基于Schwarz-Christoffel变换的圆形隧道围岩应力分布特征研究

隧道围岩应力还原到实际区域的变换方法被越来越多地应用到工程实践中，对复杂形状围岩体内的隧道围岩应力计算方法和隧道围岩稳定性的研究具有重要的实践指导意义.该文建立了不规则岩体中开挖圆形隧道的力学模型，运用复变函数理论，通过映射函数Schwarz-Christoffel变换，得到复平面单位圆到隧道所处多边形岩体的映射函数，在复变函数数域内分析了多边形岩体应力的求解步骤，运用弹性力学理论等推导了不规则岩体中圆形隧道的复变应力函数Φ(ξ)和φ(ξ)的表达式，并得出围岩体任一点应力分量σρ和σθ的解析通式.通过具体算例分析可知，岩体形状对圆形隧道的稳定性有较大的影响，隧道所处4种形状下的围岩体最大应力值分布规律为:顶底板的最大应力值从六边形、五边形、四边形、圆形围岩体依次减小，帮部的最大应力值从圆形、四边形、五边形、六边形围岩体依次减小.

Research on Surrounding Rock Stress Distributions for Circular Tunnels Based on the Schwarz-Christoffel Transformation

The transformation method with which the surrounding rock stress was mapped to an actual area, has been more and more widely applied to engineering practice, which is of practical guiding significance to the stress calculation method and the study on stability of tunnel surrounding rock. A mechanical model was established for circular tunnels in rock mass, with a mapping function obtained from a unit circle to a polygonal rock mass in the complex plane through the Schwarz-Christoffel transformation method based on the complex variable function theory. Then the solution of the stress distribution in the polygonal rock mass was studied in the complex function field, and subsequently the formulas of complex stress functions Φ(ξ) and φ(ξ) for circular tunnels in irregular rock masses were derived based on the elasticity theory. Finally, the analytical formulas of stress components σρ and σθ for any point in the surrounding rock mass were obtained. Analysis of examples indicates that the shape of the rock mass has large influence on the stability of circular tunnels. Here is the maximum stress distribution law for 4 shapes of rock masses: the maximum stresses in the roof and floor of the hexagon, the pentagon, the quadrilateral and the circle decrease in order; otherwise, those in the sidewalls of the circle, the quadrilateral, the pentagon and the hexagon decrease successively.