Many manufacturing situations involve a finite thickness plate or layer of material which is pressed against a much thicker foundation of the same or different material. One key example is a blank holder (plate) pressed against a die (foundation) in a sheet metal forming operation. In designing such a plate/foundation system the design objective often involves the contact stress distribution between the plate and foundation and the design variables are typically the thickness and modulus of the plate, the stiffness of the foundation and the applied pressure distribution on the noncontacting side of the plate. In general the problem relating the variables to the contact pressure distribution is three-dimensional and requires a complex finite element or boundary element solution. However, if the applied pressure distribution consists of sufficiently localized patches, which is often the case in applications, then an approximate 3D solution can be constructed by superposition. Specifically, the paper provides a convenient calculation procedure for the contact pressure due to a single circular patch of applied pressure on an infinite, isotropic, elastic layer which rests on a Winkler foundation. The procedure is validated by using known analytical solutions and the finite element method (FEM). Next a sensitivity study is presented for ascertaining the validity of the solution’s use in constructing solutions to practical problems involving multiple patches of loading. This is accomplished through a parametric study of the effects of loading radius, layer thickness, layer elastic properties, foundation stiffness and the form of the applied pressure distribution on the magnitude and extent of the contact pressure distribution. Finally, a procedure for determining an appropriate Winkler stiffness parameter for a foundation is presented. [S1087-1357(00)00603-1]

1.
Poulos, H. G., and Davis, E. H., 1974, Elastic Solutions for Soil and Rock Mechanics, Wiley, New York, NY.
2.
Li
,
H.
, and
Dempsey
,
J. P.
,
1990
, “
Axisymmetric Contact of an Elastic Layer Underlain by Rigid Base
,”
Int. J. Numer. Methods Eng.
,
29
, pp.
57
72
.
3.
Dempsey
,
J. P.
,
Zhao
,
Z. G.
, and
Li
,
H.
,
1991
, “
Axisymmetric Indentation of an Elastic Layer Supported by a Winkler Foundation
,”
Int. J. Solids Struct.
,
27
, No.
1
, pp.
73
87
.
4.
Hetenyi, M., 1946, Beams on Elastic Foundation; Theory with Applications in the Fields of Civil and Mechanical Engineering, University of Michigan Press, Ann Arbor, MI.
5.
Westergaard
,
H. M.
,
1926
, “
Stress in Concrete Pavements Computed by Theoretical Analysis
,”
,
7
, pp.
25
35
.
6.
Terzaghi
,
K.
,
1955
, “
Evaluation of Coefficient of Subgrade Reaction
,”
Geotechnique
,
5
, pp.
297
326
.
7.
Iyengar, K. T. S. R., and Ramu, S. A., 1979, Design Tables for Beams on Elastic Foundations and Related Problems, Elsevier Applied Science, London, England.
8.
Sneddon, I. N., 1972, The Use of Integral Transforms, McGraw-Hill, New York, NY.
9.
Vlasov, V. Z., and Leont’ev, N. N., 1966, Beams, Plates, and Shells on Elastic Foundations, Translated from Russian. NASA TT F-357, Washington, DC.
10.
Timoshenko, S., and Goodier, J. N., 1951, Theory of Elasticity, 2nd ed., McGraw-Hill, New York, NY.
11.
Vesic, A. B., 1961, “Beams on Elastic Subgrade and the Winkler’s Hypothesis,” Proceedings, 5th International Conference on Soil Mechanics and Foundation Engineering, Paris, France, pp. 845–850.
12.
Abramowitz, M., and Stegun, I. A., 1964, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, 5th printing, U.S. Government Printing Office.
13.
Fullerton, W., 1977, Function BESJ0(X), Developed in Los Alamos National Labs, NIST Guide to Available Math Software, retrieved from the internet cite: http://www.public.iastate.edu/∼math/cmlib/fnlib/besj0.