Coupled bending–torsional dynamic stiffness matrix of an axially loaded Timoshenko beam element

School of Engineering, Cardiff University, Cardiff, Wales, United Kingdom
International Journal of Solids and Structures (Impact Factor: 2.21). 03/1994; 31(6):749-762. DOI: 10.1016/0020-7683(94)90075-2


Analytical expressions for the coupled bending-torsional dynamic stiffness matrix terms of an axially loaded uniform Timoshenko beam element are derived in an exact sense by solving the governing differential equations of motion of the element. The symbolic computing package REDUCE has been used to generate an analytical expression for each of the dynamic stiffness terms in a concise form. For check purposes, numerical values of the dynamic stiffness matrix terms were obtained using the derived explicit expressions as well as by an alternative nonanalytical method based on matrix inversions and matrix multiplications. Stiffnesses obtained from both methods agreed with each other to machine accuracy. Application of the developed theory is discussed with particular reference to an established algorithm. The influence of axial force, shear deformation and rotatory inertia on the natural frequencies of a bending-torsion coupled beam with cantilever end-conditions is demonstrated by numerical results. Such results are not generally available in the literature. Therefore, results obtained by partially restricting the present theory are compared with the existing literature wherever possible. The results indicate that the method is accurate and efficient.

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    • "Furthermore, Hashemi and Richard [22] formed a DFE solution for the free vibration analysis of axially loaded bending-torsion coupled beams. An axially loaded isotropic Timoshenko beam coupled in bending and torsion was studied by Banerjee and Williams [23]. Leung [24] developed an exact DSM of a thin walled beam. "
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    Shock and Vibration 10/2014; 2014. DOI:10.1155/2014/153532 · 0.72 Impact Factor
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    • "The dynamic stiffness matrices in the frequency domain for linear structural members with distributed masses were first used by Latona [13] to validate the accuracy of lumped and consistent mass matrices. Formulations for beam members and shell elements were then obtained by Kolousěk [14], Banerjee and William [15] [16] [17], Doyle [18] [19], Papaleontiou [20], Gopalakrishnan and Doyle [21] and Yu and Roësset [22]. In this work the dynamic stiffness matrices with distributed masses were used, extending the formulation of Yu and Roësset [22] to include shear deformation, rotatory inertia and the effect of axial forces [23] "
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    • "Turning now to the problems of an exact dynamic and static stiffness matrices of the thin-walled beams, Banerjee [23] and Banerjee and Williams [24] [25] [26] derived the bending-torsional dynamic stiffness matrix for Timoshenko beam elements and recently, Kim et al. [27] presented an exact static stiffness matrices for the buckling and the elastic analyses of shear deformable thin-walled beam with non-symmetric cross-section. However, they did not consider CSD effects due to the shear forces and the restrained warping torsion. "
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    ABSTRACT: A general theory is proposed for the shear deformable thin-walled beam with non-symmetric open/closed cross-sections and its exact dynamic and static element stiffness matrices are evaluated. For this purpose, an improved shear deformable beam theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This includes the shear deformations due to the shear forces and the restrained warping torsion and due to the coupled effects between them, rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. Governing equations and force–deformation relations are derived from the energy principle and a system of linear eigenproblem with non-symmetric matrices is constructed based on 14 displacement parameters. And then explicit expressions for displacement parameters are derived and the exact dynamic and the static stiffness matrices are determined using force–deformation relationships. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with other numerical solutions available in the literature and results using the thin-walled beam element and the shell element. Particularly the influences of the coupled shear deformation on the vibrational and the elastic behavior of non-symmetric beams with various boundary conditions are investigated.
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