结构整体可靠度方法及RC框架非线性整体抗震可靠度分析
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摘要
基于性能的地震工程(Performamance-Based Earthquake Engineering, PBEE)和基于性能的抗震设计(Performance-Based Seismic Design, PBSD)是美国太平洋地震工程研究中心(Pacific Earthquake Engineering Research, PEER)提出的新一代抗震设计理念、方法与技术，得到了全球地震工程界的研究者和工程师们的热烈响应。由于地震的发生在时间、空间和强度上具有强烈的随机性，地震地面运动具有随机过程和随机场特性；另外，工程结构的地震需求和抗震能力方面都存在大量的不确定性，因此将PBEE和PBSD建立在基于结构可靠度理论的概率设计方法之上非常必要，基于可靠度理论的概率抗震性能评估也成为PBEE和PBSD的主要研究内容之一。
结构体系的抗震可靠度是对结构概率地震风险评定的定量衡量，近年来出现了利用结构整体极限状态来近似计算结构体系可靠度的趋势，新一代PBEE也将基于结构整体可靠度的地震风险作为主要的研究目标。基于上述原因，本文分别从基本变量模型和状态变量模型入手，发展了两类高效的整体可靠度方法——广义一次可靠度方法（first order reliability method, FORM）和改进高阶矩法（higher order moment method, HOMM）；然后以按我国规范设计的钢筋混凝土框架结构为研究对象，以结构整体抗震可靠度为主要研究内容，以结构整体抗震安全性评估为研究目的，分别针对承载能力、变形能力、地震损伤和连续倒塌四种整体极限状态方程，采用所提出的广义FORM和改进HOMM方法，对结构整体的抗震可靠性和鲁棒性进行了系统深入的研究。本文的主要研究内容如下：
1）将基于正态分布的Nataf变换拓展到基于Copula函数的广义Nataf变换，提出了基于广义Nataf变换的扩展一次可靠度方法（Extended FORM，EFORM），该方法可以有效地考虑随机变量之间的非线性相关性；在此基础上，提出了考虑参数不确定性的模糊一次可靠度方法（Fuzzy FORM，FFORM）；针对结构整体可靠度的基本变量模型，在MATLAB平台和OpenSees软件上实现了基于广义FORM（EFORM和FFORM）的结构整体可靠度分析。算例分析表明，本文提出的广义FORM法可以有效地考虑随机变量之间的非线性相关性和随机变量参数的不确定性。
2）基于广义Nataf变换，提出了一种考虑随机变量边缘分布和相关性信息的改进点估计法；将改进点估计法与最大熵方法相结合，提出了改进的高阶矩法（HOMM）；针对结构整体可靠度的状态变量模型，实现了基于改进HOMM的结构整体可靠度分析。算例分析表明，本文提出的改进HOMM法不仅具有与FORM相同的精度，而且计算效率比FORM高。
3）按照我国规范，在同一设防烈度下设计了三个不同高度的钢筋混凝土框架结构作为研究对象，并在OpenSees中对结构进行了有限元建模。通过与钢筋混凝土构件（柱）和结构整体的实验结果对比，验证了本文所建立OpenSees模型的合理性。对此三个结构进行了确定性的抗震性能分析，得到了其主要的抗震性态特征。
4）建立了钢筋混凝土框架结构的整体承载能力极限状态方程；将广义FORM和改进HOMM方法分别与确定性Pushover方法相结合，提出了基于FORM的设计点Pushover方法和基于改进点估计法的随机Pushover方法，对比分析表明，两种方法具有较好的一致性；将两种方法应用于钢筋混凝土框架结构整体承载能力极限状态的非线性静力抗震可靠度分析，得到了结构整体承载能力极限状态抗震可靠度指标及灵敏度指标的变化规律，获得了基于整体承载能力的静力地震易损性曲线。
5）建立了钢筋混凝土框架结构的整体变形能力极限状态方程；将广义FORM和改进HOMM方法分别与确定性能力谱法相结合，提出了基于FORM的设计点能力谱法和基于改进点估计法的随机能力谱法，对比分析表明，两种方法具有较好的一致性；将两种方法应用于钢筋混凝土框架结构整体变形能力极限状态的非线性静力抗震可靠度分析，得到了结构整体变形能力极限状态抗震可靠度指标及灵敏度指标的变化规律，获得了基于整体变形能力相应于不同破坏状态的静力地震易损性曲线。
6）建立了钢筋混凝土框架结构的整体地震损伤极限状态方程，提出了基于点估计法的结构动力反应概率密度演化方法；采用20条实际地震动和两种人工地震动，同时考虑地震动和结构的随机性，采用点估计法进行了结构整体动力反应分析及其参数灵敏度分析，得到了结构整体变形反应和整体损伤指标的概率密度演化曲线，采用改进HOMM方法、首次超越准则以及累积损伤准则，对基于整体损伤指标的地震损伤极限状态进行了非线性动力抗震可靠度分析，得到了结构整体地震损伤极限状态抗震可靠度指标的变化规律。
7）从“偶然事件”、“局部损伤”、“不成比例破坏”和“失效后果”四个方面给出了结构鲁棒性的新定义；建立了钢筋混凝土框架结构的整体连续倒塌极限状态方程；提出了基于构件抗震可靠度分析的结构最可能失效构件识别方法；将改进点估计法分别与确定性的静力Pushdown分析（PDA）和竖向增量动力分析（incremental dynamic analysis，IDA）方法相结合，提出了随机Pushdown分析方法和随机竖向IDA方法，采用此两种方法分别对完好结构和损伤结构的概率抗竖向连续倒塌能力及参数灵敏度进行了分析，采用全概率公式得到了考虑地震危险性的结构连续倒塌失效概率；在此基础上，利用基于条件可靠度和整体可靠度的鲁棒性指标，研究了结构整体抗震鲁棒性指标的变化规律，定量地揭示了结构的抗震鲁棒性与结构局部损伤之间的相互关系。
Performance-based earthquake engineering (PBEE) and performance-basedseismic design (PBSD), proposed by the Pacific Earthquake Engineering ResearchCenter (PEER), are a new-generation seismic design concept, method and technique.They have been warmly responded by the world-wide earthquake engineeringresearchers and engineers. Due to the strong randomness in occurring time, spaceand intensity of earthquakes, and also because of a lot of uncertainties in bothseismic demand and seismic capacity of civil engineering structures, it is necessaryto build a probabilistic framework of PBEE and PBSD based on structuralreliability theory. Nowadays, the probabilistic seismic performance evaluationbased on reliability has become one of the main research points of PBEE and PBSD.
Seismic reliability of structural systems is a quantitative measure ofprobabilistic seismic risk of civil infrastructures. Recently, a new trend of systemreliability analysis based on structural global limit states has emerged. Furthermore,the new-generation PBEE also takes the systematic seismic risk based on structuralglobal reliability as the main research objective. Based on the above considerations,in this dissertation, two efficient global reliability methods, i.e. the generalized firstorder reliability method (FORM) and the improved higher order moment method(HOMM), are developed from the viewpoint of the basic variable model and thestate variable model, respectively. And then, three reinforced concrete (RC) framestructures are designed according to Chinese codes, which are taken as the researchobjects of case studies in this dissertation. The seismic reliability of these RCframes is the main research contents of this dissertation, and the assessment ofseismic safety of these RC frames is the ultimate research goal of this thesis. Usingthe developed generalized FORM and improved HOMM, the global reliability androbustness of the case-study RC frames are systematically and deeply studied fromthe aspects of four global limit states, i.e., global load carrying capacity, globaldeformation capacity, global seismic damage and global progressive collapse,respectively. The main research contents of this dissertation are summarized asfollows:
1) The traditional Nataf transformation based on normal distributions isextended to the generalized Nataf transformation based on Copula functions. Thenan extended first order reliability method (EFORM) using the generalized Nataftransformation is developed to consider the nonlinear dependence among therandom variables. Furthermore, a fuzzy first order reliability method (FFORM)considering parameter uncertainties is put forward. For the basic variable model of structural global reliability, the generalized FORM (EFORM and FFORM) isrealized in the software MATLAB and the FEA platform OpenSees. It isdemonstrated through the numerical examples that the proposed generalized FORMcan effectively take into account the nonlinear dependence among the randomvariables and the epistemic uncertainty of parameters of random variables.
2) An improved point estimate method (IPEM), which can incorporate themarginal distributions and the correlation information of random variables, ispresented based on the generalized Nataf transformation. And then, an improvedhigher order moment method (IHOMM) is developed by combining the IPEM withthe maximum entropy principle. For the state variable model of structural globalreliability, the improved HOMM is realized in the software MATLAB and the FEAplatform OpenSees. It is demonstrated through the numerical examples that thedeveloped higher order moment method (IHOMM) has the same accuracy as FORM,while the former is more efficient than the latter.
3) Three RC frame buildings with different floors under the same seismicfortification intensity are designed according to the current Chinese codes. Thesestructures are modeled in the platform OpenSees. Through comparing with the testdata of RC elements (columns) and a global structure shaking table test in TsinghuaUniversity, the developed OpenSees models are verified and validated to beadequate to describe the nonlinear behavior of the case-study structures. Thedeterministic seismic performance analysis is carried out for the three structures,and their main seismic behaviors and properties are obtained.
4) The global load-carrying capacity limit state equation is established for theRC frame structures. Through the combination of the generalized FORM and theimproved HOMM with the deterministic Pushover analysis, a FORM-based design-point pushover method and a HOMM-based random pushover method aredeveloped, respectively. It is illustrated through the comparative study that the twomethods have a good agreement. The two approaches are then applied to thenonlinear static seismic reliability analysis of structural global load carryingcapacity limit state, and the variation rules of global seismic reliability indices andsensitivity factors are obtained. Furthermore, the static seismic fragility curves arederived for the global load carrying capacity.
5) The global deformation capacity limit state equation is set up for the RCframe structures. By way of the combination of the generalized FORM and theimproved HOMM with the deterministic capacity-spectrum method (CSM), theFORM-based design-point CSM and HOMM-based random CSM are presented,respectively. Comparative study shows that the two methods have a goodconsistency. The two methods are then applied to the nonlinear static seismicreliability analysis of structural global deformation capacity limit state, and the changing regularities of global seismic reliability indices and sensitivity factors areobtained. Furthermore, the static seismic fragility curves corresponding to differentdamage states are given for the global deformation capacity.
6) The global seismic damage limit state equation is established for the RCframe structures. A new probability density evolution approach for structuraldynamic responses based on the improved point estimation method (IPEM) isproposed. Using20real earthquake records and2artificial earthquake records asinputs, the global dynamic response and the global parameter sensitivity ofstructures are analyzed using the IPEM, and the probabilistic density evolutioncurves of structural global dynamic responses are derived. Employing the improvedHOMM, the first-passage criteria and the cumulative damage criteria, the nonlineardynamic seismic reliability analysis are implemented for global seismic damagelimit state of structures; and the change rules of global dynamic reliability indicesof structures are discovered.
7) A new definition of structural robustness is given from four aspects ofoccasional event, local damage, disproportional failure, and failure consequences. Aglobal progressive collapse limit state equation is established for the RC framestructures. A method for identifying the most probable failure members is proposedbased on local seismic reliability analysis of structural members. A randompushdown method and a random vertical IDA method are presented via the couplingof the IPEM with the deterministic pushdown analysis (PDA) and verticalincremental dynamic analysis (IDA). The two methods are used to analyze theprobabilistic progressive collapse resistant capacity and the parameter sensitivitiesof the intact structures and the damaged structures. With the total probabilitytheorem, the progressive collapse probability with the consideration of seismichazard is derived. On the basis of the above computations, the robustness indicesbased on the conditional reliability and the global reliability are evaluatedrespectively, and the changing rules of structural robustness are studied, in whichthe relationships between global robustness and local damage of structures arequantitatively revealed.
结构体系的抗震可靠度是对结构概率地震风险评定的定量衡量，近年来出现了利用结构整体极限状态来近似计算结构体系可靠度的趋势，新一代PBEE也将基于结构整体可靠度的地震风险作为主要的研究目标。基于上述原因，本文分别从基本变量模型和状态变量模型入手，发展了两类高效的整体可靠度方法——广义一次可靠度方法（first order reliability method, FORM）和改进高阶矩法（higher order moment method, HOMM）；然后以按我国规范设计的钢筋混凝土框架结构为研究对象，以结构整体抗震可靠度为主要研究内容，以结构整体抗震安全性评估为研究目的，分别针对承载能力、变形能力、地震损伤和连续倒塌四种整体极限状态方程，采用所提出的广义FORM和改进HOMM方法，对结构整体的抗震可靠性和鲁棒性进行了系统深入的研究。本文的主要研究内容如下：
1）将基于正态分布的Nataf变换拓展到基于Copula函数的广义Nataf变换，提出了基于广义Nataf变换的扩展一次可靠度方法（Extended FORM，EFORM），该方法可以有效地考虑随机变量之间的非线性相关性；在此基础上，提出了考虑参数不确定性的模糊一次可靠度方法（Fuzzy FORM，FFORM）；针对结构整体可靠度的基本变量模型，在MATLAB平台和OpenSees软件上实现了基于广义FORM（EFORM和FFORM）的结构整体可靠度分析。算例分析表明，本文提出的广义FORM法可以有效地考虑随机变量之间的非线性相关性和随机变量参数的不确定性。
2）基于广义Nataf变换，提出了一种考虑随机变量边缘分布和相关性信息的改进点估计法；将改进点估计法与最大熵方法相结合，提出了改进的高阶矩法（HOMM）；针对结构整体可靠度的状态变量模型，实现了基于改进HOMM的结构整体可靠度分析。算例分析表明，本文提出的改进HOMM法不仅具有与FORM相同的精度，而且计算效率比FORM高。
3）按照我国规范，在同一设防烈度下设计了三个不同高度的钢筋混凝土框架结构作为研究对象，并在OpenSees中对结构进行了有限元建模。通过与钢筋混凝土构件（柱）和结构整体的实验结果对比，验证了本文所建立OpenSees模型的合理性。对此三个结构进行了确定性的抗震性能分析，得到了其主要的抗震性态特征。
4）建立了钢筋混凝土框架结构的整体承载能力极限状态方程；将广义FORM和改进HOMM方法分别与确定性Pushover方法相结合，提出了基于FORM的设计点Pushover方法和基于改进点估计法的随机Pushover方法，对比分析表明，两种方法具有较好的一致性；将两种方法应用于钢筋混凝土框架结构整体承载能力极限状态的非线性静力抗震可靠度分析，得到了结构整体承载能力极限状态抗震可靠度指标及灵敏度指标的变化规律，获得了基于整体承载能力的静力地震易损性曲线。
5）建立了钢筋混凝土框架结构的整体变形能力极限状态方程；将广义FORM和改进HOMM方法分别与确定性能力谱法相结合，提出了基于FORM的设计点能力谱法和基于改进点估计法的随机能力谱法，对比分析表明，两种方法具有较好的一致性；将两种方法应用于钢筋混凝土框架结构整体变形能力极限状态的非线性静力抗震可靠度分析，得到了结构整体变形能力极限状态抗震可靠度指标及灵敏度指标的变化规律，获得了基于整体变形能力相应于不同破坏状态的静力地震易损性曲线。
6）建立了钢筋混凝土框架结构的整体地震损伤极限状态方程，提出了基于点估计法的结构动力反应概率密度演化方法；采用20条实际地震动和两种人工地震动，同时考虑地震动和结构的随机性，采用点估计法进行了结构整体动力反应分析及其参数灵敏度分析，得到了结构整体变形反应和整体损伤指标的概率密度演化曲线，采用改进HOMM方法、首次超越准则以及累积损伤准则，对基于整体损伤指标的地震损伤极限状态进行了非线性动力抗震可靠度分析，得到了结构整体地震损伤极限状态抗震可靠度指标的变化规律。
7）从“偶然事件”、“局部损伤”、“不成比例破坏”和“失效后果”四个方面给出了结构鲁棒性的新定义；建立了钢筋混凝土框架结构的整体连续倒塌极限状态方程；提出了基于构件抗震可靠度分析的结构最可能失效构件识别方法；将改进点估计法分别与确定性的静力Pushdown分析（PDA）和竖向增量动力分析（incremental dynamic analysis，IDA）方法相结合，提出了随机Pushdown分析方法和随机竖向IDA方法，采用此两种方法分别对完好结构和损伤结构的概率抗竖向连续倒塌能力及参数灵敏度进行了分析，采用全概率公式得到了考虑地震危险性的结构连续倒塌失效概率；在此基础上，利用基于条件可靠度和整体可靠度的鲁棒性指标，研究了结构整体抗震鲁棒性指标的变化规律，定量地揭示了结构的抗震鲁棒性与结构局部损伤之间的相互关系。
Performance-based earthquake engineering (PBEE) and performance-basedseismic design (PBSD), proposed by the Pacific Earthquake Engineering ResearchCenter (PEER), are a new-generation seismic design concept, method and technique.They have been warmly responded by the world-wide earthquake engineeringresearchers and engineers. Due to the strong randomness in occurring time, spaceand intensity of earthquakes, and also because of a lot of uncertainties in bothseismic demand and seismic capacity of civil engineering structures, it is necessaryto build a probabilistic framework of PBEE and PBSD based on structuralreliability theory. Nowadays, the probabilistic seismic performance evaluationbased on reliability has become one of the main research points of PBEE and PBSD.
Seismic reliability of structural systems is a quantitative measure ofprobabilistic seismic risk of civil infrastructures. Recently, a new trend of systemreliability analysis based on structural global limit states has emerged. Furthermore,the new-generation PBEE also takes the systematic seismic risk based on structuralglobal reliability as the main research objective. Based on the above considerations,in this dissertation, two efficient global reliability methods, i.e. the generalized firstorder reliability method (FORM) and the improved higher order moment method(HOMM), are developed from the viewpoint of the basic variable model and thestate variable model, respectively. And then, three reinforced concrete (RC) framestructures are designed according to Chinese codes, which are taken as the researchobjects of case studies in this dissertation. The seismic reliability of these RCframes is the main research contents of this dissertation, and the assessment ofseismic safety of these RC frames is the ultimate research goal of this thesis. Usingthe developed generalized FORM and improved HOMM, the global reliability androbustness of the case-study RC frames are systematically and deeply studied fromthe aspects of four global limit states, i.e., global load carrying capacity, globaldeformation capacity, global seismic damage and global progressive collapse,respectively. The main research contents of this dissertation are summarized asfollows:
1) The traditional Nataf transformation based on normal distributions isextended to the generalized Nataf transformation based on Copula functions. Thenan extended first order reliability method (EFORM) using the generalized Nataftransformation is developed to consider the nonlinear dependence among therandom variables. Furthermore, a fuzzy first order reliability method (FFORM)considering parameter uncertainties is put forward. For the basic variable model of structural global reliability, the generalized FORM (EFORM and FFORM) isrealized in the software MATLAB and the FEA platform OpenSees. It isdemonstrated through the numerical examples that the proposed generalized FORMcan effectively take into account the nonlinear dependence among the randomvariables and the epistemic uncertainty of parameters of random variables.
2) An improved point estimate method (IPEM), which can incorporate themarginal distributions and the correlation information of random variables, ispresented based on the generalized Nataf transformation. And then, an improvedhigher order moment method (IHOMM) is developed by combining the IPEM withthe maximum entropy principle. For the state variable model of structural globalreliability, the improved HOMM is realized in the software MATLAB and the FEAplatform OpenSees. It is demonstrated through the numerical examples that thedeveloped higher order moment method (IHOMM) has the same accuracy as FORM,while the former is more efficient than the latter.
3) Three RC frame buildings with different floors under the same seismicfortification intensity are designed according to the current Chinese codes. Thesestructures are modeled in the platform OpenSees. Through comparing with the testdata of RC elements (columns) and a global structure shaking table test in TsinghuaUniversity, the developed OpenSees models are verified and validated to beadequate to describe the nonlinear behavior of the case-study structures. Thedeterministic seismic performance analysis is carried out for the three structures,and their main seismic behaviors and properties are obtained.
4) The global load-carrying capacity limit state equation is established for theRC frame structures. Through the combination of the generalized FORM and theimproved HOMM with the deterministic Pushover analysis, a FORM-based design-point pushover method and a HOMM-based random pushover method aredeveloped, respectively. It is illustrated through the comparative study that the twomethods have a good agreement. The two approaches are then applied to thenonlinear static seismic reliability analysis of structural global load carryingcapacity limit state, and the variation rules of global seismic reliability indices andsensitivity factors are obtained. Furthermore, the static seismic fragility curves arederived for the global load carrying capacity.
5) The global deformation capacity limit state equation is set up for the RCframe structures. By way of the combination of the generalized FORM and theimproved HOMM with the deterministic capacity-spectrum method (CSM), theFORM-based design-point CSM and HOMM-based random CSM are presented,respectively. Comparative study shows that the two methods have a goodconsistency. The two methods are then applied to the nonlinear static seismicreliability analysis of structural global deformation capacity limit state, and the changing regularities of global seismic reliability indices and sensitivity factors areobtained. Furthermore, the static seismic fragility curves corresponding to differentdamage states are given for the global deformation capacity.
6) The global seismic damage limit state equation is established for the RCframe structures. A new probability density evolution approach for structuraldynamic responses based on the improved point estimation method (IPEM) isproposed. Using20real earthquake records and2artificial earthquake records asinputs, the global dynamic response and the global parameter sensitivity ofstructures are analyzed using the IPEM, and the probabilistic density evolutioncurves of structural global dynamic responses are derived. Employing the improvedHOMM, the first-passage criteria and the cumulative damage criteria, the nonlineardynamic seismic reliability analysis are implemented for global seismic damagelimit state of structures; and the change rules of global dynamic reliability indicesof structures are discovered.
7) A new definition of structural robustness is given from four aspects ofoccasional event, local damage, disproportional failure, and failure consequences. Aglobal progressive collapse limit state equation is established for the RC framestructures. A method for identifying the most probable failure members is proposedbased on local seismic reliability analysis of structural members. A randompushdown method and a random vertical IDA method are presented via the couplingof the IPEM with the deterministic pushdown analysis (PDA) and verticalincremental dynamic analysis (IDA). The two methods are used to analyze theprobabilistic progressive collapse resistant capacity and the parameter sensitivitiesof the intact structures and the damaged structures. With the total probabilitytheorem, the progressive collapse probability with the consideration of seismichazard is derived. On the basis of the above computations, the robustness indicesbased on the conditional reliability and the global reliability are evaluatedrespectively, and the changing rules of structural robustness are studied, in whichthe relationships between global robustness and local damage of structures arequantitatively revealed.
引文
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