Benchmark Solutions For Random Vibration Responses Of Thin Plates And Dynamic Reliability Analysis Of Nonlinear Structures

There exist uncertainties in nature widespreadly.The uncertainties of loads acting on the engineering structures in service are often modeled as a stochastic process.The accurate estimation of random vibration responses plays an important role for assessing the dynamic reliability of structures.There is an accurate and efficient method named pseudo-excitation method(PEM)for obtaining random vibration responses of linear multiple-degree-of-freedom(MDOF)system.For a general linear or nonlinear system,the probability density evolution method(PDEM)is powerful for evaluating stochastic responses and assessing dynamic reliability of structures,while the precision of numerical solution needs to be further improved.In fact,the engineering structure is a continuous system with infinite freedom of degree,and its mass,stiffness and damping are continuously distributed in the whole structures.Although the discretized models may be used approximately for the continuous systems,the continuous model can describe truly and faithfully the mechanical behavior.Therefore,the study on benchmark solutions based on the continuous model of structures is of essential significance for development of numerical methods.China is an earthquake-prone country.In recent years,with the development of infrastructure construction,some critical buildings are inevitably built in or through the near-fault zone.The near-fault ground motions have unique engineering characteristics and strong randomness,which may cause strong nonlinear behavior of structures.Therefore,random vibration and dynamic reliability analysis of nonlinear structures under near-fault ground motions will provide an important theoretical basis for seismic reliability design of buildings in near-fault areas.In order to perform random vibration and dynamic reliability analysis of structures subjected to near-fault ground motions,there are two key issues to be addressed:the one is to establish a stochastic model of ground motions that can reasonably characterize the velocity pulse of the near-fault ground motions;another is to develop an accurate and efficient method of random vibration and dynamic reliability analysis for the strong nonlinear structures.This dissertation firstly devotes to obtain the benchmark solutions of random vibration for rectangular thin plate under various random excitations.Secondly,a discontinuous Galerkin(DG)-based numerical implement of probability density evolution method is established to perform nonlinear random vibration and dynamic reliability analysis for general linear or nonlinear structures.Finally,the nonlinear random vibration and dynamic reliability analysis of building structures under the near-fault ground motion are investigated.The main contents are presented as follows:(1)An efficient semi-analytical method for stationary random vibration analysis of thin plate is proposed,and the benchmark solutions of stochastic responses of the plate subjected to various stationary random excitations are obtained.The new equivalent PSD formula of the stochastic von Mises stress that contains all of the cross-spectral density between stress components is derived.By means of pseudo-excitation method,firstly,the analytical solutions of stochastic responses for the plates are derived by introducing the exact solutions of free vibration of the plates.To take advantage of high efficiency of the pseudo-excitation method and keep the precise of results,then,a semi-analytical method for random vibration is proposed.In this method,the integration and partial derivative in spatial domain are solved analytically,and then are discretized in conjunction with the frequency domain.Some examples show that the proposed semi-analytical method can efficiently obtain the random responses consistent with those by the analytical method under various random excitations.Based on pseudo excitation method,the equivalent PSD of stochastic von Mises stress response containing all of the cross-spectral density between stress components is also obtained.(2)A time-frequency fully nonstationary power spectral density model is suggested,and the semi-analytical method for the random vibration analysis of the plate under the nonstationary random excitations is proposed to obtain the benchmark solutions of nonstationary stochastic responses.The analytical solutions are derived by combining the pseudo-excitation method and Duhamel's integral.For enhancing computational efficiency,the semi-analytical method is established by adopting analytical process in spatial domain and discretized process in frequency domain.Besides,the precise integration method is used to substitute for the Duhamel's integral in time domain.By virtue of the semi-analytical method,the benchmark solutions of time and time-frequency fully nonstationary random vibration of rectangular thin plate are obtained.By comparing with the obtained benchmark solutions of the two types of nonstationary responses,it can be observed that the time-frequency fully excitation can result in the larger root mean square of stochastic responses than time nonstationary one.(3)The benchmark solutions of random vibration for the thin plate resting on Pasternak elastic foundation subjected to accelerated moving random excitation are obtained.Analytical solutions of nonstationary random vibration of thin plate subjected to stationary moving random excitations are derived,by introducing exact solutions of free vibration of thin plate on Pasternak foundation and combining the pseudo-excitation method and Duhamel's integral.The semi-analytical method of random vibration is employed to solve the benchmark solutions of thin plates under nonstationary stochastic responses.To verify the accuracy of obtained benchmark solutions,the Monte Carlo simulation(MCS)based on precise integration method is also adopted.The results illustrate that the semi-analytical method is more efficient than the analytical method and MCS for achieving nonstationary stochastic responses of thin plate under stationary moving random excitations.(4)For the probability density evolution method,a discontinuous Galerkin(DG)method-based numerical implementation is developed to control efficiently numerical dissipation and dispersion in numerical solutions of the generalized probability density evolution equation(GPDEE).The approach utilizes a smoothed function to approximately substitute for the original discontinuous initial condition of GPDEE,which can fundamentally avoid numerical dispersion.On the other hand,the numerical dissipation can also be decreased by improving the order of the polynomial interpolation in elements.Additionally,the proposed approach is easy to construct high order scheme and reduce the mesh dependency of the finite difference method.To verify the accuracy of proposed method,the benchmark solutions of random vibration of the rectangular thin plates presented in Chapters 2,3 and 4 are also recalled.Moreover,a five-story shear frame structure with hysteretic nonlinear behavior is also adopted to illustrate the effectiveness of the present method for nonlinear structures.(5)Based on the records of near-fault ground motions,the stochastic model containing 9 basic random variables for the near-fault ground motions is established.The nonlinear random vibration and dynamic reliability analysis for nonlinear building structures under near-fault ground motions are performed.Firstly,the velocity time history with the strongest pulse is generated from the records in orthogonal directions.And long-period pulses are extracted and fitted with the single Gabor wavelet with five random parameters by the nonlinear least square method.The high-frequency components are then reproduced by random function based spectral representation method with stochastic envelope function.Furthermore,the DG-based probability density evolution method(PDEM)is utilized to efficiently perform nonlinear random vibration analysis.An absorbing boundary condition method and equivalent extreme event can be adopted to evaluate the first-passage dynamic reliability for typical nonlinear building structures.Finally,remarkable influences of fault distance and the occurrence instant of velocity pulse on dynamic reliability of buildings are scrutinized.