Open Access Open Access  Restricted Access Subscription or Fee Access

A Dynamic Finite Element for Coupled Extensional-Torsional Vibration of Uniform Composite Thin-Walled Beams

S. M. Hashemi(1*), A. Roach(2)

(1) Department of Aerospace Engineering, Ryerson University, Canada
(2) Department of Aerospace Engineering, Ryerson University, Canada
(*) Corresponding author



A Dynamic Finite Element (DFE) formulation for the free vibration analysis of extension-torsion coupled uniform composite thin-walled beams is presented. Employing the exact solutions of the differential equations governing the uncoupled vibrations of a uniform beam element, the analytical expressions for extensional and torsional dynamic trigonometric shape functions are derived. By exploiting the principle of virtual work and the frequency-dependent shape functions, the element dynamic stiffness matrix is developed. The application of the theory is demonstrated by a Circumferentially Uniform Stiffness (CUS) composite circular tube for which the influence of ply fibre-angle on the natural frequencies is studied. A variety of CUS configurations are studied and the correctness of the theory and the superiority of the proposed DFE over the conventional FEM methods are confirmed by numerical checks and the published results.
Copyright © 2016 Praise Worthy Prize - All rights reserved.


Composite Beam; Extension-Torsion Couplings; Dynamic Finite Element (DFE); Dynamic Stiffness Matrix (DSM); Finite Element Method (FEM); Materially Coupled Vibrations

Full Text:



M.W. Nixon, Extension-twist coupling of composite circular tubes with application to tilt rotor blade design. Proceedings of 28th AIAA/AHS/ASME/ASCE/ASC Structures, Structural Dynamics & Materials Conference, Montery (1987), 295-303.

O. Song and L Librescu, Free vibration and aeroelastic divergence of aircraft wings modelled as composite thin-walled beams, Proceedings of the 32nd AIAA/AHS/ASME/ ASCE/ASC Structures, Structural Dynamics & Materials Conference, Baltimore, Maryland (1991), 2128-2136.

R. Chandra and I. Chopra, Structural behavior of two-cell composite rotor blades with elastic couplings. AIAA Journal 30(12) (1992), 1914-1921.

C.M. Bothwell, R. Chandra and I. Chopra, Torsional actuation with extension-torsion composite coupling and a magnetostrictive actuator. AIAA Journal 33(4) (1995),723-729.

E.C. Smith, Vibration and flutter of stiff-inplane elastically tailored composite rotor blades. Proceedings of the 34th AIAA/AHS/ASME/ASCE/ASC Structures, Structural Dynamics & Materials (SDM) Conference, La Jolla, California (1993).

J.R. Banerjee and R. Butler, Coupled Extensional-Torsional Vibration of Composite Beams- An Exact Method, Proceedings of the 35th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, & Materials Conference, Hilton Head, S. Carolina (1994), 147-154.

J.R. Banerjee and F.W. Williams, An Exact Dynamic Stiffness Matrix for Coupled Extensional-Torsional Vibration Of Structural Members, Computers & Structures 50 (1994), 161-166.

K. Bhaskar and L. Liberscu, Buckling under axial compression of thin-walled composite beams exhibiting extension-twist coupling. Computers & structures 31 (1995), 203-212.

CES Cesnik, D.H. Hodges and M.J. Patil, Aeroelastic analysis of composite wings, Proceedings of the 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference & Exhibit, Salt Lake City, UT (1996), 1113-1123 (Technical Papers. Pt. 2).

J.B. Kosmatka and RC. Lake, Extension-twist behavior of initially twisted composite spars for tilt-rotor applications. Proceedings of the 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, & Materials Conference. Reston, VA (1996), 2175-2184 (Technical Papers. Pt. 4).

D.H. Hodges, Torsion of pretwisted beams due to axial loading. ASME Journal of Applied Mechanics 47 (1980), 393-397.

S. Krenk, The torsion-extension coupling in pretwisted elastic beams. Int. J. Solids Structures 19(1) (1983), 67-72.

G Cutri and A. Risitano, Coupled free, torsional and axial vibration of pre-twisted bars. Meccanica 14(3) (1979),157-162.

H.H. Chen and K.M. Hsiao, Coupled axial-torsional vibration of thin-walled Z-section beam induced by boundary conditions. Thin-walled Structures 45 (2007), 573-583.

K-C Liu, J. Friend and L. Yeo, Vibration analysis of pretwisted beams for the design of hybrid axial-torsional transducer. CD Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Honolulu, Hawaii (2007), 1-20.

W.H. Wittrick and F.W. Williams, A general algorithm for computing natural frequencies of elastic structures. Q. J. Mech. Appl. Math. 24 (1971), 263-284.

S.M. Hashemi and A. Roach, A Dynamic Finite Element for vibration analysis of composite circular tubes. Proceedings of the Tenth International Conference on Civil, Structural & Environmental Engineering Computing, B.H.V. Topping, (Editor), Civil-Comp Press, Stirling, United Kingdom, paper 177, 2005.

R. Chandra, A.D. Stemple and I. Chopra, Thin-walled composite beams under bending, torsional and extensional loads. Journal of Aircraft 27 (7) (1990), 619-626.

M.W. Hyer, Stress Analysis of Fibre-Reinforced Composite Materials (WCB/McGraw-Hill Company Inc., 1998)

F.W. Williams, W.P. Howson and J.R. Banerjee, Natural frequencies of members with coupled extensional torsional motion. A physical approach. Journal of Sound & Vibration 165(2) (1993), 373-375.

S.M. Hashemi and M.J. Richard, Free vibrational analysis of axially loaded bending-torsion coupled beams: a dynamic finite element, Computers & Structures 77 (2000), 711-724.

KJ. Bathe, Finite element procedure in engineering analysis. (Englewood Cliffs, NJ: Prentice-Hall, 1982).

S.M. Hashemi, M.J. Richard and G. Dhatt, A new dynamic finite element (DFE) formulation for lateral free vibrations of Euler-Bernoulli spinning beams using trigonometric shape functions, Journal of Sound & Vibration 220(4)(1999),601-629.

S.M. Hashemi and M.J. Richard, Natural frequencies of rotating uniform beams with coriolis effects. Journal of Vibration & Acoustics 123(4) (2001), 444-455.

A. Roach, Free vibration analysis of extension-torsion coupled elements using a dynamic finite element (DFE) formulation. MEng. Thesis, Ryerson University, 2005.

E.A. Armanios, A.M. Badir and V. Berdichevsky, Theory of Anisotropic Thin-Walled Closed-Cross-Section Beams. Composite Engineering 2(5-7) (1992),411-432.

A.R. Atiligan, M.V. Fulton, D.H. Hodges and L.W. Rehfield, Free-vibration analysis of composite beams. Journal of American Helicopter Society 36(3) (1991), 36-47.


  • There are currently no refbacks.

Please send any question about this web site to
Copyright © 2005-2022 Praise Worthy Prize