浅埋隧道围岩应力场的解析解

隧道围岩应力和变形分析是隧道设计的重要内容。对深埋隧道的研究已取得了很多结果。但对于受地表边界和地面荷载的影响，浅埋隧道围岩分析在数学处理上仍存在一定的困难。一般采用边界元或有限元等数值方法，未见有解析解的报导。本文采用复变函数解法研究地面荷载作用下浅埋圆形隧道围岩的平面弹性问题，该解法利用复变函数保角变换将物理平面上的研究域映射到像平面上的圆环域内。将复势函数进行罗伦级数展开，通过边界条件得到罗伦级数展开式系数的递推公式，并由复势函数确定应力分量和位移分量。最后通过算例给出了围岩应力分布和沉降曲线。所得结果适用于任意分布荷载的情况，具有通用性。

Analytical Solution for Stress Field in Surrounding Rocks of Shallow Tunnel

The stress and deformation analysis for surrounding rock of tunnels is an important task in tunnels design. Many results were obtained for the cases of deep tunnels. Due to the influence of the surface boundary and ground loads, the analysis for surrounding rocks of shallow tunnel is difficult by ma thematic treatment. Most of the solutions were obtained by using BEM or FEM. There were no analytical solutions available. In this paper, the plane elastic problem of surrounding rocks of shallow circular tunnels under ground loads was studied by means of complex variable method. The complex conformal transformation was used to map the considered region in the physical plane onto a circular ring region in the image plane. The complex potentials were expanded into Laurent Series whose coefficients could be expressed by recurrent relations. The stresses and displacements were then be determined by complex potentials. Finally, the stresses of surrounding rocks and the ground settlements are depicted through an example. The obtained results can be applied to the cases of arbitrary surface distributing loads.