Analytical solutions for elliptical tunnels in rheological rock considering excavation
WANG Hua-ning1, JIANG Ming-jing2, HE Ping2
1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China; 2. Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China
Abstract:Sequential excavation is often adopted in construction of large tunnels. The analytical solutions for deeply buried elliptical tunnels in rheological rock considering the time-dependent excavation are derived. In order to simulate the design process of excavation, the values of the major and the minor axes are assumed to be arbitrarily time-dependent functions. First, the general expressions for the stresses and the additional displacements due to excavation are derived by using the complex variable method and the corresponding principle of time-dependent boundary problem. In the derivation, the inverse mapping function is introduced to detach the time variable which should participate in the Laplace transformation. Then, the explicit analytical expressions in integral form are obtained by simulating the rock mass in the Boltzmann viscoelastic model. Finally, the parametric analysis of the stresses and displacements is then performed to investigate the influence of the excavation process and the rate for different types of rock. The results show that the stresses and displacements are greatly affected by the excavation process. The proposed solutions can be used in the mechanical analysis of excavation of elliptical tunnels, and are valuable in the preliminary design of underground tunnels.
王华宁, 蒋明镜, 何平. 流变岩体中椭圆洞室断面开挖过程的力学分析[J]. 岩土工程学报, 2013, 35(11): 1979-1987.
WANG Hua-ning, JIANG Ming-jing, HE Ping. Analytical solutions for elliptical tunnels in rheological rock considering excavation. Chinese J. Geot. Eng., 2013, 35(11): 1979-1987.
[1] SULEM J, PANET M, GUENOT A. Closure analysis in deep tunnels[J]. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstract, 1987, 24(3): 145-154. [2] HAGE CHEHADE F, SHAHROUR I. Numerical analysis of the interaction between twin-tunnels: Influence of the relative position and construction procedure[J]. Tunnelling and Underground Space Technology, 2008, 23(2): 210-214. [3] PRAZERES P G C, THOENI K, BEER G. Nonlinear analysis of NATM tunnel construction with the boundary element method[J]. Computers and geotechnics, 2012, 40: 160-173. [4] CARRANZA-TORES C, FAIRHURST C. The elastoplastic responds of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion[J]. International Journal of Rock Mechanics and Mining Science and Geomechanics Abstract, 1999, 36(6): 777-809. [5] 曹志远. 土木工程分析的施工力学与时变力学基础[J]. 土木工程学报, 2000, 33(6): 105-108. (CAO Zhi-yuan. Construction mechanics and time-varying mechanics in civil engineering[J]. China Civil Engineering Journal, 2000, 33(6): 105-108. (in Chinese)) [6] SULEM J, PANET M, GUENOT A. An analytical solution for time-dependent displacements in circular tunnel[J]. International Journal of Rock Mechanics and Mining Sience & Geomechanics Abstracts, 1987, 24(3): 155-164. [7] GOODMAN R E. Introduction to rock mechanics[M]. New York: Wiley, 1989. [8] NOMIKOS P, RAHMANNEJAD R, SOFIANOS A. Supported axisymmetric tunnels within linear viscoelastic Burgers rocks[J]. Rock Mechanics and Rock Engineering, 2011, 44: 553-564. [9] DUENSER C, BEER G. Simulation of sequential excavation with the boundary element method[J]. Computers and Geotechnics, 2012, 44: 157-166. [10] 王华宁, 曹志远, 李 悦. 考虑断面及纵向施工效应时支护圆形洞室黏弹解析分析[J]. 岩土工程学报, 2011, 33(8): 1159-1166. (WANG Hua-ning, CAO Zhi-yuan, LI Yue. Analytical research of lined circular tunnel in viscoelastic rock mass considering the effect of cross-section and longitudinal construction[J]. Chinese Journal of Geotechnical Engineering, 2011, 33(2): 1159-1166. (in Chinese)) [11] WANG H N, NIE G H. Analytical expressions for stress and displacement fields in viscoelastic axisymmetric plane problem involving time-dependent boundary regions[J]. Acta Mechanica, 2010, 210(3/4): 315-330. [12] WANG H N, LI Y, NI Q, et al. Analytical solutions for the construction of deeply buried circular tunnels with two liners in rheological rock[J]. Rock Mechanics and Rock Engineering, 2013, 46: 1481-1498. [13] LI S C, WANG M B. 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: 609-617. [14] LV 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. [15] LEE E H. Stress analysis in viscoelastic bodies[J]. Quarterly of Applied Mathematics, 1955, 13: 183-190. [16] GRAHAM G A C, SABIN G C W. The correspondence principle of linear viscoelasticity for problems that involve time-dependent regions[J]. International Journal of Engineering Science, 1973, 11(1): 123-140. [17] MUSKHELISHVILI N I. Some basic problems of the mathematical theory of elasticity[M]. Groningen: Noordhoff, 1963. [18] 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. Analytical method in Geomechanics[M]. Beijing: Coal Industry Press, 1994. (in Chinese)) [19] ZHANG Lu-qing, YANG Zhi-fa, LU Ai-zhong. Analytics study on the problem of two holes having arbitrary shapes and arrangements in plane elstostatics[J]. Science in China(Series D), 2000, 44(2):146-158. [20] 何 平, 王华宁. 深埋圆形压力隧洞施工过程的黏弹性解析解[J]. 力学季刊, 2012, 33(1): 45-52. (HE Ping, WANG Hua-ning. Analytical viscoelastic solution of deeply buried circular tunnel considering excavation process[J]. Chinese Quarterly of Mechanics, 2012, 33(1): 45-52. (in Chinese))