Analytical solution of deep buried rectangular tunnel under the action of far field shear wave
ZHANG Zhiguo1，2，3，YANG Feng1，ZHAO Qihua3
(1. School of Environment and Architecture，University of Shanghai for Science and Technology，Shanghai 200093，China；
2. Key Laboratory of Geohazard Prevention of Hilly Mountains，Ministry of Land and Resources，Fuzhou，Fujian 350002，China；
3. State Key Laboratory of Geohazard Prevention and Geoenvironment Protection，Chengdu University of Technology，
Chengdu，Sichuan 610059，China)
Abstract：The existed solutions on the interaction between the seismic responses of soil and structure have focused on the tunnels of circular cross section. The dynamic analysis for the rectangular tunnel tends to increase in complexity of calculation. An analytical solution of the seismic response of deep buried rectangular tunnel under the action of far field shear waves is derived based on the pseudo static method by using the complex analysis and conformal transformation. The shear and normal stresses in the interaction between soil and tunnel are considered in the solution of the complex function. The accuracy of calculating the conformal transformation function for rectangular tunnel is improved by introducing a correction factor. The accuracy of the analytical method is verified with the numerical model. The parameters influencing the normalized displacement ratio and the variation of the soil stress field for the rectangular tunnel are analyzed as well. The study shows that the accuracy of the analytical solution is improved by introducing the correcting coefficient，which is more close to the numerical solution. The normalized displacement ratio of the rectangular tunnel decreases with the increase of the relative stiffness ratio of tunnel to soil. The normalized displacement ratio increases as the increasing of the ratio of the cross section size. When the cross section size ratio of the rectangular tunnel and the relative stiffness ratio are unchanged，the normalized displacement ratio of the rectangular tunnel increases with the decrease of the Poisson's ratio of the soil. When the cross section size ratio is 1 (Square tunnel)，the change of Poisson?s ratio of soil has little influence on the normalized displacement ratio of the tunnel. The soil stress field around the rectangular tunnel increases with the increase of the Poisson?s ratio of soil. The soil stress field decreases with the increase of the elastic modulus of soil. The soil stress field decreases with the increase of the shear modulus of the liner.
张治国1，2，3，杨 峰1，赵其华3 . 远场剪切波作用下深埋矩形隧洞地震响应解析解[J]. 岩石力学与工程学报, 2017, 36(8): 1951-1965.
ZHANG Zhiguo1，2，3，YANG Feng1，ZHAO Qihua3 . Analytical solution of deep buried rectangular tunnel under the action of far field shear wave. , 2017, 36(8): 1951-1965.
[1] AN X，SHAWKY A，MAEKAWA K. The collapse mechanism of a subway station during the Great Hanshin earthquake[J]. Cement and Concrete Composites，1997，19(3)：241–257.
[2] 仇文革，胡 辉，赵 斌. 长大隧道穿越断层带减震结构数值解析研究[J]. 岩土工程学报，2013，35(2)：252–257.(QIU Wenge，HU Hui，ZHAO Bin. Numerical analysis of shock absorption structure in long tunnel through fault zone[J]. Chinese Journal of Geotechnical Engineering，2013，35(2)：252–257.(in Chinese))
[3] KOURETZIS G P，SLOAN S W，CARTER J P. Effect of interface friction on tunnel liner internal forces due to seismic S- and P-wave propagation[J]. Soil Dynamics and Earthquake Engineering，2013，46(4)：41–51.
[4] 晏启祥，唐茂皓，何 川，等. 基于薄壁圆柱壳理论的盾构隧道抗震拟静力分析法[J]. 岩土工程学报，2014，36(7)：1 371– 1 376.(YAN Qixiang，TANG Maohao，HE Chuan，et al. Seismic quasi-static analysis method of shield tunnel based on the theory of thin shell cylindrical shell[J]. Chinese Journal of Geotechnical Engineering，2014，36(7)：1 371–1 376.(in Chinese))
[5] FANG X Q，JIN H X，WANG B L. Dynamic interaction of two circular lined tunnels with imperfect interfaces under cylindrical P-waves[J]. International Journal of Rock Mechanics and Mining Sciences，2015，79：172–182.
[6] 刘鸿哲，黄茂松. 不同埋深矩形隧道地震响应的离心振动台试验[J]. 岩石力学与工程学报，2013，32(增2)：3 404–3 412.(LIU Hongzhe，HUANG Maosong. Centrifuge shaking table test of seismic response of rectangular tunnel with different buried depth[J]. Chinese Journal of Rock Mechanics and Engineering，2013，32(Supp.2)： 3 404–3 412.(in Chinese))
[7] SAVIN G N. Stress concentration around holes[M]. London，UK：Pergamon Press，1961：13–87.
[8] PENZIEN J. Seismically induced raking of tunnel linings[J]. Earthquake Engineering and Structure Dynamias，2000，29(5)：683–691.
[9] LEI G H，CHARLES W W，RIGBY D B. Stress and displacement around an elastic artificial rectangular hole[J]. Journal of Engineering Mechanics，2001，127(7)：880–890.
[10] HUO H，BOBET A，FERNANDEZ G，et al. Analytical solution for deep rectangular structures subjected to far field shear stresses[J]. Tunnelling and Underground Space Technology，2006，21(6)：614–624.
[11] 吕爱钟，张路青. 地下隧道力学分析的复变函数方法[M]. 北京：科学出版社，2007：50–72.(LU Aizhong，ZHANG Luqing. Complex variable function method for mechanical analysis of underground tunnel[M]. Beijing：Science Press，2007：50–72.(in Chinese))
[12] BOBET A. Drained and undrained response of deep tunnels subjected to far-field shear loading[J]. Tunnelling and Underground Space Technology，2010，25(1)：21–31.
[13] 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 Sciences，2011，53(4)：254–261.
[14] 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(增2)：3 905–3 913.(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(Supp.2)：3 905–3 913.(in Chinese))
[15] ZHU J H，JI H G. The comparison of surrounding rock stress analytical solutions for three tunnel sections[C]// Proceedings of the 2nd ISRM International Young Scholars Symposium on Rock Mechanics. Leiden：CRC Press，2011：741–744.
[16] 章建一. 矩形截面地下隧道地震作用下的拟静力分析[硕士学位论文][D]. 哈尔滨：哈尔滨工业大学，2012.(ZHANG Jianyi. Pseudo-static analysis on rectangular underground tunnel under earthquake loading[M. S. Thesis][D]. Harbin：Harbin Institute of Technology，2012.(in Chinese))
[17] DEBIASI E，GAJO A，ZONTA D. On the seismic response of shallow-buried rectangular structures[J]. Tunnelling and Underground Space Technology，2013，38(5)：99–113.
[18] 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]. 岩土力学，2014，35(1)：175–183.(ZHU Jianghong，YANG Jianhui，SHI Gaoping，et al. Calculating method for conformal mapping from exterior of unit circle to exterior of cavern with arbitrary excavation cross-section[J]. Rock and Soil Mechanics，2014，35(1)：175–183.(in Chinese))
[19] 施高萍，祝江鸿，李保海，等. 矩形巷道孔边应力的弹性分析[J]. 岩土力学，2014，35(9)：2 587–2 593.(SHI Gaoping，ZHU Jianghong，LI Baohai，et al. Elastic analysis of hole-edge stress of rectangular roadway[J]. Rock and Soil Mechanics，2014，35(9)：2 587–2 593.(in Chinese))
[20] 刘晶波，王文晖，赵冬冬，等. 地下结构抗震分析的整体式反应位移法[J]. 岩石力学与工程学报，2013，32(8)：1 618–1 624.(LIU Jingbo，WANG Wenhui，ZHAO Dongdong，et al. Integral response deformation method for seismic analysis of underground structure[J]. Chinese Journal of Rock Mechanics and Engineering，2013，32(8)：1 618–1 624.(in Chinese))
[21] ZHAO G，YANG S. Analytical solutions for rock stress around square tunnels using complex variable theory[J]. International Journal of Rock Mechanics and Mining Sciences，2015，80(3)：302–307.
[22] MUSKHELISHVILI N I. Some basic problems of the mathematical theory of elasticity[M]. Groningen，The Netherlands：Noordhoof Ltd.，1954：43–91.
[23] CHURCHILL R V. Complex variables and applications[M]. 2th ed，New York：McGraw-Hill Book Company，1960：25–69.
2017, Vol. 36 
