Keywords: three-dimensional beam finite element, finite rotations, vectorial kinematics, invariant measures.

Two vectorial finite element approaches to formulate models, for three-dimensional elastic beams, for the case of small strains in the large rotation regime are presented. These two approaches differ in the finite element local frame of reference where rigid and deformative modes are used. In both of these formulations, based on the total Lagrangian description, the use of the rotation parameters is bypassed. Complex manipulations required to obtain conservative descriptions and well-posed transformation matrices are avoided. In particular, slopes and distances are used instead of rotation parameters to compute the nonlinear representations of the strain measures in the inertial frame of reference.

In this context, classically, a co-rotational approach is used. The motion of the continuous medium is decomposed into a rigid body motion followed by a pure deformation. For this reason, finite element is studied in the linear case where, clearly drawbacks appear. Afterward, the nonlinear motion is obtained by joining the linear kinematic with a rigid body motion that is recovered by the use of orthogonal transformation matrices [2].

The interpolation of rotations to measure deformations, however, requires the use of incremental solution procedures when large rotations are considered. In effect, small rotation increments are hypothesized for the linearization of the configuration space. Consequently, small steps in the continuation process are allowed and a slow convergence is intrinsic to the formulation.

More recently non-incremental approaches have been developed where slopes are used instead of rotation parameters [3,4]. However, due to the use of a cross-sectional coordinate system, the rigid cross-section assumption is abandoned while the description of the elastic forces becomes more complex.

The aim of the proposed approaches is to avoid the use of rotation matrices while preserving the robustness and efficiency of the analysis. In our context, the current configuration of the beam element proves to be rigidly rotated and deformed according to the selected linear modes. The nonlinear rigid motion is recovered by referring to three unit and mutually orthogonal vectors. All nine components of such vectors are assumed as unknowns in the global inertial frame of reference.

In a first approach, an elemental reference formulation (ER) is followed. Rigid and deformative modes are referred to the central point of the element so that inter-elemental compatibility and constraint conditions are imposed by using Lagrange multipliers. A second one provides a compatible approach carried out by using a nodal finite element reference formulation (NR). Rigid and deformative modes are referred to the nodes of the element where we place the unknown vectors representative of the rigid rotations in the global framework. In comparison with the previous formulation, the inter-elemental compatibility is implicitly imposed.

1

T. Belytschko, B.J. Hsieh, "Non-linear transient finite element analysis with convected co-ordinates", Int. J. Numer. Meth. Engrg., 7, 255-271, 1973. doi:10.1002/nme.1620070304

2

M.A. Crisfield, "A consistent co-rotational formulation for nonlinear three-dimensional beam elements", Comput. Meth. Appl. Mech. Engrg., 81, 131-150, 1990. doi:10.1016/0045-7825(90)90106-V

3

K.E. Dufva, J.T. Sopanen, A.K. Mikkola, "Three-dimensional beam element based on a cross-sectional coordinate system approach", Nonlinear Dynamics, 43, 311-327, 2006. doi:10.1007/s11071-006-8326-7

4

J. Rhim, W. Lee, "A vectorial approach to computational modelling of beams undergoing finite rotations", Int. J. Numer. Meth. Engrg., 41, 527-540, 1998. doi:10.1002/(SICI)1097-0207(19980215)41:3<527::AID-NME297>3.3.CO;2-Z

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