Stresses in Anisotropic Rock Mass with Irregular Topography
Publication: Journal of Engineering Mechanics
Volume 120, Issue 1
Abstract
This paper presents a new analytical method for determining the state of stress in a homogeneous, general anisotropic, and elastic half‐space limited by an irregular and smooth outer boundary. The half‐space represents a rock mass with an irregular and continuous topography. The rock mass is subject to gravity and surface tractions. The stresses are determined assuming a condition of generalized plane strain, and are expressed in terms of three analytical functions following Lekhnitskii's complex function method. These analytical functions are determined using a numerical conformal mapping method and an integral equation method. As an illustrative example, it is shown how the proposed method can be used to determine the state of stress in long isolated and symmetric ridges and valleys in orthotropic or transversely isotropic rock masses. It is found that the magnitude of the stresses is of the order of the characteristic stress , where is the rock density, is the gravitational acceleration, and is the height of the ridge or depth of the valley.
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References
1.
Amadei, B., and Pan, E. (1992). “Gravitational stresses in anisotropic rock masses with inclined strata.” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 29(3), 225–236.
2.
Amadei, B., Savage, W. Z., and Swolfs, H. S. (1987). “Gravitational stresses in anisotropic rock masses.” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 24(1), 5–14.
3.
Atkinson, K. E. (1976). A survey of numerical methods for the solution of Fredholm integral equations of the second kind. Soc. for Industrial and Appl. Math., Philadelphia, Pa.
4.
Davis, P. J., and Rabinowitz, P. (1984). Methods of numerical integration. 2nd Ed., Academic Press, New York, N.Y.
5.
Kerzman, N., and Trummer, M. R. (1986). “Numerical conformal mapping via the Szegö kernel.” J. Comp. Appl. Math., 14, 111–123.
6.
Lekhnitskii, S. G. (1963). Theory of elasticity of an anisotropic elastic body. Holden‐Day, San Francisco, Calif.
7.
Liao, J. J., Savage, W. Z., and Amadei, B. (1992). “Gravitational stresses in anisotropic ridges and valleys with small slopes.” J. Geophysical Res., 97(B3), 3325–3336.
8.
Liu, L., and Zoback, M. D. (1992). The effect of topography on the state of stress in the crust: application to the site of the Cajon Pass Scientific Drilling Project, J. Geophysical Res., 97(B4), 5095–5108.
9.
McTigue, D. F., and Mei, C. C. (1981). “Gravity‐induced stresses near topography of small slope.” J. Geophys. Res., 86(B10), 9268–9278.
10.
McTigue, D. F., and Mei, C. C. (1987). “Gravity‐induced stresses near axisymmetric topography of small slope.” Int. J. Numer. Anal. Methods Geomech., 11(3), 257–268.
11.
Muskhelishvili, N. I. (1953). Some basic problems of the mathematical theory of elasticity. Noordhoof, Groningen, The Netherlands.
12.
Muskhelishvili, N. I. (1972). Singular integral equations. Noordhoff, Groningen, The Netherlands.
13.
Nehari, Z. (1952). Conformal mapping. Dover Publications, Inc., New York, N.Y.
14.
O'Donnell, S. T., and Rokhlin, V. (1989). “A fast algorithm for the numerical evaluation of conformal mappings.” SIAM J. Sci. Stat. Comput., 10(3), 475–487.
15.
Papamichael, N., and Kokkinos, C. A. (1981). “Two numerical methods for the conformal mapping of simply‐connected domains.” Comp. Methods Appl. Mech., 28, 285–307.
16.
Papamichael, N., Warby, M. K., and Hough, D. H. (1986). “The treatment of corner and pole‐type singularities in numerical conformal mapping techniques.” J. Comp. Appl. Math., 14, 163–191.
17.
Perloff, W. H., Baladi, G. Y., and Harr, M. E. (1967). “Stress distribution within and under long elastic embankments.” Hwy. Res. Rec., 181, 12–40.
18.
Sarkar, T. K., Yang, X., and Arvas, E. (1988). “A limited survey of various conjugate gradient methods for solving complex matrix equations arising in electromagnetic wave interactions.” Wave Motion, 10, 527–546.
19.
Savage, W. Z., and Swolfs, H. S. (1986). “Tectonic and gravitational stress in long symmetric ridges and valleys.” J. Geophys. Res., 91(B3), 3677–3685.
20.
Savage, W. Z., Swolfs, H. S., and Powers, P. S. (1985). “Gravitational stresses in long symmetric ridges and valleys.” Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 22(5), 291–302.
21.
Ter‐Martirosyan, Z. G., Akhpatelov, D. M., and Manvelyan, R. G. (1974). “The stressed state of rock masses in a field of body forces.” Adv. Rock Mech., 2(Part A), 569–574.
22.
Terzaghi, K., and Richart, F. E. (1952). “Stresses in rock about cavities.” Géotechnique, 3, 57–90.
23.
Trummer, M. R. (1986). “An efficient implementation of a conformal mapping method based on the Szegö kernel.” SIAM J. Numer. Anal., 23(4), 853–872.
Information & Authors
Information
Published In
Journal of Engineering Mechanics
Volume 120 • Issue 1 • January 1994
Pages: 97 - 119
Copyright
Copyright © 1994 American Society of Civil Engineers.
History
Received: Aug 24, 1992
Published online: Jan 1, 1994
Published in print: Jan 1994
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