Structural Engineering & Statics
where the load goes
Concept

Fire — where it appears

A design condition in which a structure must keep standing at temperatures that reduce both the strength and the stiffness of its materials. What a fire rating names is a duration in a standard furnace test rather than a temperature, and stiffness usually falls faster than strength.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The hour that is really a temperature. The retention factors for carbon steel against temperature: the yield stress and the elastic modulus. The modulus falls away first — at 500°C the steel has kept 78% of its strength and 60% of its stiffness — so a member's failure mode can change during a fire. A member working at 60% of its cold capacity runs out of strength at 558°C, and out of the stiffness for the same ratio at 500°C, 58 degrees earlier. There is nothing about time in any of it: a fire rating is a temperature the member must not reach, converted into the minutes a particular fire takes to get it there.

The hour that is really a temperature

A fire rating is quoted in minutes and there is no time in the physics anywhere. What decides is a temperature, and the stiffness reaches its limit sixty degrees before the strength does — so the way a member fails can change while it is burning.

materials · Fire The deflection goes on growing, and sometimes it does not stop. Second-order deflection of a sustained-loaded concrete column against age, on a log time axis. Creep takes the effective modulus down, which takes the buckling load down with it — from 11580 kN on the day to 3309 in the long term, 29 per cent of it — so the amplifier 1/(1 − N/N_cr) grows even though nothing was added to the load. At 2200 kN the column settles: 25 mm of eccentricity on the day and 60 mm at the end, a factor of 2.4 for a load that never changed. At 5294 kN — still only 46 per cent of the day-one critical load — it does not settle, and the divergence arrives at 55 days for no new reason at all.

The column that fails years later

A concrete column under sustained load goes on straining at constant stress, so its deflection grows — and because the second-order moment is the load times that deflection, the demand grows with it. There is a load below which the two settle and one above which they never do.

stability · Creep buckling The fire is one curve, and the steel in it is several. Gas temperature and steel temperature against time in a standard fire. The gas curve is the same for every member in the compartment; the three bare steel curves are section factors of 76, 160 and 323 per metre, and they reach 558 °C at 17, 11, 8 minutes — a spread of a factor of two from geometry alone. The fourth curve is the middle section with 15 mm of board on it, which reaches the same temperature at 46 minutes. The kink near 735 °C on the bare curves is not a numerical artefact: steel's specific heat spikes there as its crystal structure changes, and the member spends several minutes absorbing heat at almost constant temperature.

The temperature is a shape

Two members of the same steel in the same compartment, under the same fire and the same load ratio, fail eight minutes apart. Nothing about the material differs and nothing about the fire does. What differs is a perimeter divided by an area, and it is the only number in the whole calculation that a designer chooses.

materials · Fire A room's fire has a peak and an end, and the furnace has neither. Gas temperature in a compartment with 200 MJ/m² of fire load per square metre of its enclosing surface, linings of thermal inertia 1160 J/m²s½K and a medium growth rate, for opening factors of 0.02, 0.04, 0.08, 0.14 m½, against the standard furnace curve, dashed. At 0.02 the fire peaks at 841 °C after 120 minutes and is back at ambient by 435; at 0.04 the fire peaks at 944 °C after 60 minutes and is back at ambient by 171; at 0.08 the fire peaks at 1048 °C after 30 minutes and is back at ambient by 92; at 0.14 the fire peaks at 900 °C after 20 minutes and is back at ambient by 37, having used its fuel before the growth limit, so the fuel rather than the air decided it. A larger opening lets more air in: the fire burns hotter and gets through its fuel sooner. The furnace curve is still rising when every one of them is out.

The fire that goes out

The furnace curve rises for ever. A room's fire has a peak and an end, set by its window, its walls and its fuel, and a bigger window makes it hotter and shorter — which is worse for bare steel and better for protected steel. The protected member is hottest half an hour after the fire has started to die.

materials · Fire 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

Named alongside it

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

Critical temperatureSection factorLoad ratioBucklingCreepFire protectionHeat transferLimit stateServiceabilitySpecific heatStandard fireAmplification

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