A new modeling approach for planar beams: finite-element solutions based on mixed variational derivations

Abstract

This paper illustrates a new modeling approach for planar linear elastic beams. Referring to existing models, we first introduce the variational principles that could be adopted for the beam model derivation, discussing their relative advantages and disadvantages. Then, starting from the Hellinger-Reissner functional we derive some homogeneous and multilayered beam models, discussing some properties of their analytical solutions. Finally, we develop a planar beam finite element, following an innovative approach that could be seen as the imposition of equilibrium in the cross-section and compatibility along the axis. The homogeneous model is capable of reproducing the behavior of the Timoshenko beam, with the advantage that the shear correction factor appears naturally from the variational derivation; the multilayered beam is capable of capturing the local effects produced by boundary constraints and load distributions; the finite element is capable of predicting the cross-section stress distribution with high accuracy, and more generally the behavior of planar structural elements.

The dimensional reduction approach for 2D non-prismatic beam modelling: A solution based on Hellinger–Reissner principle

“…With respect to prismatic 2D beams (i.e., beams with constant cross-section and straight axis), Auricchio et al (2010) have presented a modeling-approach, based on the dimensional reduction, and a suitable FE approximation of the beam model. The dimensional reduction is a general mathematical procedure, initially proposed by Kantorovich and Krylov (1958), that exploits the domain geometry to reduce the problem dimension (in planar beam modeling from 2D Partial Differential Equations (PDEs) to a system of Ordinary Differential Equations (ODEs)).…”

“…This paper generalizes the procedure proposed by Auricchio et al (2010) to a non-prismatic, homogeneous, linear-elastic planar beam, with the aim to overcome the modeling limitations previously highlighted. In particular, we exploit the capability of the modeling approach described in Auricchio et al (2010) to accurately capture the cross-section stress distribution.…”

Planar Timoshenko-like model for multilayer non-prismatic beams

“…Then, the same relations as described in Auricchio et al (2015) and Balduzzi et al (2016) are obtained. Considering a multilayer prismatic beam, both standard and advanced literature states that the horizontal stress has a discontinuous distribution within the cross-section, in case of different mechanical properties between the layers, whereas the shear stress has a continuous distribution (Bareisis 2006;Auricchio et al 2010;Bardella and Tonelli 2012). In contrary, the interlayer equilibrium (12) indicates a discontinuous crosssection distribution of axial as well as shear stresses.…”

Enhanced modeling approach for multilayer anisotropic plates based on dimension reduction method and Hellinger–Reissner principle

“…In the present paper, we generalize the modeling approach presented by Auricchio et al [43] to multilayer, anisotropic plates. In Section 2, thickness shape functions with arbitrary coefficients are adopted for both displacement and stress field and, then, plate-theory partial differential equations are derived starting from the HR dual formulation.…”

“…Auricchio et al [43] introduce a new modeling approach for planar linear elastic beams based on HR principle. Specifically, they specialize the approach suggested by Alessandrini et al [35] to 2D problems and develop a planar beam multilayer models based on dimension reduction approach and HR principle.…”