Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 178 Focussed on: Computational Mechanics and Mechanics of Materials in Italy A 3D mixed frame element with multi-axial coupling for thin-walled structures with damage D. Addessi University of Rome ‘Sapienza’, Department of Structural and Geothecnical Engineering daniela.addessi@uniroma1.it P. Di Re University of Rome ‘Sapienza’, Department of Structural and Geothecnical Engineering paolo.dire@uniroma1.it A BSTRACT . A 3D mixed beam finite element is presented, modeling the warping of the cross-sections as an independent kinematic field. The beam formulation is derived on the basis of the Hu-Washizu variational principle, expressed as function of four independent fields: the standard displacements, strains and stresses and the additional warping displacement. This is interpolated along the beam axis and on the cross-section, by placing on it a regular grid of interpolation points and adopting Lagrange polynomials. The warping degrees of freedom defined at the cross-section interpolation points are condensed, thus preserving the element matrix and vector sizes. A fiber discretization of the cross-sections is adopted. The constitutive relationship at the midpoint of each fiber is based on an isotropic damage model for brittle-like materials, distinguishing between the damage variables in tension and in compression to properly describe the unilateral effect. An efficient algorithm is formulated for the element state determination, based on a consistent linearization of the governing equations. A simple numerical application on a cantilever beam with torsion in the linear elastic range is presented and two torsion tests on plain concrete beams are performed, by comparing the numerical results with the experimental outcomes. K EYWORDS . Thin-walled structures; Mixed beam formulation; Warping; Damage; Softening; Regularization. I NTRODUCTION he development of accurate and efficient finite element (FE) codes, based on enhanced beam formulations, is a significant challenge in many engineering fields and, in particular, in structural engineering. In today's professional structural applications it is often required to analyze large scale structures with irregular geometry, made from innovative composite materials, under severe loading conditions, especially in high seismicity areas. Hence, to accurately describe the global nonlinear structural response, as well as the local distribution of stresses and damaging paths, it is of great interest to formulate enhanced beam FEs, taking into account nonlinear geometric and constitutive behavior. To this end, it was widely demonstrated that the fiber approach is an efficient tool for introducing sufficiently general 3D nonlinear constitutive relationships, including plasticity, damage, crushing and so on [1, 2]. Moreover, the multi-axial coupling between the beam stress resultants is described. T