Locking‐free finite elements for shear deformable orthotropic thin‐walled beams

Abstract

Numerical models for finite element analyses of assemblages of thin-walled open-section profiles are\ud presented. The assumed kinematical model is based on Timoshenko-Reissner theory so as to take shear\ud strain effects of non-uniform bending and torsion into account. Hence, strain elastic-energy coupling terms\ud arise between bending in the two principal planes and between bending and torsion. The adopted model\ud holds for both isotropic and orthotropic beams. Several displacement interpolation fields are compared\ud with the available numerical examples. In particular, some shape functions are obtained from ‘modified’\ud Hermitian polynomials that produce a locking-free Timoshenko beam element. Analogously, numerical\ud interpolation for torsional rotation and cross-section warping are proposed resorting to one Hermitian and\ud six Lagrangian formulation. Analyses of beams with mono-symmetric and non-symmetric cross-sections\ud are performed to verify convergence rate and accuracy of the proposed formulations, especially in the\ud presence of coupling terms due to shear deformations, pointing out the decay length of end effects. Profiles\ud made of both isotropic and fibre-reinforced plastic materials are considered. The presented beam models\ud are compared with results given by plate-shell models

Static analysis of Timoshenko beam resting on elastic half-plane based on the coupling of locking-free finite elements and boundary integral

“…This FE was also proposed in [26][27][28][29]. Furthermore, Friedman and Kosmatka [30], Kosmatka [31] use this FE to solve dynamic and buckling problems, whereas in [32,33] thin-walled beam problems are studied. Alternatively, beam element in which quadratic interpolation of vertical displacement and linear interpolation of rotation may be used [35].…”

Warping Transmission in 3-D Beam Element Including Secondary Torsional Moment Deformation Effect

“…The adopted procedure is free from the assumptions of TTT. The aforementioned three systems of equations may be studied independently due to the assumption of geometrically linear conditions and the doubly symmetric shape of the cross section [16]. It is also worth noting that the nodal load vector is also readily computed analytically for frequently encountered distributions of externally applied actions along the length of the element.…”

Bars under Torsional Loading: A Generalized Beam Theory Approach

“…In all of the aforementioned research efforts warping shear stresses are not taken into account, with the exceptions of Wunderlich et al [124], Gruttmann et al [131], Nie and Zhong [128] and Wang et al [129] who exploit a stress distribution arising from the introduction of an independent warping parameter in the displacement field of the bar. However, since this theory is analogous to the Timoshenko beam theory of shear-bending loading conditions, it does not satisfy local equilibrium equations under inelastic or even elastic conditions (for relevant discussions see e.g., Simo et al [132] and Minghini et al [133]). Finally, Sapountzakis and Tsipiras in [134] presented a boundary element solution for the inelastic nonuniform torsional problem of simply or multiply connected cylindrical bars of arbitrarily shaped doubly symmetric cross section taking into account the effect of warping shear stresses.…”

“…Since then, a significant amount of relevant contributions has appeared in the literature as well [109,140,[143][144][145][146]. Since the topic at hand is analogous to the geometrically nonlinear Timoshenko beam theory of shear-bending loading conditions [78,83], it does not satisfy local equilibrium equations (for relevant discussions see e.g., Simo et al [132] and Minghini et al [133]). This problem is alleviated by introducing torsional shear correction factor at the global level [78,84,85] and suitable warping shear stress distribution at the local level [80,85,147].…”