Optimization of Conformal Mapping Functions used in Developing Closed-Form Solutions for Underground Structures with Conventional cross Sections

Document Type: Research Paper

Authors

1 School of Mining Engineering, College of Engineering, University of Tehran, Tehran, Iran

2 School of Engineering Science, College of Science, University of Tehran, Tehran, Iran

Abstract

Elastic solutions applicable to single underground openings usually suffer from geometry related simplification. Most tunnel shapes possess two axes of symmetry while a wide range of geometries used in tunneling practice involve only one symmetry axis. D-shape or horse-shoe shape tunnels and others with arched roof and floor are examples of the later category (one symmetry axis). In the present paper, with the use of conformal mapping, two methods were developed to determine the appropriate mapping functions on which an analytical elastic solution for a tunnel with one vertical axis of symmetry is based. These conformal mapping functions turn complicated geometries into a unit circle for the sake of simplification. These two approaches were introduced into a computer program used for an arbitrary tunnel cross section. Results showed that the second approach has more accuracy and is able to produce any shape, since it uses a nonlinear structure in its constitutive equations. Besides, the values for different coefficients have been presented for a variety of tunnel geometry curvature, as well as acceptable variation for the coefficients to represent tunnels with conventional shapes.

Keywords


[1] J. T. Chen, M. H. Tsai, and C. S. Liu,
"Conformal mapping and bipolar coordinate for
eccentric Laplace problems," Computer
Applications in Engineering Education, vol. 17,
pp. 314-322, 2009.
[2] S. G. Lekhnitskii, "Elasticity theory of
anisotropic bodies," ed: Nauka, Moscow, 1977.
[3] G. N. Savin, Stress concentration around holes
vol. 1: Pergamon Press, 1961.
[4] A. Verruijt, "A complex variable solution for a
deforming circular tunnel in an elastic halfplane,"
International Journal for Numerical
and Analytical Methods in Geomechanics, vol.
21, pp. 77-89, 1997.
[5] A. England, "On stress singularities in linear
elasticity," International Journal of
Engineering Science, vol. 9, pp. 571-585, 1971.
[6] H. Huo, A. Bobet, G. Fernández, and J.
Ramírez, "Analytical solution for deep 

rectangular structures subjected to far-field shear stresses," Tunnelling and underground space technology, vol. 21, pp. 613-625, 2006.
[7] H. Gerçek, "An elastic solution for stresses around tunnels with conventional shapes," International Journal of Rock Mechanics and Mining Sciences, vol. 34, pp. 96. e1-96. e14, 1997.
[8] N. I. Muskhelishvili, Some basic problems of the mathematical theory of elasticity: Springer, 1977.
[9] G. Exadaktylos and M. Stavropoulou, "A closed-form elastic solution for stresses and displacements around tunnels," International Journal of Rock Mechanics and Mining Sciences, vol. 39, pp. 905-916, 2002.
[10] V. Ukadgaonker and D. Rao, "Stress distribution around triangular holes in anisotropic plates," Composite structures, vol. 45, pp. 171-183, 1999.