Displacements and Stresses Due to a Uniform Vertical Circular Load in an Inhomogeneous Cross-Anisotropic Half-Space
Publication: International Journal of Geomechanics
Volume 6, Issue 1
Abstract
In this work, we present the solutions for displacements and stresses along the centerline of a uniform vertical circular load in an inhomogeneous cross-anisotropic half-space with its Young’s and shear moduli varying exponentially with depth. The planes of cross anisotropy are assumed to be parallel to the horizontal surface. The presented solutions can be directly integrated from the point load solution in a cylindrical coordinate system, which were derived by the writers. However, the resulting integrals of the circular solution for displacements and stresses cannot be given in closed form; hence, numerical integrations are required. For a homogeneous cross-anisotropic half-space, the numerical results agree very well with the exact solutions of Hanson and Puja, published in 1996. Two examples are given to elucidate the effect of inhomogeneity, and the type and degree of soil anisotropy on the vertical displacement and vertical normal stress in the inhomogeneous isotropic/cross-anisotropic soils subjected to a uniform vertical circular load acting on the surface. The proposed solutions can more realistically simulate the actual stratum of loading problem in many areas of engineering practice.
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Acknowledgments
The first writer wishes to thank the NSC of R.O.C. for financially supporting this research under Contract No. UNSPECIFIED93-2625-Z-239-003. All the writers would like to thank the reviewers and the editor for their constructive comments/suggestions.
References
Alexander, L. G. (1977). “Discussions to some results concerning displacement and stresses in a non-homogeneous elastic half-space.” Geotechnique, 27(2), 253–254.
Anon, B. T. W. (1960). “De spanningsverdeling in enn homogeen anisotroop elastisch half medium.” L. G. M. Mededelingen, 5(2), 33–51 (in German).
Awojobi, A. O. (1975). “The settlement of a foundation on Gibson soil of the second kind.” Geotechnique, 25(2), 221–228.
Barden, L. (1963). “Stresses and displacements in a cross-anisotropic soil.” Geotechnique, 13(3), 198–210.
Booker, J. R., Balaam, N. P., and Davis, E. H. (1985). “The behavior of an elastic non-homogeneous half-space. Part II—Circular and strip footings.” Int. J. Numer. Analyt. Meth. Geomech., 9(4), 369–381.
Brown, P. T., and Gibson, R. E. (1972). “Surface settlement of a deep elastic stratum whose modulus increases linearly with depth.” Can. Geotech. J., 9(4), 467–476.
Carrier, W. D., III, and Christian, J. T. (1973a). “Analysis of an inhomogeneous elastic half-space.” J. Soil Mech. Found. Div., 99(3), 301–306.
Carrier, W. D., III, and Christian, J. T. (1973b). “Rigid circular plate resting on a non-homogeneous elastic half-space.” Geotechnique, 23(1), 67–84.
Chowdhury, K. L. (1987). “On the axisymmetric Mindlin’s problem for a semi-space of granular material.” Acta Mech., 66(1–4), 145–160.
Chuaprasert, M. F., and Kassir, M. K. (1974). “Displacements and stresses in nonhomogeneous solid.” J. Eng. Mech. Div., Am. Soc. Civ. Eng., 100(5), 861–872.
Davis, P. J., and Rabinowitz, P. (1984). Methods of numerical integration, 2nd Ed., Academic, New York.
Doherty, J. P., and Deeks, A. J. (2003a). “Elastic response of circular footings embedded in a non-homogeneous half-space.” Geotechnique, 53(8), 703–714.
Doherty, J. P., and Deeks, A. J. (2003b). “Scaled boundary finite-element analysis of a non-homogeneous axisymmetric domain subjected to general loading.” Int. J. Numer. Analyt. Meth. Geomech., 27(10), 813–835.
Doherty, J. P., and Deeks, A. J. (2003c). “Scaled boundary finite-element analysis of a non-homogeneous elastic half-space.” Int. J. Numer. Methods Eng., 57(7), 955–973.
Gazetas, G. (1982). “Stresses and displacements in cross-anisotropic soils.” J. Geotech. Eng. Div., Am. Soc. Civ. Eng., 108(4), 532–553.
Geddes, J. D. (1975). “Vertical stress components produced by axially symmetrical subsurface loadings.” Can. Geotech. J., 12(4), 482–497.
Gerrard, C. M. (1968). “The axisymmetric deformation of a homogeneous, cross-anisotropic half-space.” Highw. Res. Rec., 5, 36–44.
Gerrard, C. M., and Harrison, W. J. (1970). “Circular loads applied to a cross-anisotropic half space.” Technical Paper 8, CSIRO.
Gerrard, C. M., and Wardle, L. J. (1973). “Solutions for point loads and generalized circular loads applied to a cross-anisotropic half-space.” Technical Paper 13, CSIRO.
Gibson, R. E. (1967). “Some results concerning displacements and stresses in a non-homogeneous elastic half-space.” Geotechnique, 17(1), 58–67.
Gibson, R. E. (1974). “The analytical method in soil mechanics.” Geotechnique, 24(2), 115–140.
Gibson, R. E., and Sills, G. C. (1969). “On the loaded elastic half-space with a depth varying Poisson’s ratio.” Z. Angew. Math. Phys., 20(5), 691–695.
Hanson, M. T., and Puja, I. W. (1996). “Love’s circular patch problem revisited: Closed form solutions for transverse isotropy and shear loading.” Q. Appl. Math., 54(2), 359–384.
Hemsley, J. A. (1998). Elastic analysis of raft foundations, Thomas Telford, London.
Hooper, J. A. (1975). “Elastic settlement of a circular raft in adhesive contact with a transversely isotropic medium.” Geotechnique, 25(4), 691–711.
Kumar, P. (1988). “Nonhomogeneous and cross-anisotropic infinite elements.” Comput. Struct., 28(3), 327–333.
Lee, K. M., and Rowe, R. K. (1989). “Deformation caused by surface loading and tunneling: The role of elastic anisotropy.” Geotechnique, 39(1), 125–140.
Lekhnitskii, S. G. (1963). Theory of elasticity of an anisotropic elastic body, Holden-Day, San Francisco.
Liao, J. J., and Wang, C. D. (1998). “Elastic solutions for a transversely isotropic half-space subjected to a point load.” Int. J. Numer. Analyt. Meth. Geomech., 22(6), 425–447.
Longman, I. M. (1956). “Note on a method for computing infinite integrals of oscillatory functions.” Proc. Cambridge Philos. Soc., 52, 764–768.
Misra, B., and Sen, B. R. (1975). “Stresses and displacements in granular materials due to surface load.” Int. J. Eng. Sci., 13(7–8), 743–761.
Nayak, M. (1973). “Elastic settlement of a cross-anisotropic medium under axisymmetric loading.” Soils Found., 13(2), 83–90.
Oner, M. (1990). “Vertical and horizontal deformation of an inhomogeneous elastic half-space.” Int. J. Numer. Analyt. Meth. Geomech., 14(9), 613–629.
Pan, E. (1989). “Static response of a transversely isotropic and layered half-space to general surface loads.” Phys. Earth Planet. Inter., 54(3–4), 353–363.
Pan, E. (1997). “Static Green’s functions in multilayered half-spaces.” Appl. Math. Model., 21(8), 509–521.
Popov, G. I. (1962). “The contact problem of the theory of elasticity for the case of a circular area of contact.” J. Appl. Math. Mech., 26(1), 207–225.
Popov, G. I. (1973). “Axisymmetric contact problem for an elastic inhomogeneous half-space in the presence of cohesion.” J. Appl. Math. Mech., 37(6), 1052–1059.
Quinlan, P. M. (1949). “A Fourier integral approach to an aelotropic medium.” PhD dissertation, California Institute of Technology, Calif.
Rajapakse, R. K. N. D., and Selvadurai, A. P. S. (1989). “Torsion of foundations embedded in a non-homogeneous soil with a weathered crust.” Geotechnique, 39(3), 485–496.
Rajapakse, R. K. N. D., and Selvadurai, A. P. S. (1991). “Response of circular footings and anchor plates in non-homogeneous elastic soils.” Int. J. Numer. Analyt. Meth. Geomech., 15(7), 457–470.
Rostovtsev, N. A. (1961). “An integral equation encountered in the problem of a rigid foundation bearing on nonhomogeneous soil.” J. Appl. Math. Mech., 25(1), 238–246.
Rowe, R. K., and Booker, J. R. (1981). “The behavior of footings resting on a non-homogeneous soil mass with a crust, part II circular footings.” Can. Geotech. J., 18(2), 265–279.
Selvadurai, A. P. S. (1996). “The settlement of a rigid circular foundation resting on a half-space exhibiting a near surface elastic non-homogeneity.” Int. J. Numer. Analyt. Meth. Geomech., 20(5), 351–364.
Suresh, S. (2001). “Graded materials for resistance to contact deformation and damage.” Science, 292, 2447–2451.
Tarn, J. Q., and Lu, C. C. (1991). “Analysis of subsidence due to a point sink in an anisotropic porous elastic half space.” Int. J. Numer. Analyt. Meth. Geomech., 15(8), 573–592.
Ter-Mkrtich’ian, L. N. (1961). “Some problems in the theory of elasticity of nonhomogeneous elastic media.” J. Appl. Math. Mech., 25(6), 1667–1675.
Wang, C. D., Tzeng, C. S., Pan, E., and Liao, J. J. (2003). “Displacements and stresses due to a vertical point load in an inhomogeneous transversely isotropic half-space.” Int. J. Rock Mech. Min. Sci., 40(5), 667–685.
Ward, W. H., Marsland, A., and Samuels, S. G. (1965). “Properties of the London clay at the ashford common shaft.” Geotechnique, 15(4), 321–344.
Watson, G. N. (1966). A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, Mass.
Yue, Z. Q., Yin, J. H., and Zhang, S. Y. (1999). “Computation of point load solutions for geo-materials exhibiting elastic non-homogeneity with depth.” Comput. Geotech., 25(2), 75–105.
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Information
Published In
International Journal of Geomechanics
Volume 6 • Issue 1 • January 2006
Pages: 1 - 10
Copyright
© 2006 ASCE.
History
Received: Jan 22, 2004
Accepted: May 3, 2005
Published online: Jan 1, 2006
Published in print: Jan 2006
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