hp-Spectral Finite Element Analysis of Shear Deformable Beams and Plates

Document Type: Research Paper

Authors

Advanced Computational Mechanics Laboratory, Department of Mechanical Engineering, Texas A&M University, College Station

Abstract

There are different finite element models in place for predicting the bending behavior of shear deformable beams and plates. Mostly, the literature abounds with traditional equi-spaced Langrange based low order finite element approximations using displacement formulations. However, the finite element models of Timoshenko beams and Mindlin plates with linear interpolation of all generalized displacements have suffered from shear locking, which has been alleviated with the help of primarily reduced/selective integration techniques to obtain acceptable solutions [1-4]. These kinds of 'fixes' have come into existence because the element stiffness matrix becomes excessively stiff with low-order interpolation functions. In this study we propose an alternative spectrally accurate hp/spectral method to model the Timoshenko beam theory and first order shear deformation theory of plates (FSDT) to eliminate shear and membrane locking. Beams and isotropic and orthotropic plates with clamped and simply supported boundary conditions are analyzed to illustrate the accuracy and robustness of the developed elements. Full integration scheme is employed for all cases. The results are found to be in excellent agreement with those published in literature.

Keywords

 [1] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells, CRC Press, Boca Raton, FL, Second Edition.

[2] Reddy J.N., 2007, Theory and Analysis of Elastic Plates and Shells, CRC Press, Boca Raton, FL, Second Edition.

[3] Reddy J.N., 2004, An Introduction to Non-Linear Finite Element Analysis, Oxford University Press, Oxford, UK.

[4] Donning B.M., Liu W.K., 1998, Meshless methods for shear-deformable beams and plates, Computer Methods in Applied Mechanical Engineering 152: 47-71.

[5] Maenghyo C., Parmerter R., 1994, Finite Element for composite bending based on efficient higher order theory, AIAA Journal 32: 2241-2248.

[6] Pontaza J.P., Reddy J.N., 2004, Mixed plate bending elements based on least squares formulations, International Journal for Numerical Methods in Engineering 60: 891-922.

[7] Urathaler Y., Reddy J.N., 2008, A mixed finite element for nonlinear bending analysis of laminated composite plates based on FSDT, Mechanics of Advanced Materials and Structures 15: 335-354.

[8] Archiniega R.A., Reddy J.N., 2007, Large deformation analysis of functionally graded shells, International Journal for Solids and Structures 44: 2036-2052.

[9] Karniadakis G.K., Sherwin S., 2004, Spectral/hp element methods for computational fluid dynamics, Oxford Science Publications, London, Second Edition.

[10] Melenk J.M., 2002, On condition numbers in hp-fem with Gauss-Lobatto-based shape functions, Journal of Computational and Applied Mathematics 139: 21-48.

[11] Maitre J.F., Pourquier O., 1996, Condition number and diagonal preconditioning: comparison of the p-version and the spectral element methods, Numerical Methods 74: 69-84.

[12] Reddy J.N., 1997, On Locking-free shear deformable beam finite elements, Computer Methods in Applied Mechanics and Engineering 149: 113-132.

[13] Reddy J.N., Wang, C.M., Lam, K.Y., 1997, Unified finite elements based on the classical and shear deformation theories of beams and axisymmetric circular plates, Communications in Numerical Methods in Engineering 13: 495-510.

[14] Reddy J.N., 2006, An Introduction to the Finite Element Method, McGraw-Hill, New York, Third Edition.

[15] Prabhakar V., Reddy, J.N., 2007, Orthogonality of modal basis in hp finite element models, International Journal for Numerical Methods in Fluids 54: 1291-1312.

[16] Osilenker B., 1999, Fourier Series in Orthogonal Polynomials, World Scientific, Singapore, Second Edition.

[17] Reddy, J.N., 2002, Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, New York.