The slab that spans both ways

A floor slab supported on all four sides has a choice about where to send its load, and the usual account is that it makes both — spanning one way and the other, sharing the work. That is true for a square panel and it stops being true remarkably quickly.

The whole of that curve comes from one equation with one unknown, and the equation is a compatibility statement rather than an equilibrium one.

One equation, and the fourth power does the rest #

Cut the panel into two families of strips at right angles. Every strip in one family crosses every strip in the other, and at the centre both pass through the same point — which must therefore deflect the same amount whichever family is asked.

A simply supported strip of span $L$ under $w$ deflects $5wL^4/384EI$ at its middle. Setting the two families equal:

$$w_x L_x^4 = w_y L_y^4 \qquad\text{with}\qquad w_x + w_y = w$$

and the share follows immediately. Nothing about the slab’s thickness, its material or its reinforcement enters, because both sides carry the same $EI$ and it cancels.

The consequence is the one this collection keeps meeting from other directions: span enters deflection as a fourth power, and a fourth power is a violent thing. A panel only 33% longer than it is wide sends three-quarters of its load the short way; the second direction is doing a quarter of the work for the same reinforcement.

Two-way action is worth having at a ratio of one and worth almost nothing by two, which is the practical rule the curve encodes and the reason panel proportions are a structural decision rather than an architectural one.

The tributary rule says something different #

An earlier essay divided the same panel with lines at 45° and handed the load to the beams around it. That division gives the short direction 62.5% of the load where the elastic calculation gives 76.0%, and the discrepancy is worth being precise about because the two rules are answering different questions.

Neither is wrong. The tributary rule is a statement about what the supporting beams should be designed for, and it is deliberately blunt: it has the property that the loads add up, it is drawable in ten seconds, and its errors are of a size the beams’ own margins absorb. The elastic share is a statement about what the slab does, and it is what decides the reinforcement in each direction.

The pattern is one worth naming, because it recurs. A rule that conserves the total can be badly wrong about the division and still produce a safe structure, provided every member designed from it has some margin — which is what makes crude load paths workable, and what makes them dangerous when the margins are trimmed.

At collapse, none of that arithmetic applies #

Everything above is elastic, and a slab does not fail elastically. Once the reinforcement yields along a line, that line becomes a hinge of constant moment, and the panel goes on carrying load until enough lines have formed to make a mechanism.

The calculation is a work equation and it takes three lines. Give the ridge a unit deflection; the external work is the load times the volume under the resulting shape; the internal work is the moment capacity times each region’s rotation times the projection of the yield lines it turns about. Equate, and the load falls out.

What makes it an upper bound is that the pattern was assumed. Any mechanism gives a load at which the slab could collapse, and the true collapse load is the lowest over all of them — so the position of the lines has to be searched rather than chosen.

The lines are not at 45° and the square panel is why everybody thinks they are #

The minimum in that search sits at 3.413 m from the short edge, and a 45° line from the corner of a 6 m-wide panel would reach the centreline at 3.000 m. The real angle is 41.3°, and the 45° mechanism gives 20.741 kN/m² against the true 20.611 — 0.63% high, which is what an upper bound being slightly too high looks like.

For a square panel the search returns 2.995 m on a 6 m panel — 45° to within the search resolution — and that is the case everybody meets first. The 45° rule is therefore another symmetry result mistaken for a geometric one, exactly as the tributary rule is, and both are exact for a square and approximate for everything else.

That last comparison is the argument for square panels stated in one number. The short span is 6 m in both cases and the short-direction reinforcement is the same; making the panel square rather than 3 : 4 buys 29% more capacity for nothing.

The corners are trying to lift, and holding them down is worth something #

There is a part of a two-way slab’s behaviour that no strip argument contains at all, and it is visible on any slab that has been loaded hard: the corners try to rise.

The reason is torsion. Near a corner where two supported edges meet, the slab is being bent in two directions at once and the twisting curvature between them is at its largest — so the corner region carries torsional moment rather than bending, and the equilibrium of that region requires a downward force at the corner. Where the corner is held down, that force is supplied and the panel is stiffer and stronger than the strip calculation says. Where it is free — a slab simply resting on walls, a precast unit — the corner lifts off, the torsional restraint is lost, and the capacity falls.

Two practical rules come out of it and both look arbitrary until the mechanism is understood. Corner reinforcement — top and bottom steel in both directions over a region about a fifth of the short span — exists to carry the torsional moments the corner is being asked for. And holding the corners down, rather than letting them lift, is worth a few per cent of capacity and considerably more stiffness.

It also makes a connection this collection has been building toward. Torsion is the internal force with no diagram, and here it is doing structural work in the least likely place — the flat corner of a floor slab, where nothing appears to be twisting at all.

What two-way action is actually worth #

The cleanest way to value it is to ask what moment capacity a slab needs for a stated load, and to compare that with the same slab spanning one way.

A 6 m one-way strip under 10 kN/m² needs $wL^2/8 = 45.0$ kNm per metre. The 6 × 8 m panel spanning both ways needs 19.4 kNm/m by the yield-line calculation above — 57% less, for the same span and the same load, because the corners are helping and the long direction is taking a share.

That saving is why two-way slabs exist, and the shape of it explains where they are used. It is largest for square panels and falls away with the aspect ratio; it needs support on all four sides, which means beams or walls in both directions; and it needs the reinforcement to run both ways, which is a detailing cost rather than a material one. Flat slabs on columns are a different structure again — supported at points rather than on lines — and their yield-line patterns are correspondingly stranger. Whichever is used, the depth available is doing most of the work: depth is the cheapest strength in a slab as much as in a beam, and a 25 mm change in thickness moves more than any amount of re-proportioning.

An upper bound is the wrong side to be on #

The weakness deserves stating plainly, because it is the opposite of the usual engineering position. A yield-line answer is a load at which the slab can collapse by the mechanism assumed, so a missed mechanism makes the answer unsafe rather than conservative — and this collection’s other plastic method, the thrust line and the safe theorem, errs the other way.

Two mechanisms are commonly missed and both are worth knowing. Corner levers: near a corner where two simply supported edges meet, the slab lifts and the yield line forks, giving a load a few per cent below the simple pattern. And fans around a concentrated load or a column head, where the lines radiate rather than run straight, which is the governing pattern for punching.

The professional response is a two-part one: search over the mechanisms that are known, and add a margin for the ones that are not — the yield-line method is conventionally used with a lower moment capacity than a strip-based elastic method would allow, and the difference is the price of being on the wrong side of a bound.

The reinforcement does not have to follow the share #

The elastic share says how much load each direction carries in a slab whose two directions are equally stiff. It does not say how the slab must be reinforced, and that distinction is the most useful thing in this essay for anybody detailing one.

A slab reinforced more heavily in one direction is stiffer in that direction, so it attracts more load, so it needs more reinforcement — a feedback that sounds unstable and is not, because the stiffness in each direction changes far less than the reinforcement does once the slab has cracked. What actually happens is that the designer chooses the load path and then provides for it, within limits set by what the slab can redistribute.

That freedom is exactly the lower-bound theorem at work. Any distribution of load between the two directions that is in equilibrium with the applied load, and for which the reinforcement is sufficient everywhere, is a safe design — regardless of what an elastic analysis says the slab would do if left alone. Hillerborg’s strip method is that observation turned into a design procedure: divide the load between the directions however is convenient, design each family of strips as a one-way slab, and the result is safe by theorem rather than by calculation.

The practical consequence is that the elastic share is a guide rather than an instruction. A designer who sends 100% of the load the short way and reinforces accordingly has designed a safe slab that will behave like a two-way slab and will crack a little more where the reality and the assumption disagree. A designer who assumes an equal share on a 1 : 2 panel and reinforces for it has done something else entirely, because the long direction will never take 50% no matter how much steel is in it — the load follows stiffness, and stiffness is not what the drawing says the steel is for.

What the picture cannot show #

The slab is isotropic in every figure. A real slab has different reinforcement in the two directions, so the moment capacity is a function of the yield line’s direction, and the search has to be over patterns and over orientations. The arithmetic is the same with an extra factor; the pictures are much less tidy.

Nothing here has a deflection in it. Yield-line analysis reports a collapse load and says nothing about serviceability, and slabs are almost always governed by deflection rather than by strength — so the method answers the question that does not decide the thickness.

The elastic share and the collapse mechanism describe different states of the same slab and neither is the whole story. A slab in service is somewhere between: cracked, partly redistributed, with a stiffness in each direction that depends on how much it has already been loaded. The two calculations bracket it, which is the usual arrangement, and the bracket is wide.

Where the ladder goes #

The first rung is the flat slab: no beams, columns only, and a structure whose yield-line pattern is a set of fans around the column heads. Punching shear is the failure that governs it and it is a local three-dimensional problem that none of the arithmetic above touches.

The second is the strip method, which is yield-line’s opposite number: instead of assuming a mechanism and getting an upper bound, it assumes a distribution of load into strips and gets a lower one, so the two together bracket the answer in the same way an arch’s thrust line and its collapse mechanism do.

The third is the one the fourth power implies. If two-way action is worth so much and disappears so fast with aspect ratio, the layout of a floor grid is a structural decision of the first order — and it is usually made by somebody drawing a column grid to suit a car park.