Advanced Steel Construction

Vol. 6, No. 3, pp. 831-851 (2010)




A.S. Shatnawi 1*, S.Z. Al-Sadder 2, M.S. Abdel-Jaber 1, R. A. Othman 3 and N. S. Ahmed 3

1 Associate Professor, Department of Civil Engineering, The University of Jordan, Amman 11942, Jordan

2 Assistant Professor, Department of Civil Engineering, Hashemite University, Zarqa 13115, Jordan

3 Assistant Professor, Department of Civil Engineering, Baghdad University, Baghdad, Iraq

*(Corresponding author: E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.)

Received: 14 January 2009; Revised: 30 October 2009; Accepted: 22 April 2010




View Article   Export Citation: Plain Text | RIS | Endnote


The large deflection of slender beams subjected to dynamic excitation is investigated in this paper. One-dimensional flexural vibration consisting of bending and shear waves is considered. A new formulation for the flexural wave behavior in geometrically non-linear beams is introduced. The formulation of the non-linear governing equations is established by considering elements under the effect of large-deflection and large-rotation while subjected to dynamic excitation. The governing equations are re-written in a numerical form using the method of characteristics. As representative examples, different types of support and load conditions are studied, e.g., cantilever and propped cantilever beams. The results showed that the method of characteristics with the proposed formulation is a suitable method to represent wave propagation phenomena in large deflection.



Slender beam, large deflection; non-linear dynamics, method of characteristics


[1] Zienkiewicz, O.Z., “The Finite Element Method”, 3rd Edition. McGraw-Hill, 1977.

[2] Ross, C.T.F., “Finite Element Method in Structural Mechanic, 1st edition, Chichester Ellis Harwood, 1985.

[3] Clough, R.W. and Penzien, J., “Dynamics of Structures”, NY : McGraw-Hill, 1975.

[4] Coates, R.C., Couiie, M.G. and Kong, F.K., “Structural Analysis”, 2nd Edition, New York: Van Nostrand Reinhold, 1986.

[5] Johnson, D. “Advanced Structural Mechanics”, London: Collins, 1986.

[6] Pany, C. and Rao, G.V., “Large Amplitude Free Vibrations of a Uniform Spring-Hinged Beam”, Journal of Sound and Vibration, 2004, Vol. 271, pp. 1163–1169.

[7] Lande, R.H. and Langley, R.S., “The Energetics of Cylindrical Bending Waves in a Thin Plate”, Journal of Sound and Vibration, 2005, Vol. 279, pp. 513–518.

[8] Pfgiffer, R., “über Die Differertialbeichung der Trasversalen Stabschwingunguen,” Z Angew. Math. Mech., 1947, No. 3, pp. 25-27.

[9] Plass, H.I., “Some Solution of Timoshenko Beam Equation for Short Pulse-Type Loading,” Journal of Applied Mechanics, 1958, Vol. 25, pp.379-385.

[10] Chou, P.C. and Mortimer, “Solution of One-dimensional Elastic Wave Problems by the (MOC),” J. of Applied Mechanics, ASME, 1967, Vol. 34, pp.745-750.

[11] Chou, P.C. and Koening, H.A., “A Unified Approach to Spherical Elastic Waves by the (MOC),” Journal of Applied Mechanics, Trans., ASME, 1966, Series 1, No. 88, pp.159-168.

[12] Vardy, A.E. and Chan, L.I., “Truss Analysis by Boundary Characteristics,” ASCE, Journal of Engineering Mechanics, 1988, Vol. 114, No. 3, pp. 520-535.

[13] Vardy, A.E. and Al-Sarraj, A.T. “Method of Characteristics Analysis of One-Dimensional Members”, Journal of Sound and Vibration, 1989, Vol. 129, No. 3, pp. 479-487.

[14] Alsarraj, A.T., Othman, R.A. and Jwad, Z.M., “Non-linear Vibration of Axially Loaded Bars,” Journal of Engineering and Technology, 1994, Vol. 12, No. 9, pp. 37-45.

[15] Al-Sadder, Samir, Othman, R., Shatnawi, Anis and Abdel-Jaber, M., “Dynamic Behavior of Slender Steel Beams under Large Deflection by the Method of Characteristics”, 6th Int. Conference on Steel and Aluminum Structural Engineering, ICSAS’07, 2007.

[16] Haisler, W.E., Stricklin, J.A. and Stebbins, F.J., “Development and Evaluation of Solution Procedures for Geometrically Nonlinear Structural Analysis,” AIAA Journal, 1972, Vol.10, No. 3, pp. 264-272.

[17] Haisler, W.E. and Stricklin, J.A. “Displacement Incrementation in Nonlinear Structure Analysis by the Self-correcting Method,” International Journal for Numerical Methods in Engineering, 1977, Vol. 11, pp. 3-10.

[18] MSC/NASTRAN for Windows 95: McNeal-Schwendler Corporation, CA, USA, 1995.