3D beam-column finite elements under tri-axial stess-strain states: non-uniform shear stress distribution and warping

In many engineering structures, the effects of shear and torsional loads are an important aspect of both the analysis and the design process. These effects are usually neglected in typical framed structures. However, in some relevant cases, such as bridges, shear walls or thin-walled frames, it is essential to account for the shear and torsional loads and their interaction with the other loading conditions to correctly reproduce the structural response. In this framework, the main task is to accurately describe the nonlinear structural response in terms of global behavior and local stress-strain distributions, reproducing the coupling of the stress components and its influence on the global response. This results even more important in large scale structures made of cementitious and/or innovative composite materials, widely adopted in nowadays professional practice. Indeed, these structures usually show degrading mechanisms and softening behavior. Hence, they require sophisticated computational models and ad hoc analysis strategies to predict the structure capacity under severe loading conditions. A standard approach to analyze these structures is the adoption of beam-column finite element (FE) models, which are often preferred with respect to two-dimensional (2D) plate/shell or three-dimensional (3D) FEs, because of their efficiency and low computational cost. However, most beam-column FE formulations are based on theclassical Euler-Bernoulli or Timoshenko theory, assuming the cross-sections to remain plane during the loading process. This assumption requires specific corrective measures, when the shear and torsion and the related warping effects are pronounced. This work discusses the simulation of RC members with a 3D 2-node beam FE that includes warping effects. The FE formulation in [1] is extended to allow the description of structural members with softening material behavior. The governing equations are derived from a four-field Hu-Washizu variational principle, with independent interpolation of the warping displacement field from the rigid section displacements, the generalized section deformations and the material stress fields. In particular, the warping of the cross-section is described by interpolating the out-of-plane displacement with the addition of a variable number of local degrees of freedom to those commonly used for the beam FE. The global nonlinear response and the local distributions of strains and stresses are described introducing a fiber cross-section discretization. Hence, the coupling of axial, flexural, shear and torsional effects in terms of material response is automatically taken into account. Focusing on RC structures, the damaging mechanisms of the concrete material is described by adopting a new 3D nonlinear constitutive relationship with plasticity and damage. This is an enhanced version of that proposed in [2] and introduces the description of the unilateral effects typically appearing in concrete-like materials, due to the crack opening and closure. A Drucker-Prager type plastic model is coupled with a two-parameter isotropic damage model, where two scalar variables are used to describe the damage in tension and compression, respectively. The localization problems and the related mesh-dependency, due to the softening material behavior, are controlled through a regularization technique based on a properly modified nonlocal integral procedure. For beam-column FEs, the nonlocal strain measures are evaluated performing the integration of the local generalized section deformations along the element axis, whereas for 2D FEs the nonlocal integration is performed considering the generalized membrane/plate deformations. The proposed model is implemented and validated through some correlation studies. These consider the numerical analysis of a series of plain concrete and RC beams subjected to torsional loads and of two RC shear walls. The results are compared with experimental measurements and with those of standard FE beam models.