Analytical solution for deep rectangular structures subjected to far-field shear stresses

Underground structures located in seismic areas have to support the static loads transferred from the surrounding ground under normal working conditions, as well as the loads imposed by any seismic event. Typically underground structures have cross section dimensions much smaller than the wave length of ground peak velocities, in which case inertial forces can be neglected and the structure can be designed using a pseudo-static analysis, where the seismic-induced loads or deformations can be approximated by a far-field shear stress or strain. Current close-form solutions for deep rectangular structures subjected to a far-fields shear stress are approximations that do not consider all the relevant variables. An analytical solution is presented in this paper for deep rectangular structures with a far-field shear stress. Complex variable theory and conformal mapping have been used to develop the solution, which is applicable to deep rectangular structures in a homogeneous, isotropic, elastic medium. The solution shows that the deformations of the structure depend on the relative stiffness between structure and the surrounding ground, and on the shape of the structure. The analytical solution has been verified by comparing its predictions with results from a finite element method and from previously published data.