Issue 29

D. Addessi et al., Frattura ed Integrità Strutturale, 29 (2014) 178-195; DOI: 10.3221/IGF-ESIS.29.16 179 The classical beam formulations, standing on Euler-Bernoulli or Timoshenko theory, assume that beam sections remain plane during the loading process. Although this results in very simple and effective formulations for describing the coupling between axial force and biaxial bending moment, an enriched kinematic description at the section level is required to include shear and torsion [3, 4]. It is worthwhile noting that the models proposed in literature, although including variable shear distributions and section warping, usually ignore the interaction between the beam element sections. Hence, these fail in describing the local response near the boundaries and the shear lag phenomenon, widely observed and discussed in literature [5, 6], which significantly influence the normal stress distributions on the cross- sections. The adoption of these formulations can lead to an incorrect underestimated evaluation of the stress values under warping constraints. In this work a 2-node 3D beam element, derived on the basis of the Hu-Washizu variational potential, is presented. A four-field mixed formulation is adopted, assuming as independent fields the standard displacements, the stresses, the strains and the warping displacements. Indeed, to take into account the warping of the cross-sections related to shear and torsion, an enriched kinematic description is introduced, as proposed in [7]. The warping field is interpolated at two levels: along the element axis and at each section, where a variable number of warping degrees of freedom (DOFs) is defined. In literature, the adoption of interpolation functions derived, analytically or numerically, by solving the linear elastic problems for specific cross-section geometries under torsion was proposed. Herein, Lagrange polynomials are adopted both for the interpolation along the axis and for describing the warping profile on the section. These result more general to describe inelastic material response and suitable for performing the required numerical integration on the cross-section area. Aiming at reproducing the degrading mechanisms typical of many engineering materials, and in particular the brittle-like materials, a nonlinear stress-strain relationship with damage is introduced [8]. Two distinct scalar damage variables are defined, reproducing the damaging mechanisms in tension and compression, and evolving according to laws defined in terms of an equivalent strain measure. The irreversibility of the two damaging processes is assumed. Furthermore, the unilateral effect is taken into account assuming the tensile damage greater than the compressive one, during the overall loading process. A fiber discretization of the cross-sections is adopted. Hence, the stress-strain constitutive relationship is defined at the material point level and integrated over the cross-section to derive the generalized stress measures. The FE formulation is defined in the basic system, i.e. in the reference system where element rigid body motions are removed, and under the assumption of small strains. The proposed beam element is implemented in the FE program FEAP [10], which is used to perform the numerical analyses. The beam FE is validated by performing a simple test on a cantilever beam in the linear elastic range, and two applications on plain concrete beams with damage experimentally analyzed, by comparing the numerically obtained results with the experimental outcomes. B EAM F ORMULATION 3D beam FE formulation, based on the Hu-Washizu mixed variational principle, including material nonlinear behavior, is introduced. Warping effects of the cross-sections are accounted for, by introducing independent DOFs. Small displacement and strain assumption holds. In Fig. 1 the FE reference configuration in the global reference system ( O , X , Y , Z ) is shown. The local coordinate system ( I , x , y , z ) is also defined (Fig. 2). The nodal displacement vector U is defined as: {             }  U U Φ U Φ T T T T T I I J J (1) where U I/J , Φ I/J are the global translation and rotation vectors at node I / J (Fig. 1). Similarly, the nodal force vector results as: {     }  P P M P M T T T T T I I J J (2) with P I/J , M I/J being the force and moment vectors at node I / J . By restraining the element nodal displacements, so as to remove all possible rigid body motions, the basic displacement vector D is introduced as: , , , , , , {    } T x J z I z J x J y I y J U  D Φ Φ Φ Φ Φ (3) A