基于复变函数理论的巷道围岩稳定性分析

Stability analysis of roadway surrounding rock based on complex variable function theory

针对双向不等压直墙半圆拱形巷道围岩塑性区的解析问题,基于Mohr-Coulomb强度准则,通过复变函数中的保角变换的方法将物理平面非圆形弹性区域映射到像平面的单位圆外域进行求解,将确定弹塑性区的交界问题转化为映射函数系数的求解问题。得到巷道围岩应力复变函数解以及巷道围岩塑性区的范围,并通过FLAC^(3D)数值模拟的手段对理论计算的准确性进行验证分析。研究结果表明:通过复变函数理论能够较为准确地得到直墙半圆拱形巷道塑性区范围的理论计算公式,结合公式可以发现影响直墙半圆拱形巷道围岩塑性区发育的影响因素为巷道断面尺寸(包括巷道宽度,直墙高度以及拱高)、垂直应力、侧压系数、黏聚力以及内摩擦角的大小,将理论应用于工程实例计算分析,采用单因素分析的方法对黏聚力和内摩擦角进行数值模拟分析,提取数值模拟试验的数据,对理论计算和数值模拟水平方向塑性区受黏聚力和内摩擦角影响的范围进行对比和分析。可以得出虽然理论计算与数值模拟的结果存在误差,但已能反映直墙半圆拱形巷道围岩塑性区分布情况,通过研究还发现,当黏聚力和内摩擦角比较小时,巷道呈现“X”形破坏,当黏聚力和内摩擦角增大时,巷道呈现类似椭圆形破坏,随着黏聚力和内摩擦角的增大,巷道塑性区范围的减幅减小,并且当水平应力大于垂直应力时,在垂直方向塑性区范围变化幅度较大。

In view of the analytical problem of the plastic zone of the surrounding rock in the semi-circular arched roadway with two-way unequal pressure straight walls,based on the Mohr-Coulomb strength criterion,the non-circular elastic zone of the physical plane was mapped to the image plane by the method of conformal transformation in the complex variable function.The problem of determining the boundary of the elastic-plastic region was transformed into the problem of solving the coefficients of the mapping function.The complex function solution of the surrounding rock stress of the roadway and the range of the surrounding plastic zone of the roadway were obtained,and the accuracy of the theoretical calculation was verified by FLAC^(3D) numerical simulation.The research results show that the theoretical calculation formula of the plastic zone range of the straight semi-circular arched roadway can be obtained accurately by the complex varia⁃ble function theory.Combined with the formula,it can be found that the influencing factors affecting the development of the surrounding rock plastic zone in the semicircular arched roadway with straight walls are the size of the roadway section(including the width of the road⁃way,the height of the straight wall and the arch height),the vertical stress,the lateral pressure coefficient,the cohesion and the size of the internal friction angle.The theory was applied to the calculation and analysis of engineering cases,and the single factor analysis method was used to carry out numerical simulation analysis on the cohesion force and internal friction angle,and the data of the numerical simula⁃tion test was extracted for comparison and analysis.It can be concluded that although there are errors in the results of theoretical calcula⁃tions and numerical simulations,it can reflect the distribution of the surrounding plastic zone of semi-circular arched roadways in straight walls.When the cohesion and internal friction angle increase,the roadway presents similar elliptical failure.With the increase of cohesion and internal friction angle,the reduction of the plastic zone of the roadway decreases,and when the horizontal stress is greater than the vertical stress,the the range of the plastic zone in the vertical direction varies greatly.