SHI Gaoping, ZHU Jianghong, YANG Jianhui. Analytical Stress Solutions Around Tunnels With Arbitrary-Excavation Cross Sections Based on Conformal Mapping From a Unit Circle to the Tunnel Exterior[J]. Applied Mathematics and Mechanics, 2020, 41(5): 541-556. doi: 10.21656/1000-0887.400220
Citation: SHI Gaoping, ZHU Jianghong, YANG Jianhui. Analytical Stress Solutions Around Tunnels With Arbitrary-Excavation Cross Sections Based on Conformal Mapping From a Unit Circle to the Tunnel Exterior[J]. Applied Mathematics and Mechanics, 2020, 41(5): 541-556. doi: 10.21656/1000-0887.400220

Analytical Stress Solutions Around Tunnels With Arbitrary-Excavation Cross Sections Based on Conformal Mapping From a Unit Circle to the Tunnel Exterior

doi: 10.21656/1000-0887.400220
  • Received Date: 2019-07-18
  • Rev Recd Date: 2019-09-21
  • Publish Date: 2020-05-01
  • The stresses around tunnels with arbitrary-excavation cross sections were studied with the stress field simplified as a uniform stress field at an angle to the horizontal axis at infinity. The conformal mapping function was introduced to express the stress functions of surrounding rock with variable ζ instead of variable z. Then the Cauchy integral operator was applied to the stress boundary equation for tunnel wall. The general analytical solutions to stress functions and analytical solutions of surrounding rock stresses were obtained for tunnels with arbitrary-excavation cross sections according to the integrand analyticity and the residue theorem. Moreover, with the stress boundary conditions for elliptical tunnels substituted into the general analytical solutions, the obtained particular solutions to stress functions consisted with the results in literatures. The surrounding rock stresses were analyzed for 2-lane road tunnels based on the explicit general analytical solutions and the results accorded with engineering rules. The stress distribution of tunnel wall was analyzed with the ABAQUS finite element method, and the results coincided with the analytical solution. The analytical calculation can be expediently carried out for underground tunnels with arbitrary-excavation cross sections as long as their conformal mapping functions are presented from the unit circle exterior to the tunnel exterior.
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  • [1]
    CARRANZA-TORRES C, FAIRHURST C. The elasto-plastic response of underground excavations in rock masses that obey the Hoek-Brown failure criterion[J]. International Journal of Rock Mechanics and Mining Sciences,1999,36(6): 777-809.
    [2]
    EXADAKTYLOS G E, STAVROPOULOU M C. A closed-form elastic solution for stresses and displacements around tunnels[J]. International Journal of Rock Mechanics and Mining Sciences,2002,39(7): 905-916.
    [3]
    BATISTA M. On the stress concentration around a hole in an infinite plate subject to a uniform load at infinity[J]. International Journal of Mechanical Science,2011,53(4): 254-261.
    [4]
    GEREK H. An elastic solution for stresses around tunnels with conventional shapes[J]. International Journal of Rock Mechanics & Mining Sciences,1997,34(3/4): 96.
    [5]
    VERRUIJT A. Deformations of an elastic half plane with a circular cavity[J]. International Journal of Solids and Structures,1998,35(21): 2795-2804.
    [6]
    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))
    [7]
    祝江鸿. 公路小净距隧道围岩稳定性与合理净距研究[D]. 博士学位论文. 北京: 北京科技大学, 2012.(ZHU Jianghong. Study on the surrounding rock stability and reasonable clear interval for road neighborhood tunnel[D]. PhD Thesis. Beijing: University of Science and Technology Beijing, 2012.(in Chinese))
    [8]
    CHAO C K, LU L M, CHEN C K, et al. Analytical solution for a reinforcement layer bonded to an elliptic hole under a remote uniform load[J]. International Journal of Solids and Structures,2009,46(14/15): 2959-2965.
    [9]
    SAVIN G N. Stress Concentration Around Holes [M]. Oxford: Pergamon Press, 1961.
    [10]
    朱大勇, 钱七虎, 周早生. 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. New method for calculating mapping function of opening with complex shape[J]. Chinese Journal of Rock Mechanics and Engineering,1999,18(3): 279-282.(in Chinese))
    [11]
    皇甫鹏鹏, 伍法权, 郭松峰, 等. 基于边界点搜索的洞室外域映射函数求解法[J]. 岩土力学, 2011,32(5): 1418-1424.(HUNAGFU Pengpeng, WU Faquan, GUO Songfeng, et al. A new method for calculating mapping function of external area of cavern with arbitrary shape based on searching points on boundary[J]. Rock and Soil Mechanics,2011,32(5): 1418-1424.(in Chinese))
    [12]
    ZHU J H, JI H G. The comparison of surrounding rock stress analytical solutions for three tunnel sections[C]// Proceeding of the 〖STBX〗2nd ISRM International Young Scholars Symposium on Rock Mechanics . Beijing, 2011.
    [13]
    祝江鸿, 杨建辉, 施高萍, 等. 单位圆外域到任意开挖断面隧洞外域共形映射的计算方法[J]. 岩石力学, 2014,35(1): 175-183.(ZHU Jianghong, YANG Jianhui, SHI Gaoping, et al. Calculating method on the conformal mapping from the exterior of unit circle to the exterior of carven with the arbitrary excavation cross section[J]. Rock and Soil Mechanics,2014,〖STHZ〗 35(1): 175-183.(in Chinese))
    [14]
    王润富. 一种保角映射法及其微机实现[J]. 河海大学学报, 1991,19(1): 86-90.(WANG Runfu. Computer solutions for complex function in elasticity[J]. Journal of Hohai University,1991,19(1): 86-89.(in Chinese))
    [15]
    DELILLO T K, ELCRAT A R, PFALZGRAFF J A. Numerical conformal mapping methods based on Faber series[J]. Journal of Computational and Applied Mathematics,1997,83(2): 205-236.
    [16]
    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))
    [17]
    刘允芳, 刘元坤, 许静. 非圆形洞室应力监测反馈分析的复变函数方法[J]. 岩土工程学报, 2012,34(10): 1925-1930.(LIU Yunfang, LIU Yuankun, XU Jing. Complex function method of stress back analysis for a non-circular underground opening[J]. Chinese Journal of Geotechnical Engineering,2012,34(10): 1925-1930.(in Chinese))
    [18]
    皇甫鹏鹏, 张路青, 伍法权. 基于高效保角变换的单个复杂洞室围岩应力场解析解研究[J]. 岩石力学与工程学报, 2011,30(S2): 3905-3913.(HUANGFU Pengpeng, ZHANG Luqing, WU Faquan. Research on analytical solution of stress field of surrounding rock of single hole with arbitrary shape based on highly-efficient conformal mapping[J]. Chinese Journal of Rock Mechanics and Engineering,2011,30(S2): 3905-3913.(in Chinese)).
    [19]
    ZHANG Z, SUN Y. Analytical solution for a deep tunnel with arbitrary cross section in a transversely isotropic rock mass[J]. International Journal of Rock Mechanics & Mining Sciences,2011,48(8): 1359-1363.
    [20]
    朱大勇, 钱七虎, 周早生, 等. 复杂形状洞室围岩应力弹性解析分析[J]. 岩石力学与工程学报, 1999,18(4): 402-404.(ZHU Dayong, QIAN Qihu, ZHOU Zaosheng, et al. Elastic solution to stresses of rock masses around openings with complex shape[J]. Chinese Journal of Rock Mechanics and Engineering,1999, 18(4): 402-404.(in Chinese))
    [21]
    张路青, 杨志法, 吕爱钟. 两平行的任意形状洞室围岩位移场解析法研究及其在位移反分析中的应用[J]. 岩石力学与工程学报, 2000,19(5): 584-589.(ZHANG Luqing, YANG Zhifa, L Aizhong. Analysis study on displacement field of surrounding rocks of two parallel tunnels with arbitrary shapes and its application to back-analysis of displacement[J]. Chinese Journal of Rock Mechanics and Engineering,2000,19(5): 584-589.(in Chinese))
    [22]
    赵凯, 刘长武, 张国良. 用弹性力学的复变函数法求解矩形硐室周边应力[J]. 采矿与安全工程学报, 2007,24(3): 361-365.(ZHAO Kai, LIU Changwu, ZHANG Guoliang. Solution for perimeter stresses of rocks around a rectangular chamber using the complex function of elastic mechanics[J]. Journal of Mining & Safety Engineering,2007,24(3): 361-365.(in Chinese))
    [23]
    施高萍, 祝江鸿, 李保海. 矩形巷道孔边应力的弹性分析[J]. 岩石力学, 2014,〖STHZ〗 35(9): 2587-2593.(SHI Gaoping, ZHU Jianghong, LI Baohai. Elastic analysis of hole-edge stress of rectangular roadway[J]. Rock and Soil Mechanics,2014,35(1): 2587-2593.(in Chinese))
    [24]
    蔡美峰, 何满潮, 刘东燕. 岩石力学与工程[M]. 北京: 科学出版社, 2002.(CAI Meifeng, HE Manchao, LIU Dongyan. Rock Mechanics and Engineering [M]. Beijing: Science Press, 2002.(in Chinese))
    [25]
    张倬元, 王士天, 王兰生. 工程地质分析原理[M]. 北京: 地质出版社, 2005.(ZHANG Zhuoyuan, WANG Shitian, WANG Lansheng. Engineering Geological Analysis Principle [M]. Beijing: Geological Publishing House, 2005.(in Chinese))
    [26]
    吕爱钟, 张路青. 地下隧洞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. Mechanics Analysis With Complex Variables Function in Tunnel Underground [M]. Beijing: Science Press, 2007.(in Chinese))
    [27]
    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. Analytic Method of Mechanical Analysis for the Surrounding Rock [M]. Beijing: Coal Industry Publishing House, 1994.(in Chinese))
    [28]
    MUSKHELISHVILI N I. Some Basic Problems of the Mathematical Theory of Elasticity [M]. Netherlands: Springer, 2009.
    [29]
    罗超文, 李海波, 刘亚群. 深埋巷道地应力测量及围岩应力分布特征研究[J]. 岩石力学与工程学报, 2010,29(7): 1418-1423.(LUO Chaowen, LI Haibo, LIU Yaqun. Study of distributing characteristics of stress in surrounding rock masses and in-situ stress measurement for deeply buried tunnels[J]. Chinese Journal of Rock Mechanics and Engineering,2010,29(7): 1418-1423.(in Chinese))
    [30]
    中华人民共和国交通部. 公路隧道设计规范: JTG D70—2004[S]. 2004.(Ministry of Communications of the People’s Republic of China. Code for Design of Road Tunnel: JTG D70—2004[S]. 2004.(in Chinese))
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