The temperature is a shape

The hour that is really a temperature replaces a fire rating with a number of degrees: a member at a load ratio of 0.6 loses its strength at 558 °C, and the fire resistance problem becomes the problem of keeping it below that. What that essay does not answer, and says so, is when the temperature arrives.

That is not a question about fire. It is a question about heat transfer into a lump of steel, and it has one geometrical parameter in it.

The three bare curves are the same steel, the same fire, the same load ratio and the same critical temperature. Everything separating them is the ratio of the surface the fire touches to the volume of metal behind it.

The one number, and where it comes from #

The section factor $A_m/V$ is the exposed perimeter per unit length divided by the cross-sectional area — a surface over a volume, so its units are one over a length. It is what appears in every heating calculation and nothing else about the shape does.

Its size is easy to feel. A section factor of 200 per metre is a section whose steel would be 5 mm thick if it were spread evenly over the surface it exposes: the whole calculation is “how thick is this member, thermally”. A thick-walled hollow section is thermally thick. A slender universal beam is a thin sheet folded into a useful shape, and it heats like one.

That spread — a factor of ten in section factor across the catalogue, a factor of nearly four in time — is the reason a fire check cannot be done on a member’s strength alone, and it is why two members with identical load ratios need different protection.

Three things about the factor are worth carrying because they are what makes it useful.

It is scale-free within a shape. Double every dimension of a section and the perimeter doubles while the area quadruples, so the factor halves. A bigger member of the same shape is always thermally thicker, which is why heavy columns are so often unprotected and light beams never are. It is the square-cube law that decides so much else about section shape, arriving in a subject that has nothing to do with strength.

It depends on the exposure. A beam under a slab is exposed on three sides rather than four, and its factor falls by roughly a quarter. A column in a wall is exposed on one or two. The factor is a property of the situation and not of the section, which is the commonest place for it to be got wrong.

And it is a geometrical decision that arrives before any of the fire engineering. The section was chosen for bending, and its section factor came with it. Choosing a slightly heavier section for its thermal thickness is a real design move, and it competes directly against protection.

Which free body produced the number #

The free body is one metre of the member, cut at both ends, treated as a single lump at one temperature.

Crossing its surface is heat, by two routes and in that order of importance: radiation from the flames and hot gases, going as the difference of the fourth powers, and convection from the moving gases, going linearly. At 800 °C radiation is about two thirds of the total and at 300 °C it is about half, so a fire calculation is an optics problem before it is a fluids one. Neither route knows anything about the material behind the surface — the modulus a stronger steel does not change is not in the heating calculation at all, and neither is the yield stress.

Inside, the heat goes entirely into raising the temperature of the steel — there is no conduction term at all, because the lump is assumed uniform.

That assumption is the one thing the free body needs and it deserves a check. Steel conducts at about 45 W/mK, and the thickest structural element here is 77 mm; the Biot number — the ratio of surface resistance to internal resistance — comes out around 0.05, which is the standard criterion for a lumped model being good to a few per cent. A steel section really is nearly isothermal in a fire, and the whole approach fails for concrete, whose conductivity is thirty times lower and whose interior stays cool for hours.

So the balance is

$$\rho\,c(T)\,\Delta T = \frac{A_m}{V}\left[\alpha(T_g - T_a) + \varepsilon\sigma(T_g^4 - T_a^4)\right]\Delta t$$

with everything on the right known and only the specific heat depending on where the member has got to. That single term is worth a section of its own.

The kink at 735 degrees #

Every bare curve on the first figure has a plateau in it, at about the temperature the members are being checked against, and it is not a numerical artefact.

Steel’s specific heat is not constant. It rises gently to about 600 °C and then spikes: at 735 °C the crystal structure changes from body-centred cubic to face-centred cubic, and the transformation absorbs latent heat. In the standard treatment the spike is folded into $c(T)$ as a term that goes to infinity at 738 °C, which means the member absorbs a large amount of heat while barely changing temperature.

The result is several minutes of delay, free, arriving exactly where fire design needs it. A member being checked at 550 °C never gets the benefit; one at 700 °C — a lightly loaded member, or a tension member at a low load ratio — gets a substantial share of its fire resistance from a metallurgical phase change nobody put in the design.

That is a rare thing in this subject: a property of the material that helps, at the temperature it is needed, for reasons entirely unconnected with anybody’s intention.

What protection actually does #

Protection does not change the destination. It changes the arrival time, and the calculation is the same balance with the surface resistance replaced by the insulation’s.

The proportionality is the practical content of the whole subject, and it can be read straight off the balance: the heat arriving is proportional to $\lambda_p/d_p$ times $A_m/V$, so holding the arrival rate constant means holding $d_p$ proportional to $A_m/V$. Which gives the form every manufacturer’s table takes — a thickness against a section factor, for each rating and each product — and explains why those tables have a column for the section factor and no column for the steel grade, the span, or the load.

Two consequences follow that are worth having in the front of the mind on a real job.

Protection is bought by the section factor, so specifying heavier steel saves protection. A 305×305×198 UC at 76 per metre needs half the board thickness of a 457×191×82 UB at 161. Whether that trade is worth making is an arithmetic question with two prices in it, and it is one of the few places where a structural decision and a fire-protection decision are the same decision.

And the load ratio moves the target rather than the rate. A member at 0.3 rather than 0.6 has a critical temperature near 700 °C rather than 558, which on the heating curve is worth another four or five minutes bare and rather more protected, because the curve is flattening. Fire resistance is bought as much by not loading a member as by covering it.

The two prices #

Once the thickness is proportional to the section factor, the choice between steel and protection becomes an arithmetic one, and it is worth doing once because the answer is not intuitive.

Take a 12 m floor beam needing sixty minutes. As a 457×191×82 UB at 161 per metre it wants 21 mm of the 0.2 W/mK board — call it 0.25 m² of surface per metre of beam, so 5 litres of material per metre. As a 610×229×113 UB it is at 128 per metre and wants 17 mm, and it also weighs 31 kg/m more.

The trade is thirty-one kilograms of steel against a fifth of the protection, per metre, and on ordinary prices the steel loses badly. Protection is nearly always cheaper than thermal mass, which is why fire resistance is bought in millimetres of board rather than in section size, and why the section factor appears in a fire calculation as a given rather than as a variable.

Where it inverts is at the two ends. A heavily loaded column at a low section factor may need no protection at all, which saves the whole line item; and a member in an aggressive environment, or one where the protection cannot be maintained or inspected, is worth oversizing to avoid depending on a coating for sixty years. Those are the two cases where the geometry is worth paying for, and both are decided before any of the arithmetic above.

The rating that is a comparison rather than a duration #

There is a persistent misreading of every number on this page and it is worth naming, because it is the same shape of error as the one the rung below exists to correct.

A sixty-minute rating is not a promise that the member survives an hour of fire. It is a statement that, in a standard furnace test following a standard temperature curve, the member reached its criterion at sixty minutes. A real compartment fire may be hotter than the standard curve early and cooler later; it may burn out in twenty minutes, or it may be a fuel-controlled fire that never reaches the furnace temperature at all.

What the rating is genuinely good for is comparison. Two members with the same rating behave similarly in the same real fire, and a member with twice the rating survives a great deal more than twice as much of it, because the standard curve’s temperature climbs steeply at the start and flattens. The number is a ranking with units attached, and the units are the reason it is so often read as a duration.

That is a general feature of standardised tests, and the same caution applies to a fatigue detail category and to a characteristic strength: the value belongs to the test programme, and its usefulness is that everybody’s members were measured against the same one.

The history, which is a furnace from 1918 #

The standard fire curve is older than most of the theory applied to it and it was never intended to describe a fire.

Furnace testing of building elements began in the 1900s, and the curve that became ASTM E119 was fixed in 1918 from what a handful of American laboratories could actually reproduce with the burners they had. ISO 834 adopted essentially the same shape in 1975, and its algebraic form — $345\log_{10}(8t+1)$ — is a fit to a furnace’s behaviour rather than a solution of anything.

The section-factor calculation is much younger. It arrives with the Swedish work of Pettersson and Magnusson in the 1960s and 1970s, which is also where the parametric curves and the whole idea of a compartment fire as a calculable object come from. So the sequence is: a reproducible test first, a physical model of the member second, and a physical model of the fire third — with the test still in every code, because it is what the products are certified against.

The result is a design method with a nineteenth-century instrument at its centre and a twentieth-century calculation wrapped round it, which is worth knowing when a fire engineer proposes to replace the curve with a real one: everything downstream of the temperature history is unaffected, and everything upstream of it changes completely.

Where the model stops #

The standard fire is a furnace, not a fire. $345\log_{10}(8t+1)$ rises forever and has no cooling branch, because it was written to describe what a test furnace does rather than what a compartment does. A real compartment fire has a growth phase, a flashover, a burning period whose duration is set by the fuel and the ventilation, and a cooling phase — during which a protected member’s temperature keeps rising for a while, because the board it is wrapped in is hotter than it is.

Nothing here is a structural calculation. The output is a temperature history, and the member’s fate depends on what its capacity does at that temperature, which is the rung below. The two halves are usually done by different people, and the interface between them is a single number.

Restraint is absent entirely. A heated member expands, and a member that cannot expand develops enormous compression — a 6 m beam at 500 °C wants to grow 36 mm, and holding it back takes far more force than any applied load. The lumped model has no mechanics in it at all, so it cannot see that, and it is the effect that decides what actually happens to a real frame in a real fire.

And the emissivity is a fitted constant. 0.7 for uncoated steel is a convention covering a surface whose real emissivity depends on its oxide layer, its paint, its soot deposit and its age. Radiation is two thirds of the heat and its coefficient is known to perhaps ±20 per cent, which is a bigger uncertainty than most of the refinements applied to the rest of the calculation.

What the pictures cannot show #

The curves show a member reaching a temperature and stop, and the two things they cannot show are what happens on either side of that instant.

Before it: the connections. The rung below records that connections are what actually run out, and a section factor is a member property with no answer for a bolted joint — a cleat and its bolts have a much higher exposed surface per unit of steel than the members they join, and they heat faster than anything else in the frame.

After it: the redistribution. A member that reaches its critical temperature does not fall down; it sheds load to whatever is cooler and stiffer, and in a real floor plate that means tensile membrane action in the slab. The whole of modern fire engineering for steel-framed buildings rests on that, and a member-by-member temperature calculation is precisely the tool that cannot see it.

The assumption the figure rests on #

That the member is at one temperature.

It is a good assumption for the section and a poor one for the member. A beam under a concrete slab is exposed on three sides and has a heat sink on the fourth, so its top flange runs several hundred degrees cooler than its bottom one — which is a thermal gradient through the depth, which is a curvature, which is a deflection with no load causing it.

That deflection is often the first thing observed in a fire test, and it arrives long before any strength has run out. It is the same object as the stress nobody restrained, at a much larger temperature difference, and the honest version of this whole calculation for a composite beam is a two- or three-zone model with a different temperature for each part of the section.

The ladder from here #

Later rungs on this anchor: parametric fire curves, where the compartment’s fuel load and ventilation replace the furnace and a cooling branch exists. The three-sided beam and the partially exposed column, where the section factor is a property of the construction rather than of the section. Intumescent coatings, whose conductivity is a function of temperature because the material is not the same material after it has swollen. Restrained thermal expansion and the compression it generates, which is the mechanics this page left out. Tensile membrane action in composite floors, and the Cardington tests that made it a design method. And concrete in fire, where the lumped model fails completely, the material spalls, and the reinforcement’s temperature depends on a cover that was specified for durability.