内压对管状双椭圆洞室围岩应力影响的解析解

Analytical solutions to the stress of surrounding rock containing tubular double-ellipse caves considering internal water pressure

以管状双椭圆高水压洞室围岩应力为研究对象,将该问题简化为混合边界的双椭圆孔平面应力计算模型。基于平面弹性复变理论,通过保角映射、Cauchy积分及留数定理等方法,求解了具有单椭圆内压孔的两个平面应力复势函数;采用Schwarz交替算法,推导了具有双椭圆内压孔无限平面内任一点的应力分量表达式。据所得表达式,编制了Matlab计算程序,研究了无限平面含双洞室或双椭圆洞室的4个算例,分析了内压及两孔相对位置对围岩应力的影响。研究结果表明:两洞室均无内压时,水平对称双椭圆洞室的左孔口0°(或右孔口180°)处有最大切向应力;各孔口的最大或最小切向应力均随内压的增大而减小;当两孔中仅一孔含内压时,两孔孔口应力随内压的增大而或增或减,具体视孔壁位置而定。

In order to solve the stress of the surrounding rock with tubular double-ellipse caves under internal high water pressure, a plane stress model for double-ellipse caves under mixed boundary condition is employed in this study. Based on the plane elastic complex variable theory, two analytical functions of the plane stress containing internal water pressure with one elliptical cave are solved by means of conformal mapping, Cauchy integral and residue theorem. Using the Schwarz＇s alternating method, the analytical expressions of component stresses in unlimited plane with two elliptical caves are obtained. According to these obtained expressions, Matlab software is applied to analyze four examples of stress in unlimited planes with two circular caves or two elliptical caves. The influences of internal pressure and the relative locations between two caves on the stress of surrounding rock are investigated. The results show that when two elliptical caves without internal pressure are arranged horizontally symmetrical, the maximum tangential orifice stress is found at 0° of left cave（or 180° of right cave）. When internal pressure increases, both the maximum and minimum tangential orifice stresses decrease; if only one of these two caves containing internal pressure, when this pressure increases, the change of tangential orifice stress depends on its location relative to cave wall.