结构系统可靠性分析的若干问题研究
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摘要
可靠性分析是对结构系统开展安全性评估、基于可靠度进行设计及优化的必要基础和前提，也是提高结构安全水平，提升工程结构经济性的有力保障和技术支撑，工程实践需求迫切。之所以至今未能广泛、全面应用于实际工程，其主要原因是结构系统可靠性基本理论中尚存部分重点、难点问题未得到完善解决。
论文主要研究了串（并）联体系可靠度计算及功能函数建立、失效模式间相性分析和主要失效模式识别等几个关键问题。通过引入新理论、改进旧方法、提出新方法等技术手段，有效提高了串（并）联体系可靠度计算精度；推导了串（并）联体系等效功能函数显示求解表达式；合理解决了失效模式相关系数计算问题；大幅度提升了识别主要失效模式的效率。重点解决了结构系统可靠性分析中，计算精度不足、计算效率低下及串（并）联体系功能函数建立困难等直接影响其工程应用的问题。
通过对现有方法的对比分析，基于等效平面思想的可靠度计算方法，相对较适合结构系统可靠性分析。但其等效过程中涉及的相关系数计算缺少理论依据，导致计算精度精度不足，应用受限。因此，在结构系统可靠性分析中引入复相关理论，解决涉及的一个单元与多个单元相关系数计算问题，提出复相关等效平面法。该法用复相关系数描述等效后平面与其余极限状态面的相关程度，合理克服了等效原则不含相关性信息的缺点，解决了等效平面法分析精度不足的问题。通过算例对比分析了该方法与数值积分、蒙特卡洛模拟的相对误差，证明对串、并联体系有较高的计算精度及运算速度；通过对典型例题计算，分析了该方法与传统结构系统可靠性分析方法的优劣，证明该方法优势明显，适合大型结构系统的可靠性分析。
对于串（并）联体系功能函数建立困难、求解效率低等问题，根据等效功能函数满足应用需求的特点，采用等效功能函数代替的解决方案。在建立的复相关等效平面法基础上，根据等效原则推导了串（并）联体系等效功能函数。给出只涉及二维积分运算的显式递推表达式。完善后的复相关等效平面法，可同步解决串（并）联体系可靠度计算和等效功能函数建立问题，也可仅求解可靠度而获得更高运算效率。通过算例分析，证明完善后的复相关等效平面法求解等效功能函数具有较高可信度。等效功能函数较为精确的反映了串、并联体系可靠度对各随机变量的敏度，满足基于可靠度的结构设计、优化等方面的应用要求。
针对结构系统可靠性分析过程中，主要失效模式相关系数求解缺少理论依据、相关系数对失效模式间相关程度描述不准确的问题，将统计分析方法中的典型相关理论应用到可靠性分析中，对失效模式间相关性问题作出合理解释；证明利用第二典型相关系数可以合理、准确描述主要失效模式间的相关程度；通过算例分析验证了其正确性和可行性，证实基于第二典型相关系数可获得较高精度的结构系统失效概率。
在对现有两大类识别主要失效模式方法进行深入分析的基础上，对非主要失效模式被识别情况较多的问题展开研究。利用力学基本理论，分析了单元失效对残余结构内力重分配的影响规律，建立了其关系式；在此基础上，证明含相同基本失效单元的失效模式线性相关；结合概率论，得出基本失效单元是组成主要失效模式必备单元的结论；对结构系统失效形式进行分析，讨论了基本单元的存在范围；采用结构的逐步搭建思想，提出包含基本单元的最小范围划分方法，理论上保证了不遗漏主要失效模式；与分支限界法结合，建立了识别桁架结构系统主要失效模式的方法；通过算例分析，证实该法合理有效，识别效率较高。
Reliability analysis of structural systems is the basis and premise of carrying outpractical engineering structure safety assessment and conducting structure design andoptimization, and it is also the strong guarantee and technical support of improving structuresafety level and upgrading engineering structure economy. The need for reliability analysis ofstructural systems is urgent in engineering practice. But the main reason why it is not widelyused in practical engineering is that parts of emphasis and difficulty in the reliability theory ofstructural systems have not been solved completely.
The paper mainly focuses on the study of the reliability calculation of serial(parallel)system, the function establishment of serial (parallel)system, the correlation analysisbetween failure modes and the effectively recognition of main failure modes, etc. Thecalculation accuracy of serial (parallel) system reliability is improved effectively byintroducing new theory, improving old method and putting forward new method, etc. Theexplicit expression of equivalent function for serial (parallel) system has been deduced; theproblem of correlation coefficients calculation between failure modes has been solvedreasonably; the recognition efficiency of main failure modes has been increased vastly. Theproblems which effects engineering application have been focused on, like poor calculationaccuracy, low computational efficiency and the difficulty in establishing function of serial(parallel) system.
Through the comparison for existing methods, the calculation method of reliability basedon equivalent plane method is relatively suitable for reliability analysis of structural systems.But for parallel system, the calculation is limited and the accuracy is not enough. The mainreason is that the calculation of correlation coefficient is lack of theoretical basis in theprocess of equivalent, specifically for parallel system. Therefore, the concept of multiplecorrelations was introduced to reliability analysis of structural systems, to solve correlationcoefficients between one element and multiple elements, and multiple correlation-equivalentplane method was presented. The method uses multiple correlation coefficients to express thecorrelation between equivalent plane and other ultimate state surfaces. It solves thenon-correlation problem in equivalent principles, and overcomes the shortcoming of greaterror when equivalent plane method was used to analyze the reliability of parallel system. By comparing the relative error of this method and numerical integration, Monte Carlo method, itproves that this method has high calculation accuracy and speed for series (parallel) system.The advantages and disadvantages were analyzed between this method and the traditionalmethod by the typical examples, which proved that this method is suitable for the reliabilityanalysis of large structural system.
Equivalent function was put forward to solve the problems of difficulty in establishingfunction of serial (parallel) system and low computational efficiency. Based on multiplecorrelation-equivalent plane method, equivalent function was deduced by equivalent principle.The explicit expression of equivalent function with constant coefficient which only involvestwo-dimension integral operation was given. Reliability of series (parallel) system andequivalent function could be solved by adequate multiple correlation-equivalent plane methodand equivalent function could be neglected to calculate reliability with high computationalefficiency. It was proved to be credible to solve equivalent function by multiplecorrelation-equivalent plane method through examples analysis. Equivalent function canreflect the acuteness of reliability for random variables accurately, and it can meet theapplication requirements of structure design and optimization based on reliability.
Canonical correlation analysis was introduced into structural reliability analysis for theproblems that calculations of correlation coefficients between the failure modes are lack oftheoretical basis and the correlation between failure modes could not be described bycorrelation coefficients in the process of solving failure probability. The correlation betweenfailure modes has been explained reasonably; it has been proved that the correlation betweenfailure modes could be described accurately by the second canonical correlation coefficients.The correctness and the feasibility have been checked by typical numerical examples; it wasproved that the calculation accuracy of structural system reliability has been improved.
On the base of analyzing the existing two kinds of identification methods of main failuremode, the problem that the reserved candidate failure elements are not all the necessaryelements of failure modes has been studied. According to the basic theory of structuralmechanics, the effect of failure elements on structure remained internal force has beenanalyzed, and the formula has been deduced; According to the formula, failure modes withthe same basic failure elements have been proved linerly correlated; Based on probabilitytheory, it can be concluded that the main failure modes only contain basic failure elements, and the necessary elements of main failure modes are the basic elements has been proved;Failure modes of structural system have been analyzed, and the failure range of structuralsystem has been discussed; Based on the thoughts of building structure gradually, the dividedmethod of minimum existence range including basic elements has been proposed to makesure main failure mode will not be left out; A new method of identifying main failure mode oftrusswork has been put forward combined with the branch bounding method. According toexamples analysis, it can be proved that this method is rational and high-efficient.
论文主要研究了串（并）联体系可靠度计算及功能函数建立、失效模式间相性分析和主要失效模式识别等几个关键问题。通过引入新理论、改进旧方法、提出新方法等技术手段，有效提高了串（并）联体系可靠度计算精度；推导了串（并）联体系等效功能函数显示求解表达式；合理解决了失效模式相关系数计算问题；大幅度提升了识别主要失效模式的效率。重点解决了结构系统可靠性分析中，计算精度不足、计算效率低下及串（并）联体系功能函数建立困难等直接影响其工程应用的问题。
通过对现有方法的对比分析，基于等效平面思想的可靠度计算方法，相对较适合结构系统可靠性分析。但其等效过程中涉及的相关系数计算缺少理论依据，导致计算精度精度不足，应用受限。因此，在结构系统可靠性分析中引入复相关理论，解决涉及的一个单元与多个单元相关系数计算问题，提出复相关等效平面法。该法用复相关系数描述等效后平面与其余极限状态面的相关程度，合理克服了等效原则不含相关性信息的缺点，解决了等效平面法分析精度不足的问题。通过算例对比分析了该方法与数值积分、蒙特卡洛模拟的相对误差，证明对串、并联体系有较高的计算精度及运算速度；通过对典型例题计算，分析了该方法与传统结构系统可靠性分析方法的优劣，证明该方法优势明显，适合大型结构系统的可靠性分析。
对于串（并）联体系功能函数建立困难、求解效率低等问题，根据等效功能函数满足应用需求的特点，采用等效功能函数代替的解决方案。在建立的复相关等效平面法基础上，根据等效原则推导了串（并）联体系等效功能函数。给出只涉及二维积分运算的显式递推表达式。完善后的复相关等效平面法，可同步解决串（并）联体系可靠度计算和等效功能函数建立问题，也可仅求解可靠度而获得更高运算效率。通过算例分析，证明完善后的复相关等效平面法求解等效功能函数具有较高可信度。等效功能函数较为精确的反映了串、并联体系可靠度对各随机变量的敏度，满足基于可靠度的结构设计、优化等方面的应用要求。
针对结构系统可靠性分析过程中，主要失效模式相关系数求解缺少理论依据、相关系数对失效模式间相关程度描述不准确的问题，将统计分析方法中的典型相关理论应用到可靠性分析中，对失效模式间相关性问题作出合理解释；证明利用第二典型相关系数可以合理、准确描述主要失效模式间的相关程度；通过算例分析验证了其正确性和可行性，证实基于第二典型相关系数可获得较高精度的结构系统失效概率。
在对现有两大类识别主要失效模式方法进行深入分析的基础上，对非主要失效模式被识别情况较多的问题展开研究。利用力学基本理论，分析了单元失效对残余结构内力重分配的影响规律，建立了其关系式；在此基础上，证明含相同基本失效单元的失效模式线性相关；结合概率论，得出基本失效单元是组成主要失效模式必备单元的结论；对结构系统失效形式进行分析，讨论了基本单元的存在范围；采用结构的逐步搭建思想，提出包含基本单元的最小范围划分方法，理论上保证了不遗漏主要失效模式；与分支限界法结合，建立了识别桁架结构系统主要失效模式的方法；通过算例分析，证实该法合理有效，识别效率较高。
Reliability analysis of structural systems is the basis and premise of carrying outpractical engineering structure safety assessment and conducting structure design andoptimization, and it is also the strong guarantee and technical support of improving structuresafety level and upgrading engineering structure economy. The need for reliability analysis ofstructural systems is urgent in engineering practice. But the main reason why it is not widelyused in practical engineering is that parts of emphasis and difficulty in the reliability theory ofstructural systems have not been solved completely.
The paper mainly focuses on the study of the reliability calculation of serial(parallel)system, the function establishment of serial (parallel)system, the correlation analysisbetween failure modes and the effectively recognition of main failure modes, etc. Thecalculation accuracy of serial (parallel) system reliability is improved effectively byintroducing new theory, improving old method and putting forward new method, etc. Theexplicit expression of equivalent function for serial (parallel) system has been deduced; theproblem of correlation coefficients calculation between failure modes has been solvedreasonably; the recognition efficiency of main failure modes has been increased vastly. Theproblems which effects engineering application have been focused on, like poor calculationaccuracy, low computational efficiency and the difficulty in establishing function of serial(parallel) system.
Through the comparison for existing methods, the calculation method of reliability basedon equivalent plane method is relatively suitable for reliability analysis of structural systems.But for parallel system, the calculation is limited and the accuracy is not enough. The mainreason is that the calculation of correlation coefficient is lack of theoretical basis in theprocess of equivalent, specifically for parallel system. Therefore, the concept of multiplecorrelations was introduced to reliability analysis of structural systems, to solve correlationcoefficients between one element and multiple elements, and multiple correlation-equivalentplane method was presented. The method uses multiple correlation coefficients to express thecorrelation between equivalent plane and other ultimate state surfaces. It solves thenon-correlation problem in equivalent principles, and overcomes the shortcoming of greaterror when equivalent plane method was used to analyze the reliability of parallel system. By comparing the relative error of this method and numerical integration, Monte Carlo method, itproves that this method has high calculation accuracy and speed for series (parallel) system.The advantages and disadvantages were analyzed between this method and the traditionalmethod by the typical examples, which proved that this method is suitable for the reliabilityanalysis of large structural system.
Equivalent function was put forward to solve the problems of difficulty in establishingfunction of serial (parallel) system and low computational efficiency. Based on multiplecorrelation-equivalent plane method, equivalent function was deduced by equivalent principle.The explicit expression of equivalent function with constant coefficient which only involvestwo-dimension integral operation was given. Reliability of series (parallel) system andequivalent function could be solved by adequate multiple correlation-equivalent plane methodand equivalent function could be neglected to calculate reliability with high computationalefficiency. It was proved to be credible to solve equivalent function by multiplecorrelation-equivalent plane method through examples analysis. Equivalent function canreflect the acuteness of reliability for random variables accurately, and it can meet theapplication requirements of structure design and optimization based on reliability.
Canonical correlation analysis was introduced into structural reliability analysis for theproblems that calculations of correlation coefficients between the failure modes are lack oftheoretical basis and the correlation between failure modes could not be described bycorrelation coefficients in the process of solving failure probability. The correlation betweenfailure modes has been explained reasonably; it has been proved that the correlation betweenfailure modes could be described accurately by the second canonical correlation coefficients.The correctness and the feasibility have been checked by typical numerical examples; it wasproved that the calculation accuracy of structural system reliability has been improved.
On the base of analyzing the existing two kinds of identification methods of main failuremode, the problem that the reserved candidate failure elements are not all the necessaryelements of failure modes has been studied. According to the basic theory of structuralmechanics, the effect of failure elements on structure remained internal force has beenanalyzed, and the formula has been deduced; According to the formula, failure modes withthe same basic failure elements have been proved linerly correlated; Based on probabilitytheory, it can be concluded that the main failure modes only contain basic failure elements, and the necessary elements of main failure modes are the basic elements has been proved;Failure modes of structural system have been analyzed, and the failure range of structuralsystem has been discussed; Based on the thoughts of building structure gradually, the dividedmethod of minimum existence range including basic elements has been proposed to makesure main failure mode will not be left out; A new method of identifying main failure mode oftrusswork has been put forward combined with the branch bounding method. According toexamples analysis, it can be proved that this method is rational and high-efficient.
引文
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