Dynamic analysis of sliding structures with unsynchronized support motions

The dynamic analysis of sliding structures is complicated due to the presence of friction. Synchronization of the kinematics of all the isolation bearings is often granted to simplify the task. This, however, may lead to inaccurate prediction of the structural responses under certain circumstances. Stepped structures or continuous bridges with seismic isolation are notable examples where unsynchronized bearing motions are expected. In this paper, a logically simple and numerically efficient procedure is proposed to solve the dynamic problem of sliding systems with unsynchronized support motions. The motion equations for the sliding and non‐sliding modes of the isolated structure are unified into a single equation that is represented as a difference equation in a discrete‐time state‐space form and the base shear forces between the sliding interfaces can be determined through simple matrix algebraic analysis. The responses of the sliding structure can be obtained recursively from the discrete‐time version of the motion equation with constant integration time step even during the transitions between the non‐sliding and sliding phases. Therefore, both accuracy and efficiency in the dynamic analysis of the highly non‐linear system can be enhanced to a large extent. Rigorous assessment of seismic structures with unsynchronized support motions has been carried out for both a stepped structure and a continuous bridge. Effectiveness of friction pendulum bearings for earthquake protection of such structures has been verified. Moreover, evident unsynchronized sliding motions of the friction bearings have been observed, confirming the necessity to deal with each of the bearings independently in the analytical model. Copyright © 2000 John Wiley & Sons, Ltd.     

Seismic response of the superstructure and attached equipment in a base-isolated building

Base isolation can be used both to protect the structure and simultaneously to reduce the response of internal equipment. The seismic response of a base-isolated structure has been studied through the shaking table test or numerical calculation before. The object of this paper is to analyse a base-isolated structure by a different analytical approach—perturbation analysis. Recognizing that the horizontal stiffness of an isolation system is much smaller than that of the superstructure, the mathematical expressions of the modal properties of base-isolated structures are derived by the perturbation method in terms of the modal properties of the superstructure and used to study the dynamic response of superstructure and attached equipment in the base-isolated building. This study shows that the first base-isolated mode not only controls the superstructural response but also dominates the response of high-frequency attachment. The contribution of higher modes to the response of base-isolated structures, which is proportional to the horizontal stiffness of isolation system, is very small.     

Non-linear seismic response analysis of soil-structure interaction systems

This paper presents an effective analysis procedure for the dynamic soil-structure interaction problem considering not only the sliding and separation phenomena but also the non-linear behaviour of soil by the finite element method. Soil is assumed to be an elasto-plastic material and the contact surface between the soil and structure is modelled by the joint element. The load transfer method is adopted to carry out dynamic non-linear response analysis. The method is applied to the response analysis of a nuclear reactor building resting on the ground surface. The effects of non-linear behaviour of soil on the safety against sliding of the structure are examined. The numerical computations reveal the following results: that the non-linear behaviour of soil reduces the response of the system and the magnitude of sliding of the structure, and that the safety against sliding obtained by the proposed method is higher than the safety obtained by classical methods. This implies the possibility of a more rational and economical design of large structures; it can be said that the proposed method provides useful information for the stability analysis of important and large structures.     

Efficient mode superposition algorithm for seismic analysis of non-linear structures

A step-by-step integration method is proposed to compute within the framework of the conventional mode superposition technique the response of bilinear hysteretic structures subjected to earthquake ground motions. The method is computationally efficient because only a few modes are needed to obtain an accurate estimate of such a response, and because it does not require the use of excessively small time steps to avoid problems of accuracy or stability. It is developed on the basis that the non-linear terms in the equations of motion for non-linear systems may be considered as additional external forces, and the fact that by doing so such equations of motion can be interpreted as the equations of motion of an equivalent linear system, excited by a modified ground motion. These linear equations are then subjected to a conventional modal decomposition and transformed, as with linear systems, into a set of independent differential equations, each representing the system's response in one of its modes of vibration. To increase the efficiency of the method and account properly for the participation of higher modes, these independent equations are solved using the Nigam-Jennings technique in conjunction with the so-called mode acceleration method. The accuracy and efficiency of the method is verified by means of a comparative study with solutions obtained with a conventional direct integration method. In this comparative study, including only a few modes, the proposed method accurately predicts the seismic response of three two-dimensional frame structures, but requiring only, on an average, about 47 per cent less computer time than when the direct integration method is used.     

An incremental mode-superposition for non-linear dynamic analysis

This paper uses an incremental mode-superposition procedure to compute the inelastic dynamic response of multi-degree-of-freedom systems. A damping matrix proportional to the instantaneous properties is used throughout the analysis. The non-linear response of several shear type plane and space frames with elastic-plastic and bilinear column properties subjected to ground excitation was computed by both the incremental mode-superposition and the direct integration of the coupled equations of motion. When all modes are considered, the responses computed by the incremental mode-superposition are identical to those from the direct integration. Fewer modes can also be used to compute the response with reasonable accuracy by performing the modal truncation for each time increment. The study shows that incorporating instantaneous damping in non-linear dynamic analysis is relatively simple and requires less computational time than the direct integration.     

A modal procedure for seismic analysis of non-linear base-isolated multistorey structures

A modal procedure for non-linear analysis of multistorey structures with high-damping base-isolation systems was proposed. Two different isolation devices were considered in the analysis: an high-damping laminated rubber bearing and a lead-rubber bearing. Starting from deformational properties verified by tests, the isolation systems were characterized using three different analytical models (an Elastic Viscous, a Bilinear Hysteretic and a Wen's Model) with parameters depending from maximum lateral strain. After non-linear modelling of isolation and lateral-force-resisting systems, the effects of material non-linearities were considered as pseudo-forces applied to the equivalent linear system (Pseudo-Force Method) and the formally linearized equations of motion were uncoupled by the transformation defined by the complex mode shapes. The modal responses were finally obtained with an extension of Nigam–Jennings technique to non-linear and non-classically damped systems, in conjunction with an iterative technique searching for non-linear contributions satisfying equations of motion and constitutive laws. Since the properties of the isolated structure usually change with maximun lateral strain of isolation bearings, the integration of a new set of governing equations was required for each design-displacement value. The procedure proposed was described in detail and then applied for the determination of modal and total seismic responses in some real cases. At first, a very good agreement between non-linear responses obtained with the proposed mode superposition and with a direct integration method was observed. Then a comparison of results obtained with the three different analytical models of the isolation bearings was carried out. At last, the exact modal response obtained with analytical models depending from the design displacement of the isolation bearings was compared with two different approximated solutions, evaluated using mode shapes and isolation properties, respectively, calculated under simplified hypothesis.© 1998 John Wiley & Sons, Ltd.     

Simplified inelastic seismic analysis of base‐isolated structures using the N2 method

In the paper a simplified nonlinear method has been applied to the analysis of base‐isolated structures. In the first part, a three‐linear idealization of the capacity curve is proposed. The initial stiffness is defined based on the first yielding point in the superstructure, whereas the secondary slope depends on the failure mechanism of the superstructure. A consequence is a much more pronounced secondary slope, which does not correspond to the presumptions used in the originally proposed N2 method. A parametric nonlinear dynamic study of single degree of freedom systems with different hardening slopes and damping has been performed for an ensemble of seven EC8 spectrum‐compatible artificial accelerograms. It was concluded that, in the long‐period range, the equal displacement rule could be assumed also for the proposed systems with non‐zero post‐yield stiffness. In the second part, the proposed idealization was used for the analysis of isolated RC frame buildings that were isolated with different (lead) rubber‐bearing isolation systems. The stiffness of the isolators was selected for three different protection levels and for three different ground motion intensities, which have resulted in elastic as well as moderately and fully damaged superstructure performance levels. Three different lateral load distributions were investigated. It was observed that a triangular distribution, with an additional force at the base, works best in the majority of practical cases. It was concluded that the N2 method can, in general, provide a reasonably accurate prediction of the actual top displacement, as well as of the expected damage to the superstructure. Copyright © 2009 John Wiley & Sons, Ltd.     

组合基础隔震在建筑工程中的应用

隔震作为一种新的抗震技术,已广泛应用于新建和加固的建筑工程,同时,许多新型式的支座得到了开发和应用。组合基础隔震是一种新的隔震设计思想,能充分应用不同类型隔震支座的特性,有效降低上部结构地震反应。本文介绍了组合基础隔震在某一工程中的应用,工程中使用的支座包括普通橡胶隔震支座、铅芯橡胶隔震支座和弹性滑板支座三种类型,对全部使用支座进行了常规检测,结构计算采用等效线性法、能量包络法和时程反应分析等方法,计算结果表明:组合基础隔震能有效降低上部结构的反应,隔震层的变形控制在安全范围之内。     

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.     

双向地震波作用下巨子型有控结构建筑的隔震设计

为了增强巨子型有控结构建筑的动力特性,提升其稳定性,设计双向地震波作用下建筑有控结构。采用3种磁流变阻尼器(MRD)与滑移隔震混合控制结构构成单体建筑有控结构,其包括巨结构和子结构,并建立该有控结构的动力分析模型。在动力分析模型中输入水平和竖向地震,得到模型的竖向和水平滑动状态运动微分方程,依据这两个方程采用自适应模糊神经网络优化动力分析模型,构建优化模型。从优化模型出发,通过实例实验分析得出,优化设计双向地震波作用下建筑有控结构时,在其上部结构层间和隔离层各安装一个MRD,可确保优化设计后的有控结构在不同双向地震工况下的地震反应控制效果最佳,且有控结构在双向地震工况2下,结构第一层、中间三层以及顶层的加速度和位移的时程曲线走向一致,且差距微小;同时有控结构的巨结构顶层侧移响应随着子结构刚度增加而提高,动力特性没有明显的变化,子结构随着其自身刚度增加顶层侧移响应表现稳定,子结构动力特性增强。     

Semi-analytical techniques for the determination of the dynamic response of equipment in structures supported on coulomb friction elements

Techniques for the analysis of equipment in structures supported on a Coulomb friction type support that is subjected to harmonic and earthquake ground motion are presented. The behaviour is governed by two phases—a sliding phase and a non-sliding phase. Since the behaviour in each of the phases is linear, an analytical expression for equipment response can be obtained in terms of the roots of an appropriate characteristic polynomial. The times of phase transition are determined by an iterative scheme. The methodology is accurate, less computationally intensive, and avoids the difficulties that can be encountered with standard numerical integration techniques for highly non-linear systems.     

Seismic response control of smart sliding isolated buildings using variable stiffness systems: an experimental and numerical study

Effectiveness of a new semiactive independently variable stiffness (SAIVS) device in reducing seismic response of sliding base isolated buildings is evaluated analytically and experimentally. Through analytical and experimental study of force—displacement behaviour of the SAIVS device, it is shown that the device can vary stiffness continuously and smoothly between minimum and maximum stiffness. Passive sliding base isolation systems reduce interstorey drifts and superstructure accelerations, but with increased base displacements, which is undesirable, under large velocity near fault pulse type earthquakes. It is a common practice to incorporate non‐linear passive dampers into the isolation system to reduce bearing displacements. Incorporation of passive dampers, however, may result in increased superstructure accelerations and drifts; while, properly designed passive dampers can be beneficial. A viable alternative is to use semiactive variable stiffness systems, which can vary the period of the sliding base isolated buildings in real time, to simultaneously reduce bearing displacements and superstructure responses further than the passive systems, which deserves investigation. This study investigates the performance of a 1:5 scaled smart sliding base isolated building model equipped with the SAIVS device analytically and experimentally, under near fault earthquakes, by developing a new moving average non‐linear tangential stiffness control algorithm for control of the SAIVS device. The SAIVS device reduces bearing displacements further than the passive cases, while maintaining isolation level forces and superstructure responses at the same level as the passive minimum stiffness case, indicating the significant potential of the SAIVS system. Copyright © 2005 John Wiley & Sons, Ltd.     

Pounding of superstructure segments in isolated elevated bridge during earthquakes

Past severe earthquakes indicate that pounding may cause considerable damage or even lead to collapse of colliding structures. The aim of this paper is to present an analysis of pounding between superstructure segments of an isolated elevated bridge induced by the propagating seismic wave. High-damping rubber bearings (HDRBs), used as isolation devices, are modelled by proposed non-linear formulation and the significance of the bearings model for pounding is indicated. The results of the study show that pounding leads to the increase or decrease of the forces acting on piers, depending on the gap size between superstructure segments. © 1998 John Wiley & Sons, Ltd.     

The applicability of the N2 method to the estimation of torsional effects in asymmetric base‐isolated buildings

The paper deals with the applicability of the extended N2 method to base‐isolated plan‐asymmetric building structures. The results obtained by the usual pushover analysis of a 3D structural model are further combined with the aid of linear dynamic (spectral) analysis to account for the dynamic effects caused by structural asymmetry. In the paper, the method has been applied to the seismic analysis of a mass‐eccentric four‐storeyed RC frame building isolated with lead rubber bearings. Three different positions of the center of isolation system (CI) with respect to the center of mass (CM) and the center of stiffness of the superstructure (CS) were considered. The response was analyzed for three different eccentricities, three different torsional to lateral frequency ratios of the superstructure, and two ground motion intensities. The stiffness of the isolators was selected for three different protection levels, which resulted in elastic as well as moderately to excessively damaged superstructure performance levels. The results are presented in terms of the top, base and relative displacements, as well as the stiff/flexible side amplification factors. A more detailed insight into the nonlinear behavior of the superstructure is given in a form of ductility factors for the flexible and stiff side frames. The results of the extended N2 method for selected lateral load distributions are compared with the average results of nonlinear dynamic analyses. It was concluded that the extended N2 method could, with certain limitations, provide a reasonable prediction of the torsional influences in minor to moderately asymmetric base‐isolated structures. Copyright © 2010 John Wiley & Sons, Ltd.     

单轴平扭耦联基础隔震结构动力简化分析

为了建立单轴平扭耦联基础隔震结构的动力简化分析方法,探讨了影响结构扭转反应的参数取值规律,首先基于层单元模型,通过假定上部结构楼层回转半径、偏心距、弹力半径相等,推导了单轴平扭耦联基础隔震结构线性化的动力计算方程;其次,运用该简化分析方程,通过一算例进行了动力响应的参数分析。结果表明:调整隔震层刚心使其与上部结构质心位置接近,可显著降低偏心隔震结构扭转反应;增大隔震层刚度半径及阻尼半径可有效减少或抑制结构扭转反应;所建简化分析方程能有效模拟偏心隔震结构动力响应。     

Analytical and experimental studies on midstory isolated buildings with modal coupling effect

In a midstory isolated building, the isolation system is incorporated into the midstory rather than the base of the building. Because of the flexibility of the substructure below the isolation system in a midstory isolated building, the contribution of higher modes to the seismic responses of the midstory isolated structure may not be negligible, especially when the coupling of higher modes exists. To investigate this modal coupling effect, a simplified three‐lumped‐mass structural model of the midstory isolated building is assumed in this study. Through the equivalent linear analysis and shaking table tests, it is found that the coupling of higher modes may lead to enlarged acceleration responses at the super‐floor and superstructure above the isolation layer. Accordingly, a simple method to prevent the midstory isolation design from the coupling of higher modes attributed to the improper design of the substructure and superstructure is proposed in this paper. Copyright © 2012 John Wiley & Sons, Ltd.     

脉冲型地震动作用下隔震结构动力响应的影响参数研究

通过对隔震结构进行非线性动力响应分析,分别研究地震动参数和支座参数对结构地震响应的影响。首先,建立铅芯橡胶支座基础隔震结构的非线性运动方程;然后,以人工合成脉冲型地震动作为输入,运用MATLAB进行编程并求解结构在脉冲型地震动作用下的地震响应;最后,分别研究速度脉冲周期、支座屈服力、屈服后与屈服前的刚度比对隔震支座最大位移和上部结构层间位移的影响。研究结果表明,脉冲周期对结构地震响应影响很大,在进行隔震设计时应使结构自振周期远离脉冲周期;支座刚度比对结构地震响应影响较大,在进行支座选型时应重点关注;支座屈服力对支座位移的影响显著,屈服力越大,支座位移越小。     

Experimental assessment of sliding materials for seismic isolation systems

Novel self-lubricating materials are currently being proposed to be used in sliding isolation systems such as the curved surface sliding isolator system, or pendulum bearing system, for the protection of buildings and structures. The current codes for anti-seismic devices are focused on the evaluation of the performance of the whole isolation system; as a consequence, a reliable procedure for the pre-assessment of the material combinations of sliding interfaces is currently missing. Therefore in this paper, an experimental methodology is proposed for the characterization of self-lubricating materials through tests on small-scale specimens performed using customized equipment able to reproduce the operational conditions of real isolation systems as per contact pressure, sliding velocity, temperature and slide path. The testing sequence has been designed in order to evaluate the sliding properties of the material in terms of static and dynamic coefficient of friction and wear resistance. Examples are reported for the assessment of two self-lubricating materials with different sliding characteristics. In order to validate the method and to confirm the reliability of extrapolating the results to real working conditions, prototypes of pendulum isolation systems incorporating the assessed materials have been tested according to the AASHTO (Guide specifications for seismic isolation design, 2nd edn. American Association of State Highway and Transportation Officials, Washington, DC, 1999) specifications and the relevant dynamic properties assessed from the Horizontal Load—Displacement loops. The experimental outcomes confirmed that the frictional characteristics provided by the proposed procedure can be reliably used in the design of seismically isolated structures.     

An imposed force summation method for non-linear dynamic analysis

A method of obtaining the response of a non-linear system is proposed, which involves imposing the solution upon the equations of motion for a linear system and solving using selected modes of vibration. Contribution from the remaining modes of the system is included in terms of their quasi-static response. The method is developed and illustrated with reference to a system with material non-linearity and special reference is given to the dynamic analysis of embankment dams.