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This book presents the bending theory of hyperelastic beams in the context of finite elasticity. The main difficulties in addressing this issue are due to its fully nonlinear framework, which makes no assumptions regarding the size of the deformation and displacement fields. Despite the complexity of its mathematical formulation, the inflexion problem of nonlinear beams is frequently used in practice, and has numerous applications in the industrial, mechanical and civil sectors. Adopting a semi-inverse approach, the book formulates a three-dimensional kinematic model in which the longitudinal bending is accompanied by the transversal deformation of cross-sections. The results provided by the theoretical model are subsequently compared with those of numerical and experimental analyses. The numerical analysis is based on the finite element method (FEM), whereas a test equipment prototype was designed and fabricated for the experimental analysis. The experimental data was acquired using digital image correlation (DIC) instrumentation. These two further analyses serve to confirm the hypotheses underlying the theoretical model. In the book’s closing section, the analysis is generalized to the case of variable bending moment. The governing equations then take the form of a coupled system of three equations in integral form, which can be applied to a very wide class of equilibrium problems for nonlinear beams.

### Chapter 1. Theoretical Analysis

Abstract
This chapter deals with the equilibrium problem of fully nonlinear beams in bending by extending the model for the anticlastic flexion of solids recently proposed in the context of finite elasticity by Lanzoni and Tarantino (J Elast 131:137–170, 2018, ). Initially, kinematics is reformulated and, subsequently, a nonlinear theory for the bending of slender beams has been 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 field, stretches and stresses for each point of the beam using both the Lagrangian and Eulerian descriptions.
Angelo Marcello Tarantino, Luca Lanzoni, Federico Oyedeji Falope

### Chapter 2. Numerical and Experimental Analyses

Abstract
The results provided by the theoretical model proposed in Chap. 1 for the bending of fully nonlinear beams are compared with those given by the numerical and experimental analyses developed in the present Chapter. The numerical model is based on the finite element method (FEM), whereas a test equipment prototype has been designed and manufactured for the experimental analysis. The experimental data have been acquired using the digital image correlation (DIC) instrumentation. The fundamental purpose of these two further analyses for the large bending of slender beams is to justify the hypotheses underlying the theoretical model.
Angelo Marcello Tarantino, Luca Lanzoni, Federico Oyedeji Falope

### Chapter 3. Generalization to Variable Bending Moment

Abstract
In this third Chapter, the theoretical model proposed in Chap. 1 for the bending of fully nonlinear beams is generalized to the case of variable bending moment. Such a generalization focuses on the local determination of curvature and bending moment along the deformed beam axis. Once the moment-curvature relationship has been derived, the equilibrium problem for nonlinear beams subjected to variable bending moment has been formulated. The governing equations assume the form of a coupled system of three equations in integral form. To solve this highly nonlinear system, an iterative numerical procedure has been proposed. Definitively, the analysis developed in this Chapter allows considering a very wide class of equilibrium problems for nonlinear beams. By way of example, the Euler beam and a cantilever beam loaded by a concentrated force of the dead or live (follower) type, applied in its free end, has been studied, showing the shape assumed by the deformed beam axis as the load multiplier increases.
Angelo Marcello Tarantino, Luca Lanzoni, Federico Oyedeji Falope