Extensibility Effects on Euler Elastica’s Stability

Abstract

Based on the conjugate point theory in calculus of variations, the extensibility effects on the stability of Euler elasticas with one clamped end and the other clamped in rotation in the post-buckling are investigated. For a slender rod, it is shown that: (1) the buckling load is a little bigger than Euler critical load, (2) the addition of extensibility to the elastica does not affect its stability in the post-buckling, in the sense that those Euler elasticas with one inflexion point are stable while those Euler elasticas with more than one inflexion point are unstable.

Appendix

Suppose $u'' = \frac{d^{2}u}{dx^{2}}$ and g(x) is a given function, then the exact solution of the initial value problem

$$ \left \{ \begin{array}{l} u'' + g ( x ) u = 0, \\[6pt] u ( 0 ) = 0,\qquad u' ( 0 ) = 1, \end{array} \right . $$

(30)

can be expressed analytically, see the book by Kamke [11],

$$ u = \rho ( x )\sin\alpha ( x ), $$

(31)

where α is a solution of the initial value problem

$$ \left \{ \begin{array}{l} \alpha' = \cos^{2}\alpha + g\sin^{2}\alpha, \\ \alpha(0) = 0, \\ \end{array} \right . $$

(32)

where $\alpha' = \frac{d\alpha}{dx}. \rho( x )$ is

(33)

In Eqs. (31), (32) and (33), we can see that the zero point of u(x) in Eq. (30) corresponds to α=0,±π,±2π,… .