随机和认知不确定性下的结构可靠性方法研究
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摘要
结构可靠性理论与方法作为保障结构系统可靠性的强有力工具，受到工业界及学术界的广泛关注，是可靠性领域的研究热点与前沿课题。由于结构可靠性理论与方法能有效处理结构各阶段所存在的各种不确定性，并且揭示了不确定性对结构可靠性影响的实质，因此，基于结构可靠性理论与方法所设计出的结构往往具有可靠性高、花费少、鲁棒性及性能好等一系列优点。目前考虑随机不确定性的结构可靠性理论与方法取得了许多理论与应用成果。
不确定性广泛存在于工程实际中，通常将其分为随机不确定性和认知不确定性两大类。随机不确定性是事物的固有属性，而认知不确定性是由于缺乏数据、信息不完备等因素所引起的。为了掌握两种不确定性对结构可靠性的影响，随机和认知不确定性下的结构可靠性理论已受到国内外众多学者的极大关注。本文基于区间理论、模糊集理论和概率论，研究了变量相关时的结构非概率可靠性问题、随机和认知不确定性下的结构可靠性分析问题、随机和认知不确定性下的结构可靠性灵敏度分析问题，拓展和完善了现有的结构可靠性理论体系。
本文的研究成果主要体现在如下几个方面：
（1）区间变量相关时的结构非概率可靠性模型。由于系统中变量间存在一定的约束限制，因此变量之间是相关的。考虑了区间变量的相关性，建立了区间变量相关时的结构非概率可靠性指标模型及优化方法。结合有限差分理论，建立了区间变量相关时的结构非概率可靠性灵敏度分析模型及优化方法。该模型考虑了区间变量的相关性，克服了传统结构非概率可靠性模型的不足，完善了现有基于区间的结构非概率可靠性模型。
（2）基于均值一阶鞍点近似的统一不确定性分析方法。当系统中的变量和参数同时含有随机和认知不确定性时，分别用随机变量和区间变量进行建模。为了避免搜索极限状态方程的MPP点（Most Probable Point, MPP）和变量非正态到正态的转换，建立了基于均值一阶鞍点近似的统一不确定性分析模型，提出了相应的求解方法,解决了现有方法在随机和认知不确定性同时存在时效率低下的问题。仿真结果验证了所提方法的精度和高效性。
（3）随机变量和模糊数下的结构可靠性分析方法。当系统中的变量和参数同时含有随机和认知不确定性时，分别用随机变量和模糊数进行建模，建立了随机变量和模糊数同时存在下的结构可靠性统一分析模型。分别基于最大熵理论和鞍点近似法，提出了两种统一不确定性求解方法：方法I和方法II，为系统中同时存在随机变量和模糊数的结构可靠性问题提供了有效的求解方法和理论指导。
（4）基于通用生成函数的多种混合变量下的结构可靠性分析方法。根据数据量的多少，分别用随机变量、P-Box（Probability Box）变量和区间变量对系统中的变量和参数进行建模。在此基础上，研究了多种混合变量的融合问题，对现有通用生成函数法进行拓展，建立了基于通用生成函数的多种混合变量下的结构可靠性分析模型，研究了相应的求解方法。该研究成果为多种混合变量下的结构可靠性分析提供了切实可行的处理方法。
（5）随机和认知不确定性下的结构可靠性灵敏度分析模型。针对系统中的变量和参数同时具有随机和认知不确定性，提出了结构可靠性灵敏度建模方法。为了避免搜索极限状态方程的MPP点和提高精度，考虑了不同样本点的权重，采用移动最小二乘法对极限状态方程进行线性化处理，建立了随机和认知不确定性下的结构可靠性灵敏度分析模型，给出了相应的求解方法，克服了现有方法只能有效处理随机不确定性下的可靠性灵敏度分析的局限性。与现有可靠性灵敏度分析方法相比，所提方法鲁棒性好，并且适合极限状态方程为隐函数的情形。
Theory of structural reliability under uncertainty, which has been a foucs of theindustry and academia, is not only a powerful tool for the protection of structuralsystems reliability, but also a hot research topic in reliability engineering. Structuralreliability theory and methods can deal with various uncertainties which come fromdifferent stages of structures effectively as well as the influence of these uncertainties onstructural reliability. Therefore, the advantages of the structures designed usingstructural reliability theory are high reiliability, less costs, more robustness and excellentperformances. Nowadays, lots of theory and application achievements have beenachieved for structural reliability theory under aleatory uncertainty.
Uncertainty widely exists in engineering practices, and it can be respectivelydivided into aleatory uncertainty and epistemic uncertainty. Aleatory uncertainty arisesfrom inherent variation while epistemic uncertainty comes from insufficient data andincomplete information. In order to grasp the essentials of these uncertainties forstructural reliability, structural reliability theory under both aleatory and epistemicuncertainties has gained increasing attention for many scholars. In this dissertation,based on the interval theory, fuzzy sets theory and probability theory, the followingproblems are studied: non-probabilistic reliability model with dependent intervalvariables; structural relialibty analysis under both aleatory and epistemic uncertainties;structural reliability sensitivity analysis under both aleatory and epistemic uncertainties.Therefore, the researches in this dissertation extend the traditional structural theory tobecomming a more mature theory.
The contributions of this dissertation are summarized as follows:
(1) Non-probabilistic reliability model with dependent interval variables. Becauseof the constraints exist in systems for many interval variables, so they often dependenteach other. The non-probabilistic reliability index model and optimization method withdependent interval variables are proposed. Furthermore, based on the finite differencetheory, non-probabilistic reliability sensitivity model and optimization methods withdependent interval variables are also developed. The dependency of interval variables is taken into account in the developed non-probabilistic reliability model to overcome thedeficiencies of the traditional non-probabilistic model, and the research has extended theexisting interval-based non-probabilistic model.
(2) Unified uncertainty analysis based on the mean value first order saddlepointapproximation (MVFOSPA-UUA). Variables and parameters are modeled by randomand interval variables under the case of systems associated with both aleatory andepistemic uncertainties. In order to avoid MPP search and non-normal to normaltransformations, a novel model, named MVFOSPA-UUA, is proposed. Furthermoer, themethod for solving the model is also developed. The low efficiency problem for theexisting methods under both aleatory and epistemic uncertainties is solved. Theaccuracy and high efficiency of the proposed method are domenstrated by simulation.
(3) Structural reliability analysis under both random variables and fuzzy numbers.Variables and parameters are modeled by random variables and fuzzy numbers underthe case of systems associated with both aleatory and epistemic uncertainties, and theunified uncertainty analysis model under both random variables and fuzzy numbers isproposed. Two unified uncertainy analysis methods, methods I and II, are developed forsolving the model. This research provides an effective reliability method and theoreticalguidance for structural systems with both random variables and fuzzy numbers.
(4) Structural reliability analysis method under mixed variables based on theuniversal generation function. Variables and parameters are respectively modeled byrandom variables, P-Box and interval variables according to the amount of data in thesystem. On this basis, the existing universal generating function is extended, and thestructural reliability analysis model under mixed variables based on the universalgenerating function is estabilished. Furthermore, a solution method is also provided.This research provides a practical method for structural reliability analysis undermultiple variables.
(5) Structural reliability sensitivity analysis model under both aleatory andepistemic uncertainities. An appropriate reliability sensitivity modeling is provided forvariables and parameters when both aleatory and epistemic uncertainties exist in thesystem. In order to avoid MPP search and improve accuracy, the weights of samples areconsidered, and the limit-state function is linearized based on the moving least squares.The reliability sensitivity model under both aleatory and epistemic uncertainties and its corresponding soulution method are estabilished, and it overcomes the limitations forthe existing methods which can only calculate reliability sensitivity under aleatoryuncertainty effectively. The proposed mthod is robustness, when compared with theexisting reliability sensitivity methods, which is suitable for the situation of thelimit-state function is an implicit function.
不确定性广泛存在于工程实际中，通常将其分为随机不确定性和认知不确定性两大类。随机不确定性是事物的固有属性，而认知不确定性是由于缺乏数据、信息不完备等因素所引起的。为了掌握两种不确定性对结构可靠性的影响，随机和认知不确定性下的结构可靠性理论已受到国内外众多学者的极大关注。本文基于区间理论、模糊集理论和概率论，研究了变量相关时的结构非概率可靠性问题、随机和认知不确定性下的结构可靠性分析问题、随机和认知不确定性下的结构可靠性灵敏度分析问题，拓展和完善了现有的结构可靠性理论体系。
本文的研究成果主要体现在如下几个方面：
（1）区间变量相关时的结构非概率可靠性模型。由于系统中变量间存在一定的约束限制，因此变量之间是相关的。考虑了区间变量的相关性，建立了区间变量相关时的结构非概率可靠性指标模型及优化方法。结合有限差分理论，建立了区间变量相关时的结构非概率可靠性灵敏度分析模型及优化方法。该模型考虑了区间变量的相关性，克服了传统结构非概率可靠性模型的不足，完善了现有基于区间的结构非概率可靠性模型。
（2）基于均值一阶鞍点近似的统一不确定性分析方法。当系统中的变量和参数同时含有随机和认知不确定性时，分别用随机变量和区间变量进行建模。为了避免搜索极限状态方程的MPP点（Most Probable Point, MPP）和变量非正态到正态的转换，建立了基于均值一阶鞍点近似的统一不确定性分析模型，提出了相应的求解方法,解决了现有方法在随机和认知不确定性同时存在时效率低下的问题。仿真结果验证了所提方法的精度和高效性。
（3）随机变量和模糊数下的结构可靠性分析方法。当系统中的变量和参数同时含有随机和认知不确定性时，分别用随机变量和模糊数进行建模，建立了随机变量和模糊数同时存在下的结构可靠性统一分析模型。分别基于最大熵理论和鞍点近似法，提出了两种统一不确定性求解方法：方法I和方法II，为系统中同时存在随机变量和模糊数的结构可靠性问题提供了有效的求解方法和理论指导。
（4）基于通用生成函数的多种混合变量下的结构可靠性分析方法。根据数据量的多少，分别用随机变量、P-Box（Probability Box）变量和区间变量对系统中的变量和参数进行建模。在此基础上，研究了多种混合变量的融合问题，对现有通用生成函数法进行拓展，建立了基于通用生成函数的多种混合变量下的结构可靠性分析模型，研究了相应的求解方法。该研究成果为多种混合变量下的结构可靠性分析提供了切实可行的处理方法。
（5）随机和认知不确定性下的结构可靠性灵敏度分析模型。针对系统中的变量和参数同时具有随机和认知不确定性，提出了结构可靠性灵敏度建模方法。为了避免搜索极限状态方程的MPP点和提高精度，考虑了不同样本点的权重，采用移动最小二乘法对极限状态方程进行线性化处理，建立了随机和认知不确定性下的结构可靠性灵敏度分析模型，给出了相应的求解方法，克服了现有方法只能有效处理随机不确定性下的可靠性灵敏度分析的局限性。与现有可靠性灵敏度分析方法相比，所提方法鲁棒性好，并且适合极限状态方程为隐函数的情形。
Theory of structural reliability under uncertainty, which has been a foucs of theindustry and academia, is not only a powerful tool for the protection of structuralsystems reliability, but also a hot research topic in reliability engineering. Structuralreliability theory and methods can deal with various uncertainties which come fromdifferent stages of structures effectively as well as the influence of these uncertainties onstructural reliability. Therefore, the advantages of the structures designed usingstructural reliability theory are high reiliability, less costs, more robustness and excellentperformances. Nowadays, lots of theory and application achievements have beenachieved for structural reliability theory under aleatory uncertainty.
Uncertainty widely exists in engineering practices, and it can be respectivelydivided into aleatory uncertainty and epistemic uncertainty. Aleatory uncertainty arisesfrom inherent variation while epistemic uncertainty comes from insufficient data andincomplete information. In order to grasp the essentials of these uncertainties forstructural reliability, structural reliability theory under both aleatory and epistemicuncertainties has gained increasing attention for many scholars. In this dissertation,based on the interval theory, fuzzy sets theory and probability theory, the followingproblems are studied: non-probabilistic reliability model with dependent intervalvariables; structural relialibty analysis under both aleatory and epistemic uncertainties;structural reliability sensitivity analysis under both aleatory and epistemic uncertainties.Therefore, the researches in this dissertation extend the traditional structural theory tobecomming a more mature theory.
The contributions of this dissertation are summarized as follows:
(1) Non-probabilistic reliability model with dependent interval variables. Becauseof the constraints exist in systems for many interval variables, so they often dependenteach other. The non-probabilistic reliability index model and optimization method withdependent interval variables are proposed. Furthermore, based on the finite differencetheory, non-probabilistic reliability sensitivity model and optimization methods withdependent interval variables are also developed. The dependency of interval variables is taken into account in the developed non-probabilistic reliability model to overcome thedeficiencies of the traditional non-probabilistic model, and the research has extended theexisting interval-based non-probabilistic model.
(2) Unified uncertainty analysis based on the mean value first order saddlepointapproximation (MVFOSPA-UUA). Variables and parameters are modeled by randomand interval variables under the case of systems associated with both aleatory andepistemic uncertainties. In order to avoid MPP search and non-normal to normaltransformations, a novel model, named MVFOSPA-UUA, is proposed. Furthermoer, themethod for solving the model is also developed. The low efficiency problem for theexisting methods under both aleatory and epistemic uncertainties is solved. Theaccuracy and high efficiency of the proposed method are domenstrated by simulation.
(3) Structural reliability analysis under both random variables and fuzzy numbers.Variables and parameters are modeled by random variables and fuzzy numbers underthe case of systems associated with both aleatory and epistemic uncertainties, and theunified uncertainty analysis model under both random variables and fuzzy numbers isproposed. Two unified uncertainy analysis methods, methods I and II, are developed forsolving the model. This research provides an effective reliability method and theoreticalguidance for structural systems with both random variables and fuzzy numbers.
(4) Structural reliability analysis method under mixed variables based on theuniversal generation function. Variables and parameters are respectively modeled byrandom variables, P-Box and interval variables according to the amount of data in thesystem. On this basis, the existing universal generating function is extended, and thestructural reliability analysis model under mixed variables based on the universalgenerating function is estabilished. Furthermore, a solution method is also provided.This research provides a practical method for structural reliability analysis undermultiple variables.
(5) Structural reliability sensitivity analysis model under both aleatory andepistemic uncertainities. An appropriate reliability sensitivity modeling is provided forvariables and parameters when both aleatory and epistemic uncertainties exist in thesystem. In order to avoid MPP search and improve accuracy, the weights of samples areconsidered, and the limit-state function is linearized based on the moving least squares.The reliability sensitivity model under both aleatory and epistemic uncertainties and its corresponding soulution method are estabilished, and it overcomes the limitations forthe existing methods which can only calculate reliability sensitivity under aleatoryuncertainty effectively. The proposed mthod is robustness, when compared with theexisting reliability sensitivity methods, which is suitable for the situation of thelimit-state function is an implicit function.
引文
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