The perimeter whose middle is not the column's

A punching check is a force divided by a perimeter, and at an internal column the perimeter is a closed loop round the column with the column at its middle. When a moment arrives, the shear runs round that loop unevenly, and one multiplier, $\beta$, says how much worse the worst point is than the average. Everything in that calculation took the middle of the loop and the middle of the column to be the same point, because at an internal column they are.

At the edge of a slab they are not, and the whole of this essay is the distance between them.

What the edge leaves of the perimeter #

A column on the slab edge has slab on three sides. The control perimeter, drawn $2d$ out from the faces, has nowhere to go on the fourth: it runs along the two side faces to the slab edge and stops. For a 400 mm square column under a 260 mm slab with an effective depth of 225 mm, the perimeter that is left is 2,614 mm. The same column inside the slab has 4,427 mm. The edge has taken 41 per cent of it.

That loss is what every account of edge punching mentions first, and it is the smaller half of the problem. The larger half is where the remaining perimeter has its middle.

The perimeter is a U. Its two legs run from the slab edge inward, and its rounded bottom and inner face lie well inside the slab. Average the position of every millimetre of it and the centroid comes out 363 mm inside the column’s centre, on the slab side. An internal column’s perimeter is symmetric about the column and its centroid is the column’s centre; an edge column’s is not, and the column sits near one end of its own perimeter rather than in the middle of it.

That would not matter if the shear arrived at the column’s centre. It does not.

Where the shear arrives #

The slab at an edge column is the end of a continuous floor. Its end span sags under load, and its end rotates as it sags — the slab surface tilting down toward the middle of the bay. The column above and below resists that rotation, and in resisting it takes a moment out of the slab: hogging, top steel in tension, the slab’s end moment.

A force and a moment on a free body are one force somewhere else. The column pushes up on the slab with the shear $V$ and twists it with the end moment $M$, and the pair is the same as $V$ alone acting a distance $e = M/V$ from the column’s centre, on the side the slab spans toward. The shear’s resultant is inside the slab, the way a cantilever’s reaction is equivalent to a single force out where its load is.

At a 7.2 m end span with columns 3.5 m long above and below, the end moment is 170 kN·m on a shear of 269 kN, and the resultant sits 631 mm inside the column’s centre — out past the inner face of the column, and past the perimeter’s centroid by 269 mm.

The end moment here comes from the simplest frame that has one: the end span on its column, held against rotation at its far support, with the fixed-end moment shared between the slab and the columns by their stiffnesses, as moment distribution shares it. The slab is taken at its gross section across a 7.2 m strip. Each of those choices moves the number, and the last section prices them; the shape of the argument survives all of them.

The eccentricity that matters is the perimeter’s #

The multiplier on the shear is EN 1992-1-1’s general expression,

$$\beta = 1 + k \, \frac{M}{V} \, \frac{u_1}{W_1},$$

with $u_1$ the perimeter, $W_1$ its first moment $\int |x| \, dl$ about the axis the moment turns about, and $k$ a fitted coefficient. At an internal column the axis passes through the column’s centre and the perimeter’s at once. At the edge it has to be one or the other, and only one of them is a statement about the perimeter.

The question $\beta$ answers is how unevenly the shear runs round the loop. A shear whose resultant passes through the loop’s own centroid can be carried evenly, every millimetre taking the same share, and the multiplier is one. A resultant off that centroid has to be carried by a lopsided distribution, and how lopsided is set by how far off it is. So the eccentricity that belongs in the expression is the resultant’s distance from the perimeter’s centroid, $e - \bar{x}$, and the moment it implies is the column’s moment less the shear times the 363 mm the perimeter’s middle has already moved. That is the reading used here, and it is the one the North American provisions write out explicitly, transferring the moment about the centroid of the critical section rather than the column.

Read that way, the slab’s end moment has a strange property. Some of it is free. The first 363 mm of eccentricity moves the resultant onto the perimeter’s centroid and makes the shear more even, not less.

At a 5 m end span the end moment is 66 kN·m and the shear 182 kN, and their ratio is 363 mm. The resultant sits on the centroid, and the edge column’s multiplier is exactly one — lower than an internal column’s would be under the same moment. Nothing about this column is favourable except that the moment its slab hands it is the moment its perimeter wants.

Shorten the span further and the resultant falls short of the centroid, on the edge side.

Now the shear crowds onto the parts of the perimeter near the edge — the two legs — and the multiplier is back up to 1.31. A short end span is not automatically the benign one. Its moment is too small to reach the middle of a perimeter that the edge has pulled inward, and the error is the same kind as too much moment, pointing the other way.

Three multipliers against one span #

That is the general expression. EN 1992-1-1 offers two other ways of handling an edge column, and both are built so that the designer does not have to compute an end moment at all.

The reduced perimeter. Where the eccentricity is toward the interior — which is the case here at every span past 5 m — EN 1992-1-1 allows the shear to be taken as uniform on a shortened perimeter $u_1^*$, whose two legs are cut back to $\min(1.5d, c_1/2)$ from the inner face. For this column that is 2,214 mm, and dividing the shear by it instead of by 2,614 is a multiplier of $u_1/u_1^* = 1.18$. It contains no moment, no span and no column stiffness.

The shortcut. Where the slab is not part of the lateral system and the spans are within a quarter of each other, a flat 1.4 may be used for an edge column. It contains nothing at all.

Against the end span the three are a V and two flat lines. The general multiplier comes down from 1.31 at 4 m to one at 5 m, where the resultant reaches the centroid, then climbs steadily as the end moment outgrows the shear: 1.34 at 6 m, 1.78 at 7.2, 2.08 at 8, 2.87 at 10. The reduced perimeter agrees with it at 5.54 m and the shortcut at 6.17 m. Below those spans both are conservative; above them both are lower, and the gap widens with every metre.

The flat lines are not wrong about their own premise. Both encode a judgement that the moment an edge column takes from its slab is limited, not by the frame’s elastic stiffness, but by what the slab’s reinforcement near the edge can actually deliver before it yields — and once a strip of slab has yielded, the moment cannot rise and the shear distributes itself. The flat-slab annex of the same standard states the limit outright: an edge column can be handed at most $0.17 \, b_e d^2 f_{ck}$, here 207 kN·m. The dashed curve applies it.

And the cap does not bind until 7.88 m. For every end span shorter than that, the elastic moment is below the moment the standard itself says the slab can transfer, so the judgement behind the flat lines — that the moment will be shed before it arrives — has nothing to shed. Past 7.88 m the capped multiplier peaks at 1.98 and falls, because the moment is held at 207 kN·m while the shear goes on rising; even at 10 m it is still 1.51, above the shortcut.

The check the shortcut passes #

A multiplier is half the check. The other half is the shear it multiplies, which also grows with the span.

Multiplied out, the three multipliers become three verdicts on one slab. At a 7.2 m end span the shear stress on the bare perimeter is 0.457 N/mm² against a resistance of 0.658. The general expression puts the check at 1.23 of its resistance; the shortcut at 0.97; the reduced perimeter at 0.82. The slab passes by both simplified routes and fails by the full one by nearly a quarter.

The crossing is where the argument becomes a design question rather than an observation. The general reading reaches one at an end span of 6.75 m, and at that span the shortcut says the column is at 91 per cent and the reduced perimeter at 76. Between 6.75 m and the 7.88 m where Annex I’s cap begins to act, the elastic end moment is fully deliverable by the standard’s own measure, the general expression says the column fails, and the shortcut says it passes with room.

None of that makes the simplified routes unsafe. They were calibrated against tests on edge connections, and a test does whatever its slab actually does, including yield. What it makes them is a different claim: that a slab will redistribute the end moment it carries before its edge connection punches. The general expression is what follows if it does not. Which of the two is true depends on whether the top steel at the slab edge yields first, and nothing on this page models yield — which is exactly why the two disagree.

A bigger column, a worse check #

At an internal column the first instinct for a failing check — more column — was the weakest of the available repairs, but it at least pointed the right way. At an edge column it does not.

A larger column has a longer perimeter, and that alone helps: the shortcut, which reads only the perimeter, falls smoothly from 1.06 at 250 mm to 0.78 at 750. But a larger column is also a stiffer one, and its stiffness goes as the fourth power of its side. In the frame that hands the slab’s end moment to the column, stiffness is what decides the share — the stiffest path takes the load, and here the load is a moment. At 300 mm the column takes 78 kN·m; at 400 mm, 170; at 550 mm, 279.

So the general reading is not monotonic at all. It is 0.81 at 300 mm, where the column’s modest moment puts the resultant 32 mm short of the perimeter’s centroid and the shear runs nearly evenly. From there it rises — 0.97 at 350, 1.23 at 400, 1.41 at 450 — and peaks at 1.53 at 550 mm, before the perimeter’s growth and the falling share of each extra millimetre of stiffness bring it slowly back down. A designer who answers a failing edge column at 400 mm with a 550 mm one has made the general check a quarter worse.

This is the same mechanism that loads a frame’s corner joint with the beam’s whole end moment and that makes a stiff element in any indeterminate structure attract the force meant for its neighbours. The punching check simply reads the result in an unusual currency, as a frame’s interior joint reads its beams’ moments in panel shear. The best edge column is not the biggest. It is the one whose stiffness hands it the moment its perimeter’s centroid wants — and that is a statement about the column, the slab and the storey height together.

What the frame is made of #

Three things set the end moment, and each moves the curve without changing its shape.

The storey height. Longer columns are softer columns. At 3 m the end moment on the 7.2 m span is 184 kN·m and the multiplier 1.91; at 4 m, 157 kN·m and 1.66; at 5 m, 137 kN·m and 1.47 — with the check at 0.99, just inside. The span at which the general reading passes the shortcut moves from 5.86 m to 6.99 m across that range.

The slab’s cracking. The figures above use the gross section of a 7.2 m strip, which is the stiffest the slab can be and therefore the reading that hands the column the least. A slab cracked to 70 per cent of its gross stiffness gives the column 203 kN·m and a utilisation of 1.48; at half its stiffness, 233 kN·m and 1.71. Cracking a slab softens it relative to the column and moves the end moment toward the column, which is the opposite of the intuition that cracking relieves.

The far support. Holding the slab’s far end against rotation is close to what an interior column of a continuous slab does under full load — a member told what its far end is doing takes a stiffness that depends on it — and an alternate-bay pattern loading would release some of it and increase the end moment further. The figures take the gentler case.

None of these is exotic. A flat slab on 3.5 m storeys and 7.2 m bays, with a 260 mm slab and 400 mm columns, is the ordinary building the shortcut was written for.

The corner, where the arithmetic runs out #

A corner column has slab on two sides only, and the control perimeter is a single quarter-loop: two legs and one rounded corner.

What the edge did on one axis, the corner does on two. The perimeter is 1,507 mm, a third of the internal column’s; its centroid sits 401 mm in along both axes; and the end moments of the two spans that meet at the corner move the resultant inward along the diagonal. The resultant reaches the centroid at a span of 4.24 m, where the multiplier is one, and the general reading passes the corner’s shortcut of 1.5 at 4.67 m.

By 6 m it is 3.12, and by 7.2 m, 4.64. Those numbers should be read as a statement about the expression rather than about the corner. The coefficient $k$ and the plastic distribution behind $W_1$ were fitted to internal columns, where the moment is a modest perturbation of a shear running round a closed loop. At a corner the loop is open at both ends, the perimeter is short, the resultant can lie outside the region the perimeter encloses, and the expression is being asked to describe a distribution of shear it was never calibrated against. What the general expression does say reliably is the direction: a corner column’s moment overwhelms its perimeter at spans where an edge column’s merely strains it, and that is why the standards cap the moment transferred to a corner rather than computing it.

The assumption every number here rests on #

Every multiplier on this page is an elastic statement: the end moment is what an uncracked or uniformly cracked frame delivers, and the shear distributes round the perimeter in the plastic pattern the general expression assumes. Real edge connections do two things that sit outside that model.

The slab near the edge yields in bending. A strip of slab of width $b_e$ beside the column carries the end moment into the column, and when its top steel yields the moment stops rising whatever the frame’s stiffness says. That is the mechanism the reduced perimeter and the shortcut implicitly rely on, and Annex I’s cap is its numerical form. Where the strip is reinforced heavily enough not to yield, the elastic moment is real.

And torsion runs along the edge. The slab’s edge between columns twists as the end span rotates, and part of the end moment reaches the column through that twisting rather than through the strip directly in front of it. Including the edge’s torsional flexibility softens the slab-column joint, as the equivalent-frame method does with its torsional members, and reduces the end moment somewhat. No edge beam or torsional member is in this model.

Both effects reduce the moment the column receives, and so both move the general reading toward the simplified ones. Neither is quantified here, and the size of the gap between the general reading and the shortcut at 7.2 m — 1.23 against 0.97 — is the size of the bet the shortcut places on them.

What the plans leave out #

It cannot show the failure. Punching is a brittle event, a cone of concrete pushed out around the column with nothing left to hang the slab from, and its strength expression is empirical — there is no mechanism in it to draw. The plans above show where the shear is concentrated; they say nothing about where the crack starts.

It cannot show the moment’s real size. The frame that produces the end moment is the simplest one that has an end moment at all, and a whole floor analysed as a plate on columns would give a different number, generally lower, with the difference concentrated at exactly the edge columns this essay is about.

And it does not show the slab beyond the edge. Every column here is flush with the slab edge. A slab that cantilevers past its edge columns by even a few hundred millimetres gives the perimeter back part of its fourth side, moves its centroid back toward the column, and puts a hogging moment from the cantilever against the span’s — a different structure, and often the better one.

What it comes to #

The edge moves the perimeter’s middle before any moment arrives. A 400 mm column at the slab edge keeps 2,614 mm of the 4,427 mm it would have inside the slab, and the centroid of what it keeps is 363 mm inside the column’s centre.

The end moment then moves the resultant, by $M/V$, and what twists the perimeter is the gap between the two. At a 5 m end span there is no gap and the multiplier is one; at 7.2 m it is 269 mm and the multiplier is 1.78.

The simplified multipliers do not read the span. The reduced perimeter gives 1.18 everywhere and the shortcut 1.4, and the general reading passes them at 5.54 and 6.17 m. At 7.2 m the three utilisations are 1.23, 0.82 and 0.97.

At the edge, a larger column draws a larger moment. On the same span the general check is 0.81 at 300 mm, 1.23 at 400 and 1.53 at 550.

Still open: the moment the slab cannot deliver #

Every number here hands the column the moment the frame’s elastic stiffness gives it, and the standard caps that moment at what a strip of slab beside the column can carry in bending. Between those two is a slab whose edge strip yields and redistributes its moment along the span, which is what the simplified multipliers rely on. Whether a typical edge strip, reinforced for the end moment the frame gives it, has the rotation capacity to shed that moment before the connection punches — and so whether the redistribution the shortcut is betting on arrives before the brittle failure it is meant to forestall — is a question about the slab’s ductility at the column, and it decides which of the two multipliers at 7.2 m is describing the building.