The hour that is really a temperature

A structural element carries a fire rating expressed in minutes: sixty, ninety, a hundred and twenty. The number looks like a material property and is not one. There is no time anywhere in what happens to steel when it is heated, and a member has no idea how long it has been in a fire.

What it has is a temperature. The rating is the number of minutes a particular standardised fire takes to bring the member to the temperature at which it can no longer carry what it is carrying, and every part of that sentence is doing work: the fire is a standard curve rather than a real one, the heating rate depends on the member’s shape, and the temperature depends on how heavily loaded it is.

The two curves separate, and that is the result #

The strength curve is flat to 400°C and then falls off a cliff. The stiffness curve starts falling immediately and is at 70% by the time the strength has moved at all.

The mechanism behind the difference is the same one that made the modulus unpurchasable at room temperature. Stiffness is a property of the interatomic bonds, and heating weakens bonds directly and continuously from the moment the temperature starts rising. Strength is a property of the obstacles that stop dislocations moving, and those obstacles — precipitates, solute atoms, grain boundaries — stay in place until the temperature is high enough to start dissolving or bypassing them, at which point the strength collapses quickly.

So the two properties that were independent at room temperature are independent in a fire as well, and they degrade on completely different schedules. That has a consequence which is the whole practical content of this page.

The failure mode can change during the fire #

A stocky column at room temperature is governed by squashing: its capacity is $Af_y$ and buckling is nowhere near. A slender one is governed by buckling: its capacity is $\pi^2EI/L_e^2$ and the yield stress is irrelevant. The crossover is at a slenderness of $\pi\sqrt{E/f_y}$.

Heat both. The buckling capacity falls with $E$, which starts dropping at once. The squash capacity falls with $f_y$, which does not move until 400°C. So the crossover slenderness moves, and it moves toward the stocky end: at 500°C, $\sqrt{E/f_y}$ has changed by $\sqrt{0.60/0.78} = 0.88$, so the crossover has come in by 12%.

That is the same mode-switch a beam meets when its span grows, arriving from a different direction: there the geometry moves the crossing, here the material does. A member whose failure mode changes during the event is an awkward object for design, and it is the reason fire design of steel columns cannot be done by simply reducing the cold capacity by the strength retention factor. The stiffness factor has to be carried through the buckling calculation separately, and for slender members it is the one that governs.

The critical temperature #

Everything above can be collapsed into one number if the question is a member’s strength rather than its stability. The load ratio is the fraction of its cold capacity that a member is actually carrying in the fire — which is usually well under one, because the load combination in a fire uses unfactored loads, and the fire is not expected to coincide with the worst wind.

The critical temperature is where the retention factor falls to the load ratio.

That is the entire structural fire calculation for a beam, and it is worth appreciating how little of it there is. One curve, one load ratio, one temperature. The complicated part is not the structure.

Which free body produced the number #

The free body is the same cross-section as everywhere else in this field, and the honest answer about where the retention curves come from is that they are not derived here — they are measured, on specimens held at temperature and pulled, and they are among the more reliable material data in the subject because the tests are straightforward and have been repeated for decades.

What follows from them is computed. The critical temperature is found by bisection on the retention curve for the given load ratio, and the two critical temperatures quoted for each ratio are found on the two different curves. The check is that the strength one always exceeds the stiffness one at every ratio — 671 against 606 at 0.3, 558 against 500 at 0.6, 526 against 400 at 0.7 — which has to be true if the stiffness curve lies below the strength curve everywhere, and would be a sign of an inverted table if it ever failed.

There is a subtlety in the strength curve worth naming. The “yield stress” in a fire is not the 0.2% proof stress used at room temperature; it is conventionally taken at 2% strain, ten times further along. That is a deliberate change of definition, made because steel at temperature has no yield point at all — the curve has become entirely rounded, so the number quoted depends even more than usual on where somebody agreed to read it, and reading it further out gives credit for the large deformations a member in a fire is permitted.

Where the model stops #

Nothing here is about the fire. The whole of the difficulty in fire engineering is on the other side: what temperature the member actually reaches. That depends on the fire’s severity, the ventilation, the fuel, the compartment, and on the member’s section factor — its heated perimeter divided by its cross-sectional area, which is a geometric property with the dimensions of one over length. A member with a lot of surface per unit of steel heats quickly; a heavy column in a corner heats slowly. Two identical sections at identical load ratios in identical fires reach different temperatures because one has three faces exposed and the other four.

And the standard fire is not a fire. The temperature-time curve behind every quoted rating is a standardised furnace curve that rises indefinitely and never decays. A real compartment fire grows, flashes over, burns out its fuel and cools, and a member that survives the cooling phase may fail during it — the structure has thermally expanded, been restrained, yielded in compression, and on cooling is pulled into a tension it was never designed for.

Steel creeps at temperature and none of this contains creep. Above about 400°C the retention factors are strain-rate dependent, and a member held near its critical temperature will fail eventually rather than immediately. The same time-dependence that concrete shows at room temperature appears in steel once it is hot, and the “critical temperature” is properly a critical temperature for a stated duration.

And restraint changes everything, in both directions, in exactly the way an imposed deformation always does. A heated member expands. Restrained, it develops enormous compression — 12 × 10⁻⁶ per degree times 210,000 gives 2.5 N/mm² per degree, so a hundred degrees of restrained expansion is 250 N/mm², which is most of the yield stress before the fire has weakened anything. That can buckle a member long before its critical temperature. On the other hand, a floor slab that has lost its bending capacity can hang in tension between supports as a membrane and carry load by a mechanism it was never designed for, which is why real buildings survive fires that member-by-member calculations say should have collapsed them.

What the picture cannot show #

There is no member on the plot. Two retention curves and a horizontal line: everything about the section’s shape, its exposure, its restraint and its connections is absent, and every one of those decides more than the curves do.

And there is no time, which is the whole point of the page and is worth stating as a limitation rather than only as a finding. A curve with temperature on its horizontal axis cannot answer the question a building regulation asks, which is how many minutes. Getting from one to the other requires the standard fire curve and a heat-transfer calculation, and both introduce more uncertainty than anything drawn here.

Nor does the plot know that the strength curve’s definition changed. The retention factor above 400°C is read at 2% strain rather than at the 0.2% proof stress, so the two ends of the same curve are not measuring the same thing. It is a defensible convention and it is invisible on the drawing.

What the numbers are worth against what they cost #

It is worth putting the retention factors beside the property they are fractions of, because a fire is the one situation in this field where a stronger grade genuinely buys something.

The retention factors are the same for every carbon steel: S275 and S460 both keep 78% of their strength at 500°C. So a member in the stronger grade at the same absolute load is at a lower load ratio, has a higher critical temperature, and survives longer — the strength that bought nothing against deflection or buckling buys real minutes here.

The catch is that nobody designs a member in a stronger grade at the same absolute load. The stronger grade is specified in order to make the member smaller, which puts the load ratio back where it was and raises the section factor as well, so the smaller member heats faster to the same critical temperature. The two effects run in opposite directions and the net result is usually worse, which is the reverse of what the retention table alone suggests.

The generalisation #

The pattern is that a rating is a physical quantity converted into a convenient unit, and the conversion loses the physics.

Minutes are convenient: they can be compared with the time to evacuate, with the fire brigade’s response, with the regulations. Temperature is what the steel responds to. The conversion between them runs through the standard fire curve and the section factor, and once it has been made the result no longer contains the load ratio, the failure mode, or the fact that a slender column and a stocky one at the same rating are protected against different things.

This site has met the shape before. The effective length factor converts a real end condition into an equivalent pinned length, and a designer who knows only the factor cannot say what happens to a case not in the table. A fire rating does the same to a temperature. Both are useful and both are lossy, and the loss is invisible until a case arrives that the conversion was not calibrated for.

A surprising place this turns up #

The most economical fire protection is often not protection at all, and the reason is on the plot.

If the critical temperature depends on the load ratio, then reducing the load ratio raises the critical temperature — and a member sized for deflection rather than strength is already at a low load ratio for free. Long-span beams governed by serviceability are typically carrying 30 to 40% of their strength capacity, so their critical temperature is over 650°C, and a proportion of them require no applied protection whatever to reach a 30-minute rating.

The corollary is uncomfortable. A design that has been efficiently optimised for strength — every member working at its full capacity, no waste — has maximised every load ratio in the structure and thereby minimised every critical temperature. Efficiency in the static design is paid for in the fire design, and the two are usually done by different people at different times.

Where the ladder goes next #

Later rungs on this anchor: the section factor and the heat-transfer calculation that turns a fire into a member temperature. The standard fire curve, the parametric curves that model real compartments, and the cooling phase. Protection materials and how their thickness is determined. Restrained thermal expansion and the compression it generates. Tensile membrane action in slabs, and the Cardington tests that established that real buildings behave far better than member calculations predict. Fire design of concrete, where the material spalls and the reinforcement’s temperature depends on its cover. Composite construction, where the concrete acts as a heat sink for the steel. And the critical temperature of a column done properly, with the stiffness retention carried through the buckling curve rather than the strength one.

Historically, structural fire engineering was for most of a century a matter of prescription rather than calculation: a table of required protection thicknesses by member type and rating, derived from furnace tests, with no structural analysis in it at all. The move to calculation began in the 1980s and accelerated after the Broadgate fire in 1990 and the Cardington full-scale tests that followed, which between them demonstrated that a steel-framed building could survive temperatures that its individual members were calculated to fail at — because the members were connected to each other, and the analysis had been of members.