A hypergeometric treatment to explain the nonlinear true behavior of redundant constraints on a straight elastic rod

TL;DR

This paper uses hypergeometric functions to precisely analyze elastic rods, revealing that reactions from redundant constraints depend on stiffness, challenging the traditional assumption of their independence.

Contribution

It provides an exact solution to elastica problems showing the true nonlinear behavior of constraints, which was previously approximated or misunderstood.

Findings

Reactions depend on flexural stiffness EJ

Traditional assumptions of independence are only first-order approximations

Exact solutions reveal nonlinear effects in elastic rods

Abstract

In theory and practice of elastic straight rods, the statically indeterminate reactions acted by perfect constraints are commonly believed not to depend on the flexural stiffness $EJ$. We solve exactly two elastica problems in order to obtain hypergeometrically (helped by Lagrange, Lauricella, Appell), the true displacements upon which the forces method is founded. As a consequence, the above reactions are found to depend on stiffness: the presumptive independence credited as general, is far from being always true, but, quite the contrary, is valid only within a first-order approximation.