In order to explore the mechanical response of a deep hydraulic tunnel in orthotropic rock mass，the analytical solutions of a non-circular tunnel under the action of the in-situ stresses and the internal water pressure，are derived. In the process，three polar coordinates are introduced and the conformal transformation method of complex function is adopted，mapping the outer region of the hole in the physical plane to the outer region of the unit circle in the image plane. Then，the stress boundary conditions on the excavation boundary are established. Finally，the stress field is obtained using the power-series method. Taking the inverted U-shaped tunnel as an example，the analytical solutions are verified by the numerical results obtained by ANSYS. The stress distributions along the boundary of a horse hoof-shaped tunnel and a circular tunnel are also analyzed. The results show that，unlike the isotropic rock mass，the stress field for an orthotropic rock mass shows obvious asymmetry. The tangential stresses along the hole boundary increase with the increase of the lateral pressure coefficients except the two corners of the horse hoof-shaped tunnel. Moreover，the internal water pressure can release the tangential stress concentration along the hole boundary，which is more obvious on the regions where the curvature is greater. But great internal pressure may also lead to tensile stress which will affect the stability of the tunnel.
张晓莉，吕爱钟，王少杰 . 正交各向异性非圆形水工隧洞的应力解析解[J]. 岩石力学与工程学报, 2017, 36(S2): 3808-3815.
ZHANG Xiaoli，LU Aizhong，WANG Shaojie. Analytic stress solutions for an orthotropic rock mass with a non-circular hydraulic tunnel. , 2017, 36(S2): 3808-3815.
EXADAKTYLOSG E，STAVROPOULOU M C. A closed-form elastic solution for stresses and displacements around tunnels[J]. International Journal of Rock Mechanics andMining Sciences，2002，39(7)：905-916.
[2]
WANG M B，LI S C. A complex variable solution for stress and displacement field around a lined circular tunnel at great depth[J]. International Journal for Numerical andAnalytical Methods in Geomechanics，2009，33(7)：939-951.
[3]
LU A Z，ZHANG N，Qin Y. Analytical solutions for the stress of a lined non-circular tunnel under full-slip contact conditions[J]. International Journal of Rock Mechanics andMining Sciences，2015，79(1)：183-192.
[4]
AMADEI B. Rock anisotropy and the theory of stress measurements[M]. Berlins：Springer，1983：267-275.
[5]
SAVIN G N. Stress concentration around holes[M]. New York：Pergamon Press，1961：150-199.
[6]
LEKHNITSKII S G，TSAI，STEPHEN W，et al. Anisotropic plates[M]. New York：Gordon and Breach，1968：157-190.
[7]
LEKHNITSKII S G，FERN P，BRANDSTATTER J J，et al. Theory of elasticity of an anisotropic elastic body[M]. Moscow：Mir Publishers，1981：215-227.
[8]
李军，柳贡慧，陈勉. 正交各向异性地层井壁围岩应力新模型[J]. 岩石力学与工程学报，2011，30(12)：2481-2485.(LI Jun，LIU Gonghui，CHEN Mian. New model for stress of borehole surrounding rock in orthotropic formation[J]. Chinese Journal of Rock Mechanics and Engineering，2011，30(12)：2481-2485.(in Chinese))
[9]
DAOUST J，HOA S V. An analytical solution for anisotropic plates containing triangular holes[J]. Composite Structures，1991，19(2)：107-130.
[10]
RAO D K N，BABU M R，REDDY K R N，et al. Stress around square and rectangular cutouts in symmetric laminates[J]. Composite Structures，2010，92(12)：2845-2859.
[11]
REZAEEPAZHAND J，JAFARI M. Stress analysis of composite plates with non-circular cutout[J]. Key Engineering Materials，2008，385-387：365-368.
[12]
SHARMA D S. Stresses around polygonal hole in an infinite laminated composite plate[J]. European Journal of Mechanics：A/Solids，2015，54：44-52.
[13]
李成，郑艳萍. 不同孔口形状对含孔复合材料板孔边应力状态影响的研究[J]. 工程力学，2007，24(10)：19-24.(LI Cheng，ZHENG Yanping. Influence of different orifice figure on hole-edge stress of composite material[J]. Engineering Mechanics，2007，24(10)：19-24.(in Chinese))
[14]
UKADGAONKER V G，RAO D K N. A general solution for stresses around holes in symmetric laminates under in-plane loading[J]. Composite Structures，2000，49(3)：339-354.
[15]
UKADGAONKER V G，KAKHANDKI V. Stress analysis for an orthotropic plate with an irregular shaped hole for different in-plane loading conditions—Part 1[J]. Composite Structures，2005，70(3)：255-274.
[16]
MANH H T，SULEM J，SUBRIN D. A closed-form solution for tunnels with arbitrary cross section excavated in elastic anisotropic ground[J]. Rock Mechanics and Rock Engineering，2015，48(1)：277-288.
[17]
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 andMining Sciences，2011，48(8)：1359-1363.
[18]
LU A Z，ZHANG N，ZHANG X L，et al. Analytic method of stress analysis for an orthotropic rock mass with an arbitrary-shaped tunnel[J].International Journal of Geomechanics，2015，15(4)：04014068
[19]
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，1994：228-229.(CHEN Ziyin. Analytical method of rock mechanics analysis[M]. Beijing：China Coal Industry Publishing House，1994：228-229.(in Chinese))
[20]
中华人民共和国行业标准编写组. DL/T 5195－2004 水工隧洞设计规范[S]. 北京：中国电力出版社，2004.(The Professional Standards Compilation Group of People¢s Republic of China. DL/T 5195－2004 Specification for design of hydraulic tunnel[S]. Beijing：China Electric Power Press，2004.(in Chinese))
[21]
吕爱钟，覃媛，陈虹宇. 马蹄形隧洞考虑支护滞后过程的应力分析[J]. 岩土力学，2014，35(增1)：42-48.(LU Aizhong，QIN Yin，CHEN Hongyu. Stress analysis of deep horseshoe tunnel considering support delay[J]. Rock and Soil Mechanics，2014，35(Supp.1)：42-48.(in Chinese))
2017, Vol. 36 
