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a b s t r a c tThis paper presents an analytical model for the study of 2D linear-elastic non-prismatic beams. Its principal aim is to accurately predict both displacements and stresses using a simple procedure and few unknown variables. The approach adopted for the model derivation is the so-called dimensional reduction starting from the Hellinger-Reissner functional, which has both displacements and stresses as independent variables. Furthermore, the Timoshenko beam kinematic and appropriate hypotheses on the stress field are considered in order to enforce the boundary equilibrium. The use of dimensional reduction allows the reduction of the integral over a 2D domain, associated with the HellingerReissner functional, into an integral over a 1D domain (i.e., the so-called beam-axis). Finally, through some mathematical manipulations, the six ordinary differential equations governing the beam structural behaviour are derived. In order to prove the capabilities of the proposed model, the solution of the six equations is obtained for several non-prismatic beams with different geometries, constraints, and load distributions. Then, this solution is compared with the results provided by an already existing, more expensive, and refined 2D finite element analysis, showing the efficiency of the proposed model to accurately predict both displacements and stresses, at least in cases of practical interest.

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.

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This paper aims at proposing a Timoshenkolike model for planar multilayer (i.e., non-homogeneous) non-prismatic beams. The main peculiarity of multilayer non-prismatic beams is a non-trivial stress distribution within the cross-section that, therefore, needs a more careful treatment. In greater detail, the axial stress distribution is similar to the one of prismatic beams and can be determined through homogenization whereas the shear distribution is completely different from prismatic beams and depends on all the internal forces. The problem of the representation of the shear stress distribution is overcame by an accurate procedure that is devised on the basis of the Jourawsky theory. The paper demonstrates that the proposed representation of cross-section stress distribution and the rigorous procedure adopted for the derivation of constitutive, equilibrium, and compatibility equations lead to Ordinary Differential Equations that couple the axial and the shear bending problems, but allow practitioners to calculate both analytical and numerical solutions for almost arbitrary beam geometries. Specifically, the numerical examples demonstrate that the proposed beam model is able to predict displacements, internal forces, and stresses very accurately and with moderate computational costs. This is also valid for highly heterogeneous beams characterized by thin and extremely stiff layers.

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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.