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.     

Implementation of configuration dependent stiffness proportional damping for the dynamics of rigid multi-block systems

The distinct element method (DEM) has been used successfully for the dynamic analysis of rigid block systems. One of many difficulties associated with DEM is modeling of damping. In this paper, new procedures are proposed for the damping modeling and its numerical implementation in distinct element analysis of rigid multi-block systems. The stiffness proportional damping is constructed for the prescribed damping ratio, based on the non-zero fundamental frequency effective during the time interval while the boundary conditions remain essentially constant. At this time interval, the fundamental frequency can be estimated without complete eigenvalue analysis. The damping coefficients will vary while the damping ratio remains the same throughout the entire analysis. A new numerical procedure is developed to prevent unnecessary energy loss that can occur during the separation phases. These procedures were implemented in the development of the distinct element method for the dynamic analyses of piled multi-block systems. The analysis results for the single-block and two-block systems were in a good agreement with the analytic predictions. Applications to the seismic analyses of piled fourblock systems revealed that the new procedures can make a significant difference and may lead to much-improved results.     

Dynamic characteristics of lateral load-deflection relationships of flexible piles

Most offshore platforms are supported on long and large-diameter piles with variable wall-thickness along the length, and soil properties varying with depth. The design and analyses of these piles are made by modelling the soil-pile system with a beam-on-Winkler foundation. Therefore, evaluation of appropriate soil-pile springs for use in such analyses is a matter of concern. Fundamental characteristics of dynamic lateral load-deflection relationships for piles were studied analytically considering the soil-pile-structure interaction under seismic loading conditions. The soil layer was assumed homogeneous, linearly elastic with hysteretic type material damping, and overlying a rigid base. A superstructure with multi-degrees of freedom was supported by a single vertical pile hinged at the rigid base. Parametric studies were carried out to identify the influence of the system parameters on the behaviour of the dynamic lateral load-deflection relationships of piles. The lateral load-deflection relationships vary considerably with depth and are influenced not only by the dynamic properties of soil but also by the structural properties of a pile and loading conditions. These lateral load-deflection relationships can be used to define the soil-pile springs for the seismic response analysis of a soil-pile-structure system, and the results can be extended to problems with soil profiles with layering and non-linearity.     

Ambient vibration survey of the fatih sultan mehmet (second Bosporus) suspension bridge

Ambient accelerations due to dynamic excitation by wind and traffic were measured on the deck, towers, cables and hangers of the Fatih suspension bridge. From these measurements it was possible to obtain natural frequencies, mode shapes and damping ratios for vertical, lateral, torsional and associated modes in the deck and tower up to a maximum of 2 Hz. The objective of the test was to validate the mathematical modelling used in seismic analyses of the bridge. The agreement between the experimental and theoretical modes was acceptable for vertical modes below 1 Hz, and for torsional modes, but it was difficult to identify the lateral modes due to low levels of response. The dynamic behaviour of this bridge and two other major European suspension bridges is discussed in relation to the differences in loading and structural design.     

Evaluation of seismic site factors in the Mississippi Embayment. I. Estimation of dynamic properties

Deep deposits of the Mississippi Embayment, overlying the New Madrid Seismic Zone, present unique challenges for the estimation of local site effects on propagated ground motion. Limited information is available on the dynamic properties of these deposits. This paper develops generalized depth dependent modulus degradation and damping curves specific to the embayment deposits. Depth dependent small strain damping is estimated using weak motion recordings during the Enola earthquake, 2001. Modulus degradation and damping curves are further constrained using limited laboratory test data of embayment soils at low confining pressure. At high confining pressures embayment specific data is unavailable and test data from outside the embayment is used in developing the dynamic properties. The representative modulus degradation and damping curves are used in non-linear and equivalent linear one-dimensional site response analyses. A companion paper describes a large-scale probabilistic seismic hazard analysis study in the Mississippi Embayment that integrates non-linear site effects.     

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.     

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.     

Spectral and fragility evaluations of retrofitted structures through strength reduction and enhanced damping

A retrofit procedure for existing buildings called the "weakening and damping technique" (WED) is presented in this paper. Weakening of structures can limit the maximum response accelerations during severe ground motions, but leads to an increase in the displacements or inter-story drifts. Added damping by using viscous dampers, on the other hand, reduces the inter-story drifts and has no significant effect on total accelerations, when structures behave inelastically. The weakening and damping technique addresses the two main causes for both structural and nonstructural damage in structures. The weakening retrofit is particularly suitable for structures that have overstressed components and weak brittle components. In this paper, the advantages of the WeD are verified by nonlinear dynamic analysis and simplified spectral approach that has been modified to fit structures with additional damping devices. A hospital structure located in the San Femando Valley in California is selected as a case study. The results from both analyses show that the retrofit solution is feasible to reduce both structural acceleration and displacement. A sensitivity analysis is also carried out to evaluate the effectiveness of the retrofitting method using different combinations of performance thresholds in accelerations and displacements through fragility analysis.     

固原重塑黄土动力特性共振柱试验研究

利用GCTS共振柱测试仪研究了干密度和固结压力对固原黄土重塑土动剪切模量和阻尼比的影响及试验数据误差的基本规律。通过设计不同工况的试验条件,对比分析结果表明:相同含水率条件下,最大动剪切模量G_(dmax)随固结压力和干密度的增大而增大。相同干密度条件下,黄土的动剪切模量比随围压的增大而增大;阻尼比随围压的增大而减小;相同围压条件下,黄土动剪切模量比随干密度的增大而增大,阻尼比随干密度的增大而减小;不同剪应变特征点的动剪切模量比和阻尼比试验数据呈现一定的离散性,且离散程度阻尼比明显高于动剪切模量比。研究成果可为该地区场地地震反应分析及工程建设提供基础资料。     

Full-scale dynamic testing of the Alamillo cable-stayed bridge in Sevilla (Spain)

Accelerations and displacements due to dynamic excitation by simulated traffic consisting of two trucks, were measured on the deck and tower of the Alamillo cable-stayed bridge. Also the dynamic testing program included the measurement with accelerometers of the free-damped vibration of the 26 cables achieved by means of quick-releasing force. From these measurements it was possible to obtain the dynamic parameters of the bridge (natural frequencies and damping ratios) and the real forces in the cables. In the paper, only the tests, results and conclusions related to dynamic parameters of the bridge are presented. The objective of the dynamic tests herein described was to validate the mathematical modelling and the wind-tunnel models used in the dynamic analysis of the bridge in front of traffic and wind-forces. As the agreement between dynamic parameters of the real bridge and theoretical and scaled models was very satisfactory, the correct dynamic behaviour of the bridge in response to traffic and wind (vortex shedding, flutter, etc.) can be deduced jointly with the correct alignment and expected internal forces in the permanent state in tower and deck.     

Insight on 2D- versus 3D-modelling of surface foundations via strength-of-materials solutions for soil dynamics

To simplify the analysis, three-dimensional soil–structure interaction problems are often modelled by considering a two-dimensional slice without changing the material properties of the soil. This procedure, although convenient, is of questionable validity because two-dimensional modelling inherently overestimates the radiation damping for translational and rocking motions. To make matters worse, two-dimensional modelling always entails an underestimation of the dynamic-spring coefficient for the translational motions. The damping ratio of the two-dimensional case, which is proportional to the ratio of the damping coefficient to the spring coefficient, will thus be even larger. Thus, reliance upon a two-dimensional analysis based on an equivalent slice of a strip foundation may result in a dangerously non-conservative design. Valuable insights into the essence of radiation damping and the difference between two-dimensional and three-dimensional models may be obtained via approximate strength-of-materials solutions based on cone–wedge models and travel-time considerations. By examining the decay of the waves along the axes of the cone–wedge models, the essence of radiation damping can be grasped. The heuristic concept of more spreading of waves in three dimensions than in two is misleading. Indeed, just the opposite is true: The less the amplitude spreads and diminishes with distance, the greater is the radiation damping. Because the damping ratio is grossly overestimated, two-dimensional modelling of a three-dimensional case cannot be recommended for actual engineering applications. It is more feasible to take the opposite approach and idealize slender soil–structure interaction problems with a radially symmetric model. As an alternative, when defining the equivalent slice of the two-dimensional strip foundation, the impedance of the soil can be changed to achieve a much better agreement of the high-frequency limits of the damping coefficients. In the low-frequency range this modified two-dimensional model also overestimates radiation damping, although to a lesser extent. As a by-product, the dimensions of the equivalent slice of a two-dimensional strip foundation are discussed; and equations for the aspect ratios determining the opening angles of the corresponding wedges are derived. Also addressed is the quite separate but related topic of the transition from square to slender rectangular foundations.     

瑞利阻尼介质有限元离散模型动力分析的数值稳定性

本文针对几种有一般阻尼的动力系数数值积分的显式方法，讨论了阻尼对稳定性的影响，并建议了瑞利阻尼介质有限元离散模型中动力分析数值稳定性的实用稳定判别方法。     

围压对粉质粘土动力学参数影响的共振柱试验研究

土的最大动剪切模量、剪应变幅和阻尼比是对土层进行动力反应分析的重要力学参数。利用英国GDS公司生产的RCA共振柱仪,研究围压对都江堰地区粉质粘土的上述动力学参数的影响。研究表明：相同围压条件下,最大动剪切模量、剪应变幅和阻尼比随重复次数基本不变,表明该试验过程具有可重复性;最大动剪切模量、剪应变幅和阻尼比均受围压影响较大,随着围压的增大,最大动剪切模量和阻尼比均逐渐增大,而剪应变幅随着围压的增大逐渐减小。据此,建立了都江堰地区粉质粘土动力学参数随围压变化的经验公式。本研究可为土层地震动力反应分析提供参考并积累基础资料。     

Seismic response of continuous span bridges through fiber-based finite element analysis

1 Introduction Older design codes based on equivalent elastic force approaches proved to be ineffective in preventing damage caused by destructive earthquakes. After recent major earthquakes (e.g. Northridge 1994, Kobe 1995, and Kocaeli 1999 etc.), the necessity for using more accurate methods, which explicitly account for geometrical nonlinearities and material inelasticity, to evaluate seismic demand on structures, became evident. Within this framework, two analysis tools are currently offe…     

正常工况下大型岸桥地震瑞利阻尼系数计算方法及试验研究

针对大型岸桥地震响应分析中不同的瑞利阻尼系数会导致计算结果存在较大偏差,基于相似模型试验数据与仿真计算结果的对比分析,并综合考虑岸桥边界约束等因素,借鉴工程中常用的瑞利阻尼系数计算方法改进了岸桥结构的阻尼系数计算方法及频率选取方式。利用仿真方法和振动台模型地震试验对比分析不同瑞利阻尼系数下岸桥结构的动力响应,结果表明：瑞利阻尼系数计算方法和频率选取对岸桥结构地震响应的影响随着地震加强而提高,当加速度峰值为0.4g、0.62g,现有的阻尼系数计算方法得到的仿真结果与试验值之间的平均误差超过20%,而采用改进后的阻尼系数计算方法,岸桥地震仿真结果与试验值误差都在10%以内。     

Modelling three‐dimensional non‐linear seismic performance of elevated bridges with emphasis on pounding of girders

Unseating of bridge girders/decks during earthquakes is very harmful to the safety and serviceability of bridges. Evidence from recent severe earthquakes indicates that in addition to damage along longitudinal direction, lateral displacement and rotation of bridge girders caused by pounding to adjacent girders can also lead to unseating. To simulate this effect, 3D modelling of the dynamic performance of whole bridge structures, including pounding, is needed strongly. This paper presents a 3D model that is practically suitable to precisely analyse pounding between bridge girders. Experiments have been conducted to verify the proposed pounding model. The 3D non‐linear modelling of steel elevated bridges is also discussed. A general‐purpose dynamic analysis program for bridges, namely dynamic analysis of bridge systems (DABS) has been developed. Seismic analyses on a chosen three‐span steel bridge are conducted for several cases including pounding as a case study. The applicability of the proposed pounding model is illustrated by the computations. The effects of poundings on the response of bridge girders are discussed and the computation results are given. Copyright © 2002 John Wiley & Sons, Ltd.     

Seismic energy dissipation for cable-stayed bridges using passive devices

This study assesses analytically the effectiveness, feasibility and limitations of elastic and hysteretic damping augmentation devices, such as elastomeric and lead–rubber bearings, with respect to the dynamic and seismic performance of cable-stayed bridges. This type of bridge, which has relatively greater flexibility, is more susceptible to undesirable vibrations due to service and environmental loadings than are conventional bridges. Therefore, damping is a very important property. Supplementary damping devices based on the plastic deformation of lead and steel are proposed at critical zones, such as the deck–abutment and deck–tower connections, to concentrate hysteretic behaviour in these specially designed energy absorbers. Inelastic behaviour in primary structural elements of the bridge can therefore be avoided, assuring the serviceability of these cable-supported bridges. Analytically, three-dimensional modelling is developed for the bridge and the damping devices, including the bridge geometrical large-displacement non-linearity and the local material and geometric non-linearities of the energy dissipation devices. The effects of various modelling and design parameters of the bridge response are also studied, including the properties, modelling accuracy and location of the devices along the bridge superstructure. It is shown that an optimum model of the seismic performance of the bridges with these passive control devices can be obtained by balancing the reduction in forces along the bridge against tolerable displacements. Appropriate locations and hysteretic energy dissipation properties of the devices can achieve a significant reduction in seismic-induced forces, as compared to the case with no dampers added, and relatively better control of displacements. In addition, proper selection of the location of the passive control systems can help redistribute forces along the structure which may provide solutions for retrofitting some existing bridges. However, caution should be exercised in simulating the device response for a reliable bridge structural performance. Moreover, while seismic response of the bridge can be significantly improved with added dampers, their degree of effectiveness also depends on the energy absorption characteristics of the dampers.     

Approximating peak responses in seismically isolated buildings using generalized modal analysis

Although the ability to simulate accurately the detailed behavior of nonlinear isolation bearings and the effects of this nonlinearity on dynamic response of the isolated building is desirable, such detailed analyses are not feasible during initial design stages when bearing properties are being selected. However, it would be very beneficial to be able to estimate accurately key engineering demand parameters at the early stages of design to understand the dynamic response characteristics of the isolated structure and to balance and optimize the bearing and structural characteristics to achieve the performance goals set for the building. Unfortunately, classical modal response spectrum analysis methods do not provide accurate results for problems with large, nonclassical damping, as is characteristic of isolated buildings. To find a method capable of predicting peak building responses even with large nonclassical damping, generalized modal response spectrum analysis is implemented. The responses of several buildings having different heights and isolated by linear viscous as well as triple friction pendulum and single friction pendulum isolation systems are investigated. Generalized modal response spectrum analysis methods were found to give significantly better predictions for all systems compared with classical methods. The behavior of buildings isolated with single friction pendulum systems exhibiting sudden changes in stiffness could not be well predicted by either general or classical modal response spectrum analysis when effective damping was increased. Copyright © 2013 John Wiley & Sons, Ltd.     

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。