Abstract
The Green's functions for an infinite elastic plate, attached respectively to a Pasternak or a Kerr base model, and subjected to a concentrated force, are obtained in terms of Bessel functions. It is shown, that for each base model, depending on the plate and base parameters, the solutions may be of different form. The method of images is then utilized to generate closed form solutions for the semi-infinite and quarter plates with simply supported boundaries. Paper also presents a generalization of Bessel functions of the Kelvin type and a discussion of their properties. They were needed for the solution of some of the equations under consideration.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Timoshenko, S. and Woinowsky-Krieger, S.,Theory of plates and shells, 2nd Ed., McGraw-Hill, New York 1959.
Kerr, A. D. and Palmer, W. T.,The deformations and stresses in floating ice plates, Acta Mech.15, 57–72 (1972).
Hertz, H.,Über das Gleichgewicht schwimmender elastischer Platten, Wiedemann's Annalen d. Physik und Chemie,22, 449–455 (1884).
Schleicher, F.,Kreisplatten auf elastischer Unterlage, Springer Verlag, Berlin 1926.
Westergaard, H. M.,Stresses in concrete pavements computed by theoretical analysis, Public Roads,7, Nr. 2, 25–35 (1926).
Koreniev, G. B.,Analysis of plates on an elastic base, (In Russian), Gos. Izd. po Stroitelstvu, Arkhitekture i Stroitelnym Materialam, Moscow 1962.
Kerr, A. D.,The bearing capacity of floating ice plates subjected to static or quasi-static loads, J. Glaciology,17, Nr. 76, 229–268 (1976).
Pasternak, P. L.,On a new method for the analysis of foundations on an elastic base using two base parameters, (In Russian), Gos Izd. po Stroitelstvu i Arkhitekture, Moscow 1954.
Kerr, A. D.,Elastic and viscoelastic foundation models, J. Appl. Mech.,31, Nr. 3, 491–498 (1964).
Kerr, A. D.,A study of a new foundation model, Acta Mech.,I/2, 135–147 (1965).
Kerr, A. D.,On the formal development of elastic foundation models, Ing. Arch.,54, 455–464 (1984).
Wyman, M.,Deflections of an infinite plate, Canadian Research,28, Sect. A, 293–302 (1950).
Bowman, F.,Introduction to Bessel functions, Dover Publ., New York 1958.
McLachlan, N. W.,Bessel functions for engineers, 2nd Ed., Clarendon Press, Oxford, England 1955.
Kerr, A. D.,Elastic plates on a liquid foundation, J. Eng. Mechanics Div., Proc. ASCE, EM3, pp. 59–71 (1963).
Uspensky, J. V.,Theory of equations, McGraw-Hill, New York 1948.
Yu, Y. Y.,On the generalized Ber, Bei, Ker, and Kei functions with applications to plate problems, Quart. J. Mech. & Appl. Maths.,10, Pt. 2, 254–256 (1957).
Koreniev, G. B.,Vvedenie v Teoriyu Besselevykh Funktsii, (Introduction to the theory of Bessel functions, in Russian), Izd. Nauka, Moscow, pp. 197–198 (1971).
Author information
Authors and Affiliations
Additional information
Research supported by National Science Foundation Grant MSM-8308919.
Rights and permissions
About this article
Cite this article
Kerr, A.D., El-Sibaie, M.A. Green's functions for continuously supported plates. Z. angew. Math. Phys. 40, 15–38 (1989). https://doi.org/10.1007/BF00945307
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00945307