Nonlinear Finite Element Analysis of Bending of Straight Beams Using hp-Spectral Approximations

Document Type: Research Paper


SiViRT Center, University of Texas, San Antonio Department of Mechanical Engineering San Antonio


Displacement finite element models of various beam theories have been developed using traditional finite element interpolations (i.e., Hermite cubic or equi-spaced Lagrange functions). Various finite element models of beams differ from each other in the choice of the interpolation functions used for the transverse deflection w, total rotation φ and/or shear strain γxz, or in the integral form used (e.g., weak form or least-squares) to develop the finite element model. The present study is concerned with the development of alternative beam finite elements using hp-spectral nodal expansions to eliminate shear and membrane locking. Both linear and non-linear analysis are carried out using both displacement and mixed finite element models of the beam theories studied. Results obtained are compared with both analytical (series) solutions and non-linear finite element solutions from literature, and excellent agreement is found for all cases.


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

[2] Severn R.T., 1970, Inclusion of shear deflection in the stiffness matrix for a beam element, Journal of Strain Analysis 5: 239-241.

[3] 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.

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

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

[6] Karniadakis G.K., Sherwin, S., 2004, Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford Science Publications, London.

[7] Bar-Yoseph P.Z., Fisher D., Gottlieb O., 1996, Spectral element methods for nonlinear spatio-temporal dynamics of Euler-Bernoulli beam, Computational Mechanics 19: 136-151.

[8] 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.

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

[10] Cook R.D., Malkus D.S., Plesha M.E., Witt R.J., 2002, Concepts and Applications of Finite Element Analysis, John Wiley and Sons Inc., NY.

[11] Edem I.B. 2006, The exact two-node Timoshenko beam finite element using analytical bending and shear rotation interdependent shape functions, International Journal for Computer Methods in Engineering Science and Mechanics 7: 425-431.

[12] 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.

[13] Reddy J.N., 2002, An Introduction to Finite Element Method, Mc.Graw Hill, NY.

[14] Osilenker B., 1999, Fourier Series in Orthogonal Polynomials, World Scientific.

[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] Reddy J.N., 2007, Non local theories for bending, buckling, and vibration of beams, International Journal of Engineering Science 45: 288-307.

[17] Reddy J.N., 1999, Theory and Analysis of Elastic Plates, Taylor and Francis, London.