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.     

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.     

基于复阻尼理论的混合结构Rayleigh阻尼模型

混合结构的阻尼矩阵不满足经典阻尼条件,导致传统的模态叠加法无法适用。复阻尼理论无法适用于时域计算,其自由振动响应中存在发散现象。针对混合结构的阻尼矩阵非比例性和复阻尼理论的时域发散性,基于频域等效原则构建了求解Rayleigh阻尼系数的数学优化模型,进而得到与复阻尼理论等效的Rayleigh阻尼运动方程。算例分析表明:依据位移时程响应和结构等效阻尼比可证明Rayleigh阻尼运动方程的正确性。基于本文研究成果,等效复阻尼理论的混合结构Rayleigh阻尼运动方程可直接采用模态叠加法,结合其确定的结构等效阻尼比,为混合结构的振型分解反应谱法提供理论依据。     

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.     

A note on classical damping matrices

An alternative damping matrix that leads to classical normal modes and that depends explicitly on a set of prescribed modal damping ratios is presented. The alternative damping matrix can be thought of as a factorized Caughey series that allows for a simple explicit solution for the coefficients of the series and thus avoids the need to solve a potentially ill‐conditioned system of algebraic equations. The relation between the proposed damping matrix and Rayleigh, Caughey and modal damping matrices is examined. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

Initial versus tangent stiffness‐based Rayleigh damping in inelastic time history seismic analyses

In the inelastic time history analyses of structures in seismic motion, part of the seismic energy that is imparted to the structure is absorbed by the inelastic structural model, and Rayleigh damping is commonly used in practice as an additional energy dissipation source. It has been acknowledged that Rayleigh damping models lack physical consistency and that, in turn, it must be carefully used to avoid encountering unintended consequences as the appearance of artificial damping. There are concerns raised by the mass proportional part of Rayleigh damping, but they are not considered in this paper. As far as the stiffness proportional part of Rayleigh damping is concerned, either the initial structural stiffness or the updated tangent stiffness can be used. The objective of this paper is to provide a comprehensive comparison of these two types of Rayleigh damping models so that a practitioner (i) can objectively choose the type of Rayleigh damping model that best fits her/his needs and (ii) is provided with useful analytical tools to design Rayleigh damping model with good control on the damping ratios throughout inelastic analysis. To that end, a review of the literature dedicated to Rayleigh damping within these last two decades is first presented; then, practical tools to control the modal damping ratios throughout the time history analysis are developed; a simple example is finally used to illustrate the differences resulting from the use of either initial or tangent stiffness‐based Rayleigh damping model. Copyright © 2013 John Wiley & Sons, Ltd.     

地震作用下多层剪切型组合结构的合理综合阻尼比取值探讨

对于组合结构,提出了一种新的基于单元瑞利阻尼模型的应变能振型阻尼比,并证明了其在特定整体阻尼矩阵下与强迫解耦法的等价性;推导了具有明确理论依据的综合阻尼比计算公式,并基于相应的应变能振型阻尼比得出了结构的刚度综合阻尼比和瑞利综合阻尼比。分别采用复振型分解法和振型分解法对算例结构进行了地震荷载作用下的弹性时程分析,结果表明,瑞利综合阻尼比对于以剪切变形为主的多层组合结构具有良好的计算精度和适用性。     

多级串联非比例阻尼隔震结构地震响应分析

建立了多级串联非比例阻尼隔震结构动力分析模型;引用分区瑞利阻尼模型将非比例阻尼矩阵分解为瑞利阻尼矩阵和体现非比例阻尼的余项阻尼矩阵,推导出结构的阻尼矩阵;并编制了MATLAB动力时程分析程序,对一实际隔震工程进行地震响应分析.结果表明:随着下部结构刚度的增加,结构的层剪力比和隔震层位移响应峰值均趋向于基础隔震结构的对应值;当下部结构为一层,且层间刚度大于上部结构底层层间刚度4～6倍时,可以近似按基础隔震结构进行动力分析.     

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.     

高速铁路客站模态特征分析与阻尼模型的注记

针对"房桥合一"高速铁路客站质量、刚度分布严重不均匀、不同阻尼比构件繁多等特点,对其模态特征和用于时程法阻尼模型的确定进行了研究。通过位能加权激励原理和位能公式的阐述、天津西站II区的模态分析与振型分解时程法的应用,并以此为标准进行了5种不同瑞利阻尼比例系数时程法的响应比较。结果表明:位能加权法激励为各模态的振型响应,求得阻尼为振型阻尼,结构模态质量累计数突变发生在第9阶,选择第1,9阶振型确定的瑞利阻尼比例系数较合理。振型分解时程法的振型阻尼可基于振型响应的位能加权法确定,直接时程法的瑞利阻尼宜选择第1阶与模态质量累计数发生突变的振型来确定,可供结构设计与分析参考。     

Damping formulation for nonlinear 1D site response analyses

Measurements and observations of ground shaking during large earthquakes have demonstrated the predominant role of site effects in the response of infrastructure during a seismic event. Despite significant efforts to model the hysteretic response and nonlinearity of soils due to medium and large ground motions, the most widely accepted nonlinear site response methods are not able to represent simultaneously the changes of stiffness and energy dissipation (damping) observed in both laboratory tests and during earthquake events. This paper presents two new soil damping formulations implemented in nonlinear one-dimensional site response analysis for small and large strains. The first formulation introduces an approach to construct a frequency-independent viscous damping matrix which reduces the over-damping at high frequencies, and therefore, the filtering at those frequencies. The second formulation introduces a reduction factor that modifies the extended Masing loading/unloading strain–stress relationship to match measured modulus reduction and damping curves simultaneously over a wide range of shear strains. A set of examples are introduced to illustrate the effect of using the two proposed formulations, separately and simultaneously, in nonlinear site response analyses.     

比例阻尼矩阵构建方法对高重力坝地震反应的影响

通过二维数值计算,讨论合理建立阻尼矩阵对高重力坝时域内进行地震反应计算的重要性。首先,以4个不同坝高的混凝土重力坝为计算对象,将三种地震波作为水平输入,解得6种不同的阻尼矩阵形式下坝体的地震反应。然后以频域内解为标准,研究各种阻尼矩阵的合理性。研究结果表明：坝高超过250 m高的重力坝在时域内进行的地震反应计算是长周期系统的动力分析问题,应重视阻尼矩阵的建模方式,不宜采用单频率参数的质量比例阻尼矩阵和刚度比例阻尼矩阵,应采用双频率参数的Rayleigh阻尼矩阵,在确定2个频率参数时除采用坝体基频外还应考虑激振地震波的频谱特性以获得合理的坝体地震反应计算结果。     

Energy dissipation models for RC members and structures

The unloading parameters of hysteretic models for RC members are given in terms of their shear-span-to-depth ratio and the viscous damping used to model other energy dissipation sources. They reflect the energy dissipation in full post-yield load cycles in 534 tests of rectangular or circular members. Pre-yield hysteretic energy dissipation—ignored if the model is elastic till yielding—amounts in the tests to a mean viscous damping around 8.5% and can be considered in nonlinear response-history analysis through a new model which combines constant elastic stiffness in virgin loading with hysteretic energy dissipation both before and after yielding. Models with linear behavior till yielding and hysteretic energy dissipation only after it come closer to the results of the new model if viscous damping is 5%.     

关于土层地震反应分析中阻尼矩阵建模方式的讨论

阻尼矩阵的选择对土层时域内地震反应计算的准确性有重要意义。本文构造了质量比例阻尼、刚度比例阻尼、Rayleigh阻尼、Caughey阻尼和Clough阻尼矩阵,研究各阻尼对土层地震反应分析精度的影响。以苏通大桥5号桥塔基础处的工程场地为例,分别以人工合成基岩波,Northridge波、Parkfield波和汶川波为输入,得到不同阻尼模型下土层的地震反应,结果表明Clough阻尼在计算精度和误差稳定性上要优于其他阻尼模型,针对Clough阻尼的不足之处提出了改进方法,算例表明改进后的Clough阻尼能在保证计算精度的前提下减少计算机的存储量和计算量。     

Nonlinear lateral interaction in pile dynamics

A model for pile lateral response to transient dynamic loading and to harmonic loading is presented allowing for nonlinear soil behaviour, discontinuity conditions at the pile-soil interface and energy dissipation through different types of damping. The approach is used to establish equivalent linear stiffness and damping parameters of single piles as well as dynamic interaction factors for approximate nonlinear analysis of pile groups. The applicability of these parameters to the pile-group analysis was examined, and a reasonable agreement with the direct analysis was found. The superposition technique may be used to analyze the response of small pile groups. Also, the dynamic stiffness of pile groups is greatly affected by both the nonlinear behavior of the soil and the slippage and gapping between the pile and soil. For a basic range of soil and pile parameters, equivalent linear stiffness and damping parameters of single piles and interaction factors for approximate nonlinear analysis are provided.     

A pseudo‐force iterative method with separate scale factors for dynamic analysis of structures with non‐proportional damping

Dynamic equilibrium equations of structural systems with non‐proportional damping are coupled through the damping terms. Such coupling invalidates application of the classical modal superposition method. In this paper, a mode‐superposition pseudo‐force method is proposed. The coupled equilibrium equations are solved by an iterative process in which the coupling terms are treated as pseudo‐forces. A scale factor for each mode of the system is obtained by optimizing the iteration convergence. Through these uniquely solved scale factors, the modified modal equations not only converge much faster but also yield results with higher accuracy. A proof of the convergence of the iterative process is also presented. Copyright © 2002 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.     

瑞利阻尼对砂土场地非线性地震响应的影响分析

在地震荷载作用下,自由场地会产生土体侧向变形和地表响应放大现象。由于土体的高度非线性,计算自由场地地震响应时,不同的阻尼比及剪切模量取值是造成其计算结果与试验结果相差较大的原因之一。目前动力计算常采用瑞利阻尼方法,其系数取值会在一定程度上影响计算结果。选用两模态简化瑞利阻尼系数计算方法,分析土体阻尼比及控制频率的取值对计算结果的影响,对比离心机模型试验,利用开源有限元平台OpenSees,采用适合于土体动力分析的多屈服面本构模型(PDMY),建立剪切梁模型模拟三维自由场地,并分析瑞利阻尼参数对自由场地地震响应和侧向变形计算结果的影响。结果表明,针对相对密度为60%的Nevada干砂,阻尼比为4%、控制频率比为5时,场地响应计算结果与试验结果较为符合。综合分析显示场地非线性响应时域计算时,应特别注意选用的瑞利阻尼参数值。