A super-convergent finite element is formulated for the dynamic flexural response of symmetric laminated composite beams subjected to various transverse harmonic forces. Based on the assumptions of Timoshenko beam theory, a one-dimensional finite beam element with two-nodes and four degrees of freedom per element is developed. The new beam element is applicable to symmetric laminated composite beams and accounts for the effects of shear deformation, rotary inertia, and Poison's ratio. The exact closed-form solution for flexural displacement functions developed in this study is employed to develop exact shape functions. Although the present finite element formulation is developed to obtain the steady state dynamic response but can be also used to capture the quasi-static analysis of the symmetric laminated composite beams. Moreover, it is also used to extract the natural frequencies and mode shapes for flexural response. The accuracy and efficiency of the present finite beam element are shown through comparisons with other established exact and Abaqus finite element solutions. The new element is demonstrated to be free from shear locking and mesh discretization errors occurring in conventional finite element solutions and illustrates an excellent agreement with those based on finite element solutions at a fraction of the computational and modeling cost.
This paper develops the exact solutions for coupled flexurallateral-torsional static response of thin-walled asymmetric open members subjected to general loading. Using the principle of stationary total potential energy, the governing differential equations of equilibrium are formulated as well as the associated boundary conditions. The formulation is based on a generalized Timoshenko-Vlasov beam theory and accounts for the effects of shear deformation due to bending and warping, and captures the effects of flexuraltorsional coupling due to cross-section asymmetry. Closed-form solutions are developed for cantilever and simply supported beams under various forces. In order to demonstrate the validity and the accuracy of this solution, numerical examples are presented and compared with well-established ABAQUS finite element solutions and other numerical results available in the literature. In addition, the results are compared against non-shear deformable beam theories in order to demonstrate the shear deformation effects. Abaqus model, the coupled static results obtained from the present solution are under-predicted 7.17%, 4.41%, 3.11% and 2.93% lower than the corresponding results based on Abaqus shell model. Again, the difference is due to the distortional effects of the cross-section, which are captured only in the Abaqus shell model.
Starting with total potential energy variational principle, the governing equilibrium coupled equations for the torsional-warping static analysis of open thin-walled beams under various torsional and warping moments are derived. The formulation captures shear deformation effects due to warping. The exact closed form solutions for torsional rotation and warping deformation functions are then developed for the coupled system of two equations. The exact solutions are subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing coupled equations. A super-convergent finite beam element is then formulated based on the exact shape functions. Key features of the beam element developed include its ability to (a) eliminate spatial discretization arising in commonly used finite elements, and (e) eliminate the need for time discretization. The results based on the present finite element solution are found to be in excellent agreement with those based on exact solution and ABAQUS finite beam element solution at a small fraction of the computational and modelling cost involved.
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