Lumped-parameter model for a rigid cylindrical foundation embedded in a soil layer on rigid rock

To represent a cylindrical rigid foundation vibrating in horizontal, vertical, rocking or torsional motions embedded in a soil layer resting on rigid rock, a lumped-parameter model is described. The coupling between the horizontal and rocking degrees of freedom is considered. For each degree of freedom eight frequency-independent real coefficients determine the springs, dashpots and the mass of the lumped-parameter model with two internal degrees of freedom. These coefficients are specified for various ratios of the radius of the foundation to the depth of the layer and lateral contact ratios. To derive the mechanical properties of the lumped-parameter model a systematic procedure of curvefitting of the dynamic-stiffness coefficient up to, in general, twice the fundamental frequency of the layer is applied, capturing the fact that below the (horizontal) fundamental frequency (cutoff frequency) no radiation of energy occurs. The lumped-parameter model can be used to represent the soil in a standard finite-element program for structural dynamics working in the time domain, whereby the structure can exhibit non-linear behaviour. Stability of the unbounded soil-layer model and of the total system is guaranteed. A hammer foundation with partial uplift of the anvil is analysed for illustration.     

Consistent lumped-parameter models for unbounded soil: Frequency-independent stiffness,damping and mass matrices

A systematic procedure to develop consistent (symmetric) stiffness, damping and mass matrices with real coefficients to represent any unbounded soil is developed. These property matrices are based on the lumped-parameter models of Reference 1. Either stiffness and damping matrices corresponding to first-order differential equations involving the internal degrees of freedom and those on the structure-soil interface result or, alternatively, in addition mass matrices are introduced, corresponding to second-order differential equations, which reduce the number of internal degrees of freedom by a factor 2. The stiffness, damping and mass matrices can easily be incorporated in a general-purpose structural dynamics program working in the time domain, whereby the structure can even be non-linear.     

Lumped-parameter model and recursive evaluation of interaction forces of semi-infinite uniform fluid channel for time-domain dam-reservoir analysis

To calculate the hydrodynamic interaction forces of the reservoir directly in the time-domain, the dynamic stiffness of each mode of the semi-infinite uniform fluid channel is either represented by a lumped-parameter model with frequency-independent real coefficients of the springs, dashpots and masses and with only a few additional internal degrees of freedom, or the interaction forces are calculated recursively. For each mode characterized by its eigenvalue, the coefficients of the lumped-parameter model and the recursive coefficients are specified, which can be used directly in a practical application. The procedures exhibit many advantages: the only approximation (replacing the rigorous dynamic stiffness by a ratio of two polynomials) can be evaluated visibly. No unfamiliar discrete-time manipulations such as the z-transformation are used. The stiffness, damping and mass matrices corresponding to the lumped-parameter model are automatically symmetrical. Stability of the procedures is also guaranteed. Combining the lumped-parameter model of the semi-infinite uniform channel with the finite-element discretization of the irregular fluid region or calculating the interaction forces recursively allows a reservoir of arbitrary shape to be analysed directly in the time domain. Non-linearities in the dam can, thus, be taken into consideration in a seismic analysis.     

Material damping for lumped-parameter models of foundations

In foundation dynamics two mechanisms of energy dissipation exist, wave radiation and material damping. Elastic continuum models of the soil capture only the radiation effect. To incorporate material damping, use is made of the fact that the dynamic-stiffness relationships of all elastic foundations may be simulated by discrete assemblages of springs, dashpots and masses. (Such lumped-parameter models are exact for simple cone models of the soil and approximate for more involved cases.) By application of the correspondence principle directly to the discrete elements of the lumped-parameter model, it is possible to introduce Voigt viscoelasticity. Each original spring is augmented by a dashpot, and each original dashpot is augmented by a mass, attached in a special way. Going a step further, more realistic non-linear-hysteretic damping is represented by replacing the augmenting dashpots and masses by frictional elements. The analysis, which is effected solely in the time domain, is illustrated by an example from earthquake engineering.     

PROPERTY MATRICES IDENTIFICATION OF UNBOUNDED MEDIUM FROM UNIT-IMPULSE RESPONSE FUNCTIONS USING LEGENDRE POLYNOMIALS: FORMULATION

A systematic procedure to construct the (symmetric) static-stiffness, damping and mass matrices representing the unbounded medium is presented addressing the unit-impulse response matrix corresponding to the degrees of freedom on the structure–medium interface. The unit-impulse response matrix is first diagonalized which then permits each term to be modelled independently from the others using expansions in a series of Legendre polynomials in the time domain. This leads to a rational approximation in the frequency domain of the dynamic-stiffness coefficient. Using a lumped-parameter model which provides physical insight the property matrices are constructed.     

Global lumped-parameter model with physical representation for unbounded medium

A systematic procedure to construct a consistent global lumped-parameter model consisting of springs, dashpots and possibly masses with frequency-independent coefficients connecting the degrees of freedom of the nodes of any structure-medium interface for the unbounded medium is presented. The dynamic-stiffness matrix is first diagonalized which then permits each term to be modelled independently from the others. Physical insight is thus provided. Alternatively, the (symmetric) static-stiffness and damping matrices and possibly mass matrix of the unbounded medium can be established directly.     

Discrete models for vertical vibrations of surface and embedded foundations

A five-parameter discrete model that approximates the dynamic force4isplacement relationship for rigid foundations undergoing vertical vibrations on a uniform elastic half-space is presented. The model involves a combination of two springs, two viscous dampers and a mass. Values of the parameters for circular, square and rectangular foundations placed on the surface or embedded in an elastic half-space are listed. The parameters are obtained by minimizing the discrepancy between the force4isplacement relation for the model and that obtained by solution of the mixed boundary-value problem of the rigid foundation on an elastic half-space. The definition of an appropriate input motion to represent wave excitation is also discussed. The input motion to the discrete model differs from the input motion that should be used in a continuum model.     

Dynamic analysis of structures with Maxwell model

A numerical method has been developed for the dynamic analysis of a tall building structure with viscous dampers. Viscous dampers are installed between the top of an inverted V‐shaped brace and the upper beam on each storey to reduce vibrations during strong disturbances like earthquakes. Analytically, it is modelled as a multi‐degree‐of freedom (MDOF) system with the Maxwell models. First, the computational method is formulated in the time domain by introducing a finite element of the Maxwell model into the equation of motion in the discrete‐time system, which is based on the direct numerical integration. Next, analyses for numerical stability and accuracy of the proposed method are discussed. The results show its numerical stability. Finally, the proposed method is applied to the numerical analysis of a realistic building structure to demonstrate its practical validity.     

Predictive control of seismic structures with semi‐active friction dampers

In this paper a predictive control method especially suitable for the control of semi‐active friction dampers is proposed. By keeping the adjustable slip force of a semi‐active friction damper slightly lower than the critical friction force, the method allows the damper to remain in its slip state throughout an earthquake of arbitrary intensity, so the energy dissipation capacity of the damper can be improved. The proposed method is formulated in a discrete‐time domain and cast in the form of direct output feedback for easy control implementation. The control algorithm is able to produce a continuous and smooth slip force for a friction damper and thus avoid exerting the high‐frequency structural response that usually exists in structures with conventional friction dampers. Using a numerical study, the control performance of a multiple degrees of freedom (DOF) structural system equipped with passive friction dampers and semi‐active dampers controlled by the proposed method are compared. The numerical case shows that by merely using a single semi‐active friction damper and a few sensors, the proposed method is able to achieve better acceleration reduction than the case using multiple passive dampers. Copyright © 2004 John Wiley & Sons, Ltd.     

Dynamic characteristics and seismic response of adjacent buildings linked by discrete dampers

Coupling adjacent buildings using discrete viscoelastic dampers for control of response to low and moderate seismic events is investigated in this paper. The complex modal superposition method is first used to determine dynamic characteristics, mainly modal damping ratio and modal frequency, of damper-linked linear adjacent buildings for practical use. Random seismic response of linear adjacent buildings linked by dampers is then determined by a combination of the complex modal superposition method and the pseudo-excitation method. This combined method can effectively and accurately determine random seismic response of non-classically damped systems in the frequency domain. Parametric studies are finally performed to identify optimal parameters of viscoelastic dampers for achieving the maximum modal damping ratio or the maximum response reduction of adjacent buildings. It is demonstrated that using discrete viscoelastic dampers of proper parameters to link adjacent buildings can reduce random seismic responses significantly. Copyright © 1999 John Wiley & Sons Ltd.     

A hybrid time frequency domain formulation for non-linear soil–structure interaction

Methods that combine frequency and time domain techniques offer an attractive alternative for solving Soil–Structure-interaction problems where the structure exhibits non-linear behaviour. In the hybrid-frequency-time-domain procedure a reference linear system is solved in the frequency domain and the difference between the actual restoring forces and those in the linear model are treated as pseudo-forces. In the solution scheme explored in this paper, designated as the hybrid-time-frequency-domain (HTFD) procedure, the equations of motion are solved in the time domain with due consideration for non-linearities and with the unbounded medium represented by frequency-independent springs and dampers. The frequency dependency of the impedance coefficients is introduced by means of pseudo-forces evaluated in the frequency domain at the end of each iteration. A criterion of stability for the HTFD approach is derived analytically and its validity is sustained numerically. As is often the case, the criterion takes the form of a limit of unity on the spectral radius of an appropriately defined matrix. Inspection of the terms in this matrix shows that convergence can be guaranteed by suitable selection of the reference impedance. The CPU times required to obtain converged solutions with the HTFD are found, in a number of numerical simulations, to be up to one order of magnitude less than those required by the alternative hybrid-frequency-time-domain approach. © 1998 John Wiley & Sons, Ltd.     

A simplified analytical model for U-shaped steel dampers considering horizontal bidirectional deformation

Hysteresis steel dampers are widely used in earthquake-resistant structures, where some of them are anisotropic and capable of sustaining earthquake-induced bidirectional deformation. In this paper, a simplified analytical model is proposed for simulating the hysteretic behavior of U-shaped steel dampers with horizontal bidirectional deformation. The proposed model is composed of a series of shear springs with different nonlinear characteristics in a radial configuration, and the Menegotto–Pinto hysteresis model is employed to represent the hysteretic characteristics of the springs. The mechanical and shape-related parameters of the hysteresis model are set according to the multi-directional deformation characteristics of steel dampers. With the aim of validating the effectiveness and applicability of the analytical model, a U-shaped steel damper was used as an example. The pseudo-static hysteretic characteristics of the steel damping element were analyzed and the elasto-plastic seismic response of a curved bridge featuring a steel hysteresis device was investigated. The results showed that the proposed model is sufficiently accurate to simulate the hysteretic behavior of U-shaped steel dampers, and thus provides a practical method to assess U-shaped steel dampers through seismic response analysis.     

Mode-based equivalent multi-degree-of-freedom system for one-dimensional viscoelastic response analysis of layered soil deposit

Discrete models such as the lumped parameter model and the finite element model are widely used in the solution of soil amplification of earthquakes. However, neither of the models will accurately estimate the natural frequencies of soil deposit, nor simulate a damping of frequency independence. This research develops a new discrete model for one-dimensional viscoelastic response analysis of layered soil deposit based on the mode equivalence method. The new discrete model is a one-dimensional equivalent multi-degree-of-freedom (MDOF) system characterized by a series of concentrated masses, springs and dashpots with a special configuration. The dynamic response of the equivalent MDOF system is analytically derived and the physical parameters are formulated in terms of modal properties. The equivalent MDOF system is verified through a comparison of amplification functions with the available theoretical solutions. The appropriate number of degrees of freedom (DOFs) in the equivalent MDOF system is estimated. A comparative study of the equivalent MDOF system with the existing discrete models is performed. It is shown that the proposed equivalent MDOF system can exactly present the natural frequencies and the hysteretic damping of soil deposits and provide more accurate results with fewer DOFs.     

转动弹簧宏模型模拟剪力墙受力性能

本文将剪力墙结构根据精度需要划分成若干6节点矩形单元,每个单元由2根斜置正交弹簧及联结的上下水平刚性横杆组成,2根斜置弹簧方向分别与外力作用下的主拉、压应力方向一致。本文引入单元位移场线性假定及斜压场理论计算弹簧内力,根据变形协调条件由单元结点位移计算弹簧变形,由弹簧刚度及倾角推导了该宏模型单元刚度矩阵,迭代计算使结构内力及位移逼近真值。算例分析表明,该模型能较好地反映剪力墙的受力性能。     

全埋入单桩基础横向振动模态正交性研究

采用Timoshenko梁模型作为挠曲波在全埋入单桩基础中的传播模型,根据频域中全埋入单桩基础横向振动时的运动微分方程,结合各节点的力平衡、位移协调方程及局部对偶坐标系中内力及位移的对偶变换关系,严格推导全埋入单桩基础横向振动的模态正交性条件。以回传射线矩阵法为基础,求解全埋入单桩基础横向振动的自振频率、衰减系数及不同自振频率所对应的模态,并通过具体算例验证全埋入单桩基础横向振动模态的正交性条件公式推导的正确性。     

Approximate dynamic model of embedded foundation in time domain

A discrete model to represent the unbounded soil (halfspace) in a soil–structure interaction analysis in the time domain is developed. For each dynamic degree of freedom of the foundation node, the discrete model consists of a mass M₀ which is attached to a rigid support with a spring K and with a damper C₀. In addition, a free node with the mass M₁ is introduced, which is connected to the foundation node with a damper C₁. All coefficients are frequency-independent. The discrete model is semi-empirical. It is based on a semi-infinite truncated cone, whereby, after enforcing the static stiffness, the remaining parameters are modified to achieve an optimal fit of the dynamic-stiffness coefficient in the frequency domain. The spring K is equal to the static stiffness. The coefficients appearing in the equations for the dampers C₀, C₁ and the masses M₀, M₁ are specified (assuming a homogeneous halfspace) for the disc, the embedded cylinder, the rectangle (also embedded) and the strip. A square on a layer whose stiffness increases with depth resting on a homogeneous halfspace is also treated. For an embedded foundation, eccentricities arise. Material damping increases the damper C₀ and the mass M₀.     

MR阻尼器对斜拉索共振峰频率漂移的影响

本文采用Spencer提出的MR模型,对斜拉索和MR阻尼器组成的系统进行了动力分析,考察了MR阻尼器型号、安装位置、施加的电压以及斜拉索基频(张力、索长、质量)、激励荷载(类型、频率、大小)等各种因素对斜拉索共振峰频率漂移的影响。进而对钱江三桥南岸154m长的斜拉索进行了现场试验。斜拉索自由振动的衰减信号通过阶跃激励的方法获得,经小波变换、Hilbert变换识别出系统的共振峰频率,并与斜拉索在安装油阻尼器和无阻尼器两种情况进行了比较。计算了杭州湾大桥330m长的索在被动控制、半主动控制下的各阶共振峰频率。研究表明,斜拉索在安装MR阻尼器后共振峰频率略微增大。其影响程度比安装油阻尼器时要大。因此MR阻尼器的制振效果除主要来自耗能外,还有部分来自调频作用。     

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.     

Discrete model for dynamic through-the-soil coupling of 3-D foundations and structures

An efficient discrete model for predicting the dynamic through-the-soil interaction between adjacent rigid, surface foundations supported by a homogeneous, isotropic and linear elastic half-space is presented. The model utilizes frequency-independent springs and dashpots, and the foundation mass, for the consideration of soil–foundation interaction. The through-the-soil coupling of the foundations is attained by frequency-independent stiffness and damping functions, developed in this work, that interconnect the degrees of freedom of the entire system of foundations. The dynamic analysis of the resulting coupled system is performed in the time domain and includes the time lagging effects of coupled dynamic input due to wave propagation using an appropriate modification of the Wilson-θ method. The basic foundation interaction model is also extended to the evaluation of coupled building-foundation systems. © 1998 John Wiley & Sons, Ltd.