Non-classical damping in dynamic analysis of base-isolated structures with internal equipment

It has been shown that the use of base isolation not only attenuates the response of a primary structural system but also reduces the response of a secondary system mounted on or within the main structure. The isolation system, superstructure and equipment may be made of different materials with significantly different energy dissipation characteristics such that the damping matrix for the combined system is non-classical and can only be approximately expressed by modal damping ratios if the classical mode method is used for analysis. The object of this paper is to evaluate the accuracy of this procedure in approximating the responses of base-isolated structures and internal equipment. The complex mode method can provide exact solutions to problems with non-classical damping and is used here to find the exact response of the isolation-superstructure-equipment system. The entire system is assumed to be linear elastic with viscous damping and the superstructure is assumed to be proportionally damped so that the deformation of the superstructure can be expressed in terms of its classical modes. Recognizing that the ratio of the equipment mass to the structural mass and the ratio of the stiffness of the isolation system to the superstructural stiffness are both small, perturbation methods are used to find the response. This study shows that the response of base-isolated structures can be determined by the classical mode method to some degree of accuracy, but the higher frequency content is distorted. The equipment response derived by the classical mode method is much smaller than the exact solution so that the complex mode method should be applied to find equipment response.     

基于不同参数的基础隔震结构非线性概率密度演化

为了研究不同设计参数条件下基础隔震结构非线性响应的概率密度演化特征,采用两质点模型来模拟基础隔震结构,隔震层与上部结构分别采用Bouc-Wen模型与刚度退化的Bouc-Wen模型来描述其非线性特征,运用概率密度演化理论,进行隔震结构非线性随机地震响应的概率密度演化分析。采用基于物理的随机地震动模型生成人工地震动,提出基础隔震结构非线性随机地震响应的概率密度演化分析的基本步骤。通过改变基础隔震结构的设计参数,同时考虑激励的随机性,研究基础隔震结构非线性随机地震响应的概率密度演化规律。结果表明,基础隔震结构的阻尼比、周期比和屈重比取合理范围,能使隔震结构上部和下部的位移可控。     

隔震结构中非经典阻尼影响及最佳阻尼比分析

本文采用双自由度非比例阻尼振动模型描述基础隔震体系,用拉普拉斯变换方法获得其地震时域响应的近似解析解,借助于应谱理论分析了非比例阻尼对隔震体系中的上部结构层间最大剪力及隔震层最大位移等响应值的影响,探寻了隔震系统最佳阻尼比的取值范围。     

Seismic response of the superstructure and attached equipment in a base-isolated building

Base isolation can be used both to protect the structure and simultaneously to reduce the response of internal equipment. The seismic response of a base-isolated structure has been studied through the shaking table test or numerical calculation before. The object of this paper is to analyse a base-isolated structure by a different analytical approach—perturbation analysis. Recognizing that the horizontal stiffness of an isolation system is much smaller than that of the superstructure, the mathematical expressions of the modal properties of base-isolated structures are derived by the perturbation method in terms of the modal properties of the superstructure and used to study the dynamic response of superstructure and attached equipment in the base-isolated building. This study shows that the first base-isolated mode not only controls the superstructural response but also dominates the response of high-frequency attachment. The contribution of higher modes to the response of base-isolated structures, which is proportional to the horizontal stiffness of isolation system, is very small.     

Seismic response evaluation of base-isolated reinforced concrete buildings under bidirectional excitation

This paper reports on an investigation of the seismic response of base-isolated reinforced concrete buildings, which considers various isolation system parameters under bidirectional near-fault and far-fault motions. Three-dimensional models of 4-, 8-, and 12-story base-isolated buildings with nonlinear effects in the isolation system and the superstructure are investigated, and nonlinear response history analysis is carried out. The bounding values of isolation system properties that incorporate the aging effect of isolators are also taken into account, as is the current state of practice in the design and analysis of base-isolated buildings. The response indicators of the buildings are studied for near-fault and far-fault motions weight-scaled to represent the design earthquake (DE) level and the risk-targeted maximum considered earthquake (MCE_R) level. Results of the nonlinear response history analyses indicate no structural damage under DE-level motions for near-fault and far-fault motions and for MCE_R-level far-fault motions, whereas minor structural damage is observed under MCE_R-level near-fault motions. Results of the base-isolated buildings are compared with their fixed-base counterparts. Significant reduction of the superstructure response of the 12-story base-isolated building compared to the fixed-base condition indicates that base isolation can be effectively used in taller buildings to enhance performance. Additionally, the applicability of a rigid superstructure to predict the isolator displacement demand is also investigated. It is found that the isolator displacements can be estimated accurately using a rigid body model for the superstructure for the buildings considered.     

Effect of supplemental damping on LRB and FPS seismic isolators under near-fault ground motions

Numerical simulations are performed to assess the effects of near-fault ground motions on base-isolated buildings that consist of either lead-rubber (LRB) or friction-pendulum system (FPS) bearings in addition to supplemented viscous dampers. While LRB and FPS isolation systems have been applied for a number of years, the addition of supplemental damping devices is being currently considered for strong ground motions to reduce the isolator displacements. However, the main problem in this case is that the addition of damping may increase both internal deformation and absolute accelerations of the superstructure and thus may defeat many of the gains for which base isolation is intended. In the present paper, a detailed and systematic investigation on the performance of LRB and FPS isolation systems, provided with supplemental viscous damping under the effect of near-fault ground motions, has been carried out by using commercial finite element software.     

Effects of the improper modeling of viscous damping on the first-mode and higher-mode dominated responses of base-isolated buildings

In many finite element platforms, a classical global damping matrix based on the elastic stiffness of the system (including isolators) is usually developed as part of the solution to the equations of motion of base-isolated buildings. The conducted analytical and numerical investigations illustrate that this approach can lead to the introduction of unintended damping to the first and higher vibration modes and the spurious suppression of the respective structural responses. A similar shortcoming might be observed even when a nonclassical damping model (ie, an assembly of the superstructure and isolation system damping sub-matrices) is used. For example, the use of Rayleigh damping approach to develop the superstructure damping sub-matrix can lead to the undesired addition of damping to the isolated mode arising from the mass-proportional component of the superstructure damping. On the other hand, the improper use of nonclassical stiffness-proportional damping (eg, determining the proportional damping coefficient, β_k , based on the first mode) can result in assigning significant damping to the higher-modes and the unintended mitigation of the higher-mode responses. Results show that a nonclassical stiffness-proportional model in which β_k is determined based on the second modal period of a base-isolated building can reasonably specify the intended damping to the higher modes without imparting undesirable damping to the first mode. The nonclassical stiffness-proportional damping can be introduced to the numerical model through explicit viscous damper elements attached between adjacent floors. In structural analysis software such as SAP2000®, the desired nonclassical damping can be also modeled through specifying damping solely to the superstructure material.     

Seismic response of base-isolated buildings including soil–structure interaction

This study investigates the effect of soil–structure interaction (SSI) on the response of base-isolated buildings. The equations of motion are formulated in the frequency domain, assuming frequency-independent soil stiffness and damping constants. An equivalent fixed-base system is developed that accounts for soil compliance and damping characteristics of the base-isolated building. Closed-form expressions are derived, followed by a thorough parametric study involving the pertinent system parameters. For preliminary design, the methodology can serve as a means to assess effective use of base isolation on building structures accounting for SSI. This study concludes that the effects of SSI are more pronounced on the modal properties of the system, especially for the case of squat and stiff base-isolated structures.     

Optimum damping in linear isolation systems

Optimum isolation damping for minimum acceleration response of base-isolated structures subjected to stationary random excitation is investigated. Three linear models are considered to account for the energy dissipation mechanism of the isolation system: a Kelvin element, a linear hysteretic element and a standard solid linear element, commonly used viscoelastic models for isolation systems comprising natural rubber bearings and viscous dampers. The criterion selected for optimality is the minimization of the mean-square floor acceleration response. The effects of the frequency content of the excitation and superstructure properties on the optimum damping and on the mean-square acceleration response are addressed. The study basically shows that the attainable reduction in the floor acceleration largely depends on the energy dissipation mechanism assumed for the isolation system as well as on the frequency content of the ground acceleration process. Special care should be taken in accurately modelling the mechanical behaviour of the energy dissipation devices.     

MODAL ANALYSIS OF SOFT-SOIL SITES INCLUDING RADIATION DAMPING

For the one-dimensional analysis of soft-soil layers on an elastic half-space, a general form of analytical solution is developed for converting radiation damping due to energy leaking back to the half-space into equivalent modal damping, allowing the modal analysis technique to be extended to a site where radiation damping has to be accounted for. Closed-form solutions for equivalent modal damping ratios and effective modal participation factors are developed for a single layer with a shear wave velocity distribution varying from constant to linearly increasing with depth. Compact and recursive forms of solutions for equivalent modal damping ratios are developed for a system with an arbitrary number of homogeneous layers on an elastic half-space. Comparisons with numerical solutions show that the modal solutions are accurate. The nominal frequency of a site, i.e. the inverse of four times the total shear wave travel time through the layers, is an important parameter for estimating the high mode frequencies. A parameter study shows that for the same impedance ratio of the bottom layer to the elastic half-space, a system of soil layers with an increasing soil rigidity with depth has, in general, larger peak modal amplifications at the ground surface than does a single homogeneous layer on an elastic half-space, while a system with a decreasing soil rigidity with depth has smaller modal peak amplifications. © 1997 by John Wiley & Sons, Ltd.     

地基半空间等效集总体系的比拟法与实测分析

先略述莱斯默比拟法的形成;再由半空间理论等效为质弹体系,得出辐射阻尼比、刚度及参振土质量,并论述两体系的结合;最后经实测、分析和使用,考虑土体非匀质性折减阻尼比以作修正,使其更为实用。这有助于消除在我国长期认为阻尼比大而不安全、不便使用的疑虑,以便推动半空间理论在我国的实用化。     

Seismic response of heavily damped base isolation systems

The development of an efficient energy-dissipating mechanism that works in conjunction with laminated elastomeric bearings in order to reduce the lateral deformation of the isolation system has always been a goal of base isolation research. Theoretically, this deformation will be reduced to the minimum if damping augmentation of the isolation system can reach a critical value. However, augmenting the isolation damping may cause some unwanted side effects. The purpose of this paper is to study the influence of isolation damping on the seismic response of heavily damped base-isolated buildings. The base isolation system is assumed to be linearly viscoelastic and is analysed using the complex mode method. Solutions derived by using perturbation techniques for a two-degree-of-freedom system and the computer simulation for a multiple-degree-of-freedom system reveal that augmenting the isolation damping can reduce efficiently the deformation of the isolation system, but at the price of increasing the high-frequency vibration in the superstructure. When the damping ratio of the isolation system is beyond some level, increasing the isolation damping will enlarge the extreme values of the base and superstructural accelerations. It is also found that approximate solutions derived from the use of classical damping and classical modes of vibration in the seismic analysis of heavily damped base isolation systems can be substantially in error.     

Seismic response of segmental buildings

This paper proposes an aseismic design concept in which the superstructure of a base-isolated building is divided into several segments. Each segment may comprise a few storeys and is interconnected by additional vibrational isolation systems. The dynamic characteristics of the segmental buildings are investigated. The optimum parameters of the vibration isolation systems are determined by minimizing the mean square acceleration response. The seismic response of a typical segmental building subjected to the N—S component of the 1940 El Centro earthquake input is evaluated and compared with the responses of the corresponding fixed-base and conventional base-isolated buildings. The comparisons show that, when the superstructure is segmented, while the acceleration response in the superstructure remains as small as that in the conventional base-isolated building, the displacement across the base isolation system at foundation level is substantially reduced.     

The role of damping in seismic isolation

In the current code requirements for the design of base isolation systems for buildings located at near-fault sites, the design engineer is faced with very large design displacements for the isolators. To reduce these displacements, supplementary dampers are often prescribed. These dampers reduce displacements, but at the expense of significant increases in interstorey drifts and floor accelerations in the superstructure. An elementary analysis based on a simple model of an isolated structure is used to demonstrate this dilemma. The model is linear and is based on modal analysis, but includes the modal coupling terms caused by high levels of damping in the isolation system. The equations are solved by a method that avoids complex modal analysis. Estimates of the important response quantities are obtained by the response spectrum method. It is shown that as the damping in the isolation system increases, the contribution of the modal coupling terms due to isolator damping in response to the superstructure becomes the dominant term. The isolator displacement and structural base shear may be reduced, but the floor accelerations and interstorey drift are increased. The results show that the use of supplemental dampers in seismic isolation is a misplaced effort and alternative strategies to solve the problem are suggested. Copyright © 1999 John Wiley & Sons, Ltd.     

Out-of-plane (SH) soil-structure interaction: a shear wall with rigid and flexible ring foundation

Soil-structure interaction (SSI) of a building and shear wall above a foundation in an elastic half-space has long been an important research subject for earthquake engineers and strong-motion seismologists. Numerous papers have been published since the early 1970s; however, very few of these papers have analytic closed-form solutions available. The soil-structure interaction problem is one of the most classic problems connecting the two disciplines of earthquake engineering and civil engineering. The interaction effect represents the mechanism of energy transfer and dissipation among the elements of the dynamic system, namely the soil subgrade, foundation, and superstructure. This interaction effect is important across many structure, foundation, and subgrade types but is most pronounced when a rigid superstructure is founded on a relatively soft lower foundation and subgrade. This effect may only be ignored when the subgrade is much harder than a flexible superstructure: for instance a flexible moment frame superstructure founded on a thin compacted soil layer on top of very stiff bedrock below. This paper will study the interaction effect of the subgrade and the superstructure. The analytical solution of the interaction of a shear wall, flexible-rigid foundation, and an elastic half-space is derived for incident SH waves with various angles of incidence. It found that the flexible ring (soft layer) cannot be used as an isolation mechanism to decouple a superstructure from its substructure resting on a shaking half-space.     

Seismic response of base-isolated liquid storage tanks considering fluid–structure–soil interaction in time domain

The seismic response analysis of a base-isolated liquid storage tank on a half-space was examined using a coupling method that combines the finite elements and boundary elements. The coupled dynamic system that considers the base isolation system and soil–structure interaction effect is formulated in time domain to evaluate accurately the seismic response of a liquid storage tank. Finite elements for a structure and boundary elements for liquid are coupled using equilibrium and compatibility conditions. The base isolation system is modeled using the biaxial hysteretic element. The homogeneous half-space is idealized using the simple spring-dashpot model with frequency-independent coefficients. Some numerical examples are presented to demonstrate accuracy and applicability of the developed method.Consequently, a general numerical algorithm that can analyze the dynamic response of base-isolated liquid storage tanks on homogeneous half-space is developed in three-dimensional coordinates and dynamic response analysis is performed in time domain.     

脉冲型地震动作用下隔震结构动力响应的影响参数研究

通过对隔震结构进行非线性动力响应分析,分别研究地震动参数和支座参数对结构地震响应的影响。首先,建立铅芯橡胶支座基础隔震结构的非线性运动方程;然后,以人工合成脉冲型地震动作为输入,运用MATLAB进行编程并求解结构在脉冲型地震动作用下的地震响应;最后,分别研究速度脉冲周期、支座屈服力、屈服后与屈服前的刚度比对隔震支座最大位移和上部结构层间位移的影响。研究结果表明,脉冲周期对结构地震响应影响很大,在进行隔震设计时应使结构自振周期远离脉冲周期;支座刚度比对结构地震响应影响较大,在进行支座选型时应重点关注;支座屈服力对支座位移的影响显著,屈服力越大,支座位移越小。     

Seismic response of base-isolated structures with vertical-rocking coupling

A base-isolated building is liable to have a small horizontal eccentricity between the centre of mass of the superstructure and the centre of rigidity of the supporting bearings. In seismic analysis, the structure is modelled as a rigid block with tributary masses supported on massless elastomeric rubber bearings placed at a constant elevation below the centre of mass. This simplified system has three degrees of freedom: two translations and one rotation in the vertical plane. The investigation of the dynamic behaviour of a base-isolated building is carried out for both the detuned and the perfectly tuned cases. In the detuned case, the natural frequencies of the system are assumed to be well separated. In the perfectly tuned case, the uncoupled rocking frequency is assumed to be identical to the vertical translational frequency, which may result from an unusual mass distribution and/or an extreme aspect ratio of the superstructure. Perturbation methods are implemented in finding the dynamic characteristics for both cases. However, the dynamic response of the perfectly tuned case is the major concern in this investigation. The Green's functions for the displacement response of the three-degree-of-freedom system are derived for both the undamped and damped conditions. The response spectrum modal superposition method is used in estimating the maximum acceleration response. A simple method, accounting for the effect of closely spaced modes, is proposed for combining modal maxima and results in an approximate solution corresponding to a single-degree-of-freedom system. This approximate solution may be used for the preliminary design of a base-isolated structure. Numerical results for a base-isolated building subjected to the vertical component of the El Centro earthquake of 1940 were carried out for comparison with these analytical results. The proposed modal combination method showed superiority over the conventional Square Root of the Sum of the Squares method in estimating maximum responses. The results also indicated that the approximate single-degree-of-freedom system yields accurate estimations. It is shown that the effect of rocking coupling on the vertical response of base-isolated structures subjected to transient loadings, such as earthquake motions, can generally be neglected as a result of the combined effects of the time lag between the maximum translational and rotational responses and the influence of damping in the isolation system, which for elastomeric bearings can be as high as 8 to 10 per cent of critical.     

A comparative study of performances of various base isolation systems,part I: Shear beam structures

A comparative study of performances of different base isolators for shear beam type structures is carried out. Several leading base isolation systems, including the laminated rubber bearing with and without lead plug, the resilient-friction base isolator with and without sliding upper plate, and the EDF system are considered. Displacement and acceleration response spectra for a shear beam structure subject to the accelerograms of the N00W component of El Centro 1940 and the N90W component of Mexico City 1985 earthquakes and their magnified forms are evaluated. A series of parametric studies is carried out and advantages and disadvantages of various base isolation systems are identified. Comparisons of the results with the response spectra of a fixed-base structure show that the base isolation systems are, in general, highly effective in reducing the peak acceleration transmitted to the superstructure. Thus, the deflections and stresses generated in a base-isolated structure are significantly lower than those of a fixed-base one. Furthermore, the results of the study also show that the friction-type base isolators are less sensitive to severe variations in frequency content and amplitude of the ground acceleration.     

基于动力可靠度的隔震结构参数模糊优化

分别选取Bouc—Wen和退化Bouc—Wen模型描述隔震层及上部结构滞变回复力，采用虚拟激励法进行隔震结构随机响应分析。用各层最大层间位移响应和累计疲劳损伤指数建立双参数的功能状态方程，用一次二阶矩理论计算隔震结构失效概率。选用一个隔震框架作为数值算例，探讨了低频过滤器、隔震阻尼比和隔震刚度对隔震结构各子系统条件失效概率的影响。建立了隔震结构参数多目标优化模型，用失效概率确定单因素评判的隶属度，并采用最大隶属度法对隔震参数进行模糊优化。