Similar articles

Abstract: A tensor-based updated Lagrangian (UL) formulation for the geometrically nonlinear analysis of 2D beam-column structures is developed by using curvilinear coordinates, which has considered the effects of the deformed curvature. Between the known configuration C₁ and the desired configuration C₂, a configuration C₂^* derived by rigid-body motion of C₁ is introduced to eliminate the element-end transverse displacements between C₂^* and C₂. A stiffness matrix is obtained in C₂^*; and then by a transformation defined by the element-end displacements, the stiffness matrix in C₂^* is transformed into that in C₁. Comparing the stiffness matrix with that in the conventional UL formulation for a 2D beam element, the initial displacement stiffness matrix emerges, which results from the deformed curvature within the element. Numerical examples have verified the accuracy and efficiency of the present formulation, and the results show that the deformed curvatures have significant effects when deformations are large.

[1] Bathe, K.J., 1996. Finite Element Procedures. Englewood Cliffs. Prentice Hall, NJ.

[2] Bathe, K.J., Bolourchi, S., 1979. Large displacement analysis of three-dimensional beam structures. International Journal for Numerical Methods in Engineering, 14(7):961-986.

[3] Crisfield, M.A., 1991. Non-linear Finite Element Analysis of Solids and Structures, vol. 1: Essentials. Wiley, Chichester.

[4] Gjelsvik, A., Bodner, S.R., 1962. The energy criterion and snap buckling of arches. Journal of Engineering Mechanics Division, 88(5):87-134.

[5] Hsiao, K.M., Hou, F.Y., 1987. Nonlinear finite element analysis of elastic frames. Computers and Structures, 26(4):693-701.

[6] Hsiao, K.M., Lin, J.Y., Lin, W.Y., 1999. A consistent co-rotational finite element formulation for geometrically nonlinear dynamic analysis of 3-D beams. Computer Methods in Applied Mechanics and Engineering, 169(1-2):1-18.

[7] Izzuddin, B.A., 1996. Quartic formulation for elastic beam-columns subject to thermal effects. Journal of Engineering Mechanics, 122(9):861-871.

[8] Izzuddin, B.A., 2001. Conceptual issues in geometrically nonlinear analysis of 3D framed structures. Computer Methods in Applied Mechanics and Engineering, 191(8-10):1029-1053.

[9] Meek, J.L., Hoon Swee, T., 1984. Geometrically nonlinear analysis of space frames by an incremental iterative technique. Computer Methods in Applied Mechanics and Engineering, 47(3):261-282.

[10] Meek, J.L., Xue, Q., 1996. A study on the instability problem for 2D-frames. Computer Methods in Applied Mechanics and Engineering, 136(3-4):347-361.

[11] Pi, Y.L., Trahair, N.S., 1998. Non-linear buckling and postbuckling of elastic arches. Engineering Structures, 20(7):571-579.

[12] Przemieniecki, J.S., 1968. Theory of Matrix Structural Analysis. McGraw-Hill, New York.

[13] Saleeb, A.F., Chang, T.Y.P., Gendy, A.S., 1992. Effective modelling of spatial buckling of beam assemblages, accounting for warping constraints and rotation-dependency of moments. International Journal for Numerical Methods in Engineering, 33(3):469-502.

[14] Stolarski, H., Belytschko, T., 1982. Membrane locking and reduced integration for curved elements. Journal of Applied Mechanics, 49(1):172-176.

[15] Teh, L.H., Clarke, M.J., 1998. Co-rotational and Lagrangian formulations for elastic three-dimensional beam finite elements. Journal of Constructional Steel Research, 48(2-3):123-144.

[16] Washizu, K., 1964. Some considerations on a naturally curved and twisted slender beam. Journal of Mathematics and Physics, 43(2):111-116.

[17] Washizu, K., 1982. Variational Methods in Elasticity and Plasticity. Pergamon Press Ltd.

[18] Williams, F.W., 1964. An approach to the non-linear behaviour of the members of a rigid jointed plane framework with finite deflections. The Quarterly Journal of Mechanics and Applied Mathematics, 17(4):451-469.

[19] Yang, Y.B., McGuire, W., 1986. Stiffness matrix for geometric nonlinear analysis. Journal of Structural Engineering, 112(4):853-877.

[20] Yang, Y.B., Yau, J.D., Leu, L.J., 2003. Recent developments in geometrically nonlinear and postbuckling analysis of framed structures. Applied Mechanics Reviews, 56(4):431-449.

[21] Yang, Y.B., Lin, S.P., Leu, L.J., 2007. Solution strategy and rigid element for nonlinear analysis of elastically structures based on updated Lagrangian formulation. Engineering Structures, 29(6):1189-1200.