Structural Engineering & Statics
where the load goes
Series

Stability energy — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Four guesses at one buckling mode. A pin-ended column, with four assumed shapes and the load each of them gives. The reference is a ten-term Ritz expansion solved as an eigenvalue problem, at 9.8696 EI/L² — which is π², as it must be. a half sine gives 9.870, its own sag shape gives 9.882, a mid-span sag gives 10.000, a parabola gives 12.000. Every one of them is high and none of them is low, because an assumed shape is a constraint on the column and a constraint can only stiffen it.

    Guessing the shape, and getting the load anyway

    A column's buckling load can be had from a shape that is wrong everywhere, because the energy criterion is stationary at the true mode. The error in the load is the square of the error in the shape, and it is always high.

    part 1 · stability
  2. Two quotients from one guessed shape. The critical load of a pin-ended column whose outer quarters keep 10% of the middle's flexural rigidity from four guessed shapes, each worked two ways: Rayleigh's quotient, strain energy of the guess's own curvature over the work of the load, and Timoshenko's, which uses the curvature the guess's bending moment would cause instead. The exact load is 3.225 EI₀/L². a half sine: 8.256 by Rayleigh and 3.745 by Timoshenko; its own sag shape: 8.041 by Rayleigh and 3.712 by Timoshenko; a mid-span sag: 8.875 by Rayleigh and 3.847 by Timoshenko; a parabola: 6.600 by Rayleigh and 3.492 by Timoshenko. Both are upper bounds, and from the same shape the second is never the worse of the two.

    The bound from underneath

    Rayleigh's quotient turns a guessed shape into a buckling load that is always too high. Divide the same guess differently — use the curvature its bending moment would cause instead of its own — and a parabola that was 21.6 per cent high is 1.3 per cent high. Add one more number that needs no guess at all and the load is caught from below as well, which is the only side of it an amplifier can safely use.

    part 2 · stability

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