Analytical calculation of the half space stress field around tunnels under seismic loading of SV waves

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Analytical Solutions are proposed for the calculation of the stress and deformation fields around cavities, of various shapes, in the linear elastic Half Plane Space, due to seismic excitation of SV Waves.The analytical model, is based both on the expansion of infinite series of Bessel & Hankel type, for the estimation of the wave potentials, both from the cavity and the ground surface of incident, reflected and scattered, combined with the Complex Variable Method.The close form Solution, is developed in two stages. In the first, by applying the Complex Mapping Theory, a mapping is proposed so as the region in the plane w consisting of the half space with a cavity of various shapes (circular, square, hoop) is mapped conformally onto two vertical circles in the plane ζ. With the application of this mapping the problem of the free stress boundary conditions at the surface is solved.In the second stage, the complex Wave Potentials are calculated in the transformed plane ζ and represented by their Bessel and Hankel series expansion.Thus the calculation of the stress field as well as the deformations of the sections of underground structures, subjected to seismic excitation of shear wave, of SV type is achieved with the implementation of the boundary conditions both in the surface of the cavity and the upper boundary of the half space assuming trr=trθ=0. The language of technical computing, MATLAB, is applied for the composing of the relevant computing codes.Finally in order to compare the results with previous approximations, an analysis without taken into account the scattered wave potentials from the free surface (Relaxed Boundaries) has been developed, for various cavity shapes. This assumption can be considered because the values of these potentials are small. By omitting these wave potentials this problem becomes a full space case, under incident and reflected SV waves. Comparing the proposed analytical methodology with that with Relaxed Boundaries the former calculated stresses and displacements are larger than those of the latter. Although the values of the scattered wave potentials, from the free surface are small, their contribution to the stress field is significant.This proposed analytical methodology solved the problem of the calculation of the stress field around cavities in case of half space under seismic loading of SV waves.