Optimal weight absorber designs for vibrating structures exposed to random excitations

Optimal mass ratios that minimize the response of a laminated beam with an attached absorber are tabulated for various values of beam damping. The beam is treated as an equivalent one degree of freedom (1DOF) main system vibrating in the fundamental mode. The beam is subjected to Gaussian white noise force and Gaussian white noise base frame acceleration. Optimal absorber frequency ratios and absorber damping ratios have been tabulated by others; the results for the classical 1DOF main system with attached absorber suggest that the optimized non-dimensional response decreases monotonically as the mass ratio increases. However, to generalize this monotonic relation may lead to inappropriate conclusions. If we define a constraint such that an increase in absorber mass leads to a proportional decrease in available beam construction material, i.e. effectively the combined mass of the beam and absorber is minimized, then variations in the mass ratio will affect the beam's parameters such as mass, stiffness and damping. Since some of these parameters are used for non-dimensionalising the response, inspection of non-dimensional responses may in some cases lead to inappropriate conclusions. This paper shows the optimal mass ratios for minimizing the response of a structure exposed to earthquake or fluid flow type random excitations.     

Generation of critical stochastic earthquakes

A stochastic critical excitation is defined as that excitation with a given variance that maximizes the variance in the dynamic response of a system. A non-stationary filtered shot noise is used to develop a stochastic critical excitation model of an earthquake ground motion process, and the response statistics for a linear system are determined in both time and frequency domains. The sensitivity of response to several assumed earthquake pulse arrival rate functions is examined. Responses to recorded strong ground motion and to stochastic critical excitations with the same total energy are compared to assess the degree of conservatism in the procedure. An application of the procedure to seismic qualification of equipment is presented.     

确定土层地震最大反应的简化方法

本文将水平土层简化为剪切型多自由度线性体系,根据对土层地震反应的振型迭加法所得结果的统计分析,确定体系的传递函数。参考概率理论中有关求极值的办法,给出与功率谱密度相应的反应谱。同时,根据振型均方根的办法,给出土层的加速度、相对位移、剪应变和剪应力的最大反应值沿深度的分布。分析结果表明,文中所提供的方法与精确法所得的结果有很好的吻合。      

结构随机地震响应分析的复模态法

本文对多自由度基础平动结构随机地震响应问题进行了系统研究。针对用第1振型近似代表上部结构所得方程为非经典阻尼和非对称结构情况。用复模态法解耦。获得了以第1振型表示的结构地震响应的解析解。对单自由度体系。此解即为结构响应的精确解。本文方法也可用于带TMD减震结构等的随机地震响应分析与优化设计。     

The optimum design of a base isolation system with frictional elements

A linear multistorey structure with a seismic base isolation system consisting of rubber bearings and frictional elements is considered. The non-linear equations of motion are derived for the first mode vibration and the stochastic response to a white noise ground acceleration is determined. Based on this response, suitable objective functions are defined and the optimum design of the isolation system is performed. It is shown that a small amount offriction increases the effectiveness of the system compared with the same system but without frictional elements.     

Optimum absorber parameters for minimizing vibration response

Optimum parameters are determined for absorbers, which, when attached to one mass of a main system with two degrees of freedom, minimize the harmonic response of that mass. Comparison is made with the absorber parameters that are determined by treating the main system as an equivalent one degree-of-freedom system and using classical results. Close agreement is obtained if the ratio of the two natural frequencies of the main system is reasonably large. This is in agreement with the author's recent work on optimum absorber parameters which minimize the response of elastic bodies. The extension of the method to multi degree-of-freedom main systems is outlined. The conditions for which different values of these parameters are predicted when the response is minimized over narrow and broad frequency bands are determined.     

基础隔震结构随机地震响应分析的复模态法

本文对多自由度基础隔震结构的随机地震响应问题进行了系统研究，首先建立了运动方程，然后用第一振型将上部结构展开，针对所得方程为非经典阻尼、非对称质量和非对称刚度情况，用复模态法解耦，获得了以第一振型表示的结构地震响应的解析解，对单自由度体系，此解即为结构响应的精确解，从而建立了两自由度体系在任意非经典阻尼与非对称质量和刚度情况下随机地震响应解析解分析的一般方法。本文方法也可用于带TMD减震结构、无损伤“加层减震”加固结构的随机地震响应分析与优化设计。     

Asymmetric one‐storey elastic systems with non‐linear viscous and viscoelastic dampers: Simplified analysis and supplemental damping system design

Investigated is the accuracy in estimating the response of asymmetric one‐storey systems with non‐linear viscoelastic (VE) dampers by analysing the corresponding linear viscous system wherein all non‐linear VE dampers are replaced by their energy‐equivalent linear viscous dampers. The response of the corresponding linear viscous system is determined by response history analysis (RHA) and by response spectrum analysis (RSA) extended for non‐classically damped systems. The flexible and stiff edge deformations and plan rotation of the corresponding linear viscous system determined by the extended RSA procedure is shown to be sufficiently accurate for design applications with errors generally between 10 and 20%. Although similar accuracy is also shown for the ‘pseudo‐velocity’ of non‐linear VE dampers, the peak force of the non‐linear VE damper cannot be estimated directly from the peak damper force of the corresponding linear viscous system. A simple correction for damper force is proposed and shown to be accurate (with errors not exceeding 15%). For practical applications, an iterative linear analysis procedure is developed for determining the amplitude‐ and frequency‐dependent supplemental damping properties of the corresponding linear viscous system and for estimating the response of asymmetric one‐storey systems with non‐linear VE dampers from the earthquake design (or response) spectrum. Finally, a procedure is developed for designing non‐linear supplemental damping systems that satisfy given design criteria for a given design spectrum. Copyright © 2003 John Wiley & Sons, Ltd.     

Seismic response of linear and non‐linear asymmetric systems with non‐linear fluid viscous dampers

This investigation is concerned with the seismic response of one‐story, one‐way asymmetric linear and non‐linear systems with non‐linear fluid viscous dampers. The seismic responses are computed for a suite of 20 ground motions developed for the SAC studies and the median values examined. Reviewed first is the behaviour of single‐degree‐of‐freedom systems to harmonic and earthquake loading. The presented results for harmonic loading are used to explain a few peculiar trends—such as reduction in deformation and increase in damper force of short‐period systems with increasing damper non‐linearity—for earthquake loading. Subsequently, the seismic responses of linear and non‐linear asymmetric‐plan systems with non‐linear dampers are compared with those having equivalent linear dampers. The presented results are used to investigate the effects of damper non‐linearity and its influence on the effects of plan asymmetry. Finally, the design implications of the presented results are discussed. Copyright © 2005 John Wiley & Sons, Ltd.     

CRITICAL CROSS POWER SPECTRAL DENSITY FUNCTIONS AND THE HIGHEST RESPONSE OF MULTI-SUPPORTED STRUCTURES SUBJECTED TO MULTI-COMPONENT EARTHQUAKE EXCITATIONS

The highest response of multi-supported structures subjected to partially specified multi-component earthquake support motions is considered. The seismic inputs are modelled as incompletely specified vector Gaussian random processes with known autospectral density functions but unknown cross spectral densities and these unknown functions are determined such that the steady state response variance of a given linear system is maximized. The resulting cross power spectral density functions are shown to be dependent on the system properties, autospectra of excitation and the response variable chosen for maximization. It emerges that the highest system response is associated neither with fully correlated support motions, nor with independent motions, but, instead, specific forms of cross power spectral density functions are shown to exist which produce bounds on the response of a given structure. Application of the proposed results is demonstrated by examples on a ground based extended structure, namely, a 1578 m long, three span, suspension cable bridge and a secondary system, namely, an idealized piping structure of a nuclear power plant.     

Optimum absorber parameters for simple systems

In the classical problem a damped one degree-of-freedom absorber system is attached to a main system, which has one degree of freedom and is undamped. The optimum values of absorber stiffness and damping, which will minimize the resonant response of the main mass, are well known. In this paper the effect on these optimum conditions of light damping in the main system is studied. The authors show that optimum parameters for absorbers, which are attached to beams and plates, can be obtained simply and accurately from those for an equivalent one degree-of-freedom main system. This depends upon the concept of an effective mass for the elastic body and the representation of its response by the single relevant mode. It will be shown in a later paper that for more complex elastic bodies such as cylindrical shells, for which the natural frequencies are more closely spaced, these simple concepts do not predict accurately optimum absorber parameters.     

Earthquake response of elastic SDF systems with non‐linear fluid viscous dampers

The steady‐forced and earthquake responses of SDF systems with a non‐linear fluid viscous damper (FVD) are investigated. The energy dissipation capacity of the FVD is characterized by the supplemental damping ratio ζsd and its non‐linearity by a parameter designated α. It is found that the structural response is most effectively investigated in terms of ζ_sd and α because (1) these two parameters are dimensionless and independent, and (2) the structural response varies linearly with the excitation intensity. Damper non‐linearity has essentially no influence on the peak response of systems in the velocity‐sensitive spectral region, but differences up to 14% were observed in the other spectral regions. The structural deformation is reduced by up to 25% when ζ_sd= 5%; and by up to 60% when ζ_sd= 30%. Non‐linear FVDs are advantageous because they achieve essentially the same reduction in system responses but with a significantly reduced damper force. For practical applications, a procedure is presented to estimate the design values of structural deformation and forces for a system with non‐linear FVD directly from the design spectrum. It is demonstrated that the earthquake‐induced force in a non‐linear FVD can be estimated from the damper force in a corresponding system with linear FVD, its peak deformation, and peak relative velocity; however, the relative velocity should not be approximated by the pseudo‐velocity as this approximation introduces a large error in the damper force. Finally, a procedure is presented to determine the non‐linear damper properties necessary to limit the structural deformation to some design value or the structural capacity for a given design spectrum. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

Critical earthquake input power spectral density function models for engineering structures

The problem of determining optimal power spectral density models for earthquake excitation which satisfy constraints on total average power, zero crossing rate and which produce the highest response variance in a given linear system is considered. The solution to this problem is obtained using linear programming methods. The resulting solutions are shown to display a highly deterministic structure and, therefore, fail to capture the stochastic nature of the input. A modification to the definition of critical excitation is proposed which takes into account the entropy rate as a measure of uncertainty in the earthquake loads. The resulting problem is solved using calculus of variations and also within linear programming framework. Illustrative examples on specifying seismic inputs for a nuclear power plant and a tall earth dam are considered and the resulting solutions are shown to be realistic.     

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.     

Investigations into critical earthquake load models within deterministic and probabilistic frameworks

This paper deals with the determination of critical earthquake load models for linear structures subjected to single‐point seismic inputs. The primary objective of this study is to examine the realism in critical excitations and critical responses vis a vis the framework adopted for the study and constraints that these excitations are taken to satisfy. Two alternative approaches are investigated. In the first approach, the critical earthquake is expressed in terms of a Fourier series that is modulated by an enveloping function that imparts transient nature to the inputs. The Fourier coefficients are taken to be deterministic and are constrained to satisfy specified upper and lower bounds. Estimates on these bounds, for a given site, are obtained by analysing past earthquake records from the same site or similar sites. The unknown Fourier coefficients are determined such that the response of a given structure is maximized subjected to these bounds and additional constraints on intensity, peak ground acceleration, peak ground velocity and peak ground displacement. In the second approach, the critical earthquake is modelled as a partially specified non‐stationary Gaussian random process which is defined in terms of a stationary random process of unknown power spectral density (psd) function modulated by a deterministic envelope function. The input is constrained to possess specified variance and average zero crossing rate. Additionally, a new constraint in terms of entropy rate representing the expected level of disorder in the excitation is also imposed. The unknown psd function of the stationary part of the input is determined so that the response of a given structure is maximized. The optimization problem in both these approaches is solved by using sequential quadratic programming method. The procedures developed are illustrated by considering the seismic response of a tall chimney and an earth dam. It is concluded that the imposition of lower and upper bounds on Fourier coefficients in the first approach and constraints on amount of disorder in the second approach are crucial in arriving at realistic critical excitations. Copyright © 2002 John Wiley & Sons, Ltd.     

Maximum response of asymmetric structures subject to A multicomponent earthquake

A simple and general technique to obtain a six-component earthquake input motion of a rigid foundation using three-linear-components earthquake record is presented. Using the multicomponent input, the maximum structural response to an arbitrarily oriented earthquake is derived. The method is applied to determine the response spectrum for a two-d.o.f. laterally and torsionally coupled linear system. The effects of foundation geometry, structural properties and apparent wave velocity on the structural response are investigated. It is found that increasing the foundation size reduces the peak values of the input motion; this reduction is more pronounced for a square foundation than for a circular one.     

Earthquake response of a steel frame building

A study has been made of the response, during the San Fernando earthquake 9 February 1971, of the nine-storey steel frame Building 180, located at the California Institute of Technology, Jet Propulsion Laboratory, Pasadena. The analysis throws light on the actual dynamical properties of the building during the earthquake, and also demonstrates that it is possible, when the ground motion is specified, to make accurate predictions of building motions during moderate earthquakes by using a linear viscously damped model. Methods of evaluating the lower mode periods and damping ratios from the earthquake records are described and the values obtained are compared with results from dynamic testing before and after the earthquake and with the periods computed from computer models of the building. Although no structural damage occurred and computed stresses in the steel frame were less than yield stresses, the periods measured by an ambient vibration test after the earthquake were of the order of 10 per cent higher than the pre-earthquake values. The maximum periods during the earthquake were found to be about 30 per cent higher than the post-earthquake periods.     

高层建筑结构的抗震可靠度分析与优化设计

本文根据我们在文献２中给出的等效随机地震静力作用模型，紧密结合规范和利用我们在文献６中提出的结构体系可靠度分析的最弱失效模式法，提出了结构构件和体系“小震不坏”和“大震不倒”及结构体系在设计基准期内的抗震可靠度分析方法；重新校准了结构构件的目标可靠度指标；综合考虑结构造价和损失期望，提出了结构体系抗震目标可靠度的优化决策方法；分别给出了满足构件抗震目标可靠指标与同时满足构件和体系抗震目标可靠指标的     

Evaluation of modal pushover analysis using generic frames

An Erratum has been published for this article in Earthquake Engineering and Structural Dynamics 2003; 32:1795. The recently developed modal pushover analysis (MPA) has been shown to be a significant improvement over the pushover analysis procedures currently used in structural engineering practice. None of the current invariant force distributions accounts for the contribution of higher modes—higher than the fundamental mode—to the response or for redistribution of inertial forces because of structural yielding. By including the contributions of a sufficient number of modes of vibration (generally two to three), the height‐wise distribution of responses estimated by MPA is generally similar to the ‘exact’ results from non‐linear response history analysis (RHA). Although the results of the previous research were extremely promising, only a few buildings were evaluated. The results presented below evaluate the accuracy of MPA for a wide range of buildings and ground motion ensembles. The selected structures are idealized frames of six different heights: 3, 6, 9, 12, 15, and 18 stories and five strength levels corresponding to SDF‐system ductility factor of 1, 1.5, 2, 4, and 6; each frame is analysed for 20 ground motions. Comparing the median values of storey‐drift demands determined by MPA to those obtained from non‐linear RHA shows that the MPA predicts reasonably well the changing height‐wise variation of demand with building height and SDF‐system ductility factor. Median and dispersion values of the ratios of storey‐drift demands determined by MPA and non‐linear‐RHA procedures were computed to measure the bias and dispersion of MPA estimates with the following results: (1) the bias and dispersion in the MPA procedure tend to increase for longer‐period frames and larger SDF‐system ductility factors (although these trends are not perfect); (2) the bias and dispersion in MPA estimates of seismic demands for inelastic frames are usually larger than for elastic systems; (3) the well‐known response spectrum analysis (RSA), which is equivalent to the MPA for elastic systems, consistently underestimates the response of elastic structures, e.g. up to 18% in the upper‐storey drifts of 18‐storey frames. Finally, the MPA procedure is simplified to facilitate its implementation in engineering practice—where the earthquake hazard is usually defined in terms of a median (or some other percentile) design spectrum for elastic systems—and the accuracy of this simplified procedure is documented. Copyright © 2003 John Wiley & Sons, Ltd.