The Bending of Beams in Finite Elasticity

Mechanics of High-Flexible Beams Under Live Loads

“…The beam model proposed in [27] for the uniform inflexion was then developed to study the general case with variable curvature [1]. Such generalization allowed to derive a completely nonlinear moment-curvature relationship for the beam axis.…”

“…This relationship has a fairly complex expression. However, for the class of slender beams, it has been shown in [1] that, for a wide range of curvatures, the relationship can be simplified by assuming a law of proportionality between the external bending moment and the curvature itself. This linearity is lost due to high curvatures, such as when the beam is inflexed until to form a tight loop, or when particular constitutive laws, characterized by a strong nonlinear behavior already for low stress values, are adopted.…”

“…In the conditions specified above, the axis of an inflexed beam in the context of the general theory is bent according to a law very similar to that of Elastica. Thus, although the structural models considered are different, the study carried out in [1] constitutes an indirect demonstration of validity of the celebrated hypothesis of Jacob Bernoulli [2]. Furthermore, the applicability limits of this hypothesis are also established.…”

“…The analysis is based on the recent model proposed by L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity. [1]. In this model the complete three-dimensional kinematics of the beam is taken into account, both deformations and displacements are considered large and a nonlinear constitutive law in assumed.…”

“…External static loads dependent on deformation (live loads) are also considered. The governing equations are derived on the basis of a moment-curvature relationship obtained in [1]. The governing equations take the form of a highly nonlinear coupled system of equations in integral form, which is solved through an iterative numerical procedure.…”