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| Analytic method for elasto-plastic analysis of circular tunnels under non-axisymmetric stresses#br# |
| LU Aizhong，ZHANG Xiaoli，WANG Shaojie |
| (Institute of Hydroelectric and Geotechnical Engineering，North China Electric Power University，Beijing 102206，China) |
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Abstract The elasto-plastic analysis for surrounding rock mass of circular tunnels has always been a concerned problem of rock mechanics. Analytical solutions for determination of the plastic zone in the vicinity of a circular tunnel subject to non-axisymmetric stresses are presented based on the Mohr-Coulomb failure criterion. The conformal transformation method of complex variable function is employed to map the non-circular elastic regions in the physical plane to the outer regions of the unit circle in the image plane，and the problem of determining the elasto-plastic interface is transformed into solving the coefficients of the mapping function. The key point is to expand the complex combined stress expressions of the elastic-plastic interface into the Fourier series. The coefficients of the series are determined by solving the complex linear equations. Then，the nonlinear equations with the coefficients of the mapping function are established. The optimization method of the mixed penalty function is used to determining the elasto-plastic interface. The correctness of the method is verified by examples，and the rule for the impact of the cohesions，the internal friction angles and the lateral pressure coefficients on the shape of plastic zone is analyzed.
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[1] KASTNER H. 隧道与坑道静力学[M]. 同济大学翻译组译. 上海：上海科学技术出版社，1980：56–61.(KASTNER H. Statik des tunnel-und stollenbaues[M]. Translated by Tongji University. Shanghai：Shanghai Science and Technology Publishers，1980：56–61.(in Chinese))
[2] 刘夕才. 轴对称巷道变形的弹塑性理论分析[J]. 力学与实践，1994，16(5)：20–22.(LIU Xicai. Analytical elasto-plastic analysis of axisymmetric roadway deformation[J]. Mechanics in Engineering，1994，16(5)：20–22.(in Chinese))
[3] 袁文伯，陈 进. 软化岩层中巷道的塑性区与破碎区分析[J]. 煤炭学报，1986，(3)：77–86.(YUAN Wenbo，CHEN Jin. Analysis of plastic zone and loose zone around opening in softening rockmass[J]. Journal of China Coal Society，1986，(3)：77–86.(in Chinese))
[4] 蒋斌松，张 强，贺永年，等. 深部圆形巷道破裂围岩的弹塑性分析[J]. 岩石力学与工程学报，2007，26(5)：982–986.(JIANG Binsong，ZHANG Qiang，HE Yongnian，et al. Elastoplastic analysis of cracked surrounding rocks in deep circular openings[J]. Chinese Journal of Rock Mechanics and Engineering，2007，25(5)：982–986.(in Chinese))
[5] FLORENCE A L，SCHWER L E. Axisymmetric compression of a Mohr-Coulomb medium around a circular hole[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1978，2(4)：367–379.
[6] REED M B. The influence of out-of-plane stress on a plane strain problem in rock mechanics[J]. International Journal for Numerical and Analytical Methods in Geomechanics，1988，12(2)：173–181.
[7] PARK K H，KIM Y J. Analytical solution for a circular opening in an elastic-brittle-plastic rock[J]. International Journal of Rock Mechanics and Mining Sciences，2006，43(4)：616–622.
[8] LU A Z，XU G S，SUN F，et al. Elasto-plastic analysis of a circular tunnel including the effect of the axial in situ stress[J]. International Journal of Rock Mechanics and Mining Sciences，2010，47(1)：50–59.
[9] BROWN E T，BRAY J W，LADANYI B，et al. Ground response curves for rock tunnels[J]. Journal of Geotechnical Engineering，1983，109(1)：15–39.
[10] WANG Y. Ground response of circular tunnel in poorly consolidated rock[J]. Journal of Geotechnical Engineering，1996，122(9)：703–708.
[11] CARRANZA-TORRES C，FAIRHURST C. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion[J]. International Journal of Rock Mechanics and Mining Sciences，1999，36(6)：777–809.
[12] CARRANZA-TORRES C. Elasto-plastic solution of tunnel problems using the generalized form of the Hoek-brown failure criterion[J]. International Journal of Rock Mechanics and Mining Sciences，2004，41(Supp.1)：629–639.
[13] SHARAN S K. Elastic-brittle-plastic analysis of circular openings in Hoek-Brown media[J]. International Journal of Rock Mechanics and Mining Sciences，2003，40(6)：817–824.
[14] SHARAN S K. Exact and approximate solutions for displacements around circular openings in elastic-brittle-plastic Hoek-Brown rock[J]. International Journal of Rock Mechanics and Mining Sciences，2005，42(4)：542–549.
[15] SHARAN S K. Analytical solutions for stresses and displacements around a circular opening in a generalized Hoek-Brown rock[J]. International Journal of Rock Mechanics and Mining Sciences，2008，45(1)：78–85.
[16] 陈立伟，彭建兵，范 文，等. 基于统一强度理论的非均匀应力场圆形巷道围岩塑性区分析[J]. 煤炭学报，2007，32(1)：20–23.(CHEN Liwei，PENG Jianbing，FAN Wen，et al. Analysis of surrounding rock mass plastic zone of round tunnel under non-uniform stress field based on the unified strength theory[J]. Journal of China Coal Society，2007，32(1)：20–23.(in Chinese))
[17] 于学馥，郑颖人，刘怀恒，等. 地下工程围岩稳定分析[M]. 北京：煤炭工业出版社，1983：156–171.(YU Xuefu，ZHENG Yingren，LIU Huaiyuan，et al. The stability analysis of the surrounding rock mass in underground engineering[M]. Beijing：China Coal Industry Publishing House，1983：156–171.(in Chinese))
[18] 孙广忠. 岩体力学基础[M]. 北京：科学出版社，1983：158–163.(SUN Guangzhong. Fundamental of rock mechanics[M]. Beijing：Science Press，1983：158–163.(in Chinese))
[19] 孙金山，卢文波. 非轴对称荷载下圆形隧洞围岩弹塑性分析解析解[J]. 岩土力学，2007，28(增1)：327–332.(SUN Jinshan，LU Wenbo. Analytical elastoplastic solutions to supporting rock masses of circular tunnels under asymmetric load[J]. Rock and Soil Mechanics，2007，28(Supp.1)：327–332.(in Chinese))
[20] 魏悦广. 两向不等压作用下圆形巷道弹塑性分析摄动解[J]. 岩土程学报，1990，12(4)：11–20.(WEI Yueguang. Perturbation solution for elasto-plastic analysis of circular tunnel under unequal compression in two directions[J]. Chinese Jounal of Geotechnical Engineering，1990，12(4)：11–20.(in Chinese))
[21] 侯公羽，李晶晶，赵伟伟，等. 两向不等压圆形巷道弹塑性摄动解[J]. 岩石力学与工程学报，2014，33(增2)：3 639–3 647.(HOU Gongyu，LI Jingjing，ZHAO Weiwei，et al. Perturbation solutions for elasto- plastic problems of circular tunnel under unequal compression[J]. Chinese Journal of Rock Mechanics and Engineering，2014，33(Supp.2)：3 639–3 647.(in Chinese))
[22] GALIN L A. Plane elstic-plastic problem： plastic regions around circular holes in plates and beam[J]. Journal of Applied Mathematics and Mechanics，1946，(10)：365–386.
[23] 严克强. 不对称荷载作用下圆洞围岩塑性区的估算方法[J]. 岩土工程学报，1980，2(2)：74–79.(YAN Keqiang. Estimation method for plastic zone of circular tunnel under unsymmetrical load[J]. Chinese Journal of Geotechnical Engineering，1980，2(2)：74–79.(in Chinese))
[24] DETOURNAY E. An approximate statical solution of the elastoplastic interface for the problem of Galin with a cohesive-frictional material[J]. International Journal of Solids and Structures，1986，22(12)：1 435–1 454.
[25] DETOURNAY E，FAIRHURST C. Two-dimensional elastoplastic analysis of a long， cylindrical cavity under non-hydrostatic loading[J]. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts，1987，24(4)：197–211.
[26] TOKAR G. Generalization of Galin?s problem to frictional materials and discontinuous stress fields[J]. International Journal of Solids and Structures，1990，26(2)：129–147.
[27] MUSKHELISHVILI N I. Some basic problems of the mathematical theory of elasticity[M]. Groningen：Noordhoff International Publishing，1977：105–166.
[28] 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，1994：14–16.(CHEN Ziyin. Analytical method of rock mechanics analysis[M]. Beijing：China Coal Industry Publishing House，1994：14–16.(in Chinese))
[29] LU A Z，ZHANG L Q，ZHANG N. Analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay[J]. International Journal of Rock Mechanics and Mining Sciences，2011，48(3)：514–519.
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2018, Vol. 37 
