An adaptive response surface method for reliability analysis of structures with multiple loading sequences

The response surface method (RSM) has been widely used in conjunction with non-linear finite element (NLFE) analysis to predict the reliability levels of structures that do not have explicit failure functions. In the present study, the solution of the reliability analysis initially diverged when the loading was applied in sequence in the NLFE analysis. A case study was performed in order to investigate the cause of the divergence. It was found that the divergence was mainly due to the non-smoothness of the response surface, and the quality and validity of the experimental design for numerical or NLFE analysis. An adaptive design approach is proposed to overcome these problems in reliability analysis, and several suggestions are made to improve the robustness of the RSM. Three numerical examples have been chosen to demonstrate the proposed method, which was verified by an independent Monte Carlo Simulation.     

Discussion on: moment methods for structural reliability

A few comments on moment method for reliability analysis advocated by Zhao and Ono are presented in the paper. It is demonstrated using a number of numerical examples that the first four moments of a distribution, particularly when they are crudely estimated, do not always allow good estimates of tail probabilities, and that some different equivalent forms of performance function that describe the same structural problem may give totally incorrect values for the reliability when the Zhao and Ono methodology is applied to them. Furthermore, the computation method for the first four moments proposed by the same authors may be incorrect so that the reliability result is unreasonable accordingly.     

基于HDMR响应面的结构可靠度计算方法

介绍了响应面法在可靠度分析中的应用进展,提出了一种基于高维模型表示(HDMR)扩展的迭代响应面法。在详述采用HDMR响应面法计算结构可靠度的基本原理后,采用Matlab语言对若干数值算例进行计算,同时给出了其它计算方法的结果以对比分析。结果表明:本文方法具有较高的效率和可靠的精度,在结构可靠度分析中具有一定的实用性。     

Comparison of response surface and neural network with other methods for structural reliability analysis

The evaluation of the failure probability and safety levels of structural systems is of extreme importance in structural design, mainly when the variables are eminently random. Some examples of random variables on real structures are material properties, loads and member dimensions. It is necessary to quantify and compare the importance of each one of these variables in the structural safety. Many researchers studied structural reliability problems and nowadays there are several approaches for these problems. Two recent approaches, the Response Surface (RS) and the Artificial Neural Network (ANN) techniques, have emerged attempting to solve complex and more elaborated problems. In this work, these two techniques are presented, and comparison are carried out using the well known First Order Reliability Method (FORM), Direct Monte Carlo Simulation and Monte Carlo Simulation with Adaptive Importance Sampling technique with approximated and exact limit state functions. Problems with simple limit state functions (LSF) and closed form solutions of the failure probability are solved in order to highlight the advantages and shortcomings using these techniques. Some remarks are outlined regarding the fact that RS and ANN techniques have presented equivalent precision levels. It is observed that in problems where the computational cost of structural evaluations (looking for the failure probability and safety levels) is high, these two techniques may turn feasible the evaluation of the structural reliability through simulation techniques.     

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.     

High-order limit state functions in the response surface method for structural reliability analysis

The stochastic response surface method (SRSM) is a technique for the reliability analysis of complex systems with low failure probabilities, for which Monte Carlo simulation (MCS) is too computationally intensive and for which approximate methods are inaccurate. Typically, the SRSM approximates a limit state function with a multi-dimensional quadratic polynomial by fitting the polynomial to a number of sampling points from the limit state function. This method can give biased approximations of the failure probability for cases in which the quadratic response surface can not conform to the true limit state function’s nonlinearities. In contrast to recently proposed algorithms which focus on the positions of sample points to improve the accuracy of the quadratic SRSM, this paper describes the use of higher order polynomials in order to approximate the true limit state more accurately. The use of higher order polynomials has received relatively little attention to date because of problems associated with ill-conditioned systems of equations and an approximated limit state which is very inaccurate outside the domain of the sample points. To address these problems, an algorithm using orthogonal polynomials is proposed to determine the necessary polynomial orders. Four numerical examples compare the proposed algorithm with the conventional quadratic polynomial SRSM and a detailed MCS.     

Application of kriging method to structural reliability problems

Approximation methods are widely used to alleviate the computational burden of engineering analyses. For structural reliability analyses, the common approach is to use the response surface method (RSM) based on the least square regression. However, another approximation method based on kriging has gained popularity especially in the field of deterministic optimization. However, the application of the kriging method to structural reliability problems has not been realized until recently. Therefore, this paper investigates the use of the kriging method for structural reliability problems by comparing it with the most common RSM. The effects of the kriging parameters are also examined on the basis of the β computation and fitting behavior. It can be deduced from the results given in this paper that using the most common approach in the literature to find the kriging parameters does not guarantee a good result for the structural reliability problems. As a result, some advantages as well as disadvantages of the kriging method are reported, based on the results obtained from the application of the kriging method to the examples from the literature. Finally, this paper suggests the points for which the kriging model could be improved to get better results for structural reliability problems.     

Computational intelligence methods for the efficient reliability analysis of complex flood defence structures

With the continual rise of sea levels and deterioration of flood defence structures over time, it is no longer appropriate to define a design level of flood protection, but rather, it is necessary to estimate the reliability of flood defences under varying and uncertain conditions. For complex geotechnical failure mechanisms, it is often necessary to employ computationally expensive finite element methods to analyse defence and soil behaviours; however, methods available for structural reliability analysis are generally not suitable for direct application to such models where the limit state function is only defined implicitly. In this study, an artificial neural network is used as a response surface function to efficiently emulate the complex finite element model within a Monte Carlo simulation. To ensure the successful and robust implementation of this approach, a genetic algorithm adaptive sampling method is designed and applied to focus sampling of the implicit limit state function towards the limit state region in which the accuracy of the estimated response is of the greatest importance to the estimated structural reliability. The accuracy and gains in computational efficiency obtainable using the proposed method are demonstrated when applied to the 17th Street Canal flood wall which catastrophically failed when Hurricane Katrina hit New Orleans in 2005.     

大跨度悬索桥颤振可靠度分析的改进响应面法

介绍一种新的计算悬索桥颤振可靠度方法——改进响应面法。该方法利用传统响应面法将极限状态方程近似表达为简单的多项式形式,有效地解决FORM和SORM无法求解隐式极限状态方程的可靠度问题。另外,改进响应面法的使用还能有效地利用现有的确定性颤振分析软件。通过引入有限元方法,改进响应面法可应用于悬索桥颤振可靠度问题。通过使用重要抽样技术,提高改进响应面法的计算效率和计算精度,有效地解决传统响应面法在结构可靠度较大或失效概率较低时出现的迭代不收敛问题。数值算例验证该方法的效率和精度。最后,用该方法计算江阴长江大桥的颤振可靠度,结果表明基于经验公式的改进响应面法会过高地估计大跨度悬索桥的颤振可靠度。实际的悬索桥颤振可靠度应该采用基于有限元法的改进响应面法进行计算。     

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.     

CQ2RS: a new statistical approach to the response surface method for reliability analysis

Assessing the reliability of a complex structure requires a deal between reliability algorithms and numerical methods used to model the mechanical behavior. The Response Surface Method (RSM) represents a convenient way to achieve this purpose. The interest of such a method is that the user is allowed to choose and check the computed mechanical experiments. Nevertheless, this choice in an optimal way turns out to be not always an easy task. This paper proposes a response surface method named CQ2RS (Complete Quadratic Response Surface with ReSampling) allowing to take into account the knowledge of the engineer on one hand and to reduce the cost of the reliability analysis using a statistical formulation of the RSM problem on the other hand. Some academic and industrial examples illustrate the efficiency of the method.     

基于高斯过程回归的响应面法及边坡可靠度分析

提出了一种新型高斯过程响应面法(GPRSM),通过高斯过程回归算法构建随机变量与功能函数响应值之间的关系。该方法相较多项式响应面法对于功能函数为高维和高度非线性的可靠度问题,具有更高的精度和计算效率。此方法可以通过新增加训练点以动态更新响应面函数。与此同时,为了模拟岩土参数的空间变异性,通过KL展开构建随机场,并与极限平衡法结合进行边坡稳定性分析。使用提出的GPRSM构建替代模型并用于蒙特卡洛模拟求解边坡失稳概率,在保障计算精度的同时减少了对边坡稳定性分析程序的调用。最后将所提出的方法分别应用于功能函数为显式和隐式两个案例,并与其他论文中的方法对比,证明了该方法的有效性和适用性。     

结构可靠度分析的向量型层递响应面法

结构可靠度分析的传统响应面法和随机响应面法都属于标量型响应面法,为克服其局限性,利用预处理Krylov子空间法研究建立了结构可靠度分析的向量型层递响应面法。首先将随机刚度矩阵和节点荷载向量线性展开,通过选取合适的预处理器建立了预处理Krylov子空间及层递基向量,据此将随机节点位移向量进行层递展开,形成向量型的层递响应面。进而,利用混沌多项式展开层递基向量和层递响应面,根据混沌多项式确定层递响应面的样本点选取方法。在此基础上,根据可靠指标的几何含义,利用响应量的层递展开式建立了结构可靠度分析的迭代算法。算例分析表明,该方法与传统响应面法或随机响应面法相比,能够取得更高的计算效率和计算精度。     

结构可靠度分析的全局响应面法研究

对功能函数不能明确表达的可靠度分析问题,常采用响应面法。本文提出把响应面区分为局部响应面和全局响应面。常用的二次多项式响应面及常规的神经网络响应面均属于局部响应面,其仅在验算点附近(±σ)处与真实极限状态曲面符合较好。本文提出的全局神经网络及模糊神经网络响应面、改进全局神经网络及模糊神经网络响应面则属于全局响应面,其在全局范围均与真实极限状态曲面符合良好,故其对全局有较好预测效果。文中对以上各种响应面法计算效果及其对全局预测效果通过算例进行了对比分析,其中改进全局神经网络及模糊神经网络响应面法计算精度较好,进行有限元分析次数最少,该方法用于大型复杂结构的可靠度分析,可相应提高工作效率和解题质量。     

地下结构工程中的几种可靠度分析方法

介绍了基于两种计算模式下地下结构工程可靠度研究特点,并分析了当前研究方法的进展,包括响应面法、随机有限元法、模糊判别法、人工神经网络、耗散结构理论等,提出地下结构工程可靠度分析的广阔前景。     

基于顺序组件法的系统可靠度敏感性分析方法

岩土工程一般是由多个失效模式组成的复杂系统,且参数的不确定性对岩土工程的失效模式及可靠指标具有重要影响.为了分析参数不确定性对系统可靠度的影响,提出基于顺序组件法的系统可靠度敏感性分析方法(SCMSA).SCMSA利用SCM组合元件的原理,在计算两元件并联和串联的简单系统可靠度及敏感性的基础上,进一步计算两元件组成的组...     

Monte-Carlo抽样法在边坡可靠性分析中的应用研究

针对复杂边坡可靠性分析时,功能函数具有高度非线性和隐式表达的特性,为克服传统边坡可靠性分析解决此类问题的不足,将Monte-Carlo重要抽样与响应面法相结合,利用响应面法实现边坡极限状态函数的显式表达,并确定设计验算点,以该点为新的抽样中心应用Monte-Carlo重要抽样计算边坡的失效概率。通过两个算例进行了分析,结果表明:该方法在保证精度的同时提高计算效率,适于解决含有大量随机变量的复杂边坡工程可靠性分析。     

A preliminary design formula for the strength of stiffened curved panels by design of experiment method

In bridge construction, the use of stiffened plates for box-girder or steel beams is common day to day practice. The advantages of the stiffening from the economical and mechanical points of view are unanimously recognized. For curved steel panels, however, applications are more recent and the literature on their mechanical behaviour including the influence of stiffeners is therefore limited. Their design with actual finite element software is possible but significantly time-consuming and this reduces the number of parameters which can be investigated to optimise each panel. The present paper is thus dedicated to the development of a preliminary design formula for the determination of the ultimate strength of stiffened cylindrical steel panels. This approximate formula is developed with the help of a design of experiment method which has been adapted from the current statistical knowledge. This method is first presented, and its feasibility as well as its efficiency are illustrated through an application to the reference case of unstiffened curved panels. Then, the case of stiffened curved panels is investigated and a preliminary design formula is developed. The ease of use of this formula for preliminary design is finally illustrated in a cost optimisation problem.     

First order vs. second order reliability analysis of series structures

The paper presents an extension of the first order reliability method for computation of failure probabilities. The improvement is achieved by a better approximation to the limit state surface. The analysis is demonstrated for individual failure modes and for series structures. The results obtained are very close to ‘exact’ results obtained by simulation. The method is seen as a supplement to first order reliability methods for special cases. It is not meant to completely replace these methods.     

Artificial neural network-based response surface methods for reliability analysis of pre-stressed concrete bridges

Two artificial neural network (ANN)-based response surface methods for reliability analysis of pre-stressed concrete bridges are presented. The first method is the traditional ANN-based response surface method, originally introduced by Papadrakakis et al. in 1996 (Papadrakakis, M., Papadopoulos, V. and Lagaro, N., 1996. Structural reliability analysis of elastic-plastic structures using neural network and Monte carlo simulation. Computer Methods in Applied Mechanics and Engineering, 136, 145–163), which is applied here to the reliability analysis of pre-stressed concrete bridges. The second method is an improved ANN-based response surface method developed recently by the authors for the reliability analysis of truss structures, in which the key idea is that the uniform design method (UDM) is adopted to select training data for establishing an ANN model, thereby greatly improving the quality of training datasets for establishing an ANN model and dramatically reducing the required number of training datasets. There are two main objectives of the present work. Firstly, an attempt is made to extensively examine the performance of the traditional ANN-based response surface method since no detailed study has been carried out to investigate the effectiveness of this ANN-based response surface method on the basis of the reliability analysis of complicated structures such as pre-stressed concrete bridges. Secondly, the recently developed ANN-based response surface method is extended to the reliability analysis of pre-stressed concrete bridges. A detailed numerical investigation is carried out to compare the performance of the two methods.