The tension a fire leaves behind

A room’s fire has a peak and an end, set by its window, its walls and its fuel, and a steel member in it follows the gas temperature up and then back down. The earlier essay found that the end matters to how hot the steel gets: a bigger window makes the fire hotter and shorter, and a member that survives the peak has, as far as its strength goes, survived the fire.

As far as its strength goes. A beam in a building is not free to grow when it is heated; its ends are framed into columns and floors that resist its expansion, and the force that resistance produces is not strength — it is a movement nobody applied, held back. The earlier essay ended on the forces that arrive as the fire cools. They turn out to be larger than the ones that arrived as it heated.

A beam that cannot grow #

Take a bare S355 steel beam with a section factor of 160 per metre — an ordinary floor beam, unprotected — in a compartment fire with an opening factor of 0.04 m^½ and 200 MJ/m² of fuel. The steel heats to 940 °C at 60 minutes and cools back to the room’s temperature over the next three hours.

Its ends are held by the structure around it. The holding is a spring: the columns bend, the floor beyond gives a little, and the connections slip, as every real connection does somewhere between pinned and rigid. Its axial stiffness, as a share of the beam’s own cold axial stiffness, is the one number this essay turns on; for a beam in an ordinary frame a tenth is a fair middle value, and the essay sweeps it.

The arithmetic at each moment is a balance. The beam wants to be longer by its thermal strain, the spring says how much of that it gets, and the beam’s stiffness, which falls as it heats, says how hard it pushes. Where the push would exceed the beam’s yield stress at that temperature, which is also falling, the beam yields — it gets shorter, permanently, by whatever it cannot carry.

Out in compression, back in tension #

The path is a loop with a twist in it. On the way up, the beam pushes against its ends with a stress that grows as the steel expands: 44 N/mm² at 200 °C, 108 at 450. Meanwhile its yield stress is falling — it holds its cold value to 400 °C and then drops, to 78 per cent at 500 °C and 47 at 600. The compression peaks at 134 N/mm² at 628 °C, the two meet a few degrees later, and from there the beam can push no harder than its yield stress lets it. It follows the yield stress down as the steel heats on to 940 °C, the compression falling to 19 N/mm², and every millimetre of expansion it cannot push out against its ends is a millimetre it yields by.

On the way down, the thermal strain comes back out. The yield the beam took does not. A beam that has yielded short by 1.02 per cent of its length is, when cold, shorter than the gap it sits in by that much less whatever the spring lets it pull in, and it is held stretched across that gap. It ends the fire at 195 N/mm² of tension, at 20 °C.

When the tension arrives #

In time the story is one that a fire-resistance rating does not see. The compression rises through the first three quarters of an hour and is then capped by the falling yield stress. After the peak at 60 minutes the stress drops through zero at 96 minutes, by which time the gas is well below the steel and the fire is, by any ordinary measure, over. The tension then grows for hours as the steel cools, reaching 195 N/mm², 55 per cent of the beam’s yield stress, by the time it is cold.

The force that a fire leaves behind is larger than the force it caused, and it arrives after the fire has gone out. A fire rating measures the minutes to the critical temperature — an hour that is really a temperature — and the cooling tension happens after that clock has stopped, in a building that may already have been re-entered.

How much the restraint decides #

The restraint is the variable. A beam held weakly — 2 per cent of its own stiffness — barely notices: it reaches 39 N/mm² of compression, yields late and little, and is left at 28 N/mm² of tension. At 10 per cent, 134 and 195. At 20 per cent the beam reaches 220 N/mm² of compression before it yields and is left at 355 N/mm² — its full yield stress in tension, the spring having pulled it back far enough to yield it the other way.

From 4 per cent restraint upwards, the tension at the end is larger than the compression at the height. The asymmetry is the yield stress itself. On the way up the compression is capped by a yield stress that the heat has lowered; on the way down the tension is capped by a yield stress that the cooling has restored. The fire limits how hard the beam can push; nothing limits how hard the cold beam can be pulled except its own full strength. It is the same asymmetry as a bar yielded one way and then the other, with temperature instead of a reversal moving the yield stress between the two halves of the cycle.

Which fires leave tension at all #

Across forty-eight fires, every combination of eight fuel loads and six openings, the tension left behind falls on one curve of the steel’s peak temperature. A fire whose steel stays below about 600 °C leaves nothing: the beam never yields, and everything it expanded it gives back. Above the temperature at which the beam first yields, the tension climbs steadily with the peak — 23 N/mm² from a fire that peaks at 661 °C, 135 at 800, 202 at 950 — and levels off near 270 for the hottest fires. A beam taken that hot starts to pull while it is still hot and weak on the way down, yields in tension there, and gives back part of the shortening it took — so the very hottest fires leave a little less shortening than the line through the cooler ones would predict, and the same tension.

The curve is the same whatever produced the peak — a slow fire with a great deal of fuel, or a fast one through a big window — because the tension is made of the yielding, and the yielding is made of how far past its first yield the steel was taken. The fire’s duration enters only through the peak it reaches.

The tension is the shortening times the restraint #

The whole result collapses to one line. Back at 20 °C with no further yielding, the beam is short by its plastic strain $\varepsilon_p$ and held by a spring of stiffness $\kappa$ times its own, and the tension is

$$\sigma = \frac{\kappa E\,\varepsilon_p}{1 + \kappa},$$

capped at the yield stress. Every tenth of a per cent of plastic shortening is worth 19 N/mm² of tension at a tenth’s restraint. The dots for the forty-eight fires lie on that line exactly until the cap, which is the check that the simulation contains nothing but the shortening and the spring.

So the design question is not how hot the beam got. It is how much it yielded, and how stiffly it is held while it cools.

What the connections are asked for #

The tension does not stay in the beam. It goes to the connections at each end, and most beam-to-column connections in a floor are detailed to carry shear, at a point the beam diagram treats as having no size, though a connection is not a point. Put a size on the stress: a 457 mm deep universal beam of 67 kg/m has a section of about 8,550 mm², and 195 N/mm² across it is 1,670 kN of tension. The robustness rules ask the same connection to carry a tie force of a few hundred kilonewtons — the force sized for the floor to hang across a lost column, as a cable pays back the dynamic factor — — about 270 kN for an 8 m beam at 6 m centres carrying 7 kN/m², by the usual expression — and a fin plate or a partial-depth end plate detailed for that is the connection the cooling beam pulls on with six times as much.

That is the mechanism by which connections fail in the cooling phase of real fires: bolts sheared or end plates torn after the fire is out, in a frame that stood through the peak. The beam’s own strength is not the problem — it is back to its full cold strength — and the column, cold or nearly so, is fine. What fails is the one component sized for a load that was never going to be the largest.

And the force has another end. The tension that pulls on the connection pulls the column too, sideways at floor level, toward the beam that cooled. A column that carried the expansion’s push outward during the fire is pulled back inward afterwards, by more, and an edge column with a fire-affected bay on only one side is pulled one way only. The beam is the member that yielded; the column is the member that is bent by the consequence, at a moment when nobody is checking either.

What protection buys #

Fire protection is specified by the minutes a member must survive the standard fire, and in that currency it postpones the critical temperature. Read against the curve of tension and peak temperature, it does something more definite. A fire whose steel stays below the temperature at which the restrained beam first yields leaves no tension at all — for a tenth’s restraint, somewhere about 600 °C — and protection that keeps the beam’s peak below that in the compartment’s real fire removes the cooling tension entirely, not partly.

That is a different design target from the rating. A protected beam rated for an hour in the standard fire may still reach 700 °C in a compartment fire with a large fuel load and a small window, and the cooling tension it is left with is the one the curve gives for 700 °C, whatever the rating said. A beam whose protection is sized to the compartment’s peak — which the parametric fire gives — can be sized so that the restraint never yields it.

And it is a target the restraint moves. A beam held more stiffly yields earlier — at 20 per cent restraint the compression meets the falling yield stress near 550 °C rather than 640 — so the stiffer the frame around a beam, the cooler the beam must be kept for its connections to be spared.

The beam cut at its connections #

The free body is the beam alone, cut at its two connections, with the restraint replaced by the axial force it exerts. Equilibrium is trivial: the force is the same at both ends. Everything is in compatibility — the beam’s length, made of its thermal strain, its elastic strain and its plastic strain, must equal the gap the restraint leaves it, which is the original span less whatever the spring has given.

The plastic strain is the memory. Every step of yielding while hot is a permanent change to the beam’s length that the compatibility condition carries forward, and when the thermal strain returns to zero it is the plastic strain alone that the restraint has to stretch across. A beam that never yields has no memory, and returns to exactly where it started.

The 10 per cent beam by hand #

With a tenth’s restraint and $E = 210{,}000$ N/mm², the spring’s share is $21{,}000$ N/mm² per unit strain. The fire left the beam 1.02 per cent short. Cold, the beam is stretched by the share of that the spring cannot give: $\sigma = 21{,}000 \times 0.0102/1.1 = 195$ N/mm².

The compression on the way up is the restrained thermal stress until it meets the falling yield stress: at 628 °C the beam’s modulus has fallen to about a quarter of its cold value and its thermal strain is about 0.9 per cent, and the spring in series with the softened beam gives $21{,}000 \times 54{,}600 \times 0.0089/75{,}600 = 134$ N/mm², against a yield stress of 143 there that has fallen to meet it by 640 °C.

What the model assumes #

A spring that does not change. The restraint is the surrounding structure, and in a real fire the columns and floors beside the beam heat too and soften, so the restraint itself falls during heating and partly returns on cooling. A softer restraint while hot means less yielding and less tension afterwards; a restraint that recovers its stiffness fully on cooling, as cold columns do, means the tension is as computed.

Straight members. A beam in compression at high temperature bows sideways or sags, and a beam that sags under its own load while hot shortens its chord without yielding axially — which adds to the shortening the cooling must take back. Catenary sag while hot is a large effect in real floors and makes the cooling tension larger.

And no creep. Steel above 400 °C creeps under sustained stress, and creep is a plastic strain that accrues with time as well as temperature — the hot counterpart of a strain imposed and a stress that leaks away. A long fire therefore leaves more shortening than the instantaneous model gives, and more tension. The model’s numbers are a floor on the shortening a slow fire leaves, not an estimate of it, which is the direction that matters for a connection.

What the curves cannot show #

They cannot show the connection’s own heating. A connection is a mass of steel and bolts that heats more slowly than the beam’s web, and its strength on the way up may be higher than the beam’s; on the way down it is back to its cold strength along with everything else, and the tension finds whatever it was cold-designed for.

They cannot show the floor. A composite floor slab, cast onto the beam, carries part of the restraint and part of the tension, and its own thermal gradient bows it in a way the beam alone does not.

And they cannot show what was done about it. Connections with slotted holes, or end plates thin enough to deform rather than tear, let the beam’s ends draw in on cooling without the force building up — a deliberate softening of exactly the restraint the essay’s figures turn on.

What it comes to #

A restrained beam yields in compression as a fire heats it, near 640 °C after peaking at 134 N/mm² for a tenth’s restraint, because its strength falls while its expansion does not.

The yield is kept; the expansion is not. The beam ends the fire short by 1.02 per cent of its length.

Cold, that shortening is tension. 195 N/mm², 55 per cent of the beam’s yield stress — more than the compression at the height — from 4 per cent restraint upwards the end pulls harder than the peak pushed.

It arrives after the fire is out, on connections designed for shear. For a 457 mm beam, about 1,670 kN against a tie force of about 270.

Still open: the floor that sags while it is hot #

Every beam here stays straight and carries its restraint axially. A real floor beam heated under its load sags, by hundreds of millimetres at 700 °C, and a sagging beam held at its ends is a catenary — it pulls in on its connections while still hot, the same direction the cooling will pull. When it cools it does not straighten fully, so the length it shortened by sagging is added to the length it shortened by yielding. Whether the hot catenary’s tension and the cooling’s tension add in the connection, so that the force the connection must survive is their sum, or whether the sag on the way up relieves enough of the compression that less yielding is left to take back on the way down, is the question a beam with a load on it asks of everything this page has computed for a beam without one.