Elastic Subsurface Stress Analysis for Circular Foundations. I
Publication: Journal of Engineering Mechanics
Volume 124, Issue 5
Abstract
In part I of this analysis an attempt is made to determine a simple estimate of the stresses resulting from a circular foundation subjected to concentric or eccentric loading. It is assumed that the foundation loading can be modeled as combinations of uniform, linear, and quadratic tractions applied over a circular area on the surface of an elastic half space. The present analysis for quadratic and linear loading are combined with a uniform loading solution (normal or shear traction), previously derived by the authors, to provide the requisite loading conditions and resulting internal stress fields. The current analysis consists of using potential functions to derive closed form expressions for the elastic field in the half space. The half space is taken as cross-anisotropic (transversely isotropic), where the planes of isotropy are parallel to the free surface. The x- and y-axes are taken in the plane of the surface with z directed into the half space. Hence the boundary conditions within the circular loaded area are on the shear stress components τxz, τyz, and normal stress σzz. The solutions presented within actually comprise seven different boundary value problems for the transversely isotropic half space. The analytical solutions for a point normal or shear force are first used to write the solution for distributed loading over a circle in the form of a double integral over the loaded area. It is shown, with the aid of Hankel transform analysis, that the integrals appearing in the elastic field have been previously evaluated in terms of complete elliptic integrals. The necessary integral evaluations are provided in Appendix I. The limiting form of the expressions for an isotropic material are also included.
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References
1.
Boussinesq, J. (1885). Application des potentiels. Gauthier-Villars, Paris, France.
2.
Bowles, J. E. (1982). Foundation analysis and design. McGraw-Hill Book Co. Inc., New York, N.Y., 70–74.
3.
Eason, G., Noble, B., and Sneddon, I. N. (1955). “On certain integrals of Lipschitz-Hankel type involving products of Bessel functions.”Philosophical Trans. Royal Soc., London, U.K., A247, 529–551.
4.
Elliot, H. A.(1948). “Three-dimensional stress distributions in hexagonal aeolotropic crystals.”Proc., Cambridge Philosophical Soc., 44, 522–533.
5.
Fabrikant, V. I. (1988). “Elastic field around a circular punch.”J. Appl. Mech., 55, 604–610, Appendix A, Equation (AA2).
6.
Fabrikant, V. I. (1989). Applications of potential theory in mechanics: a selection of new results. Kluwer Academic Publishers, Dordrecht, The Netherlands, 71–79.
7.
Fabrikant, V. I. (1991). Mixed boundary value problems of potential theory and their applications in engineering. Kluwer Academic Publishers, Dordrecht, The Netherlands, 355–356.
8.
Gerrard, C. M., and Harrison, W. J. (1970). “Circular loads applied to a cross-anisotropic half space.” CSIRO Australian Div. Appl. Geomech. Tech. Paper No. 8, Commonwealth Scientific and Industrial Research Organization.
9.
Gradshteyn, I. S., and Ryzhik, I. M. (1980). Table of integrals, series, and products. Academic Press, Inc., San Diego, Calif., 904–909.
10.
Hanson, M. T. (1992a). “The elastic field for conical indentation including sliding friction for transverse isotropy.”J. Appl. Mech., 59, S123–S130.
11.
Hanson, M. T.(1992b). “The elastic field for spherical Hertzian contact including sliding friction for transverse isotropy.”J. Tribology, 114, 606–611.
12.
Hanson, M. T., and Puja, I. W.(1996). “Love's circular patch problem revisited: closed form solutions for transverse isotropy and shear loading.”Quarterly of Appl. Mathematics, 54(2), 359–384.
13.
Hanson, M. T., and Puja, I. W.(1997). “The evaluation of certain infinite integrals involving products of Bessel functions: a correlation of formula.”Quarterly of Appl. Mathematics, 55(3), 505–524.
14.
Hanson, M. T., and Puja, I. W.(1998). “Elastic Subsurface Stress Analysis for Circular Foundations. II.”J. Engrg. Mech., ASCE, 124(5), 547–555.
15.
Hanson, M. T., and Wang, Y.(1997). “Concentrated ring loadings in a full space or half space: solutions for transverse isotropy and isotropy.”Int. J. Solids and Struct., 34(11), 1379–1418.
16.
Love, A. E. H. (1929). “The stress produced in a semi-infinite solid by pressure on part of the boundary.”Philosophical Trans. Royal Soc., London, U.K., A228, 377–420.
17.
Muki, R. (1960). “Asymmetric problems of the theory of elasticity for a semi-infinite solid and a thick plate.”Progress in solid mech., I. N. Sneddon and R. Hill, eds., North-Holland Publishing Co., Amsterdam, The Netherlands, 399–439.
18.
Newmark, N. M. (1942). “Influence charts for computation of stresses in elastic soils.”Bull. 38, University of Illinois Engineering Experiment Station, Urbana, Ill.
19.
Poulos, H. G., and Davis, E. H. (1974). Elastic solutions for soil and rock mechanics. John Wiley & Sons, Inc., 43–49, 338–397.
20.
Sneddon, I. N. (1951). Fourier transforms. McGraw-Hill Book Co. Inc., New York, N.Y., 450–473.
21.
Sowers, G. F. (1979). Introductory soil mechanics and foundations: geotechnical engineering. Macmillan Publishing Co., New York, N.Y., 462–466.
22.
Terazawa, K. (1916). “On the elastic equilibrium of a semi-infinite solid ....”J. Coll. Sci., Tokyo, Japan, 37, article 7.
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Copyright © 1998 American Society of Civil Engineers.
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Published online: May 1, 1998
Published in print: May 1998
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