不确定结构可靠性分析与优化设计研究
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摘要
工程实际结构中存在着诸多不确定因素。可靠性方法是处理结构不确定因素的有效途径之一。本文从不同角度和层面对结构可靠性问题进行了有益的探索和研究。研究主要涉及结构非概率可靠性模型、结构非概率可靠性度量与安全系数、混合结构可靠性分析与优化设计、混合桁架结构可靠性拓扑优化设计等内容。主要研究内容如下:
1.提出了一种基于椭球凸集的结构非概率可靠性模型,采用结构安全域的体积与基本变量域的体积之比来度量结构非概率可靠性。证明了基于凸集的非概率可靠性模型与概率可靠性模型的兼容性。在椭球确定区间向量的条件下,对比分析了基于凸集与基于区间的非概率可靠性模型,揭示了两者的联系和差异。
2.分别研究了结构安全系数与非概率可靠性度量。建立了中心、非概率与区间三种安全系数的函数关系,对比分析了非概率可靠性指标和非概率集合可靠度两种非概率可靠性度量。探讨了安全系数与非概率可靠性度量的关系,构建了两者的桥梁。
3.对比研究了非概率安全系数法和非概率可靠性方法。证明了非概率安全系数和非概率可靠性指标在度量结构安全上的等价性。从设计思想、度量方法和表现形式等方面分析了两种方法的联系和区别。优化实例分析结果表明,非概率安全系数法和非概率可靠性方法在结构重量和截面尺寸分配上存在着一定差异。
4.提出了模糊-非概率混合结构可靠性分析与优化设计方法。基于能度可靠性理论,建立了混合结构可靠性模型,以最小模糊可靠性指标和最大失效可能度来度量混合结构可靠性。提出了混合结构可靠性分析方法,并以最大失效可能度为约束条件构建了混合结构可靠性优化模型,通过算例验证了所提方法的有效性和可行性。
5.提出了概率-非概率混合桁架结构可靠性拓扑优化方法。采用混合可靠性指标来度量结构可靠性,给出了混合可靠性指标的求解方法,建立了以结构重量为优化目标、以位移、应力混合可靠性指标为约束条件的桁架结构可靠性拓扑优化模型,给出了混合可靠性指标对设计变量的灵敏度计算公式,通过算例验证了所提方法的有效性和可行性。
6.研究了具有有界不确定参数的多学科系统的不确定性分析问题。结合泰勒级数和全局敏度方程,分别采用区间分析法和凸模型方法推导了系统输出和连接变量的计算公式,从理论及实例对两种方法的求解结果进行了比较。结果表明,区间分析法比凸模型方法的解区间更接近蒙特卡罗仿真法的解区间。
Uncertainties exist in engineering structures. Reliability method is an effective tool to deal with uncertainties of structures. The exploring research is made in order to solve the problem of structural reliability. The research includes non-probabilistic reliability model, non-probabilistic reliability measures and safety factors, hybrid structural reliability analysis and reliability-based optimization, reliability-based topology optimization of hybrid truss structures and so on. The main content is as follows:
1. A structural non-probabilistic reliability model is proposed, in which uncertain parameters of structures is described as ellipsoidal convex model. The ratio of the volume of safe region to the total volume of the region constructed by the basic variables is utilized as the measurement of structural reliability. The compatibility of the non-probabilistic reliability model with the probabilistic reliability model is proved. In the case of interval vector determined from the ellipsoid of the uncertain parameter vector, a comparison between the two non-probabilistic reliability models based on ellipsoidal model and interval model is presented, in which their interrelation and difference illustrated.
2. Safety factors and non-probabilistic reliability measures are investigated respectively. The functions of three kinds of safety factors named central, non-probabilistic and interval factors are established. A comparison between the two kinds of non-probabilistic reliability measures, which are reliability index and set-theoretic reliability measure, is presented. The connections between safety factors and non-probabilistic reliability measures are discussed. A bridge between safety factors and non-probabilistic reliability measures is made.
3. A comparison between the non-probabilistic safety factor and non-probabilistic reliability methods for structural design is presented. The equivalence of non-probabilistic safety factor and non-probabilistic reliability index in the measurement of structural safety is proved. The differences and relations of design concept, measure method and representation formation, between the two non-probabilistic methods are investigated. The result from a numerical example shows that the differences of the two non-probabilistic methods in structure weight and bar sectional dimension.
4. A reliability analysis and reliability-based optimization method of fuzzy and non- probabilistic hybrid structures is proposed. Based on possibility reliability theory, a reliability model of hybrid structures is established. The minimum fuzzy reliability index and the maximum failure possibility are utilized as the measurement of hybrid structural reliability. A reliability analysis method of hybrid structures is presented. By regarding the maximum failure possibility as constrains, a reliability-based optimization model of hybrid structures is developed. Three numerical examples show the method proposed is effective and feasible.
5. A reliability-based topology optimization method of probabilistic and non- probabilistic hybrid truss structures is proposed. The hybrid reliability index is utilized as the measurement of structural reliability. A reliability-based optimization model of probabilistic and non-probabilistic hybrid truss structures is developed, in which the structure weight is taken as objective function, both the hybrid reliability index of structural displacement and bar stress are taken as constraint functions. The sensitivity of hybrid reliability index for design variables is derived. A numerical example illustrates the feasibility and validity of the method proposed.
6. An uncertainty analysis method for multidisciplinary systems with uncertain-but-bounded parameters is presented. Based on Taylor series and global sensitivity equations, the numerical algorithms for the linking variables and the outputs of the systems are deduced using interval analysis method and convex models respectively. The comparison of the solved results between interval analysis method and convex models is performed through the mathematical proof and the numerical examples. The results show that the interval of solution obtained by interval analysis method is closer to Monte Carlo simulation than that obtained by convex models.
1.提出了一种基于椭球凸集的结构非概率可靠性模型,采用结构安全域的体积与基本变量域的体积之比来度量结构非概率可靠性。证明了基于凸集的非概率可靠性模型与概率可靠性模型的兼容性。在椭球确定区间向量的条件下,对比分析了基于凸集与基于区间的非概率可靠性模型,揭示了两者的联系和差异。
2.分别研究了结构安全系数与非概率可靠性度量。建立了中心、非概率与区间三种安全系数的函数关系,对比分析了非概率可靠性指标和非概率集合可靠度两种非概率可靠性度量。探讨了安全系数与非概率可靠性度量的关系,构建了两者的桥梁。
3.对比研究了非概率安全系数法和非概率可靠性方法。证明了非概率安全系数和非概率可靠性指标在度量结构安全上的等价性。从设计思想、度量方法和表现形式等方面分析了两种方法的联系和区别。优化实例分析结果表明,非概率安全系数法和非概率可靠性方法在结构重量和截面尺寸分配上存在着一定差异。
4.提出了模糊-非概率混合结构可靠性分析与优化设计方法。基于能度可靠性理论,建立了混合结构可靠性模型,以最小模糊可靠性指标和最大失效可能度来度量混合结构可靠性。提出了混合结构可靠性分析方法,并以最大失效可能度为约束条件构建了混合结构可靠性优化模型,通过算例验证了所提方法的有效性和可行性。
5.提出了概率-非概率混合桁架结构可靠性拓扑优化方法。采用混合可靠性指标来度量结构可靠性,给出了混合可靠性指标的求解方法,建立了以结构重量为优化目标、以位移、应力混合可靠性指标为约束条件的桁架结构可靠性拓扑优化模型,给出了混合可靠性指标对设计变量的灵敏度计算公式,通过算例验证了所提方法的有效性和可行性。
6.研究了具有有界不确定参数的多学科系统的不确定性分析问题。结合泰勒级数和全局敏度方程,分别采用区间分析法和凸模型方法推导了系统输出和连接变量的计算公式,从理论及实例对两种方法的求解结果进行了比较。结果表明,区间分析法比凸模型方法的解区间更接近蒙特卡罗仿真法的解区间。
Uncertainties exist in engineering structures. Reliability method is an effective tool to deal with uncertainties of structures. The exploring research is made in order to solve the problem of structural reliability. The research includes non-probabilistic reliability model, non-probabilistic reliability measures and safety factors, hybrid structural reliability analysis and reliability-based optimization, reliability-based topology optimization of hybrid truss structures and so on. The main content is as follows:
1. A structural non-probabilistic reliability model is proposed, in which uncertain parameters of structures is described as ellipsoidal convex model. The ratio of the volume of safe region to the total volume of the region constructed by the basic variables is utilized as the measurement of structural reliability. The compatibility of the non-probabilistic reliability model with the probabilistic reliability model is proved. In the case of interval vector determined from the ellipsoid of the uncertain parameter vector, a comparison between the two non-probabilistic reliability models based on ellipsoidal model and interval model is presented, in which their interrelation and difference illustrated.
2. Safety factors and non-probabilistic reliability measures are investigated respectively. The functions of three kinds of safety factors named central, non-probabilistic and interval factors are established. A comparison between the two kinds of non-probabilistic reliability measures, which are reliability index and set-theoretic reliability measure, is presented. The connections between safety factors and non-probabilistic reliability measures are discussed. A bridge between safety factors and non-probabilistic reliability measures is made.
3. A comparison between the non-probabilistic safety factor and non-probabilistic reliability methods for structural design is presented. The equivalence of non-probabilistic safety factor and non-probabilistic reliability index in the measurement of structural safety is proved. The differences and relations of design concept, measure method and representation formation, between the two non-probabilistic methods are investigated. The result from a numerical example shows that the differences of the two non-probabilistic methods in structure weight and bar sectional dimension.
4. A reliability analysis and reliability-based optimization method of fuzzy and non- probabilistic hybrid structures is proposed. Based on possibility reliability theory, a reliability model of hybrid structures is established. The minimum fuzzy reliability index and the maximum failure possibility are utilized as the measurement of hybrid structural reliability. A reliability analysis method of hybrid structures is presented. By regarding the maximum failure possibility as constrains, a reliability-based optimization model of hybrid structures is developed. Three numerical examples show the method proposed is effective and feasible.
5. A reliability-based topology optimization method of probabilistic and non- probabilistic hybrid truss structures is proposed. The hybrid reliability index is utilized as the measurement of structural reliability. A reliability-based optimization model of probabilistic and non-probabilistic hybrid truss structures is developed, in which the structure weight is taken as objective function, both the hybrid reliability index of structural displacement and bar stress are taken as constraint functions. The sensitivity of hybrid reliability index for design variables is derived. A numerical example illustrates the feasibility and validity of the method proposed.
6. An uncertainty analysis method for multidisciplinary systems with uncertain-but-bounded parameters is presented. Based on Taylor series and global sensitivity equations, the numerical algorithms for the linking variables and the outputs of the systems are deduced using interval analysis method and convex models respectively. The comparison of the solved results between interval analysis method and convex models is performed through the mathematical proof and the numerical examples. The results show that the interval of solution obtained by interval analysis method is closer to Monte Carlo simulation than that obtained by convex models.
引文
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