Nonstationary seismic responses of structure with nonlinear stiffness subject to modulated Kanai–Tajimi excitation

A simple structure under earthquake excitation is modeled as a single‐degree‐of‐freedom system with nonlinear stiffness subject to modulated Kanai–Tajimi excitation. The nonstationary responses including the nonstationary probability densities of the system responses and the statistical moments are obtained in semi‐analytical form. By applying the stochastic averaging method based on the generalized harmonic functions, the averaged Fokker–Planck–Kolmogorov(FPK) equation governing the nonstationary probability density of the amplitude is derived. Then, the solution of the FPK equation is approximately expressed by a series expansion in terms of a set of properly selected basis functions with time‐dependent coefficients. According to the Galerkin method, the time‐dependent coefficients are solved from a set of linear first‐order differential equations. Thus, the nonstationary probability densities of the amplitude and the state responses as well as the statistic moments of the amplitude are obtained. Finally, two types of the modulating functions, i.e. constant function and exponential function, are considered to give some semi‐analytical formulae. The proposed procedures are checked against the Monte Carlo simulation. The effects of the structure natural frequency and the intensity of the excitation as well as the ground stiffness on the system responses are discussed. It should be pointed out that the proposed method is good for broadband excitation and light damping. Copyright © 2011 John Wiley & Sons, Ltd.     

A stochastic approach to nonlinear unconfined flow subject to multiple random fields

In this study, the KLME approach, a moment-equation approach based on the Karhunen–Loeve decomposition developed by Zhang and Lu (Comput Phys 194(2):773–794, 2004), is applied to unconfined flow with multiple random inputs. The log-transformed hydraulic conductivity F, the recharge R, the Dirichlet boundary condition H, and the Neumann boundary condition Q are assumed to be Gaussian random fields with known means and covariance functions. The F, R, H and Q are first decomposed into finite series in terms of Gaussian standard random variables by the Karhunen–Loeve expansion. The hydraulic head h is then represented by a perturbation expansion, and each term in the perturbation expansion is written as the products of unknown coefficients and Gaussian standard random variables obtained from the Karhunen–Loeve expansions. A series of deterministic partial differential equations are derived from the stochastic partial differential equations. The resulting equations for uncorrelated and perfectly correlated cases are developed. The equations can be solved sequentially from low to high order by the finite element method. We examine the accuracy of the KLME approach for the groundwater flow subject to uncorrelated or perfectly correlated random inputs and study the capability of the KLME method for predicting the head variance in the presence of various spatially variable parameters. It is shown that the proposed numerical model gives accurate results at a much smaller computational cost than the Monte Carlo simulation.     

Earthquake response of hysteretic mass-column using non-Gaussian closure

A non-Gaussian closure scheme based on the Edgeworth expansion of the probability density function is used to study the response of a hysteretic structure under random parametric excitation. The system considered consists of a weightless mass supporting a concentrated mass and it is subjected to the vertical and horizontal components of the ground acceleration modeled as nonstationary Gaussian white noise processes. The material of the structure exhibits bilinear hysteretic behaviour. The equation governing the motion of the system is transformed into an Itô stochastic differential equation. A set of ordinary differential equations governing the response statistics are obtained. These form an infinite hierarchy of equations which must be truncated in order to solve for moments of any order. The Edgeworth expansion of the joint density is used to truncate this infinite hierarchy. Such a closure scheme appears desirable since for hysteretic systems an explicit expression of the probability density is required. A frequently used closure scheme based on Gaussian assumption underestimates the response. The non-Gaussian density can be used in reliability studies.     

Non-stationary random seismic responses of non-uniform beams

In this paper, a direct Ritz method is presented for analysing the responses of non-uniform shear beams subjected to the action of an evolutionary random seismic excitation. Because of the use of a pseudo ground acceleration, time-dependent random responses can conveniently be calculated by solving a set of deterministic equations of motion. These responses include power spectral densities, variances and higher spectral moments of any required responses of the beam. A non-uniform beam example was solved and the time-dependent variances and second spectral moments of both the beam-top displacement and the beam-base shear force are given to show the simplicity and effectiveness of the method, which can also easily be applied to other continuous elastic structures.     

Semigroup solutions to stochastic unsteady groundwater flow subject to random parameters

Two methods for the solution of partial differential equations (PDE) for the general case of random in time physical parameters are presented and their application to the solution of unsteady regional groundwater flow equations are illustrated. The first method is the semigroup approach which directly offers a solution without resorting to closure approximations (hierarchy techniques), perturbation techniques, or Montecarlo simulation techniques. The semigroup approach can also handle the general stochastic problem when randomness also appears as initial conditions, boundary conditions or forcing terms. The second method is an approximation scheme to obtain the semigroup solution in complex cases and permits the solution of equations with more than one random coefficient.     

Propagation of partially coherent non-stationary random waves in a viscoelastic layered half-space

This paper presents a highly accurate method based on the precise integration method (PIM) and on the pseudo excitation method (PEM). The method computes the propagation behaviour of partially coherent non-stationary random waves in a viscoelastic, transversely isotropic solid, which consists of a multi-layered soil resting on a homogeneous semi-infinite space. The excitation source is a local rupture between two layers, which causes a partially coherent non-stationary random field. The analysis of non-stationary random wave propagation is transformed into that for deterministic waves by using PEM. The resulting governing equations in the frequency-wavenumber domain are linear ordinary differential equations, which are solved very precisely by using PIM. The evolutionary power spectral densities of the ground level responses are investigated and some typical earthquake phenomena are explained.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.     

A faster simulation method for the stochastic response of hysteretic structures subject to earthquakes

A semi-analytical forward-difference Monte Carlo simulation procedure is proposed for the determination of the lower order statistical moments and the joint probability density function of the stochastic response of hysteretic non-linear multi-degree-of-freedom structural systems subject to nonstationary gaussian white noise excitation, as an alternative to conventional direct simulation methods. The method generalizes the so-called Ermak-Allen algorithm developed for simulation applications in molecular dynamics to structural hysteretic systems. The proposed simulation procedure rely on an assumption of local gaussianity during each time step. This assumption is tantamount to various linearizations of the equations of motion. The procedure then applies an analytical convolution of the excitation process, hereby reducing the generation of stochastic processes and numerical integration to the generation of random vectors only. Such a treatment offers higher rates of convergence, faster speed and higher accuracy. The procedure has been compared to the direct Monte Carlo simulation procedure, which uses a fourth-order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process. The considered system was a multi-dimenensional hysteretic shear frame, where the constitutive equation of the hysteretic shear forces are described by a bilinear hysteretic model. The comparisons show that significant savings in computer time and accuracy can be achieved.     

A numerical method of moments for solute transport in physically and chemically nonstationary formations: linear equilibrium sorption with random K<Subscript>d</Subscript>

A Lagrangian perturbation method is applied to develop a method of moments for solute flux through a three-dimensional nonstationary flow field. The flow nonstationarity stems from medium nonstationarity and internal and external boundaries of the study domain. The solute flux is described as a space-time process where time refers to the solute flux breakthrough through a control plane (CP) at some distance downstream of the solute source and space refers to the transverse displacement distribution at the CP. The analytically derived moment equations for solute transport in a nonstationarity flow field are too complicated to solve analytically, a numerical finite difference method is implemented to obtain the solutions. This approach combines the stochastic model with the flexibility of the numerical method to boundary and initial conditions. The developed method is applied to study the effects of heterogeneity and nonstationarity of the hydraulic conductivity and chemical sorption coefficient on solute transport. The study results indicate all these factors will significantly influence the mean and variance of solute flux.     

Generation of multivariate autoregressive sequences with emphasis on initial values

Certain aspects of data generation are studied through multivariate autoregressive (AR) models. The main emphasis is on the preservation of certain desired moments and the effect of initial values on these moments. The problem of preservation of moments is approached in a nontraditional way by starting with the initial values. For this purpose, general AR processes with a random start and with time-varying parameters are introduced to lay a foundation for the analysis of all types of AR processes, including the periodic cases. It is shown that an AR process with a random start and with parameters obtained from the moment equations is capable of generating jointly multivariate normal vectors with any specified means and covariance matrices, and with any specified autocovariance matrices up to a given lag. With a random start, there is no transition period involved for achieving these moments. A simple solution is proposed for matrix equations of the form BB^T = A which appear in the moment equations. The aggregation properties of general AR process are also studied.

A more detailed analysis is given for the two-period first-order periodic autoregressive model, PAR₂(1). For the PAR₂(1) process with a random start and with parameters obtained from the moment equations, it is shown that the autocovariance function depends only on the period and the lag, and therefore the process is periodic (covariance) stationary. The PAR₂(1) process with a fixed start is also studied. It is shown that the moments of this process depend on the absolute time, in addition to the period and the lag, and therefore the process is not periodic stationary. This dependence diminishes with time, and periodic stationarity is realized if the AR parameters satisfy certain conditions. In that case, the PAR₂(1) process with a fixed start converges to that with a random start, but only after a certain transition period. This proves the superiority of a random start over a fixed start.     

Automated first and second order moment equations for a set of stochastic differential equations of type AŻ+BZ=C(t)

The generation of the second and higher order moment equations for a set of stochastic differential equations based on Ito's differential lemma is difficult, even for small system of equations. From the knowledge of the statistical properties of the Gaussian white noises associated with the parameters and input coefficients of a set of stochastic differential equations of typeA.+B.Z=C(t), a way to automatically generate the second order moment equations in a computer is presented in this paper. The resulting set of first and second order moment equations is also presented in the same state-space form of the original set of stochastic differential equations through a vectorization of the correlation matrix, which takes advantage of its symmetry. The procedure involved here avoids the inversion of matrixA to apply Ito's differential lemma. Therefore, the presented numerical implementation reduces the computational effort required in the formulation and solution of the moment equations. Moreover, other robust and efficient numerical deterministic integration schemes can be equally applied to the solution of the moment equations.     

Response analysis of hysteretic multi-storey frames under earthquake excitation

Local incremental stiffness relations are formulated for a class of elasto-plastic beam elements. The earthquake acceleration is modelled as a filtered white noise process. The Itǒ differential equations of the integrated system made up of the structural system and the excitation process are then formulated. Instead of the original system an equivalent nonlinear system is considered, in which the drift vector is given by a series expansion of order n ≧ 1, where n = 1 represents the well-known case of equivalent linearization. Only components of the drift vector representing the non-analytieal constitutive equations are replaced by a polynomial expansion. The coefficients of this expansion are determined from a least mean square criterion, and are sequentially updated. Especially an equivalent system with a cubic expansion to the drift vector is investigated. The hierarchy of statistical moment equations is closed by a cumulant neglect closure scheme. The method has been applied to a two-storey frame. The results are compared to those of numerical simulation, and provide substantial improvements compared to equivalent linearization.     

Wavelet-vaguelette decomposition of spatiotemporal random fields

A wavelet-based orthogonal decomposition of the solution to stochastic differential/pseudodifferential equations of parabolic type is derived in the cases of random initial conditions and random forcing. The family of spatiotemporal models considered can represent anomalous diffusion processes when the spatial operator involved is a fractional or multifractional pseudodifferential operator. The results obtained are applied to the generation of the sample paths of Gaussian spatiotemporal random fields in the family studied.     

Seismic spatial effects on long-span bridge response in nonstationary inhomogeneous random fields

The long-span bridge response to nonstationary multiple seismic random excitations is investigated using the PEM (pseudo excitation method). This method transforms the nonstationary random response analysis into ordinary direct dynamic analysis, and therefore, the analysis can be solved conveniently using the Newmark, Wilson-θ schemes or the precise integration method. Numerical results of the seismic response for an actual long-span bridge using the proposed PEM are given and compared with the results based on the conventional stationary analysis. From the numerical comparisons, it was found that both the seismic spatial effect and the nonstationary effect are quite important, and that both stationary and nonstationary seismic analysis should pay special attention to the wave passage effect.     

非平稳地震激励下隔震曲线梁桥振动控制研究

曲线梁桥由于其平面不规则性导致结构在地震激励下产生弯扭耦合效应,使得隔震曲线梁桥的地震响应更加复杂。目前常用的控制方法是将隔震技术与附加减震装置相结合对曲线梁桥进行控制。本文将地震动考虑为一均匀调制非平稳随机过程,针对隔震曲线梁桥长周期、低频率的特点,选取Clough-Pension平稳地震动功率谱模型作为随机地震动输入模型,对无控(NON-C)、经典线性最优控制(COC)以及序列最优控制算法(SOC)三种状态下的曲线桥梁进行随机响应分析。通过建立曲线梁桥在随机地震动作用下的运动方程,求出减震控制结构的位移谱密度、加速度谱密度响应及时变方差。分析结果表明:序列最优控制算法(SOC)在使隔震层位移得到减小的同时,可以更有效地控制上部结构的地震响应,具有更好的控制效果。     

Effective linear damping and stiffness coefficients of nonlinear systems for design spectrum based analysis

A stochastic approach for obtaining reliable estimates of the peak response of nonlinear systems to excitations specified via a design seismic spectrum is proposed. This is achieved in an efficient manner without resorting to numerical integration of the governing nonlinear equations of motion. First, a numerical scheme is utilized to derive a power spectrum which is compatible in a stochastic sense with a given design spectrum. This power spectrum is then treated as the excitation spectrum to determine effective damping and stiffness coefficients corresponding to an equivalent linear system (ELS) via a statistical linearization scheme. Further, the obtained coefficients are used in conjunction with the (linear) design spectrum to estimate the peak response of the original nonlinear systems. The cases of systems with piecewise linear stiffness nonlinearity, along with bilinear hysteretic systems are considered. The seismic severity is specified by the elastic design spectrum prescribed by the European aseismic code provisions (EC8). Monte Carlo simulations pertaining to an ensemble of nonstationary EC8 design spectrum compatible accelerograms are conducted to confirm that the average peak response of the nonlinear systems compare reasonably well with that of the ELS, within the known level of accuracy furnished by the statistical linearization method. In this manner, the proposed approach yields ELS which can replace the original nonlinear systems in carrying out computationally efficient analyses in the initial stages of the aseismic design of structures under severe seismic excitations specified in terms of a design spectrum.     

Conditional simulation of spatially variable seismic ground motions based on evolutionary spectra

This paper proposes a computational procedure for the conditional simulation of spatially variable seismic ground motions for long span bridges with multiple supports. The seismic ground motions, with part of their time histories measured at some supports, are regarded as zero‐mean nonstationary random processes characterized by predefined evolutionary power spectral density. To conditionally simulate unknown seismic ground motion time histories at other supports, the Kriging method is first described briefly for the conditional simulation of a random vector comprised of zero‐mean Gaussian variables. The multivariate oscillatory processes characterized by the evolutionary power spectral density matrix are then introduced, and the Fourier coefficients of the oscillatory processes and their covariance matrix are derived. By applying the Kriging method to the random vector of the Fourier coefficients and using the inverse Fourier transform, unknown nonstationary seismic ground motion time histories can be simulated. A numerical example is selected to demonstrate capabilities of the proposed simulation procedure, and the results show that the procedure can ensure unbiased time‐varying correlation functions, especially the cross correlation between known and unknown time histories. The procedure is finally applied to the Tsing Ma suspension bridge in Hong Kong to generate ground accelerations at its multiple supports using limited seismic records. Copyright © 2012 John Wiley & Sons, Ltd.     

二维横各向同性弹性随机介质中的波场特征

本文通过交错网格有限差分正演．模拟了平面地震波在二维横各向同性弹性随机介质模型中的传播及其自激自收时间记录．为研究横各向同性弹性随机介质模型中的波场特征,我们在五个不同的时间区段上,分别计算剖面的三个统计特征(横向中心频率、纵向中心频率、波场能量相对值)．这样,对应每一个横各向同性弹性随机介质模型．均可计算得到15个不同的波场特征量．我们通过在二维横各向同性弹性随机介质中的正演模拟．研究当自相关长度以及介质的各向异性系数变化时,对应的上述波场特征量的变化特点．证实了在随机介质模型中．各向异性系数的变化会引起波场记录上的某些统计特征的变化,归纳得出了若干结论．     

平稳随机地震地面运动过程模型及其统计特征

地震地面运动过程具有强烈的随机性,应用随机理论对实际工程结构进行地震可靠性分析和抗震设计与加固时都需要建立合理的随机地震地面运动模型,本文选择3种典型的随机地震动模型,即理想白噪声模型、金井清模型和改进的金井消模型,分析了它们的物理概念、频域特征以及适用范围。引入状态向量,建立状态方程．通过复振型叠加法分析了地震地面运动过程的时域统计特性,推导出3种随机地震动模型的相关函数的解析表达式．这些结果可为结构随机地震反应时域分析和抗震可靠性评估提供基础。