Correct Formulation of the Stability Problem for Timoshenko Beam

Abstract:

This article is about the nonlinear problems of the theory of elastic Cosserat – Timoshenko’s rods in the material (Lagrangian) description with energy conjugated vectors of forces, moments and strains. The variational formulations of static problems was given. The differential equations of the plane stability problems were obtained from the second variation of the Lagrangian functional. The exact solutions of the stability problems for basic types of the end fixities of the rod were obtained for the Timoshenko’s rod (taking into account only bending and shear stiffness). It appears that classical well-known equilibrium stability functional and stability equations for the Timoshenko’s rod are incorrect. Also well-known Engesser formula (with bending and shear stiffness) is incorrect. The numerical solution of the stability problems for hinged Timoshenko’s rod with rigid support was obtained. Also, simplified formula for this problem was derived using asymptotic analysis.

• Periodical: Applied Mechanics and Materials (Volumes 725-726)
• Pages: 854-862
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.725-726.854
• Online since: January 2015
• Authors: Vladimir Lalin, Daria Kushova*
• Keywords: Energy Conjugated Forces, Equations of Stability, Functional of Variational Formulation, Large Displacements, Large Rotation, Moments, Strain

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