Structural Engineering & Statics
where the load goes
Concept

Localisation — where it appears

The concentration of deformation into a small zone — a neck, a plastic hinge, a shear band — after which the rest of the member unloads. It ends the uniform behaviour every average measurement assumes, and it is why a strain measured over a length becomes a statement about the length as well as the material.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

The ductility number depends on the ruler. Elongation after fracture against the gauge length it was measured over, in units of √S₀. A tensile specimen extends uniformly until the ultimate load and then localises into a neck, so the total is a strain (16 per cent here) plus a length (7.2 mm), and dividing a length by the gauge length is what makes the curve fall. At the two standard gauges the same steel reports 27.5 per cent over 5.65√S₀ and 21.8 over 11.3√S₀ — a ratio of 1.264, and 42 per cent of the first number is a property of the specimen rather than of the material.

The ductility that depends on the ruler

Percentage elongation after fracture is the most quoted ductility measure in the subject and one of the least well defined. A specimen stretches uniformly until the ultimate load and then localises, so the number is a strain plus a length — and dividing a length by the gauge length makes the answer a property of the specimen.

materials · Gauge length The curve the machine drew and the curve the material was on. The same tensile test twice. Force over the ORIGINAL area against extension over the original length is the engineering curve, which peaks at 431 MPa and 24.6 per cent and then falls. Force over the ACTUAL area against the natural logarithm of the length ratio is the true curve, which passes 538 MPa at the same instant and keeps rising. Nothing softens anywhere on this figure: the descending branch is the original area still being divided by, and the material is hardening the whole way. The two separate at the very first plastic strain and the gap between them is exactly e^ε, which is 1.25 at the ultimate load. Past that instant the deformation stops being uniform, so the engineering curve is drawn dashed: the real one falls faster than this, because the extension is now happening in one short length of the bar and the strain axis is still dividing it by the whole gauge.

The curve was rising the whole time

Every tensile curve ever printed turns over and falls, and nothing about the material does. The fall is the original area still being divided by after the specimen has stopped having it — and correcting the two denominators turns a strength, a ductility and a failure into one exponent.

materials · Gauge length What a crack makes of a notch, against how sharp the notch is. The fatigue notch factor against the root radius, at a fixed elastic factor of 3 in a 430 MPa steel. K_t is a property of the shape and does not move along this axis at all — the dashed line — while the factor fatigue actually feels climbs toward it from below. At a 1 mm root the answer is 2.40, which is 30 per cent of the notch relieved; at 0.1 mm it is 1.38, and a notch that concentrates by 3 elastically is barely felt. Nothing has changed about the stress field: what has changed is that the peak is confined to a smaller volume than the material's own process size, so the crack starts against an average rather than against a maximum.

The notch a crack does not feel in full

The elastic concentration factor is a property of shape and knows nothing about size, which is what makes it so useful and so misleading. A fatigue crack starts against an average over a volume the material owns, so two notches with the same factor and different radii have different fatigue strengths — and the stronger the steel, the less of that relief it gets.

materials · Stress concentration 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

Named alongside it

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

DuctilityNeckingAluminiumHeat-affected zoneSize effectTensile strengthWeldGauge lengthStrain hardeningUniform elongationCharacteristic strengthConstraint

All concepts