Appropriate viscous damping for nonlinear time‐history analysis of base‐isolated reinforced concrete buildings

There is no consensus at the present time regarding an appropriate approach to model viscous damping in nonlinear time‐history analysis of base‐isolated buildings because of uncertainties associated with quantification of energy dissipation. Therefore, in this study, the effects of modeling viscous damping on the response of base‐isolated reinforced concrete buildings subjected to earthquake ground motions are investigated. The test results of a reduced‐scale three‐story building previously tested on a shaking table are compared with three‐dimensional finite element simulation results. The study is primarily focused on nonlinear direct‐integration time‐history analysis, where many different approaches of modeling viscous damping, developed within the framework of Rayleigh damping are considered. Nonlinear direct‐integration time‐history analysis results reveal that the damping ratio as well as the approach used to model damping has significant effects on the response, and quite importantly, a damping ratio of 1% is more appropriate in simulating the response than a damping ratio of 5%. It is shown that stiffness‐proportional damping, where the coefficient multiplying the stiffness matrix is calculated from the frequency of the base‐isolated building with the post‐elastic stiffness of the isolation system, provides reasonable estimates of the peak response indicators, in addition to being able to capture the frequency content of the response very well. Furthermore, nonlinear modal time‐history analyses using constant as well as frequency‐dependent modal damping are also performed for comparison purposes. It was found that for nonlinear modal time‐history analysis, frequency‐dependent damping, where zero damping is assigned to the frequencies below the fundamental frequency of the superstructure for a fixed‐base condition and 5% damping is assigned to all other frequencies, is more appropriate, than 5% constant damping. Copyright © 2013 John Wiley & Sons, Ltd.     

Modeling viscous damping in nonlinear response history analysis of buildings for earthquake excitation

The Rayleigh damping model, which is pervasive in nonlinear response history analysis (RHA) of buildings, is shown to develop ‘spurious’ damping forces and lead to inaccurate response results. We prove that a viscous damping matrix constructed by superposition of modal damping matrices—irrespective of the number of modes included or values assigned to modal damping ratios—completely eliminates the ‘spurious’ damping forces. This is the damping model recommended for nonlinear RHA. Replacing the stiffness‐proportional part of Rayleigh damping by the tangent stiffness matrix is shown to improve response results. However, this model is not recommended because it lacks a physical basis and has conceptual implications that are troubling: hysteresis in damping force–velocity relationship and negative damping at large displacements. Furthermore, the model conflicts with the constant‐damping model that has been the basis for fundamental concepts and accumulated experience about the inelastic response of structures. With a distributed plasticity model, the structural response is not sensitive to the damping model; even the Rayleigh damping model leads to acceptable results. This perspective on damping provides yet another reason to employ the superior distributed plasticity models in nonlinear RHA. OpenSees software has been extended to include a damping matrix defined as the superposition of modal damping matrices. Although this model leads to a full populated damping matrix, the additional computational demands are demonstrated to be minimal. Copyright © 2015 John Wiley & Sons, Ltd.     

Effect of viscous damping devices on the response of seismically isolated structures

Viscous and other damping devices are often used as elements of seismic isolation systems. Despite the widespread application of nonlinear viscous systems particularly in Japan (with fewer applications in the USA and Taiwan), the application of viscous damping devices in isolation systems in the USA progressed intentionally toward the use of supplementary linear viscous devices due to the advantages offered by these devices. This paper presents experimental results on the behavior of seismically isolated structures with low damping elastomeric (LDE) and single friction pendulum (SFP) bearings with and without linear and nonlinear viscous dampers. The isolation systems are tested within a six‐story structure configured as moment frame and then again as braced frame. Emphasis is placed both on the acquisition of data related to the structural system (drifts, story shear forces, and isolator displacements) and on non‐structural systems (floor accelerations, floor spectral accelerations, and floor velocities). Moreover, the accuracy of analytical prediction of response is investigated based on the results of a total of 227 experiments, using 14 historic ground motions of far‐fault and near‐fault characteristics, on flexible moment frame and stiff braced frame structures isolated with LDE or SFP bearings and linear or nonlinear viscous dampers. It is concluded that when damping is needed to reduce displacement demands in the isolation system, linear viscous damping results in the least detrimental effect on the isolated structure. Moreover, the study concludes that the analytical prediction of peak floor accelerations and floor response spectra may contain errors that need to be considered when designing secondary systems. Copyright © 2014 John Wiley & Sons, Ltd.     

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。     

Discussion of ‘Modelling viscous damping in nonlinear response history analysis of buildings for earthquake excitation’ by Anil K. Chopra and Frank McKenna

This discussion deals with recommendations in the paper on appropriate damping formulations for use in nonlinear response history analysis of buildings. Concern over potentially excessive damping forces and moments should extend beyond the damping moments produced by the stiffness proportional part of Rayleigh damping that corresponds to rotational springs used to explicitly model plastic hinges. The key to an appropriate damping formulation for nonlinear analysis is a realistic mechanism that allows all damping forces and moments to be meaningfully assessed. Then features can be added to keep these forces and moments within reasonable bounds. Copyright © 2016 John Wiley & Sons, Ltd.     

Higher-mode effect on the seismic responses of buildings with viscoelastic dampers

In conventional modal analysis procedures, usually only a few dominant modes are required to describe the dynamic behavior of multi-degrees-of-freedom buildings. The number of modes needed in the dynamic analysis depends on the higher-mode contribution to the structural response, which is called the higher-mode effect. The modal analysis approach, however, may not be directly applied to the dynamic analysis of viscoelastically damped buildings. This is because the dynamic properties of the viscoelastic dampers depend on their vibration frequency. Therefore, the structural stiffness and damping contributed from those dampers would be different for each mode. In this study, the higher-mode effect is referred to as the response difference induced by the frequency-dependent property of viscoelastic dampers at higher modes. Modal analysis procedures for buildings with viscoelastic dampers distributed proportionally and non-proportionally to the stiffness of the buildings are developed to consider the higher-mode effect. Numerical studies on shear-type viscoelastically damped building models are conducted to examine the accuracy of the proposed procedures and to investigate the significance of the higher-mode effect on their seismic response. Two damper models are used to estimate the peak damper forces in the proposed procedures. Study results reveal that the higher-mode effect is significant for long-period viscoelastically damped buildings. The higher-mode effect on base shear is less significant than on story acceleration response. Maximum difference of the seismic response usually occurs at the top story. Also, the higher-mode effect may not be reduced by decreasing the damping ratio provided by the viscoelastic dampers. For practical application, it is realized that the linear viscous damping model without considering the higher-mode effect may predict larger damper forces and hence, is on the conservative side. Supported by: Science Council, Chinese Taipei, grant no. 88-2625-2-002-006     

宿迁市文体馆基础隔震非线性时程分析研究

宿迁市文体馆4500座位，约13000m^2,位于8度抗震设防区，主体结构为钢筋混凝土结构空间框架，钢网壳屋盖。该工程采用基础隔震技术设计，在桩基顶面与上部结构之间设置架空层，用作安置设备管道及隔震层。隔震层由叠层橡胶隔震支座和粘滞阻尼器组成。对主体结构基础隔震采用空间模型非线性时程分析方法进行了详细分析，结果表明：采用基础隔震措施可显著降低结构地震作用，上部结构水平地震作用减震系数可按0．25采用；设置附加粘滞阻尼器能较好地解决降低地震作用和限制隔震层位移之间的矛盾，对提高隔震体系的性能具有重要作用。技术经济比较表明，本工程采用基础隔震措施，具有明显的社会、经济效益。     

Predominant period and equivalent viscous damping ratio identification for a full‐scale building shake table test

The predominant period and corresponding equivalent viscous damping ratio, also known in various loading codes as effective period and effective damping coefficient, are two important parameters employed in the seismic design of base‐isolated and conventional building structures. Accurate determination of these two parameters can reduce the uncertainty in the computation of lateral displacement demands and interstory drifts for a given seismic design spectrum. This paper estimates these two parameters from data sets recorded from a full‐scale five‐story reinforced concrete building subjected to seismic base excitations of various intensities in base‐isolated and fixed‐base configurations on the outdoor shake table at the University of California, San Diego. The scope of this paper includes all test motions in which the yielding of the reinforcement has not occurred and the response can still be considered ‘elastic’. The data sets are used with three system identification methods to determine the predominant period of response for each of the test configurations. One of the methods also determines the equivalent viscous damping ratio corresponding to the predominant period. It was found that the predominant period of the fixed‐base building lengthened from 0.52 to 1.30 s. This corresponded to a significant reduction in effective system stiffness to about 16% of the original stiffness. The paper then establishes a correlation between predominant period and peak ground velocity. Finally, the predominant periods and equivalent viscous damping ratios recommended by the ASCE 7‐10 loading standard are compared with those determined from the test building. Copyright © 2017 John Wiley & Sons, Ltd.     

Study on effects of damping in laminated rubber bearings on seismic responses for a ⅛ scale isolated test structure

The effects of damping in various laminated rubber bearings (LRB) on the seismic response of a ?‐scale isolated test structure are investigated by shaking table tests and seismic response analyses. A series of shaking table tests of the structure were performed for a fixed base design and for a base isolation design. Two different types of LRB were used: natural rubber bearings (NRB) and lead rubber bearings (LLRB). Three different designs for the LLRB were tested; each design had a different diameter of lead plug, and thus, different damping values. Artificial time histories of peak ground acceleration 0.4g were used in both the tests and the analyses. In both shaking table tests and analyses, as expected, the acceleration responses of the seismically isolated test structure were considerably reduced. However, the shear displacement at the isolators was increased. To reduce the shear displacement in the isolators, the diameter of the lead plug in the LLRB had to be enlarged to increase isolator damping by more than 24%. This caused the isolator stiffness to increase, and resulted in amplifying the floor acceleration response spectra of the isolated test structure in the higher frequency ranges with a monotonic reduction of isolator shear displacement. Copyright © 2002 John Wiley & Sons, Ltd.     

Evaluation of Rayleigh damping and its influence on engineering demand parameter estimates

The effects of Rayleigh damping model on the engineering demand parameters of two steel moment‐resisting frame buildings were evaluated. Two‐dimensional models of the buildings were created and response history analysis were conducted for three different hazard levels. The response history analysis results indicate that mass‐proportional damping leads to high damping forces compared with restoring forces and may lead to overestimation of floor acceleration demands for both buildings. Stiffness‐proportional damping, on the other hand, is observed to suppress the higher‐mode effects in the nine‐story building resulting in lower story drift demands in the upper floors compared with other damping models. Rayleigh damping models, which combine mass‐proportional and stiffness‐proportional components, that are anchored at reduced modal frequencies lead to reasonable damping forces and floor acceleration demands for both buildings and does not suppress higher‐mode effects in the nine‐story building. Copyright © 2012 John Wiley & Sons, Ltd.     

铅芯橡胶支座基础隔震体系参数优化配置研究

探讨了铅芯橡胶支座胶支座（LRB）用于基础隔震体系时参数的优化配置问题，对一算例采用非线性时程分析法研究了不同地震波激励下的地震反应。结果表明，对于具体工程控震指标要求，可以通过优选LRB参数来实现。     

Inelastic response of RC frame buildings with seismic isolation

A comprehensive parametric study on the inelastic seismic response of seismically isolated RC frame buildings, designed for gravity loads only, is presented. Four building prototypes, with 23 m × 10 m floor plan dimensions and number of storeys ranging from 2 to 8, are considered. All the buildings present internal resistant frames in one direction only, identified as the strong direction of the building. In the orthogonal weak direction, the buildings present outer resistant frames only, with infilled masonry panels. This structural configuration is typical of many existing RC buildings, realized in Italy and other European countries in the 60s and 70s. The parametric study is based on the results of extensive nonlinear response‐time history analyses of 2‐DOF systems, using a set of seven artificial and natural seismic ground motions. In the parametric study, buildings with strength ratio (F_y/W) ranging from 0.03 to 0.15 and post‐yield stiffness ratio ranging from 0% to 6% are examined. Three different types of isolation systems are considered, that is, high damping rubber bearings, lead rubber bearings and friction pendulum bearings. The isolation systems have been designed accepting the occurrence of plastic hinges in the superstructure during the design earthquake. The nonlinear response‐time history analyses results show that structures with seismic isolation experience fewer inelastic cycles compared with fixed‐base structures. As a consequence, although limited plastic deformations can be accepted, the collapse limit state of seismically isolated structures should be based on the lateral capacity of the superstructure without significant reliance on its inherent hysteretic damping or ductility capacity. Copyright © 2012 John Wiley & Sons, Ltd.     

Effect of viscous,viscoplastic and friction damping on the response of seismic isolated structures

In this paper the efficiency of various dissipative mechanisms to protect structures from pulse‐type and near‐source ground motions is examined. Physically realizable cycloidal pulses are introduced, and their resemblance to recorded near‐source ground motions is illustrated. The study uncovers the coherent component of some near‐source acceleration records, and the shaking potential of these records is examined. It is found that the response of structures with relatively low isolation periods is substantially affected by the high‐frequency fluctuations that override the long duration pulse. Therefore, the concept of seismic isolation is beneficial even for motions that contain a long duration pulse which generates most of the unusually large recorded displacements and velocities. Dissipation forces of the plastic (friction) type are very efficient in reducing displacement demands although occasionally they are responsible for substantial permanent displacements. It is found that the benefits by hysteretic dissipation are nearly indifferent to the level of the yield displacement of the hysteretic mechanism and that they depend primarily on the level of the plastic (friction) force. The study concludes that a combination of relatively low friction and viscous forces is attractive since base displacements are substantially reduced without appreciably increasing base shears and superstructure accelerations. Copyright © 2000 John Wiley & Sons, Ltd.     

The cause of unproportionately large higher mode contributions in the inelastic seismic responses of high‐rise core‐wall buildings

In the conventional seismic design of high‐rise reinforced concrete core‐wall buildings, the design demands such as design shear and bending moment in the core wall are typically determined by the response spectrum analysis procedure, and a plastic hinge is allowed to form at the wall base to limit the seismic demands. In this study, it is demonstrated by using a 40‐story core‐wall building that this conventional approach could lead to an unsafe design where the true demands—the maximum inelastic seismic demands induced by the maximum considered earthquake—could be several times greater than the design demands and be unproportionately dominated by higher vibration modes. To identify the cause of this problem, the true demands are decomposed into individual modal contributions by using the uncoupled modal response history analysis procedure. The results show that the true demands contributed by the first mode are reasonably close to the first‐mode design demands, while those contributed by other higher modes are much higher than the corresponding modal design demands. The flexural yielding in the plastic hinge at the wall base can effectively suppress the seismic demands of the first mode. For other higher modes, however, a similar yielding mechanism is either not fully mobilized or not mobilized at all, resulting in unexpectedly large contributions from higher modes. This finding suggests several possible approaches to improve the seismic design and to suppress the seismic demands of high‐rise core‐wall buildings. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of seismic demands on isolators in asymmetric buildings using non‐linear analysis

A procedure based on rigorous non‐linear analysis is presented that estimates the peak deformation among all isolators in an asymmetric building due to strong ground motion. The governing equations are reduced to a form such that the median normalized deformation due to an ensemble of ground motions with given corner period T_d depends primarily on four global parameters of the isolation system: the isolation period T_b, the normalized strength η, the torsional‐to‐lateral frequency ratio Ω_θ, and the normalized stiffness eccentricity e_b/r. The median ratio of the deformations of the asymmetric and corresponding symmetric systems is shown to depend only weakly on T_b, η, and Ω_θ, but increases with e_b/r. The equation developed to estimate the largest ratio among all isolators depends only on the stiffness eccentricity and the distance from the center of mass to the outlying isolator. This equation, multiplied by an earlier equation for the deformation of the corresponding symmetric system, provides a design equation to estimate the deformations of asymmetric systems. This design equation conservatively estimates the peak deformation among all isolators, but is generally within 10% of the ‘exact’ value. Relative to the non‐linear procedure presented, the peak isolator deformation is shown to be significantly underestimated by the U.S. building code procedures. Copyright © 2003 John Wiley & Sons, Ltd.     

An alternative approach for the seismic rehabilitation of existing RC buildings using seismic isolation

Usually, buildings with seismic isolation are designed to comply with an operational building performance level after strong earthquakes. This approach, however, may limit the use of seismic isolation for the seismic rehabilitation of existing buildings with low lateral strength or substandard details, because it often requires invasive strengthening measures in the superstructure or the use of expensive custom‐made devices. In this paper, an alternative approach for the seismic rehabilitation of existing buildings with seismic isolation, based on the acceptance of limited plastic deformations in the superstructure under strong earthquakes, is proposed and then applied to a real case study, represented by a four‐storey RC frame building. Nonlinear response‐time histories analyses of an accurate model of the case‐study building are carried out to evaluate the seismic performances of the structure, comparing different rehabilitation strategies with and without seismic isolation. Initial costs of the intervention and possible (future) repair costs are then estimated. Based on the results of this study, values of the behavior factor (i.e. response modification factor) higher than those adopted in the current codes for base‐isolated buildings are tentatively proposed. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of near‐fault ground motions on the nonlinear dynamic response of base‐isolated r.c. framed buildings

Near‐fault ground motions are characterized by long‐period horizontal pulses and high values of the ratio between the peak value of the vertical acceleration, PGA_V, and the analogous value of the horizontal acceleration, PGA_H, which can become critical for base‐isolated (BI) structures. The objective of the present work is to check the effectiveness of the base isolation of framed buildings when using High‐Damping‐Rubber Bearings (HDRBs), taking into consideration the combined effects of the horizontal and vertical components of near‐fault ground motions. To this end, a numerical investigation is carried out with reference to BI reinforced concrete buildings designed according to the European seismic code (Eurocode 8). The design of the test structures is carried out in a high‐risk region considering (besides the gravity loads) the horizontal seismic loads acting alone or in combination with the vertical ones and assuming different values of the ratio between the vertical and horizontal stiffnesses of the HDRBs. The nonlinear seismic analysis is performed using a step‐by‐step procedure based on a two‐parameter implicit integration scheme and an initial‐stress‐like iterative procedure. At each step of the analysis, plastic conditions are checked at the potential critical sections of the girders (i.e. end sections of the sub‐elements in which a girder is discretized) and columns (i.e. end sections), where a bilinear moment–curvature law is adopted; the effect of the axial load on the ultimate bending moment (M‐N interaction) of the columns is also taken into account. The response of an HDRB is simulated by a model with variable stiffness properties in the horizontal and vertical directions, depending on the axial force and lateral deformation, and linear viscous damping. Copyright © 2011 John Wiley & Sons, Ltd.     

An investigation into the effects of damping and nonlinear geometry models in earthquake engineering analysis

The growing emphasis of considering the behavior of structures at extreme performance states, such as collapse, has necessitated the characterization of the effects of varying attributes of the structural model. One source of variability that has not previously been considered is variability in the mathematical model. This study investigated the effects of changing the geometric nonlinearity approach and damping model on a four‐story buckling restrained braced frame, a four‐story steel moment resisting frame, and an eight‐story steel moment resisting frame. The variations in behavior are quantified using the maximum interstory drift ratio as the performance metric and qualified by comparing the relative displaced shapes and component response histories at the collapse performance state. Copyright © 2015 John Wiley & Sons, Ltd.     

多向地震作用下隔震配电建筑物的非线性动力反应分析

本文结合我国首例将隔震技术应用于变电建筑物的工程———西安市330kV西北郊变电站110kV配电楼工程,研究隔震结构在多向地震作用下的非线性动力反应,并把计算结果与单向地震作用的结果进行分析比较。结果表明,本工程采用隔震技术是可行的,采用基础隔震措施可有效地控制结构的偏心扭转效应,使屋面层及楼面层的运动规律趋于一致,建筑物整体趋于平动,同时还可显著降低上部结构及电气设备的地震作用,并且在多向地震作用下的结果比单向地震作用时的隔震效果更加合理。目前该工程已竣工投产,本文工作将为隔震技术在电力系统的更广泛应用打下一个良好的基础。