Journal of Engineering

Volume 2017 (2017), Article ID 6145375, 12 pages

https://doi.org/10.1155/2017/6145375

## Dynamic Response of a Circular Tunnel in an Elastic Half Space

Faculty of Civil Engineering, Yıldız Technical University, Davutpaşa Campus, Esenler, 34210 İstanbul, Turkey

Correspondence should be addressed to İrfan Coşkun

Received 17 November 2016; Revised 1 March 2017; Accepted 9 March 2017; Published 26 April 2017

Academic Editor: Mohamed Ichchou

Copyright © 2017 İrfan Coşkun and Demirhan Dolmaseven. 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

The vibration of a circular tunnel in an elastic half space subjected to uniformly distributed dynamic pressure at the inner boundary is studied in this paper. For comparison purposes, two different ground materials (soft and hard soil) are considered for the half space. Under the assumption of plane strain, the equations of motion for the tunnel and the surrounding medium are reduced to two wave equations in polar coordinates using Helmholtz potentials. The method of wave expansion is used to construct the displacement fields in terms of displacement potentials. The boundary conditions associated with the problem are satisfied exactly at the inner surface of the tunnel and at the interface between the tunnel and surrounding medium, and they are satisfied approximately at the free surface of the half space. A least-squares technique is used for satisfying the stress-free boundary conditions at the half space. It is shown by comparison that the stresses and displacements are significantly influenced by the properties of the surrounding soil, wave number (i.e., the frequency), depth of embedment, and thickness of the tunnel wall.

#### 1. Introduction

The dynamic behavior of underground structures such as tunnels and pipelines is an important engineering problem in the field of dynamic soil-structure interactions. Compared to the large volume of literature on the dynamic response of structures in infinite media, the corresponding problem in a half space has not received much attention. Even so, this problem needs analysis, as half spaces are always present in metropolitan areas. It is assumed that this limitation is mainly due to the difficulties in satisfying boundary conditions at the free surface of the ground. Thiruvenkatachar and Viswanathan [1] investigated the dynamic response of an elastic half space with a cylindrical cavity at a finite depth subjected to time-dependent surface tractions on the boundary of the cavity using a series of wave functions and the method of successive approximations. El-Akily and Datta [2, 3] studied the response of a circular cylindrical shell to disturbances in an elastic half space using matched asymptotic expansions and a successive reflection technique. Datta et al. [4] studied the dynamic response of a cylindrical pipe with a circular cross-section lying in a concentric cylindrical region buried in an elastic half space. In this study, the fields within each of the regions were expanded in a series of wave functions, and the unknown coefficients appearing in the solution were obtained by considering a finite number of terms in the series. In another study, Wong et al. [5] considered the two-dimensional response of a tunnel with a noncircular cross-section embedded in an elastic half space using a technique involving wave function expansion in the half space combined with a finite element representation of the tunnel and its vicinity. Balendra et al. [6] studied the in-plane vibrations of a tunnel-soil-building system in a viscoelastic half space. The displacement fields were formulated by the method of wave function expansion, and the boundary conditions were satisfied only at a finite number of points along the traction-free surface, tunnel-soil interface, and soil-foundation interface. Therefore, the unknown coefficients of the wave function were obtained using a least-squares approach. Lee and Karl [7] studied the scattering and diffraction of plane waves by underground, circular, and cylindrical cavities at various depths in a half space. In this study, the authors used the Fourier-Bessel series and approximated the half space as a large-diameter elastic cylinder. Luco and de Barros [8, 9] studied the seismic response of a cylindrical shell embedded in a layered viscoelastic half space using a technique that combined an indirect integral representation for the exterior domain with a simplified shell theory for tunnel representation. Guan and Moore [10] studied the dynamic response of multiple cavities deeply buried in a viscoelastic medium that were subjected to moving or seismic loading using the Fourier-Bessel series. Bayıroğlu [11] studied the dynamic response of an elastic half space with a circular cylindrical shell using the finite element method. Stamos and Beskos [12] studied the dynamic response of an infinitely long cylindrical tunnel buried in a half space that was subjected to plane harmonic waves using a special boundary element method in the frequency domain. Davis et al. [13] studied the transverse response of cylindrical cavities and pipes embedded in a half space that were subjected to incident plane SV waves. The solution was obtained using the Fourier-Bessel series and a convex approximation of the half-space free surface. Yang et al. [14] studied the ground vibrations caused by trains moving in tunnels embedded in half space using the finite element method. Liang et al. [15] investigated the diffraction of incident plane SV waves by a circular cavity in a saturated poroelastic half space using a wave function series and a downward concave approximation of the half-space free surface. Jiang et al. [16] studied the scattering of plane waves by a cylindrical cavity with lining in a poroelastic half plane using the complex variable function method. Zhou et al. [17] studied the scattering of the elastic waves by a circular pipeline in a poroelastic medium using the wave function expansion method. Gupta et al. [18] studied the generation and propagation of vibrations from underground railways by performing a parametric study on the soil and tunnel parameters. Lin et al. [19] studied the dynamic response of a circular underground tunnel in an elastic half space subjected to incident plane P-waves using an analytical solution scheme. Coşkun et al. [20] studied the vibration of an elastic half space with a cylindrical cavity subjected to a uniform harmonic pressure at the inner surface using the method of wave function expansion. Liu and Wang [21] studied the dynamic response of twin circular tunnels in a full space subjected to plane harmonic excitation using the complex variable method. Liu et al. [22] studied the scattering of plane waves by a shallow lined circular tunnel in an elastic half space using the complex variable method and the image technique. Hamad et al. [23] studied the dynamic interaction of two parallel tunnels embedded in a half space using a fully coupled approach and a superposition approach. More recently, Huang et al. [24] studied the nonlinear dynamic responses of circular tunnels buried in normal fault ground using the finite element method.

In this study, the dynamic response of a circular cylindrical tunnel embedded in an elastic half space is analyzed. The tunnel lies parallel to the plane free surface of the medium at a finite depth and is subjected to a harmonic normal pressure at the inner surface. By introducing potentials, the governing equations for the tunnel and surrounding medium are decoupled and reduced to Helmholtz equations, satisfied by the potentials. The series solution for these equations is obtained via the wave function expansion method. The boundary conditions at the inner surface of the tunnel and at the tunnel-soil interface are satisfied exactly; they are satisfied only approximately along the traction-free surface of the half space using the least-squares method. Once the unknown wave function coefficients are determined numerically, the displacements and stresses at any point in both the tunnel and surrounding medium can be calculated in a straightforward manner.

#### 2. Formulation of the Problem

Consider an infinitely long circular tunnel with inner and outer radii and* b*, respectively, buried in an elastic half space at a depth below the free surface and subjected to harmonic pressure that is uniformly distributed as at the inner surface (Figure 1). As the loading and geometry of the tunnel are assumed to be -independent, the problem is two-dimensional along the - and -axis and corresponds to the plane strain case.