Large Deformation Analyses of Space-Frame Structures., Using Explicit Tangent Stiffness Matrices, Based on the Reissner variational principle and a von Karman Type Nonlinear Theory in Rotated Reference Frames

摘要

This paper presents a simple finite element method, based on assumed moments and rotations, for geometrically nonlinear large rotation analyses of space frames consisting of members of arbitrary cross-section. A von Karman type nonlinear theory of deformation is employed in the updated Lagrangian co-rotational reference frame of each beam element, to account for bending, stretching, and torsion of each element. The Reissner variational principle is used in the updated Lagrangian co-rotational reference frame, to derive an explicit expression for the (12×12) symmetric tangent stiffness matrix of the beam element in the co-rotational reference frame. The explicit expression for the finite rotation of the axes of the co-rotational reference frame, from the global Cartesian reference frame is derived from the finite displacement vectors of the 2 nodes of each finite element. Thus, the explicit expressions for the tangent stiffness matrix of each finite element of the beam, in the global Cartesian frame, can be seen to be derived as text-book examples of nonlinear analyses. When compared to the primal (displacement) approach wherein C~1 continuous trial functions (for transverse displacements) over each element are necessary, in the current approach the trial functions for the transverse bending moments and rotations are very simple, and can be assumed to be linear within each element. The present (12×12) symmetric tangent stiffness matrices of the beam, based on the Reissner variational principle and the von Karman type simplified rod theory, are much simpler than those of many others in the literature. The present approach does not involve such numerical procedures as selective reduced integration or suppression of attendant Kinematic modes. The present methodologies can be extended to study the very large deformations of plates and shells as well. Metal plasticity may also be included, through the method of plastic hinges, etc. This paper is a tribute to the collective genius of Theodore von Karman (1881-1963) and Eric Reissner (1913-1996).