A Generic Analytical Elastic Solution for Excavation Responses of an Arbitrarily Shaped Deep Opening under Biaxial In Situ Stresses

Abstract

This paper presents a generic analytical solution for stresses and displacements developed around an arbitrarily shaped underground opening excavated in an elastic soil/rock mass with biaxial in situ stresses. A series of separable functions satisfying the governing biharmonic equation is employed as the basic stress function for the problem considered. With the basic stress function and the stress rotation equation, the normal stress and the shear stress on the opening boundary surface are expressed in terms of separable function series with unknown coefficients to make use of the boundary condition. Based on the variational theory, the unknown coefficients are determined by minimizing the difference between the real stress boundary condition and the assumed stress function on the opening surface. The excavation responses around different shapes of openings, including elliptical-shaped, ovaloid-shaped, horseshoe-shaped, and D-shaped openings, are investigated and compared with those from the finite-element simulations. The results show that both the stress field and the displacement field agree well with those from the corresponding numerical model, which verify the proposed solving technique. The proposed solution provides an easy, generic, and feasible approach to assess the stresses and displacements developed around an arbitrarily shaped opening, which is simpler and more straightforward than the complex theory–based solutions.