fast Fourier transform. The accuracy and applicability of the approximation method are investigated based on the estimation error of the cumulant and using numerical examples. Applications of the proposed method to various aspects of structural reliability analysis and design are illustrated.