Damped resonant appendages to increase inherent damping in buildings

It is demonstrated that the addition of a tuned mass-spring-dashpot system with a relatively small mass and a high damping ratio can be an effective way to increase the inherent damping characteristics of buildings and reduce, thus, their response to earthquake excitations. The demonstration is based on a theoretical formulation and on numerical and experimental studies that confirm this formulation. In the theoretical formulation, it is shown first that, if certain conditions are satisfied, the damping ratios in two of the modes of the system that is formed by a building and an appendage in resonance are approximately equal to the average of the corresponding damping ratios of the building and the appendage. Based on this finding, it is then shown that an attached appendage with a high damping ratio and tuned to the fundamental frequency of a building may increase the damping ratio in the fundamental mode of the building to a value close to half the damping ratio of the appendage. In the numerical study, the response of a ten-storey shear building is analysed under two different earthquake ground motions with and without the proposed resonant appendages. Appendages with damping ratios of 20 and 30 per cent are considered. In this study, it is found that under one of the ground motions the maximum displacement of the building's roof is reduced 30 per cent with the appendage with 20 per cent damping and 39 per cent with the one with 30 per cent damping. Similarly, with these two appendages the building's base shear is reduced 31 and 41 per cent, respectively. In the experimental study, a wooden three-storey structural model is tested in a shaking table with and without an appendage designed and constructed to have a damping ratio of 53-5 per cent. The test is conducted under random and sinusoidal base excitations. In the shaking table test under random excitation, the attached appendage reduces the response of the model 38-6 per cent, while in that under sinusoidal vibration 45-2 per cent.     

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.     

Viscous damping formulation and high frequency motion propagation in non-linear site response analysis

Non-linear time domain site response analysis is widely used in evaluating local soil effects on propagated ground motion. This approach has generally provided good estimates of field behavior at longer periods but has shortcomings at relatively shorter periods. Viscous damping is commonly employed in the equation of motion to capture damping at very small strains and employs an approximation of Rayleigh damping using the first natural mode only. This paper introduces a new formulation for the viscous damping using the full Rayleigh damping. The new formulation represents more accurately wave propagation for soil columns greater than 50 m thick and improves non-linear site response analysis at shorter periods. The proposed formulation allows the use of frequency dependent viscous damping. Several examples, including a field case history at Treasure Island, California, demonstrate the significant improvement in computed surface response using the new formulation.     

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.     

结构动力响应计算结果对应于不同阻尼的修正方法

本文提出了一种当结构系统采用Rayleigh阻尼且阻尼参数发生变化时，对系统动力反应结果直接进行修正的方法并通过算例对其精度进行了验证。这一方法可以提高大型工程结构动力计算的效率。     

A dynamic stiffness formulation for the analysis of secondary systems

Formulation of a frequency-domain substructure approach for the analysis of secondary systems is presented. The total system contemplated includes the primary structure, the secondary system, and the foundation medium, which is also treated as a substructure. A dynamic stiffness matrix in physical co-ordinates characterizes each one of the substructures. Elimination of the internal degrees of freedom of the primary structure prior to assembly of the equations for the coupled system is carried out with the aid of a truncated set of unconstrained normal modes. Accounting for the residual static flexibility of the truncated modes obviates potential problems of rank deficiency resulting from modal truncation. The formulation contemplates an arbitrary multi-component scattered motion at the soil–structure interface and imposes no limitations on the configuration of the primary or the secondary system. Connectivity between the systems is treated as an arbitrary linear relation between selected co-ordinates in each substructure. This feature is shown to be useful for modelling the commonly encountered situation where secondary systems are attached to torsionally eccentric structures. Copyright © 1999 John Wiley & Sons, Ltd.     

Interpretation and nonuniqueness of CTRW transition distributions: Insights from an alternative solute transport formulation

The continuous time random walk (CTRW) has both an elegant mathematical theory and a successful record at modeling solute transport in the subsurface. However, there are some interpretation ambiguities relating to the relationship between the discrete CTRW transition distributions and the underlying continuous movement of solute that have not been addressed in existing literature. These include the exact definition of “transition”, and the extent to which transition probability distributions are unique/quantifiable from data. Here, we present some theoretical results which address these uncertainties in systems with an advective bias. Simultaneously, we present an alternative, reduced parameter CTRW formulation for general advective transport in heterogeneous porous media, which models early- and late-time transport by use of random transition times between sparse, imaginary planes normal to flow. We show that even in the context of this reduced-parameter formulation there is nonuniqueness in the definitions of both transition lengths and waiting time distributions, and that neither may be uniquely determined from experimental data. For practical use of this formulation, we suggest Pareto transition time distributions, leading to a two-degree-of-freedom modeling approach. We then demonstrate the power of this approach in fitting two sets of existing experimental data. While the primary focus is the presentation of new results, the discussion is designed to be pedagogical and to provide a good entry point into practical modeling of solute transport with the CTRW.     

结构动力分析阻尼模型研究

提出了结构动力时程分析修正的Ｒａｙｌｅｉｇｈ阻尼模型和一个基于单元特性的阻尼参数方法,可用于合理确定复杂阻尼结构的阻尼参数,工程应用实例表明,阻尼参数的计算对时程响应计算结果的影响是十分明显的,必须在分析结构体系振动特性的基础上慎重确定阻尼参数,以避免为为放大或滤掉主要振型,造成虚假的结论。     

Nonlinear dynamic analysis of multi-base seismically isolated structures with uplift potential I： formulation

The complexity of modern seismically isolated structures requires the analysis of the structural system and the isolation system in its entirety and the ability to capture potential discontinuous phenomena such as isolator uplift and their effects on the superstructures and the isolation hardware. In this paper, an analytical model is developed and a computational algorithm is formulated to analyze complex seismically isolated superstructures even when undergoing highly-nonlinear phenomena such as uplift. The computational model has the capability of modeling various types of isolation devices with strong nonlinearities, analyzing multiple superstructures (up to five separate superstructures) on multiple bases (up to five bases), and capturing the effects of lateral loads on bearing axial forces, including bearing uplift. The model developed herein has been utilized to form the software platform 3D-BASIS-ME-MB, which provides the practicing engineering community with a versatile tool for analysis and design of complex structures with modern isolation systems.     

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.     

两结构高效阻尼控制系统随机动力反应分析

对由高效阻尼装置(HEDMS)连接的两单自由度相邻结构在过滤白噪声激励下的相对位移响应进行了随机分析。在对阻尼装置滞变位移等价线性化的基础上，采用虚拟激励法计算结构响应的谱密度和互谱密度，研究了该阻尼装置各参数对位移响应方差的影响以及该装置的适用范围。     

Equating incremental dynamic analysis with static nonlinear analysis at near-field excitation

In an incremental dynamic analysis(IDA) using a set of ground motion records,nonlinear time history analysis needs to be performed on structures.It is well recognized that IDA calls for high computational efforts and the results are highly sensitive to selected ground motions.As a result,alternative static methods are needed.This study aims to introduce a new double-stage(N1- N2) static method to estimate capacity curves of MR frames.The technique is regulated to resemble IDA results with specific emphasis on near-field ground motions.Using an ensemble of 56 near-field earthquake records,required ID As have been carried out for SAC-Los Angeles 3-,9- and 20-story buildings and an additional 15-story building.The results of the proposed static method are compared with those from IDA,displacement-based adaptive procedure(DAP),and multimodal procedure(MMP).The results indicate that in addition to enhanced accuracy,very little time is required in the case of N1-N2 method.Thus,for the 3-story structure,the time required is less than 1 minute.The proposed N1-N2 method shows the best accuracy in terms of lateral mechanisms for the 15-story frame while for the other cases,the first mode load pattern leads to the best accuracy.     

Ritz method for dynamic analysis of large discrete linear systems with non-proportional damping

Real and complex Ritz vector bases for dynamic analysis of large linear systems with non-proportional damping are presented and compared. Both vector bases are generated utilizing load dependent vector algorithms that employ recurrence equations analogous to the Lanczos algorithm. The choice of static response to fixed spatial loading distribution, as a starting vector in recurrence equations, is motivated by the static correction concept. Different phases of dynamic response analysis are compared with respect to computational efficiency and accuracy. It is concluded that the real vector basis approach is approximately eight times more efficient than the complex vector basis approach. The complex vector basis has some advantages with respect to accuracy, if the excitation is of piecewise linear form, since the exact solution can be utilized. In addition, it is demonstrated that both Ritz vector bases, real and complex, possess superior accuracy over the adequate eigenvector bases.     

复阻尼多自由度系统动力分析的模态叠加法

工程实际中,复阻尼多自由度系统的瞬态响应过去一直是通过频域方法求解的。频域方法的一个突出问题是求传递函数矩阵的计算工作量过大。本文给出一种有效而实用的时域解法———实模态叠加法,此外还介绍了复模态叠加法。实模态叠加法是基于一个事实,即n维复向量空间中的复向量可以在n维实向量空间中的一组线性无关的实向量构成的基下表出;另外,就此方法还讨论了复阻尼多自由度系统初始运动条件的给出和转换问题。复模态叠加法则是通过变量替换的方法,变其中的复特征值问题为形式上的一个实特征值问题来解决的。     

Effective linear damping and stiffness coefficients of nonlinear systems for design spectrum based analysis

A stochastic approach for obtaining reliable estimates of the peak response of nonlinear systems to excitations specified via a design seismic spectrum is proposed. This is achieved in an efficient manner without resorting to numerical integration of the governing nonlinear equations of motion. First, a numerical scheme is utilized to derive a power spectrum which is compatible in a stochastic sense with a given design spectrum. This power spectrum is then treated as the excitation spectrum to determine effective damping and stiffness coefficients corresponding to an equivalent linear system (ELS) via a statistical linearization scheme. Further, the obtained coefficients are used in conjunction with the (linear) design spectrum to estimate the peak response of the original nonlinear systems. The cases of systems with piecewise linear stiffness nonlinearity, along with bilinear hysteretic systems are considered. The seismic severity is specified by the elastic design spectrum prescribed by the European aseismic code provisions (EC8). Monte Carlo simulations pertaining to an ensemble of nonstationary EC8 design spectrum compatible accelerograms are conducted to confirm that the average peak response of the nonlinear systems compare reasonably well with that of the ELS, within the known level of accuracy furnished by the statistical linearization method. In this manner, the proposed approach yields ELS which can replace the original nonlinear systems in carrying out computationally efficient analyses in the initial stages of the aseismic design of structures under severe seismic excitations specified in terms of a design spectrum.     

钢筋混凝土剪力墙非线性动力分析模型

介绍了目前常用的钢筋混凝土剪力墙的几种非线性动力分析模型,并对模型涉及的各元件的滞回特性及有关参数的取值作了简要讨论。提供的力学模型和滞回特性可供钢筋混凝土剪力墙、框－墙结构和、钢－混凝土混合结构的非线性动力分析时采用。     

不同阻尼特性材料组合结构的弹塑性动力时程响应计算

本文提出一种不必形成与存储总阻尼矩阵的非线性结构动力时程分析的算法,用于计算非比例、非经典阻尼结构弹塑性动力响应。该算法的软件实现简单,对现有比例阻尼结构分析程序稍加修改即可。基于此算法的计算机程序满足了由两种阻尼特性材料建造的建筑结构非线性动力时程分析的需要。     

A flexible nonlinear modelling framework for nonstationary generalized extreme value analysis in hydroclimatology

Parameters in a generalized extreme value (GEV) distribution are specified as a function of covariates using a conditional density network (CDN), which is a probabilistic extension of the multilayer perceptron neural network. If the covariate is time or is dependent on time, then the GEV‐CDN model can be used to perform nonlinear, nonstationary GEV analysis of hydrological or climatological time series. Owing to the flexibility of the neural network architecture, the model is capable of representing a wide range of nonstationary relationships. Model parameters are estimated by generalized maximum likelihood, an approach that is tailored to the estimation of GEV parameters from geophysical time series. Model complexity is identified using the Bayesian information criterion and the Akaike information criterion with small sample size correction. Monte Carlo simulations are used to validate GEV‐CDN performance on four simple synthetic problems. The model is then demonstrated on precipitation data from southern California, a series that exhibits nonstationarity due to interannual/interdecadal climatic variability. Copyright © 2009 Her Majesty the Queen in right of Canada. Published by John Wiley & Sons, Ltd.     

A modal superposition pseudo-force method for dynamic analysis of structural systems with non-proportional damping

A modal superposition pseudo-force method for the dynamic analysis of structural systems with non-proportional damping is presented. The method combines the advantages of the classical modal superposition method and the pseudo-force method. When the system damping is non-proportional, the dynamic equilibrium equations in generalized coordinates are coupled through the damping terms. In the present method, these coupled equations are solved by an iterative process in which the coupling terms are treated as pseudo-forces. A proof of the convergence of the iterative process is given. Numerical examples show the good convergence characteristics of the process and the good accuracy of the obtained results.     

A method for nonlinear dynamic response analysis of semi-infinite foundation soil

This paper presents development of a special finite difference method for the nonlinear dynamic response analysis of semi-infinite foundation soil. Semi-infinite domain is mapped into the finite domain using special mapping. For the region of engineering interest, mapping is isometrical, and for far field, shrink mapping which transforms an infinite interval into a finite interval is adopted. Using linear and nonlinear constitutive models, the responses of semi-infinite foundation soil are computed using a proposed method with a small mesh model and an extensive mesh model. Surface loadings or incident earthquake waves are applied to the models in the computations. Good agreements were obtained among the theoretical and computed results of the two models and the effectiveness of the proposed method was demonstrated.