基于复变理论的双向不等压圆形巷道围岩应力及塑性区分析

Analysis of surrounding rock stress and plastic zone of two-way unequal pressure circular roadway based on complex variable theory

为了解决双向不等压条件下圆形巷道的非轴对称弹塑性问题,给定圆形巷道应力和塑性区的解析解,本文基于复变理论,利用复变函数中保角变换的方法,推导出圆形巷道在双向不等压应力场中其周边围岩的切向应力方程,再结合Mohr-Coulomb强度准则,进一步得到圆形巷道在双向不等压条件下的塑性区计算公式,由上述两式可分别计算不同侧压系数下圆形巷道的应力分布状态和塑性区大小及形状。通过代入煤矿具体地质参数,分析不同围岩力学性质对巷道周边切向应力分布和塑性区的影响规律,并采用数值模拟软件FLAC 3D 5.0进行验证。分析结果表明,数学公式计算与数值模拟结果相吻合,从而验证了该计算方法的准确性;侧压系数与埋深是影响巷道周边围岩切向应力分布的主要因素,侧压系数大于1时,圆形巷道从顶板围岩开始旋转360°其切向应力值呈正弦函数分布,且侧压系数越大曲线振幅越大;侧压系数、巷道半径与围岩力学参数(黏聚力、内摩擦角)是影响圆形巷道塑性区形状与大小的主要因素,其中侧压系数为塑性区形状的决定性因素,侧压系数为1时,塑性区形状为圆形,侧压系数不为1时,圆形巷道由圆形过渡为椭圆形,最后变为两头等大的“葫芦”形,且其长轴总平行于地应力较小的方向;对塑性区大小的影响四者分别存在一个临界值,使圆形巷道不形成封闭的塑性区。

In order to solve the non-axisymmetric elastoplastic problem of circular roadway under bidirectional unequal pressure conditions,the analytical solution of stress and plastic zone of circular roadway is determined.Based on the complex theory,this paper uses the method of conformal transformation in complex variable function to derive the tangential stress equation of surrounding rock in circular roadway in two-way unequal stress field.Furthermore,combined with the Mohr-Coulomb strength criterion,the plastic zone calculation formula of the circular tunnel under the bidirectional unequal pressure condition is obtained and the stress distribution state and the size and shape of the plastic zone under different lateral pressure coefficients are calculated.Based on the specific geological parameters of coal mines,the influence of mechanical properties of different surrounding rocks on the tangential stress distribution and plastic zone around the roadway is analyzed and verified by numerical simulation software FLAC 3D 5.0.The analysis results show that the mathematical formula is consistent with the numerical simulation results,which verifies the accuracy of the calculation method.The lateral pressure coefficient and buried depth are the main factors affecting the tangential stress distribution of surrounding rock around the roadway.When the lateral pressure coefficient is greater than 1,the circular tunnel starts to rotate 360°from the surrounding rock of the roof.The tangential stress value is distributed as a sine function,and the lateral pressure coefficient is larger,the curve amplitude is larger.The lateral pressure coefficient,roadway radius and surrounding rock mechanical parameters(cohesion,internal friction)are the main factors affecting the shape and size of the plastic zone of the circular roadway.The lateral pressure coefficient is the decisive factor of the shape of the plastic zone.When the lateral pressure coefficient is 1,the shape of the plastic zone is circular,and the lateral pressure coefficient is not 1.The circular roadway transitions from a circular shape to an elliptical shape,and finally becomes a two-headed“hoist”shape,and its long axis is always parallel to the direction in which the ground stress is small.There is a critical value for the influence of the size of the plastic zone,so that the circular roadway does not form a closed plastic zone.