Numerical technique for dynamic analysis of structures with friction devices

In recent years a number of studies on employing friction elements for the seismic protection of buildings has demonstrated conclusively that such devices can markedly reduce earthquake-induced vibrations. Any numerical estimate of the effectiveness of such isolation systems implies a correct solution of the pertinent nonlinear equations of motion. In direct integration algorithms, the phase transitions between adherence and sliding, or the sliding phase may be accompanied by marked high-frequency oscillation of the relative velocity difference. The paper presents a numerical technique for overcoming these problems, thus leading to increased accuracy of the solutions of equations of motions with Coulomb damping. Since only the damping matrix and the loading vector are involved, the procedure is also computationally efficient. In order to validate the proposed numerical technique, an experimental study of a friction system has been carried out. The dynamic response of a four-storey braced frame with friction devices is presented as an example for the practical application of the proposed numerical technique.     

A model of triple friction pendulum bearing for general geometric and frictional parameters

Current models describing the behavior for the triple friction pendulum (TFP) bearing are based on the assumption that the resultant force of the contact pressure acts at the center of each sliding surface. Accordingly, these models only rely on equilibrium in the horizontal direction to arrive at the equations describing its behavior. This is sufficient for most practical applications where certain constraints on the friction coefficient values apply as a direct consequence of equilibrium. This paper presents a revised model of behavior of the TFP bearing in which no assumptions are made on the location of the resultant forces at each sliding surface and no constraints on the values of the coefficient of friction are required, provided that all sliding surfaces are in full contact. To accomplish this, the number of degrees of freedom describing the behavior of the bearing is increased to include the location of the resultant force at each sliding surface and equations of moment equilibrium are introduced to relate these degrees of freedom to forces. Moreover, the inertia effects of each of the moving parts of the bearing are accounted for in the derivation of the equations describing its behavior. The model explicitly calculates the motion of each of the components of friction pendulum bearings so that any dependence of the coefficient of friction on the sliding velocity and temperature can be explicitly accounted for and calculations of heat flux and temperature increase at each sliding surface can be made. This paper presents (a) the development of the revised TFP bearing model, (b) the analytic solution for the force–displacement relations of two configurations of the TFP bearing, (c) a model that incorporates inertia effects of the TFP bearing components and other effects that are useful in advanced response history analysis, and (d) examples of implementation of the features of the presented model. Copyright © 2016 John Wiley & Sons, Ltd.     

Characterizing friction in sliding isolation bearings

The force–displacement behavior of the Friction Pendulum? (FP) bearing is a function of the coefficient of sliding friction, axial load on the bearing and effective radius of the sliding surface. The coefficient of friction varies during the course of an earthquake with sliding velocity, axial pressure and temperature at the sliding surface. The velocity and axial pressure on the bearing depend on the response of the superstructure to the earthquake shaking. The temperature at an instant in time during earthquake shaking is a function of the histories of the coefficient of friction, sliding velocity and axial pressure, and the travel path of the slider on the sliding surface. A unified framework accommodating the complex interdependence of the coefficient of friction, sliding velocity, axial pressure and temperature is presented for implementation in nonlinear response‐history analysis. Expressions to define the relationship between the coefficient of friction and sliding velocity, axial pressure, and temperature are proposed, based on available experimental data. Response‐history analyses are performed on FP bearings with a range of geometrical and liner mechanical properties and static axial pressure. Friction is described using five different models that consider the dependence of the coefficient of friction on axial pressure, sliding velocity and temperature. Frictional heating is the most important factor that influences the maximum displacement of the isolation system and floor spectral demands if the static axial pressure is high. Isolation system displacements are not significantly affected by considerations of the influence of axial pressure and velocity on the coefficient of friction. Copyright © 2014 John Wiley & Sons, Ltd.     

Hysteretic models for sliding bearings with varying frictional force

The friction pendulum system is a sliding seismic isolator with self‐centering capabilities. Under severe earthquakes, the movement may be excessive enough to cause the pendulum to hit the side rim of the isolator, which is provided to restrain the sliding. The biaxial behavior of a single friction pendulum, in which the slider contacts the restrainer, is developed using a smooth hysteretic model with nonlinear kinematic hardening. This model is extended to simulate the biaxial response of double and triple friction pendulums with multiple sliding surfaces. The model of a triple friction pendulum is based on the interaction between four sliding interfaces, which in turn is dependent upon the force and displacement conditions prevailing at these interfaces. Each of these surfaces are modeled as nonlinear biaxial springs suitable for a single friction pendulum, using the yield surface, based on the principles of the classical theory of plasticity, and amended for varying frictional yield force, due to variation in vertical load and/or velocity‐dependent friction coefficient. The participation of the nonlinear springs is governed by stick‐slip conditions, dictated by equilibrium and kinematics. The model can simulate the overall force‐deformation behavior, track the displacements in individual sliding surfaces, and account for the ultimate condition when the sliders are in contact with their restrainers. The results of this model are verified by comparison to theoretical calculations and to experiments. The model has been implemented in programs IDARC2D and 3D‐BASIS, and the analytical results are compared with shake table experimental results. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical modeling of MDOF structures with sliding supports using rigid‐plastic link

In this paper, the responses of multi‐degree‐of‐freedom (MDOF) structures on sliding supports subjected to harmonic or random base motions are investigated. Modeling of the friction force under the foundation raft is accomplished by using a fictitious rigid link which has a rigid–perfectly plastic material. This will result in identical equations of motion for the sliding structure, both in the sliding and non‐sliding (stick) phases which greatly simplifies the implementation of the method into a numerical algorithm. In this model the phase transition times are determined with high accuracy. This has two advantages: first, it prevents the so‐called high‐frequency oscillation of the relative velocity at the end of the sliding phase, and second, the time steps can be selected so that each falls exactly within one phase of motion. In this case, the stiffness matrix of the structure remains constant throughout each phase and thus any method for solving the non‐linear differential equations of motion (e.g. Newmark method) can be used without iteration. The proposed method, besides its simplicity, is numerically very efficient and considerably reduces the required analysis time compared with most of the other methods. Copyright © 2001 John Wiley & Sons, Ltd.     

A new friction law for sliding rigid blocks under cyclic loading

A rigid block sliding down an inclined plane under the action of gravity was monitored with accelerometers and an LVDT to investigate how the transition from static to kinetic friction develops. Once the transition patterns were identified, experiments were carried out to study the response of a dynamically excited rigid block sliding down the inclined plane of a shaking table. Harmonic time series were used as input motions. The laboratory results allowed the definition of a continuous friction law to model the continuous variation of the friction from its static to kinetic condition. This law was implemented into the commercial 3D distinct element code 3DEC to numerically reproduce the experiments carried out, thus validating the friction law. Afterwards, the friction law was used to evaluate the sliding due to the kinetics of the block. Three cases were analyzed: constant friction coefficient, Coulomb friction law and the proposed friction law. These computations were compared to laboratory measurements. It is found that permanent displacements computed by considering the new law are in better agreement with laboratory measurements.     

基于速度-状态摩擦本构定律的三维DDA方法

原有三维非连续变形分析(DDA)方法采用常摩擦系数的Mohr-Coulomb定律作为切向破坏准则,然而当描述更大尺度构造块体的运动与变形时,常摩擦系数不再适用.速度-状态摩擦本构定律能够定量描述地震周期各阶段断层面剪应力变化,解释发震断层行为.本文将速度-状态摩擦定律与三维DDA方法相结合,首先推导了计算摩擦系数的实用公式,随后通过滑动-保持-滑动实验与速度步进实验算例对改进的三维DDA方法进行了验证.结果表明,应用速度-状态摩擦本构定律的三维DDA方法能够比较准确地模拟静摩擦的时间依赖性与动摩擦的速度依赖性,解决了将三维DDA方法在地学中应用的基本问题.     

Investigation of the sliding behavior between steel and mortar for seismic applications in structures

The friction developed between a steel base plate and a mortar base contributes shear resistance to the building system during a seismic event. In order to investigate the possible sliding behavior between the base plate and the mortar, a shake table study is undertaken using a large rigid mass supported by steel contact elements which rest on mortar surfaces connected to the shake table. Horizontal input accelerations are considered at various magnitudes and frequencies. The results provide a constant friction coefficient during sliding with an average value of approximately 0.78. A theoretical formulation of the friction behavior is also undertaken. The theoretical equations show that the sliding behavior is dependent on the ratio of the friction force to the input force. The addition of vertical accelerations to the system further complicates the sliding behavior as a result of the varying normal force. This results in a variable friction resistance which is a function of the amplitude, phase, and frequency of the horizontal and vertical input motions. In general, this study showed a consistent and reliable sliding behavior between steel and mortar. Copyright © 2009 John Wiley & Sons, Ltd.     

Periodic response of a sliding oscillator system to harmonic excitation

This paper deals with the periodic response of an oscillating system which is supported on a frictional interface. The base excitation is assumed harmonic and the frictional force is assumed to be of the Coulomb type. Though each segment of the motion of such a system is described by linear equations, its complete response is highly non-linear and varied. The most fundamental periodic solutions are derived analytically and numerically. The results indicate that such a system has several subharmonic resonant frequencies and that while the friction reduces the peak response of the system when it is excited at its ‘fixed-base’ natural frequency, ω_n, the sliding can induce considerably higher levels of response, when compared with those of a non-sliding, fixed-base system, for frequencies less than ω_n. The results obtained herein may find application in the area of vibration isolation.     

Dynamic loading tests and seismic response analysis of a stud-type damper composed of multiple friction units with disc springs

This paper presents an experimental study on the performance of a shear-sliding stud-type damper composed of multiple friction units with high-tension bolts and disc springs. A numerical evaluation of the response reduction effects achieved by the stud-type damper is also presented. In dynamic loading tests, the behavior of stud-type multiunit friction damper specimens was investigated. Three different full-scale damper specimens, which were composed of five, six, or seven friction units with two or four sliding surfaces, were incorporated into loading devices for testing. The stud-type friction dampers demonstrated stable rigid-plastic hysteresis loops without any remarkable decrease in the sliding force even when subjected to repetitive loading, in addition to showing no unstable behavior such as lateral buckling. The damper produced a total sliding force approximately proportional to the number of sliding surfaces and friction units. The total sliding force of the stud-type damper can thus be estimated by summing the contributions of each friction unit. In an earthquake response simulation, the control effects achieved by stud-type dampers incorporated into an analytical high-rise building model under various input waves, including long-period, long-duration and pulse-like ground motions, were evaluated. A satisfactory response reduction was obtained by installing the developed stud-type dampers into the main frame without negatively impacting usability and convenience in terms of building planning.     

Dynamics of resilient-friction base isolator (R-FBI)

The Resilient-Friction Base Isolator (R-FBI) is composed of a set of flat rings which can slide on each other with a central rubber core and/or peripheral rubber cores. In this base isolator the interfacial friction force acts in parallel with the elastic force in the rubber. It combines the beneficial effect of friction damping with that of the resiliency of rubber. The rubber cores distribute the sliding displacement and velocity along the height of the R-FBI. They do not carry any vertical loads and are not vulcanized to the sliding rings. The system's analytical model and the computer experimental results for both horizontal and vertical components of recorded ground motions and various levels of friction and damping are presented. These results clearly demonstrate the R-FBI's potential as an effective aseismic base isolator.     

Seismic response of simple structures using spring-assisted sliding slabs with coulomb damping

Coulomb damping can be utilized effectively to reduce the dynamic response of structures subjected to seismic ground motions. To activate this damping, some parts of a vibrating structure are allowed to slide at rough interfaces. The dynamic response of structures provided with sliding interfaces at the base, between a floor slab and frame and in the cross bracings of a frame has been examined recently. In this paper, a simple slab sliding system provided with a spring to introduce a recovery mechanism and to reduce the sliding displacement requirement for low frequency structures has been examined. The equations of motion for this system are developed. An approach is presented to solve these coupled equations for earthquake induced ground motions. Structures with varying frequency and friction characterisics are considered and the numerical results are presented in response spectrum form. It is observed that, in low frequency structures, provision of a rather weak spring can reduce the sliding displacement requirements without significantly increasing the forces in the supporting frame and the acceleration input to supported secondary systems.     

浮放设备地震滑移反应数值模拟研究

以前人研究建立的浮放设备滑移反应分析理论为基础,给出了浮放设备在水平和竖向地震输入下滑移反应运动微分方程,选取连续型的库仑摩擦力模型,采用Runge-Kutta法求解浮放设备地震滑移反应运动微分方程,可以得到浮放设备在地震作用下的绝对加速度、相对速度和相对位移反应时程。采用编制的计算程序,进行了浮放设备地震滑移反应参数影响研究,结果表明:浮放设备水平滑移反应随着水平地震地面输入或楼层反应输入的增大而增大,随着设备与支撑面之间摩擦系数的增大而减小;当水平向输入加速度峰值大于0.3g时,需要考虑竖向激励的影响。     

Proposed design models for the asymmetric friction connection

The Asymmetric Friction Connection (AFC) remains elastic during moderate earthquake shaking but slides and dissipates energy through friction during severe earthquake shaking. The sliding friction forces developed are dependent on the clamping force in the connection which is provided by fully tensioned bolts which pass through slotted holes. During sliding these bolts are subject to moment and shear as well as axial force. Moment–shear–axial force interaction reduces the clamping axial force on the sliding interfaces thereby reducing the sliding shear resistance (V_ss). Two methods to evaluate the moment–shear–axial force interaction have been proposed so that the sliding shear strength can be quantified, but as yet, these methods are not robust. This paper describes the results of 60 tests undertaken to improve the two methods, namely the moment–shear–axial force bolt model and the effective coefficient of friction method, for AFCs with high hardness steel shims. The bolts were M16 to M30 bolts and cleat thicknesses ranged from 12 mm to 25 mm. It is shown that either method may be used in design as the results obtained are similar. Copyright © 2014 John Wiley & Sons, Ltd.     

新型滑移隔震结构模型的振动台试验研究

利用实体软钢棒作为消能限位装置,将一种摩擦性能优良的二硫化钼材料作为隔震支座的滑移材料,提出并制作了一种可以应用于框架结构既能隔震又可以消能的新型摩擦滑移隔震装置。探讨了其设计方法和应用方法,并对安装了该新型摩擦滑移隔震装置的一相似比为1:5的5层框架结构模型进行了振动台试验,测试了框架结构在单向地震波作用下的地震反应规律,分析了摩擦滑移隔震结构的加速度反应、层间剪力反应、隔震层滑移量及隔震层剪力的变化规律。结果表明:一般情况下当设防烈度为8度,Ⅱ类场地时,该隔震结构的加速度响应可降低50%左右,层间剪力响应可降低50%左右,减震效果比较明显。另外,只要确定合理的构造方案和实施方案,这种新型摩擦滑移隔震装置就能满足框架结构的隔震减震要求,可应用于实际工程结构中。     

MODULATED HOMOGENEOUS FRICTION: A SEMI-ACTIVE DAMPING STRATEGY

A control strategy for semi-active friction devices leading to efficient hysteretic dissipaters is proposed. The control algorithm makes the contact force between the sliding surfaces of the damper proportional to the absolute value of the prior local peak of the damper deformation. This control logic leads to a non-linear force–deformation relation that satisfies homogeneity of degree one; this means that, like in a linear viscoelastic damping model, when the deformation is scaled by a constant, the force results are scaled by the same constant. The closed-loop system shows rectangular hysteresis loops which enclose an area proportional to the square of the deformation of the damper. Some characteristics of the dynamic response of structures incorporating this type of semi-active damper are investigated. It is demonstrated that in the case of single-degree-of-freedom models, the period of vibration and decay ratio are independent of the amplitude of vibration. In the case of multi-degree-of-freedom models with this type of nonlinearity, the free-vibration response can exhibit natural modes of vibration. A linearization method is proposed and modelling tools for the delay associated with actuator dynamics and for the flexibility of the brace connecting the damper to the structure are presented. © 1997 by John Wiley & Sons, Ltd.     

竖向地震动对滑动摩擦隔震桥梁结构地震反应的影响

利用接触摩擦单元建立了滑动摩擦隔震桥梁支座在竖向地震动作用下的有限元模型,通过算例讨论了竖向地震动对不同支座摩擦系数和不同刚度(墩径)滑动摩擦隔震桥梁地震反应的影响,初步探讨了竖向地震动的激励方向对分析结果的影响,分析结果表明,竖向地震动对滑动摩擦隔震桥梁结构的地震反应有较大的影响,在竖向地震动较为明显地区的滑动摩擦隔震桥梁结构设计时,应予以考虑。     

Incremental dynamic analysis of concrete gravity dams including base and lift joints

The growth in computer processing power has made it possible to use time-consuming analysis methods such as incremental dynamic analysis(IDA) with higher accuracy in less time.In an IDA study,a series of earthquake records are applied to a structure at successively increasing intensity levels,which causes the structure to shift from the elastic state into the inelastic state and finally into collapse.In this way,the limit-states and capacity of a structure can be determined.In the present research,the IDA of a concrete gravity dam considering a nonlinear concrete behavior,and sliding planes within the dam body and at the dam-foundation interface,is performed.The influence of the friction angle and lift joint slope on the response parameters are investigated and the various limit-states of the dam are recognized.It is observed that by introducing a lift joint,the tensile damage can be avoided for the dam structure.The lift joint sliding is essentially independent of the base joint friction angle and the upper ligament over the inclined lift joint slides into the upstream direction in strong earthquakes.     

弹性滑移支座力学性能的试验研究

介绍了弹性滑移支座的原理、构造和特点;通过对其在不同工况下的性能试验,研究了竖向荷载、位移幅值以及加载频率对弹性滑移支座力学性能的影响,并给出了试验值与理论计算值之间的对比。研究结果表明：弹性滑移支座具有良好的工作性能,滞回曲线饱满,耗能能力强;竖向荷载和加载频率对弹性滑移支座的力学性能有一定的影响,而位移幅值对其影响较小;弹性滑移支座的恢复力模型,可以用考虑速度的指数摩擦力模型来描述,并且模拟得较为准确。