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Fixed-Pole Concept in 3D Beam Finite Elements – Relationship to Standard Approaches and Analysis of Different Interpolations

A family of geometrically exact spatial beam finite elements based on the fixed-pole approach is developed and presented in this thesis. All the proposed elements are derived from the principle of virtual work by interpolating the virtual spins (or virtual configurational spins). The fixed-pole element is derived from the configuration-tensor approach. By utilising the standard Galerkin approach this element turns out to have non-standard six-dimensional system unknowns. In order to reap the benefits of the fixed-pole approach, but still be able to have standard system unknowns, a modification of the fixed-pole approach is proposed: the non- standard and standard virtual quantities are related at the nodal level. This results in a family of modified-fixed pole elements, consisting of three interpolation options. The fixed-pole and the modified fixed-pole formulations are tested with respect to their accuracy, strain-invariance, path-independence and robustness on a number of planar and spatial numerical examples. The results show that non-invariance and path-dependence is present in all formulations (in some formulations even in the planar case). In the case of the modified-fixed pole formulation, these problems are overcome by (i) interpolating only the relative rotations between two nodes and (ii) implementing the generalised shape-functions. The repeated numerical tests show that results are strain- invariant, path-independent and that the solution procedure is more robust. Although the results obtained by employing Interpolation option 3 are somewhat improved in comparison to those before this intervention, they still exhibit some non-invariant and path-dependent behaviour. In the case of the fixed-pole formulation, strain-invariance and path-independence is achieved by (i) interpolating only the relative configurations between two nodes and (ii) by deriving and implementing six-dimensional shape functions. The results of the repeated numerical tests confirm that this intervention leads to strain-invariant, path-independent and more robust elements.

geometrically exact beam theory; fixed-pole approach; strain invariance; path dependence

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