The edge that cannot hold the arch

The force nobody put in the model is the thrust that appears in a slab strip whose ends cannot move apart. As the strip deflects, the chord between its ends wants to shorten; held, it cannot, and the strip is forced into compression — an arch inside its own depth, pushing on its supports, carrying load by the thrust times a lever arm that no bending calculation contains. For a 4 m strip 200 mm thick, whose yield-line load is 30 kN/m, the arch is worth nearly four times as much, at a deflection of a few centimetres.

That essay held the ends rigidly, and said in a sentence that real ends are held by the surrounding slab, that an edge panel has less restraint and a corner panel less still. It also noted that with the end springs made soft the peak falls back to the flexural value. Both statements are about a stiffness, and the question they leave is how much: how stiff must the thing at the end of the strip be before the arch is worth counting?

A strip between two springs #

The strip is the same: 4 m long, 200 mm thick, a metre wide, uncracked concrete for its compression, its plastic moment 30 kN·m per metre. Its ends are now held by springs, and the natural measure of a spring’s stiffness is the strip’s own axial stiffness $EA/L$ — 1,500 kN/mm for this strip — because the strip and the springs share the job of resisting the arch’s demand for a shorter chord.

Held rigidly, the strip peaks at 116.8 kN/m, 3.89 times its flexural load, when it has deflected a quarter of its thickness, 48 mm. With springs of five times its own axial stiffness it gets most of that: 108.6 kN/m. With springs of half its axial stiffness it gets 74.8, two and a half times the flexural load. With springs of a twentieth it gets 45.3, half as much again as flexure — and it gets it only after sagging by two-thirds of its thickness, 133 mm, a deflection nobody would accept and at which a real slab would long since have cracked through.

The arch is only as strong as what holds its ends, and what holds them has to be stiff in the same terms as the strip itself.

Half the gain for half the stiffness #

The reason is a pair of springs in series. The arch needs the chord between the strip’s ends to stay the length it was; the chord shortens by the strip’s own compression and by the ends’ movement outward, and both are in series against the same thrust. A restraint as stiff as the strip halves the thrust that a given deflection produces.

On a logarithmic scale the enhancement is an S-curve across four decades of stiffness. A hundredth of the strip’s axial stiffness buys 17 per cent. Half the full gain needs 0.56 times $EA/L$; nine tenths needs 5.6 times. The thing at the end of the strip has to be comparable in axial stiffness to the strip itself before the arch is mostly there, and an order of magnitude stiffer before it is nearly all there.

That is the same arithmetic the stiffest path takes the load runs between members, and the same one that decides how much thrust an arch loses to its own shortening: wherever a force has to pass through two elastic things in series, the softer decides, and a restraint is just a second elastic thing in series with the strip.

A soft restraint makes the arch wait #

The softer restraint does not just lower the peak; it moves it.

A rigidly held strip pushes back as soon as it deflects, and its thrust builds fast; it peaks at a quarter of its thickness, while it still has most of its depth to arch in. A softly held strip lets its ends move out first, so it sags further before the thrust is large, and by the time the thrust is large the strip has used up its lever arm: the arch’s depth is the thickness less the deflection. Below about a twentieth of $EA/L$ the peak is pinned at two-thirds of the thickness, the deflection at which the crushing block and the shrinking lever arm meet, and the enhancement that arrives there is small.

So a soft restraint costs twice: less thrust for a given deflection, and less depth left when the thrust comes. Restraint that is not stiff enough does not deliver the arch late; it delivers a smaller arch late, which is the difference between a reserve and a hope.

What the restraint has to carry #

Stiffness is the first requirement; strength is the second, and it is large.

At the rigid peak the strip pushes on its supports with 2,280 kN per metre of width — 2.3 meganewtons on every metre of the panel’s edge. Whatever provides the restraint has to carry that as well as be stiff, and has to carry it without moving, because moving is what softens it. In the middle of a floor the force is balanced by the panel on the other side, which is pushing back with its own thrust: a slab is stiff and strong in its own plane, and an interior panel is held by a plate that is itself held. At an edge there is nothing on the other side.

Held by the floor, but only against one panel #

“Balanced by the panel on the other side” is true when one panel is overloaded and its neighbours are not, which is the situation the arch is usually counted for: a column removed beneath one bay, a fire in one compartment, one panel carrying a concentrated load. The neighbours are then lightly loaded plates, wide in plan, and the thrust spreads into them as it would into a deep beam; their in-plane stiffness against a push along one edge is many times one strip’s $EA/L$, and the overloaded panel sits far up the S-curve.

When every panel is loaded at once the arithmetic changes. Each strip’s thrust pushes on the next strip, which pushes back with an equal thrust of its own, so the interior supports do not move — but the line of strips as a whole pushes on the edges of the floor, and the thrust is the same all the way along it. The interior panels are held only as stiffly as the two ends of the line are held. If the line is $n$ panels long and each end is an edge band of stiffness $k$, the band’s movement is shared out among the $n$ strips, and each strip behaves as though its own ends were held by springs of $n$ times $k$. With the 2 m band at 8 m column centres, one panel between two bands reaches 1.51 times its flexural load, a line of five reaches 2.01, and a line of eight 2.34. A uniformly loaded floor is an edge panel at the scale of the floor: it gets the arch only as far as its perimeter can hold the summed thrust, and the perimeter is the softest thing in it.

That is the reason the enhancement is credited to a local event and not to a floor’s general capacity. Overload one panel and the floor around it is the restraint; overload all of them and there is no floor around them, only an edge.

The same restraint, read from the other side #

The stiffness that a slab’s arch needs is the stiffness that a fire turns against a beam. A steel beam held at its ends while it heats is pushed into compression by its own expansion, yields, and ends the fire in tension — and the restraint that does that is the same surrounding structure, measured in the same units, as a fraction of the member’s own axial stiffness. A restraint at a tenth of a beam’s $EA/L$ is enough to yield it in a fire; the same tenth of a slab strip’s $EA/L$ gives that strip a quarter of the arch it could have.

The two problems are mirror images. The arch wants the ends held so that the chord cannot shorten; the heated beam wants them free so that the member can lengthen. Any edge stiff enough to give a panel its arch is stiff enough to put a hot beam along the same line into compression, and a designer who counts on one is committed to the other — through the whole of a compartment fire’s rise and decay, and through the cooling that reverses the force on connections designed for neither.

The edge panel #

An edge panel is held on its inner side by the floor and on its outer side by whatever runs along the edge — here, a band of the same slab, bending in its own plane between columns, as a horizontal beam spanning from one column to the next and loaded by the strip’s thrust.

The band’s stiffness against a uniform push is the deflection of a simply supported beam turned on its side: $384EI/5L^4$ per unit length, with $I$ its thickness times its width cubed over twelve and $L$ the column spacing. A band 2 m wide between columns 8 m apart is 75 kN/mm per metre of strip — a twentieth of the strip’s own 1,500 — and the edge panel peaks at 1.69 times its flexural load, against 3.89 for an interior one. Columns 6 m apart make the band nearly three times stiffer and give 2.17; 10 m apart, 1.46.

Widening the band helps quickly, because its stiffness goes as its width cubed: a band 4 m wide — a fifth of a slab panel’s width given over to stiffening the edge in its plane — gives 3.41 at 6 m and 2.81 at 8 m. But a 4 m band is not an edge detail; it is most of an adjoining panel, and it is itself the edge of something. An edge band of ordinary width is soft beside the strip it is asked to hold, which is the arithmetic behind the earlier essay’s warning that the enhancement is smallest at the edges, and a measure of how small.

A corner panel is held softly on two sides, and its arch has less still. The arch of a floor is distributed the opposite way to its demand for robustness: largest in the middle, where a slab that loses a support has neighbours all round, and least at the edges and corners, where a lost column is also a lost restraint.

That is the inverse of where the reserve is needed. A structure survives losing a member when the remaining parts can find another path, and the columns most exposed to being lost — to a vehicle, to a blast at the façade, to a fire that starts at a window — are the perimeter columns. The panel that a perimeter column used to carry is an edge panel, held on its outer side by a band that has just lost one of its supports and now spans twice as far: the band’s stiffness falls by the fourth power of its span, to a sixteenth, and the arch falls to almost nothing. What carries that panel is catenary tension in ties, at deflections many times larger, and not the compression that the interior panels would have offered.

Longer strips gain less, however they are held #

The span of the strip enters twice, and the two effects pull in opposite directions.

Held rigidly, a longer strip gains less: 4.37 at 3 m, 2.67 at 8 m. Its chord shortens by the square of the deflection over the span, so a longer strip has to sag further before the shortening is enough to drive the arch, and it sags into its own lever arm. Held by an edge band, a longer strip is held relatively more stiffly, since the band’s stiffness does not depend on the strip’s span while the strip’s own $EA/L$ falls with it — a twenty-fifth of $EA/L$ at 3 m, a tenth at 8 m — but the longer strip had less to gain in the first place, and ends at 1.22. The arch is a short-span phenomenon in the middle of a floor, and a small one anywhere else.

The edge band, by hand #

The band’s stiffness, per millimetre of edge, is

$$k = \frac{384\,E\,I}{5\,L^4} = \frac{384 \times 30{,}000 \times (200 \times 2000^3/12)}{5 \times 8000^4} = \frac{384 \times 4.0 \times 10^{15}}{5 \times 4.10 \times 10^{15}} = 75\ \text{N/mm per mm},$$

75 kN/mm for a metre of strip. The strip’s own axial stiffness is $E\,b\,t/L = 30{,}000 \times 1000 \times 200/4000 = 1.5 \times 10^6$ N/mm, 1,500 kN/mm. The band is a twentieth of it.

At the other end the floor is stiff, so the strip and the band are the only two things in series: the chord shortening the arch resists is taken up by $L/EA + 1/k$, and the band’s term is twenty times the strip’s. The thrust a given deflection produces is a twenty-first of what rigid ends would give — and the figure’s 1.69 against 3.89 is the result of that, on a curve that also runs out of lever arm.

The column spacing enters at the fourth power, which is why it moves the answer more than anything else on the edge. Bring the columns in from 8 m to 6 m and the band’s stiffness rises by $(8/6)^4$, a little over three, to 237 kN/mm: still a sixth of the strip’s own, and worth 2.17 rather than 1.69. Take them out to 10 m and it falls by $(10/8)^4$, about two and a half, to 31 kN/mm, a fiftieth of the strip, worth 1.46. Widening the band works through its cube and spacing the columns through its fourth power, and neither is a free variable once a building has been planned.

Uncracked, elastic, one-way #

The calculation rests on choices that limit it.

The strip and the band are uncracked. A slab that has been loaded to several times its flexural load is cracked through at its hinges, and an edge band that is bending in its own plane under two meganewtons is cracked on its outer face. Both are softer than their uncracked stiffness, the band proportionately more, which makes the edge panel’s enhancement smaller still. How much softer is not a property of any one section: between the cracks the concrete still carries tension, and the stiffness averaged along the band sits somewhere between the cracked and uncracked values, in a place set by the least reliable property of the concrete.

The strip is one-way. A real panel spans both ways and arches both ways, with a compression dome in its middle and a ring of tension round its edge: the two-way arch, which Bailey’s method computes for composite floors in fire, can hold itself in with its own reinforcement where a one-way strip needs something outside it.

The restraint is a linear spring. A real edge is a column, a beam and a slab connected by things that slip, and a gap of a few millimetres at a joint is a spring of zero stiffness until it closes — which, for an arch that peaks at 48 mm of deflection, may be most of the travel.

Nothing creeps. A thrust of two meganewtons per metre held for a fire’s duration, or for the hours after a column is lost, is a sustained load on concrete, and the restraint’s stiffness decays with it. Creep belongs to the member that is loaded, and here both the strip and the band are loaded hard in compression: over hours the chord shortens, the restraint gives, and the arch relaxes towards the lower curve of the figures above.

Still open: the slab that holds itself in #

A strip needs something outside it to push against. A two-way panel does not, in principle: its middle can arch in compression while its edges carry the thrust round as a ring in tension, so the panel is a self-equilibrating dome that needs its supports only for vertical load. That is how a composite floor with no edge restraint at all can carry far more than its yield-line load in a fire. Whether a concrete edge panel, held only vertically along its free side, can develop enough ring tension in its own reinforcement to recover the arch the edge band cannot provide — and how much reinforcement that ring needs at the slab’s edge, where the panel is also most likely to be damaged — is the question of whether the arch has to be held at all, or only closed.