Explicit analysis of large transformation of a Timoshenko beam: post-buckling solution, bifurcation, and catastrophes

This paper exposes full analytical solutions of a plane, quasi-static but large transformation of a
Timoshenko beam. The problem is first re-formulated in the form of a Cauchy initial value problem where
load (force and moment) is prescribed at one end and kinematics (translation, rotation) at the other one. With
such formalism, solutions are explicit for any load and existence, and unicity and regularity of the solution
of the problem are proven. Therefore, analytical post-buckling solutions were found with different regimes
driven explicitly by two invariants of the problem. The paper presents how these solutions of a Cauchy initial
value problem may help tackle (i) boundary problems, where physical quantities (of load, position or section
orientation) are prescribed at both ends and (ii) problems of quasi-static instabilities. In particular, several
problems of bifurcation are explicitly formulated in case of buckling or catastrophe.