Keywords: compressional and frictional, deep beams, elastic, non-uniform, numerical analysis, Winkler foundation.

This research presents a numerical solution to evaluate the response of deep beams on non-uniform elastic foundations. In the present study, the Winkler model is used to model the elastic foundations. This model assumes the soil medium to consist of a system of independent spring elements. The displacement occurs immediately under the loaded area and outside this region; the displacement is zero [1].

The finite element method is used to solve the problem of deep beams on both vertical and horizontal modulus of subgrade reactions. The finite element method in Cartesian coordinates is formulated using one dimensional isoparametric elements to model the beam and elastic vertical and horizontal springs with different values at each location to model the soil.

A computer program coded in Fortran77 for the analysis of beams on elastic foundations is developed in this study based on the computer program of Hinton and Owen [2]. The results from this method are plotted with other methods reported in the literature to compare and check the accuracy of the solutions. Good agreement is found in this method, with the results obtained from analytical solution using Green's function formulation [3].

Typical results are presented in graphical form for different parameters. A parametric study is performed to show the influence of several important factors on the behaviour of beams resting on elastic foundations and subjected to static loads. For example, the effects of vertical subgrade reaction K_z, horizontal subgrade reaction K_x and beam depth on the behaviour of beams resting on elastic foundations.

From this study, many main conclusions are obtained. The results from the Green's and finite element solutions check the accuracy of the method used in this paper in which good agreements are obtained including:

• The effect of non-uniformity of frictional restraint at the beam-foundation interface is found to be smaller on the behaviour (deflection, moment and shear) than the effect of non-uniformity of compressional restraint.
• The effect of beam depth on the behaviour of beams is found to be similar for uniform and non-uniform subgrade reactions cases.

1

M. Hetenyi, "Beams on Elastic Foundations", the University of Michigan Press, Ann Arbor, 1974.

2

E. Hinton, D.R.J. Owen, "Finite Element Programming", Academic Press, London, U.K., 1977.

3

Y. Guo, Y. Weitsman, "Solution Method for Beams on Non-uniform Elastic Foundations", Journal of Engineering Mechanics, 592-594, May 2000.

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