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

崔建斌; 姬安召; 熊贵明

doi:10.21656/1000-0887.390326

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

doi: 10.21656/1000-0887.390326

¹陇东学院数学与统计学院, 甘肃庆阳 745000;²陇东学院能源工程学院, 甘肃庆阳 745000;³太原理工大学矿业工程学院, 太原 030024

基金项目: 甘肃省自然科学基金(17JR5RM355)

详细信息

作者简介:
崔建斌(1972—)，男，教授，硕士(通讯作者. E-mail: cuijb0658@163.com).

中图分类号: O241
计量
- 文章访问数: 1244
- HTML全文浏览量: 189
- PDF下载量: 369
- 被引次数: 0
出版历程
- 收稿日期: 2018-11-26
- 修回日期: 2019-03-13
- 刊出日期: 2019-10-01

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

¹School of Mathematics and Statistics, Longdong University, QingYang, GanSu 745000, P.R.China;²School of Energy Engineering, Longdong University, QingYang, GanSu 745000, P.R.China;³College of Mining Engineering, Taiyuan University of Technology, Taiyuan 030024, P.R.China

摘要

摘要: 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σ^ρ和σ^θ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.
- 多边形围岩隧道 /
- SchwarzChristoffel变换 /
- 平面应力 /
- 围岩应力分布
Abstract: 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 σ^ρ and σ^θ 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.
- polygonal surrounding rock tunnel /
- Schwarz-Christoffel transformation /
- plane stress /
- stress distribution of surrounding rock

HTML全文

参考文献(18)

[1]	田尤, 杨为民, 黄晓, 等. 天水市麦积区幅黄土滑坡发育分布特征及其孕灾因素分析[J]. 地质力学学报, 2016,22(1): 25-38.(TIAN You, YANG Weimin, HUANG Xiao, et al. Distribution characteristics and inducing factors of loess landslide in Maiji mappable unit, Tianshui[J]. Journal of Geomechanics,2016,22(1): 25-38.(in Chinese))
[2]	石玲, 王涛, 辛鹏. 陕西省宝鸡市地质灾害发育特征[J]. 地质力学学报, 2013,19(4): 351-363.(SHI Ling, WANG Tao, XIN Peng. Development characteristics of the geo-harzards in Baoji city, Shaanxi province[J]. Journal of Geomechanics,2013,19(4): 351-363.(in Chinese))
[3]	FAHIMIFAR A, TEHRANI F M, HEDAYAT A, et al. Analytical solution for the excavation of circular tunnels in a visco-elastic Burger’s material under hydrostatic stress field[J]. Tunnelling and Underground Space Technology,2010,25(4): 297-304.
[4]	GAO G Y, CHEN Q S, ZHANG Q S, et al. Analytical elasto-plastic solution for stress and plastic zone of surrounding rock in cold region tunnels[J]. Cold Regions Science and Technology,2012,72: 50-57.
[5]	ZHANG J U, HOU D Q, CHEN X P. Displacement analytical solution of a deep elliptical tunnel in transversely isotropic rock mass[J]. Advanced Materials Research,2012,402: 593-597.
[6]	陈子荫. 围岩力学分析中的解析方法[M]. 北京: 煤炭工业出版社, 1994.(CHEN Ziyin. Analytical Method for Mechanics Analysis of Surrounding Rock [M]. Beijing: China Coal Industry Publishing House, 1994.(in Chinese))
[7]	祝江鸿. 隧洞围岩应力复变函数分析法中的解析函数求解[J]. 应用数学和力学, 2013,34(4): 345-354.(ZHU Jianghong. Analytic functions in stress analysis of the surrounding rock for caverns with the complex variable theory[J].Applied Mathematics and Mechanics,2013,34(4): 345-354.(in Chinese))
[8]	吕爱钟, 张路青. 地下隧道力学分析中的复变函数方法[M]. 北京: 科学出版社, 2007.(L Aizhong, ZHANG Luqing. Method of Complex Variable Function in Mechanical Analysis of Underground Tunnel [M]. Beijing: Science Press, 2007.(in Chinese))
[9]	朱大勇, 钱七虎, 周早生, 等. 复杂形状洞室映射函数的新解法[J]. 岩石力学与工程学报, 1999,18(3): 279-282.(ZHU Dayong, QIAN Qihu, ZHOU Zaosheng, et al. New method for mapping function of complex shaped cavern[J]. Chinese Journal of Rock Mechanics and Engineering,1999,18(3): 279-282.(in Chinese))
[10]	郑志强. 单位圆到任意曲线保角变换的近似计算方法[J]. 应用数学和力学, 1992,13(5): 449-457.(ZHENG Zhiqiang. An approximate method on the conformal mapping from a unit circle to an arbitrary curve[J]. Applied Mathematics and Mechanics,1992,13(5): 449-457.(in Chinese))
[11]	胡龙飞, 刘全坤, 冯秋红. 基于共形映射的壁板型材挤压模具型腔建模[J]. 机械工程学报, 2008,44(8): 180-184.(HU Longfei, LIU Quankun, FENG Qiuhong. Modeling for shape extrusion of aluminum alloy flat-plate based on conformal mapping[J]. Chinese Journal of Mechanical Engineering,2008,44(8): 180-184.(in Chinese))
[12]	李成, 郑艳萍, 李大磊. 积分方程法对含复杂孔形复合材料板孔边应力分布的研究[J]. 机械强度, 2006,28(6): 931-936.(LI Cheng, ZHENG Yanping, LI Dalei. Research on hole-edge stress distribution of composite materials plate with complex holes by integral equations method[J]. Journal of Mechanical Strength,2006,28(6): 931-936.(in Chinese))
[13]	MUSKHELISHVILI N I. Some Basic Problems of the Mathematical Theory of Elasticity [M]. Groningen: P Noordhoff Ltd, 1953.
[14]	闻国椿. 共形映射和边值问题[M]. 北京: 高等教育出版社, 1985.(WEN Guochun. Conformal Mapping and Boundary Value Problems [M]. Beijing: Higher Education Press, 1985.(in Chinese))
[15]	徐趁肖, 齐红元, 朱衡君. 任意边界域映射建模理论及模具设计应用[J]. 机械工程学报, 2002,38(9): 83-86.(XU Chenxiao, QI Hongyuan, ZHU Hengjun. Modeling theory of complicated contact region conformal mapping and application on diedesign[J]. Chinese Journal of Mechanical Engineering,2002,38(9): 83-86.(in Chinese))
[16]	徐趁肖, 朱衡君, 齐红元. 复杂边界单连通域共形映射解析建模研究[J]. 工程数学学报, 2002,19(4): 135-138.(XU Chenxiao, ZHU Hengjun, QI Hongyuan. Analytically modeling of complicated boundary simply-connected region conformal mapping[J]. Chinese Journal of Engineering Mathematics,2002,19(4): 135-138.(in Chinese))
[17]	崔建斌, 姬安召, 王玉风, 等. 单位圆到任意多边形区域的Schwarz Christoffel变换数值解法[J]. 浙江大学学报(理学版), 2017,44(2): 161-167.(CUI Jianbin, JI Anzhao, WANG Yufeng, et al. Numerical solution method for Schwarz Christoffel transformation from unit circle to arbitrary polygon area[J]. Journal of Zhejiang University(Science Edition),2017,44(2): 161-167.(in Chinese))
[18]	万世文. 深部大跨度隧道失稳机理与围岩控制技术研究[D]. 博士学位论文. 南京: 中国矿业大学, 2011.(WAN Shiwen. Research on collapse mechanism and surrounding rock control of long-span roadway in deep[D]. PhD Thesis. Nanjing: China University of Mining and Technology, 2011.(in Chinese))