On Analytic Solutions of Elastic Net Displacements around a Circular Tunnel

Hazrati Aghchai, M.; Moarefvand, P.; Salari Rad, H.

doi:10.22044/jme.2020.9252.1821

Document Type : Original Research Paper

Authors

Department of Mining and Metallurgical Engineering, Amirkabir University of Technology, Tehran, Iran

https://doi.org/10.22044/jme.2020.9252.1821

Abstract

Displacements around a tunnel, occurring as a result of excavation, consist of the elastic and plastic parts. In this paper, we discuss the elastic part of displacements as a result of excavation, called net displacement. In general, the previous analytical solutions presented for determining the displacements around a circular tunnel in an elastic medium do not give the net displacements directly. The well-known Kirsch solution is the most widely used method for determining the induced stresses and net displacements around a circular opening in a biaxially-loaded plate of homogeneous, isotropic, continuous, linearly elastic material. However, the complete solution for obtaining the net displacements has not been presented or highlighted in the available literature. Using the linear elasticity, this paper reviews and presents three different analytical methods for determining the net displacements directly as well as induced stresses around a circular tunnel. The three solution methods are the Lame' method, airy stress function method, and complex variable method. The tunnel is assumed to be situated in an elastic, continuum, and isotropic medium in the plane strain condition. The solutions are presented for both the hydrostatic and non-hydrostatic in situ stresses in the 2D biaxial loading condition along with an internal pressure. Loading and unloading in tunneling occurring as a result of excavation and stress differences between the induced and initial ones are considered to evaluate the net displacements directly. Finally, some examples are given to demonstrate the complete solution and show the difference between the net elastic displacements as a result of excavation and total elastic displacements that are not real.

Keywords

Main Subjects

Rock Mechanics

References

[1]. Kirsch, C. (1898). Die theorie der elastizitat und die bedurfnisse der festigkeitslehre. Zeitschrift des Vereines Dtsch. Ingenieure; 42:797–807.

[2]. Obert, L. and Duvall, W.I. (1967). Rock mechanics and the design of structures in rock.

[3]. Bray, J. (1978). Analytical and computational methods in engineering rock mechanics. Taylor & Francis.

[4]. Hoek, E. and Brown, E.T. (1980). Underground excavations in rock. CRC Press.

[5]. Goodman, R.E. (1989). Introduction to rock mechanics. Wiley New York.

[6]. Hudson, J.A. and Harrison, J.P. (2000). Engineering rock mechanics: an introduction to the principles. Elsevier.

[7]. Brady, B.H.G. and Brown, E.T. (2013). Rock mechanics: for underground mining. Springer Science & Business Media.

[8]. Savin, G.N. (1961). Stress concentration around holes.

[9]. Poulos, H.G. and Davis, E.H. (1974). Elastic solutions for soil and rock mechanics. John Wiley.

[10]. Popov, E.P. and Balan, T.A. (1990). Engineering mechanics of solids. Prentice Hall Englewood Cliffs, NJ.

[11]. Ameen, M. (2005). Computational elasticity. Harrow Alpha Sci. Int. Ltd.

[12]. Jaeger, J.C., Cook, N.G.W. and Zimmerman, R. (2009). Fundamentals of rock mechanics. John Wiley & Sons.

[13]. Sadd, M.H. (2009). Elasticity: theory, applications, and numerics. Academic Press.

[14]. Nam, S.W. and Bobet, A. (2007). Radial deformations induced by groundwater flow on deep circular tunnels. Rock Mech. Rock Eng. 40 (1):23.

[15]. Timoshenko, S.P. and Goodier, J.N. (1970). Theory of Elasticity, 3rd Edn McGraw-Hill. New York.

[16]. Bobet, A. (2009). Elastic solution for deep tunnels. Application to excavation damage zone and rockbolt support. Rock Mech. Rock Eng. 42 (2):147–74.

[17]. Oreste, P.P. and Peila, D. (1996). Radial passive rockbolting in tunnelling design with a new convergence-confinement model. Int. J. rock Mech. Min. Sci. Geomech. Abstr. 33 (5):443–54.

[18]. Carranza-Torres, C. and Fairhurst, C. (1999). The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci. 36(6):777–809.

[19]. Carranza-Torres, C. and Fairhurst, C. (2000). Application of the convergence-confinement method of tunnel design to rock masses that satisfy the Hoek-Brown failure criterion. Tunn. Undergr. Sp. Technol. 15 (2):187–213.

[20]. Sharan, S.K. (2003). Elastic–brittle–plastic analysis of circular openings in Hoek–Brown media. Int. J. Rock Mech. Min. Sci. 40 (6):817–24.

[21]. Sharan, S.K. (2005). Exact and approximate solutions for displacements around circular openings in elastic–brittle–plastic Hoek–Brown rock. Int. J. Rock Mech. Min. Sci. 42(4):542–9.

[22]. Sharan, S.K. (2008). Analytical solutions for stresses and displacements around a circular opening in a generalized Hoek–Brown rock. Int. J. Rock Mech. Min. Sci. 1(45):78–85.

[23]. Bobet, A. (2006). A simple method for analysis of point anchored rockbolts in circular tunnels in elastic ground. Rock Mech. Rock Eng. 39 (4):315.

[24]. Oreste, P. (2008). Distinct analysis of fully grouted bolts around a circular tunnel considering the congruence of displacements between the bar and the rock. Int. J. Rock Mech. Min. Sci. 45 (7):1052–67.

[25]. Carranza-Torres, C. (2009). Analytical and numerical study of the mechanics of rockbolt reinforcement around tunnels in rock masses. Rock Mech. Rock Eng. 42 (2):175–228.

[26]. Bobet, A. (2011). Einstein HH. Tunnel reinforcement with rockbolts. Tunn. Undergr. Sp. Technol. 26 (1):100–23.

[27]. Tan, C.H. (2016). Difference solution of passive bolts reinforcement around a circular opening in elastoplastic rock mass. Int. J. Rock Mech. Min. Sci. 81:28–38.

[28]. Tan, C.H. (2016). Passive bolts reinforcement around a circular opening in strain-softening elastoplastic rock mass. Int. J. Rock Mech. Min. Sci. (88):221–34.

[29]. Lazemi, H.A., Marji, M.F., Bafghi, A.R.Y. and Goshtasbi, K. (2013). Rock Failure Analysis of the Broken Zone Around a Circular Opening/Analiza pęknięcia skały w strefie naruszonej wokół otworu kolistego. Arch. Min. Sci. 58 (1):165–88.

[30]. Abdollahi, M.S., Najafi, M., Bafghi, A.Y. and Marji, M.F. (2019). A 3D numerical model to determine suitable reinforcement strategies for passing TBM through a fault zone, a case study: Safaroud water transmission tunnel, Iran. Tunn. Undergr. Sp. Technol. 88:186–99.

[31]. Einstein, H.H. and Schwartz, C.W. (1979). Simplified analysis for tunnel supports. J. Geotech. Geoenvironmental Eng. 105 (ASCE 14541).

[32]. Kreyszig, E. (2011). Advanced engineering mathematics. United States of America: Jhon Wiley & Sons. INC.

[33]. Simmons, G.F. (2016). Differential equations with applications and historical notes. CRC Press.

[34]. Brown, E.T., Bray, J.W., Ladanyi, B. and Hoek, E. (1983). Ground response curves for rock tunnels. J. Geotech. Eng. 109(1):15–39.

[35]. Airy, G.B.I.V. (1863). On the strains in the Interior of beams. Philos. Trans. R. Soc. London. 153:49–79.

[36]. Little, R.W. (1973). Elasticity Prentice-Hall. Englewood Cliffs, NJ. 88–94.

[37]. Michell, J.H. (1899). On the direct determination of stress in an elastic solid, with application to the theory of plates. Proc. London Math. Soc. 1 (1):100–24.

[38]. Kolosov, G.V. (1909). On the Application of the Theory of Functions of a Complex Variable to a Plane problem in the Mathematical Theory of Elasticity. Printing-house K Mattisena, Juriev.

[39]. Muskhelishvili, N.I. (1966). Some basic problems of the mathematical theory of elasticity. Nauka, Moscow.

[40]. Rocscience. Phase 2, www.rocscience.com. (2004).

Journal of Mining and Environment

On Analytic Solutions of Elastic Net Displacements around a Circular Tunnel

References

References

Volume 11, Issue 2
April 2020
Pages 419-432

On Analytic Solutions of Elastic Net Displacements around a Circular Tunnel

References

References

Volume 11, Issue 2April 2020Pages 419-432

Volume 11, Issue 2
April 2020
Pages 419-432