A mathematical model to predict the inelastic response of a steel frame: Establishment of parameters from shaking table experiments

The purpose of this research is to use data from experiments to formulate a mathematical model that will predict the non-linear response of a single-storey steel frame to an earthquake input. The process used in this formulation is system identification. In experiments performed on a shaking table, the frame was subjected to two earthquake motions at several intensities. In each case the frame underwent severe inelastic deformation. A computer program which incorporates the concepts of system identification makes use of the recorded data to establish four parameters in a non-linear mathematical model. When different amounts of data are used in the program, parameter sets are established which give the best model response for that amount of test data. The resulting sets of parameters reflect the way in which the properties of the structure change during the excitation. However, when the full durations of the different excitations are used, the sets of parameters are almost identical. For each of these sets of parameters, the correlation of the computed accelerations with the measured is excellent, and the shape of the computed displacement response compares very well with the measured response, although the permanent offset of the displacements is not computed exactly. Suggestions are given on how to overcome this deficiency in the mathematical model.     

俯冲带板内地震竖向位移谱阻尼修正系数模型研究

基于日本K-NET和KiK-net台网中4695条俯冲板内地震记录的竖向分量,建立了位移谱阻尼修正系数(DMF)模型.采用基于场地周期的场地分类方法,并通过固定效应法推导出模型系数.该DMF模型考虑了谱周期、阻尼比和场地条件的影响,可以用来调整与震源和距离无关的设计反应谱.利用阻尼比对数的二次方程式对原数据进行拟合回归...     

Analysis of the observed seismic response of a highway bridge

Strong-motion accelerograms obtained on the San Juan Bautista 156/101 Separation Bridge during the 6 August 1979 Coyote Lake, California, earthquake are used to examine the response of this multiple-span bridge to moderate levels of earthquake loading. Although the bridge was not damaged, the records are of significant engineering interest as they are the first to be recorded on a highway bridge structure in North America. A technique of system identification is used to determine optimal modal parameters for linear models which can closely replicate the observed time-domain seismic response of the bridge. Time variations in frequency and damping in the horizontal response are identified using a moving-window analysis. A three-dimensional finite element model is developed to study the bridge response in detail. The first two horizontal modal frequencies computed from this model are in excellent agreement with information obtained during the system identification analysis provided the finite element model's expansion joints are locked, preventing relative translational motions from occurring across the joints. Locking is confirmed by the observed seismic deformations of the structure in the fundamental mode. Fundamental vertical frequencies of the individual spans, predicted by the finite element model, are in very good agreement with ambient vibration test data.     

Modal identification of a centrifuge soil model using subspace state space method

In this paper, modal parameters of a layered soil system comprising of a soft clay layer overlying a dense sand layer are identified from accelerometer recordings in a centrifuge test. For the first time, the subspace state space system identification (4SID) method was employed to identify the natural frequencies, damping ratios, and complex valued mode shapes while considering the non-proportional damping in a soil system. A brief review of system identification concepts needed for application of the 4SID techniques to structural modal identification is provided in the paper. The identified natural frequencies were validated against those estimated by transfer function spectra. The computed normal mode shapes were compared with closed-form solutions obtained from the one-dimensional shear wave propagation equation. The identified modal parameters were then employed to synthesize state space prediction models which were subsequently used to simulate the soil response to three successive base motions. The identified models captured acceleration time-histories and corresponding Fourier spectra reasonably well in the small and moderate shaking events. In the stronger third shaking event, the model performed well at greater soil depths, but was less accurate near the surface where nonlinearities dominated.     

水平分层土层系统等效阻尼比的简化计算方法

在实际工程场地中,很多土层可视为水平分层,各层土的物理和力学性质存在差异,其中包括土的振动阻尼比。本文讨论水平分层土层系统的等效阻尼比的近似计算方法,基于5个不同的加权函数推导了10种等效阻尼比的计算公式。通过2个算例,分别以等效阻尼比为参数计算水平分层土层的地震反应,并与准确解相比较,分析了不同等效阻尼比近似计算方法的计算精度。数值结果表明,若等效阻尼比计算方法选择不恰当,会导致土层地震反应的计算结果出现较大误差。针对2种不同类型的水平分层土层,建议采用基于三角形分布的加权函数来计算土层系统的等效阻尼比。     

俯冲带板间地震竖向加速度谱的阻尼修正系数模型

我国台湾省和南海东部地区处于地质构造复杂和地震频发的俯冲带区域,这一地区的结构抗震设计影响着该地区的开发建设进程。为建立适用于该地区的俯冲带板间地震竖向加速度谱的阻尼修正系数模型,本文选用日本K-NET与KiK-net台网的3 552条俯冲带板间地震记录,分四类场地建立以阻尼比和谱周期为参数的阻尼修正系数模型,并分析其它未纳入模型的参数对误差的影响。结果表明：采用阻尼比对数的三次多项式表达式即可实现良好拟合;模型在短周期内的误差主要受场地效应与路径效应影响,在长周期和高阻尼比时误差主要受震源效应影响;分场地建立阻尼修正系数模型可有效降低场地效应产生的误差。本文的研究成果可用于求解俯冲带地区无震源效应与路径效应的竖向设计反应谱。     

Optimum tuned-mass dampers for minimizing steady-state response of support-excited and damped systems

A numerical searching procedure to find the optimum tuning frequency and damping ratio of the tuned-mass damper which can reduce the steady-state response of damped main systems to a minimum level is developed and applied to the two different harmonic excitation sources, support motion of fixed-displacement amplitude and support motion of fixed-acceleration amplitude. The explicit formulae for these optimum parameters are then derived through a sequence of curve-fitting schemes. It has been found that, as the error of the explicit formulae is negligible, they provide a convenient tool to compute the optimum parameters in engineering applications. The numerical results show that the tuned-mass damper is less effective in reducing the system's response when there is a high level of damping incorporated into the system. It is also found that the optimum tuning frequency is strongly influenced by the damping level of a system, especially in regard to the fixed-acceleration support motion, but the optimum damping ratio of the tuned-mass damper is not sensitive to the damping level of a system. The response of the damped system using the undamped optimum value as the damping of the tuned-mass damper is not much different from the response using the damped optimum value.     

Response‐only modal identification of structures using strong motion data

Dynamic characteristics of structures — viz. natural frequencies, damping ratios, and mode shapes — are central to earthquake‐resistant design. These values identified from field measurements are useful for model validation and health‐monitoring. Most system identification methods require input excitations motions to be measured and the structural response; however, the true input motions are seldom recordable. For example, when soil–structure interaction effects are non‐negligible, neither the free‐field motions nor the recorded responses of the foundations may be assumed as ‘input’. Even in the absence of soil–structure interaction, in many instances, the foundation responses are not recorded (or are recorded with a low signal‐to‐noise ratio). Unfortunately, existing output‐only methods are limited to free vibration data, or weak stationary ambient excitations. However, it is well‐known that the dynamic characteristics of most civil structures are amplitude‐dependent; thus, parameters identified from low‐amplitude responses do not match well with those from strong excitations, which arguably are more pertinent to seismic design. In this study, we present a new identification method through which a structure's dynamic characteristics can be extracted using only seismic response (output) signals. In this method, first, the response signals’ spatial time‐frequency distributions are used for blindly identifying the classical mode shapes and the modal coordinate signals. Second, cross‐relations among the modal coordinates are employed to determine the system's natural frequencies and damping ratios on the premise of linear behavior for the system. We use simulated (but realistic) data to verify the method, and also apply it to a real‐life data set to demonstrate its utility. Copyright © 2012 John Wiley & Sons, Ltd.     

Seismic response of bridge piers on pile groups for different soil damping models and lumped parameter representations of the foundation

This paper presents a wide parametric study aimed at elucidating the influence, on the computed seismic response of bridge piers, of two related aspects of the model: (1) the adoption of the classical hysteretic or the causal Biot's damping models for the soil and (2) the use of two different lumped parameter models of different complexity and accuracy to approximate the impedances of the pile foundation. A total of 2072 cases, including different superstructures, pile foundations, soil deposits, and seismic input signals, are studied. The results are presented so that the influence of the different parameters involved in the analysis can be assessed. From an engineering point of view, both lumped parameter models provide, in general, sufficiently low errors. The choice of the most adequate model for each case will depend not only on the configuration of the structure and the soil-foundation system but also on the assumed soil damping model, whose influence on the computed seismic responses is relevant in many cases. The nonphysical behaviour provided by the classical hysteretic damping model for the soil at zero frequency generates issues in the process of fitting the impedance functions. It is also found that larger deck displacements are predicted by Biot's model due to the higher damping at low frequencies provided by the classical hysteretic damping model.     

基于复模态的有限元模型修正算法

针对地下结构地震响应分析中无限地基辐射阻尼问题，引入复模态情况下的具有非简化的堆积阻尼矩阵的阻尼模型，并针对具有集中质量阵的阻尼模型提出了合并与质量有关的阻尼和堆积阻尼的思想，并据此提出了一种修正此类有限元模型的两步法，首先从复模态参数中提取实模态参数，采用基于模态残余力的识别算法修正刚度矩阵，然后根据复模态参数和已得的刚度矩阵来识别阻尼模型中的刚度参与系数和质量阻尼堆积阻尼联合矩阵。     

Control law for variable damping device defined by a non-linear differential equation

This paper proposes a non-linear control law for a variable damping device (VDD) aimed at reducing structural seismic responses. The VDD is attached to the structure by an auxiliary spring element composing a non-linear Maxwell element. The VDD's damping coefficient is adjusted to control the reactive internal force in the non-linear Maxwell element. A large controlled force is thus produced with little external power required to adjust the VDD's damping coefficient. The proposed control law defines the rate or increment of the VDD's damping coefficient at a certain moment by a differential equation or its discretized form. The controlled force vs. deformation relation plots parallelogram-like hysteretic curves, which indicates quick action and energy dissipation. Fundamental characteristics of an SDOF model with the VDD controlled by the proposed law are examined for impulse, sin and seismic excitations. The law for the SDOF model is extended to one for an MDOF model. The control effect for a 3DOF model is examined by numerical experiments. Copyright © 1999 John Wiley & Sons, Ltd.     

Experimental characterization,modeling and identification of the NEES‐UCSD shake table mechanical system

This paper proposes a simple conceptual mathematical model for the mechanical components of the NEES‐UCSD large high‐performance outdoor shaking table and focuses on the identification of the parameters of the model by using an extensive set of experimental data. An identification approach based on the measured hysteresis response is used to determine the fundamental model parameters including the effective horizontal mass, effective horizontal stiffness of the table, and the coefficients of the classical Coulomb friction and viscous damping elements representing the various dissipative forces in the system. The effectiveness of the proposed conceptual model is verified through a comparison of analytical predictions with experimental results for various tests conducted on the system. The resulting mathematical model will be used in future studies to model the mechanical components of the shake table in a comprehensive physics‐based model of the entire mechanical, hydraulic, and electronic system. Copyright © 2007 John Wiley & Sons, Ltd.     

A non-linear numerical model of the tuned liquid damper

The Tuned Liquid Damper (TLD) is modelled numerically as an equivalent tuned mass damper with non-linear stiffness and damping. These parameters are derived from extensive experimental results described in References 1 and 2. This Non-linear Stiffness and Damping (NSD) model captures the behaviour of the TLD system adequately under a variety of loading conditions. In particular, the NSD model incorporates the stiffness hardening property of the TLD under large amplitude excitation. Copyright © 1999 John Wiley & Sons, Ltd.     

基于土-结构体系对建筑结构响应的减震效果评估

提出一种基于土-结构体系地震记录的土-结构相互作用(SSI)的减震评估方法。该方法采用简化的SSI模型,通过系统辨识确定模型参数。将上部建筑结构地震反应的SSI减震效应分解为惯性相互作用和运动相互作用,同时还提出由惯性相互作用和运动相互作用单独降低结构响应的方法。将2011年东北地震太平洋沿岸期间两栋中层建筑用此方法进行分析,结果表明:当建筑物结构响应进入非弹性范围时,惯性相互作用的减震效果降低。     

Pipeline response to random ground motion by discrete model

The response of buried pipelines to random excitation by earthquake forces is obtained using a lumped mass model. The earthquake is considered as a stationary random process characterized by a power spectral density function (PSDF). The cross spectral density function between two random inputs along the length of the pipe is defined with the help of the local earthquake PSDF which is the same for all points, and a frequency dependent exponentially decaying (with distance) function. Soil resistance to dynamic excitation along the pipelength is obtained in an approximate manner with the help of frequency independent impedance functions derived from half-space analysis and Mindlin's static stresses within the soil due to point loads. The proposed method has the advantage that it can take into consideration the cross terms in soil stiffness and damping matrices and can consider any boundary condition that needs to be satisfied at the ends of the pipe. A parametric study is also made to show the influence of cross terms in the soil stiffness and damping matrices on the response of the pipe.     

小应变硬化模型(HSS)在Rayleigh波作用下场地响应分析中的应用

为研究冲击荷载或地震作用下产生的,以Rayleigh波为主的面波对浅层地表土体动力响应特征以及数值模拟中土层阻尼的设置方法,以厦门地区浅层的素填土及粉质黏土为研究对象,采用有限元动力分析,土体本构采用小应变硬化模型(HSS),利用模型本身的滞回环特性,输入变化的小应变参数,考察HSS模型的小应变参数对场地动力响应的影响,并与土体采用摩尔-库伦模型结合Rayleigh阻尼("MC+Rayleigh阻尼")的计算结果进行对比。研究表明:当采用带有滞回环的HSS模型时,波速随初始剪切模量Gref0的增大而增大,但振幅减小,残余变形量也有所减小;小应变参数γ0.7对波的影响较小;HSS模型能够给出残余变形量,而"MC+Rayleigh阻尼"由于本构模型为理想弹塑性模型,在卸载重加载条件下表现为纯弹性行为,无法反映出卸载重加载过程中塑性应变的积累及其累积阻尼效应;但HSS模型还不能够全面反映循环加载作用下塑性体积应变的累积,因此在考虑滞回阻尼的基础上,仍然建议借助Rayleigh阻尼来更加全面地模拟土体的实际阻尼特性。     

Identification of modal parameters of non‐classically damped linear structures under multi‐component earthquake loading

A method for parametric system identification of classically damped linear system in frequency domain is adopted and extended for non‐classically damped linear systems subjected up to six components of earthquake ground motions. This method is able to work in multi‐input/multi‐output (MIMO) case. The response of a two‐degree‐of‐freedom model with non‐classical damping, excited by one‐component earthquake ground motion, is simulated and used to verify the proposed system identification method in the single‐input/multi‐output case. Also, the records of a 10 storey real building during the Northridge earthquake is used to verify the proposed system identification method in the MIMO case. In this case, at first, a single‐input/multi‐output assumption is considered for the system and modal parameters are identified, then other components of earthquake ground motions are added, respectively, and the modal parameters are identified again. This procedure is repeated until all four components of earthquake ground motions which are measured at the base level of the building are included in the identification process. The results of identification of real building show that consideration of non‐classical damping and inclusion of the multi‐components effect of earthquake ground motions can improve the least‐squares match between the finite Fourier transforms of recorded and calculated acceleration responses. Copyright © 2006 John Wiley & Sons, Ltd.     

An identification procedure of soil profile characteristics from two free field acclerometer records

In this paper, an analytical, numerical and experimental approach for identifying soil profile characteristics by using system identification and free field records, is presented. First, a theoretical soil amplification function for two sites is defined and expressed in terms of the different parameters of the layers constituting the soil profiles (thickness, damping ratio, shear wave velocity and unit weight). Then, this function is smoothed with an analogous function obtained from experimental data by using the least squares minimization technique. The identification of the parameters is performed by solving, numerically, a non-linear optimisation problem. To demonstrate the numerical efficiency and the validity of this approach, two examples are treated. The first one consists in the identification of characteristics of a given uniform soil layer. The second example consists in the experimental validation of this approach with the data recorded within the Garner Valley Down Hole Array (GVDA). Finally, this approach is applied to identify, simultaneously, soil profile characteristics of sites from only a single soil acceleration record at free surface of each site. This procedure is utilised to identify soil profile characteristics of sites by using strong ground motions data recorded during the recent Boumerdes earthquake of May 21, 2003.     

基于脉冲响应数据的ARMA法建模以及模态参数识别

本文提出了基于脉冲响应数据的ARMA法建模以及模态参数识别的新方法。该方法利用单位脉冲响应函数与ARMA模型G reen函数等价的特点,通过脉冲响应函数来估计系统响应的自相关系数,然后建立推广的Yu le-W alker方程以求得ARMA模型自回归系数并进行参数识别。最后通过算例,对一个模拟系统进行了参数识别,对采样频率、识别精度、模型阶数之间的影响规律进行了分析。