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.

本文公开了 Timoshenko 梁的平面、准静态但大变换的完整解析解。该问题首先以柯西初值问题的形式重新表述，其中在一端规定载荷（力和力矩），另一端规定运动学（平移、旋转）。有了这种形式主义，对于任何负载和存在，解都是明确的，并且证明了问题解的唯一性和规律性。因此，在由问题的两个不变量明确驱动的不同机制中找到了分析后屈曲解。本文介绍了柯西初值问题的这些解决方案如何帮助解决 (i) 边界问题，其中物理量（载荷、位置或截面方向）在两端规定，以及 (ii) 准静态不稳定性问题。特别是，