Keywords: beams, deep, finite element, foundation, three dimensions, Winkler model.

This paper deals with linear elastic behaviour of deep beams resting on elastic foundations. Winkler models for both the compressional and frictional resistances are used to model the elastic foundations. This model assumes that the base consists of closely spaced independent linear springs. The displacement occurs immediately under the loaded area and outside this region; the displacement is zero [1].

The finite element method is an approximate method for the analysis structures. The basic philosophy of this method is that the structures or the continuum is divided into small elements of various shapes and types, which are assembled together to form an approximate mathematical model.

In this paper, the finite element method in Cartesian coordinate is used to solve the problems of deep beams resting on Winkler type elastic foundations with both normal and frictional restraints. The three dimensional isoparametric brick elements are used, each node have three degrees of freedom (the deflection w and the displacements u and v). In the present study, the computer program presented in the text by Hinton and Owen [2]. This program is modified in this work to be capable of solving the problem of beams on elastic foundations. Three-dimensional analysis is performed. The numerical results obtained from the finite element analysis have been compared with available exact and other analytical and numerical results [3,4]. The modified program is coded with Fortran77 language. Also, the Winkler-type foundations are included in the program.

Several important parameters are incorporated in the analysis, namely, the vertical subgrade reaction, horizontal subgrade reaction and beam depth to trace their effects on deflections, bending moments and shear forces.

From this study, many main conclusions are obtained. The results obtained from the exact, finite difference and finite element solutions check the accuracy of the method used in this paper in which they are in good agreement.

The effect of friction at the beam-foundation interface is found to be small on the deflection, moment and shear. The effect of varying vertical modulus of elastic foundation on deflection, moment and shear is significant.

1

E. Winkler, "Die Lehre Von Elasticitaet and Festigkeit", (H. Dominic us), Prague, 182-184, 1867.

2

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

3

M. Hetenyi, "Beams on Elastic Foundations", the University of Michigan Press, Ann Arbor, 1946.

4

A.A. Al-Azzawi, M.A. Al-Ani, "Finite Difference Analysis of Beams on Winkler Foundation", Engineering and Technology Journal, University of Technology, Baghdad, Iraq, 23, 578-592, 2004.

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