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.     

An unconditionally stable time-stepping procedure with algorithmic damping: a weighted integral approach using two general weight functions

An accurate algorithm for the integration of the equations of motion arising in structural dynamics is presented. The algorithm is an unconditionally stable single-step implicit algorithm incorporating algorithmic damping. The displacement for a Single-Degree-of-Freedom system is approximated within a time step by a function which is cubic in time. The four coefficients of the cubic are chosen to satisfy the two initial conditions and two weighted integral equations. By considering general weight functions, eight additional coefficients arise. These coefficients are selected to (i) minimize the difference between exact and approximate solutions for small time steps, (ii) incorporate specified algorithmic damping for large time steps, (iii) ensure unconditional stability and (iv) minimize numerical operations in forming the amplification matrix. The accuracy of the procedure is discussed, and the solution time is compared with a widely used algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

Optimal weakening and damping using polynomial control for seismically excited nonlinear structures

This paper presents an approach for the optimal design of a new retrofi t technique called weakening and damping that is valid for civil engineering inelastic structures.An alternative design methodology is developed with respect to the existing ones that is able to determine the locations and the magnitude of weakening and/or softening of structural elements and adding damping while insuring structural stability.An optimal polynomial controller that is a summation of polynomials in nonlinear states is used...     

Conditional spectrum‐based ground motion record selection using average spectral acceleration

The use of a seismic intensity measure (IM) is paramount in decoupling seismic hazard and structural response estimation when assessing the performance of structures. For this to be valid, the IM needs to be sufficient;that is, the engineering demand parameter (EDP) response should be independent of other ground motion characteristics when conditioned on the IM. Whenever non‐trivial dependence is found, such as in the case of the IM being the first‐mode spectral acceleration, ground motion selection must be employed to generate sets of ground motion records that are consistent vis‐à‐vis the hazard conditioned on the IM. Conditional spectrum record selection is such a method for choosing records that are consistent with the site‐dependent spectral shape conditioned on the first‐mode spectral acceleration. Based on a single structural period, however the result may be suboptimal, or insufficient, for EDPs influenced by different period values, for example, peak interstory drifts or peak floor accelerations at different floors, potentially requiring different record suites for each. Recently, the log‐average spectral acceleration over a period range, AvgSA, has emerged as an improved scalar IM for building response estimation whose hazard can be evaluated using existing ground motion prediction equations. Herein, we present a recasting of conditional spectrum record selection that is based on AvgSA over a period range as the conditioning IM. This procedure ensures increased efficiency and sufficiency in simultaneously estimating multiple EDPs by means of a single IM. Copyright © 2017 John Wiley & Sons, Ltd.     

A method for the direct estimation of floor acceleration spectra for elastic and inelastic MDOF structures

This paper deals with floor acceleration spectra, which are used for the seismic design and assessment of acceleration‐sensitive equipment installed in buildings. In design codes and in practice, not enough attention has been paid to the seismic resistance of such equipment. An ‘accurate’ determination of floor spectra requires a complex and quite demanding dynamic response history analysis. The purpose of the study presented in this paper is the development of a direct method for the determination of floor acceleration spectra, which enables their generation directly from the design spectrum of the structure, by taking into account the structure's dynamic properties. The method is also applicable to inelastic structures, which can greatly improve the economic aspects of equipment design. A parametric study of floor acceleration spectra for elastic and inelastic single‐degree‐of‐freedom (SDOF) and multiple‐degree‐of‐freedom structures was conducted by using (non)linear response history analysis. The equipment was modelled as an elastic single‐degree‐of‐freedom system. The proposed method was validated by comparing the results obtained with the more accurate results obtained in a parametric study. Due to its simplicity, the method is an appropriate tool for practice. In the case of inelastic structural behaviour, the method should be used in combination with the N2 method, or another appropriate method for simplified nonlinear structural analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Performance spectra based method for the seismic design of structures equipped with passive supplemental damping systems

A new direct performance‐based design method utilizing design tools called performance‐spectra (P‐Spectra) for low‐rise to medium‐rise frame structures incorporating supplemental damping devices is presented. P‐Spectra are graphic tools that relate the responses of nonlinear SDOF systems with supplemental dampers to various damping parameters and dynamic system properties that structural designers can control. These tools integrate multiple response quantities that are important to the performance of a structure into a single compact graphical format to facilitate direct comparison of different potential solutions that satisfy a set of predetermined performance objectives under various levels of seismic hazard. An SDOF to MDOF transformation procedure that defines the required supplemental damping properties for the MDOF structure to achieve the response defined by the target SDOF system is also presented for hysteretic, linear viscous and viscoelastic damping devices. Using nonlinear time‐history analyses of idealized shear structures, the accuracy of the transformation procedure is verified. A seismic performance upgrade design example is presented to demonstrate the usefulness of the proposed method for achieving design performance goals using supplemental damping devices. Copyright © 2012 John Wiley & Sons, Ltd.     

A note on the Hertz contact model with nonlinear damping for pounding simulation

Structural pounding during earthquakes has been recently investigated extensively by using different models of impact force. In this paper, reexamination of the Hertz contact model with nonlinear damping is made. Based on this reexamination, the formula used to determine the damping constant in terms of the spring stiffness, the coefficient of restitution and relative approaching velocity of two colliding bodies is found to be incorrect for pounding simulation in structural engineering. In order to correct this problem, a more accurate approximating formula for the damping constant is theoretically derived. The correctness of the derived analytical formula has been confirmed through numerical simulations. Copyright © 2008 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.     

Simplified analysis of asymmetric structures with supplemental damping

This study investigated the effects of neglecting off‐diagonal terms of the transformed damping matrix on the seismic response of non‐proportionally damped asymmetric‐plan systems with the specific aim of identifying the range of system parameters for which this simplification can be used without introducing significant errors in the response. For this purpose, a procedure is presented in which modal damping ratios computed by neglecting off‐diagonal terms of the transformed damping matrix are used in the traditional modal analysis. The effects of the simplification are evaluated first by comparing the aforementioned modal damping ratios with the apparent damping ratios obtained from the complex‐valued eigenanalysis. The variation of a parameter that was defined by Warburton and Soni as an indicator of the errors introduced by the simplification is examined next. Finally, edge deformations obtained from the simplified procedure are compared with those obtained from the direct integration of the equations of motion. It is found that the simplified procedure may be used without introducing significant errors in response for most practical values of the system parameters. Furthermore, estimates of the edge deformations, in general, tend to be on the conservative side. Copyright © 2001 John Wiley & Sons, Ltd.     

Nonlinear performance of classical damping

The performance of a classical damping matrix, constructed either from the use of initial structural properties or current structural properties, in the step-by-step solution of a nonlinear multiple degree of freedom (MDOF) system is analytically evaluated. The analytical results are confirmed by numerical examples. Consequently, some conclusions are drawn from these analytical results that might be considered as rough guidelines for practical applications. It is found that a classical damping matrix constructed from initial structural properties is adequate for practical applications, since it has approximately the same damping effect as obtained by current structural properties and is more efficient in terms of computing.     

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.     

高层建筑筒体结构动力分析的新方法

主要研究简体结构动力分析的新方法，先利用QR法建立了简体结构的新动力模型，然后利用5次样条加权残数法建立了求解动力模型的新算法，最后提出了高层与超高层建筑结构动力分析的新方法，文中附有算例。     

含随机参数粘弹性结构的有限元法分析

本文在空间上采用有限元法。在时域上采用离散的方法。建立了用扩阶随机有限元方法求解含随机参数粘弹性问题的计算模型，并给出了算例。结果表明，本文解和Monte-Carlo模拟解吻合。     

俯冲带板间地震水平加速度谱阻尼修正系数研究

基于日本K-net和Kik-net台站的俯冲带地区板间地震的数据,分SCⅠ~SCⅣ四种场地分别建立了水平向加速度反应谱阻尼修正系数模型。利用最小二乘法进行回归拟合,拟合公式采用对数三次项提高拟合精度。通过统计检验和残差分析,验证模型的拟合效果。研究表明:低于5%与高于5%阻尼比的阻尼修正系数变化趋势差异明显,分别在周期点0.3~0.5s区间达到极大值和极小值;不同场地类别的阻尼修正系数统计上显著不同,建立不同场地类别的模型是必要的;模型残差的标准差分布在0.012~0.712之间,建立的阻尼修正系数模型较好的拟合了阻尼修正系数随谱周期和阻尼比的变化趋势。     

Displacement–acceleration control via stiffness–damping collaboration

An approach is presented to stiffness–damping simultaneous optimization for displacement–acceleration simultaneous control. To make a shear building model stiffer, the sum of mean-square interstorey drifts to stationary random excitations is minimized or the mean-square top-floor absolute acceleration is maximized subject to the constraints on total storey stiffness capacity and total damper capacity. Optimality conditions are derived and a two-step optimization method using the optimality conditions is devised. In the first step, the optimal design is found for a specified set of total storey stiffness capacity and total damper capacity. In the second step, a series of optimal designs is found with respect to a varied set of total storey stiffness capacity and total damper capacity. While increase of total stiffness capacity and increase of total damper capacity are both effective in reduction of deformation, only increase of total damper capacity is effective in reduction of acceleration. Acceleration control is carried out in the second step via increase of total damper capacity. It is shown through numerical examples that the proposed method is efficient and reliable. Copyright © 1999 John Wiley & Sons, Ltd.     

结构动力方程的显式级数积分格式

将Newmark-β法中常平均加速度法的基本假定与精细指数算法结合,根据指数矩阵的Taylor级数展开式,提出了动力方程的显式级数解,并设计了相应的时程积分算法.该算法的精度可根据Taylor级数展开式的项数进行灵活控制.算例的结果表明:在满足稳定性条件的前提下,随着时间步长的增加,其精度优于传统的时程积分法.通过稳定性的分析,指出其稳定性条件是显然满足的.     

Evaluation of damping reduction factors for estimating elastic response of structures with high damping

The damping reduction factors are used in a few building codes in order to estimate the elastic response spectrum with high damping ratios from its 5%‐damped counterpart. At present, much research regarding this factor has been published. The purpose of this paper is to evaluate the accuracy of five different models of damping reduction factors. In addition, the damping reduction factors adopted in the recent U.S.A. seismic codes are also evaluated. Statistical studies are carried out according to a total of 216 earthquake ground motions recorded on firm sites in California. Considered here are elastic single‐degree‐of‐freedom (SDOF) systems with natural vibrating periods between 0.1 and 6.0 s. Mean ratios and dispersion of approximate to exact maximum elastic displacement demands corresponding to various damping reduction factors are computed as functions of the natural vibrating period and the viscous damping ratio. Copyright © 2005 John Wiley & Sons, Ltd.     

A computationally efficient algorithm for the solution of eigenproblems for large structures with non-proportional damping using Lanczos method

In this paper, a solution method is presented to solve the eigenproblem arising in the dynamic analysis of non-proportional damping systems with symmetric matrices. The method is based on the Lanczos method to generate one pair of Krylov subspaces consisting of trial vectors, which is then used to reduce a large eigenvalue problem into a much smaller one. The method retains the n order quadratic eigenproblem, without employing the method of matrix augmentation traditionally used to cast the problem as a linear eigenproblem of order 2n. In this process, the method preserves the sparseness and symmetry of the system matrices and does not invoke complex arithmetic; thus making it very economical for use in solving large problems. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

Active viscous damping system with amplifying braces for control of MDOF structures

The development and the applications of an active controlled viscous damping device with amplifying braces are described. The system of the dampers, defined as active viscous damping system (AVDS), connected to an amplifying brace (AB) is presented herein. Instantaneous control theory with velocity and acceleration feedback is used to obtain the control forces at each time step during an excitation. Control of the damping forces is possible due to the mechanical structure of the proposed AVDS, and the connection to the AB. The proposed system can be efficiently used to enhance the damping of a structure without modifying its stiffness. The added damping forces can be adjusted in a wide range. The efficiency of the presented system is demonstrated by a numerical simulation of a seven‐storey building subjected to earthquakes. The simulation shows a considerable reduction of control forces required for control to the AVDS with AB, compared to the same system without AB. Copyright © 2002 John Wiley & Sons, Ltd.     

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.