Structural Engineering & Statics
where the load goes
Concept

Aluminium — where it appears

A structural metal with a third of steel's modulus and no yield plateau, whose strength is defined by a 0.2 per cent proof stress. Welding anneals a strip of it either side of the weld, and its curved stress–strain law is usually written in Ramberg–Osgood form.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

The strength was bought in a furnace and the welder gives it back. Proof stress as delivered and beside a weld, for four aluminium alloys. The heat-treated alloys lose half of it: the strength of a 6xxx extrusion is in precipitates formed by an ageing treatment, and the arc dissolves them for 32 mm either side of the weld, permanently. The work-hardened tempers lose nearly as much, because the heat undoes exactly the work. The annealed ones lose nothing at all, because there is nothing left in them to anneal. The consequence is the crossover: 5083-H22 is 1.04 times 6061-T6 as delivered and 0.92 times it once welded, so the stronger alloy is the weaker member. The 240 mm member drawn, with two longitudinal welds, keeps 87% of its parent capacity — a weld along a member softens a strip and leaves a section, and the same weld across it softens the whole of one.

The strength the welder gives back

A 6082-T6 extrusion is twice as strong as a 5083-H111 plate and, welded across, the two are within a few per cent of each other. The heat of the arc anneals the metal for thirty millimetres either side, permanently, and the strength that was bought in a furnace is given back at the first joint.

materials · Heat-affected zone Three routes to one deflection, and the one that is wrong. Two 2500 mm bars of 250 MPa proof stress meeting at a loaded apex, the drop of the apex against the load. The line is geometry: each bar's extension from its own stress, divided by the sine of its slope. The dots are the derivative of the total complementary energy with respect to the load, and lie on it. The dashed line is the derivative of the strain energy — Castigliano's theorem applied to a material that is not linear — which leaves the truth by ten per cent at 99 kN and at 170 kN gives 308 mm for a deflection of 46.0.

The other area under the curve

Castigliano's theorem says a deflection is the derivative of the strain energy with respect to the load, and it is true only while the material is linear. Past that, the right energy is the area on the other side of the stress–strain curve. On two aluminium bars at their proof stress the strain energy gives a deflection four times too large, and on a redundant truss minimising it picks a set of forces in perfect equilibrium that no deformed shape can produce.

deflection · Strain energy 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 Ductile until it was welded. A 2.0 m 6082-T6 member pulled in tension, as supplied and with one weld across it whose heat-affected zone is 60 mm long: the load per unit of parent section against the member's overall strain, to the maximum load, where the weaker part begins to neck. The plain member reaches 310 N/mm² at 7.0% strain, 140 mm of stretch. The welded one reaches 185 N/mm² at 0.62%, 12.3 mm — 8.8% of the plain member's, and the parent never reaches its proof stress of 260 N/mm². (uniform elongations assumed: 7.0% for the parent, 12% for the zone)

The soft zone that takes all the stretch

The metal beside a weld in heat-treated aluminium is weaker than the rest of the member and more ductile, and it is tempting to read the second fact as a consolation. It is not. A member welded across stretches only where it is weak, and the weak zone is sixty millimetres long; a two-metre 6082-T6 tie that would extend 140 mm before necking extends 12 mm once welded, and the parent metal never yields at all. What decides it is one ratio, the zone's ultimate strength over the parent's, and the ductility returns only when that ratio reaches one.

materials · Heat-affected zone A few per cent between two welds is most of the second one's stretch. For a 2.0 m 6082-T6 member with two welds whose heat-affected zones are each 60 mm long, the stretch beyond the elastic at maximum load as a multiple of one weld's (solid), against how much stronger the second zone is than the first; dashed, the share of its own uniform plastic strain the stronger zone reaches when the weaker peaks, (1/(1 + gap))ⁿ with the zone's hardening exponent n = 10.4. Identical zones give 2.00 times one weld's stretch; 1.74 at a gap of 3 per cent, 1.23 at 15. The stronger zone gives half its stretch at a gap of 7.1 per cent. (uniform elongation of the zone assumed at 12%, and each zone's proof stress and ultimate scaled together)

Two welds, and the one that decides

A member welded twice has two soft zones in series. If they were identical, both would reach their ultimate strength at the same load and the member would stretch twice as far as with one weld before either necked. They are never identical, and the flat top of an aluminium zone's stress–strain curve means that a second zone five per cent stronger than the first gives only sixty per cent of its stretch. Every weld added makes the member weaker, by the statistics of its weakest link, and more ductile, by less than it would if the welds matched.

materials · Heat-affected zone The weakest weld is not the one straight across. The ultimate strength of a member of 6082-T6 (parent ultimate 310 N/mm², zone 185) whose heat-affected zone crosses it at every angle from along the load (0) to straight across it (90 degrees), as load over the gross section. Across the load the zone's neighbours stop it contracting, and it carries 214 N/mm², 15 per cent above its own uniaxial 185. At 54.7 degrees, the angle a free sheet necks along, they give it nothing and the member carries 185. Below 24.2 degrees the zone's own neck would need more than a section through the parent can carry, and the member reaches 278 — the strength it has with the zone, 25 per cent of the section, running along it. At 30 degrees, 237; at 70, 196.

The weld that is weakest at fifty-five degrees

A soft heat-affected zone that the load crosses fails by necking along its own line, and it can do that only if nothing along the line has to stretch. Straight across the load, the parent on either side stops the zone contracting and lends it 15 per cent. At 54.7 degrees — the angle a free sheet necks along — it lends nothing, and the member carries the zone's own strength. Below 24 degrees the zone cannot neck alone at all, and a weld running along the member costs a tenth of its strength while adding to its stretch.

materials · Heat-affected zone 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.

DuctilityHeat-affected zoneProof stressWeldLocalisationNeckingNet sectionTangent modulusWork hardeningAnnealingCastiglianoColumn curve

All concepts