马尔科夫链方法在可靠性分析中的应用研究

1、西 北 工 业 大 学博 士 学 位 论 文学位研究生题目：马尔科夫链方法在可靠性分析中的应用研究作 者： 袁 修 开 学科专业： 飞行器设计 指导教师： 吕 震 宙 2021年11月Application Research of Markov Chain Simulation in Reliability AnalysisA Thesis Submitted for the Degree of Doctor of PhilosophyYuan XiukaiAcademic Advisor: Professor Lu ZhenzhouNorthwestern Polytechnical Uni

2、versityNovember 2021摘要马尔科夫链方法由于具有自适应性以及较高的模拟效率，在许多领域中得到了广泛的应用。本文将针对可靠性分析中遇到的各种问题，包括非正态变量问题，小失效概率问题，可靠性灵敏度问题，扩展可靠性问题以及动态可靠性问题等，探讨了马尔科夫链在可靠性分析中的应用，开展了一系列的可靠性分析方法，主要内容如下：(1)基于对失效域样本的高效马尔可夫链模拟方法和鞍点估计法，提出了两种可快速分析小失效概率情况下含非正态变量的非线性极限状态函数的可靠性方法。第一种方法采用高效的改良的马尔科夫链方法来获得近似设计点，然后在近似设计点处将极限状态函数线性化，最后采用鞍点估计法求解失效

3、概率。所提方法相对于基于均值点处线性化的鞍点估计方法，仅需增加少量计算量即可以提高近似求解的精度。第一种方法那么在非正态空间中，将所求失效概率转化为线性极限状态函数的失效概率与一个特征比例因子的乘积。线性极限状态函数是通过马尔可夫链模拟非线性极限状态函数失效域中的样本而获得的，它与非线性极限状态函数具有近似相同的设计点。而概率论中的乘法定理是获取特征比例因子的理论依据，它反映了非线性极限状态函数失效概率与线性极限状态函数失效概率的关系。线性极限状态函数的失效概率可以由鞍点估计法求得，而特征比例因子可以由马尔可夫链快速模拟线性与非线性失效域中的样本而近似算得。定性分析和定量的算例比照分析说明：所

4、提算法具有较广的适用范围，并且它的实现过程较为简单，计算精度和效率均较高。(2)针对实际工程中可靠性分析设计的极限状态方程为隐式的情况，提出了两种基于马尔科夫链模拟的支持向量机可靠性分析方法。所提的第一种方法方法采用改良的马尔科夫链来产生极限状态重要区域上的样本点，再采用支持向量机方法求得相应的函数替代模型来进行可靠性分析。由于马尔科夫链能够自适应的模拟极限状态重要失效区域附近的样本，并且由于采用马尔科夫链备选样本点而非状态点作为训练样本，因而所提方法能够高效快速逼近对失效概率奉献较大区域的极限状态方程，并且充分利用了模拟过程产生的有用信息。所提方法还采用了一种渐变方差的模拟策略，改善了马尔科

5、夫链模拟样本的质量。另外所提方法分别采用支持向量机分类方法和回归方法来构建函数替代模型，能够实现风险最小化的极限状态方程的替代，使得失效概率可以高效高精度地被逼近。最后给出了数值算例和工程算例，说明本文所提方法在计算效率和精度上具有较好的性能。所提出的第二种方法是基于筛选的马尔科夫链模拟的支持向量机可靠性分析方法，该方法将马尔科夫链与支持向量机特有的边界性质相结合来产生训练样本，既充分利用了马尔科夫链的自适应性，使抽样的样本点落在重要区域，又利用了支持向量机分类面的边界的渐近逼近性，使训练样本以较高效率和精度来产生，这样就保证了拟合的支持向量机在失效概率的近似上具有较高的精度。所提方法能够很好

6、的解决原有传统响应面存在的两个根本问题，并通过数值算例和工程算例证实了所提方法的精度、效率。(3)提出了一种基于马尔可夫链模拟的可靠性灵敏度分析方法，该方法将原极限状态灵敏度转化成线性超平面灵敏度与两个修正失效域灵敏度的代数和的形式，通过两次马尔可夫链模拟得到超平面的灵敏度解析解及修正失效域灵敏度的估计，进而得到原极限状态的灵敏度解。文中给出了可靠性灵敏度估计值及估计值方差和变异系数的计算公式。所提方法采用快捷的马尔可夫链模拟，因而效率远比Monte Carlo模拟要高。将所提方法运用到粉末冶金涡轮盘的低周疲劳寿命可靠性分析上，得到影响涡轮盘的寿命可靠度的各参数的灵敏度，为工程设计提供了指导。

7、(4)可靠性分析中根本变量分布参数为区间均匀变量时，失效概率及可靠性灵敏度为分布参数的函数，基于条件概率马尔科夫链模拟提出了一种失效概率函数及其灵敏度的求解方法，并提出了新的度量指标，它为失效概率函数和可靠性灵敏度在分布参数空间上的统计特征值。所提方法的主要思路是首先采用贝叶斯公式将失效概率函数转化成全局失效概率与参数后验密度的表达式。再通过条件概率模拟方法来求解全局失效概率，还采用了三阶最大熵法来得到失效域样本的条件密度分布，最终得到所求的失效概率函数，而可靠性灵敏度函数为失效概率函数的导数。文中结合算例探讨了所提方法的精度、效率和适用性。(5)针对临界水平和变量分布参数同时变化的情况，极限

8、状态函数的累积分布及其导数均为二维函数，其求解较困难。本文提出基于贝叶斯公式的累积分布及其导数的求解方法。所提方法首先采用贝叶斯公式将所求二维函数转化成全局失效概率及后验密度的表述式，再采用Monte Carlo方法及条件概率模拟方法来进行求解。所求方法与传统的求解方法相比可以给出二维扩展累积分布函数的表达式，且仅需一次可靠性分析即可求解。(6)针对随机鼓励下随机结构系统的动力可靠性分析问题，探讨了本文所提方法在动力学问题中的应用。首先将随机鼓励随机结构系统的动力可靠性问题转化成一般的可靠性问题，然后采用前文提出的方法来进行求解，得到了相应的失效概率、失效概率函数及灵敏度函数等，论证了前文所提

9、方法的可行性，也为工程中动力可靠性分析设计提供了可选方法。关键词：可靠性，灵敏度分析，马尔科夫链，鞍点估计，支持向量机，失效概率函数，累积分布函数，最大熵法，蒙特卡洛，动态可靠性，随机鼓励AbstractFor the high adaptability and high simulation efficiency, Markov chain algorithm has wide application. The problems, including nonnormal variables, small failure probability, reliability sensitivity

10、, augment reliability and dynamical reliability, are frequently encountered in practical reliability analysis. For these problems, the solutions are investigated with Markov chain algorithm, and some novel reliability analysis methods are presented. The main contributions are listed as follows:(1)Ba

11、sed on fast Markov chain simulation for the samples distributed in failure region and Saddlepoint Approximation (SA) technique, two efficient reliability analysis methods are presented for evaluating the small failure probability of non-linear limit state function (LSF) with non-normal variables. Th

12、e first method firstly utilizes the advanced Markov chain simulation to efficiently obtain the approximate design point, and then linearizes the LSF at the design point, at last uses the SA to compute the failure probability with the linear limit state function. Compared with the SA based on mean po

13、int, the proposed method improves the precision by just increasing a little computation. In the second presented method, the failure probability of the non-linear LSF is transformed into a product of the failure probability of a linear LSF and a feature ratio factor. The linear LSF is obtained with

14、the samples distributed in the failure region, which are generated by the fast Markov chain simulation. The maximum likelihood point in the failure region of the linear LSF is approximately the same as that of the non-linear LSF. The feature ratio factor, which can be evaluated on the basis of multi

15、plicative rule of probability, exposes the relation between the failure probability of the non-linear LSF and that of the linear LSF. The failure probability of the linear LSF can be calculated by SA technique, and the feature ratio factor can be fast computed by the samples distributed in the failu

16、re regions of the non-linear LSF and those of the linear LSF. Qualitative analysis and quantitative computation demonstrate that the presented method has wide application, and it can be easily implemented and possesses high precision and high efficiency.(2)For the implicit LSF usually encountered in

17、 engineering reliability analysis and design, two support vector machine (SVM) reliability analysis methods are proposed based on fast Markov chain simulation. In the proposed method, Markov chain is used to simulate the samples in the important region defined by the LSF, and the SVM is employed to

18、obtain the solver surrogate by use of these samples. Since Markov chain can adaptively simulate the samples of the important region, and the candidate state but not Markov state is used as the training samples, the proposed method can well approximate the limit state function in the zone surrounding

19、 the design points, and can make full use of the information provided by Markov chain simulation. In addition, gradual change on variance in simulation process is adopted to improve the quality of the Markov chain samples. Moreover, the proposed method uses the SVM regression method and classificati

20、on method to construct the solver surrogate, which can automatically apply the Structural Risk Minimization (SRM) inductive principle in approximating the limit state equation, and thus approximate the failure probability with high precision. Finally numerical and engineering examples illustrate tha

21、t the proposed method owns good performance in calculating efficiency and precision. In the second presented method, an advanced SVM reliability analysis method is proposed on filtered Markov chain simulation. This method combines the Markov chain simulation and the margin property of the SVM to gen

22、erate training samples, it utilizes the adaptability of the Markov chain which makes the samples fall in the important region of the LSF, and utilizes asymptotic margin property of the SVM classification method which makes the samples generate in high precision. The proposed method solves the two pr

23、oblems which are encountered in traditional response surface method. The numerical and engineering examples demonstrate the precision and efficiency of the proposed method. (3)A decomposed arithmetic of reliability sensitivity analysis is proposed on the basis of Markov chain simulation. The propose

24、d method decomposes the reliability sensitivity of the actual limit state into algebra sum of reliability sensitivities of a linear failure region and modified failure regions, which can be rapidly solved by two Markov chain simulations. The variance of the reliability sensitivity estimation is anal

25、yzed as well. Markov chain can simulate the samples of the specified failure region and own higher efficiency than Monte Carlo method, thus the proposed method based on it also has high efficiency; this conclusion is demonstrated by the numerical examples and the fatigue life reliability sensitivity

26、 results of the powder metallurgy turbine disk.(4)In case that the distribution parameters of the basic variables are uniformly distributed interval variables in reliability analysis, the failure probability and the reliability sensitivity are the functions of the distribution parameters. The condit

27、ional probability Markov chain simulation method is proposed to obtain the failure probability function, reliability sensitivity function and the new reliability measures, which are the statistics characteristic values of the failure probability function and the reliability sensitivity function in t

28、he space of the distribution parameters. The key idea of the proposed method is that the failure probability function is transformed into the expression of the global failure probability and the posterior density of the distribution parameter, then the global failure probability is computed by condi

29、tional probability simulation method based on Markov chain algorithm, and the posterior density is obtained by the third order maximum entropy method based on failure samples, finally the failure probability function is obtained and the reliability sensitivity function is the derivative of the failu

30、re probability function. The accuracy, efficiency and applicability of the proposed method are demonstrated with several examples. (5)When the threshold value and the distribution parameter are both variables, the cumulative distribution function of the LSF and its derivatives are two-dimension func

31、tions, which are difficult to obtain in reliability analysis. The Bayes formula is adopted to transform the two-dimension function into the expression of global failure probability and the posterior distribution of the variables, and then Monte Carlo simulation and the conditional probability simula

32、tion method are adopted to analyze. Compared with the traditional analysis method, the proposed method can give the expression of the two-dimension augment cumulative distribution function, and only one reliability analysis is needed. (6)The methods proposed in this paper are applied to the dynamica

33、l reliability problem of random structure subjected to stochastic excitation. It firstly transforms the dynamical reliability problem into traditional one, and then uses the methods proposed in this paper to obtain the failure probability, the failure probability function and the reliability sensiti

34、vity function of the distribution parameters of the basic random variables. The feasibilities of the proposed methods, which can provide alternative solutions for the dynamical reliability, are demonstrated. Keyword: Reliability; sensitivity analysis; Markov chain; Saddlepoint Approximation (SA); su

35、pport vector machine (SVM); failure probability function; cumulative distribution function (CDF); maximum entropy method; Monte Carlo; dynamical reliability; stochastic excitation目录 TOC o 1-3 h z u HYPERLINK l _Toc246387131 摘要 PAGEREF _Toc246387131 h V HYPERLINK l _Toc246387132 Abstract PAGEREF _Toc

36、246387132 h IX HYPERLINK l _Toc246387133 目录 PAGEREF _Toc246387133 h XIII HYPERLINK l _Toc246387134 第一章 绪论 PAGEREF _Toc246387134 h 1 HYPERLINK l _Toc246387135 1.1.可靠性分析的近似解析法和数字模拟法 PAGEREF _Toc246387135 h 1 HYPERLINK l _Toc246387136 1.2.可靠性分析的函数替代方法 PAGEREF _Toc246387136 h 3 HYPERLINK l _Toc246387137

37、 1.3.可靠性优化中失效概率函数的求解 PAGEREF _Toc246387137 h 4 HYPERLINK l _Toc246387138 1.4.可靠性灵敏度的分析求解 PAGEREF _Toc246387138 h 5 HYPERLINK l _Toc246387139 1.5.论文主要工作 PAGEREF _Toc246387139 h 6 HYPERLINK l _Toc246387140 第二章 基于马尔科夫链模拟与鞍点估计的非正态变量可靠性分析方法 PAGEREF _Toc246387140 h 7 HYPERLINK l _Toc246387141 2.1.基于马尔科夫链的

38、一次鞍点估计可靠性分析方法 PAGEREF _Toc246387141 h 8 HYPERLINK l _Toc246387142 2.1.1.根本思路 PAGEREF _Toc246387142 h 8 HYPERLINK l _Toc246387143 2.1.2.马尔科夫链模拟 PAGEREF _Toc246387143 h 8 HYPERLINK l _Toc246387144 2.1.3.鞍点估计法 PAGEREF _Toc246387144 h 11 HYPERLINK l _Toc246387145 2.1.4.所提方法的根本步骤 PAGEREF _Toc246387145 h

39、15 HYPERLINK l _Toc246387146 2.1.5.算例 PAGEREF _Toc246387146 h 16 HYPERLINK l _Toc246387147 2.1.6.本节小结 PAGEREF _Toc246387147 h 17 HYPERLINK l _Toc246387148 2.2.基于条件概率马尔科夫链模拟的可靠性分析方法 PAGEREF _Toc246387148 h 18 HYPERLINK l _Toc246387149 2.2.1.条件概率方法的根本思路 PAGEREF _Toc246387149 h 18 HYPERLINK l _Toc24638

40、7150 2.2.2.线性极限状态函数失效概率求解的鞍点估计法 PAGEREF _Toc246387150 h 20 HYPERLINK l _Toc246387151 2.2.3.特征因子的求解 PAGEREF _Toc246387151 h 20 HYPERLINK l _Toc246387152 2.2.4.失效概率的估计值和近似方差分析 PAGEREF _Toc246387152 h 21 HYPERLINK l _Toc246387153 2.2.5.基于条件概率马尔科夫链模拟的可靠性分析方法步骤 PAGEREF _Toc246387153 h 24 HYPERLINK l _Toc

41、246387154 2.2.6.所提方法的进一步讨论 PAGEREF _Toc246387154 h 24 HYPERLINK l _Toc246387155 2.2.7.算例 PAGEREF _Toc246387155 h 25 HYPERLINK l _Toc246387156 2.2.8.本节小结 PAGEREF _Toc246387156 h 33 HYPERLINK l _Toc246387157 2.3.本章小结 PAGEREF _Toc246387157 h 34 HYPERLINK l _Toc246387158 第三章 基于马尔科夫链模拟的支持向量机方法 PAGEREF _T

42、oc246387158 h 35 HYPERLINK l _Toc246387159 3.1.可靠性分析的函数替代方法 PAGEREF _Toc246387159 h 35 HYPERLINK l _Toc246387160 3.1.1.响应面方法 PAGEREF _Toc246387160 h 35 HYPERLINK l _Toc246387161 3.1.2.支持向量机方法 PAGEREF _Toc246387161 h 37 HYPERLINK l _Toc246387162 3.2.基于马尔科夫链模拟的支持向量机可靠性分析方法 PAGEREF _Toc246387162 h 38 H

43、YPERLINK l _Toc246387163 3.2.1.改良的马尔科夫链模拟产生训练样本点 PAGEREF _Toc246387163 h 39 HYPERLINK l _Toc246387164 3.2.2.支持向量机回归方法 PAGEREF _Toc246387164 h 40 HYPERLINK l _Toc246387165 3.2.3.支持向量机分类方法 PAGEREF _Toc246387165 h 42 HYPERLINK l _Toc246387166 3.2.4.基于马尔科夫链模拟的支持向量机方法的步骤 PAGEREF _Toc246387166 h 45 HYPERL

44、INK l _Toc246387167 3.2.5.算例 PAGEREF _Toc246387167 h 46 HYPERLINK l _Toc246387168 3.2.6.本节小结 PAGEREF _Toc246387168 h 55 HYPERLINK l _Toc246387169 3.3.基于筛选马尔科夫链模拟的支持向量机分类可靠性分析方法 PAGEREF _Toc246387169 h 56 HYPERLINK l _Toc246387170 3.3.1.支持向量机分类方法的边界特性 PAGEREF _Toc246387170 h 56 HYPERLINK l _Toc246387

45、171 3.3.2.基于支持向量机边界的筛选马尔科夫链模拟方法 PAGEREF _Toc246387171 h 57 HYPERLINK l _Toc246387172 3.3.3.基于筛选马尔科夫链的支持向量机分类方法 PAGEREF _Toc246387172 h 58 HYPERLINK l _Toc246387173 3.3.4.算例 PAGEREF _Toc246387173 h 60 HYPERLINK l _Toc246387174 3.3.5.本节小结 PAGEREF _Toc246387174 h 63 HYPERLINK l _Toc246387175 3.4.本章小结 P

46、AGEREF _Toc246387175 h 64 HYPERLINK l _Toc246387176 第四章 可靠性灵敏度分析的分解算法 PAGEREF _Toc246387176 h 65 HYPERLINK l _Toc246387177 4.1.可靠性灵敏度及其抽样模拟求解方法 PAGEREF _Toc246387177 h 65 HYPERLINK l _Toc246387178 4.2.可靠性灵敏度的分解算法 PAGEREF _Toc246387178 h 66 HYPERLINK l _Toc246387179 4.2.1.分解算法的根本思路 PAGEREF _Toc246387

47、179 h 66 HYPERLINK l _Toc246387180 4.2.2.马尔科夫链模拟特定失效域上的样本 PAGEREF _Toc246387180 h 67 HYPERLINK l _Toc246387181 4.2.3.线性超平面的可靠性灵敏度的求解 PAGEREF _Toc246387181 h 68 HYPERLINK l _Toc246387182 4.2.4.及的求解 PAGEREF _Toc246387182 h 69 HYPERLINK l _Toc246387183 4.2.5.可靠性灵敏度估计及其方差和变异系数 PAGEREF _Toc246387183 h 70

48、 HYPERLINK l _Toc246387184 4.3.算例 PAGEREF _Toc246387184 h 71 HYPERLINK l _Toc246387185 4.4.循环载荷作用下涡轮盘寿命可靠性灵敏度分析 PAGEREF _Toc246387185 h 76 HYPERLINK l _Toc246387186 4.5.本章小结 PAGEREF _Toc246387186 h 78 HYPERLINK l _Toc246387187 第五章 失效概率函数与灵敏度函数的求解方法 PAGEREF _Toc246387187 h 79 HYPERLINK l _Toc24638718

49、8 5.1.失效概率函数及其特征指标的求解方法 PAGEREF _Toc246387188 h 79 HYPERLINK l _Toc246387189 5.1.1.区间分布参数下的失效概率函数及其指标 PAGEREF _Toc246387189 h 80 HYPERLINK l _Toc246387190 5.1.2.失效概率函数及其指标求解的Monte Carlo 直接法 PAGEREF _Toc246387190 h 81 HYPERLINK l _Toc246387191 5.1.3.失效概率函数及其指标求解的FOSM法 PAGEREF _Toc246387191 h 82 HYPER

50、LINK l _Toc246387192 5.1.4.基于贝叶斯公式的Monte Carlo模拟求解方法 PAGEREF _Toc246387192 h 83 HYPERLINK l _Toc246387193 5.1.5.基于贝叶斯公式的条件概率马尔科夫链模拟求解方法 PAGEREF _Toc246387193 h 87 HYPERLINK l _Toc246387194 5.1.6.算例 PAGEREF _Toc246387194 h 90 HYPERLINK l _Toc246387195 5.1.7.本节小结 PAGEREF _Toc246387195 h 101 HYPERLINK

51、l _Toc246387196 5.2.可靠性灵敏度函数及其特征指标的求解 PAGEREF _Toc246387196 h 103 HYPERLINK l _Toc246387197 5.2.1.区间分布参数下的全局灵敏度及其指标 PAGEREF _Toc246387197 h 103 HYPERLINK l _Toc246387198 5.2.2.全局灵敏度指标求解的一次二阶矩(FOSM)法 PAGEREF _Toc246387198 h 104 HYPERLINK l _Toc246387199 5.2.3.全局灵敏度指标求解的Monte Carlo法 PAGEREF _Toc246387

52、199 h 105 HYPERLINK l _Toc246387200 5.2.4.全局灵敏度指标求解的条件概率马尔科夫链模拟方法 PAGEREF _Toc246387200 h 106 HYPERLINK l _Toc246387201 5.2.5.算例 PAGEREF _Toc246387201 h 108 HYPERLINK l _Toc246387202 5.2.6.本节小结 PAGEREF _Toc246387202 h 115 HYPERLINK l _Toc246387203 5.3.本章小结 PAGEREF _Toc246387203 h 116 HYPERLINK l _To

53、c246387204 第六章 极限状态扩展累积分布函数及其导数的分析方法 PAGEREF _Toc246387204 h 117 HYPERLINK l _Toc246387205 6.1.扩展累积分布函数和灵敏度函数 PAGEREF _Toc246387205 h 117 HYPERLINK l _Toc246387206 6.1.1.扩展累积分布函数 PAGEREF _Toc246387206 h 117 HYPERLINK l _Toc246387207 6.1.2.扩展累积分布函数的导函数 PAGEREF _Toc246387207 h 118 HYPERLINK l _Toc2463

54、87208 6.1.3.极限状态函数的扩展概率密度对参数的灵敏度函数 PAGEREF _Toc246387208 h 118 HYPERLINK l _Toc246387209 6.2.Monte Carlo 直接求解方法 PAGEREF _Toc246387209 h 118 HYPERLINK l _Toc246387210 6.3.正态变量线性极限状态函数情况下的FOSM求解方法 PAGEREF _Toc246387210 h 119 HYPERLINK l _Toc246387211 6.4.基于贝叶斯公式的求解方法 PAGEREF _Toc246387211 h 120 HYPERL

55、INK l _Toc246387212 6.4.1.求解扩展函数、及的贝叶斯公式 PAGEREF _Toc246387212 h 120 HYPERLINK l _Toc246387213 6.4.2.基于贝叶斯公式的Monte Carlo模拟求解方法 PAGEREF _Toc246387213 h 123 HYPERLINK l _Toc246387214 6.4.3.基于贝叶斯公式的条件概率模拟求解方法 PAGEREF _Toc246387214 h 124 HYPERLINK l _Toc246387215 6.5.算例 PAGEREF _Toc246387215 h 127 HYPER

56、LINK l _Toc246387216 6.6.本章小结 PAGEREF _Toc246387216 h 142 HYPERLINK l _Toc246387217 第七章 随机鼓励下随机结构动力可靠性分析 PAGEREF _Toc246387217 h 143 HYPERLINK l _Toc246387218 7.1.随机结构系统的动力可靠性分析 PAGEREF _Toc246387218 h 143 HYPERLINK l _Toc246387219 7.1.1.平稳随机响应分析的虚拟鼓励法 PAGEREF _Toc246387219 h 143 HYPERLINK l _Toc246

57、387220 7.1.2.随机结构系统的动力可靠性分析 PAGEREF _Toc246387220 h 144 HYPERLINK l _Toc246387221 7.2.单自由度振子体系动力可靠性问题 PAGEREF _Toc246387221 h 145 HYPERLINK l _Toc246387222 7.3.十五杆结构动力可靠性分析 PAGEREF _Toc246387222 h 148 HYPERLINK l _Toc246387223 7.4.地震随机鼓励下随机钢框架结构的动力可靠性分析 PAGEREF _Toc246387223 h 151 HYPERLINK l _Toc24

58、6387224 7.5.本章小结 PAGEREF _Toc246387224 h 153 HYPERLINK l _Toc246387225 第八章 结论与展望 PAGEREF _Toc246387225 h 155 HYPERLINK l _Toc246387226 参考文献 PAGEREF _Toc246387226 h 159 HYPERLINK l _Toc246387227 发表论文及科研 PAGEREF _Toc246387227 h 165 HYPERLINK l _Toc246387228 获奖情况 PAGEREF _Toc246387228 h 168 HYPERLINK l

59、 _Toc246387229 致谢 PAGEREF _Toc246387229 h 169 HYPERLINK l _Toc246387230 学位论文知识产权声明书 PAGEREF _Toc246387230 h 170第一章 绪论在工程中，由于不确定因素的广泛存在，可靠性被引入到产品的平安分析和设计中，并逐渐成为科学和工程中一个非常重要的概念。概率论和随机过程理论的开展，为随机可靠性理论体系的建立和开展奠定了坚实的根底。目前基于可靠性的分析设计方法已经广泛应用于航空、航天、汽车、船舶和土木工程等领域。在可靠性理论日益广泛应用的同时，工程结构机构的复杂性，多样性对可靠性分析设计方法也提出了更

60、高的要求。目前仍然存在一些难点问题，传统的可靠性分析方法并没有很好的解决。这些问题包括：1非正态变量问题2高维问题3隐式极限状态问题4可靠性优化设计。对于这几方面的问题，传统的可靠性分析方法还有待进一步的完善和开展。可靠性分析的近似解析法和数字模拟法可靠性分析方法简单的分，一般可分为近似解析法和数字模拟法。可靠性分析中传统的近似解析方法具有计算快捷和计算效率高的优势，因此在工程上得到了广泛的应用。一次可靠度方法 REF _Ref242193636 r h * MERGEFORMAT 1- REF _Ref243016031 r h * MERGEFORMAT 9是工程常用的方法，它采用将极限状