The force nobody put in the model

A yield-line calculation is an upper bound on the load a slab can carry, and it is usually a good one. It is also, for a slab whose edges are held by the rest of the floor, wrong by a factor of about four — in the safe direction, which is the reason it has stood for seventy years without anybody minding very much. What the calculation leaves out is not a subtlety about the reinforcement. It is a force in the plane of the slab that nobody applied and nobody drew.

Two mechanisms are on that curve, they have opposite signs, and the same restraint produces both.

Which free body produced the number #

Take half the strip, cut at mid-span, and draw what crosses the cut.

Before anything moves, the answer is a shear and a moment, and the mechanism analysis proceeds from there. Now let the strip deflect by $w$ at mid-span and ask a purely geometric question: how long is the strip’s own centreline now?

For a parabolic deflected shape it is longer than the straight chord by

$$\Delta = \int_0^L \tfrac{1}{2}(v')^2\,dx = \frac{8w^2}{3L}$$

At $w = 48$ mm on a 4 m span that is 1.54 mm. The strip has to accommodate 1.54 mm of extra length between two supports that have not moved, so it must shorten by that amount — and a strip that is being shortened against its will is a strip in compression.

That compression is not applied by anybody. It is generated by the geometry, resisted by the strip’s own axial stiffness and by whatever the surrounding floor supplies, and it is exactly the same mechanism that makes an arch an arch: a member that cannot spread must thrust.

The arch inside the depth #

Once the thrust exists, the free body at each hinge carries a compression as well as a moment, and the compression has a lever arm.

At mid-span the compression sits near the top of the section; at each end hinge it sits near the bottom. The vertical distance between the two thrust positions is the arch’s rise, and it is limited by the depth the strip has left after deflecting:

$$r = \frac{t - w}{2}, \qquad N \le \frac{f_c\,b\,(t-w)}{2}$$

so the load the arch carries is $8Nr/L^2$, which for a plain strip with no reinforcement at all is $2f_c b(t-w)^2/L^2$. A restrained strip with no plastic moment whatsoever carries 86.6 per unit width, against a yield-line load of exactly zero. That single comparison is the cleanest statement of what membrane action is: it does not need the section to have any bending capacity, because an arch does not.

The whole enhancement therefore lives in the depth the strip has left, and the quickest way to see how completely is to take some of that depth away. The same 4 m span, the same bars, the same plastic moment — and 150 mm of concrete instead of 200.

Nothing in the flexural calculation moved: the flat line sits at 30.0 in both pictures. What moved is the height of an arch nobody drew, and it moved with the square of the depth left after deflecting. That is the sense in which membrane action is a property of the section’s spare geometry rather than of its strength — and it is why the reserve is largest in the thick transfer slabs where the strength was never in doubt.

The peak is where two opposing trends cross. At small $w$ the thrust is small because the geometric shortening is small; as $w$ grows the thrust grows quadratically but the lever arm falls linearly, and the crush limit takes over. The product peaks at $w = 48$ mm — 0.24 of the thickness, one part in 83 of the span — where the thrust has reached 2,280 kN per metre of width and the lever arm is 76 mm.

The deflection is smaller than anybody would notice #

That number deserves to be sat with, and it is worth reading the whole of the rising branch rather than only its top. A 4 m span is allowed about 16 mm at its serviceability limit, and at 16 mm the restrained strip is already carrying 1.39 times its yield-line load. At 24 mm it is at 1.84 times, at 32 mm at 2.43, and the peak of 3.89 arrives at 48 mm — three times the serviceability deflection, and still only a quarter of the slab’s own thickness.

So the enhancement is not something a slab reaches only in extremis; a third of it is present before a floor has moved further than a code allows. That is one reason load tests on real floors so consistently outperform their own calculations — the same gap the slab essay records and attributes elsewhere. The other explanations — strain hardening, tensile strength ignored in the analysis, two-way action counted conservatively — are all real and none of them is worth a factor of four.

It is worth being precise about what the flat line in those two figures is. A yield-line load is an upper bound on the load that mechanism can carry, computed from a hinge count and a work equation. Being an upper bound on one mechanism makes it a lower bound on nothing whatever once the ends are held, because the restrained strip is not carrying by that mechanism at all — it is carrying by a second one the work equation never wrote down.

The restraint that has to exist #

Everything above needs the ends held, and the honest question is what holds them.

In a real floor, the surrounding slab does. An interior panel is restrained by the panels around it, which are themselves restrained, and the whole plate is a self-equilibrating raft in which the thrust from one panel is the reaction of another. An edge panel has much less restraint and a corner panel less still, so the enhancement is largest exactly where the reinforcement is heaviest and smallest at the edges, which is the wrong way round from a designer’s point of view. It is the stiffest path taking the load applied to a floor plate rather than to a set of beams.

The generator refuses to draw the unrestrained case as anything but the flexural load: with the end springs made soft, the peak falls to 30.0, which is the yield-line value to the last figure. That is not a result — it is the check that the model has not put anything in by hand.

Two consequences follow. Membrane action is a property of a floor, not of a slab, so it cannot be computed from a panel in isolation. And it is destroyed by the things that destroy in-plane continuity: a movement joint, a large opening, a demolition, an edge beam that is itself free to spread.

Where the arch runs out #

Past the peak the lever arm shrinks faster than the thrust can grow, and at $w = t$ there is no rise left at all. The arch branch has fallen back to exactly the yield-line load, which is the check the model makes at that station: with $r = 0$ the arching term is zero and only the plastic moments remain.

But the strip has not been sitting still while that happened. The same geometric extension that generated the thrust is still growing, and once the section has cracked through, the material available to resist it is the reinforcement. At $w = 0.6t$ the demand has exceeded what 800 mm²/m of 500 N/mm² bar can supply, the tie has yielded, and it is carrying its full 400 kN per metre.

So the two branches overlap. The catenary overtakes the arch at 0.6 of the thickness, when the arch still has 40% of its lever arm; switching between them at $w = t$ would put a step in the curve, and the honest picture is where the two cross. The minimum on the curve is 54.0 — still 1.8 times the yield-line load, and the strip never falls back to it.

The cable that a floor becomes #

Beyond the crossing the strip is a cable. Its capacity is $8Tw/L^2$ with $T$ the tie’s yield force, so it rises linearly with deflection and has no maximum at all — it fails when the reinforcement fractures or when its anchorage pulls out, whichever comes first.

That is a robustness argument rather than a design one, and the number that makes it so is where the catenary gets back to the arch’s peak: 2.17 thicknesses of deflection, which is 434 mm, one part in 9 of the span. Nobody designs for that. A floor that has dropped a ninth of its span is a floor that has visibly failed, and it is still holding.

The one thing that moves that number is the tie itself, and it moves it a long way. Doubling the reinforcement leaves the arching branch untouched, since the arch is carried by concrete in compression and does not care what steel is in the section, and halves the sag the catenary needs.

Half the sag for twice the steel is a poor return on a design curve and an excellent one on a robustness argument, because the quantity that matters after a column has gone is how far the floor has to fall before it catches itself. The bars are not being asked to make the floor stronger; they are being asked to shorten the drop.

The essay on robustness names catenary action as the mechanism a building falls back on when a column goes, and this is that mechanism computed. Its two requirements are both about detailing rather than about capacity: continuity of reinforcement through supports, so that a tie exists at all, and anchorage able to develop it, so that the tie can reach yield rather than pulling out of a lap.

The same curve, elsewhere #

The comparison is exact in form and instructive in its difference. A shallow frame that snaps through has a maximum load, a falling branch and a region where it can carry nothing; a restrained slab strip has a maximum load, a falling branch, and a rising branch beyond. Both are geometric-nonlinear problems in which the load path changes kind as the structure moves, and neither is visible to any analysis that assumes small displacements.

That is the general lesson. A structure whose geometry is part of its load path has a capacity that depends on how far it has moved, and the linear analysis that computes forces from an undeformed shape has no way to represent it — in either direction.

There is a second lesson in the frame, and it is about the approximations that get made in place of the exact path. The standard closed form for a snapping frame’s maximum load, $2EA(h/a)^3/3\sqrt{3}$, is derived by assuming the frame is nearly flat at every step.

That is the shape of every geometric approximation in this subject, membrane action included. The arithmetic on this page assumes the strip is shallow enough for its chord shortening to be a quadratic in the deflection, and that assumption is best at the peak — 0.24 of a thickness, a two-hundredth of the span — and worst out on the catenary branch, where the strip has moved a ninth of its span and nothing about it is shallow any more. The reserve at the peak is computed; the reserve at the recovery is an argument.

How much of it survives being asked for #

Membrane action is largest on short spans and shallowest on long ones, and the trend is worth tabulating because it is the reverse of what a designer would guess.

The arching term goes as $t^2/L^2$ and the flexural term as $1/L^2$, so the ratio would be independent of span if the thrust were crush-limited throughout — and it is not, because on a longer span the geometric shortening at a given fraction of the thickness is smaller, so the thrust is demand-limited over more of the range and the peak moves outward. The effect is worth most where a slab is thick relative to its span, which is to say in exactly the flat-slab and transfer-slab construction where nobody is short of capacity.

The general form of the argument is one this collection has met before, under a different cause. A member prevented from changing length develops force instead of movement, and the size of that force is set by the restraint rather than by the member.

The parallel with thermal restraint is exact and is the most useful way to file this. In both cases a length change is demanded by something other than a load, the restraint refuses it, and a force appears whose magnitude is set by the restraint’s stiffness rather than by anything in the loading. Where they differ is the sign of the consequence: a restrained thermal expansion is a nuisance, and a restrained deflection is a gift.

What the thrust does to everything else #

A force of 2,280 kN per metre of width does not confine itself to the mechanism that generated it. Two things happen to the rest of the slab, and they point in opposite directions.

The shear capacity rises. Spread over a 200 mm thickness, the peak thrust is 11.4 N/mm² of mean in-plane compression, which is a large fraction of the concrete’s design strength. Compression closes the diagonal cracks that a shear failure has to open, and every code’s shear expression carries a term that adds capacity in proportion to the axial stress for exactly that reason. So the arching state is one in which the slab is stronger in shear as well as in bending.

But the bending capacity rises much faster. Flexure gains a factor of 3.89 and shear gains a fraction of that, so the ratio between them moves — and it moves toward shear. A slab whose yield-line calculation said it would form hinges and rotate may, once the arch is counted, reach its shear capacity first. That is the part worth being careful about: the ductile mechanism has been enhanced into a brittle one. The reserve computed on this page exists only if the slab survives to use it, and a punching failure at a column head takes the whole curve — arch, catenary and all — off the table at once.

And the thrust has to be resisted somewhere. It leaves the panel horizontally at both ends, and at an interior panel it is absorbed by the next panel’s equal and opposite thrust, which is why the raft is self-equilibrating and why nothing outside it ever sees the force. At the boundary of the floor plate it is not absorbed by anything, which is the honest reason edge panels arch so poorly — not that their restraint is soft, but that the reaction has nowhere to go. Where a perimeter beam or a column line does take it, it takes it as a horizontal force of a magnitude comparable with the column’s vertical load, applied at floor level, and appearing in no analysis anybody ran.

That last observation is the one to keep. The enhancement is real, it is large, and its equal and opposite is a force delivered into members that were sized without it. A reserve that is invisible in the calculation is invisible on both sides of the equals sign.

Where the model stops #

The strip is rigid-plastic and one-dimensional. A real slab spans two ways, its membrane forces are a field rather than a line, and the compressive arch in one direction interacts with the tensile ring around the outside. The two-way case has a compression zone in the middle and a tension ring at the edges, and its enhancement is larger than a one-way analysis gives.

The restraint is taken as rigid. With finite restraint the thrust is smaller, the peak lower and further out, and the whole curve flattens. Real restraint is neither rigid nor knowable, and this is the largest uncertainty in the subject by a wide margin.

No concrete crushing softening is included. The arch is capped by a rectangular stress block at $f_c$, and a real compression zone at that strain is already spalling. The peak is therefore optimistic and the falling branch steeper than drawn.

Nothing here is a design method. Codes generally ignore membrane action for exactly the reasons above, and the honest position is that it is a large, real, unquantifiable reserve rather than a capacity to be spent.

And the enhancement disappears at fire. A heated slab expands, which relieves the very geometric shortening that generated the thrust, and then loses stiffness so that the deflection grows — so a fire moves a floor along this curve from the arching branch to the catenary one, which is why a floor in a fire hangs rather than arching and why the tensile branch is the one fire engineering is interested in.

And the catenary branch has no plateau to find. A cable resists load by changing shape, so its stiffness depends on how far it has already sagged, and the tensile branch of every curve above rises with deflection instead of levelling off. Nothing about that shape looks like a strength limit, which is precisely why it is unusable as one: the load a catenary carries is whatever the deflection it has been allowed to reach will support.

What the pictures cannot show #

The load-deflection curve is drawn as a single line, and a real restrained slab does not follow one: it cracks, and each crack is a small drop and a redistribution. The curve is an envelope of a jagged process and its smoothness is an artefact of a rigid-plastic idealisation.

Nothing in the figure shows the in-plane force directly, which is the whole subject. The thrust is 2,280 kN per metre of width at the peak — a force comparable with what a column carries — travelling horizontally through a slab that is drawn as a line. There is no axis on the figure it could be plotted against, and it is invisible for the same reason a bimoment is: the drawing shows a mechanism, and this is a stress resultant the mechanism did not ask for.

And the horizontal axis is a deflection in thicknesses, which flatters the picture. Reading it in spans instead: the peak is at $L/83$, the crossing at $L/33$, and the recovery at $L/9$. Only the first of those three is a structure anybody would still be standing under.

The ladder from here #

Later rungs on this anchor: the two-way case, with its compression dome and its peripheral tension ring, and the Bailey method that computes it. The restraint stiffness as a design variable, and what an edge beam has to be to supply it. Membrane action in composite steel-concrete floors at fire, which is where the whole subject became a design method rather than a curiosity. Tie force requirements in codes, and the fact that they are specified as forces with no analysis behind them precisely because the analysis is intractable. The dynamic version, where a column is removed suddenly and the floor has to reach its catenary state through an impact rather than a slow deflection. And the negative case: the slab that is restrained on three sides and free on the fourth, where the thrust has nowhere to go and the whole enhancement disappears.

The finding worth carrying furthest is not the factor of four. It is that a structure’s real capacity can depend on a force that appears in no drawing, is applied by nobody, is generated by the structure’s own movement, and is destroyed by anybody who cuts a joint through it.