The problem of structural isolation from ground transmitted vibrations by open or infilled trenches under conditions of plane strain is numerically studied. The soil medium is assumed to be linear elastic or viscoelastic, homogeneous and isotropic. Horizontally propagating Rayleigh waves or waves generated by the motion of a rigid foundation or by surface blasting are considered in this work. The formulation and solution of the problem is accomplished by the boundary element method in the frequency domain for harmonic disturbances or in conjunction with Laplace transform for transient disturbances. The proposed method, which requires a discretisation of only the trench perimeter, the soil-foundation interface and some portion of the free soil surface on either side of the trench appears to be better than either finite element or finite difference techniques. Some parametric studies are also conducted to assess the importance of the various geometrical, material and dynamic input parameters and provide useful guidelines to the design engineer.

Vibration isolation using open or filled trenches

The isolation of structures from ground transmitted waves by open and infilled trenches in a three-dimensional context is numerically studied. The soil medium is assumed to be elastic or viscoelastic, homogeneous and isotropic. Waves generated by the harmonic motion of a surface rigid machine foundation are considered in this work. The formulation and solution of the problem is accomplished by the boundary element method in the frequency domain. The infinite space fundamental solution is used requiring discretization of the trench surface, the soil-foundation interface and some portion of the free soil surface. The proposed methodology is first tested for accuracy by solving three characteristic wave propagation problems with known solutions and then applied to several vibration isolation problems involving open and concrete infilled trenches. Three-dimensional graphic displays of the surface displacement pattern around the trenches are also presented.

Vibration isolation using open or filled trenches Part 2: 3-D homogeneous soil

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.

Approximate dynamic model of embedded foundation in time domain

Based on the approximation by polynomial-fraction, a series of systematic lumped-parameter models are developed in this paper for efficiently representing the dynamic behaviour of unbounded soil. Concise formulation is first employed to represent the dynamic flexibility function of foundation with a ratio of two polynomials. By defining an appropriate quadratic error function, the optimal coefficients of the polynomials can be directly solved from a system of linear equations. Through performing partial-fraction expansion on this polynomial-fraction and designing two basic discrete-element models corresponding to the partial fractions, systematic lumped-parameter models can be conveniently established by connecting these basic units in series. Since the systematic lumped-parameter models are configured without introducing any mass, the foundation input motion can be directly applied to these models for their applications to the analysis of seismic excitation. The effectiveness of these new models is strictly validated by successfully simulating a semi-infinite bar on an elastic foundation. Subsequently, these models are applied for representing the dynamic stiffness functions for different types of foundation. Comparison of the new models with the other existing lumped-parameter models is also made to illustrate their advantages in requiring fewer parameters and featuring a more systematic expansion. Copyright © 2002 John Wiley & Sons, Ltd.

Systematic lumped-parameter models for foundations based on polynomial-fraction approximation

This paper presents how soil–structure interaction affects the seismic performance of Tuned Mass Dampers (TMD) when installed on flexibly based structures. Previous studies on this subject have led to inconsistent conclusions since the soil and structure models employed considerably differ from each other. A generic frequency-independent model is used in this paper to represent a general soil–structure system, whose parameters cover a wide spectrum of soil and structural characteristics. The model structure is subjected to a stationary random excitation and the root-mean-square responses of engineering interest are used to measure the TMD's performance. Extensive parametric studies have shown that strong soil–structure interaction significantly defeats the seismic effectiveness of TMD systems. As the soil shear wave velocity decreases, TMD systems become less effective in reducing the maximum response of structures. For a structure resting on soft soil, the TMD system can hardly reduce the structural seismic response due to the high damping characteristics of soil–structure systems. The model structure is further subjected to the NS component of the 1940 El Centro, California earthquake to confirm the TMD's performance in a more realistic environment. Copyright © 1999 John Wiley & Sons Ltd.

Seismic effectiveness of tuned mass dampers considering soil–structure interaction

The frequency-independent foundation impedances, commonly used in soil-structure dynamic interaction problems, are developed for a circular footing resting on a homogeneous halfspace. As they ignore the structure attached to the foundation, the error introduced in the structural response may be 50 per cent or more in the neighbourhood of the fundamental frequency of the soil-structure system.



The present study proposes a new method developed for most dynamic soil-structure interaction problems. The key idea is to retain for the frequency-independent impedances values computed for the fundamental frequency of the soil-structure system; thus these values include the dynamic characteristics of the whole soil-structure system and lead to a satisfactory approximation of the exact solution over a wide frequency range. The method is developed here for the horizontal and rocking modes of a structure with a circular base resting on a homogeneous halfspace.



Numerical applications are given for a simple linear oscillator in order to make possible a thorough parametric study. The response of some idealized building-foundation systems to harmonic excitation or to a seismic input is next examined in order to illustrate the efficiency of the proposed model.

Frequency‐independent impedances of soil‐structure systems in horizontal and rocking modes

An extension of the boundary element method to heterogeneous domains composed of horizontal layers is here proposed. It includes a numerical computation of the corresponding Green's functions, thanks to an inverse Hankel transform of the closed form solutions obtained in the spectral domain with suitable variables derived from displacements and stress vectors to obtain the decoupling between P–SV and SH waves. Transmission and reflection operators are introduced to avoid the problem of overflowing exponentials met with in Thomson–Haskell matrices.



Applications are given in the soil–structure interaction field to compute the impedances of surface and embedded circular foundations resting on a viscoelastic halfspace.

François Chapel

Papers

Frequency‐independent impedances of soil‐structure systems in horizontal and rocking modes

Boundary element method applied to linear soil-structure interaction on a heterogeneous soil