Variational Principles of Extensible Elastica

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  • Kondo Kyohei
    防衛大学校システム工学群航空宇宙工学科

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  • 伸張エラスティカの変分原理
  • シンチョウ エラスティカ ノ ヘン ブン ゲンリ

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Abstract

An extensible elastica is a rigorous mathematical model of the Bernoulli-Euler beam whose cross-sections remain plane and normal to the axis after deformations. The principle of virtual work for the extensible elastica expressed in terms of the normal strain and rotation of the axis is derived from the principle of virtual work in the three-dimensional elasticity. And it is shown that the derived principle yields the exact equilibrium equations for a beam in the large deformations and rotations. Utilizing linear constitutive equations, we get the theorem of stationary potential energy expressed also in terms of the axial strain and rotation. And, from the Trefftz criterion on the second variation of the potential energy, we get the buckling equations for the extensible elastica, which give the buckling load higher than the Euler load for a cantilever elastica subjected to compressive end load.

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