Solution of random structural system subject to nonstationary excitation: transforming the equation with random coefficients to one with deterministic coefficients and random initial conditions

This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator.     

Order statistics of functionals of strong ground motion for a class of renewal processes

In this paper, probability distribution functions are derived for the order statistics of various functionals of strong ground motion at a site. These functionals can be: Modified Mercalli Intensity (MMI), peak ground acceleration (PGA), Fourier spectral amplitudes of acceleration, response spectrum amplitudes (spectral displacement, pseudo-spectral velocity and pseudo-spectral acceleration), and amplitudes of the peaks (local maxima and local minima) in the time historyof the response of SDOF and MDOF structures at the site. Three parameters of the response of a structure are considered: displacement, shear force and bending moment at each level (storey) of the structure. The earthquake sources contributing to the risk of ground motion at the site are a number of point, area or volume sources, each with defined frequency of occurence-magnitude relationship. The magnitudes of the possible events at these sources are discretized, and the occurrence of events of different magnitudes are assumed to be statistically independent. For each magnitude, it is assumed that the eartquakes occur in a Poissonian sequence or in a renewal process which is a generalization of the Poissonian. For these assumptions, the probability distribution functions are presented for the number of earthquakes, n, during which a given level of site or structural response is exceeded during the exposure time, and for the return period of the exceedances. For example, for single-degree- of-freedom: (SDOF) or multi-degree-of-freedom structures, (MDOF) n can be the number of earthquakes during which the response of a storey will exceed a given level at least m times(m = 1, 2, 3,…) during the exposure time. These probability distribution functions can be used to extend the concept of uniform probability functionals to more than one exceedance. A more important application is to generalize the uniform probability functionals method of site response (uniform probability Fourier or response spectra) to uniform probability envelopes of displacement, shears and bending moments of a given structure. The uniform probability envelopes can be for exceedance at least once during at least one earthquake, or, in general, for exceedance at least m times per earthquake (m = 1, 2,…) during at least n earthquakes. In other words, during at least n earthquakes at least m peaks in the response can be higher than the specified level. Such uniform probability envelopes can be used (1) to define new design guidelines for building codes based on cost-benefit analysis; (2) to construct more refined probability distribution functions for the damage and total economic losses caused by earthquakes; and (3) to develop planning and decision strategies on strengthening and retrofitting existing buildings.     

Evaluation of the structural response under seismic actions using non‐linear static methods

The paper investigates the degree of accuracy achievable when some non‐linear static procedures based on a pushover analysis are used to evaluate the seismic performance. In order to assess the significance of different sources of errors, three types of structural systems are analysed: (i) single‐degree‐of‐freedom (SDOF) systems with different hysteretic behaviour; (ii) shear‐type multi‐degree‐of‐freedom (MDOF) systems with elastic–perfect plastic (EPP) shear force–interstorey drift relationships; (iii) a steel moment‐resisting frame with rigid joints and EPP moment–curvature relationship. In SDOF systems, the source of approximation comes only from the calibration of the demand spectrum, while in MDOF systems some further errors are introduced by the schematization with an equivalent SDOF system. The non‐linear static procedures are compared with rigorous time‐history analyses carried out by considering ten generated earthquake ground motions compatible with the Eurocode 8 elastic spectra. It was found that SDOF systems with longer periods satisfy the equal displacement approximation regardless of the hysteretic model, while hysteresis loops with smaller energy dissipated indicate lower response for shorter periods. This is the opposite of what predicted by the ATC‐40 capacity spectrum method, which underestimates and overestimates, respectively, the actual response of low‐ and high‐ductility systems. Conversely, the inelastic spectrum method proposed by Vidic, Fajfar and Fischinger leads to the most accurate results for all types of structural systems. The analyses carried out on EPP shear‐type frames point out a large concentration of the ductility demand on some storeys. However, such a concentration markedly reduces when some hardening is accounted for. Copyright © 2006 John Wiley & Sons, Ltd.     

Simplified analysis of frame structures with viscoelastic dampers considering the effect of soil-structure interaction

In this study, simplified numerical models are developed to analyze the soil-structure interaction (SSI) effect on frame structures equipped with viscoelastic dampers (VEDs) based on pile group foundation. First, a single degree-of-freedom (SDOF) oscillator is successfully utilized to replace the SDOF energy dissipated structure considering the SSI effect. The equivalent period and damping ratio of the system are obtained through analogical analysis using the frequency transfer function with adoption of the modal strain energy (MSE) technique. A parametric analysis is carried out to study the SSI effect on the performance of VEDs. Then the equilibrium equations of the multi degree-of-freedom (MDOF) structure with VEDs considering SSI effect are established in the frequency domain. Based on the assumption that the superstructure of the coupled system possesses the classical normal mode, the MDOF superstructure is decoupled to a set of individual SDOF systems resting on a rigid foundation with adoption of the MSE technique through formula derivation. Numerical results demonstrate that the proposed methods have the advantage of reducing computational cost, however, retaining the satisfactory accuracy. The numerical method proposed herein can provide a fast evaluation of the efficiency of VEDs considering the SSI effect.     

Output only modal identification and structural damage detection using time frequency ＆ wavelet techniques

The primary objective of this paper is to develop output only modal identifi cation and structural damage detection.Identif ication of multi-degree of freedom(MDOF) linear time invariant(LTI) and linear time variant(LTV—due to damage) systems based on Time-frequency(TF) techniques—such as short-time Fourier transform(STFT),empirical mode decomposition(EMD),and wavelets—is proposed.STFT,EMD,and wavelet methods developed to date are reviewed in detail.In addition a Hilbert transform(HT) approach to determine ...     

基于结构能量分析的抗震设计新方法的研究

概要介绍了多自由度结构能量分析方法，根据单自由度体系得到的输入能量谱，研究分析了多自由度与单自由度输入能量和耗能的等效性问题，并对一座钢筋混凝土框架结构算例进行了能量计算，对计算结果进行了分析，为能量分析设计方法应用于结构抗震设计进行有益的探索。     

行波效应对大跨度空间结构随机地震响应的影响

深入研究了行波效应对大跨度空间结构随机地震响应的影响,进一步完善了大跨度空间结构随机地震响应分析理论。推导了双支座、单自由度体系地震响应功率谱密度函数的解析表达式,研究了不同频率体系的响应峰值随地面视波速的变化规律,分析了多支撑点、多自由度体系的地震响应功率谱矩阵的特点,发现多自由度体系地震响应随地面视波速的变化规律与单自由度体系相似。数值模拟了某体育馆网壳结构在不同地面视波速情况下的随机地震响应,结果表明,考虑地震动行波效应后,结构地震响应随地面视波速的变化而显著变化,当视波速较低时其变化规律很复杂;且支撑点附近、受拟静力位移影响较大的部分杆件的地震响应明显增大,远离支撑点处、受拟静力位移影响较小的部分杆件的地震响应稍有减小。由此得出结论,对于大跨度空间结构的随机地震响应分析,必须考虑地震动的行波效应,尤其当受拟静力位移影响较大的部分杆件对结构抗震设计起控制作用时;且应对可能出现的地面视波速进行全面分析,作为结构抗震设计依据。     

Probabilistic analysis of peak response of MDOF systems with uncertain PSD function

The use of uniform hazard spectra which have the same probability of exceedance at different frequencies has been proposed for the future version of the National Building Code of Canada. Commonly used combination rules to estimate the peak responses of multi‐degree‐of‐freedom (MDOF) systems are the square root of sum of squares rule and the complete quadratic combination rule. However, the probability that the peak response of a MDOF system exceeds the one estimated by using these rules with the peak modal responses from the uniform hazard spectra cannot be inferred directly. The assessment of the probability of exceedance of the peak response of MDOF systems is presented by considering that the uncertainty in seismic excitation due to all potential earthquakes can be lumped in the power spectral density function of the ground acceleration with uncertain model parameters. This probability is evaluated based on the random vibration of linear systems and the first‐order reliability method. It is found that the under‐ or over‐estimations are less than about 5 or 10% if the modal contributions are not within 10–90% of, or not within 20–80% of, the absolute sum of the effective modal peak responses, respectively. Otherwise, severe under‐ or over‐estimation could result. Copyright © 2002 John Wiley & Sons, Ltd.     

Prediction of earthquake energy input from smoothed fourier amplitude spectrum

This paper attempts to show analytically that the energy-input spectra of damped SDOF systems and undamped MDOF systems excited by an earthquake motion can be predicted by smoothing the Fourier amplitude spectrum of the base acceleration. The spectral window for smoothing in the frequency domain for a damped SDOF system is identical with the probability density function of the time-variant or instantaneous vibration frequency resulting from non-linear hysteresis. The spectral window for an undamped MDOF system is identical with the set of squared participation factors associated with vibration modes. It was found that the increase in damping factor and the increase in participation of higher modes provide wider spectral windows, resulting in more flattened or unaltered energy-input spectra due to enhanced smoothing effects.     

The complete SRSS modal combination rule

The complete Square‐Root‐of‐Sum‐of‐Squares (c‐SRSS) modal combination rule is presented. It expresses the structural response in terms of uncoupled SDOF modal responses, yet accounting fully for modal response variances and cross‐covariances. Thus, it is an improvement over the classical SRSS rule which neglects contributions from modal cross‐covariances. In the c‐SRSS rule the spectral moments of the structural response are expressed rigorously in terms of the spectral moments of uncoupled modal responses and of some coefficients that can be computed straightforwardly as a function of modal frequencies and damping, without involving the computation of cross‐correlation coefficients between modal responses. An example shows an application of the c‐SRSS rule for structural systems with well separated and closely spaced modal frequencies, subjected to wide‐band and narrow‐band excitations. Comparisons with response calculations using the SRSS and the Complete Quadratic Combination rules are given and discussed in detail. Based on the c‐SRSS rule a response spectrum formulation is introduced to estimate the maximum structural response. An example considering a narrow‐band excitation from the great Mexico earthquake of September 19, 1985, is given and the accuracy of the response spectrum formulation is examined. Copyright © 2010 John Wiley & Sons, Ltd.     

双线性单自由度体系强迫动力反应的Hilbert谱与本征振动模态:输入简谐波频率的影响

系统地研究了双线性单自由度体系在简谐波输入下表现出非线性力学行为时,输入简谐波频率对体系动力响应的Hilbert谱、Hilbert边缘谱及Fourier幅值谱的影响.研究结果表明,如果体系输入简谐波频率为f,那么体系动力响应Hilbert边缘谱的能量分布在f附近一个较宽的频带上,该频带的产生是体系动力响应Hilbert谱中所蕴含的波内调制的必然结果,它源自于体系某个本征振动模态瞬时频率的波动,而这种瞬时频率的波动描述了体系屈服与卸载的非线性力学行为;体系动力响应的Fourier幅值谱自3f起,每隔2f就会出现一个幅值明显高出周围其它分量的Fourier“伪”谐波分量,这也是体系非线性力学行为所造成的结果.     

A nonlinear SDOF model for the simplified evaluation of the displacement demand of low-rise URM buildings

The determination of displacement demands for masonry buildings subjected to seismic action is a key issue in the performance-based assessment and design of such structures. A technique for the definition of single-degree-of-freedom (SDOF) nonlinear systems that approximates the global behaviour of multi-degree-of-freedom (MDOF) 3D structural models has been developed in order to provide useful information on the dependency of displacement demand on different seismic intensity measures. The definition of SDOF system properties is based on the dynamic equivalence of the elastic properties (vibration period and viscous damping) and on the comparability with nonlinear hysteretic behaviour obtained by cyclic pushover analysis on MDOF models. The MDOF systems are based on a nonlinear macroelement model that is able to reproduce the in-plane shear and flexural cyclic behaviour of pier and spandrel elements. For the complete MDOF models an equivalent frame modelling technique was used. The equivalent SDOF system was modelled using a suitable nonlinear spring comprised of two macroelements in parallel. This allows for a simple calibration of the hysteretic response of the SDOF by suitably proportioning the contributions of flexure-dominated and shear-dominated responses. The comparison of results in terms of maximum displacements obtained for the SDOF and MDOF systems demonstrates the feasibility and reliability of the proposed approach. The comparisons between MDOF and equivalent SDOF systems, carried out for several building prototypes, were based on the results of time-history analyses performed with a large database of natural records covering a wide range of magnitude, distance and local soil conditions. The use of unscaled natural accelerograms allowed the displacement demand to be expressed as a function of different ground motion parameters allowing for the study of their relative influence on the displacement demand for masonry structures.     

Application of wavelet theory to identify yielding in seismic response of bi‐linear structures

The time–frequency and the time‐scale analysis methods are used in this paper to identify the dynamic characteristics of non‐linear seismic response of structural systems with single degree of freedom (SDOF) and multiple degrees of freedom (MDOF). Based on the floor acceleration response time histories of bi‐linear SDOF and MDOF structures, the current study compares the results of system identification using the short‐time Fourier transform (STFT), continuous wavelet transform (CWT) and discrete wavelet transform (DWT) methods. The aim is to identify the frequency variations and the time at on‐set of yielding and unloading of a bi‐linear structural system during seismic response. The results demonstrate that the CWT method is better than the STFT method in both time and frequency resolutions, and that the DWT method is the best at detecting the time at on‐set of yielding and unloading. Combining the results of CWT and DWT methods therefore provides accurate information of both frequency variations and yielding time in non‐linear seismic response. To alleviate the problems associated with noise‐contaminated signals, e.g. seismic response data recorded on site, the study suggests that low‐pass filtering be carried out before applying the DWT method to decompose the signals into multiple levels of details. Copyright © 2001 John Wiley & Sons, Ltd.     

云南漾濞地震大理某高层建筑结构地震响应观测数据初步分析

2021年5月21日21时21分至22时32分,云南漾濞先后发生了M5.6级、M6.4级、M5.0级和M5.2级地震,位于大理的某高层建筑结构地震反应观测台阵获取了这4次地震的结构动力响应,观测数据同步性好,数据质量高。该高层建筑为框架剪力墙结构,地上26层,地下1层,三分量加速度测点共8个分别位于建筑的第1层、4层、7层、10层、13层、17层、20层和25层,数据实时传输至中国地震局工程力学研究所燕郊数据中心。本文对结构台阵的观测记录进行了初步分析,绘制建筑结构观测楼层的三向绝对加速度及其傅里叶幅值谱,检验数据同步性和质量,通过滤波和积分得到相对速度和相对位移,利用功率谱方法分析得到频率响应函数,并利用复模态指数函数方法得到两水平方向前三阶模态频率和振型。通过4次地震结构模态频率和振型的初步对比结果表明:主体结构基本完好,这与现场调查结果吻合。该结构台阵获取的前震、主震和余震反应记录,为后续开展深入的模态参数分析、地震损伤识别以及研究框架剪力墙结构的振动特性和抗震性能提供了宝贵数据。     

延性需求谱在基于性能的抗震设计中的应用

基于性能的抗震设计理论涉及如何简便而合理地确定结构在指定强度地震下的弹塑性位移需求。本文给出了利用延性需求谱求解结构位移需求的一般步骤:借助模态Pushover分析将多自由度体系分解为几个非线性单自由度体系,以考虑各阶振型的影响;利用延性需求谱计算对应模态的等效单自由度体系的延性及位移需求,并以一定方式组合转化为多自由度体系位移需求。最后,通过算例分析表明:利用延性需求谱求解结构位移需求是一种具有一定精度可为工程接受的简便方法,在基于性能的抗震设计中具有较好的应用前景。     

线性SDOF体系共振动力响应的Hilbert谱分析

利用HHT方法分析了线性SDOF体系在共振状态下的强迫动力响应,系统地研究了线性SDOF体系在简谐波、线性调频波与正弦调频波输入下处于共振状态时结构动力响应的H ilbert谱与本征模态函数的特征,并且将H ilbert谱、Morlet小波谱及Fourier幅值谱所蕴含的物理意义及分辨率特性进行了比较。研究结果表明,与Fourier幅值谱相比,H ilbert谱在刻画非平稳时程的某些时变特征方面更加直观,其物理意义更加明确,而与Morlet小波谱相比H ilbert谱则具有较高的分辨率。     

Simulation of the response of base-isolated buildings under earthquake excitations considering soil flexibility

The accurate analysis of the seismic response of isolated structures requires incorporation of the flexibility of supporting soil.However,it is often customary to idealize the soil as rigid during the analysis of such structures.In this paper,seismic response time history analyses of base-isolated buildings modelled as linear single degree-of-freedom(SDOF) and multi degree-of-freedom(MDOF) systems with linear and nonlinear base models considering and ignoring the flexibility of supporting soil are conducted.The flexibility of supporting soil is modelled through a lumped parameter model consisting of swaying and rocking spring-dashpots.In the analysis,a large number of parametric studies for different earthquake excitations with three different peak ground acceleration(PGA) levels,different natural periods of the building models,and different shear wave velocities in the soil are considered.For the isolation system,laminated rubber bearings(LRBs) as well as high damping rubber bearings(HDRBs) are used.Responses of the isolated buildings with and without SSI are compared under different ground motions leading to the following conclusions:(1) soil flexibility may considerably influence the stiff superstructure response and may only slightly influence the response of the flexible structures;(2) the use of HDRBs for the isolation system induces higher structural peak responses with SSI compared to the system with LRBs;(3) although the peak response is affected by the incorporation of soil flexibility,it appears insensitive to the variation of shear wave velocity in the soil;(4) the response amplifications of the SDOF system become closer to unit with the increase in the natural period of the building,indicating an inverse relationship between SSI effects and natural periods for all the considered ground motions,base isolations and shear wave velocities;(5) the incorporation of SSI increases the number of significant cycles of large amplitude accelerations for all the stories,especially for earthquakes with low and moderate PGA levels;and(6) buildings with a linear LRB base-isolation system exhibit larger differences in displacement and acceleration amplifications,especially at the level of the lower stories.     

The non‐stationary cross‐correlation coefficients in the seismic analysis of MDOF structural systems

A method concerning the evaluation, in a very compact form, of the non‐stationary modal cross‐correlation coefficients of MDOF structural systems subjected to seismic excitations is presented. It is available both in the case when the excitation is considered as a white‐noise process and when it is considered as a filtered process. The evaluation of these coefficients is required when a transient seismic analysis is performed by the use of the modal response spectrum approach. This is necessary when the strong‐motion phase of the earthquake is significantly short with respect to the fundamental period of the structure. Copyright © 2003 John Wiley & Sons, Ltd.     

Mode-based equivalent multi-degree-of-freedom system for one-dimensional viscoelastic response analysis of layered soil deposit

Discrete models such as the lumped parameter model and the finite element model are widely used in the solution of soil amplification of earthquakes. However, neither of the models will accurately estimate the natural frequencies of soil deposit, nor simulate a damping of frequency independence. This research develops a new discrete model for one-dimensional viscoelastic response analysis of layered soil deposit based on the mode equivalence method. The new discrete model is a one-dimensional equivalent multi-degree-of-freedom (MDOF) system characterized by a series of concentrated masses, springs and dashpots with a special configuration. The dynamic response of the equivalent MDOF system is analytically derived and the physical parameters are formulated in terms of modal properties. The equivalent MDOF system is verified through a comparison of amplification functions with the available theoretical solutions. The appropriate number of degrees of freedom (DOFs) in the equivalent MDOF system is estimated. A comparative study of the equivalent MDOF system with the existing discrete models is performed. It is shown that the proposed equivalent MDOF system can exactly present the natural frequencies and the hysteretic damping of soil deposits and provide more accurate results with fewer DOFs.     

Performance spectra based method for the seismic design of structures equipped with passive supplemental damping systems

A new direct performance‐based design method utilizing design tools called performance‐spectra (P‐Spectra) for low‐rise to medium‐rise frame structures incorporating supplemental damping devices is presented. P‐Spectra are graphic tools that relate the responses of nonlinear SDOF systems with supplemental dampers to various damping parameters and dynamic system properties that structural designers can control. These tools integrate multiple response quantities that are important to the performance of a structure into a single compact graphical format to facilitate direct comparison of different potential solutions that satisfy a set of predetermined performance objectives under various levels of seismic hazard. An SDOF to MDOF transformation procedure that defines the required supplemental damping properties for the MDOF structure to achieve the response defined by the target SDOF system is also presented for hysteretic, linear viscous and viscoelastic damping devices. Using nonlinear time‐history analyses of idealized shear structures, the accuracy of the transformation procedure is verified. A seismic performance upgrade design example is presented to demonstrate the usefulness of the proposed method for achieving design performance goals using supplemental damping devices. Copyright © 2012 John Wiley & Sons, Ltd.