考虑非正定相关性的测量不确定度蒙特卡洛评定方法

当采用蒙特卡洛法评定测量不确定度考虑输入量相关性时,需基于Nataf逆变换产生服从任意边缘概率分布的相关多维随机变量。为了解决Nataf逆变换过程中输入量相关系数矩阵非正定时,无法产生线性变换矩阵的问题,提出了基于Barzilai-Borwein梯度法的迭代修正算法。进而探讨了输入量服从非正态分布且相关的蒙特卡洛法实施步骤。最后,采用提出的迭代修正算法并基于Nataf逆变换的蒙特卡洛方法,对高速轮轨试验台轮轨纵向蠕滑率不确定度进行了评定,验证了该算法的可行性及有效性。     

Reliability sensitivity analysis with random and interval variables

In reliability analysis and reliability‐based design, sensitivity analysis identifies the relationship between the change in reliability and the change in the characteristics of uncertain variables. Sensitivity analysis is also used to identify the most significant uncertain variables that have the highest contributions to reliability. Most of the current sensitivity analysis methods are applicable for only random variables. In many engineering applications, however, some of uncertain variables are intervals. In this work, a sensitivity analysis method is proposed for the mixture of random and interval variables. Six sensitivity indices are defined for the sensitivity of the average reliability and reliability bounds with respect to the averages and widths of intervals, as well as with respect to the distribution parameters of random variables. The equations of these sensitivity indices are derived based on the first‐order reliability method (FORM). The proposed reliability sensitivity analysis is a byproduct of FORM without any extra function calls after reliability is found. Once FORM is performed, the sensitivity information is obtained automatically. Two examples are used for demonstration. Copyright © 2009 John Wiley & Sons, Ltd.     

Reliability analysis of structures based on a probability‐uncertainty hybrid model

The traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.     

基于极限状态法的岸桥结构抗震可靠性分析

针对集装箱码头岸边装卸桥(简称岸桥)在地震作用下的安全可靠性问题,对岸桥在极限状态下的抗震可靠性开展了研究。通过对地震参数与材料力学性能的随机性描述,获得了地震作用下各随机变量的概率分布规律和统计参数;采用有限元软件ANSYS建立了岸桥结构的力学模型,用时间历程法对地震作用下岸桥承载能力进行了计算;根据可靠性理论建立了地震作用下岸桥的可靠度计算模型,用 Monte-Carlo法计算了岸桥在极限状态下的抗震可靠度。结果表明在8级烈度地震作用下,岸桥结构整体上比较安全,但门框横梁与陆侧立柱的连接处的可靠度指标比较低,存在失效可能性;与许用应力法相比,极限状态法更具合理性。     

Sensitivity based reduced approaches for structural reliability analysis

In the reliability analysis of a complex engineering structures a very large number of system parameters can be considered to be random variables. The difficulty in computing the failure probability increases rapidly with the number of variables. In this paper, a few methods are proposed whereby the number of variables can be reduced without compromising the accuracy of the reliability calculation. Based on the sensitivity of the failure surface, three new reduction methods, namely (a) gradient iteration method, (b) dominant gradient method, and (c) relative importance variable method, have been proposed. Numerical examples are provided to illustrate the proposed methods.     

基于Nataf变换的桥梁结构地震易损性分析

地震易损性分析是美国太平洋地震工程研究中心(PEER)提出的“性能化地震工程”理论框架的重要一环. 目前所采用的易损性分析方法大多未考虑结构的不确定性及结构随机参数之间的相关性. 为此, 引入均匀设计考虑结构参数的随机性, 同时结合Nataf变换, 处理结构随机参数之间的相关性. 提出一种综合考虑地震动随机性、结构参数随机性及结构随机参数相关性的桥梁结构地震易损性分析方法. 以一座三跨连续梁为例, 分别在考虑结构随机参数相关性和不考虑结构参数相关性的前提下计算其地震易损性. 分析结果表明:提出的方法能够有效考虑结构随机参数相关性;忽视结构随机参数相关性会导致高估桥梁结构地震易损性.     

考虑载荷作用次数的机械零部件可靠性灵敏度分析方法

 为了研究载荷多次作用时,机械零部件的可靠度及可靠性灵敏度变化规律,从灵敏度角度修改零件的设计参数,降低制造成本,建立一种可靠性模型结合了随机摄动法、Edgeworth级数技术,并考虑了载荷的作用次数．摄动法和Edgeworth级数可以在基本随机参数的前4阶矩已知的情况下,研究具有任意分布参数的机械零件的可靠性灵敏度设计问题,顺序统计量理论考虑了载荷作用次数在可靠度和灵敏度计算中的影响．使用这种模型计算出的可靠度会随着载荷作用次数而变化,这与静态的可靠度计算方法存在差别．以某一型号的螺栓为算例,应用此模型计算了其可靠度、随机变量均值和方差的可靠性灵敏度．由提出的方法得到了可靠度和可靠性灵敏度值及其随载荷作用次数变化的曲线．可靠度及可靠性灵敏度随载荷作用次数变化的规律是：载荷作用次数增加,可靠度值降低,变化趋势单调;载荷作用次数达到最大时,可靠度达到最小;随机变量均值和方差的灵敏度随载荷作用次数变化出现不同的变化趋势,其中螺栓截面直径的均值和方差灵敏度随载荷作用次数的变化最大,随作用次数的增加,螺栓截面直径的参数将对螺栓的可靠性起主要的决定作用．     

失效相关的齿轮一次四阶矩可靠性分析

齿轮是机械传动的关键零件之一,为了分析其可靠性,在改进的验算点一次二阶矩可靠性方法基础上,应用Taylor级数展开和Hermite多项式近似等方法,推导了齿轮功能函数的验算点前四阶矩,分析了具有齿根断裂和齿面点蚀两种主要相关失效模式的齿轮传动可靠性,给出了其相关系数和齿轮的可靠度。另一方面,因影响齿轮失效的因素繁杂众多,计算齿轮的可靠度时,不管是用一次二阶矩可靠性方法还是四阶矩可靠性方法,其计算量均偏大,且容易出错。针对该问题,提出了一种分类变差系数验算点一次四阶矩可靠性分析方法,该方法对齿轮功能函数的不同基本随机变量进行分类综合,减少了设计变量,使计算量明显减小,解决了用矩方法分析结构可靠性时计算量偏大且易出错的难题。最后应用所提分类变差系数验算点一次四阶矩可靠性分析法估计了某车床主轴箱一对传动齿轮的可靠性,计算结果显示该对齿轮齿根弯曲疲劳强度和齿面接触疲劳强度有一定的相关性,所用一次四阶矩方法因包含偏度、峰度等更高阶的统计信息可进一步提高估计精度。     

A practical engineering method for fuzzy reliability analysis of mechanical structures

The fuzzy sets theory in reliability analyses is studied. The structure stress is related to several other variables, such as structure sizes, material properties, external loads; in most cases, it is difficult to be expressed in a mathematical formula, and the related variables are not random variables, but fuzzy variables or other uncertain variables which have not only randomness but also fuzziness. In this paper, a novel approach is presented to use the finite element analysis as a “numerical experiment” tool, and to find directly, by fuzzy linear regression method, the statistical property of the structure stress. Based on the fuzzy stress–random strength interference model proposed in this paper, the fuzzy reliability of the mechanical structure can be evaluated. The compressor blade of a given turbocharger is then introduced as a realistic example to illustrate the approach.     

New reliability modeling methods for structural systems with hybrid uncertainty

The present study investigates the hybrid reliability modeling of structures in which the inputs contain both random variables and interval variables. Hybrid uncertainty is divided into three categories, including random variables mixed with random variables, interval variables mixed with interval variable, and random variables mixed with interval variables. In order to perform the reliability analysis of structural systems, first, the Bayes method is proposed in the present study to obtain distribution parameters of random variables. Moreover, the self-sample method is introduced to obtain the interval boundaries based on the least available measuring data. Then, the reliability models are established for three situations and the reliability indices are defined and derived accordingly. The abovementioned three types of reliability indices outline the general situation of structural systems. Finally, the specific calculation process is described in details through different examples. Furthermore, the accuracy and efficiency of the proposed method is discussed by comparing the results obtained from the Monte Carlo simulation and those of other methods. The obtained results indicate that the performance of the proposed model in solving reliability modeling problems is better.     

Probability density evolution analysis of stochastic seismic response of structures with dependent random parameters

Performance evaluation and reliability assessment of real-world structures under earthquakes is of paramount importance. Generally, different mechanical property parameters of a structure are usually not independent, nor completely dependent, but partly dependent or correlated. Therefore, how to reasonably characterize such partial dependency and whether such partial dependency real matters in the stochastic response and reliability of structures under earthquakes are crucial issues. For this purpose, in the present paper, a novel physically-guided data-driven methodology of capturing the correlation configuration of basic random variables and the probability density evolution method are synthesized. The physically-guided data-driven methodology is firstly outlined. In this methodology, the underlying physical mechanism between dependent random variables is firstly involved to establish a random function model, and then the available observed data are adopted to identify the parameters in this model. What is more, physical constraints are also revealed for the initial modulus of elasticity and compressive strength of concrete. The probability density evolution method is then adopted, and the point selection by minimizing the GF-discrepancy is adjusted according to the correlation configuration and physical constraints. A reinforced concrete frame structure subjected to earthquake input is studied. It is found that when the structure is in the strongly nonlinear stage, the correlation configuration has considerable effects on the standard deviation of the stochastic responses, by a factor of nearly 2. In addition, whether the mechanical parameters in different floors are independent or not has great effects on the stochastic responses as well. Problems to be further studied are also outlined.     

Homotopy approach for random eigenvalue problem

A novel approach, referred to as the homotopy stochastic finite element method, is proposed to solve the eigenvalue problem of a structure associated with some amount of uncertainty based on the homotopy analysis method. For this approach, an infinite multivariate series of the involved random variables is proposed to express the random eigenvalue or even a random eigenvector. The coefficients of the multivariate series are determined using the homotopy analysis method. The convergence domain of the derived series is greatly expanded compared with the Taylor series due to the use of an approach function of the parameter h. Therefore, the proposed method is not limited to random parameters with small fluctuation. However, in practice, only single‐variable and double‐variable approximations are employed to simplify the calculation. The numerical examples show that with a suitable choice of the auxiliary parameter h, the suggested approximations can produce very accurate results and require reduced or similar computational efforts compared with the existing methods.     

工程结构可靠度指标计算的混沌搜索方法

基于可靠度指标b的几何涵义提出了一种用混沌搜索工程结构可靠度指标和设计验算点的新方法该方法利用混沌内在的随机性与遍历性进行求解最终获得全局最优算例结果表明该方法简单实用性能良好能够处理基本随机变量的非正态分布和变量之间的相关性是解决非线性功能函数可靠度问题的有效途径     

A time-variant reliability analysis method for structural systems based on stochastic process discretization

In this paper, we propose a new method for analyzing time-variant system reliability based on stochastic process discretization, which provides an effective tool for reliability design of many relatively complex structures considering the whole lifecycle. Within a design lifetime, the stochastic process is discretized into a series of random variables, and meanwhile, we can derive a time-invariant limit-state function in each time interval; the discretized random variables from the stochastic processes and the original random variables are transformed to the independent normal space, and a conventional time-invariant system reliability problem is derived through the linearization to each discretized limit-state functions; by solving this time-invariant system reliability problem, we can obtain the structural reliability or failure probability within the design lifetime. Finally, in this paper, we provide four numerical examples to verify the effectiveness of the method.     

An efficient approach for reliability-based topology optimization

This article presents an efficient approach for reliability-based topology optimization (RBTO) in which the computational effort involved in solving the RBTO problem is equivalent to that of solving a deterministic topology optimization (DTO) problem. The methodology presented is built upon the bidirectional evolutionary structural optimization (BESO) method used for solving the deterministic optimization problem. The proposed method is suitable for linear elastic problems with independent and normally distributed loads, subjected to deflection and reliability constraints. The linear relationship between the deflection and stiffness matrices along with the principle of superposition are exploited to handle reliability constraints to develop an efficient algorithm for solving RBTO problems. Four example problems with various random variables and single or multiple applied loads are presented to demonstrate the applicability of the proposed approach in solving RBTO problems. The major contribution of this article comes from the improved efficiency of the proposed algorithm when measured in terms of the computational effort involved in the finite element analysis runs required to compute the optimum solution. For the examples presented with a single applied load, it is shown that the CPU time required in computing the optimum solution for the RBTO problem is 15–30% less than the time required to solve the DTO problems. The improved computational efficiency allows for incorporation of reliability considerations in topology optimization without an increase in the computational time needed to solve the DTO problem.     

任意分布参数的机械零件的可靠性稳健设计（一）：理论部分

 将可靠性优化设计理论、可靠性灵敏度技术和稳健设计方法相结合，讨论了具有任意分布参数的机械零件的可靠性稳健设计问题，提出了可靠性稳健设计的计算方法．把可靠性灵敏度融入可靠性优化设计模型之中，将可靠性稳健设计归结为满足可靠性要求的多目标优化问题．在基本随机参数的前四阶矩已知的情况下，通过计算机程序可以实现具有任意分布参数的机械零件的可靠性稳健设计，迅速准确地得到具有任意分布参数的机械零件的可靠性稳健设计信息。     

Analysis of complex system reliability with correlated random vectors

The analysis of reliability of complex engineering systems remains a challenge in the field of reliability. It will be even more difficult if correlated random vectors are involved, which is generally the case as practical engineering systems invariably contain parameters that are mutually correlated. A new method for transforming correlated distributions, involving the Nataf transformation, is proposed that avoids the solution of integral equations; the method is based on the Taylor series expansion of the probability density function (PDF) of a bivariate normal distribution resulting in an explicit polynomial equation of the equivalent correlation coefficient. The required numerical results can be obtained efficiently and accurately.The proposed method for transformation of correlated random vectors is useful for developing a method for system reliability including complex systems with correlated random vectors. Based on the complete system failure process (originally defined as the development process of nonlinearity) and the fourth-moment method, the analysis of system reliability for elastic-plastic material avoids the identification of the potential failure modes of the system and their mutual correlations which are required in the traditional methods. Finally, four examples are presented – two examples to illustrate the potential of the new method for transformation of correlated random vectors, and two examples to illustrate the application of the proposed more effective method for system reliability.     

Non-stationary random response analysis of structures with uncertain parameters

This paper proposes a non-stationary random response analysis method of structures with uncertain parameters. The structural physical parameters and the input parameters are considered as random variables or interval variables. By using the pseudo-excitation method and the direct differentiation method (DDM), the analytical expression of the time-varying power spectrum and the time-varying variance of the structure response can be obtained in the framework of first order perturbation approaches. In addition, the analytical expression of the first-order and second-order partial derivative (e.g., time-varying sensitivity coefficient) for the time-varying power spectrum and the time-varying variance of the structure response expressed via the uncertainty parameters can also be determined. Based on this and the perturbation technique, the probabilistic and non-probabilistic analysis methods to calculate the upper and lower bounds of the time-varying variance of the structure response are proposed. Finally the effectiveness of the proposed method is demonstrated by numerical examples compared with the Monte Carlo solutions and the vertex solutions.     

Seismic reliability of non-linear frames with PR connections using systematic RSM

An effective, efficient, and robust reliability analysis algorithm is proposed for non-linear structures, where seismic loading can be applied in the time domain. The method is developed specifically for steel frame structures considering all major sources of non-linearity, including geometry, material, and partially restrained (PR) connections. The non-linearity due to PR connections is modeled by moment-relative rotation curves using the four-parameter Richard model. For seismic excitation, the loading, unloading, and reloading behavior at PR connections is modeled using moment-relative rotation curves and the Masing rule. The proposed algorithm intelligently integrates the response surface method, the finite element method, the first-order reliability method, and an iterative linear interpolation scheme. The uncertainties in all the random variables including the four parameters of Richard model are considered. Two unique features of the proposed algorithm are that (1) actual earthquake time histories can be used to excite structures in the presence of major sources of non-linearity and uncertainty and (2) it is possible to estimate the risk corresponding to both the serviceability and strength limit states. The algorithm is verified using the Monte Carlo simulation technique. The verified algorithm is first used to study the reliability of a frame structure in the presence of PR connections with different degrees of flexibility. Then the algorithm is used to estimate the reliability of a frame structure excited by 13 actual recorded earthquake time histories, 12 of them recorded during the Northridge earthquake of 1994. As expected, the reliabilities of the frame are found to be quite different, when excited by several time histories of the Northridge earthquake.     

基于Kriging改进响应面法的桥梁地震动力可靠度研究

鉴于桥梁结构对地震响应的非线性和复杂性等问题,提出了基于Kriging改进响应面法的桥梁结构地震动力可靠度分析方法。首先利用Kriging模型的优越模拟性能,将其作为响应面函数,并采取自适应策略加以改进,而后采取线性过滤器脉冲响应法对地震随机激励荷载进行了离散,并基于首次超越问题的定义建立了动力可靠度极限状态方程,最后对桥梁结构的地震可靠度问题进行了分析。计算分析结果表明:提出的基于Kriging模型的改进响应面法能有效完成桥梁地震动力可靠度的计算分析(包括结构参数随机性对桥梁动力可靠度的影响),且结果具有较高的准确性和高效性。