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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.

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 [

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.

Consider an infinitely long circular tunnel with inner and outer radii

Circular tunnel embedded in a half space.

Therefore, using polar coordinates, the displacement vector

The boundary conditions for the problem will be determined at the inner surface of the tunnel, at the interface between the tunnel and the surrounding medium, and at the free surface of the half space. It will be assumed that all contacts are perfect such that the displacements and the tractions are continuous across the interface between the tunnel and the surrounding medium. Given that the tunnel is subjected to an internal pressure

Equations (

Numerical computations are presented for a concrete circular tunnel of outer radius

Figures

Radial displacement

Tangential displacement

Radial stress

Tangential stress

One can see from Figure

Figure

Radial displacement

Radial

Figures

Radial displacement

Radial stress

Tangential stress

Figures

Radial displacement

Radial displacement

Tangential displacement

Figures

Tangential stress

Tangential stress

Figure

Tangential stress

Tangential stress

Finally, the effect of the wall thickness on the free-surface vertical displacements

Vertical displacement

In this paper, the dynamic response of a circular tunnel buried in an elastic half space is discussed. Two types of ground material (hard and soft soil) were considered for the half space for comparison purposes. The effects of the soil type, frequency, and depth on the stresses and displacements at the outer surface of the tunnel wall and the effects of the tunnel thickness on the stresses at the tunnel wall and the vertical displacements at the free surface of the half space have been presented in the figures. It was found that, generally, larger stresses and displacements occurred in the soft-soil case than in the hard-soil case. Additionally, the radial displacements and tangential stresses in both soils were significantly larger than the corresponding tangential displacements and radial stresses, respectively.

In the range of frequencies considered, the maximum radial stresses in the hard soil and the maximum tangential stresses in the soft soil first decreased with frequency and then increased. However, the maximum values of the tangential stress in the hard soil and radial stress in the soft soil increased first with frequency and then decreased. For the hard soil, it was observed that the maximum radial stresses occurred at the points nearest and farthest from the free surface; the maximum tangential stresses occurred at points along the

Variations of stress and displacement at the outer surface of the tunnel wall strongly depended on the depth of embedment. The maximum radial stresses decreased with increasing depth for both soils. The maximum tangential stress increased with depth in the hard soil but increased, decreased, and then increased again in the soft soil. As in the case of the variations with frequency, the maximum displacements (like the maximum stresses) also increased and/or decreased significantly with depth for both soils.

The thickness of the tunnel also affected the stresses and displacements considerably. Both the maximum stress and displacement in the tunnel and the maximum vertical displacement at the free surface (which occurs at the point nearest to the top of the tunnel) decreased as the thickness of the tunnel increased. Additionally, the amplitudes of the vertical displacement along the free surface decreased gradually as one moved away from the tunnel, as expected.

From the preceding remarks, it can be concluded that the properties of the surrounding ground, the frequency, the thickness of the tunnel wall, and the depth of embedment have a considerable effect on the dynamic response of the tunnel.

The authors declare that there are no conflicts of interest regarding the publication of this paper.