横观各向同性岩体中任意形状巷道的位移解析解

Displacement Analytic Solution of a Deep Tunnel with Arbitrary Cross Section in Transversely Isotropic Rock Mass

在采矿工程中遇到大量的层状岩体稳定性问题。对于层状岩体,在力学上可将其处理成横观各向同性体。根据横观各向同性弹性力学的平衡方程、物理方程和几何方程,推导了横观各向同性平面应变问题的相容方程;根据应力函数的复变函数表示,推导了横观各向同性平面应变问题中位移分量的复变函数表示;利用保角映射,推导了横观各向同性岩体中任意形状巷道的位移解析解。虽然在推导中假设巷道横断面平行于各向同性面,但位移解析解表明,围岩径向位移不仅与各向同性面上的弹性模量和泊松比有关,而且与垂直各向同性面方向的弹性模量和泊松比有关。

In mining engineering a large number of stability problems of layered rock mass is are often encountered.Layered rock mass is generally considered to be transversely isotropic solid in mechanics.Based on the equilibrium equations,generalized Hooke＇s law equations and geometry equations of transversely isotropic elasticity mechanics,compatibility equation of plane strain of transversely isotropic body is deduced.Based on the complex variable expression of stress function,the complex variable expression of displacement components of transversely isotropic plane strain problem is deduced.Finally the displacement analytic solution of a deep tunnel with arbitrary cross section in transversely isotropic rock mass is derived by conformal mapping method.Although the cross section of the tunnel is assumed to be parallel to the isotropic plane in the deduction,the displacement analytic solution indicates that radial displacement of surrounding rock is related not only to elastic modulus and poisson ratio on the isotropic plane but also to that in the direction normal to the isotropic plane.