Explicit analysis of large transformation of a Timoshenko beam : Post-buckling solution, bifurcation and catastrophes
Résumé
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, 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 (1) boundary problem, where physical quantities (of load, position or section orientation) are prescribed at both ends (2) problem of quasi-static instabilities. In particular several problems of bifurcation are explicitly formulated in case of buckling or catastrophe.
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