Abstract
This paper deals with the equilibrium problem of fully nonlinear beams in bending by extending the model for the anticlastic flexion of solids recently proposed by Lanzoni and Tarantino (2018) in the context of finite elasticity. In the first part of the paper it is shown, through a parametric analysis, that some geometrical parameters of the displacement field lose importance when slender beams are considered. Therefore, kinematics is reformulated and, subsequently, a fully nonlinear theory for the bending of slender beams is developed. In detail, no hypothesis of smallness is introduced for the deformation and displacement fields, the constitutive law is considered nonlinear and the equilibrium is imposed in the deformed configuration. Explicit formulas are obtained which describe the displacement fields of the inflexed beam, the stretches and the stresses for each point of the beam using both the Lagrangian and Eulerian descriptions. All these formulas are linearized by retrieving the classical formulae of the infinitesimal bending theory of beams. In the second part of the paper the theoretical results are compared with those provided by numerical and experimental analyses developed for the same equilibrium problem with the aim of justify the hypotheses underlying the theoretical model. The numerical model is based on the finite element (FE) method, whereas a test equipment prototype is designed and manufactured for the experimental analysis.
Highlights
4 linear analysis of this classic problem seems to escape to a convincing modelling
By varying the geometrical and constitutive parameters, it was shown by means of 116 a numerical analysis that these equilibrium equations are substantially satisfied at each point of the beam
The equilibrium problem of hyperelastic slender beams under 888 anticlastic bending has been investigated in the context of finite elasticity
Summary
4 linear analysis of this classic problem seems to escape to a convincing modelling. This is basically due to the nonlinearities involved in the problem formulation,. 36 Shield [9] studied the problem of the bending of a beam by assuming small strains but large displacements He retrieved the linear Lamb solution [10] for 38 the deflection of the middle surface of the beam. All the aforementioned works address the bending problem in a two-dimensional 42 context, systematically neglecting the pure deformation of the cross sections In this way, the modelling of the problem was substantially simplified, since the 44 displacement field is assumed to be plane. By varying the geometrical and constitutive parameters, it was shown by means of 116 a numerical analysis that these equilibrium equations are substantially satisfied at each point of the beam This preliminary study shows that, as 118 the beam becomes more and more slender, some geometrical quantities of the displacement field lose their physical meaning.
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