The fire that goes out

The time a steel member takes to reach its critical temperature was worked out against the standard furnace curve, $345\log_{10}(8t+1)$. That curve was fixed in 1918 from what test furnaces could reproduce, and it has one property no real fire shares: it never stops rising. A fire in a room runs out of fuel, or of air, and then goes out. How hot it gets and how long it lasts are properties of the room.

Three properties of a room #

The compartment fire in Annex A of EN 1991-1-2 takes three numbers from the room and none from anywhere else.

The opening factor, $O = A_v\sqrt{h_{eq}}/A_t$: the area of the windows, times the square root of their height, divided by the total area of the room’s enclosing surfaces. It is the room’s ventilation. A fully developed fire is starved of air rather than of fuel, and Kawagoe’s measurements in the 1950s showed that it then burns at a rate set by the air flowing through its openings, which goes as $A_v\sqrt{h}$ — the height because a taller opening lets hot gas out at the top and cold air in at the bottom at once.

The thermal inertia of the linings, $b = \sqrt{\rho c \lambda}$: how readily the walls, floor and ceiling take heat out of the gas. Dense concrete soaks heat up and has a $b$ near 2,000. Lightweight insulating board does not and has a $b$ of a few hundred. A room lined with the second keeps its heat in the fire.

The fire load, $q_{t,d}$, in megajoules per square metre of the enclosing surface: how much there is to burn.

The square root in the opening factor is the physics of a hot room. Hot gas leaves through the top of an opening and cold air enters through the bottom, driven by the difference in density between the gas inside and the air outside. That pressure difference grows with the height over which it acts, the velocity it drives goes as the square root of the pressure, and so the mass of air through a window goes as its area times the square root of its height. A tall narrow window ventilates a fire better than a low wide one of the same area. Dividing by the total enclosing area $A_t$ spreads that air over the surfaces the fire has to heat, so a large room with the same window is a less ventilated room.

The fire load is per square metre of those enclosing surfaces, not of floor, which makes the numbers look smaller than an occupancy’s survey value. An office floor is typically assessed at a characteristic 511 MJ per square metre of floor. A room 10 m square and 3 m high has 100 m² of floor and 320 m² of enclosing surface, so that becomes about 160 MJ/m² of enclosing surface before any factors for sprinklers or the risk of ignition are applied. The 200 MJ/m² used throughout this page is an ordinary office, not a warehouse.

The first two combine into one time scale, $\Gamma = (O/b)^2/(0.04/1160)^2$, which stretches or compresses the fire’s heating. A room with $O = 0.04$ and $b = 1160$ has $\Gamma = 1$, and its fire heats very nearly as the furnace does — within 25 °C at ten minutes and within 10 °C from twenty minutes to sixty. The furnace, read this way, is one particular room, and not a very unusual one. The third number decides when that heating stops: the fire burns for $t_{max} = 0.2\times10^{-3}\,q_{t,d}/O$ hours, and then cools at a rate fixed by how long it was.

Which free body produced the number #

The free body is the gas in the room, and the balance on it explains why the opening factor and the linings appear together as $O/b$.

Heat enters the gas from the burning fuel, at a rate set by the air the openings admit, so in proportion to $A_v\sqrt{h}$. It leaves by three routes: out through the openings with the hot gas, into the walls, floor and ceiling, and into anything else in the room, steel included. The walls are the largest sink for most of the fire, and a wall of thermal inertia $b$ exposed to a sudden rise in temperature absorbs heat at a rate proportional to $b$ and to the square root of time. Dividing the heat supplied per unit of enclosing surface by the wall’s ability to absorb it leaves the ratio $O/b$, and squaring it turns the wall’s square-root-of-time behaviour into a scale on time itself. Γ is the ratio of how fast the room is fed to how fast its walls can eat, and the fire’s temperature is whatever balances the two.

The steel is not in that balance. A member’s heat capacity is small next to the linings’, so the gas temperature is calculated first and handed to the section-factor calculation afterwards, exactly as the furnace curve was. The member responds to the fire and does not affect it.

A larger window, a hotter and shorter fire #

The first figure holds the fuel and the walls fixed and changes only the window.

The time scale is worth reading as a clock before looking at the curves. The heating branch is a single function of stretched time, $\Gamma t$, so a room with $\Gamma = 4$ goes through the furnace’s history four times as fast. The fire with an opening factor of 0.08 has exactly that: after 15 minutes its gas is at 944 °C, which is the furnace at one hour, and after 30 minutes it is at 1,048 °C, which is the furnace at two hours, to within a degree. A well-ventilated room does not have a different kind of fire. It has the furnace’s fire, run fast, and stopped early.

With a small opening, 0.02, the fire is starved of air. It burns slowly, peaks at 841 °C, and takes two hours to use its fuel; the gas is not back to ambient for more than seven hours. Double the opening and the fire burns twice as fast: 944 °C after an hour, out by 171 minutes. Double it again and it reaches 1,048 °C in half an hour and is out in an hour and a half. More air makes the same fuel burn hotter and sooner, and the total heat released is the same in every case, because the fuel is the same.

At 0.14 something changes. The ventilation-controlled duration would be 17 minutes, which is shorter than the growth limit of 20 minutes that the Annex sets for a medium-growth fire: the fuel is gone before the air could have been the constraint. The fire is fuel-controlled, the Annex switches to a limiting time scale, and the peak falls to 900 °C at 20 minutes, with the gas at ambient by 37. Beyond an opening factor of 0.12, for this fire load, the window stops mattering and the fuel decides.

The steel inside one fire #

A member in that room does not see the gas temperature. It sees what reaches it through its surface, and the section factor and the protection decide how fast that is.

The bare member is a thin sheet of steel folded into a beam — the shape that makes it efficient in bending is what gives it so much surface per kilogram — and it follows the gas so closely that its peak, 940 °C, is four degrees below the fire’s and a fraction of a minute later. The protected member is a different object. Its board has a low conductivity and a real heat capacity, so it lets heat through slowly and stores some of it. When the gas starts to cool at 60 minutes, the steel is at about 650 °C and the board around it is much hotter than that. The heat already in the board keeps flowing inward until the gas has fallen far enough that the flow reverses. That happens at 88 minutes, and at that moment the gas is at 709 °C and the steel at 714: the member peaks at about the instant the fire around it has cooled to its own temperature.

A furnace test cannot show that half hour, because the furnace never cools. It is the part of a real fire in which a protected member does the most dangerous thing it will do, and the fire engineer has to find it by calculation.

Two members, two worst fires #

If each of those fires is run against the same two members, the answer to “which fire is worst” depends on which member is asked.

The bare member’s worst fire is the hot one and the protected member’s is the long one. A bare section has no delay to exploit: it reaches whatever the gas reaches, so the fire with the highest peak is the one that threatens it, and more ventilation is worse up to the point where the fuel takes over. A protected section has half an hour or more of delay, and a fire that is over in forty minutes has cooled before the board has let most of its heat through. The fire that threatens it is the one that stays hot for long enough to soak the board through — the badly ventilated fire, cooler at its peak and far longer.

The two curves cross the critical temperature on opposite sides. Every one of these fires takes the bare member well past 558 °C. None of the fuel-controlled fires takes the protected member there, and the most starved of the ventilation-controlled ones takes it to 734 °C. Choosing the design fire therefore means choosing it for the member: a single “worst case” for the room does not exist.

What the board must be, for the whole fire #

The design question for a protected member is how much board lets it see a given fire out, and the furnace answer and the room’s answer are not the same number.

The most instructive point on that curve is at 0.04. That fire is, for its first hour, very nearly the furnace. A member given its 60-minute rating, 20.9 mm, reaches 558 °C at sixty minutes in the furnace, which is what the rating means. In the room the fire then begins to cool, and a member with 20.9 mm goes on heating past 558 °C during the cooling. Seeing that fire out takes 31.4 mm, half as much again as the rating, and every millimetre of the difference is for the part of the fire after its peak.

The rating is not wrong. It answers the question it was built for: how members compare in a standard test. What it does not do is say what survives a particular room, and the room can ask for less — 8.3 mm for the well-ventilated fire that burns out its fuel — or for nearly two and a half times more.

More fuel, a longer fire #

The third property of the room is how much there is to burn.

Through the same window, doubling the fuel doubles the duration and raises the peak by only about a hundred degrees each time, because the heating curve is flattening by the time a long fire reaches its end. The protected member is far more sensitive than the gas is, because the extra duration is exactly what the board cannot outlast: its peak rises by 168 °C from 100 to 200 MJ/m², and by 210 °C from 200 to 400.

Fire loads are statistical quantities, taken as a high fractile of surveyed values for an occupancy, and they belong to a use rather than a building. A room designed as offices and later used as an archive has changed the most sensitive input in this calculation without anybody touching the structure.

Walls that give heat back #

The linings are the property of a room that a structural engineer is least likely to be asked about and that moves the answer most.

A concrete room takes a large share of the fire’s heat into its walls, floor and ceiling, and the gas stays cooler. A room lined with lightweight insulation — the kind installed to save energy, and increasingly to meet acoustic and thermal standards — hands almost all of it back, and the same fuel burning through the same window reaches 1,265 °C instead of 767 °C. The protected member follows, from 629 °C to 857 °C, which is the difference between a member that survives and one that does not. Improving a building’s thermal envelope is a change to its fire, and it is made by people with no reason to think so.

Why the furnace is still in every code #

A parametric fire needs to know the room, and the furnace does not, which is both its weakness and why it survives.

A product certified in the furnace has a number that is the same in every building. A parametric calculation has a number that belongs to one room, with one window area and one lining and one assumed fire load, and it is wrong the day any of the three changes. A standardised test earns its place by being the same test for everybody, and a fire engineer proposing a parametric design is proposing to trade that portability for accuracy. The trade is usually worth making for a large or unusual building and rarely for an ordinary one.

The two approaches also disagree in a consistent direction, and it is worth stating plainly. For bare steel and well-ventilated rooms, the furnace is conservative: real fires burn out before the furnace would stop. For protected steel in poorly ventilated rooms with plenty of fuel, it is not, because the furnace has no cooling phase and the protected member’s worst half hour is in the cooling.

Where the model stops #

The fire is fully developed and uniform. Annex A describes a compartment after flashover, with one gas temperature everywhere. It covers rooms up to 500 m² of floor and 4 m high; a large open-plan floor burns as a travelling fire, moving across the space, and no single temperature describes it.

The openings are fixed. Windows are assumed to have broken and to stay broken. A glazed façade that holds for twenty minutes and then fails gives a different fire in each of its phases.

The member is a single lump and the board is a constant. The heating calculation is the lumped model of the section factor, with the board’s conductivity fixed, which is a reasonable account of a board and a poor one of an intumescent coating, whose conductivity changes as it swells.

And temperature is not structure. The member’s peak temperature decides its strength at that moment, not whether the frame stands. A member restrained against expansion builds compression as it heats and, if it has yielded, pulls on its connections as it cools, so the cooling phase that these figures show for temperature is also a phase of new forces, and they arrive at the joints. A connection is neither a pin nor a rigid joint even at room temperature, and its behaviour is spread over a region rather than a point; in a cooling frame that region is where the shortening beam pulls. Several connection failures in fire tests have happened after the fire had gone out.

Still open: the forces that arrive as it cools #

The cooling phase is where the open questions begin. A restrained steel beam that yielded in compression while it heated, followed through the cooling phase of a parametric fire until its shortening tears the bolts from its end plate. The travelling fire across a large floor, where every member sees a short hot fire at a different time and no one room exists to take an opening factor from. Intumescent coatings, whose protection is a property of the fire’s heating rate as well as its peak. And concrete, where the lumped model fails and the cooling phase brings spalling and a loss of strength that does not recover when the room is cold again.