结构可靠度计算方法及灵敏度分析研究
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摘要
可靠度理论在结构设计与安全评估中发挥着重要作用。然而,传统可靠性理论应用于复杂工程结构时,计算精度、效率以及稳定性等方面往往不能满足现实要求,适用范围存在一定局限。本论文针对可靠度分析中的隐式非线性、复合随机以及可靠性灵敏度等问题进行了系统研究并提出了相关解决方法,这对完善可靠度理论和指导工程实际具有重要意义。本论文主要研究成果包括：
(1)针对一次二阶矩法处理强非线性功能函数不能收敛的问题,提出了旋转梯度算法。该算法迭代方向由迭代点处的负梯度方向与上一步迭代方向的线性组合确定,并通过旋转率来识别极限状态方程的非线性程度,进而确定旋转系数以及迭代方向计算公式。这样在保证算法收敛性的同时还能保证收敛速度。因不需要试算或人为设定参数值,该方法适用于复杂结构的可靠度分析。
(2)系统地建立了复杂结构可靠度分析的Kriging方法。首先基于Kriging模型建立了可靠度分析的krgiging(?)向应面法：然后利用Kriging模型可直接获得预测点方差并能根据各试验点与待测点距离不同而赋予不同权重的特点,提出了只增加有效点策略来重构极限状态函数的改进Kriging(?)向应面,进一步与旋转梯度算法结合,可以有效提高复杂结构非线性极限状态方程可靠度分析的精度和效率；最后,为进一步提高强非线性极限状态函数的可靠度计算精度,以验算点为抽样中心建立了Kriging重要抽样方法。
(3)对比分析了线性结构复合随机可靠度的三种方法。首先,基于泰勒展开法推导了复合随机可靠度计算的二阶泰勒解析表达式；其次,基于统计概念给出了Kriging抽样方法的计算步骤；第三,基于首次超越破坏准则建立了Kriging迭代法,将结构随机参数在可靠指标的迭代求解中处理。其中Kriging(?)向应面的数值抽样法通过响应面来拟合动力可靠度与结构随机参数的非线性关系,可以方便地利用有限元程序直接计算,避免了泰勒展开等理论推导方法的繁琐和困难。同时,对影响结构动力分析和可靠度计算精度和效率的结构随机响应进行研究,探讨了基于动态刚度阵法的平稳和非平稳随机响应计算的简便方法。
(4)针对多失效模式灵敏度分析情形,将失效模式相关度概念引入到多失效模式系统灵敏度分析中,并根据二维条件失效模式近似公式,提出了适于多维条件失效模式分析的相关系数公式。利用该公式可以将复杂的体系灵敏度求解转换成各失效模式失效概率相关系数的求导过程,从而降低分析难度,减少计算工作量。
(5)在传统可靠性灵敏度因子基础上,利用标准差灵敏度值构造了一种新灵敏度因子。新灵敏度因子不仅能反映各随机因素对结构失效概率影响的重要性程度,而且因子的数值大小还能表征将单个变量作为确定性变量处理或将多个变量同时作为确定性变量处理时所引起的可靠指标误差大小。采用新灵敏度因子进行灵敏度分析时,只需要进行一次分析就能识别出各随机变量对结构失效整体影响的相关信息,可显著提高计算效率。
Reliability theory plays a vital role in safety assessment and design by analysis. However, the traditional theory cannot take computational efficiency> accuracy and stability into consideration simultaneously or satisfy the practical demands when applied to complex engineering structures, thus the scope of application exits some limitations. This paper studies on a series of key theories of structural reliability, involving highly nonlinear problem, combined random vibration problem as well as reliability sensitivity analysis problem etc.,and provides some new theories and methods. Further research on these problems has important significance for perfecting structural reliability theory and guiding engineering practice. The main contributions of the dissertation are as follows:
(1) A new method named rotation gradient algorithm (RGA) is proposed to overcome the divergence problem in reliability analysis when solving high nonlinear performance function by first order reliability method. The new iteration direction is determined by the linear combination of the negative gradient direction and former iteration direction. And the value of rotation angle is determined by the rotation coefficient and the relative rotaion rate is used to identify the different nonlinear degree of limited state function. So the algorithm can ensure convergence stability and convergence rates. No trial calculation and artificial parameters need preset, so it is suitable for reliability analysis of complex engineering structure.
(2) Reliability calculation method suitable to complex structure based on the Kriging model is established systematically. Taking advantage of the characteristics of different weight of the test point with each experimental point according to different distance, an improved Kriging response surface method is proposed to reconstruct performance function by the strategy of increasing efficient points only. In order to further impove reliability calculation accuracy when dealing with high nonlinear performance function, Kriging importance sampling method is established by locating the sampling center at the design points.
(3) Three methods are presented for the combined random reliability problem of linear structure. Firstly, derives the second order perturbation analytical expression for calculating the combined random reliability problem based on perturbation expansion method; Secondly, the calculation process of Kriging sampling method based on statistical conception are also given; and thirdly, the Kriging iteration method is constructed based on the first passage theory, where the structure random parameter is solved in the process of reliability index iteration. Meanwhile, this paper studies the structural random response, which affects the accuracy and efficiency of structural dynamic analysis and reliability calculation., and investigates the calculation method of non-stationary random response based on dynamic stiffness matrix method.
(4) The correlation degree concept is introduced into sensitivity analysis of system with multiple failure modes and correlation coefficient formula is presented for multidimension conditional failure modes based on bi-normal conditional failure probabilities. Then the complex solving process of system sensitivity is carried out by computing the weight coefficient derivation process of failure probabilities that corresponding to all failure modes. Thus the analysis difficulty and workload is decreased but also the calculation process is more easily realized.
(5) The paper constructs a new reliability sensitivity factor by the standard deviation sensitivities based on analysis and derivation of traditional reliability sensitivity factors. Different from the traditional sensitivity factor, the new one not only can reflect the importance difference of all variables to structural failure probability, but also represent the error caused by one or more variables considered as constant. According to the conclusion, the related influence information of each random variable to structural failure probability can be obtained by primary analysis of reliability sensitivity. Thus this method can improve the calculation efficiency and reduce the analysis difficulty greatly.
(1)针对一次二阶矩法处理强非线性功能函数不能收敛的问题,提出了旋转梯度算法。该算法迭代方向由迭代点处的负梯度方向与上一步迭代方向的线性组合确定,并通过旋转率来识别极限状态方程的非线性程度,进而确定旋转系数以及迭代方向计算公式。这样在保证算法收敛性的同时还能保证收敛速度。因不需要试算或人为设定参数值,该方法适用于复杂结构的可靠度分析。
(2)系统地建立了复杂结构可靠度分析的Kriging方法。首先基于Kriging模型建立了可靠度分析的krgiging(?)向应面法：然后利用Kriging模型可直接获得预测点方差并能根据各试验点与待测点距离不同而赋予不同权重的特点,提出了只增加有效点策略来重构极限状态函数的改进Kriging(?)向应面,进一步与旋转梯度算法结合,可以有效提高复杂结构非线性极限状态方程可靠度分析的精度和效率；最后,为进一步提高强非线性极限状态函数的可靠度计算精度,以验算点为抽样中心建立了Kriging重要抽样方法。
(3)对比分析了线性结构复合随机可靠度的三种方法。首先,基于泰勒展开法推导了复合随机可靠度计算的二阶泰勒解析表达式；其次,基于统计概念给出了Kriging抽样方法的计算步骤；第三,基于首次超越破坏准则建立了Kriging迭代法,将结构随机参数在可靠指标的迭代求解中处理。其中Kriging(?)向应面的数值抽样法通过响应面来拟合动力可靠度与结构随机参数的非线性关系,可以方便地利用有限元程序直接计算,避免了泰勒展开等理论推导方法的繁琐和困难。同时,对影响结构动力分析和可靠度计算精度和效率的结构随机响应进行研究,探讨了基于动态刚度阵法的平稳和非平稳随机响应计算的简便方法。
(4)针对多失效模式灵敏度分析情形,将失效模式相关度概念引入到多失效模式系统灵敏度分析中,并根据二维条件失效模式近似公式,提出了适于多维条件失效模式分析的相关系数公式。利用该公式可以将复杂的体系灵敏度求解转换成各失效模式失效概率相关系数的求导过程,从而降低分析难度,减少计算工作量。
(5)在传统可靠性灵敏度因子基础上,利用标准差灵敏度值构造了一种新灵敏度因子。新灵敏度因子不仅能反映各随机因素对结构失效概率影响的重要性程度,而且因子的数值大小还能表征将单个变量作为确定性变量处理或将多个变量同时作为确定性变量处理时所引起的可靠指标误差大小。采用新灵敏度因子进行灵敏度分析时,只需要进行一次分析就能识别出各随机变量对结构失效整体影响的相关信息,可显著提高计算效率。
Reliability theory plays a vital role in safety assessment and design by analysis. However, the traditional theory cannot take computational efficiency> accuracy and stability into consideration simultaneously or satisfy the practical demands when applied to complex engineering structures, thus the scope of application exits some limitations. This paper studies on a series of key theories of structural reliability, involving highly nonlinear problem, combined random vibration problem as well as reliability sensitivity analysis problem etc.,and provides some new theories and methods. Further research on these problems has important significance for perfecting structural reliability theory and guiding engineering practice. The main contributions of the dissertation are as follows:
(1) A new method named rotation gradient algorithm (RGA) is proposed to overcome the divergence problem in reliability analysis when solving high nonlinear performance function by first order reliability method. The new iteration direction is determined by the linear combination of the negative gradient direction and former iteration direction. And the value of rotation angle is determined by the rotation coefficient and the relative rotaion rate is used to identify the different nonlinear degree of limited state function. So the algorithm can ensure convergence stability and convergence rates. No trial calculation and artificial parameters need preset, so it is suitable for reliability analysis of complex engineering structure.
(2) Reliability calculation method suitable to complex structure based on the Kriging model is established systematically. Taking advantage of the characteristics of different weight of the test point with each experimental point according to different distance, an improved Kriging response surface method is proposed to reconstruct performance function by the strategy of increasing efficient points only. In order to further impove reliability calculation accuracy when dealing with high nonlinear performance function, Kriging importance sampling method is established by locating the sampling center at the design points.
(3) Three methods are presented for the combined random reliability problem of linear structure. Firstly, derives the second order perturbation analytical expression for calculating the combined random reliability problem based on perturbation expansion method; Secondly, the calculation process of Kriging sampling method based on statistical conception are also given; and thirdly, the Kriging iteration method is constructed based on the first passage theory, where the structure random parameter is solved in the process of reliability index iteration. Meanwhile, this paper studies the structural random response, which affects the accuracy and efficiency of structural dynamic analysis and reliability calculation., and investigates the calculation method of non-stationary random response based on dynamic stiffness matrix method.
(4) The correlation degree concept is introduced into sensitivity analysis of system with multiple failure modes and correlation coefficient formula is presented for multidimension conditional failure modes based on bi-normal conditional failure probabilities. Then the complex solving process of system sensitivity is carried out by computing the weight coefficient derivation process of failure probabilities that corresponding to all failure modes. Thus the analysis difficulty and workload is decreased but also the calculation process is more easily realized.
(5) The paper constructs a new reliability sensitivity factor by the standard deviation sensitivities based on analysis and derivation of traditional reliability sensitivity factors. Different from the traditional sensitivity factor, the new one not only can reflect the importance difference of all variables to structural failure probability, but also represent the error caused by one or more variables considered as constant. According to the conclusion, the related influence information of each random variable to structural failure probability can be obtained by primary analysis of reliability sensitivity. Thus this method can improve the calculation efficiency and reduce the analysis difficulty greatly.
引文
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