浅埋矩形顶管求解的复变函数实践

A complex variable solution for rectangle pipejacking in elastic half-plane

同时考虑洞周边界与地表边界,采用复变函数法求解浅埋矩形顶管施工引起的土体应力场、位移场。根据黎曼存在定理、复变函数的三角插值理论,提出计算含矩形的半无限域到同心圆环域共形映射的计算方法。在此基础上,将边界条件等式两边都展成洛朗级数,采用幂级数解法求得了复应力函数的系数。从该解法与有限元解的对比来看,在大部分点处结果相差不大,误差在2%左右。结果表明:(1)提出的保角映射函数形式及系数的求解方法适用于浅埋矩形隧洞;(2)提出的复变函数解法具有步骤清晰、收敛快、操作简单等特点。

Considering the boundary and the surface conditions,the stress field and displacement field caused by the construction of the shallow rectangular pipe jacking are solved by the complex function method.According to the Riemann’s existence theorem and basic complex variable theory,a conformal mapping function which can transform the half-plane with a rectangle cavity into the concentric ring is established.The both sides of boundary conditions equation are expanded into Laurent series,and then the coefficients of complex stress function are solved by power series method.The derived solution is applied to an example and a comparison is made using FEM method to show the accuracy of the methods,the results of this paper are almost the same as those of the FEM method,and the error is about2%.The result shows:(1)The method presented in this paper is applicable to a shallow buried rectangular tunnel;(2)The complex function method proposed in this paper is characterized by clear steps,fast convergence and simple operation.