Reliability-based optimization of structural systems by adaptive target safety – Application to RC frames

Reliability-based design optimization (RBDO) aims to find the best compromise between cost reduction and safety assurance. Traditionally, component optimization is defined by minimizing the structural cost under a prescribed reliability target for a single limit state. However, as structural failure is rarely devoted to only one component, the system approach becomes necessary to deal with realistic applications. In this paper, a methodology for system reliability-based design optimization (SRBDO) is proposed. Instead of specifying identical predefined component targets, the method is based on adaptive target reliabilities for structural components. An updating procedure is included in the optimization process to ensure the required system reliability. The proposed method aims to find the best compromise between satisfying the target system reliability and optimizing the component performance. The application to reinforced concrete structures shows the interest of the adaptive target reliabilities as well as the efficiency of the updating procedure.     

Reliability-based design optimization with discrete design variables and non-smooth performance functions: AB-PSO algorithm

The purpose of reliability-based design optimization (RBDO) is to find a balanced design that is not only economical but also reliable in the presence of uncertainty. Practical applications of RBDO involve discrete design variables, which are selected from commercially available lists, and non-smooth (non-differentiable) performance functions. In these cases, the problem becomes an NP-complete combinatorial optimization problem, which is intractable for discrete optimization methods. Moreover, the non-smooth performance functions would hinder the use of gradient-based optimizers as gradient information is of questionable accuracy. A framework is presented in this paper whereby subset simulation is integrated with a new particle swarm optimization (PSO) algorithm to solve the discrete and non-smooth RBDO problem. Subset simulation overcomes the inefficiency of direct Monte Carlo simulation (MCS) in estimating small failure probabilities, while being robust against the presence of non-smooth performance functions. The proposed PSO algorithm extends standard PSO to include two new features: auto-tuning and boundary-approaching. The former feature allows the proposed algorithm to automatically fine tune its control parameters without tedious trial-and-error procedures. The latter feature substantially increases the computational efficiency by encouraging movement toward the boundary of the safe region. The proposed auto-tuning boundary-approaching PSO algorithm (AB-PSO) is used to find the optimal design of a ten-bar truss, whose component sizes are selected from commercial standards, while reliability constraints are imposed by the current design code. In multiple trials, the AB-PSO algorithm is able to deliver competitive solutions with consistency. The superiority of the AB-PSO algorithm over standard PSO and GA (genetic algorithm) is statistically supported by non-parametric Mann-Whitney U tests with the p-value less than 0.01.     

A general procedure for first/second-order reliabilitymethod (FORM/SORM)

First/second-order reliability method (FORM/SORM) is considered to be one of the most reliable computational methods for structural reliability. Its accuracy is generally dependent on three parameters, i.e. the curvature radius at the design point, the number of random variables and the first-order reliability index. In the present paper, the ranges of the three parameters for which FORM/SORM is accurate enough are investigated. The results can help us to judge when FORM is accurate enough, when SORM is required and when an accurate method such as the inverse fast Fourier transformation (IFFT) method is required. A general procedure for FORM/SORM is proposed which includes three steps: i.e. point fitting limit state surface, computation of the sum of the principal curvatures K_s and failure probability computation according to the range of K_s. The procedure is demonstrated by several examples.     

地震时饱和砂土的失效概率

用Excel 2 0 0 0中的规划求解软件对大量的液化和未液化数据做了可靠度计算。由可靠性分析所得的液化概率 ,本文将其定义为失效概率Pf。以这些分析结果为基础讨论了加速度变量的分布和它的变异系数对失效概率的影响。对于给定的概率分布 ,失效概率在很大程度上与它的变异系数有关 ;但是 ,似乎与其分布关系不大 (随机变量的变异系数保持不变 )。同样 ,也以这些分析结果为基础 ,讨论了失效概率函数Pf=f(Fs)。Fs 为安全系数。本文着重指出 ,基于饱和砂土极限状态函数的可靠度分析 ,如果对随机变量的不确定性做了合理的考虑 ,则由可靠度分析所得的失效概率函数Pf=f(Fs)和Juang等人用Bayesian定理所得的液化概率函数PL =f(Fs)几乎是一样的。液化概率函数可为场地液化势的合理评估和工程风险设计决策提供依据。     

Reliability sensitivity method by line sampling

Reliability sensitivity refers to the derivative of the failure probability with respect to the distribution parameter of basic random variable. Conventionally, this requires repetitive evaluations of the failure probability for different distribution parameters, which is a direct but computationally expensive task. An efficient simulation algorithm is presented to perform reliability sensitivity analysis using the line sampling technique, which gives a good failure probability evaluation for high-dimensional problems and still presents a comparative one for low-dimensional problems. On the basis of the line sampling procedure for failure probability analysis, the concept and implementation are presented for reliability sensitivity. It is shown that the desired information about reliability sensitivity can be obtained by a very limited increase of computation effort based on the failure probability analysis by the line sampling technique. The presented reliability sensitivity algorithm is more efficient than the one based on the direct Monte Carlo technique, especially for cases where the failure probability is low and the number of random variables is large, which is illustrated by several examples. Additionally, limitations of the line sampling based reliability sensitivity method are demonstrated by a numerical example as well.     

Plasticity effects on frame member reliability

This paper proposes a computational procedure to correctly construct the nonlinear limit state and estimate the reliability of members in ductile framed structures. Due to the uncertainties in loads and structural properties, the structure could be in one of many different damage states before failure occurs in the member of interest. Since the enumeration of all these damage states and the computation of their effect on member reliability is tedious, the proposed method uses three criteria based on the statistical relationships among the members to efficiently identify those damage states that significantly affect the reliability of the desired member. This is used to determine the polyhedral envelope of the entire failure domain, corresponding to various levels of cumulative damage. The failure probability is then estimated through the union of failure domains defined by each of the linear segments of this polyhedral surface, using second-order bounds.     

Shape design of arch dams under load uncertainties with robust optimization

Due to an increased need in hydro-electricity, water storage, and flood protection, it is assumed that a series of new dams will be build throughout the world. The focus of this paper is on the non-probabilistic-based design of new arch-type dams by applying means of robust design optimization (RDO). This type of optimization takes into account uncertainties in the loads and in the material properties of the structure. As classical procedures of probabilistic-based optimization under uncertainties, such as RDO and reliability-based design optimization (RBDO), are in general computationally expensive and rely on estimates of the system’s response variance, we will not follow a full-probabilistic approach but work with predefined confidence levels. This leads to a bi-level optimization program where the volume of the dam is optimized under the worst combination of the uncertain parameters. As a result, robust and reliable designs are obtained and the result is independent from any assumptions on stochastic properties of the random variables in the model. The optimization of an arch-type dam is realized here by a robust optimization method under load uncertainty, where hydraulic and thermal loads are considered. The load uncertainty is modeled as an ellipsoidal expression. Comparing with any traditional deterministic optimization method, which only concerns the minimum objective value and offers a solution candidate close to limit-states, the RDO method provides a robust solution against uncertainty. To reduce the computational cost, a ranking strategy and an approximation model are further involved to do a preliminary screening. By this means, the robust design can generate an improved arch dam structure that ensures both safety and serviceability during its lifetime.     

Design and sensitivity analysis using the probability-safety-factor method. An application to retaining walls

This paper presents a new method for designing engineering works that makes the classical approach, based on safety factors, and the modern, probability-based, approach compatible, and includes a sensitivity analysis. The method consists of a sequence of classical designs, based on given safety factors, that (a) minimize cost or optimize an alternative objective function, (b) calculate the different failure mode probabilities or their upper bounds, and (c) update the safety factors to satisfy both the safety factors and the failure probability requirements. The process is repeated until convergence. As a result, an automatic design of the engineering work, the safety factors and the corresponding probabilities of failure for all failure modes are obtained. A double safety check is used and the correspondence between safety factors and probabilities of failure for the different modes are easily understood. An advantage of this approach is that the optimization procedure and the reliability calculations are decoupled. In addition, a sensitivity analysis is performed using a method that consists of transforming the data parameters into artificial variables and using the dual associated problem. The method is illustrated by its application to a retaining wall design.     

Reliability-based optimization of steel truss arch bridges

A reliability-based optimization approach is developed and applied to minimize the weight of steel truss arch bridges subject to probabilistic (the overall probability failure of the structure) and deterministic (stress and deflection) constraints. The method intelligently integrates the genetic algorithm (GA), the finite element method and the first order reliability method. A real-coded/integer-coded method is used to realistically represent the values of the design variables. Three GA operators consisting of constraint aggregate selection procedure, arithmetic crossover, and non-uniform mutation are proposed. The finite element method (FEM) and the first order reliability method are used to compute the value of the probabilistic and deterministic constraint functions. A numerical example involving a detailed computational model of a long span steel arch bridge with a main span of 550 m is presented to demonstrate the applicability and merits of the present method. Finally, several important parameters in the present method are discussed.     

Investigation of reliability method formulations in DAKOTA/UQ

Reliability methods are probabilistic algorithms for quantifying the effect of simulation input uncertainties on response metrics of interest. In particular, they compute approximate response function distribution statistics (probability, reliability and response levels) based on specified input random variable probability distributions. In this paper, a number of algorithmic variations are explored for both the forward reliability analysis of computing probabilities for specified response levels (the reliability index approach (RIA)) and the inverse reliability analysis of computing response levels for specified probabilities (the performance measure approach (PMA)). These variations include limit state linearizations, probability integrations, warm starting and optimization algorithm selections. The resulting RIA/PMA reliability algorithms for uncertainty quantification are then employed within bi-level and sequential reliability-based design optimization approaches. Relative performance of these uncertainty quantification and reliability-based design optimization algorithms are presented for a number of computational experiments performed using the DAKOTA/UQ software.     

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。     

火灾时管片可靠度计算及优化研究

受火灾高温的影响,隧道衬砌结构可靠性降低。笔者通过自由变形理论,解析计算可靠度优化模型中的结构内力,明确了计算中的随机变量;从结构可靠度指标几何意义出发,基于优化思想确定了可靠度计算公式;考虑火灾下衬砌管片的损伤特点,提出了火灾下管片的功能函数,建立了管片高温下可靠度计算优化模型;最后通过案例分析并利用非线性规划求解优化问题,得到火灾下衬砌可靠度的变化规律,并对管片截面厚度进行了优化设计。研究表明,常规设计管片各处可靠度指标差异较大,隧道拱顶可靠度最低,10 min后失效概率大于0.023,通过优化设计可使火灾持续30 min后满足基本安全。     

Reliability-based design optimization using convex linearization and sequential optimization and reliability assessment method

In this study, an effective method for reliability-based design optimization (RBDO) is proposed enhancing sequential optimization and reliability assessment (SORA) method by convex linearization. In SORA, reliability estimation and deterministic optimization are performed sequentially. And the sensitivity and function value of probabilistic constraint at the most probable point (MPP) are obtained in the process of finding reliability information. In this study, the convex linearization is constructed by utilizing the sensitivity and function value of the probabilistic constraint at the MPP. So no additional evaluation of the probabilistic constraint is required in the deterministic optimization in SORA. The proposed RBDO method is applied to numerical examples and compared to various RBDO methods. It is shown that the proposed method is very efficient with similar accuracy.     

Component and system reliability analysis of nonlinear reinforced concrete grids with multiple failure modes

The bending moment redistribution is an inherent behavior in reinforced concrete grids, which increases the number of possible critical cross-sections susceptible to reach a limit state. Despite the fact that its influence could be important, the high number of failure cross-sections is not often considered in reinforced concrete reliability analyses. In this paper, a local approach of reliability analysis applied to grid structures is developed by taking into account the dominant failure modes. This approach is based on random sampling coupled with finite element analysis, through the use of a localized response surface technique. A mode selection strategy has been developed to capture the individual failure modes in order to classify their importance in the global system reliability. For large-scale systems, this procedure intends to reduce the global computational time in the reliability analysis. Numerical applications aim to show the effect of internal force redistribution, as well as the efficiency of the proposed approach and the interest of considering multiple failure modes.     

巨-子型有控结构体系的可靠度分析及其优化

采用首次超越破坏机制,定义结构层间位移为极限状态控制指标,确定了基于概率密度演化方法的巨-子型有控结构体系的抗震动力可靠度的计算方法;基于可靠度并结合"等可靠度准则"对其进行优化。编制SAP2000与MATLAB接口程序将SAP2000中的分析结果带入到MATLAB中进行概率密度演化分析。结果表明:巨-子型有控结构在弹塑性阶段依然有很好的响应控制效果;巨–子型有控结构比巨型框架结构具有更高的抗震可靠度;经过优化后MSCSS的响应控制能力及可靠度都得到了显著提高。     

Safety analysis of structures with probability and evidence theory

In safety analysis of structures, classical probabilistic analysis has been a popular approach in engineering. However, it is not always to obtain sufficient information to model all uncertain parameters of structures system by probability theory, especially at early stage of design. Under this circumstance, probability theory (used to model random uncertainty) combined with evidence theory (used to model epistemic uncertainty) may be utilized in safety analysis of structures. This paper proposed a novel method for safety analysis of structures based on probability and evidence theory. Firstly, Bayes conversion method is used as the way for precision of evidence body, and the mean and variance of epistemic uncertain variables is defined. Then epistemic uncertainty variables is transformed to normal random variables by reflection transformation method, and the checking point method (J-C method) is used to solve most probability point and reliability. A numerical example and two engineering examples are given to demonstrate the performance of the proposed method. The results show both precision and computational efficiency of the method is high. Moreover, the proposed method provides basis for reliability-based optimization with the hybrid uncertainties.     

A Refined Methodology for Durability‐Based Service Life Estimation of Reinforced Concrete Structural Elements Considering Fuzzy and Random Uncertainties

Abstract: A reliable method for service life estimation of the structural element is a prerequisite for service life design. A new methodology for durability‐based service life estimation of reinforced concrete flexural elements with respect to chloride‐induced corrosion of reinforcement is proposed. The methodology takes into consideration the fuzzy and random uncertainties associated with the variables involved in service life estimation by using a hybrid method combining the vertex method of fuzzy set theory with Monte Carlo simulation technique. It is also shown how to determine the bounds for characteristic value of failure probability from the resulting fuzzy set for failure probability with minimal computational effort. Using the methodology, the bounds for the characteristic value of failure probability for a reinforced concrete T‐beam bridge girder has been determined. The service life of the structural element is determined by comparing the upper bound of characteristic value of failure probability with the target failure probability. The methodology will be useful for durability‐based service life design and also for making decisions regarding in‐service inspections.     

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.     

Analysis of approximations for multinormal integration in system reliability computation

In the context of first-order reliability analysis, the computation of multivariate normal integrals is a key step in the analysis of the system probability of failure. Several approximate methods for multinormal integration have been developed, since the direct numerical integration in large dimensions (30–50) is not feasible. The product of conditional marginal (PCM) method was proposed as a simple and effective method for system reliability computation. Although PCM is fairly accurate in computing parallel system reliability, it can result in a large overestimation of the failure probability of series systems with highly reliable elements. The paper presents an improved version, referred to as I-PCM, to eliminate this shortcoming of the original method. The I-PCM method employs a simple modification of bivariate integrals based on the additive law of probability. Detailed error analyses and numerical examples are presented in the paper to illustrate the improved accuracy and efficiency of the proposed I-PCM method.