单位圆外域-洞室外域共形映射下任意开挖断面隧洞围岩应力解析解

Analytical Stress Solutions Around Tunnels With Arbitrary-Excavation Cross Sections Based on Conformal Mapping From a Unit Circle to the Tunnel Exterior

以地下任意开挖断面隧洞为研究对象,将原岩应力场归化为无穷远处与水平轴呈一角度的均匀应力场,引入单位圆外域-洞室外域的共形映射函数,把目标域中以变量z表示的围岩应力函数转化为映射域中的变量ζ表示.将洞壁的应力边界条件方程转化成Cauchy积分方程,利用被积函数的解析性和留数理论,获得了任意开挖断面隧洞围岩应力函数解析通式和围岩应力解析解.将椭圆形隧洞应力边界条件代入应力函数解析通式,求得的φ(z)和ψ(z)与文献结果一致;以公路两车道隧道为研究对象,利用围岩应力解析解开展围岩应力计算和分析,所获结果符合工程规律;利用ABAQUS软件对隧道洞壁应力进行对比分析,分布规律与解析解结果一致.研究表明,只要给出任意开挖断面隧洞的单位圆外-洞室外域共形映射函数,利用研究成果可方便地开展围岩应力解析计算分析.

The stresses around tunnels with arbitrary-excavation cross sections were studied with the stress field simplified as a uniform stress field at an angle to the horizontal axis at infinity.The conformal mapping function was introduced to express the stress functions of surrounding rock with variableζinstead of variable z.Then the Cauchy integral operator was applied to the stress boundary equation for tunnel wall.The general analytical solutions to stress functions and analytical solutions of surrounding rock stresses were obtained for tunnels with arbitrary-excavation cross sections according to the integrand analyticity and the residue theorem.Moreover,with the stress boundary conditions for elliptical tunnels substituted into the general analytical solutions,the obtained particular solutions to stress functions consisted with the results in literatures.The surrounding rock stresses were analyzed for 2-lane road tunnels based on the explicit general analytical solutions and the results accorded with engineering rules.The stress distribution of tunnel wall was analyzed with the ABAQUS finite element method,and the results coincided with the analytical solution.The analytical calculation can be expediently carried out for underground tunnels with arbitrary-excavation cross sections as long as their conformal mapping functions are presented from the unit circle exterior to the tunnel exterior.