A Comparative Study of Least-Squares and the Weak-Form Galerkin Finite Element Models for the Nonlinear Analysis of Timoshenko Beams

Subject Areas : Engineering

W Kim ¹ , J.N Reddy ²

1 - Department of Mechanical Engineering, Korea Army Academy at Yeong cheon, Yeong cheon, 770-849 South Korea
2 - Department of Mechanical Engineering, Texas A&M University, College Station

Received: 2010-06-19 Accepted : 2010-07-17 Published : 2010-06-30

Keywords: Finite element models, Least-squares model, Weak-form Galerkin model, Geometric nonlinearity, Timoshenko Beam Theory,

Abstract :

In this paper, a comparison of weak-form Galerkin and least-squares finite element models of Timoshenko beam theory with the von Kármán strains is presented. Computational characteristics of the two models and the influence of the polynomial orders used on the relative accuracies of the two models are discussed. The degree of approximation functions used varied from linear to the 5th order. In the linear analysis, numerical results of beam bending under different types of boundary conditions are presented along with exact solutions to investigate the degree of shear locking in the newly developed mixed finite element models. In the nonlinear analysis, convergences of nonlinear finite element solutions of newly developed mixed finite element models are presented along with those of existing traditional model to compare the performance.

References:

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

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

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

[4] Wang C.M., Reddy J.N., Lee K.H., 2000, Shear Deformable Beams and Plates: Relationships with Classical Solutions, Elsevier, New York.

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

[6] Pontaza J.P., 2005, Least-squares variational principles and the finite element method: Theory, formulations, and models for solids and fluid mechanics, Finite Elements in Analysis and Design 41: 703-728.

[7] Bochev P.B., Gunzburger M.D., 2009, Least-squares Finite Element Methods, Springer, New York.

[8] Jiang B.-N., 1998, The least-squares Finite Element Method, Springer, New York.

[9] Jou J., Yang S.-Y., 2000, Least-squares finite element approximations to the Timoshenko beam problem, Applied Mathematics and Computation 115: 63-75.

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