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

Abstract

a b s t r a c tThe present paper considers a non-prismatic beam i.e., a beam with a cross-section varying along the beam axis. In particular, we derive and discuss a model of a 2D linear-elastic non-prismatic beam and the corresponding finite element. To derive the beam model, we use the so-called dimensional reduction approach: from a suitable weak formulation of the 2D linear elastic problem, we introduce a variable cross-section approximation and perform a cross-section integration. The satisfaction of the boundary equilibrium on lateral surfaces is crucial in determining the model accuracy since it leads to consider correct stress-distribution and coupling terms (i.e., equation terms that allow to model the interaction between axial-stretch and bending). Therefore, we assume as a starting point the Hellinger-Reissner functional in a formulation that privileges the satisfaction of equilibrium equations and we use a cross-section approximation that exactly enforces the boundary equilibrium.The obtained beam-model is governed by linear Ordinary Differential Equations (ODEs) with non-constant coefficients for which an analytical solution cannot be found, in general. As a consequence, starting from the beam model, we develop the corresponding beam finite element approximation. Numerical results show that the proposed beam model and the corresponding finite element are capable to correctly predict displacement and stress distributions in non-trivial cases like tapered and arch-shaped beams.

Planar Timoshenko-like model for multilayer non-prismatic beams

“…In contrary, the interlayer equilibrium (12) indicates a discontinuous crosssection distribution of axial as well as shear stresses. Furthermore, generalizing the results already discussed by Auricchio et al (2015) and Balduzzi et al (2016), the horizontal stress r x could be seen as the independent variable that completely defines the stress state on the interlayer surfaces. Finally, generalizing the results discussed by Boley (1963) and Hodges et al (2008Hodges et al ( , 2010, the shear stress jumps within the crosssection depend on the variation of the mechanical properties of the material-determining the jumps of horizontal stress-and on the slopes of the interlayer surfaces h 0 i x ð Þ.…”

“…To the author's knowledge, the most enhanced modeling approaches that seem capable to overcome all the so far discussed limitations have been presented by Rubin (1999), Hodges et al (2008Hodges et al ( , 2010, Auricchio et al (2015), Beltempo et al (2015a), and Balduzzi et al (2016). In greater detail, Rubin (1999), Hodges et al (2008Hodges et al ( , 2010 limit their investigations to planar tapered beams whereas Auricchio et al (2015), Beltempo et al (2015a), and Balduzzi et al (2016) consider more complex geometries.…”

“…In greater detail, Rubin (1999), Hodges et al (2008Hodges et al ( , 2010 limit their investigations to planar tapered beams whereas Auricchio et al (2015), Beltempo et al (2015a), and Balduzzi et al (2016) consider more complex geometries. On the one hand, the beam model proposed by Rubin (1999) seems to achieve the best compromise between simplicity and effectiveness.…”

“…On the one hand, the beam model proposed by Rubin (1999) seems to achieve the best compromise between simplicity and effectiveness. On the other hand, both the derivation procedure and the resulting models proposed by Auricchio et al (2015) and Beltempo et al (2015a) seem sometimes scarcely manageable and computationally expensive. Finally, Balduzzi et al (2016) propose a simple and effective modeling approach capable to describe the behavior of a large class of nonprismatic homogeneous beam bodies using the independent variables usually adopted in prismatic Timoshenko beam models.…”

“…Assuming also that all the stress components vanish outside the beam domain X, Equation (12) recovers also the stress constraints coming from boundary equilibrium (7d). 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).…”