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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 88
PROCEEDINGS OF THE NINTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping and M. Papadrakakis
Paper 210

Equilibrium Configurations of Heavy Elastica Beams Under Unilateral Contact Constraints

M. Abdel-Jaber1, S. Al-Sadder2, A. Shatnawi1 and M. Mahdi2

1Department of Civil Engineering, The University of Jordan, Amman, Jordan
2Faculty of Engineering, Hashemite University, Zarka, Jordan

Full Bibliographic Reference for this paper
M. Abdel-Jaber, S. Al-Sadder, A. Shatnawi, M. Mahdi, "Equilibrium Configurations of Heavy Elastica Beams Under Unilateral Contact Constraints", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 210, 2008. doi:10.4203/ccp.88.210
Keywords: heavy elastica beam, equilibrium configurations, unilateral contact constraint.

Summary
A problem that involves large deflections and moving bounday nonlinearities has been studied in this work. An inextensible heavy elastica simply supported beam is elevated from a smooth horizontal rigid ground of a prescribed elevation. The beam was left to take its final equilibrium configuration in contact with the ground. The elastica theory is used to obtain the large deflection equations that are governed by a non-dimensional parameter and the prescribed elevation.

The solution is obtained by solving the resulting nonlinear differential equations using a sixth-order Runge-Kutta numerical integration scheme in connection with a one or two-parameter shooting technique. A new scheme for incremental loading has been suggested in this paper to overcome the convergence difficulties associated with these types of problems.

Little research is available concerning the behavior of slender beams under large deflections and having moving boundaries. Kooi and Kuipers [1] studied the problem of an infinite heavy elastica lifted from one point from a horizontal rigid ground. The governing differential equations with the required fixed and moving boundary conditions were solved using incremental finite differences in connection with the conventional Newton-Raphson iteration technique. Kooi and Kuipers [2] revisited the previous problem, but used large displacement finite elements in connection with the gap elements (to represent unilateral contact between the heavy elastica and the ground). Randy [3] established a new procedure called moving finite elements for solving constrained contact problems of beams subject to small deflections. The analysis dealt with differential and interface equations (moving boundary equations) with a fixed number of conventional two-node elements.

This paper presents a numerical solution for equilibrium configurations of heavy elastica simply supported slender beams under unilateral contact constraint which involves large deflections and moving boundary nonlinearities. The beam was left to take its final equilibrium configuration to become in contact with the smooth ground. Solution are obtained for different values of elevation and non-dimensional beam weight parameter by solving the resulting nonlinear differential equations using a sixth-order Runge-Kutta numerical integration scheme in connection with a one or two-parameter shooting technique. The ANSYS package is used to check the validity and accuracy of the method of solution presented.

The effect of the non-dimensional beam weight and the elevation on the equilibrium configurations is significant and it can be readily caculated using the methodologies developed. Excellent agreement has been obtained in the present study with that obtained using ANSYS.

References
1
B.W. Kooi, M. Kuipers, "A Unilateral Contact Problem With The Heavy Elastica", Int. J. Non-Linear Mech., Vol. 19, No.4, pp. 309-321, 1984. doi:10.1016/0020-7462(84)90059-3
2
B.W. Kooi, Kuipers, "A Unilateral Contact Problem With The Heavy elastica solve by use of finite element", Int. J. Non-Linear Mech., Vol. 21, No.1/2, pp. 95-103, 1985. doi:10.1016/0045-7949(85)90233-0
3
J.G. Randy, "Moving Finite Element Analysis for The Elastic Beams in Contact Problem", Int. J. Comp. and Struct., Vol.24, No. 4, pp.571-579, 1986. doi:10.1016/0045-7949(86)90195-1

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