Symplectic Elasticity Analysis of Stress in Surrounding Rock of Elliptical Tunnel

Abstract

A large number of non-circular tunnels are widely used in engineering practice. The application of these geometric structures not only makes the boundary conditions of mechanical problems more complex, but also makes the theoretical solution more difficult. The introduction of curvilinear coordinate system can easily transform the elliptical region into a circular region, and the physical relationship can also be adjusted accordingly. In the state space, the transformed problem can be solved by using the dual variables of Hamiltonian mechanics, and the special solution equation can be obtained. With regard to the non-homogeneous boundary conditions of general engineering problems, Fourier expansion is used to transform them into algebraic sums. According to the special solution equation, the relationship between the direction eigenvalues and the angle coefficients of each order of the expansion term is found. Then the analytical solution of the non-homogeneous problem is obtained by superposing the special solutions of the expansion terms of each order. By comparing the calculation results of the boundary stress around the elliptical tunnel, it can be seen that the results are completely consistent with the finite element calculation results, which verifies the correctness and reliability of the method in this paper. This paper analyzes the influence of internal pressure, lateral pressure coefficient and elliptical shape coefficient on the circumferential stress around the elliptical hole, and obtains the following conclusions: increasing the internal pressure of the elliptical hole can effectively reduce the circumferential compressive stress at the hole edge; when the lateral pressure coefficient approaches 1, the circumferential stress distribution around the elliptical hole tends to be uniform; the minimum value of the circumferential stress around the elliptical hole is independent of the shape coefficient, while the maximum value is closely related to the shape coefficient. The research results can not only provide specific theoretical guidance for the construction and construction of elliptical tunnel, but also expand the application scope of the symplectic elasticity method.