Structural Engineering & Statics
where the load goes
Concept

Thermal strain — where it appears

The strain a temperature change asks of a material: its coefficient of expansion times the change, about twelve millionths per degree for steel. It produces no stress in a member free to move, and stress in proportion to how much of it the member is prevented from taking.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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 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 designFirePlastic strainCatenaryCritical temperatureImposed deformationPlastic hingeRestraintRobustnessSection factorThermal restraint

All concepts