A KINEMATICALLY EXACT SPACE FINITE STRAIN BEAM MODEL - FINITE-ELEMENTFORMULATION BY GENERALIZED VIRTUAL WORK PRINCIPLE

The present paper presents a novel finite element formulation for stat ic analysis of linear elastic spatial frame structures extending the f ormulation given by Simo and Vu-Quoc [A geometrically-exact rod model incorporating shear and torsion-warping deformation, Int. J. Solids St ructures 27 (3) (1991) 371-393], along the lines of the work on the pl anar beam theory presented by Saje [A variational principle for finite planar deformation of straight slender elastic beams, Internat. J. So lids and Structures 26 (1990) 887-900]. We apply exact non-linear kine matic relationships of the space finite-strain beam theory, assuming t he Bernoulli hypothesis and neglecting the warping deformations of the cross-section. Finite displacements and rotations as well as finite e xtensional, shear, torsional and bending strains are accounted for in the formulation. A deformed configuration of the beam is described by the displacement vector of the deformed centroid axis and an orthonorm al moving frame, rigidly attached to the cross-section of the beam. Th e position of the moving frame relative to a fixed reference frame is specified by an orthogonal matrix, parametrized by the rotational vect or which rotates the moving frame from an arbitrary position into the deformed configuration in one step. Also, the incremental rotational v ector is introduced, which rotates the moving frame from the configura tion obtained at the previous iteration step into the current configur ation of the beam. Its components relative to the fixed global coordin ate system are taken to be the rotational degrees of freedom at nodal points. Because in 3-D space both the axial and the follower moments a re non-conservative, not the variational principle but the principle o f virtual work has been introduced as a basis for the finite element d iscretization. Here we have proposed the generalized form of the princ iple of virtual work by including exact kinematic equations by means o f a procedure. similar to that of Lagrangian multipliers. This makes p ossible the elimination of the displacement vector field from the prin ciple, so that the three components of the incremental rotational vect or field remain the only functions to be approximated in the finite el ement implementation of the principle. Other researchers, on the other hand, employ the three components of the incremental rotational vecto r field and the three components of the incremental displacement vecto r field. As a result, more accurate and efficient family of beam finit e elements for the non-linear analysis of space frames has been obtain ed. A one-field formulation results in the fact that in the present fi nite elements the locking never occurs. Any combination of deformation states is described equally precisely. This is in contrast with the e lements developed in literature, where, in order to avoid the locking, a reduced numerical integration has to be applied, which unfortunatel y, diminishes the accuracy of the solution. Polynomials have been chos en for the approximation of the components of the rotational vector. I n this case the order of the numerical integration can rationally be e stimated and the computer program can be coded in such a way that the degree of polynomials need not be limited to a particular value. The N ewton method is used for the iterative solution of the non-linear equi librium equations. In an non-equilibrium configuration, the tangent st iffness matrix, obtained by the linearization of governing equations u sing the directional derivative, is non-symmetric even for conservativ e loadings. Only upon achieving an equilibrium state, the tangent stif fness matrix becomes symmetric. Thus, obtained tangent stiffness matri x can be symmetrized without affecting the rate of convergence of the Newton method. For non-conservative loadings, however, the tangent sti ffness matrix is always non-symmetric. The numerical examples demonstr ate capability of the present formulation to determine accurately the non-linear behaviour of space frames. In numerical examples the out-of -plane buckling loads are determined and the whole pre-and post-critic al load-displacement paths of a cantilever and a right-angle frame are traced. These, in the analysis of space beams standard verification e xample problems, show excellent accuracy of the solution even when emp loying only one element to describe the displacements of the size of t he structure itself, the rotations of 2 pi, and extensional strains mu ch beyond the realistic values of linear elastic material.