Structural Engineering & Statics
where the load goes
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Inelastic buckling — the series

2 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Why the curve sags, and why the two axes are not the same column. The same column curve with the sag computed rather than drawn. A hot-rolled section carries a residual compression of 30% of yield at its flange tips before anything is applied, so the tips yield first and what is left resisting a change of shape is the elastic core. About the major axis the stiffness follows the core's width; about the minor axis it follows its cube. The worst loss is 27% at λ = 74 about the minor axis against 23% about the major, and the whole effect lives between λ = 75 and λ = 89 — outside that band nothing has yielded, or everything has. No imperfection appears anywhere in this figure.

    The column that had yielded before it was loaded

    A real column sits below both of the two straight answers over the whole middle of the slenderness range, and the usual explanation — that it was not straight — is only half of it. The other half is that the flange tips had already yielded when it left the rolling mill.

    part 1 · stability
  2. A column curve with no plateau. The tangent-modulus buckling stress of perfect pin-ended columns of an aluminium alloy of 0.2 per cent proof stress 250 N/mm² and modulus 70,000, as a share of the proof stress, against the slenderness λ̄ — the square root of the proof stress over the Euler stress — for Ramberg–Osgood exponents of 5, 10, 20 and 40; dashed, the sharp envelope a material with a plateau at the proof stress would give, the lesser of the proof stress and Euler's. n = 5: 1.54 at λ̄ = 0.2, 0.66 at 1, 0.41 at 1.5; n = 10: 1.16 at λ̄ = 0.2, 0.74 at 1, 0.45 at 1.5; n = 20: 1.04 at λ̄ = 0.2, 0.82 at 1, 0.45 at 1.5; n = 40: 1.00 at λ̄ = 0.2, 0.88 at 1, 0.45 at 1.5. Stocky columns buckle above the proof stress, because the curve goes on rising past it; slender ones follow Euler; in between every curve sags below the envelope.

    The column with no plateau

    A steel column's curve is the lesser of two lines, its yield stress and Euler's, because mild steel keeps its whole stiffness up to yield and then has a plateau to stop on. Aluminium and stainless steel have no plateau: their stiffness starts falling well below the proof stress and never stops. Their column curves are not the steel curve moved; they are a different shape — four tenths below the corner for a stainless steel, above the proof stress for a stocky column — and the shape is set by two numbers, only one of which describes the knee.

    part 2 · stability

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