EQUIVALENT LINEARIZATION OF GENERALLY PINCHING HYSTERETIC,DEGRADING SYSTEMS

Many types of structural systems that undergo cycles of inelastic deformation under severe natural hazard loadings exhibit ‘pinching’ of hysteresis loops. In this paper, a generally pinching hysteretic restoring force model—an extension of the Bouc–Wen differential hysteresis model—is used in stochastic equivalent linearization of single-degree-of-freedom structural systems. The severity and rate of pinching are controlled by the hysteretic energy dissipation and the pinching level can be specified to match experimental data. Under white noise excitations, estimates of reponse statistics from linearization are shown to compare favourably with those from Monte Carlo simulation. Numerical studies on the sensitivity of the accuracy of response statistics obtained by linearization to changes in the hysteresis parameters showed that, for a range of practical cases, the linearization method can be used in lieu of simulation and that, in low-frequency systems, some hysteresis parameters may be set to a constant value a priori to reduce the number of model parameters that needs to be estimated or identified, and to simplify further random vibration analysis and/or performance evaluation studies.     

Fuzzy random vibration of multi-degree-of-freedom hysteretic systems subjected to earthquakes

Earthquake excitation not only has evident randomness but also has strong fuzziness owing to the uncertainties in the definition of earthquake intensity and site soil classification. In this paper, the seismic ground motion is simulated as a stationary filtered white noise with fuzzy parameters, an analytical procedure is proposed to analyse the fuzzy random vibration of multi-degree-of-freedom (MDOF) hysteretic systems, and the covariance matrix of the fuzzy random responses is derived by the equivalent linearization technique. Finally, some numerical results for a two-storey shear hysteretic frame are presented to demonstrate the validity and effectiveness of the procedure proposed.     

高速列车-桥梁竖向随机振动的时域分析方法

提出时间相关多维有色噪声形式的轨道不平顺激励下列车-桥梁耦合系统协方差响应的时域递推方法。用白噪声滤波法生成轨道不平顺有色噪声过程,在宽频带内识别滤波器参数以同时实现滤波成型和波长截断功能。提出基于高阶Pade近似的累次时滞系统,以实现列车多轮对下轨道不平顺激励的大时滞再现;再结合成型滤波器构造列车下轨道不平顺激励的一致白噪声模型。建立列车-桥梁垂向振动的状态方程,将其与激励模型联立得到一致白噪声激励下的列车-桥梁扩阶状态方程。将方差递推法推广到时变系统,求解列车-桥梁系统的随机振动。分析结果与Monte Carlo模拟法符合良好,表明了方法的正确性。     

Nonstationary stochastic response of structural systems equipped with nonlinear viscous dampers under seismic excitation

Nonlinear viscous dampers are supplemental devices widely used for enhancing the performance of structural systems exposed to seismic hazard. A rigorous evaluation of the effect of these damping devices on the seismic performance of a structural system should be based on a probabilistic approach and take into account the evolutionary characteristics of the earthquake input and of the corresponding system response. In this paper, an approximate analytical technique is proposed for studying the nonstationary stochastic response characteristics of hysteretic single degree of freedom systems equipped with viscous dampers subjected to a fully nonstationary random process representing the seismic input. In this regard, a stochastic averaging/linearization technique is utilized to cast the original nonlinear stochastic differential equation of motion into a simple first‐order nonlinear ordinary differential equation for the nonstationary system response variance. In comparison with standard linearization schemes, the herein proposed technique has the significant advantage that it allows to handle realistic seismic excitations with time‐varying frequency content. Further, it allows deriving a formula for determining the nonlinear system response evolutionary power spectrum. By this way, ‘moving resonance’ effects, related to both the evolutionary seismic excitation and the nonlinear system behavior, can be observed and quantified. Several applications involving various system and input properties are included. Furthermore, various response parameters of interest for the seismic performance assessment are considered as well. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed technique. Copyright © 2014 John Wiley & Sons, Ltd.     

Deterministic and stochastic analyses of a nonlinear system with a Biot visco‐elastic element

In the paper the Biot hysteretic model involving an infinite collection of elements whose dynamic characteristics are specified through a probability density‐like function is re‐examined; in the limit case, the Biot model yields a dynamic system with ideal hysteretic damping which is known to be problematic for random vibration analysis. It is shown that bona fide Monte Carlo simulations can be conducted for the Biot model by treating, cautiously, the integrodifferential equation which is involved; this is based on recursive calculations of double integrals encountered in the representation of the system dynamics. The numerical results which pertain to the Monte Carlo studies are further used to assess the accuracy of a statistical linearization procedure adopted in determining the response of the hysteretically damped system to white noise. Published in 2001 by John Wiley & Sons, Ltd.     

Improvement of decentralized random decrement technique for data processing in wireless sensor network

The Random Decrement Technique (RDT), based on decentralized computing approaches implemented in wireless sensor networks (WSNs), has shown advantages for modal parameter and data aggregation identification. However, previous studies of RDT-based approaches from ambient vibration data are based on the assumption of a broad-band stochastic process input excitation. The process normally is modeled by filtered white or white noise. In addition, the choice of the triggering condition in RDT is closely related to data communication. In this project, research has been conducted to study the nonstationary white noise excitations as the input to verify the random decrement technique. A local extremum triggering condition is chosen and implemented for the purpose of minimum data communication in a RDT-based distributed computing strategy. Numerical simulation results show that the proposed technique is capable of minimizing the amount of data transmitted over the network with accuracy in modal parameters identification.     

Response uncertainty and time-variant reliability analysis for hysteretic MDF structures

Response uncertainty evaluation and dynamic reliability analysis corresponding to classical stochastic dynamic analysis are usually restricted to the uncertainties of the excitation. The inclusion of the parameter uncertainties contained in structural properties and excitation characteristics has become an increasingly important problem in many areas of dynamics. In the present paper, a point estimate procedure is proposed for the evaluation of stochastic response uncertainty, and a response surface approach procedure in standard normal space is proposed for analysis of time-variant reliability analysis for hysteretic MDF structures having parameter uncertainties. Using the proposed procedures, the response uncertainties and time-variant reliability can be easily obtained by several repetitions of stochastic response analysis under given parameters without conducting sensitivity analysis, which is considered to be one of the primary difficulties associated with conventional methods. In the time-variant reliability analysis, the failure probability can be readily obtained by improving the accuracy of the first-order reliability method using the empirical second-order reliability index. The random variables are divided into two groups, those with CDF and those without CDF. The latter are included via the high-order moment standardization technique. A numerical example of a 15F hysteretic MDF structure that takes into account uncertainties of four structural parameters and three excitation characteristics is performed, based on which the proposed procedures are investigated and the effects of parameter uncertainties are discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

Response of a non-linear system with restoring forces governed by fractional derivatives—Time domain simulation and statistical linearization solution

In this paper, the random response of a non-linear system comprising frequency dependent restoring force terms is examined. These terms are accurately modeled in seismic isolation and in many other applications using fractional derivatives. In this context, an efficient numerical approach for determining the time domain response of the system to an arbitrary excitation is first proposed. This approach is based on the Grunwald–Letnikov representation of a fractional derivative and on the well-known Newmark numerical integration scheme for structural dynamic problems. Next, it is shown that for the case of a stochastic excitation, in addition to the time domain solutions, a frequency domain solution can be readily determined by the method of statistical linearization. The reliability of this solution is established in a Monte Carlo simulation context using the herein adopted time domain solution scheme. Furthermore, several related parameter studies are reported.     

Midbroken reinforced concrete shear frames due to earthquakes. A hysteretic model to quantify damage at the storey level

A nonlinear hysteretic model for the response and local damage analyses of reinforced concrete shear frames subject to earthquake excitation is proposed, and, the model is applied to analyse midbroken reinforced concrete (RC) structures due to earthquake loads. Each storey of the shear frame is represented by a Clough and Johnston hysteretic oscillator with degrading elastic fraction of the restoring force. The local damage is numerically quantified in the domain [0,1] using the maximum softening damage indicators which are defined in closed form based on the variation of the eigenfrequency of the local oscillators due to the local stiffness and strength deterioration. The proposed method of response and damage analyses is illustrated using a sample 5 storey shear frame with a weak third storey in stiffness and/or strength subject to sinusoidal and simulated earthquake excitations for which the horizontal component of the ground motion is modeled as a stationary Gaussian stochastic process with Kanai-Tajimi spectrum, multiplied by an envelope function.     

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.     

高斯白噪声作用下刚性滑移结构稳态响应的概率密度函数

利用能量包线随机平均法，导出了库伦滑动摩擦刚性结构在高斯白噪声地面作用下响应的稳态联合概率密度函数的解析解，它与文献[2]的等效非线性系统法所得之解完全相同。与蒙特卡洛模拟结果对比表明，当库伦阻尼较小时，该解析解能够满意地预测研究系统响应的概率分布。     

随机结构动力可靠度分析的极值概率密度方法

提出了随机结构动力可靠度分析的极值概率密度方法。基于概率密度演化的基本思想,构造一个虚拟随机过程,使得随机结构动力反应的极值为该虚拟随机过程的截口随机变量。进而．采用概率密度演化方法,建立概率密度演化方程并求解给出随机结构动力反应的极值分布。在安全域内积分即可给出结构动力可靠度,当安全界限为随机变量时,采用这一方法几乎不增加额外的工作量,与随机模拟结果的比较表明,本文建议方法具有良好的精度和效率。     

Correlation failure analysis of an uncertain hysteretic vibration system

In this paper, a numerical method for correlation sensitivity analysis of a nonlinear random vibration system is presented. Based on the first passage failure model, the probability perturbation method is employed to determine the statistical characteristics of failure modes and the correlation between them. The sensitivity of correlation between failure modes with respect to random parameters characterizing the uncertainty of the hysteretic loop is discussed. In a numerical example, a two-DOF shear structure with uncertain hysteretic restoring force is considered. The statistical characteristics of response, failure modes and the sensitivity of random hysteretic loop parameters are provided, and also compared with a Monte Carlo simulation.     

Stochastic response of combined primary-secondary structures under seismic input

A technique for non-stationary stochastic analysis of linear combined primary and secondary subsystems subjected to a zero-mean Gaussian base excitation is presented. The proposed technique, based on the use of the Taylor's expansion in evaluating the operators which appear in the step-by-step procedure, does not require the evaluation of the complex eigenproperties of the combined system. Operating in this way, even though the numerical procedure is a conditionally stable one, appears to be more efficient than existing methods to evaluate the dynamic response of such composite systems. It is also shown that the proposed procedure is available whether the seismic input is idealized as a filtered white noise or it is defined by means of its autocorrelation function.     

基于自然激励技术和FT时移特性的模态阻尼比识别方法

基于环境激励下结构动力响应信号分析与处理识别结构的模态参数,是结构健康监测和损伤诊断的一个重要环节,目前为止,要得到较为可靠的识别结果仍有一定困难,尤其是模态阻尼比。基于自然激励技术和傅里叶变换的时移特性,提出了一种新的结构模态阻尼比估算方法,通过理论推导和仿真算例验证了该方法的可行性,进而利用一刚构-连续组合梁桥在环境激励下的动力测试数据,通过该方法对其阻尼比进行了识别,并将识别结果与数据驱动随机子空间法的识别结果进行了对比。结果表明:提出的方法可以减轻噪声影响,得到可接受的识别结果,可为大型工程结构阻尼比的识别提供一个方便和有效的途径。     

Modeling groundwater velocity uncertainty in nonstationary composite porous media

Despite the intensive research over the past decades in the field of stochastic subsurface hydrology, our ability to analyze and model heterogeneous groundwater systems remains limited. Most existing theories are either too restrictive to handle practical complexity or too expensive to be applied to realistic problem sizes. In this paper we present approximate, closed-form equations that allow modeling 2D nonstationary flows in statistically inhomogeneous aquifers, including composite aquifers containing multiple zones characterized by different statistical models. The composite representation has the effect of decreasing the variance of deviations from the mean, relaxing the limitation of the small-perturbation assumption. The simple formulas are illustrated with a number of examples and compared with a corresponding first-order nonstationary numerical analysis and Monte Carlo simulation. The results show that, despite the gross simplifications, the closed-form equations are robust and able to capture complex variance dynamics, reproducing surprisingly well the first-order numerical solutions and the Monte Carlo simulation even in highly nonstationary, variable situations.     

Bivariate Gaussian mixture–based equivalent linearization method for stochastic seismic analysis of nonlinear structures

To address challenges in stochastic seismic analysis of nonlinear structures, this paper further develops a recently proposed Gaussian mixture–based equivalent linearization method (GM‐ELM). The GM‐ELM uses a Gaussian mixture distribution model to approximate the probabilistic distribution of a nonlinear system response. Using properties of the Gaussian mixture model, GM‐ELM can decompose the non‐Gaussian response of a nonlinear system into multiple Gaussian responses of linear single–degree of freedom oscillators. With the set of the equivalent linear systems identified by GM‐ELM, response statistics as crossing rate and first‐passage probability can be computed conveniently using theories of linear random vibration analysis. However, the original version of GM‐ELM may lead to an inaccurate estimate because of the heuristic parameters of the linear system introduced to supplement insufficient information. To overcome this limitation and define unique equivalent linear systems, this paper proposes a further developed version of GM‐ELM, which uses a mixture of bivariate Gaussian densities instead of univariate models. Moreover, to facilitate the use of elastic response spectra for estimating the mean peak responses of a nonlinear structure, a new response spectrum combination rule is proposed for GM‐ELM. Two numerical examples of hysteretic structural systems are presented in this paper to illustrate the application of the bivariate GM‐ELM to nonlinear stochastic seismic analysis. The analysis results obtained by the bivariate GM‐ELM are compared with those obtained by the univariate GM‐ELM, the conventional equivalent linearization method, the tail equivalent linearization method, and Monte Carlo simulation. The supporting source code and data are available for download at https://github.com/yisangri/GitHub‐bGM‐ELM‐code.git     

Simulation of orthogonal horizontal ground motion components for specified earthquake and site characteristics

A method for generating an ensemble of orthogonal horizontal ground motion components with correlated parameters for specified earthquake and site characteristics is presented. The method employs a parameterized stochastic model that is based on a time‐modulated filtered white‐noise process with the filter having time‐varying characteristics. Whereas the input white‐noise excitation describes the stochastic nature of the ground motion, the forms of the modulating function and the filter and their parameters characterize the evolutionary intensity and nonstationary frequency content of the ground motion. The stochastic model is fitted to a database of recorded horizontal ground motion component pairs that are rotated into their principal axes, a set of orthogonal axes along which the components are statistically uncorrelated. Model parameters are identified for each ground motion component in the database. Using these data, predictive equations are developed for the model parameters in terms of earthquake and site characteristics and correlation coefficients between parameters of the two components are estimated. Given a design scenario specified in terms of earthquake and site characteristics, the results of this study allow one to generate realizations of correlated model parameters and use them along with simulated white‐noise processes to generate synthetic pairs of horizontal ground motion components along the principal axes. The proposed simulation method does not require any seed recorded ground motion and is ideal for use in performance‐based earthquake engineering. Copyright © 2011 John Wiley & Sons, Ltd.     

Stochastic earthquake response of secondary systems in base-isolated structures

This paper studies the stochastic responses of secondary systems in base-isolated shear beam structures. A number of base isolation systems such as the laminated rubber bearing (LRB), the resilient-friction base isolator (R-FBI) with or without sliding upper plate, and the EDF system are considered. The stochastic models for the El Centro 1940 and the Mexico City 1985 earthquakes which preserve the non-stationary evolutions of amplitude and frequency content of ground accelerations are used as earthquake excitations. The technique of equivalent linearization is utilized and the mean-square response statistics of secondary systems and primary structure are evaluated. The accuracy of the linearization scheme is verified by comparison with the Monte Carlo simulation results. Statistically estimated peak responses of the secondary system are evaluated and the results are compared with the response spectra for actual earthquake accelerograms. It is shown that the use of base isolation systems, generally, provides considerable protection for structural contents. In particular, the LRB system is remarkably effective in reducing responses of secondary systems. Results for the Mexico City earthquake show that the base-isolated structures are sensitive to long period ground excitations.