Structural Engineering & Statics
where the load goes
Concept

Tangent modulus — where it appears

The slope of a stress-strain curve at the stress a member has reached, which is the stiffness available to resist any further change. Substituting it for the elastic modulus is what turns Euler's formula into an inelastic column curve, and it explains why residual stress lowers a real column's capacity.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A I-section at 80% of its plastic moment. The same I-section drawn three ways: the shape, the strain across its depth, and the stress that strain produces in mild steel. The strain diagram is a straight line, because plane sections stay plane whatever the material is doing. The stress diagram is not: 0% of the area has yielded, working inward from both faces, and the neutral axis sits at 70.9 mm against a centroid at 100.0 mm. The compression resultant is 322.5 kN and the tension resultant 322.5 kN, on a lever arm of 180.1 mm, which multiplies back to the 58.1 kNm the section is carrying. A rolled residual stress pattern of ±30% of yield is locked in before any load arrives.

The stress that was there before the load

A rolled steel section leaves the mill carrying eighty N/mm² of stress with nothing applied to it, in a pattern that sums to no force and no moment. It is invisible to every calculation and it is the knee in every column curve.

materials · Residual stress 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.

stability · Inelastic buckling The windward guy tightens, the leeward one gives way. A 120 m mast on three guy levels, at a wind of 3 N/mm, with the deflection drawn 0.54 times its true size. The guys start at 160 kN each and end at 259 against 95, 299 against 83, 221 against 112 kN. The leeward guys still carry a real force — the lowest keeps 28 per cent of its partner's tension — and supply almost none of the restraint, because their tangent modulus has fallen to 54 per cent of the steel's. The mast top moves 100 mm, its worst bending moment is 554 kNm at 40 m, and it is carrying 801 kN of axial load that nothing but the guys put there.

Held by something that goes soft

A guy is a cable, so it has no stiffness of its own — what resists a mast's movement is the guy's geometry changing, and how much of that there is depends on the tension already in it. Wind pushes the mast towards the leeward guy, which is the one losing tension.

structures · Guyed mast 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.

stability · Inelastic buckling Two theories, one plate, and a column beside it. The buckling stress of a long plate of an austenitic stainless steel of proof stress 230 N/mm² and modulus 200,000 (Ramberg–Osgood exponent 6), simply supported on its edges and compressed along them, as a share of the proof stress, against its slenderness λ̄p, the square root of the proof stress over the elastic plate's buckling stress. Flow theory: 2.30 at λ̄p = 0.6 and 0.86 at 1. Deformation theory: 0.99 and 0.71. A column of the same material at the same slenderness, by its tangent modulus: 0.76 and 0.58. Dashed, the sharp envelope, the lesser of the proof stress and the elastic stress. Flow theory runs far above the proof stress on stocky plates — 5.29 at 0.4 — which no plate test has reached.

The plate the wrong theory gets right

A column of stainless steel buckles by its tangent modulus, the one stiffness its rounded curve has left. A plate needs three — along the load, across it, and in twist — and the two classical theories of plasticity give it different ones. Flow theory, the one whose physics is right, keeps most of the plate's elastic stiffness and puts a stocky stainless plate's buckling stress at twice its proof stress. Deformation theory, whose physics is wrong, softens every direction and agrees with the tests. Two thirds of the gap is the twist alone, and either way a rounded curve costs a plate much less than it costs a column.

stability · Inelastic buckling

Named alongside it

The objects these essays reach for when they reach for this one.

Column curveAluminiumBucklingImperfectionProof stressResidual stressSelf-equilibratingSlendernessAnchorBeam-columnCableCritical load

All concepts