A new simplified approach for assessing nonlinear seismic response of plan-asymmetric structures considering soil-structure interaction

An innovative approximate method is presented to consider the plan asymmetry, nonlinear structural behaviour and soil-structure interaction (SSI) effects simultaneously. The proposed method so-called Flexible base 2DMPA (F2MPA) is an extension of 2 degrees of freedom modal pushover analysis (2DMPA) approach to consider foundation flexibility in seismic response analysis of plan asymmetric structures which itself were developed based on Uncoupled Modal Response History Analysis method for inelastic fixed-base asymmetric structures. In F2MPA for each mode shape using 2DMPA procedure, the elastic and inelastic properties of 2DOF modal systems corresponding to the fixed-base structure are initially derived. Then in each time step, displacements and inelastic restoring forces of the superstructure are computed from modal equations of the flexibly-supported structure. In each time step, the nonlinear secant stiffness matrix corresponding to the n-th MDOF modal equations of soil-structure system is updated using the corresponding modal 2DOF system of fixed-base structure. To update the transformed modal stiffness matrix of the SSI system, this matrix is partitioned and it is assumed that the non-linear variation of the superstructure can be estimated from the variation of modal stiffness matrix of the fixed-base structure. Accuracy of the proposed method was verified on an 8-story asymmetric-plan building under different seismic excitations. The results obtained from F2MPA method were compared with those obtained by nonlinear response history analysis of the asymmetric soil-structure system as a reference response. It was shown that the proposed approach could predict the results of the nonlinear time history analysis with a good accuracy. The main advantage of F2MPA is that this method is much less time-consuming and useful for the practical aims such as massive analysis of a nonlinear structure under different records with multiple intensity levels.     

Improved dynamic correction method in seismic analysis of both classically and non‐classically damped structures

A step‐by‐step approximate procedure taking into consideration high‐frequency modes, usually neglected in the modal analysis of both classically and non‐classically damped structures, is presented. This procedure can be considered as an extension of traditional modal correction methods, like the mode‐acceleration method and the dynamic correction method, which are very effective for structural systems subjected to forcing functions described by analytical laws. The proposed procedure, herein called improved dynamic correction method, requires two steps. In the first step, the number of differential equations of motion are reduced and consequently solved by using the first few undamped mode‐shapes. In the second step, the errors due to modal truncation are reduced by correcting the dynamic response and solving a new set of differential equations, formally similar to the original differential equations of motion. The difference between the two groups of differential equations lies in the forcing vector, which is evaluated in such a way as to correct the effects of modal truncation on applied loads. Copyright © 2001 John Wiley & Sons, Ltd.     

非经典振型分解法求混凝土房屋及其上钢塔的地震响应

钢结构与混凝土结构阻尼比不同,混凝土房屋与其顶上钢塔组成了非比例阻尼结构系统。本文用非经典振型分解法求解该类结构系统的线弹性地震响应,发现只用前几阶振型响应迭加的结果即可逼近直接积分法的精确度。     

Efficient mode superposition algorithm for seismic analysis of non-linear structures

A step-by-step integration method is proposed to compute within the framework of the conventional mode superposition technique the response of bilinear hysteretic structures subjected to earthquake ground motions. The method is computationally efficient because only a few modes are needed to obtain an accurate estimate of such a response, and because it does not require the use of excessively small time steps to avoid problems of accuracy or stability. It is developed on the basis that the non-linear terms in the equations of motion for non-linear systems may be considered as additional external forces, and the fact that by doing so such equations of motion can be interpreted as the equations of motion of an equivalent linear system, excited by a modified ground motion. These linear equations are then subjected to a conventional modal decomposition and transformed, as with linear systems, into a set of independent differential equations, each representing the system's response in one of its modes of vibration. To increase the efficiency of the method and account properly for the participation of higher modes, these independent equations are solved using the Nigam-Jennings technique in conjunction with the so-called mode acceleration method. The accuracy and efficiency of the method is verified by means of a comparative study with solutions obtained with a conventional direct integration method. In this comparative study, including only a few modes, the proposed method accurately predicts the seismic response of three two-dimensional frame structures, but requiring only, on an average, about 47 per cent less computer time than when the direct integration method is used.     

The extended N2 method taking into account higher mode effects in elevation

The N2 method has been extended in order to take into account higher mode effects in elevation. The extension is based on the assumption that the structure remains in the elastic range when vibrating in higher modes. The seismic demand in terms of displacements and storey drifts can be obtained by enveloping the results of basic pushover analysis and the results of standard elastic modal analysis. The approach is consistent with the extended N2 method used for plan‐asymmetric buildings. The proposed procedure was applied to three variants of three steel frame buildings used in the SAC project. The structural response was investigated for two sets of ground motions. Different ground motion intensities were used in order to investigate the influence of the magnitude of plastic deformations. The N2 results were compared with the results of nonlinear response‐history analysis, two other pushover‐based methods (modal pushover analysis (MPA) and modified MPA (MMPA)), and pushover analysis without consideration of higher modes. It was found that a considerable influence of higher modes on storey drifts is present at the upper part of medium‐and high‐rise structures. This effect is the largest in the case of elastic behaviour and decreases with ground motion intensity. The higher mode effects also depend on the spectral shape. The approximate methods (extended N2, MPA and MMPA) are able to provide fair estimates of response in the case of the test examples. Accuracy decreases with the height of the building, and with the intensity of ground motion. The N2 results are generally conservative. Copyright © 2011 John Wiley & Sons, Ltd.     

Towards a seismic design method for plane steel frames using equivalent modal damping ratios

A method for seismic design of plane steel moment resisting frames based on the use of equivalent modal damping ratios is developed. The method determines the design base shear of the structure through spectrum analysis using rationally obtained equivalent modal damping ratios instead of the crude strength reduction (behavior) factor. An equivalent linear structure, which retains the mass and initial stiffness of the original non-linear structure and takes into account geometrical non-linearity and inelasticity in the form of equivalent, time-invariant, modal damping ratios is established. The equivalent damping ratios for the first few significant modes are numerically computed by first iteratively forming a frequency response transfer function modulus until it satisfies certain smoothness criteria and then by solving a set of non-linear algebraic equations. Thus, design equations providing equivalent damping ratios as functions of period and allowable deformation and damage are constructed using extensive numerical data coming from plane steel moment resisting frames excited by various seismic motions. These equations can be used in conjunction with a design spectrum, appropriately constructed for high damping values, and modal synthesis tools to calculate the seismic design forces of the structure. The proposed method is illustrated by numerical examples. It is concluded that unlike the usual approach of seismic codes employing a single common value of the strength reduction factor value for all modes, the proposed approach working with deformation and damage dependent equivalent modal damping ratios leads to more accurate and deformation and damage controlled results.     

关于结构振型参与系数和振型贡献的分析

采用振型分解反应谱法求解多自由度弹性体系的地震反应时,为了在满足所需计算精度的前提下减少工作量,需要对振型数量进行合理的选择,而振型数的确定主要取决于结构各阶振型对总体反应的贡献。通过数学推导,对振型贡献及振型数量的选择问题进行了研究。首先,讨论了振型参与系数的性质,在此基础上给出了能够反映结构振型贡献参数的数学表达式,对这些参数的力学含义进行了解释,并给出了相关证明;其次,对有效质量法、振型位移控制法等基于不同振型贡献标准的确定振型数的方法进行了分析比较,指出了其合理性和不足。本文研究对进一步理解结构振型贡献和振型数的选择问题具有一定的理论意义。     

Simplified seismic analysis of asymmetric building systems

The paper reviews the uncoupled modal response history analysis (UMRHA) and modal pushover analysis (MPA) procedure in the analysis of asymmetric structures. From the pushover curves in ADRS format, showing the relationships of base shear versus roof translation and base torque versus roof rotation, a bifurcating characteristic of the pushover curves of an asymmetric structure is observed. A two‐degree‐of‐freedom (2DOF) modal stick is constructed using lump mass eccentrically placed at the end of beam which is connected with the column by a rotational spring. By converting the equation of motion of a whole structure into 2DOF modal equations, all of the elastic properties in the 2DOF modal sticks can be determined accurately. A mathematical proof is carried out to demonstrate that the 2DOF modal stick is consistent with the single‐degree‐of‐freedom (SDOF) modal stick at elastic state. The bifurcating characteristic of modal pushover curves and the interaction of modal translation and rotation can be considered rationally by this 2DOF modal stick. In order to verify the effectiveness of this proposed 2DOF modal stick, a two‐storey asymmetric building structure was analysed by the UMRHA procedure incorporating this novel 2DOF modal sticks (2DMPA) and conventional SDOF modal sticks (SDMPA), respectively. The analytical results are compared with those obtained by nonlinear response history analysis (RHA). It is illustrated that the accuracy of the rotational response histories obtained by 2DMPA is much better than those obtained by SDMPA. Consequently, the estimations of translational response histories on flexible side (FS) and stiff side (SS) of the building structure are also improved. Copyright © 2006 John Wiley & Sons, Ltd.     

A response spectrum method for random vibration analysis of mdf systems

A response spectrum method for stationary random vibration analysis of linear, multi-degree-of-freedom systems is developed. The method is based on the assumption that the input excitation is a wide-band, stationary Gaussian process and the response is stationary. However, it can also be used as a good approximation for the response to a transient stationary Gaussian input with a duration several times longer than the fundamental period of the system. Various response quantities, including the mean-squares of the response and its time derivative, the response mean frequency, and the cumulative distribution and the mean and variance of the peak response are obtained in terms of the ordinates of the mean response spectrum of the input excitation and the modal properties of the system. The formulation includes the cross-correlation between modal responses, which is shown to be significant for modes with closely spaced natural frequencies. The proposed procedure is demonstrated for an example structure that is subjected to an ensemble of earthquake-induced base excitations. Computed results based on the response spectrum method are in close agreement with simulation results obtained from time-history dynamic analysis. The significance of closely spaced modes and the error associated with a conventional method that neglects the modal correlations are also demonstrated.     

弹性能量半径演化谱及其在多自由度体系地震反应计算中的应用

目前用于结构抗震设计的反应谱仅能反映峰值反应,无法体现反应值随时间的变化。文中提出一种弹性能量半径演化谱,可反映线性单自由度体系弹性能量(即动能与弹性势能之和)随地震持时的变化,且其峰值近似等于结构峰值位移。文中给出了利用地震动演化功率谱得到该演化谱的方法并进一步发展了一种计算线性多自由度体系地震位移反应的新方法。通过两座框架结构的地震反应计算,将新方法与传统振型组合法及时程分析法的计算结果进行对比,发现对于振型稀疏结构,新方法计算结果与SRSS法接近;而对于振型密集结构,新方法计算结果较CQC法更精确,且避免了CQC法相关系数的复杂计算。     

Modal time history analysis of non-classically damped structures for seismic motions

The step-by-step modal time history integration methods are developed for dynamic analysis of non-classically damped linear structures subjected to earthquake-induced ground motions. Both the mode displacement and mode acceleration-based algorithms are presented for the calculation of member and acceleration responses. The complex-valued eigenvectors are used to effect the modal decoupling of the equations of motion. However, the recursive step-by-step algorithms are still in terms of real quantities. The numerical results for the acceleration response and floor response spectra, obtained with these approaches, are presented. The mode acceleration approach is observed to be decidedly better than the mode displacement approach in as much as it alleviates the so-called missing mass effect, caused by the truncation of modes, very effectively. The utilization of the mode acceleration-based algorithms is, thus, recommended in all dynamic analyses for earthquake-induced ground motions.     

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 2: Complex modes

A method, based on the Hilbert–Huang spectral analysis, has been proposed by the authors to identify linear structures in which normal modes exist (i.e., real eigenvalues and eigenvectors). Frequently, all the eigenvalues and eigenvectors of linear structures are complex. In this paper, the method is extended further to identify general linear structures with complex modes using the free vibration response data polluted by noise. Measured response signals are first decomposed into modal responses using the method of Empirical Mode Decomposition with intermittency criteria. Each modal response contains the contribution of a complex conjugate pair of modes with a unique frequency and a damping ratio. Then, each modal response is decomposed in the frequency–time domain to yield instantaneous phase angle and amplitude using the Hilbert transform. Based on a single measurement of the impulse response time history at one appropriate location, the complex eigenvalues of the linear structure can be identified using a simple analysis procedure. When the response time histories are measured at all locations, the proposed methodology is capable of identifying the complex mode shapes as well as the mass, damping and stiffness matrices of the structure. The effectiveness and accuracy of the method presented are illustrated through numerical simulations. It is demonstrated that dynamic characteristics of linear structures with complex modes can be identified effectively using the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

An incremental mode-superposition for non-linear dynamic analysis

This paper uses an incremental mode-superposition procedure to compute the inelastic dynamic response of multi-degree-of-freedom systems. A damping matrix proportional to the instantaneous properties is used throughout the analysis. The non-linear response of several shear type plane and space frames with elastic-plastic and bilinear column properties subjected to ground excitation was computed by both the incremental mode-superposition and the direct integration of the coupled equations of motion. When all modes are considered, the responses computed by the incremental mode-superposition are identical to those from the direct integration. Fewer modes can also be used to compute the response with reasonable accuracy by performing the modal truncation for each time increment. The study shows that incorporating instantaneous damping in non-linear dynamic analysis is relatively simple and requires less computational time than the direct integration.     

The extended N2 method considering higher mode effects in both plan and elevation

The extended N2 method has been developed, which takes into account higher mode effects both in plan and in elevation. The extension is based on the assumption that the structure remains in the elastic range when vibrating in higher modes. The seismic demand in terms of displacements and storey drift can be obtained by combining the results of basic pushover analysis and those of standard elastic modal analysis. In the paper, the proposed procedure was summarized and applied to a test example which represents an actual 8-storey reinforced concrete building. The results obtained by the extended N2 method were compared with the results of nonlinear response history analysis and basic N2 analysis without the consideration of higher modes. The extended N2 method was able to provide fair, conservative estimates of response in the case of the test example. In comparison to the basic N2 method, the prediction of seismic demand was greatly improved in the upper part of the building and at the flexible edges.     

Earthquake analysis of gravity dams including hydrodynamic interaction

A general procedure for analysis of the response of gravity dams, including hydrodynamic interaction and compressibility of water, to the transverse horizontal and vertical components of earthquake ground motion is presented. The problem is reduced to one in two dimensions considering the transverse vibration of a monolith of a dam, and the material behaviour is assumed to be linearly elastic The complete system is considered as composed of two substructures—the dam, represented as a finite element system, and the reservoir, as a continuum of infinite length in the upstream direction governed by the wave equation. The structural displacements of the dam (including effects of water) are expressed as a linear combination of the modes of vibration of the dam with the reservoir empty. The effectiveness of this analytical formulation lies in its being able to produce excellent results by considering only the first few modes. The complex frequency response for the modal displacements are obtained first. The responses to arbitrary ground motion are subsequently obtained with the aid of the Fast Fourier Transform algorithm An example analysis is presented to illustrate results obtained from this method. It is concluded that the method is very effective and efficient and is capable of producing results to any desired degree of accuracy by including the necessary number of modes of vibration of the dam.     

Seismic response analysis of highway overcrossings including soil–structure interaction

This paper presents a systematic procedure for the seismic response analysis of highway overcrossings. The study employs an elementary stick model and a more sophisticated finite element formulation to compute response quantities. All dynamic stiffnesses of approach embankments and pile groups are approximated with frequency‐independent springs and dashpots that have been established elsewhere. A real eigenvalue analysis confirms the one‐to‐one correspondence between modal characteristics obtained with the three‐dimensional finite element solutions and the result of the simpler stick‐model idealization. A complex eigenvalue analysis yields modal damping values in the first six modes of interest and shows that modal damping ratios assume values much higher than those used by Caltrans. The validity of the proposed method is examined by comparing the computed time response quantities with records from the Meloland Road and Painter Street overcrossings located in southern and northern California, respectively. The proposed procedure allows for inexpensive parametric analysis that examines the importance of considering soil–structure interaction at the end abutments and centre bent. Results and recommendations presented by past investigations are revisited and integrated in comprehensive tables that improve our understanding of the dynamic characteristics and behaviour of freeway overcrossings. The study concludes with a step‐by‐step methodology that allows for a simple, yet dependable dynamic analysis of freeway overcrossings, that involves a stick model and frequency‐independent springs and dashpots. Copyright © 2002 John Wiley & Sons, Ltd.     

Modal damping for torsionally coupled buildings on elastic foundation

A method to determine the approximate normal modes and the modal damping for torsionally coupled buildings on an elastic foundation is presented. The modal damping is determined by an iterative procedure which matches the approximate normal mode solution with the rigorous solution. The response quantity to be matched is selected in a consistent and logical manner. The normal modes and the damping ratios thus found are then used to determine the seismic response of the interaction system by the response spectrum technique.     

A modal pushover analysis procedure to estimate seismic demands for unsymmetric‐plan buildings

An Erratum has been published for this article in Earthquake Engng. Struct. Dyn. 2004; 33:1429. Based on structural dynamics theory, the modal pushover analysis (MPA) procedure retains the conceptual simplicity of current procedures with invariant force distribution, now common in structural engineering practice. The MPA procedure for estimating seismic demands is extended to unsymmetric‐plan buildings. In the MPA procedure, the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by non‐linear static analysis using the inertia force distribution for each mode, which for unsymmetric buildings includes two lateral forces and torque at each floor level. These ‘modal’ demands due to the first few terms of the modal expansion are then combined by the CQC rule to obtain an estimate of the total seismic demand for inelastic systems. When applied to elastic systems, the MPA procedure is equivalent to standard response spectrum analysis (RSA). The MPA estimates of seismic demand for torsionally‐stiff and torsionally‐flexible unsymmetric systems are shown to be similarly accurate as they are for the symmetric building; however, the results deteriorate for a torsionally‐similarly‐stiff unsymmetric‐plan system and the ground motion considered because (a) elastic modes are strongly coupled, and (b) roof displacement is underestimated by the CQC modal combination rule (which would also limit accuracy of RSA for linearly elastic systems). Copyright © 2004 John Wiley & Sons, Ltd.     

Efficient modal analysis for structures with soil-structure interaction

An efficient methodology is presented which uses modal analysis implemented in the frequency domain to obtain the structural response of a system with soil-structure interaction. The interaction effects are represented using a free-field ground motion modification factor, derived for each mode of vibration and used in the determination of structural response. Applying this algorithm, the advantages of the modal superposition method are fully exploited, and the interaction problem can be solved easily and effectively within the framework of the conventional frequency domain analysis for a fixed-base structure. In addition, this method produces accurate approximation with less computational effort due to consideration of only the first few vibration modes of the structure.     

Pseudodynamic response of torsionally unbalanced two‐storey test structure

Two one‐way eccentric, two‐storey, one‐by‐one‐bay reinforced concrete (RC) structures are pseudodynamically tested under unidirectional ground motions. Theoretical considerations about the effect of torsional coupling on modal periods and shapes agree with modal results of the test structure, considering member stiffness is equal to the secant stiffness to yielding in skew‐symmetric bending. Modal periods of such an elastic structure are in fair agreement with effective periods inferred from the measured response at the beginning of a test of a thoroughly cracked structure and at the end of the test. A time‐varying stiffness matrix and a non‐proportional damping matrix fitted to the test results may be used to reproduce the measured response approximately by modal superposition and identify the role of the four time‐varying modes. Flexible side columns sustained very large drift demands simultaneously in the two transverse directions and suffered significant but not heavy, damage at lap‐splices. RC‐jacketing of the flexible side columns practically eliminated the static eccentricity between the floor centres of twist and mass as well as the torsional response. Inelastic time‐history analysis with point‐hinge member models, using as elastic stiffness the secant stiffness to yielding and neglecting post‐ultimate‐strength cyclic degradation of resistance in members with plain bars and poor detailing, predicted fairly well the response until the peak displacements and member deformations occurred. After that, it underestimated displacement peaks and the lengthening of the apparent period and missed the gradual drifting of the response towards a permanent offset. Copyright © 2007 John Wiley & Sons, Ltd.