隧洞围岩应力复变函数分析法中的解析函数求解

利用复变函数理论进行地下任意开挖断面隧洞围岩应力分析的前提，是根据围岩应力边界条件方程推导出两个解析函数．从Harnack定理出发，将隧洞围岩应力边界条件方程转化成积分方程；把Laurent级数有限项表示的映射函数引入积分方程中，将以任意开挖断面为边界条件的解析函数求解转化成以单位圆周线为边界条件的求解问题．对积分方程中各被积函数在讨论域内的解析性进行了分析，在此基础上利用留数理论求解了方程中各项积分值，并获得了用来表示任意开挖断面隧道围岩应力的两个解析函数通式．给出了圆形和椭圆形隧道的两个解析函数求解算例，所获得的结果与文献中的结果一致．利用留数理论推导出的两个解析函数通式，适用于任意开挖断面隧洞的围岩应力解析解的计算，且计算过程更为简单，计算结果更为精确．

Analytic Functions in Stress Analysis of the Surrounding Rock for Caverns With the Complex Variable Theory

Two analytic functions must be derived from the stress equation for the boundary condition, when the surrounding rock stress is analyzed in the tunnel with arbitrary excavation cross section with the complex variable theory. Based on the Harnack theorem, the stress equation for the boundary condition was transformed into an integral equation in the surrounding rock. The mapping function expressed with the finite Laurent series was introduced into the integral equation, and then the boundary with arbitrary excavation cross section was change into the boundary with the unit circle line for solving two analytic functions. The analyticity was studied for each integrand in the integral equation in discussed domain. Using above result, each integral term of the equation was solved with the Residue theorem and the general formulas of two analytic functions were obtained to calculate the surrounding rock stress in tunnel with arbitrary excavation cross section. Moreover, study examples of the analytic functions were provide for the circular tunnel and the elliptic tunnel and two general formulas for analytic function. The results were accord with the literatures. General formulas of two analytic function obtained with the Residue theorem are adapt to calculate the surrounding rock stress in the caverns with arbitrary excavation cross section. Compared to other methods, it is much simpler to obtain the analytical solutions and the calculating results are more precise for the actual caverns.