Research on Dynamic Reliability Estimation Methods of Structures Subjected to Random Earthquake Excitations
|School||Beijing Jiaotong University|
|Keywords||dynamic reliability random variables in high dimensions small failure probability Monte-Carlo method random earthquake process multi-grid composite wall structure first excursion damage accumulation|
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 dimensionsMethods, 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 dimensionsWhen 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 parametersFor 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 <code for seismic design of buildings>. 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.