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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 75
PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping and Z. Bittnar
Paper 18

A Rectangular Finite Element for Plane Elasticity with In-Plane Rotation

A.I. Mousa and S.M. Tayeh

Department of Civil and Structural Engineering, Islamic University of Gaza, Palestine

Full Bibliographic Reference for this paper
A.I. Mousa, S.M. Tayeh, "A Rectangular Finite Element for Plane Elasticity with In-Plane Rotation", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 18, 2002. doi:10.4203/ccp.75.18
Keywords: strain based, plane elasticity, statical condensation, equilibrium, in-plane rotation.

Summary
The development of displacement fields by the use of strain based approach was first applied to curved elements. It was revealed that to obtain satisfactory converged results, the finite elements based on independent polynomial displacement functions require the curved structures to be divided into a large number of elements, references [1] and [2]. This work shows that there are two essential components to any displacement field. The first of which relates to rigid body modes of displacements while the second component is due to the straining of the element and these are approximately represented by assumed independent polynomial terms for the various component of strains in so far as it is allowed by the compatibility equations.

In reference [3] it was shown that the above approach is not confined to the development of curved elements only but also to plane elements for the analysis of plane elasticity problems. Several such rectangular elements were developed, notably a rectangular element based on linear variation of the direct strains and constant shearing strain. This element has the two essential degrees of freedom at each of the four corner nodes, and it was shown to produce converged results when applied to several elasticity problems without the use of large number of elements. This element is referred to as (SBRIE).

In reference [4] Sfendji developed a rectangular element based on linear variation of all the three strain components. It has two degrees of freedom at each comer node and at an internal node. This element is referred to as (SBRIE1).

In the present paper the strain based approach is applied to development of a new rectangular finite element for the general plane elasticity. This rectangular finite element has three degrees of freedom (Two general external degrees of freedom and the in-plane rotation) at each of the four corner nodes. This element is based on linear variations of all the three components of strain. The new element is referred to as SBREIR

This element is used to obtain solutions to two dimensional elasticity problems where the contribution of the shear stress on deformation can be significant. The performance of the element is investigated by applying it to two well-known plane elasticity problems, a cantilever beam loaded at the free end and a simply supported beam loaded by a point load at the midpoint. Convergence curves are plotted for vertical deflection at the points of application of load as well as for bending stress at points at the upper tension fibre and shearing stress at internal points in each problem.

It is found that the solutions obtained using this element are satisfactory for results of deflection, bending stress and the shearing stress even when only a small number of elements is used in the finite element solution. The new element gives very good results and faster convergence to the analytical solution than the bilinear rectangular element. It also has less discontinuities in the corner stresses than the bilinear rectangular element.

References
1
Ashwell, D.G., Sabir A.B. and Roberts, T.M., "Limitations of certain finite elements when applied to arches", Int. J. Mech. Sci., Vol. 13, 1971. doi:10.1016/0020-7403(71)90017-8
2
Sabir, A.B., "THE NODAL solution routine for large number of linear simultaneous equations in finite element analysis of plates and shells", Finite elements for thin shells and curved members, Ashwell and Gallagher, Willey, 1976.
3
Sabir, A.B., "Stiffness matrices for general deformation (out of plane and in- plane) of curved beams based on independent strain functions", The mathematics of finite elements and applications II. Academic press, 1975.
4
A. Sfendji, "Finite elements for plane elasticity problems". M.Sc. Thesis, University of Wales, U.K., 1988.
5
Timoshenko S. and Goodier J.N., "Theory of Elasticity", Third Edition, McGraw Hill, New York, 1985.

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