Structural Engineering & Statics
where the load goes
Concept

Plastic strain — where it appears

The part of a deformation that remains when the load is removed, produced by slip rather than by stretching bonds. It is what a ductile structure spends to redistribute force, and its capacity is the property that separates a warning from a collapse.

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

Yielding one way makes it easier to yield the other. Mild steel taken to a strain of 1.20% and then pushed back the other way. The stress falls by 550 N/mm² before it yields again, against a yield stress of 275 — the elastic range is twice the yield stress and not once it, which is the Bauschinger effect and is a consequence of the yield surface sliding rather than growing.

Yielding one way, and then the other

A material that has yielded in tension yields earlier in compression than it did the first time, and by an amount that is exactly what makes its elastic range twice its yield stress rather than once it. That is a property no monotonic test reports and every reversing structure depends on.

materials · Ductility 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 Out in compression, back in tension. The axial stress in a bare S355 beam (section factor 160 per metre) whose ends are held by the structure around it with 10 per cent of its own axial stiffness, against its temperature, through a compartment fire with an opening factor of 0.04 m^½ and 200 MJ/m² of fuel: heating (solid) and cooling (dashed), with the yield stress at each temperature dotted either side. Heating, the beam pushes against its ends and reaches 134 N/mm² of compression at 628 °C, where the yield stress has fallen to meet it; it then yields, shortening plastically as it follows the falling yield stress up to its peak of 940 °C. Cooling, the thermal expansion comes back out and the plastic shortening does not: the beam passes through zero and ends at 20 °C in 195 N/mm² of tension.

The tension a fire leaves behind

A steel beam held at its ends by the structure around it is pushed into compression as a fire heats it, and yields, because its strength is falling while its expansion is not. When the fire goes out the expansion comes back and the yield does not. Held at a tenth of its own stiffness, a beam that peaked at 134 N/mm² of compression on the way up ends the fire at 195 N/mm² of tension — more than it was ever pushed, arriving hours after the fire was out, and on connections that were designed to carry shear.

materials · Fire Shorter after every cycle, at the same load. The shortening of a column hinge at the end of each cycle, in yield strains, for an I-section of 3,000 mm² and 200 mm depth and the rectangle of the same area, each carrying a constant 30% of its squash load while its curvature cycles between ±3.0 times its first-yield curvature, in elastic–perfectly plastic steel. The I-section shortens by 8.14 yield strains every cycle, 97.1 after 12; the rectangle by 1.60, 19.4. Neither settles: the load has not changed, and the column is shorter after every cycle.

The column that shortens on every cycle

A column hinge bent back and forth by an earthquake carries its share of the building's weight the whole time. In elastic–perfectly plastic steel an I-section at three tenths of its squash load, cycled to three times its first-yield curvature each way, comes back from every cycle shorter by eight yield strains, and goes on doing so: it never settles. The flange that yields in compression shortens, and the other flange, carrying its part of the column's load, never reaches yield in tension to pull it back. A rectangle of the same area shortens a fifth as fast, and the overturning that swings the column's load up and down hardly matters. The weight and the curvature make the ratchet between them.

materials · Moment-curvature The more it carries, the less it pulls when the fire is out. For a 9 m 457 × 191 × 67 steel beam held at its ends at 10 per cent of its own axial stiffness through the fire: the tension pulling on its ends once it has cooled (solid) and the largest tension while it is still hot (dashed), against the load it carries as a share of its cold plastic moment; dotted, the straight beam's 1,669 kN. At 0.10, 1,017 kN cold and 41 hot; at 0.39, 613 and 293; at 0.54, 477 and 397. From 0.56 the beam can find no sag that carries its load near the fire's peak and runs away.

The beam that pays the fire back in sag

A steel beam held at its ends and kept straight through a fire yields short on the way up and pulls on its connections with its plastic shortening on the way down. A beam carrying a floor does not stay straight. Squeezed by its own expansion, it forms a hinge at about 500 °C and bows out of the way, hangs as a catenary while still hot, and keeps a metre of sag when it cools — length it no longer has to be pulled for. It ends the fire pulling a third as hard as the straight beam, nearly whatever holds it, and the hard pull comes earlier instead: at 800 °C, on connections that are as hot as it is.

materials · Fire

Named alongside it

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

Connection designDuctilityFireHysteresisPlastic hingeThermal strainAxial shorteningBauschinger effectBuckling-restrainedCatenaryCritical temperatureCyclic loading

All concepts