Abstract
In this paper the mathematical formulation of the equilibrium problem of high-flexible beams in the framework of fully nonlinear structural mechanics is presented. The analysis is based on the recent model proposed by L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. 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. After having illustrated and discussed the peculiar mechanical aspects of this special class of structures, the criteria and methods of analysis have been addressed. A classification of the structures based on the degree of kinematic constraints has been proposed, distinguishing between isogeometric and hypergeometric structures. 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 L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. 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. Finally, the proposed analysis is applied to some popular structural systems subjected to dead and live loads. The results are compared and discussed.
Highlights
During the period from the end of the seventeenth century to the second half of the eighteenth century the theory of elastic curves was one of the most studied themes, sometimes not without controversy and rivalry, by the major exponents of mathematical and mechanical culture.As a result of its important developments in the analysis of elastic structures, the determination of the curvature of an inflexed beam was undoubtedly the topic of principal interest
A distributed live load is characterized by a density per unit length of the beam in the deformed configuration g : L × V × U × U → V, which in general depends on the curvilinear abscissa s, on the deformation function f (s) and on its derivatives f (1)(s) and f (2)(s)
The analysis is based on a recent solution for the bending of nonlinear beams obtained by Lanzoni and Tarantino [1] in the context of finite elasticity
Summary
During the period from the end of the seventeenth century to the second half of the eighteenth century the theory of elastic curves was one of the most studied themes, sometimes not without controversy and rivalry, by the major exponents of mathematical and mechanical culture. The mathematical formulation of the Elastica model is based on three fundamental hypotheses: The displacements are considered large, the deformations small and the classic constitutive law of the elastic linear theory is taken into account. In these circumstances, the nonlinearities of the governing equation are due solely to large displacements, which require to impose the equilibrium conditions with respect to the deformed configuration.
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