Prediction of the largest peak nonlinear seismic response of asymmetric buildings under bi-directional excitation using pushover analyses

A simplified procedure is proposed to predict the largest peak seismic response of an asymmetric building to horizontal bi-directional ground motion, acting at an arbitrary angle of incidence. The main characteristics of the proposed procedure is as follows. (1) The properties of two independent equivalent single-degree-of-freedom models are determined according to the principal direction of the first modal response in each nonlinear stage, rather than according to the fixed axis based on the mode shape in the elastic stage; the principal direction of the first modal response in each nonlinear stage is determined based on pushover analysis results. (2) The bi-directional horizontal seismic input is simulated as identical spectra of the two horizontal components, and the contribution of each modal response is directly estimated based on the unidirectional response in the principal direction of each. (3) The drift demand at each frame is determined based on four pushover analyses considering the combination of bi-directional excitations. In the numerical example, nonlinear time-history analyses of six four-story torsionally stiff (TS) asymmetric buildings are carried out considering various directions of seismic inputs, and these results are compared with the predicted results. The results show that the proposed procedure satisfactorily predicts the largest peak response displacement at the flexible-side frame of a TS asymmetric building.     

Analysis of three-dimensional strong ground motions along principal axes,San Fernando earthquake

An orthogonal set of principal axes is defined for earthquake ground motions. These principal axes are obtained such that the corresponding variances of motion have maximum, minimum and intermediate values and the covariances equal zero. This indicates that the corresponding components of motion along the principal axes are uncorrelated with respect to each other. Since real earthquake accelerograms are assumed to be reasonably well represented by Gaussian random processes, the three components of motion along the principal axes are statistically independent of each other. Using these principal axes and applying the moving-window technique to the ground accelerograms recorded during the San Fernando earthquake of 9 February 1971, time-dependent characteristics of three-dimensional ground motions along principal axes are determined. Results of the analysis indicate significant correlation between directions of principal axes and directions from the recording stations to the fault slip zone. It is concluded that three components of ground motion can be generated stochastically with statistical independence being maintained, provided they are assumed to be directed along principal axes.     

随机地震动场多点激励下大跨度桥梁地震反应分析方法

地震输入问题一直是工程结构抗震研究关注的焦点。对大跨度桥梁结构，考虑随机地震动场的多点激励而进行地震反应分析较为合理。本文结合大跨度桥梁抗震设计，系统地介绍了随机地震动场的模型以及随机地震动场多点激励下大跨度桥梁地震反应分析的方法。     

An envelope for Mohr's circle in seismically excited three‐dimensional structures

A previously developed response‐spectrum‐based procedure for computing the envelope that bounds the time‐varying realizations of Mohr's circle at any prescribed location within a two‐dimensional structure is extended for use with three‐dimensional structures subjected to as many as three translational components of ground acceleration. The proposed envelope, which is completely defined by quantities that are routinely used and calculated in conventional response spectrum analyses, is developed for the general case in which the principal directions of the earthquake, along which the ground accelerations are uncorrelated, are unknown. The accuracy of the proposed envelope is evaluated by comparing it to the results of an ensemble of time‐history analyses performed on a concrete arch dam using simulated accelerograms. It is found that the proposed envelope has a level of accuracy that is suitable for structural design and analysis. The largest observed difference between the simulated and predicted mean envelopes is less than 5%. Copyright © 2004 John Wiley & Sons, Ltd.     

Principal axes of M-DOF structures Part II: Dynamic loading

This paper is the second in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and response modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended response modification devices/systems, and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading. In this two-part series, the first paper is concerned with static loading, which provides definitions and fundamental formulations, with the conclusion that cross effects of a statically loaded M-DOF structure resulting from the lack of principal axes are of insignificant magnitude. However, under dynamic or earthquake loading, a relatively small amount of energy transferred across perpendicular directions is accumulated, which may result in significant enlargement of the structural response. This paper deals with a formulation to define the principal axes of M-DOF structures under dynamic loading and develops quantitative measures to identify cross effects resulting from the non-existence of principal axes.     

Principal axes of M-DOF structures Part Ⅱ: Dynamic loading

This paper is the second in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and response modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended response modification devices/systems, and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading. In this two-part series, the first paper is concerned with static loading, which provides definitions and fundamental formulations, with the conclusion that cross effects of a statically loaded M-DOF structure resulting from the lack of principal axes are of insignificant magnitude. However, under dynamic or earthquake loading, a relatively small amount of energy transferred across perpendicular directions is accumulated, which may result in significant enlargement of the structural response. This paper deals with a formulation to define the principal axes of M-DOF structures under dynamic loading and develops quantitative measures to identify cross effects resulting from the non-existence of principal axes.     

Analysis of bridge structures crossing strike-slip fault rupture zones: A simple method for generating across-fault seismic ground motions

This article presents a simple and effective method for generating across-fault seismic ground motions for the analysis of ordinary and seismically isolated bridges crossing strike-slip faults. Based on pulse models available in the literature, two simple loading functions are first proposed to represent the coherent (long-period) components of ground motion across strike-slip faults. The loading functions are then calibrated using actual near-fault ground-motion records with a forward-directivity velocity pulse in the fault-normal direction and a fling-step displacement in the fault-parallel direction. The effectiveness of the proposed method is demonstrated by comparing time history responses and seismic demands of ordinary and seismically isolated bridges obtained from nonlinear response history analyses using the actual ground-motion records and the calibrated loading functions. A comprehensive methodology is also presented for selecting the input parameters of the loading functions based on empirical equations and practical guidelines. Finally, an analysis procedure for bridge structures crossing strike-slip faults is introduced based on the proposed method for generating across-fault ground motions and the parameter selection methodology for the loading functions.     

Evaluation of combination rules for peak response calculation in three‐component seismic analysis

The responses, r_e, given by several multicomponent combination rules used in seismic codes for determining peak responses to three ground motion components are evaluated for elastic systems and compared with the critical response r_cr; this is defined as the largest response for all possible incident angles of the seismic components and obtained by means of the CQC3‐rule when a principal seismic component is vertical, or the GCQC3‐rule when it departs from the vertical direction. The combination rules examined are the SRSS‐, 30%‐, 40%‐ and IBC‐rules, considering different alternatives for the design horizontal spectrum. Assuming that a principal seismic component is along the vertical direction, the upper and lower bounds of the ratio r_e/r_cr for each combination rule are determined as a function of the spectral intensity ratio of the horizontal seismic components and of the responses to one seismic component acting alternately along each structural axis. Underestimations and overestimations of the critical response are identified for each combination rule and each design spectrum. When a component departs from the vertical direction, the envelopes of the bounds of the ratio r_e/r_cr for each combination rule are calculated, considering all possible values of the spectral intensity ratios. It is shown that the inclination of a principal component with respect to the vertical axis can significantly reduce the values of r_e/r_cr with respect to the case when the component is vertical. Copyright © 2003 John Wiley & Sons, Ltd.     

Characteristics of horizontal ground motion measures along principal directions

<正>Ground motion records are often used to develop ground motion prediction equations(GMPEs) for a randomly oriented horizontal component,and to assess the principal directions of ground motions based on the Arias intensity tensor or the orientation of the major response axis.The former is needed for seismic hazard assessment,whereas the latter can be important for assessing structural responses under multi-directional excitations.However,a comprehensive investigation of the pseudo-spectral acceleration(PSA) and of GMPEs conditioned on different axes is currently lacking.This study investigates the principal directions of strong ground motions and their relation to the orientation of the major response axis, statistics of the PSA along the principal directions on the horizontal plane,and correlation of the PSA along the principal directions on the horizontal plane.For these,three sets of strong ground motion records,including intraplate California earthquakes,inslab Mexican earthquakes,and interface Mexican earthquakes,are used.The results indicate that one of the principal directions could be considered as quasi-vertical.By focusing on seismic excitations on the horizontal plane,the statistics of the angles between the major response axis and the major principal axis are obtained;GMPEs along the principal axes are provided and compared with those obtained for a randomly oriented horizontal component;and statistical analysis of residuals associated with GMPEs along the principal directions is carried out.     

Evaluation of combination rules for maximum response calculation in multicomponent seismic analysis

The existing rules for combining peak response to individual components of ground motion are evaluated. The response values r_e to two horizontal components of ground motion estimated by four multicomponent combination rules—SRSS‐, 30%‐, 40%‐ and simplified‐SRSS‐rules—are compared with the critical response, r_cr, obtained by the CQC3‐rule, which takes into account the direction of the principal ground components with respect to the structural axes and provides the largest response over all possible seismic incident angles. The following results are obtained in the first part of the paper and are valid for any elastic structure and any earthquake design response spectrum: For realistic values of the ratio γ of the design spectra for the two principal components of ground motion the SRSS‐rule estimate lies between 0.79r_cr and 1.00r_cr, the Simplified‐SRSS‐rule estimate lies between 1.00r_cr and 1.26r_cr, the 40%‐rule estimate lies between 0.99r_cr and 1.25r_cr, and the 30%‐rule estimate lies between 0.92r_cr and 1.16r_cr. None of the multicomponent combination rules account for the increase in response of systems if the vibration periods of the two modes that contribute most to the response to the x‐ and y‐components of ground motion are close to each other. Evaluated in the second part of the paper is the accuracy of the multicomponent combination rules in estimating the response of a range of one‐storey systems with (a) symmetrical plan and (b) unsymmetrical plan, and of two multistorey buildings. The SRSS‐rule underestimates the response by up to 16% and the other three rules overestimate it by up to 18%. Although these errors appear to be smaller than the many approximations inherent in structural design, they can be eliminated with very little additional computation by using an explicit formula for the critical response based on the CQC3 rule. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

Peak response of 2-DOF uncoupled oscillator under two-directional base motion

This paper investigates the peak response of a 2-DOF uncoupled linear oscillator under arbitrarily oriented two-component horizontal earthquake motion. Input data consist of 22 two-component acceleration records from the magnitude 6.6, October 15, 1979 Imperial Valley earthquake. One-directional response spectra for each principal direction of the oscillator, and two-directional response spectra (amplitude of the peak resultant response and its direction with respect to principal directions) are calculated by considering all angles of incidence of the excitation. The results indicate that, for oscillators that are not too flexible (i.e. with natural frequencies greater than 0.8 Hz), the two-directional peak response is controlled by the weaker direction of the oscillator. For flexible oscillators the two-directional peak response is controlled by the direction whose natural frequency is closer to dominant ground motion frequencies. The results also show that when the two-directional peak response is expressed as a weighted sum of one-directional peak responses, the weighting factors are not constants, but functions of oscillator and dominant ground motion frequencies.     

Response to orthogonal components of ground motion and assessment of percentage combination rules

The effects of horizontal components of ground motion on the linear response of torsionally stiff and torsionally flexible systems, on soft and firm soil conditions, are examined. A one‐story, two‐way asymmetric structural system is used, subjected to uncorrelated ground motion components along their principal directions. Spectral densities for ground accelerations in firm and soft soils are modeled based on recorded data from large intensity Mexican earthquakes. It is shown that for firm soils, in general, these effects are important in the case of torsionally flexible systems that are stiff under translation, or for torsionally stiff systems that are flexible in translation. The percentage combination rules usually specified in seismic design codes are assessed against the dynamic response. Such combination rules can result in overly conservative design forces or underestimated design forces, particularly for torsionally flexible structures. Given the relative magnitude of the response to each ground motion component, it was found that using different percentage values in the combination rules has no significant effect on improving the estimation of the total response. Copyright © 2003 John Wiley & Sons, Ltd.     

Inelastic seismic behavior of soil–pile raft–structure system under bi-directional ground motion

Performance based design of structure requires a reasonably accurate prediction of displacement or ductility demand. Generally, displacement demand of structure is estimated assuming fixity at base and considering base motion in one direction. In reality, ground motions occur in two orthogonal directions simultaneously resulting in bidirectional interaction in inelastic range, and soil–structure interaction (SSI) may change structural response too. Present study is an attempt to develop insight on the influence of bi-directional interaction and soil–pile raft–structure interaction for predicting the inelastic response of soil–pile raft–structure system in a more reasonably accurate manner. A recently developed hysteresis model capable to simulate biaxial interaction between deformations in two principal directions of any structural member under two orthogonal components of ground motion has been used. This study primarily shows that a considerable change may occur in inelastic demand of structures due to the combined effect of such phenomena.     

Principal axes of M-DOF structures part I: Static loading

This paper is the first in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and control modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended control devices/systems, and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects - of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading. This first paper deals with quantitative definitions of principal axes and “cross effects” of three-dimensional structures under static load by using linear algebra. It shows theoretically that, for three-dimensional structures, such principal axes rarely exist. Under static loading conditions, the cross effect is typically small and negligible from the viewpoint of engineering applications. However, it provides the theoretical base for subsequent quantification of the response couplings under dynamic loads, which is reported in part II of this series.     

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.     

空间网格结构多维多点随机地震响应分析的高效算法

将林家浩教授提出的“虚拟激励法”进一步推广应用于空间网格结构多维多点非平稳随机地震响应分析,推导了多维虚拟激励随机振动分析方法的理论公式,给出了峰值响应估计方法,并讨论了多维地震动的随机模型及参数选取,通过编制的专用计算机程序分析了网壳结构的随机地震响应。本方法自动包含了参振振型间及各输人地震分量间的相关项,计算精确、快速,非常适合分析频率密集型空间网格结构的随机地震响应,是一种高效的随机振动分析算法。     

双向地震动作用的拟等延性系数谱

首先建立了以强度折减系数表述的恢复力特性满足二维屈服面模型的理想弹塑性单质点系统(它在2个相互垂直的主轴方向上分别具有水平平动自由度)在双向地震动作用下的归一化运动方程。然后引入单向地震动作用下等延性系数的强度折减系数谱,给出了双向地震动作用的拟等延性系数谱(定义为系统分别承受双向和单向地震动作用,在同一主轴方向上的最大位移反应之比)最后通过硬土场地10组双向地震动记录拟等延性系数谱的统计平均结果,分析了结构周期、位移延性系数和阻尼等因素对谱值及结构双向地震反应的影响。结果表明,双向地震动作用与单向地震动作用相比主要增加结构较长周期方向的最大位移反应。若在基于位移的抗震设计中降低结构较短周期方向的设计位移延性系数,可在一定程度上降低双向地震动的不利影响。因定义的谱为比值形式,阻尼对其影响不大。     

Principal axes of m-DOF structures PartⅠ:Static loading

This paper is the first in a two-part series that discusses the principal axes of M-DOF structures subjected to static and dynamic loads. The primary purpose of this series is to understand the magnitude of the dynamic response of structures to enable better design of structures and control modification devices/systems. Under idealized design conditions, the structural responses are obtained by using single direction input ground motions in the direction of the intended control devices/systems,and by assuming that the responses of the structure is decoupleable in three mutually perpendicular directions. This standard practice has been applied to both new and retrofitted structures using various seismic protective systems. Very limited information is available on the effects of neglecting the impact of directional couplings (cross effects - of which torsion is a component) of the dynamic response of structures. In order to quantify such effects, it is necessary to examine the principal axes of structures under both static and dynamic loading.This first paper deals with quantitative definitions of principal axes and "cross effects" of three-dimensional structures under static load by using linear algebra. It shows theoretically that, for three-dimensional structures, such principal axes rarely exist. Under static loading conditions, the cross effect is typically small and negligible from the viewpoint of engineering applications. However, it provides the theoretical base for subsequent quantification of the response couplings under dynamic loads, which is reported in part Ⅱ of this series.     

Analysis of the spatial variation of seismic waves and ground movements from smart-1 array data

Specially designed arrays of strong motion seismographs located near earthquake sources are required for engineering studies of the near-source properties and the spatial variation of seismic waves. The SMART-1 array in Taiwan provides good records for this type of study. Careful study of the observed strong motion data permits the identification of wave types, directions and apparent wave velocities. In this paper, a principal direction ratio R (f,α) is defined; this indicates the principal direction of the motion (along a nearly straight line) within the range 0 < R < 1. Vertical motion of the ground is also included in this study. Orbit spectrum analysis is used to verify the identification of wave directions and wave types. The spatial variation of seismic waves along the principal direction is studied. From frequency-domain analysis, mathematical models of the spatial variation of ground displacement are developed using a wave-number spectrum and the cross-spectral density function between two spatial coordinates; these models in turn can provide two alternative models for the random vibration analysis of extensive structures subject to multiple point seismic excitation. The SMART-1 array data gathered during the January 29, 1981 earthquake also are used to demonstrate calculation of the ground strains and differential movements of the array site. From time-domain analysis, the spatial variation of seismic waves is defined for ground motion along the identified principal direction. The time variation of evolutionary spectra characterized by frequency-dependent parameters is used for this formulation. The SMART-1 array data again form the basis for discussion of the spatial variation of model parameters.