倾斜坡体中圆孔扩张的弹性应力分析

ELASTIC STRESS ANALYSIS OF CIRCULAR CAVITY EXPANSION UNDER A SLOPE

针对实际工程中遇到的斜坡边界下圆形孔扩张问题,提出了相应的计算方法。在考虑坡体自重应力的情况下,通过坐标变换将其转为水平边界半无限体中圆孔扩张,采用复变函数共形映射的方法得到应力解析。当圆形孔埋置较深时,将问题简化为无限体中圆孔的扩张,通过弹性力学的基本叠加得到最终应力解答。以一斜坡边界下圆形孔洞扩张为例,求解了圆孔周围径向正应力σr、环向正应力σφ与切应力τrφ的分布。计算结果表明:斜坡倾角对坡体中应力分布影响显著,某点应力随距圆心距离的增大而减小,超过约4倍圆孔半径远处逐渐趋于稳定,其值接近于初始地应力场。当圆孔为深埋的情况时,与简化为无限平面的情况进行了对比,两种方法计算结果接近,距圆心2.5R远处岩体径向正应力σr和环向正应力σφ均为负值,而切应力τrφ正负值间隔分布,各应力极值分布与斜坡倾向呈一定的相关性。

This paper presents methods for calculating circular cavity expansion under a slope. The slope boundary is turned to be horizontal with a coordinate transformation technique. Using the complex variable method and mapping the hole and half-plane to a circular ring offer an analytical solution for stress, considering the initial gravity stress. When the circular cavity is deep enough, the influence of the ground surface is ignored and the problem is simplified as cavity expansion in an infinite space. Then, the solutions are attained by superposition of results from stress components. The radial stress, σr, hoop stress, σφ and shear stress, τrφ, are analyzed and the results show that the stresses at a certain point in a slope are affected by the slope dip and the distance from the cavity center. Specifically, with a distance from the cavity center greater than 4 times the cavity radius R, the stress caused by a cavity at a point can be safely neglected, which means the final value is close to original gravity stresses. Meanwhile, results calculated by superposition method for the deep cavity agree with those from complex variables method. At a point, specifically a distance of 2.5R from the cavity center, values of σr and σφ are all negative. However, the positive and negative values of τrφ are cross distributed. The distribution of stress extremum is related to the slope dip.