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.     

大跨度斜拉桥横桥向减震研究

斜拉桥在横桥向采用塔-梁、墩-梁固结的约束体系,导致其整体刚度增加,地震惯性力增大,给边墩及其基础的抗震设计造成困难。分别采用位移相关型(方案1)和速度相关型(方案2)两类减震装置对一座斜拉桥的横桥向进行了减震研究。方案1在边墩-主梁间设置位移相关型减震装置,并对其屈服荷载进行了参数分析;方案2对速度相关型减震装置的安装位置和数量进行了优化分析,并对其参数取值进行了参数分析;对横桥向固结体系和减震体系的地震反应进行了对比。结果表明:地震作用下两类减震装置发生滞回变形,延长了结构在横桥向的周期,有效降低了边墩的地震剪力和弯矩反应;横桥向墩-梁间的相对位移会增大,可通过减震装置参数的选取将其控制在合理的范围内;塔底的地震剪力和弯矩反应变化不明显。2种方案均可用于斜拉桥横向减震。     

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.     

Elemental damping formulation: an alternative modelling of inherent damping in nonlinear dynamic analysis

To date, nonlinear dynamic analysis for seismic engineering predominantly employs the classical Rayleigh damping model and its variations. Though earlier studies have identified issues with the use of this model in nonlinear seismic analysis, it still remains the popular choice for engineers as well as for software providers. In this paper a new approach to modelling damping is initiated by formulating the damping matrix at an elemental level. To this regard, two new elemental level discrete damping models adapted from their global counterparts are proposed for its application in nonlinear dynamic analysis. Implementation schemes for these newly proposed models using Newmark incremental method and revised Newmark total equilibrium method is outlined. The performance of these proposed models, compared to existing models, is illustrated by conducting nonlinear dynamic analyses on a four story RC frame designed to Eurocodes. The incremental dynamic analysis study presented in the paper illustrates the fact that both the proposed models seem to produce more reliable results from an engineering perspective in comparison to the global models. It is also shown that the proposed elemental damping models lead to smaller and more realistic damping moments in the plastic hinges. Furthermore, these models could be easily included in existing software frameworks without adding noticeably to the computational effort. The computation time required for these models is approximately equivalent to the one required when using the tangent Rayleigh damping matrix with constant coefficients.     

Numerical artifacts associated with Rayleigh and modal damping models of inelastic structures with massless coordinates

The paper presents a detailed reexamination of the effects of three damping models on the inelastic seismic response of structures with massless degrees of freedom. The models considered correspond to (a) Rayleigh damping based on current properties (tangent stiffness), (b) Rayleigh damping based on initial properties, and (c) modal damping. The results suggest that some nonzero damping forces/moments at massless DOFs obtained in multistory frames for the case of Rayleigh damping with tangent stiffness may be numerical artifacts rather than a deficiency of the damping model. The results also indicate that significant artificial numerical oscillations in the velocities of the secondary components of MDOF structures are introduced when modal damping or mass-proportional damping is used.     

Implementation and application of the unconditionally stable explicit parametrically dissipative KR‐α method for real‐time hybrid simulation

In real‐time hybrid simulations (RTHS) that utilize explicit integration algorithms, the inherent damping in the analytical substructure is generally defined using mass and initial stiffness proportional damping. This type of damping model is known to produce inaccurate results when the structure undergoes significant inelastic deformations. To alleviate the problem, a form of a nonproportional damping model often used in numerical simulations involving implicit integration algorithms can be considered. This type of damping model, however, when used with explicit integration algorithms can require a small time step to achieve the desired accuracy in an RTHS involving a structure with a large number of degrees of freedom. Restrictions on the minimum time step exist in an RTHS that are associated with the computational demand. Integrating the equations of motion for an RTHS with too large of a time step can result in spurious high‐frequency oscillations in the member forces for elements of the structural model that undergo inelastic deformations. The problem is circumvented by introducing the parametrically controllable numerical energy dissipation available in the recently developed unconditionally stable explicit KR‐α method. This paper reviews the formulation of the KR‐α method and presents an efficient implementation for RTHS. Using the method, RTHS of a three‐story 0.6‐scale prototype steel building with nonlinear elastomeric dampers are conducted with a ground motion scaled to the design basis and maximum considered earthquake hazard levels. The results show that controllable numerical energy dissipation can significantly eliminate spurious participation of higher modes and produce exceptional RTHS results. Copyright © 2014 John Wiley & Sons, Ltd.     

基于动力BNWF法的冻土-桩相互作用模型研究

基于水平循环荷载作用下不同负温冻土环境中单桩动力特性模型试验结果,在已有分析桩-土-结构相互作用的动力BNWF模型的基础上,提出改进的冻土-桩基动力相互作用非线性反应分析模型。在该模型中,利用改进的双向无拉力多段屈服弹簧考虑桩侧冻土的水平非线性力学特性,同时兼顾桩侧与冻土间的竖向非线性摩擦效应、桩尖土的挤压与分离作用以及远场土体阻尼对桩基动力特性的影响。其中桩侧水平多段屈服弹簧参数根据冻土非线性p-y关系获得,该关系曲线以三次函数曲线段及常值函数段共同模拟,并由室内冻土压缩试验结果确定。最后基于改进的动力BNWF模型,提取动位移荷载作用下该桩顶力-位移滞回曲线及桩身不同埋深处的弯矩动响应数值分析结果,并与相应的模型试验结果对比,二者具有较好的拟合度,表明本文所提出的改进模型在分析冻土-桩动力相互作用时有较好的适用性。     

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.     

Modal decomposition method for stationary response of non-classically damped systems

The stationary response of multi-degree-of-freedom non-classically damped linear systems subjected to stationary input excitation is studied. A modal decomposition procedure based on the complex eigenvectors and eigenvalues of the system is used to derive general expressions for the spectral moments of response. These expressions are in terms of cross-modal spectral moments and explicitly account for the correlation between modal responses; thus, they are applicable to structures characterized with significant non-classical damping as well as structures with closely spaced frequencies. Closed form solutions are presented for the important case of response to white-noise input. Various quantities of response of general engineering interest can be obtained in terms of these spectral moments. These include mean zero-crossing rate and mean, variance and distribution of peak response over a specified duration. Examples point out several instances where non-classical damping effects become significant and illustrate the marked improvement of the results of this study over conventional analysis based on classical damping approximations.     

Damping/global energy balance in FE model of bridge foundation lateral response

A global energy analysis is presented of three static unloading–reloading foundation lateral loading cycles, calculated using the nonlinear finite element (FE) program DYNAFLOW. This simulates seismic action on an offshore pier foundation in the Rion-Antirion Bridge in Greece, located in deep-sea water (65 m). A cyclic horizontal force is applied at a height of 30 m to a rigid raft 78 m in width placed on the surface of an idealized 2-layer soil profile consisting of a 3.5 m man-made gravel layer over soft deep natural clay, with elastic vertical steel inclusions reinforcing the soil. Results of the two-dimensional FE run are used for the energy analysis. It is verified that for the three cycles, the sum of energies associated with the external forces and moments, mostly dissipated through hysteresis loops, is about equal to the sum of the total internal energies dissipated or stored in the system. For the smaller loops almost all energy is dissipated in the soil, while for the largest loop about half of the energy is dissipated by horizontal sliding at the raft-soil interface. Global damping ratios obtained from the areas of the horizontal and rocking moment hysteresis loops are about double of those computed from the corresponding static backbone curves using the Masing criterion.     

Spectral analysis and design approach for high force‐to‐volume extrusion damper‐based structural energy dissipation

High force‐to‐volume extrusion damping devices can offer significant energy dissipation directly in structural connections and significantly reduce seismic response. Realistic force levels up to 400 kN have been obtained experimentally validating this overall concept. This paper develops spectral‐based design equations for their application. Response spectra analysis for multiple, probabilistically scaled earthquake suites are used to delineate the response reductions due to added extrusion damping. Representative statistics and damping reduction factors are utilized to characterize the modified response in a form suitable for current performance‐based design methods. Multiple equation regression analysis is used to characterize reduction factors in the constant acceleration, constant velocity, and constant displacement regions of the response spectra. With peak device forces of 10% of structural weight, peak damping reduction factors in the constant displacement region of the spectra are approximately 6.5 ×, 4.0 ×, and 2.8 × for the low, medium, and high suites, respectively. At T = 1 s, these values are approximately 3.6 ×, 1.8 ×, and 1.4 ×, respectively. The maximum systematic bias introduced by using empirical equations to approximate damping reduction factors in design analyses is within the range of +10 to ?20%. The seismic demand spectrum approach is shown to be conservative across a majority of the spectrum, except for large added damping between T = 0.8 and 3.5 s, where it slightly underestimates the demand up to a maximum of approximately 10%. Overall, the analysis shows that these devices have significant potential to reduce seismic response and damage at validated prototype device force levels. Copyright © 2007 John Wiley & Sons, Ltd.     

A two‐step direct method for estimating the seismic response of nonlinear structures equipped with nonlinear viscous dampers

The insertion of fluid viscous dampers in building structures is an innovative technology that can improve significantly the seismic response. These devices could be very useful also in the retrofit of existing buildings. The effect of this typology of damping system is usually identified with an equivalent supplemental damping ratio, which depends on the maximum displacement of the structure, so that iterative procedures are required. In this paper, a simplified direct assessment method for nonlinear structures equipped with nonlinear fluid viscous dampers is proposed. The method proposed in this study is composed by two steps. The first one yields the direct estimate of the supplemental damping ratio provided by nonlinear viscous dampers in presence of a linear elastic structural response. The second step extends the procedure to structures with nonlinear behavior. Both graphical and analytical approaches have been developed. The proposed method has then been verified through several applications and comparisons with nonlinear dynamic analyses. Moreover, an investigation has been performed with regard to the influence of the relations that define the damping reduction factor and the hysteretic damping. Copyright © 2014 John Wiley & Sons, Ltd.     

Damping‐adjusted combination rule for the response spectrum analysis of base‐isolated buildings

The classical response spectrum method continues to be the most popular approach for designing base‐isolated buildings, therefore avoiding computationally expensive nonlinear time‐history analyses. In this framework, a new method for the seismic analysis and design of building structures with base isolation system (BIS) is formulated and numerically validated, which enables one to overcome the main shortcomings of existing techniques based on the response spectrum method. The main advantages are the following: first, reduced computational effort with respect to an exact complex‐valued modal analysis, which is obtained through a transformation of coordinates in two stages, both involving real‐valued eigenproblems; second, effective representation of the damping, which is pursued by consistently defining different viscous damping ratios for the modes of vibration of the coupled BIS‐superstructure dynamic system; and third, ease of use, because a convenient reinterpretation of the combination coefficients leads to a novel damping‐adjusted combination rule, in which just a single response spectrum is required for the reference value of the viscous damping ratio. The proposed approach is specifically intended for design situations where (i) the dynamic behaviour of seismic isolators can be linearised and (ii) effects of nonproportional damping, as measured by modal coupling indexes, are negligible in the BIS‐superstructure assembly. Copyright © 2012 John Wiley & Sons, Ltd.     

Modelling of elastic damping in nonlinear time-history analyses of cantilever RC walls

Damping constitutes a major source of uncertainty in dynamic analysis and an open issue to experimental and analytical research. After a thorough review of the current views and approaches existing in literature on damping and its appropriate modelling, this paper focuses on the implications of the available modelling options on analysis. As result of a series of considerations, a damping modelling solution for nonlinear dynamic analyses of cantilever RC walls is suggested within the frame of Direct Displacement-Based Design, supported by comparative analyses on wall structures.     

Response prediction,experimental characterization and P‐spectra design of frames with viscoelastic–plastic dampers

Viscoelastic–plastic (VEP) dampers are hybrid passive damping devices that combine the advantages of viscoelastic and hysteretic damping. This paper first formulates a semi‐analytical procedure for predicting the peak response of nonlinear SDOF systems equipped with VEP dampers, which forms the basis for the generation of Performance Spectra that can then be used for direct performance assessment and optimization of VEP damped structures. This procedure is first verified against extensive nonlinear time‐history analyses based on a Kelvin viscoelastic model of the dampers, and then against a more advanced evolutionary model that is calibrated to characterization tests of VEP damper specimens built from commercially available viscoelastic damping devices, and an adjustable friction device. The results show that the proposed procedure is sufficiently accurate for predicting the response of VEP systems without iterative dynamic analysis for preliminary design purposes. A design method based on the Performance Spectra framework is then proposed for systems equipped with passive VEP dampers and is applied to enhance the seismic response of a six‐storey steel moment frame. The numerical simulation results on the damped structure confirm the use of the Performance Spectra as a convenient and accurate platform for the optimization of VEP systems, particularly during the initial design stage. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution of random structural system subject to nonstationary excitation: transforming the equation with random coefficients to one with deterministic coefficients and random initial conditions

This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator.     

Problems encountered from the use (or misuse) of Rayleigh damping

Rayleigh damping is commonly used to provide a source of energy dissipation in analyses of structures responding to dynamic loads such as earthquake ground motions. In a finite element model, the Rayleigh damping matrix consists of a mass‐proportional part and a stiffness‐proportional part; the latter typically uses the initial linear stiffness matrix of the structure. Under certain conditions, for example, a non‐linear analysis with softening non‐linearity, the damping forces generated by such a matrix can become unrealistically large compared to the restoring forces, resulting in an analysis being unconservative. Potential problems are demonstrated in this paper through a series of examples. A remedy to these problems is proposed in which bounds are imposed on the damping forces. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct finite element method for nonlinear earthquake analysis of concrete dams: Simplification,modeling, and practical application

A direct finite element (FE) method for nonlinear response history analysis of semi-unbounded dam-water-foundation systems has recently been presented. The analysis procedure employs standard viscous-damper absorbing boundaries to model the semi-unbounded foundation and fluid domains and specifies the seismic input as effective earthquake forces—determined from a control motion defined at the foundation surface—at these boundaries. Presented in this paper are several simplifications to this direct FE method that greatly facilitates its implementation in commercial FE software. Also addressed is the modeling of the principal nonlinear mechanisms for concrete dams, calibration of damping in the numerical model to ensure consistency with values measured at actual dams, and practical procedures for implementation of the direct FE method with a commercial FE program.     

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.     

Direct finite element method for nonlinear analysis of semi‐unbounded dam–water–foundation rock systems

A direct finite element method is presented for nonlinear earthquake analysis of interacting dam–water–foundation rock systems. The analysis procedure applies viscous damper absorbing boundaries to truncate the semi‐unbounded fluid and foundation‐rock domains and specifies at these boundaries effective earthquake forces determined from the design ground motion defined at a control point on the free surface. The analysis procedure is validated numerically by computing the frequency response functions and transient response of an idealized dam–water–foundation rock system and comparing with results from the substructure method. Because the analysis procedure is applicable to nonlinear systems, it allows for modeling of concrete cracking, as well as sliding and separation at construction joints, lift joints, and at concrete–rock interfaces. Implementation of the procedure is facilitated by commercial finite element software with nonlinear material models that permit modeling of viscous damper boundaries and specification of effective earthquake forces at these boundaries. Copyright © 2017 John Wiley & Sons, Ltd.