Shock and Vibration

Volume 2018, Article ID 7382392, 12 pages

https://doi.org/10.1155/2018/7382392

## A Simple Model for Vertical Dynamic Interactions among a Group of Strip Footings Rested on Homogeneous Half-Space

^{1}College of Mechanical & Electrical Engineering, Hohai University, Changzhou 213022, China^{2}College of Civil Engineering, Nanjing Tech University, Nanjing 211816, China

Correspondence should be addressed to Jue Wang; moc.liamg@tujntajw

Received 10 April 2018; Revised 19 June 2018; Accepted 28 June 2018; Published 25 July 2018

Academic Editor: Roberto Nascimbene

Copyright © 2018 Jue Wang and Ding Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple model for vertical dynamic interactions among a group of strip footings rested on half-space is presented in this paper. An analytical method is presented to obtain the contact pressures and the impedance matrix for a group of surface strip footings. In order to conveniently solve the unknown contact pressures between the soil and footings, the soil-footing interfaces are discretized into a series of strip elements. The Green function for each element under uniform harmonic force is derived and calculated by the piecewise integration and Cauchy principal value integral. The influences of footing and soil parameters on contact pressures and vertical dynamic impedances of footing groups are discussed in detail. The SSSI effect between adjacent footings increases with the decrease of the distance ratio *S*/*L*. For three footings in a group, the middle footing experiences greater cross-interactional effect than the side ones. The present method has high accuracy, which is not only simple but also suitable for the high-frequency analysis.

#### 1. Introduction

The soil-structure interaction (SSI) has been paid comprehensive attention over the past few decades [1–3]. The substructure method has been widely applied in SSI research due to that the footing and the half-space can be analyzed separately by using the respectively suitable methods. In the substructure method, the reaction of the soil against the footing can be described by the frequency-dependent stiffness and damping coefficients which are commonly called as dynamic impedance. Therefore, it is a key step to obtain the impedance function of the footing in the analysis of SSI.

Reissner [4] derived the first analytical solution for the vertical vibration of a circular plate subjected to a harmonic uniform force and marked the beginning of the elastic half-space theory about SSI. Sung [5] presented three kinds of supposed contact pressure distributions (static rigidity, uniform, and parabolic) beneath the footing. The dynamic impedances for some representative cases [6, 7] were obtained based on these assumed distributions. However, the assumed distribution could not yield a constant surface displacement of the supporting medium beneath the rigid footing as demanded from physical considerations; it is then necessary to find the mean value for the footing displacement through various averaging techniques. On the contrary, the footing impedances were analyzed more rigorously and treated as a mixed boundary-value problem. Luco and Westmann [8] presented the impedance for a massless rigid strip footing by letting the Cauchy singular dual integral equations reduce to the second kind of Fredholm integral equations. Ma et al. [9] transformed the dual integral equations into a set of linear equations using an infinite series of orthogonal Jacobi polynomials for the rocking impedance of the rigid strip footing. As the contact stresses cannot be expressed by the elementary functions, there are certain mathematical limitations to solve the mixed boundary-value problem. Therefore, some semianalytical approaches were presented to obtain the impedance more conveniently. Jiang and Song [10] investigated the impedance of a massless rigid strip footing by the thin layered method [11], that is, the analytical solution in the horizontal direction and the finite element discretization in the vertical direction. Lin et al. [12] studied the similar problem based on the precise integration method [13].

In the aforementioned literature, dynamic interaction between single footing and elastic half-space was considered. However, the construction of the metros and high-speed trains is becoming prevalent with the acceleration of urbanization. In such a situation, the footings associate together through soil ground. This results in the structure-soil-structure interaction (SSSI) under the externally vertical exciting loads [14, 15]. Taking the advantage in the SSI model proposed by Parmelee [16], Warburton et al. [17] derived governing equations for the response of two geometrically identical cylindrical bodies attached to the surface of an elastic half-space, which initiates the SSSI study. Liou [18] presented an analytical solution for the dynamic stiffness matrix of adjacent surface rigid footings, based on the assumed contact stress distribution which linearly varies in the radius direction in the cylindrical coordinate system. Currently, analyses of dynamic interactions between multiple footings are mainly through numerical methods [19–22] such as the finite element method and the boundary element method due to the rapid progress in computer technique. However, these methods are far too time-consuming and complicated for actual engineering and designers. The analytical method for the impedance matrix of footing group based on the elastic half-space theory is still very rare, while it is efficient and has great significance for solving the seismic response of structure groups and assessing structure safety.

In this paper, the Green function of uniform harmonic vertical force is derived and numerically calculated by the piecewise integrations and the Cauchy principal value integral. Contact pressures and the impedance matrix of multiple strip footings considering the SSSI effect are obtained by combining the Green function with the element discretization technique. In contrast with the mixed boundary-value method, the present method avoids the directly solving of contact pressures which cannot be expressed by the elementary functions. The validity and wide applicability of the present method has been verified by the comparative studies. The SSSI effect on the contact pressures and impedances for a group of surface strip footings are illustrated by the parametric studies.

#### 2. General Formulation of SSSI Problem

For the strip footings such as track footings, dams, or building footings with high ratio of length to width, it is reasonable to consider the problem as a plane strain case with a coordinate system (*x*-*z*) where the *z*-axis is normal to the space surface. Consider a group of strip footings with different width *L*_{m} (*m* = 1, … , *M*) and different separation distance *S*_{m}, as shown in Figure 1. *M* footings rest on a linear elastic half-space, and the *m*th footing is excited by the vertical harmonic line excitations *T*_{m} exp(*iωt*) (*m* = 1, … , *M*).