Tech Science Press, CMES, vol.57, no.2, pp.175-204, 2010

Locking-free Thick-Thin Rod/Beam Element Based on a von Karman Type Nonlinear Theory in Rotated Reference Frames For Large Deformation Analyses of Space-Frame Structures

Author(s): H.H. Zhu, Y.C. Cai, J.K. Paik and S.N. Atluri

Abstract :
This paper presents a new shear flexible beam/rod element for large deformation analyses of space-frame structures, comprising of thin or thick beams. The formulations remain uniformly valid for thick or thin beams, without using any numerical expediencies such as selective reduced integrations, etc. A von Karman type nonlinear theory of deformation is employed in the co-rotational reference frame of the present beam element, to account for bending, stretching, torsion and shearing of each element. Transverse shear strains in two independant directions are introduced as additional variables, in order to eliminate the shear locking phenomenon. An assumed displacement approach is used to derive an explicit expression for the (16x16) symmetric tangent stiffness matrix of the beam element in the co-rotational reference frame. Numerical examples demonstrate that the present element is free from shear locking and is suitable for the large deformation analysis of spaced frames with thick/thin members. Significantly, this paper provides a text-book example of an explicit expression for the (16x16) symmetric tangent stiffness matrix of a finitely deforming beam element, which can be employed in very simple analyses of large deformations of space-frames. The present methodologies can be extended to study the very large deformations of plates and shells as well, and can be shown to be theoretically valid for thick or thin plates and shells, without using selective reduced integrations and without the need for stabilizing the attendant zero-energy modes.