The bending of fully nonlinear beams. Theoretical, numerical and experimental analyses

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.