Analytic Functions in Stress Analysis of the Surrounding Rock for Caverns With the Complex Variable Theory

ZHU Jiang-hong

doi:10.3879/j.issn.1000-0887.2013.04.003

Volume 34 Issue 4

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ZHU Jiang-hong. 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. doi: 10.3879/j.issn.1000-0887.2013.04.003

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Analytic Functions in Stress Analysis of the Surrounding Rock for Caverns With the Complex Variable Theory

doi: 10.3879/j.issn.1000-0887.2013.04.003

ZHU Jiang-hong

School of Civil Engineering and Architecture, Zhejiang University of Science and Technology, Hangzhou 310023, P.R.China

Received Date: 2012-02-22
Rev Recd Date: 2013-04-03
Publish Date: 2013-04-15

Abstract

Abstract

Two analytic functions must be derived from the stress equation for the boundary condition, when the surrounding rock stress is analyzed in the tunnel with arbitrary excavation cross section with the complex variable theory. Based on the Harnack theorem, the stress equation for the boundary condition was transformed into an integral equation in the surrounding rock. The mapping function expressed with the finite Laurent series was introduced into the integral equation, and then the boundary with arbitrary excavation cross section was change into the boundary with the unit circle line for solving two analytic functions. The analyticity was studied for each integrand in the integral equation in discussed domain. Using above result, each integral term of the equation was solved with the Residue theorem and the general formulas of two analytic functions were obtained to calculate the surrounding rock stress in tunnel with arbitrary excavation cross section. Moreover, study examples of the analytic functions were provide for the circular tunnel and the elliptic tunnel and two general formulas for analytic function. The results were accord with the literatures. General formulas of two analytic function obtained with the Residue theorem are adapt to calculate the surrounding rock stress in the caverns with arbitrary excavation cross section. Compared to other methods, it is much simpler to obtain the analytical solutions and the calculating results are more precise for the actual caverns.
- surrounding rock stress,
- analytic function,
- excavation cross section,
- Laurent series,
- residue theorem

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References(25)

References

[1]	WANG Shui-lin, WU Zhen-jun, GUO Ming-wei, GE Xiu-run. Theoretical solutions of a circular tunnel with the influence of axial in situ stress in elasticbrittleplastic rock[J].Tunnelling and Underground Space Technology, 2012, 30: 155-168.
[2]	LI Shu-cai, WANG Ming-bin. Elastic analysis of stressdisplacement field for a lined circular tunnel at great depth due to ground loads and internal pressure[J].Tunnelling and Underground Space Technology, 2008, 23(6): 609-617.
[3]	Gao G Y, Chen Q S, Zhang Q S, Chen G Q. 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.
[4]	Muskhelishvili N I.Some Basic Problems of the Mathematical Theory of Elasticity： Fundamental Equations, Plane Theory of Elasticity, Torsison, and Bending [M]. Groningen: P Noordhoof, 1953.
[5]	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 Zi-yin.Analytic Method of Mechanical Analysis for the Surrounding Rock [M]. Beijing: Coal Industry Publishing House, 1994. (in Chinese))
[6]	Zhu J H, Ji H G. The comparison of surrounding rock stress analytical solutions for three tunnel sections[C]// Proceeding of the 2nd ISRM International Young Scholars’Symposium on Rock Mechanics, 2011: 741-744.
[7]	皇甫鹏鹏, 伍法权, 郭松峰, 熊峥. 基于边界点搜索的洞室外域映射函数求解法[J].岩石力学, 2011, 32(5)：1418-1424. (HUANGFU Peng-peng, WU Fa-quan, GUO Song-feng, XIONG Zheng. 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))
[8]	朱大勇, 钱七虎, 周早生, 徐文焕.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 Da-yong, QIAN Qi-hu, ZHOU Zao-sheng, XU Wen-huan. 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))
[9]	蔡晓鸿, 蔡勇斌, 蔡勇平, 甘晓荣, 孙富学. 二向不等围压和内压作用下椭圆形洞室的计算[J]. 地下空间与工程学报, 2008, 4(3): 453-459.(CAI Xiao-hong, CAI Yong-bin, CAI Yong-ping, GAN Xiao-rong, SUN Fu-xue. Computation of elliptic tunnel under the combined action of twodimensional unequal adjoining rock pressure and internal pressure[J].Chin J Undergr Sp Eng, 2008, 4(3): 453-459. (in Chinese))
[10]	卢文超, 仲政, 王旭. 浅埋隧道围岩应力场的解析解[J].力学季刊, 2003, 24(1):5054. (LU Wen-chao, ZHONG Zheng, WANG Xu. Analytical solution for stress field in surrounding rocks of shallow tunnel[J].Chin Quart Mech, 2003, 24(1), 50-54. (in Chinese))
[11]	王明斌, 李术才, 李树忱, 王兆清. 圆形隧道围岩附加应力场的解析解答[J]. 岩土力学, 2006, 27(增刊):207210. (WANG Ming-bin, LI Shu-cai, LI Shu-chen, WANG Zhao-qing. Analytical solution of subsidiary stress field for circular tunnel[J].Rock Soil Mech, 2006, 27(Supp):207210. (in Chinese))
[12]	朱大勇, 钱七虎, 周早生, 徐文焕. 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 Da-yong, QIAN Qi-hu, ZHOU Zao-sheng, XU Wen-huan. 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))
[13]	吕爱钟. 地下洞室最优开挖形状的确定方法[J].岩石力学与工程学报, 1996, 15(3):275281.(Lü Ai-zhong. The method for optimum shapes of tunnels and cavities[J].Chinese Journal of Rock Mechanics and Engineering,1996, 15(3): 275-281. (in Chinese))
[14]	吕爱钟.高地应力区地下硐室断面形状的选择方法[J]. 煤炭学报,1997, 22（5）：495498.(Lü Ai-zhong. Selection method of section configration of underground chamber in high strata stress zone[J].Journal of China Coal Society, 1997, 22(5):495-498. (in Chinese))
[15]	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 Lu-qing, YANG Zhi-fa, Lü Ai-zhong. Analysis study on displacement field of surrounding rocks of two parallel tunnels with arbitrary shapes[J].Chinese Journal of Rock Mechanics and Engineering,2000, 19(5): 584-589. (in Chinese))
[16]	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 Chang-wu, ZHANG Guo-liang. 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))
[17]	汤澄波, 沈其良. 煤矿深部天幕巷道围岩应力分析和围岩稳定性[J].煤炭学报, 1994, 19（5）：531-540. (TANG Cheng-bo, SHEN Qi-liang. Stresses and stability analysis of rocks surrounding a tunnel in cyclorama arch section in deep mines[J].Journal of China Coal Society, 1994, 19(5): 531540. (in Chinese))
[18]	汤澄波, 范广勤. 高地应力区天幕线拱形硐室断面的解法[J].煤炭工程, 1989（1）：16-18. (TANG Cheng-bo, FAN Guang-qin. Determination of the mapping function for the exterior domain of a non-circular opening by means of the multiplication of tree absolutely convergent series[J].Coal Engineering, 1989(1): 16-18. (in Chinese))
[19]	王润富.弹性力学的复变函数计算机解[J].河海大学学报, 1991, 19(2):71-74. (WANG Run-fu. Computer solutions for complex function in elasticity[J].Journal of Hohai University, 1991, 19(2): 71-74. (in Chinese))
[20]	刘金高, 王润富.马蹄形孔口和梯形孔口的应力集中问题[J].岩土工程学报, 1995, 17（5）：57-63. (LIU Jin-gao, WANG Run-fu. The stress concentration of U-shaped and laddershaped holes[J].Chinese Journal of Geotechnical Engineering, 1995, 17(5): 57-63. (in Chinese))
[21]	Zhang Z Z, Sun Y Z. Analytical solution for a deep tunnel with arbitrary cross section in a transversely isotropic rock mass[J].Int J Rock Mech Min, 2011, 48(8): 1359-1363.
[22]	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 Mei-feng, HE Man-chao, LIU Dong-yan.Rock Mechanics and Engineering[M]. Beijing: Science Press, 2002. (in Chinese))
[23]	张倬元, 王士天, 王兰生.工程地质分析原理[M]. 第2版.北京：地质出版社, 2005. (ZHANG Zhuo-yuan, WANG Shi-tian, WANG Lan-sheng.Principles of Engineering Geological Analysis [M]. 2nd ed. Beijing: Geological Publishing House, 2005. (in Chinese))
[24]	祝江鸿.公路小净距隧道围岩稳定性与合理净距研究[D]. 博士论文. 北京：北京科技大学, 2012. (ZHU Jiang-hong. Study on the surrounding rock stability and reasonable clear interval for road neighborhood tunnel[D]. Ph D thesis. Beijing: University of Science and Technology, 2012. (in Chinese))
[25]	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. (Lü Ai-zhong, ZHANG Lu-qing.Mechanics Analysis With Complex Variables Function in Tunnel Underground [M]. Beijing: Science Press, 2007. (in Chinese))