Modal analysis of circular flexible foundations under vertical vibration

A methodology using modal analysis is presented to evaluate dynamic displacements of a circular flexible foundation on soil media subjected to vertical vibration. The interaction effects between the foundation and the underlying soil are represented using modal soil impedance functions determined by an efficient procedure developed. The displacements of the foundation can then be easily solved by modal superposition. Comparing with existing solutions, the presented method is found to provide accurate results with less computational effort using only a few vibration modes. In addition, parametric studies for modal responses of the flexible foundation indicate that the response of the foundation are significantly influenced by relative stiffness between the foundation and the soil medium, load distributions, vibration frequency range, and the foundation mass. Besides, justification for flexible foundations to be considered as rigid are investigated.     

任意刚性基础竖向动阻抗的简化计算

对于均质弹性半空间上的任意形状的刚性明置和埋置基础,其动刚度和阻尼系数的确定,已有很多这方面的研究。通常基础的任意形状用其外包的规则几何形状代替原有的不规则基础形状,以达到确定动刚度和阻尼系数的目的,而且这两个参数的确定仅仅是对单独刚性基础的,无法考虑相邻基础对其产生的影响。针对上述两方面不完善之处作了进一步探讨,引入相邻基础动力相互作用因子的概念,并利用地基为平面应变假定以求之。推荐的方法经验证,非常准确。     

Energy-based dynamic parameter identification for Pasternak foundation model

Parameter identification of Pasternak foundation models(PFM) is never satisfactory, which discourages the application and popularization of PFM. In the present study, an energy-based model to predict the dynamic foundation coefficients was proposed using the vibration kinetic energy and potential energy of a Pasternak foundation-rigid plate system. On the basis of the Pasternak foundation, the relationship among the natural frequency, dynamic foundation coefficients, rigid plate configuration, and vibrating soil equivalent mass per unit area was considered. To obtain the natural frequencies of the Pasternak foundation-rigid plate system, dynamic tests were performed. Using two or more dynamic test results of various rigid plates on a foundation, a set of equations of dynamic foundation coefficients was set up to directly identify the foundation coefficients and equivalent mass per unit area of vibrating soil. The feasibility of the proposed method was verified by comparing it with the outdoor and indoor test results and finite element analysis results. When the proposed method is used to obtain the dynamic parameters, PFM can be generalized and applied more widely in engineering practice.     

Simplified analysis of dynamic response of rigid foundations with arbitrary geometries

A simple, efficient numerical procedure is presented for the analysis of the steady-state dynamic response of rigid foundations of arbitrary shape on a homogeneous, isotropic elastic half-space. The contact area of the foundation and soil medium is discretized into square subregions. The influence of the rigid square subregion is approximated by that of an equivalent rigid circular base with the same contact area. This approximation is a better representation of a rigid square subregion than a uniformly stressed subregion used in earlier approaches. Simple expressions are used to determine the influence of the subregion on itself and on the rest of the subregions, without the expense of numerical integration procedures. The present method is demonstrated to converge rapidly. The accuracy of this method is examined by comparison with available published theoretical and experimental results.     

Dynamic plate-soil interaction - finite and infinite, flexible and rigid plates on homogeneous, layered or Winkler soil

An integral method to calculate the solution of a homogeneous or layered soil due to a harmonic point load is described. An infinite plate at the surface of the soil can be introduced in this integration in wavenumber domain, too. Finite structures on the soil are calculated by a combined finite element and boundary element method, which makes use of the point load solution of the soil. The compliance functions for a vertical point load and some vibration modes are calculated for realistic parameters of the plate and the soil and for a wide range of frequencies. The influence of the stiffness of the soil and the foundation is investigated, showing that the soil mainly affects the low-frequent response whereas the structural properties are more important at higher frequencies. A rigid approximation of flexible plates is only found at low frequencies, if the elastic length is used as the radius of a rigid disk. At higher frequencies, a characteristic behaviour of the flexible plate of approximately is observed, what is in clear contrast to the compliance of rigid foundations. A plate on a visco-elastic support (Winkler soil) shows similar displacements as a plate on a homogeneous half-space, but the maximal stresses between the plate and the soil are considerably smaller which is found to be more realistic for a plate on a layered soil. For practical applications, a normalized diagram and some explicit formulas of the exact and the approximate solutions of an infinite plate on a homogeneous half-space are given, which is a useful model to represent the soil-structure interaction of flexible foundations.     

Soil–structure interaction for a SDOF oscillator supported by a flexible foundation embedded in a half-space: Closed-form solution for incident plane SH-waves

A closed-form analytical solution is presented for the dynamic response of a SDOF oscillator, supported by a flexible foundation embedded in an elastic half-space, and excited by plane SH waves. The solution is obtained by the wave function expansion method. The solution is verified for the special case of a rigid foundation by comparison with published results. The model is used to investigate the effect of the foundation flexibility on the system response. The results show that the effect is significant for both foundation response and structural relative response. For a system with more flexible foundation, the radiation damping is smaller, the foundation response is larger, especially for obliquely incident waves, while the structural relative response is smaller, and the system frequency shifts towards lower frequencies. This simple model may be helpful to obtain insight into the effects of soil–structure interaction for a slim structure on an extended flexible foundation.     

Dynamic analysis of 3-D flexible embedded foundations by a frequency domain BEM-FEM

A study on the dynamic response of three-dimensional flexible foundations of arbitrary shape, embedded in a homogenous, isotropic and linear elastic half-space is presented. Both massive and massless foundations are considered. The soil-foundation system is subjected to externally applied forces, and/or to obliquely incident seismic waves. The numerical method employed is a combination of the frequency domain Boundary Element Method, which is used to simulate the elastic soil medium, and the Finite Element Method, on the basis of which the stiffness matrix of the foundation is obtained. The foundation and soil media are combined by enforcing compatibility and equilibrium conditions at their common interface. Both relaxed and completely bonded boundary conditions are considered. The accuracy of the proposed methodology is partially verified through comparison studies with results reported in the literature for rigid embedded foundations.     

Dynamics of rigid strip foundations embedded in orthotropic elastic soils

This study is concerned with the dynamic response of an arbitrary shaped rigid strip foundation embedded in an orthotropic elastic soil. The foundation is subjected to time-harmonic vertical, horizontal and moment loadings. The boundary-value problem related to an embedded foundation is analysed by using the indirect boundary integral equation method. The kernel functions of the integral equations are displacement and traction Green's functions of an anisotropic elastic half plane. Exact analytical solutions are used for the Green's functions. The boundary integral equation is solved by using numerical techniques. Selected numerical results are presented for the impedances of rectangular and semi-circular rigid strip foundations embedded in four types of anisotropic soils. A discussion on the influence of soil anisotropy and frequency of excitation on the impedances is presented. The versatility of the analysis is demonstrated by considering the through soil interaction between two semi-circular strip foundations.     

Dynamic response of flexible strip-foundations by boundary and finite elements

A time domain Boundary Element-Finite method is employed to determine the dynamic response of flexible surface two-dimensional foundations under conditions of plane strain placed on an elastic soil medium and subjected either to transient external forces or to obliquely incident seismic waves. The elastic, isotropic, and homogeneous soil medium is treated by the time domain Direct Boundary Element Method, while the flexible foundation is treated by the Finite Element Method. The two methods are appropriately combined through equilibrium and compatibility considerations at the soil-foundation interface. Parametric studies examining the effect of the relative stiffness between the foundation and the soil and the spatial distribution of the dynamic disturbances on the foundation response are presented.     

Boundary differential equation method: simplified dynamic soil stiffnesses for embedded rigid foundations

With a simplified model and Galerkin's weighted residual procedure, two simple differential equations of dynamic behavior of a bounded rectangular medium are established along the boundaries in the x- and y-direction in the medium. Solutions of these equations yield closed form expressions of soil stiffnesses for various cases of a partially embedded rigid foundation, including the stiffnesses per depth of foundation with rectangular base area and the stifnesses of strip foundation. The developed procedure provides the definition of the weight functions, which are used in Galerkin's method for weighted residual. In addition to these weight functions, their conjugators are also suitable for weight functions. When the soil depth is finite, the original weight functions fail to produce physically meaningful results in some frequency range but the conjugators do not fail at any frequencies. The developed equations to compute soil stiffnesses for embedded foundations are simple yet capable of calculating the responses close to those computed by the much more elaborated finite element method.     

Rocking response of flexible circular foundations on layered media

The analysis of the response of a flexible circular foundation on layered media due to an arbitrarily distributed vertical loading is presented. The analysis is based on the ‘ring method’ approach, i.e. discretization of the foundation in a set of concentric rings. The arbitrarily distributed loading is expanded in the circumferential direction in a Fourier series. The influence coefficient matrix of soil for each element of the series is evaluated utilizing the stiffness matrix approach. The stiffness matrix of the foundation is obtained from the finite difference energy method approach. Numerical examples illustrate the influence of several soil-foundation parameters on the rocking response of a foundation. Results are presented in terms of displacement and soil reaction distributions and impedance functions point to significantly different responses of flexible and rigid foundations.     

Base uplifting analysis of flexibly supported liquid-storage tanks

An approximate method is presented for the analysis of earthquake induced uplifting in base plates of unanchored liquid-storage tanks that are supported directly on flexible soil foundations. The method takes due account of the non-linearities arising from the continuous variation of base contact area, membrane action and plastic yielding in the base plate. First, an ‘exact’ solution is presented for a uniformly loaded flexibly supported prismatic beam, uplifted by a concentrated force applied at one end. A series of such beams are later used in an approximate model of the base plate. The method is highly efficient and believed to be accurate for the level of uplifts encountered in practice. The results are presented for a range of foundation stiffness, representing loose sandy soils to rigid concrete mat. It is shown that, for the same overturning base moment, the flexibly supported tanks, in comparison with the rigidly supported tanks, experience significantly smaller axial compressive stress, but larger base uplift, foundation penetration, plastic rotation at plate boundary and hoop compressive stress in the tank wall.     

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.     

CALCULATING THE DYNAMIC STIFFNESS MATRIX OF 2-D FOUNDATIONS BY DISCRETE WAVE NUMBER INDIRECT BOUNDARY ELEMENT METHODS

Apart from some special cases, calculating the dynamic stiffness matrix of foundations on a layered half-space, especially for embedded foundations, is computationally expensive. An efficient method for two-dimensional foundations in a horizontally layered soil media is presented in this paper. This method is based on indirect boundary element methods and uses discrete wave number solution methods for calculating Green's functions for displacements and analytical methods for the integrations over the boundary. For surface foundations, the present method applies at all frequencies. For embedded foundations or for constructing energy transmitting boundaries, because the free-field part is modelled by boundary elements and the excavated part is modelled by finite elements, the present method applies only at low frequencies for the spring coefficients (the real parts of the dynamic stiffness matrix) but applies at all frequencies for the damping coefficients (the imaginary part of the dynamic stiffness matrix) for undamped sites. The novelty of the method can be used for three-dimensional foundations. © 1997 by John Wiley & Sons, Ltd.     

基础柔性对土-结构相互作用系统响应影响的一个解析解

采用波函数展开法,通过SH波入射均匀半空间中二维埋置半圆形刚柔复合基础-单质点模型,推导土-刚柔复合基础-上部结构动力相互作用的解析解,并验证解的正确性。研究表明:基础柔性对于系统响应峰值与系统频率有较大影响。考虑基础柔性后,上部结构相对响应峰值相比全刚性基础结果均有一定减小,且系统频率也会产生向低频偏移的现象。     

Diffraction effects between foundations due to incident rayleigh waves

The dynamic behaviour of a finite number of rigid, adjacent foundations on the surface of a linear-elastic, isotropic and homogeneous halfspace due to a far-field excitation of the Rayleigh type is the subject of the present work. The dynamic behaviour of this system differs from that of a single foundation subjected to the same excitation due to the existence of the natural coupling between adjacent foundations caused by the wave propagation through the underlying soil. For the determination of the diffraction forces acting on the foundations an analytical procedure is followed. The stresses at the interfaces between the foundations and subsoil are approximated by series expansions of orthogonal polynomials. It is interesting to notice that, apart from the loads appearing in the direction of incidence, additional loads perpendicular to the given incidence direction act on the foundations due to scattered waves. Their intensity depends on the excitation frequency and the distance between the foundations. The method followed here can be applied for the determination of the dynamic loads acting on fixed foundations in the case of a seismic excitation of relatively long duration.     

Vertical vibration of three-dimensional rigid foundations on layered media

A numerical method of analysis is presented for the determination of the steady-state vertical vibration of rigid foundations with arbitrary three-dimensional geometries resting on the surface of a layered soil medium. The method utilizes the flexibility concept applied to steady-state periodic problems and it is solved in the frequency domain. The accuracy of the method is verified by comparison with several published solutions for massless, smooth rigid rectangular foundations on a homogeneous, isotropic elastic half-space. Parametric solutions are presented to study the dynamic behaviour of massless, smooth rigid rectangular foundations on a homogeneous, elastic stratum.     

A simple boundary element formulation and its application to wavefield excited soil-structure interaction

A simple boundary element formulation which is based directly on the point load solutions for an elastic full-space is presented. It is integrated in a finite element program to calculate dynamic soil-structure interaction problems. The combined boundary and finite element method is applied to structures which are excited by horizontally propagating waves in the soil. For three different types of flexible structure-elastic beams, low and high (square) shear walls-and the corresponding rigid structures the vibration modes and the soil-structure transfer functions have been investigated. The flexible foundations display the same wave pattern as the exciting free-field of the soil, but the amplitudes are reduced with increasing frequency, depending on the stiffness or wave resistance of the structure. Rigid structures show, in part, quite different behaviour, giving free-field reductions caused by kinematic and inertial soil-structure interaction.     

Vertical and rocking impedances for rigid rectangular foundations on soils with bounded non-homogeneity

The vertical and rocking response of rigid rectangular foundations resting on a linear-elastic, compressible, non-homogeneous half-space soil model is studied. The non-homogeneity is described by a continuous yet bounded increase of shear modulus with depth. The mixed boundary value problem is solved by means of the semi-analytical method of the subdivision of the foundation/soil contact area whereby the influence functions for the sub-regions are determined by integration of the corresponding surface-to-surface Green's functions for the particular soil model. Impedance functions are given for representative values of the non-homogeneity parameters, the Poisson's ratio and the foundation geometry over a wide range of frequencies. Significant features associated with the soil non-homogeneity are pointed out. Copyright © 1999 John Wiley & Sons Ltd.     

Seismic earth pressures on rigid and flexible retaining walls

While limiting-equilibrium Mononobe–Okabe type solutions are still widely used in designing rigid gravity and flexible cantilever retaining walls against earthquakes, elasticity-based solutions have been given a new impetus following the analytical work of Veletsos and Younan [23]. The present paper develops a more general finite-element method of solution, the results of which are shown to be in agreement with the available analytical results for the distribution of dynamic earth pressures on rigid and flexible walls. The method is then employed to further investigate parametrically the effects of flexural wall rigidity and the rocking base compliance. Both homogeneous and inhomogeneous retained soil is considered, while a second soil layer is introduced as the foundation of the retaining system. The results confirm the approximate convergence between Mononobe–Okabe and elasticity-based solutions for structurally or rotationally flexible walls. At the same time they show the beneficial effect of soil inhomogeneity and that wave propagation in the underlying foundation layer may have an effect that cannot be simply accounted for with an appropriate rocking spring at the base.