随机地震作用下结构动力可靠度计算方法研究
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摘要
目前结构抗震设计的发展趋势是基于性能的抗震设计,基于性能的抗震设计方法通常需要评价给定的设计是否满足设定目标的要求,这其中包括以设定的可靠度目标来评价结构的性能。
地震发生的时间、空间和强度都具有明显的随机性,地震动特性也具有明显的随机性,即地震动的随机过程性,因此结构反应也具有随机性,应用随机振动理论,把结构性能设计建立在可靠度理论基础上是必要的。
动力可靠度分析中,广泛应用的基于跨越过程理论的方法需要得到联合概率密度函数的解析解,并对跨越事件的性质进行假定,这些限制使得基于跨越过程理论的结构动力可靠度分析精度很难得到保证,且适用范围有限；而Monte-Carlo法是计算可靠度的一般性方法,适用于各种情况,但对于工程中感兴趣的失效概率较小的问题,该法需要大量的样本来进行结构分析。
因此,本文主要研究当地震随机过程样本表示为高维随机变量时,以结构反应时程分析和Monte-Carlo法思想为基础的动力可靠度数值模拟方法；目的在于如何减少所需输入样本数,但同时不影响计算精度,解决小失效概率计算效率低下的问题；针对线性结构、非线性结构和随机结构分别展开研究。
另外,针对新型结构-密肋复合墙结构,本文首先对密肋复合墙体进行了基于首超破坏准则的动力可靠度分析,随后以密肋复合墙体为研究对象提出了基于累积损伤破坏准则的动力可靠度高效计算方法,最后提出了密肋复合墙结构基于动力可靠度计算的设计方法,所提方法适用于各种结构,对促进随机振动理论的工程应用有一定的意义。
本文主要完成的工作有：
1.线性结构高维小失效概率动力可靠度计算方法研究
对比分析了高维随机变量条件下,线性结构小失效概率的计算方法：区域分解法和重要抽样法,提出了概率简单叠加法则法,其中重要抽样法包括单元失效域计算法,直接方差放大系数法,功率谱法。
(1)对比分析了线性结构动力可靠度计算的重要抽样法-单元失效域计算法和区域分解法,给出了受非平稳随机地震作用的区域分解法的计算步骤,结果表明计算小失效概率时,两种方法计算效率同样高效。
(2)对比分析了线性结构动力可靠度计算的三种重要抽样法：单元失效域计算法,方差放大系数法及功率谱法,其中提出了依据权重函数随机选取失效域下标用来模拟属于此失效域的随机样本的方法,结果表明三种方法对比Monte-Carlo计算效率均有显著提高,而基于计算单元失效域的重要抽样法计算效率最高。
(3)提出了线性结构高维小失效概率计算的概率简单叠加法则法：首先引入构成总失效域的由持时内的时间点划分出的单元失效域,其次引入构成总失效域的互斥的单元失效域,最后根据两种单元失效域的关系和概率简单叠加法则计算总失效概率,结果表明概率简单叠加法则法是计算小失效概率的高效算法。
2.非线性结构高维小失效概动力可靠度计算方法研究
研究了将随机过程样本表示为高维随机变量条件下的非线性结构动力可靠度计算方法：子集模拟法和修正的子集模拟法(基于分裂路径的子集模拟法、混合子集模拟法、序列重要抽样法、基于重要抽样法的子集模拟法),提出了修正的条件反应法和基于超越概率等价的非线性体系小失效概率计算新方法。
(1)给出了非线性结构高维小失效概率计算的子集模拟法的计算步骤,确定了计算条件失效概率时所需随机变量样本的具体数目,计算了受平稳高斯白噪声作用的单自由度Duffing体系和采用三线型恢复力模型的三自由度结构的动力可靠度,其中通过迭代的数值逐步积分法计算Duffing非线性体系的反应。
(2)提出了修正的条件反应法,该法将基于反应时程分解的子集模拟法代替子集模拟法应用到条件反应法中求解失效概率,可通过线性反应来估计非线性反应,或通过单自由度结构的线性反应来估计多自由度结构的线性反应,计算效率较基于反应时程分解的子集模拟法有了进一步提高,是计算小失效概率的高效方法。
(3)提出了基于超越概率等价的计算Duffing弱非线性体系小失效概率的计算方法,该法利用基于平均超越概率等价线性化法,结合基于计算单元失效域的重要抽样法,来进行动力可靠度计算。
3.随机结构动力可靠度计算方法研究
对于随机结构,对比分析了动力可靠度计算的极值分布-泰勒展开法、渐进展开法和重要抽样法,提出了基于功率谱计算的重要抽样法。
(1)对比分析了随机结构动力可靠度计算的极值分布-泰勒展开法,渐进展开法和重要抽样法。
(2)提出了基于功率谱计算的重要抽样法,该法利用将结构反应功率谱用于重要抽样法的思想,根据激励样本计算结构时程反应,并将其变换为频率和功率谱密度的曲线,应用离散的反应功率谱密度值增大幅值的均值和方差,或者对于平稳随机反应功率谱,可利用反应功率谱的相似形——频域内反应的绝对值平方曲线的期望值增大幅值的方差,结果表明该法是高效求解线性随机结构动力可靠度的有效方法。
4.密肋复合墙结构的动力可靠度分析
(1)首次计算了受非平稳随机地震作用的密肋复合墙体的基于首次超越破坏准则的动力可靠度。根据均匀调制非平稳地震动模型得出了结构随机反应动力可靠度的具体计算公式,并与同样采用有限元实体模型的同模型尺寸的框架及剪力墙进行了对比分析,结果表明密肋复合墙体的动力可靠度介于框架及剪力墙之间。
(2)提出了密肋复合墙体基于累积损伤破坏准则的动力可靠度计算方法。首先根据密肋复合墙体试验研究得出墙体的刚度、开裂和屈服荷载、屈服和极限位移、及损伤因子的表达式；随后生成随机地震动过程样本；对层间光滑型滞变恢复力模型,利用等价线性化法,并通过对状态方程的求解,得到反应的统计量；针对密肋复合墙体的损伤模型,给出了基于累积损伤破坏准则的动力可靠度的计算公式；分别计算了墙体在多遇地震和罕遇地震下,砌块开裂和墙体达极限状态两种情况下的动力可靠度；所提方法的计算结果与Monte-Carlo法计算结果吻合较好,计算效率较高。
(3)提出了一种可用于非线性密肋复合墙结构的基于动力可靠度计算的设计方法。首先,根据三角级数模型生成平稳高斯随机地震过程,由时间包络函数可得非平稳随机地震过程,其中功率谱密度函数采用Clough-Penzien模型,其参数采用与《建筑抗震设计规范》对应的地震动随机模型参数；其次,根据密肋复合墙结构的力学性能试验,建立密肋复合墙体的恢复力模型并确定其参数；随后对密肋复合墙结构进行动力分析,应用子集模拟法对其进行动力可靠度计算,并给出基于动力可靠度计算设计方法的执行框图；最后,应用所提方法对6层密肋复合墙结构进行了计算,结果表明密肋复合墙结构基于动力可靠度计算的设计方法是可行的。
The current trend of structural seismic design is toward performance-based design, which requires assessing whether a given design satisfies specified engineering performance objectives. Given reliability objectives can be used as a way to assess structural performance.
The time, locations and intensity of earthquakes and the characteristics of earthquake ground motions are random. The safety analysis of structures under earthquake loads is a typical example in random vibration since the excitation, namely, ground motion induced by an earthquake, is random in general. Taking account of reliability theory in structural performance-based seismic design is essential.
Pioneered by Rice, early work on the first excursion problem was focused on out-crossing theory to give an analytical approximation. For the out-crossing theory, analytical expressions of joint probability density functions of structural responses have to be derived, and the out-crossing characteristics of events need to be assumed. These limitations make the analytical solutions from out-crossing theory approximate and applicable only for limited cases. Monte-Carlo simulations offer a robust methodology well suited to solving such high-dimensional reliability problems. The efficiency and accuracy of Monte-Carlo simulations do not depend on the geometry of the failure domain or the number of random parameters involved. Instead, but only depend on the failure probability and the number of simulation samples. For many practical applications where one deals with small failure probabilities, the computational effort required by Monte-Carlo simulations is prohibitively high.
Simulation methods offer a feasible alternative for the numerical solution of first-excursion problems with larger state-space dimensions. The aim of the thesis is to improve the efficiency, namely decrease the number of input samples, for small failure probability calculations in high dimensions of linear and nonlinear structures, and at the same time the accuracy of simulation methods should not be decreased. The basic idea of simulation methods in this thesis is based on structural response history calculations and Monte-Carlo method.
As a new structural system, multi-grid composite wall structure has three lines of defence against earthquakes. In this thesis, dynamic reliability analysis for multi-grid composite walls based on first-excursion law is made. And a method for dynamic reliability estimations based on damage accumulation of multi-grid composite walls is proposed. Finally a design method based on dynamic reliability estimations for multi-grid composite wall structure is proposed, which is meaningful for the practical applications of random vibration theory. The proposed methods are applicable to other structures with proper modification.
The main work of the thesis includes four parts as follows:
1. Research on dynamic reliability methods of linear structures especially for small failure probabilities in high dimensions
Methods, which include domain decomposition method, importance sampling method consisting of elementary failure regions method, amplificatory factor of variance method and power spectrum method, and simple additive rules of probability, for small failure probability calculations in high dimensions of linear structures are compared.
(1) Importance sampling method with elementary failure regions calculations and domain decomposition method for dynamic reliability estimations of linear structures are compared. Detail procedures of domain decomposition method under nonstationary random earthquakes are given. Results show that for small failure probability calculations, the efficiency of the two methods is almost the same.
(2) Three importance sampling methods including elementary failure regions method, amplificatory factor of variance method and power spectrum method, for dynamic reliability calculation of linear structures are compared. The method of generating samples belonging to given failure domains is proposed according to failure domain subscripts chosen through weight functions. The results show that the calculation efficiency of each of the three approaches is much higher than Monte-Carlo method, and the elementary failure regions method is the most efficient.
(3) Method of simple additive rules of probability for small failure probability calculations in high dimensions of linear structures is proposed. Firstly elementary failure regions corresponding to the failure of a particular output response at a particular instant are proposed. Secondly mutual exclusive sets are proposed. Finally according to the relation between elementary failure regions and mutual exclusive sets, and simple additive rules of probability, the failure probability can be expressed.
2. Research on dynamic reliability methods of nonlinear structures especially for small failure probabilities in high dimensions
When the input random process sample is expressed as random variables in high dimensions, dynamic reliability methods for nonlinear structures are studied, which include subset simulation method, modified subset simulation method consisting of subset simulation method with splitting, hybrid subset simulation method, sequential importance sampling method, and subset simulation method based on importance sampling method, modified response conditioning method, and a new small failure probability calculation method for Duffing nonlinear system.
(1) The calculation procedures of subset simulation method for small failure probability calculations in high dimensions of nonlinear structures are proposed. The number of samples for calculating conditional probabilities is determined. The efficiency of subset simulation method is demonstrated by calculating the first excursion probabilities of two examples. One is a Duffing system with single degree-of-freedom, and the other is a three degree-of-freedom structure with three lines restoring model. And the two structures are both subjected to stationary white noise excitation. Iterative numerical integration is used for calculations of Duffing nonlinear system.
(2) A modified response conditioning method is proposed by combining subset simulation with splitting and response conditioning method, where linear response can be used to estimate nonlinear response, or response of single degree-of-freedom structure can be used to estimate response of multi-degree-of-freedom structure. Failure probability results of modified response conditioning method are compared with those of subset simulation with splitting. The proposed method is efficient for small failure probability calculations.
(3) Considering the Duffing oscillator subjected to random noise for investigation, a new method for dynamic reliability calculation of non-linear systems is proposed. The new method adopts a linearization principle named mean upcrossing rate linearization and an importance sampling procedure based on calculation of elementary failure regions.
3. Research on dynamic reliability methods of structures with uncertain structural parameters
For dynamic reliability calculations of stochastic structures, three methods including asymptotic expansions method, importance sampling method, and a method based on extreme value distribution and Taylor series expansion are compared, and importance sampling method based on response power spectrum is proposed.
(1) Three methods including asymptotic expansions method, importance sampling method, and a method based on extreme value distribution and Taylor series expansion are compared.
(2) Importance sampling method based on response spectrum for structural dynamic reliability estimation under random excitation is proposed, In order to improve the calculation efficiency of dynamic reliability estimations, increasing the variances of input random excitation amplitudes using power spectrum with importance sampling method is proposed. According to random vibration theory, the power spectral density of stationary random response is similar to expectation on square of absolute value of response in frequency domain, which can be easily calculated by Fourier transform and can also be used to increase the variances of input random excitation amplitudes.
4. Dynamic reliability analysis of multi-grid composite wall structure
(1) First-excursion dynamic reliability analysis of multi-grid composite wall subjected to non-stationary random earthquake excitations is done. The detailed reliability formulation based on time-envelope non-stationary stochastic ground motion is proposed. With the same size, multi-grid composite wall, frame and shear wall, idealized as solid models in finite element modeling are considered for reliability analysis under frequent earthquakes and rare earthquakes. Results show that the dynamic reliability of multi-grid composite wall is between the reliability of frame and the reliability of shear wall.
(2) Based on damage accumulation of multi-grid composite walls, a method of dynamic reliability estimations is proposed. The equations including stiffness, shear forces at filling blocks cracking and multi-grid composite walls yielding, ultimate displacement, and damage index are obtained through tests of multi-grid composite walls. Employing these equations in reliability calculations, procedures of dynamic reliability estimations based on damage accumulation of multi-grid composite walls subjected to random earthquake excitations are proposed. Finally the proposed method is applied to the standard tested composite wall subjected to random earthquake excitations which can be specified by a finite number of input random variables. The dynamic reliability estimates, when filling blocks crack under small earthquakes and when the composite wall reach limit state under large earthquakes, are obtained using the proposed method by taking damage indexes as thresholds. The results from the proposed method which show good agreement with those from Monte-Carlo simulations demonstrate the proposed method is effective.
(3) A design method of non-linear multi-grid composite wall structure based on dynamic reliability computation is proposed. Gaussian stationary random earthquake processes are generated through trigonal series models firstly. Nonstationary random earthquake processes can be obtained by multiplying time envelop function with stationary processes. Clough-Penzien model is selected for power spectral density function. The random earthquake model parameters are determined corresponding to
地震发生的时间、空间和强度都具有明显的随机性,地震动特性也具有明显的随机性,即地震动的随机过程性,因此结构反应也具有随机性,应用随机振动理论,把结构性能设计建立在可靠度理论基础上是必要的。
动力可靠度分析中,广泛应用的基于跨越过程理论的方法需要得到联合概率密度函数的解析解,并对跨越事件的性质进行假定,这些限制使得基于跨越过程理论的结构动力可靠度分析精度很难得到保证,且适用范围有限；而Monte-Carlo法是计算可靠度的一般性方法,适用于各种情况,但对于工程中感兴趣的失效概率较小的问题,该法需要大量的样本来进行结构分析。
因此,本文主要研究当地震随机过程样本表示为高维随机变量时,以结构反应时程分析和Monte-Carlo法思想为基础的动力可靠度数值模拟方法；目的在于如何减少所需输入样本数,但同时不影响计算精度,解决小失效概率计算效率低下的问题；针对线性结构、非线性结构和随机结构分别展开研究。
另外,针对新型结构-密肋复合墙结构,本文首先对密肋复合墙体进行了基于首超破坏准则的动力可靠度分析,随后以密肋复合墙体为研究对象提出了基于累积损伤破坏准则的动力可靠度高效计算方法,最后提出了密肋复合墙结构基于动力可靠度计算的设计方法,所提方法适用于各种结构,对促进随机振动理论的工程应用有一定的意义。
本文主要完成的工作有：
1.线性结构高维小失效概率动力可靠度计算方法研究
对比分析了高维随机变量条件下,线性结构小失效概率的计算方法：区域分解法和重要抽样法,提出了概率简单叠加法则法,其中重要抽样法包括单元失效域计算法,直接方差放大系数法,功率谱法。
(1)对比分析了线性结构动力可靠度计算的重要抽样法-单元失效域计算法和区域分解法,给出了受非平稳随机地震作用的区域分解法的计算步骤,结果表明计算小失效概率时,两种方法计算效率同样高效。
(2)对比分析了线性结构动力可靠度计算的三种重要抽样法：单元失效域计算法,方差放大系数法及功率谱法,其中提出了依据权重函数随机选取失效域下标用来模拟属于此失效域的随机样本的方法,结果表明三种方法对比Monte-Carlo计算效率均有显著提高,而基于计算单元失效域的重要抽样法计算效率最高。
(3)提出了线性结构高维小失效概率计算的概率简单叠加法则法：首先引入构成总失效域的由持时内的时间点划分出的单元失效域,其次引入构成总失效域的互斥的单元失效域,最后根据两种单元失效域的关系和概率简单叠加法则计算总失效概率,结果表明概率简单叠加法则法是计算小失效概率的高效算法。
2.非线性结构高维小失效概动力可靠度计算方法研究
研究了将随机过程样本表示为高维随机变量条件下的非线性结构动力可靠度计算方法：子集模拟法和修正的子集模拟法(基于分裂路径的子集模拟法、混合子集模拟法、序列重要抽样法、基于重要抽样法的子集模拟法),提出了修正的条件反应法和基于超越概率等价的非线性体系小失效概率计算新方法。
(1)给出了非线性结构高维小失效概率计算的子集模拟法的计算步骤,确定了计算条件失效概率时所需随机变量样本的具体数目,计算了受平稳高斯白噪声作用的单自由度Duffing体系和采用三线型恢复力模型的三自由度结构的动力可靠度,其中通过迭代的数值逐步积分法计算Duffing非线性体系的反应。
(2)提出了修正的条件反应法,该法将基于反应时程分解的子集模拟法代替子集模拟法应用到条件反应法中求解失效概率,可通过线性反应来估计非线性反应,或通过单自由度结构的线性反应来估计多自由度结构的线性反应,计算效率较基于反应时程分解的子集模拟法有了进一步提高,是计算小失效概率的高效方法。
(3)提出了基于超越概率等价的计算Duffing弱非线性体系小失效概率的计算方法,该法利用基于平均超越概率等价线性化法,结合基于计算单元失效域的重要抽样法,来进行动力可靠度计算。
3.随机结构动力可靠度计算方法研究
对于随机结构,对比分析了动力可靠度计算的极值分布-泰勒展开法、渐进展开法和重要抽样法,提出了基于功率谱计算的重要抽样法。
(1)对比分析了随机结构动力可靠度计算的极值分布-泰勒展开法,渐进展开法和重要抽样法。
(2)提出了基于功率谱计算的重要抽样法,该法利用将结构反应功率谱用于重要抽样法的思想,根据激励样本计算结构时程反应,并将其变换为频率和功率谱密度的曲线,应用离散的反应功率谱密度值增大幅值的均值和方差,或者对于平稳随机反应功率谱,可利用反应功率谱的相似形——频域内反应的绝对值平方曲线的期望值增大幅值的方差,结果表明该法是高效求解线性随机结构动力可靠度的有效方法。
4.密肋复合墙结构的动力可靠度分析
(1)首次计算了受非平稳随机地震作用的密肋复合墙体的基于首次超越破坏准则的动力可靠度。根据均匀调制非平稳地震动模型得出了结构随机反应动力可靠度的具体计算公式,并与同样采用有限元实体模型的同模型尺寸的框架及剪力墙进行了对比分析,结果表明密肋复合墙体的动力可靠度介于框架及剪力墙之间。
(2)提出了密肋复合墙体基于累积损伤破坏准则的动力可靠度计算方法。首先根据密肋复合墙体试验研究得出墙体的刚度、开裂和屈服荷载、屈服和极限位移、及损伤因子的表达式；随后生成随机地震动过程样本；对层间光滑型滞变恢复力模型,利用等价线性化法,并通过对状态方程的求解,得到反应的统计量；针对密肋复合墙体的损伤模型,给出了基于累积损伤破坏准则的动力可靠度的计算公式；分别计算了墙体在多遇地震和罕遇地震下,砌块开裂和墙体达极限状态两种情况下的动力可靠度；所提方法的计算结果与Monte-Carlo法计算结果吻合较好,计算效率较高。
(3)提出了一种可用于非线性密肋复合墙结构的基于动力可靠度计算的设计方法。首先,根据三角级数模型生成平稳高斯随机地震过程,由时间包络函数可得非平稳随机地震过程,其中功率谱密度函数采用Clough-Penzien模型,其参数采用与《建筑抗震设计规范》对应的地震动随机模型参数；其次,根据密肋复合墙结构的力学性能试验,建立密肋复合墙体的恢复力模型并确定其参数；随后对密肋复合墙结构进行动力分析,应用子集模拟法对其进行动力可靠度计算,并给出基于动力可靠度计算设计方法的执行框图；最后,应用所提方法对6层密肋复合墙结构进行了计算,结果表明密肋复合墙结构基于动力可靠度计算的设计方法是可行的。
The current trend of structural seismic design is toward performance-based design, which requires assessing whether a given design satisfies specified engineering performance objectives. Given reliability objectives can be used as a way to assess structural performance.
The time, locations and intensity of earthquakes and the characteristics of earthquake ground motions are random. The safety analysis of structures under earthquake loads is a typical example in random vibration since the excitation, namely, ground motion induced by an earthquake, is random in general. Taking account of reliability theory in structural performance-based seismic design is essential.
Pioneered by Rice, early work on the first excursion problem was focused on out-crossing theory to give an analytical approximation. For the out-crossing theory, analytical expressions of joint probability density functions of structural responses have to be derived, and the out-crossing characteristics of events need to be assumed. These limitations make the analytical solutions from out-crossing theory approximate and applicable only for limited cases. Monte-Carlo simulations offer a robust methodology well suited to solving such high-dimensional reliability problems. The efficiency and accuracy of Monte-Carlo simulations do not depend on the geometry of the failure domain or the number of random parameters involved. Instead, but only depend on the failure probability and the number of simulation samples. For many practical applications where one deals with small failure probabilities, the computational effort required by Monte-Carlo simulations is prohibitively high.
Simulation methods offer a feasible alternative for the numerical solution of first-excursion problems with larger state-space dimensions. The aim of the thesis is to improve the efficiency, namely decrease the number of input samples, for small failure probability calculations in high dimensions of linear and nonlinear structures, and at the same time the accuracy of simulation methods should not be decreased. The basic idea of simulation methods in this thesis is based on structural response history calculations and Monte-Carlo method.
As a new structural system, multi-grid composite wall structure has three lines of defence against earthquakes. In this thesis, dynamic reliability analysis for multi-grid composite walls based on first-excursion law is made. And a method for dynamic reliability estimations based on damage accumulation of multi-grid composite walls is proposed. Finally a design method based on dynamic reliability estimations for multi-grid composite wall structure is proposed, which is meaningful for the practical applications of random vibration theory. The proposed methods are applicable to other structures with proper modification.
The main work of the thesis includes four parts as follows:
1. Research on dynamic reliability methods of linear structures especially for small failure probabilities in high dimensions
Methods, which include domain decomposition method, importance sampling method consisting of elementary failure regions method, amplificatory factor of variance method and power spectrum method, and simple additive rules of probability, for small failure probability calculations in high dimensions of linear structures are compared.
(1) Importance sampling method with elementary failure regions calculations and domain decomposition method for dynamic reliability estimations of linear structures are compared. Detail procedures of domain decomposition method under nonstationary random earthquakes are given. Results show that for small failure probability calculations, the efficiency of the two methods is almost the same.
(2) Three importance sampling methods including elementary failure regions method, amplificatory factor of variance method and power spectrum method, for dynamic reliability calculation of linear structures are compared. The method of generating samples belonging to given failure domains is proposed according to failure domain subscripts chosen through weight functions. The results show that the calculation efficiency of each of the three approaches is much higher than Monte-Carlo method, and the elementary failure regions method is the most efficient.
(3) Method of simple additive rules of probability for small failure probability calculations in high dimensions of linear structures is proposed. Firstly elementary failure regions corresponding to the failure of a particular output response at a particular instant are proposed. Secondly mutual exclusive sets are proposed. Finally according to the relation between elementary failure regions and mutual exclusive sets, and simple additive rules of probability, the failure probability can be expressed.
2. Research on dynamic reliability methods of nonlinear structures especially for small failure probabilities in high dimensions
When the input random process sample is expressed as random variables in high dimensions, dynamic reliability methods for nonlinear structures are studied, which include subset simulation method, modified subset simulation method consisting of subset simulation method with splitting, hybrid subset simulation method, sequential importance sampling method, and subset simulation method based on importance sampling method, modified response conditioning method, and a new small failure probability calculation method for Duffing nonlinear system.
(1) The calculation procedures of subset simulation method for small failure probability calculations in high dimensions of nonlinear structures are proposed. The number of samples for calculating conditional probabilities is determined. The efficiency of subset simulation method is demonstrated by calculating the first excursion probabilities of two examples. One is a Duffing system with single degree-of-freedom, and the other is a three degree-of-freedom structure with three lines restoring model. And the two structures are both subjected to stationary white noise excitation. Iterative numerical integration is used for calculations of Duffing nonlinear system.
(2) A modified response conditioning method is proposed by combining subset simulation with splitting and response conditioning method, where linear response can be used to estimate nonlinear response, or response of single degree-of-freedom structure can be used to estimate response of multi-degree-of-freedom structure. Failure probability results of modified response conditioning method are compared with those of subset simulation with splitting. The proposed method is efficient for small failure probability calculations.
(3) Considering the Duffing oscillator subjected to random noise for investigation, a new method for dynamic reliability calculation of non-linear systems is proposed. The new method adopts a linearization principle named mean upcrossing rate linearization and an importance sampling procedure based on calculation of elementary failure regions.
3. Research on dynamic reliability methods of structures with uncertain structural parameters
For dynamic reliability calculations of stochastic structures, three methods including asymptotic expansions method, importance sampling method, and a method based on extreme value distribution and Taylor series expansion are compared, and importance sampling method based on response power spectrum is proposed.
(1) Three methods including asymptotic expansions method, importance sampling method, and a method based on extreme value distribution and Taylor series expansion are compared.
(2) Importance sampling method based on response spectrum for structural dynamic reliability estimation under random excitation is proposed, In order to improve the calculation efficiency of dynamic reliability estimations, increasing the variances of input random excitation amplitudes using power spectrum with importance sampling method is proposed. According to random vibration theory, the power spectral density of stationary random response is similar to expectation on square of absolute value of response in frequency domain, which can be easily calculated by Fourier transform and can also be used to increase the variances of input random excitation amplitudes.
4. Dynamic reliability analysis of multi-grid composite wall structure
(1) First-excursion dynamic reliability analysis of multi-grid composite wall subjected to non-stationary random earthquake excitations is done. The detailed reliability formulation based on time-envelope non-stationary stochastic ground motion is proposed. With the same size, multi-grid composite wall, frame and shear wall, idealized as solid models in finite element modeling are considered for reliability analysis under frequent earthquakes and rare earthquakes. Results show that the dynamic reliability of multi-grid composite wall is between the reliability of frame and the reliability of shear wall.
(2) Based on damage accumulation of multi-grid composite walls, a method of dynamic reliability estimations is proposed. The equations including stiffness, shear forces at filling blocks cracking and multi-grid composite walls yielding, ultimate displacement, and damage index are obtained through tests of multi-grid composite walls. Employing these equations in reliability calculations, procedures of dynamic reliability estimations based on damage accumulation of multi-grid composite walls subjected to random earthquake excitations are proposed. Finally the proposed method is applied to the standard tested composite wall subjected to random earthquake excitations which can be specified by a finite number of input random variables. The dynamic reliability estimates, when filling blocks crack under small earthquakes and when the composite wall reach limit state under large earthquakes, are obtained using the proposed method by taking damage indexes as thresholds. The results from the proposed method which show good agreement with those from Monte-Carlo simulations demonstrate the proposed method is effective.
(3) A design method of non-linear multi-grid composite wall structure based on dynamic reliability computation is proposed. Gaussian stationary random earthquake processes are generated through trigonal series models firstly. Nonstationary random earthquake processes can be obtained by multiplying time envelop function with stationary processes. Clough-Penzien model is selected for power spectral density function. The random earthquake model parameters are determined corresponding to
. And then according to the mechanical performance tests of multi-grid composite walls, restoring force model is proposed and parameters of the model are determined. Dynamic analysis of multi-grid composite wall structure is carried out and subset simulation method is adopted for dynamic reliability computation. Subset simulation method is an efficient numerical simulation method for small failure probability computation in high dimensions of non-linear structures. The procedures of design method based on dynamic reliability computation are presented. Finally, a six-storey multi-grid composite wall structure is used as an example to demonstrate the proposed method is effective.
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