 Solution

ID: past-exam-of-the-elasticity-and-plasticity-course-of-the-university-of-oxford/2008/3/b/solution

ξ = s / L, λ = P L^{2} / (π^{2} E I)

. The endpoint interval becomes

[0, 1]

, and multiplication of the dimensional equation by

L^{2} / E I

gives

θ_{ξξ} + π^{2} λ sin θ = 0, θ (0) = θ (1) = 0.

The energy in these variables is

U = (E I / L) \int_{0}^{1} [θ_{ξ}^{2} /2 - π^{2} λ (1 - cos θ)] d ξ

. Thus

λ

measures load relative to the first linearized eigenvalue for the angular endpoint conditions actually imposed.

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