Structural Engineering & Statics
where the load goes
Internal forces

Turn the column, and the slab passes

A flat slab that is comfortable under gravity fails its punching check the moment a moment arrives at the column, and nothing about the load has changed. The fix is not more concrete. It is the column's plan shape and, at equal area, which way round it is turned — worth more than adding half again as much column.

Assumes A check made on a perimeter, not on a section and The strength with no mechanism in it.

15 min read 8 figures Geometry beats materialWhich failure arrives firstThe load must go somewhere

A punching check is a force divided by a perimeter, and everything in it is geometry except the concrete strength.

The hero is that check failing. A 7.2 m bay of a 225 mm slab on a 400 × 400 mm column, carrying 12 kN/m² — 604 kN across a control perimeter of 4,427 mm — at a shear stress of 0.697 N/mm² against a resistance of 0.658. It is over by six per cent, and the only thing separating it from a slab that passes is an eccentricity of 300 mm.

The moment enters as one number

The load leaning 300 mm off the column’s centre is a transferred moment of 181 kN·m, and it does not enter the check as a moment. It enters as a multiplier.

β=1+ke e u1W1\beta = 1 + k_e \, e \, \frac{u_1}{W_1}

Three things in that expression are worth separating, because they behave differently.

ee is the eccentricity, M/VM/V, and the demand is linear in it. u1u_1 is the control perimeter, the same length the shear is divided by. And W1W_1 is the perimeter’s own first moment about its centre — a shape property with dimensions of area, which measures how well the perimeter is arranged to carry an uneven shear.

kek_e is a fitted coefficient depending on the column’s plan ratio, and it is the part of the expression with no derivation behind it.

A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658.
Fig. 1 The same slab with no eccentricity at all. β=1\beta = 1, the shear stress is 0.606 N/mm² against the same 0.658, and the check is at 92 per cent. Everything that follows is the distance between this figure and the one at the top of the page.
A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.642 N/mm² against a resistance of 0.658.
Fig. 2 120 mm of eccentricity — 72 kN·m, a modest unbalanced moment from spans differing by half a metre. β\beta is 1.060, the stress is 0.642, and the check is at 97.6 per cent. Nothing has been added to the slab and nothing has been taken away.

Push the eccentricity out to what a column in a lateral system actually transfers, and the check stops being marginal and starts being a different design.

A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.787 N/mm² against a resistance of 0.658.
Fig. 3 And 600 mm, which is what a column in a frame resisting wind or an earthquake will see. β=1.299\beta = 1.299, the stress is 0.787 against 0.658, and the slab is twenty per cent over. The vertical load has been 604 kN throughout.

Four figures, one load, and the check has moved from 92 per cent to 120. That is the whole reason this rung exists: the quantity that decides a punching check most often is the one least likely to have been in the gravity model it was checked from.

Which free body produced the number

The quantity β\beta multiplies is not a stress in any section, and it is worth being precise about that before treating it as one.

Cut the slab along a closed surface standing 2d2d out from the column face, through the full depth of the slab. Lift out the column and the disc of slab inside that surface. What crosses the cut is the shear from everything outside it — 604 kN of it here — and what is inside carries no shear across it at all, which is why the shaded area in every figure on this page is subtracted from the load.

That is a free body, and it is the whole derivation. The 0.606 N/mm² is 604 kN divided by 4,427 mm of perimeter times 225 mm of depth: a force over an area, with no assumption about how it is distributed and no bending stress anywhere in it.

Which is why the moment cannot be handled the way a moment usually is. There is no section to compute a bending stress on and no neutral axis to take it about — the resistance itself has no mechanism behind it, and neither does the demand. The perimeter is a length, the shear is a force, and a moment is not a thing either of them can absorb.

So the moment is handled by asking a different question: if the shear must run round this perimeter unevenly, how much larger is the largest value than the average? β\beta is that ratio, and multiplying the whole demand by it is the statement that the check is on the worst point of the perimeter rather than on its mean.

That reading explains why β\beta is insensitive to almost everything. It is a ratio of a peak to a mean on one closed curve, and it depends on the curve’s shape and on the eccentricity and on nothing else — not on the load, not on the concrete, not on the reinforcement. A check written as a ratio of two of its own quantities cannot be improved by making either of them better.

Which shape carries an uneven shear better

The obvious response to a failing punching check is more column. The arithmetic says something more specific.

A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 450 × 450 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4627 mm long against 1800 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 602 kN across 4627 × 225 mm, a shear stress of 0.664 N/mm² against a resistance of 0.658.
Fig. 4 The column taken from 400 mm square to 450 — 27 per cent more concrete. The perimeter grows from 4,427 mm to 4,627, β\beta is essentially unchanged at 1.148, and the utilisation falls from 1.059 to 1.009. Still failing, for a column a size and a half larger.
A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 250 × 800 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4927 mm long against 2100 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 601 kN across 4927 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658.
Fig. 5 The same area — 200,000 mm² against the square’s 202,500 — reproportioned to 250 × 800 with its short dimension in the direction the load leans. The perimeter is 4,927 mm, β\beta falls to 1.118, and the utilisation is 0.921. The slab passes, on the same amount of concrete that failed as a square.

Two things moved and both moved the right way. The perimeter is 300 mm longer, because a rectangle of a given area has more edge than a square does. And β\beta is smaller, because kek_e falls as the column gets short in the direction of the lean.

That is the opposite of the general statement usually made about this — that a long thin column is worse for moment transfer than a square one. The statement is not wrong about its own mechanism; it is wrong about the sum, because the perimeter term is larger than the shape term and points the other way.

A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 800 × 250 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4927 mm long against 2100 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 601 kN across 4927 × 225 mm, a shear stress of 0.641 N/mm² against a resistance of 0.658.
Fig. 6 And here it is turned through ninety degrees, with the long dimension in the direction of the lean. The perimeter is the same 4,927 mm and the load the same 601 kN, but kek_e has gone from 0.45 to 0.8 and β\beta from 1.118 to 1.183. The utilisation is 0.974 — still passing, and six per cent worse than the same column the other way round.

Six per cent for a decision that costs nothing. A rectangular column has an orientation, the orientation is drawn by whoever set out the grid, and until this check is run there is no structural reason for it to be one way rather than the other.

What it costs to fix it with size instead

A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 600 × 600 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 5227 mm long against 2400 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 597 kN across 5227 × 225 mm, a shear stress of 0.581 N/mm² against a resistance of 0.658.
Fig. 7 The brute-force repair: 600 mm square, 2.25 times the concrete of the column that failed. The perimeter is 5,227 mm, β\beta is 1.144, and the utilisation is 0.882 — comfortably passing, at a cost in floor area, in formwork and in the architecture of every storey the column runs through.

Set the four repairs side by side at the same 300 mm of eccentricity and the ranking is unambiguous.

Reproportion at constant area — 250 × 800 — takes 1.059 to 0.921, and costs nothing. Turn the same rectangle the right way — 800 × 250 to 250 × 800 — is worth 0.974 to 0.921 on its own. Enlarge the square by a size and a half takes 1.059 to 1.009, and does not fix it. Enlarge it to 600 square takes it to 0.882 for 2.25 times the concrete.

There is a fifth, and it belongs in a different category.

A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 5370 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 595 kN across 5370 × 300 mm, a shear stress of 0.412 N/mm² against a resistance of 0.615.
Fig. 8 The original 400 mm column with the slab thickened from 225 mm to 300. The perimeter grows to 5,370 mm because it stands 2d out from the face, the stress is divided by a larger depth as well, and the demand falls to 0.412 against a resistance of 0.615 — a utilisation of 0.670, from a change that touches no column at all.

Depth beats everything and it is the expensive one, because it is 75 mm of concrete over the whole floor plate rather than at one column, and because it raises the slab’s self-weight, the building’s height and the mass in every seismic calculation. That is why the local repairs are worth ranking at all: a designer reaching for depth is spending a global quantity to fix a local check.

Why the material levers do not help the same way

Two obvious moves change the resistance rather than the demand, and it is worth recording what they are worth.

Doubling the flexural reinforcement ratio over the column — 0.75 per cent to 1.5 — raises the resistance from 0.658 N/mm² to 0.829, a gain of 26 per cent, because the punching expression takes the cube root of ρfck\rho f_{ck}. That is a real repair and it is the standard one, and its limit is that reinforcement crowds: the top steel over a column in a flat slab is already the densest region in the floor.

Raising the concrete from C30 to C50 raises the resistance to 0.780, a gain of 19 per cent — for a strength increase of 67 per cent, again because of the cube root.

Both act on the resistance. Neither touches β\beta, which is a property of the geometry and the loading, so both leave the sensitivity to eccentricity exactly where it was. A slab repaired by reinforcement is a slab whose punching capacity still falls by 30 per cent when a moment arrives that was not in the gravity model.

That asymmetry is the practical content of the whole essay. The geometric levers change the demand and are therefore proof against the uncertainty; the material levers change the resistance and merely raise the bar the uncertainty has to clear.

The two ways an uneven shear is actually resisted

The physical picture behind β\beta is a split, and the code’s version of it is one of the more visible approximations in concrete design.

A moment transferred from a slab to a column goes over in two parts. Some of it is carried in bending, by the slab reinforcement crossing a strip either side of the column — a flexural transfer, resisted by steel that is there anyway. The rest is carried by eccentric shear, running round the perimeter more heavily on one side than the other.

The split is not derived; it is assigned. Codes give the flexural share as a fraction depending on the column’s plan proportions — which is exactly what kek_e is, read from the other direction — and hand the remainder to the shear check. So the coefficient this essay measures a six per cent difference across is a partition rule rather than a physical quantity, and it is the same class of object as the assumption that a joint detailed as simple carries no moment: a share of something, decided in advance, and then designed for as though it were true.

Two consequences follow.

The flexural half has to be delivered. The reinforcement carrying it must be concentrated within a narrow band across the column — roughly the column width plus 1.5d each side — and reinforcement placed evenly across the bay does not deliver it. A slab detailed with uniform top steel is one where the flexural transfer assumed by the split does not exist, and the eccentric shear is larger than β\beta says.

And the split assumes redistribution. The flexural share arrives only after the slab has cracked and rotated enough to develop it, which is a serviceability event on the way to an ultimate one. On a slab where the plan is irregular enough that the moment arrives at one column rather than being shared, that rotation may be concentrated at exactly the column with the least capacity to allow it.

Where the eccentricity comes from, and why it is late

The four sources of an unbalanced moment at an internal column are worth naming, because they arrive at four different points in a project.

Unequal spans. Two bays differing by half a metre put a permanent eccentricity into the column, and it is in the geometry from the first sketch.

Pattern loading. One bay loaded and the next not is a load case that must be considered and is frequently not run for punching, because the gravity punching check is done on the full-load case where the moments balance.

Lateral load. The frame that resists wind or seismic action transfers moment into the slab at every column it uses, and the lateral system is usually designed after the floor plate.

And construction. A column out of position by 25 mm, a bay poured before its neighbour, a temporary prop line — each of these is an eccentricity that no drawing records.

The first is in the model. The second is a load case somebody has to remember. The third arrives after the slab is sized, and it is the one that produces the largest numbers. The fourth is invisible.

This is why punching is the check that is failed rather than the check that is close, and it is the reason to prefer geometric fixes: they are decided early, they cost nothing, and they are robust to a quantity that arrives late.

The same argument at a transfer column

Everything above is about an eccentricity of a few hundred millimetres. There is a case where it is a metre or more, and it is worth naming because it is where this check most often decides a scheme.

A column that stops and is picked up by the floor below it delivers its whole axial load into that floor at a point, and almost never exactly over the column that continues down. The offset is architectural — a grid change between the residential floors and the podium, a car-park module that does not align with an apartment module — and it is an eccentricity in exactly the sense of this essay, measured in metres rather than millimetres.

β\beta is linear in the eccentricity, so a 1.5 m offset on the same slab and column gives β\beta around 1.75 and a utilisation of 1.6. The punching check is not close; it is out by a factor, and no reinforcement ratio and no concrete grade will recover it.

Which is why transfer at a flat slab is not done. The load goes into a transfer beam or a transfer plate — a member deep enough to carry the offset in bending — and the depth is chosen by that requirement rather than by the span. This is stiffness-is-not-strength arriving from a third direction: the element is sized by a geometric requirement of the load path, and its own bending capacity is a consequence rather than a design.

The general shape is the one this whole rung is about. When a check is a demand divided by a length, the ways to improve it are to shorten the load path’s eccentricity, to lengthen the perimeter, or to move the whole problem to a member built for it. Making the material better is the option that does the least.

What is at stake if it does fail

Punching failure has no warning and no residual capacity. The slab drops off the column, its top reinforcement pulls out of the concrete it was anchored in, and the load that column was carrying arrives at its neighbours as an impact.

That is the mechanism behind the disproportionate collapse of a flat-slab building: the failure is local, brittle, and it loads the adjacent columns beyond their own punching capacity in the same instant. A flat slab has no beams to hang from and no alternative path, which is why the integrity reinforcement — bottom bars carried continuously through the column — is required by code and is the only thing standing between one punched column and a floor-by-floor collapse.

The integrity steel is not a strength provision. It does nothing for the punching check, contributes nothing to the resistance in any figure on this page, and exists only for the state after the check has been failed.

What to carry away

The moment enters as a multiplier and dominates the check. 604 kN gives 92 per cent utilisation at no eccentricity and 120 per cent at 600 mm.

The column’s proportions beat its area. At equal area, a 250 × 800 rectangle passes where a 450 × 450 square fails, because it is 300 mm longer round the control perimeter.

And the orientation is worth six per cent for nothing. The same rectangle turned the wrong way is 0.974 instead of 0.921 — a decision made by whoever drew the grid.

Depth beats all of it and is charged globally. 75 mm of slab takes the utilisation to 0.670 and adds its own weight to every other calculation in the building.

Where the model stops

The column is internal. An edge column loses a third of its control perimeter and a corner column two thirds, and both have an unavoidable eccentricity because the slab only reaches them on two sides or one. Every number here is the favourable case.

kek_e is a fitted table and not a derivation. It is a piecewise interpolation on the plan ratio, and the six per cent this essay measures between two orientations is six per cent of a coefficient somebody drew through test data.

There is no shear reinforcement anywhere on this page. Studs, links and shear ladders raise the resistance substantially and bring a second perimeter into the check — the outer one beyond the reinforced zone — which is a different calculation with its own governing case.

And the slab is uncracked in the model and will not be in the building. The flexural cracking over a column is what the reinforcement ratio in the resistance expression is standing in for, and the expression is empirical precisely because there is no mechanism to write down.

The ladder from here

Later rungs on this anchor: edge and corner columns, where the perimeter is truncated and the eccentricity is structural rather than accidental. Shear reinforcement and the outer perimeter, and why a slab with studs has three checks rather than one. Openings near columns, which cut the perimeter and are the commonest way a punching check is quietly invalidated on site. Drop panels and column heads, which are the same argument answered by moving the perimeter into thicker slab. And the punching check under a raft, where the load arrives upwards and the arithmetic inverts.

The coefficient at the centre of this essay is younger than the failure it prevents. Flat slabs were built for sixty years with punching checks that took no account of moment transfer at all, and the eccentricity term entered the codes in the 1970s after a sequence of collapses in buildings whose gravity checks had all been satisfied. What the collapses had in common was not a heavy floor. It was a column carrying a moment nobody had divided by anything.

Where this goes next

A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658. A check made on a perimeter, not on a section Every shear check in this collection is made on a plane cut through a member. A slab sitting on a column has no such plane, because the shear leaves in every direction at once — so the check is made on a closed line, and a line grows with the column while the load grows with the square of the bay. Two centres, and the distance between them is a torque. A storey 30 by 18 m with its walls drawn heavy, pushed in one direction by 1000 kN. The force acts through the centre of mass and the storey turns about the centre of rigidity — the stiffness-weighted centroid of the walls, at x = 15.0 m — and the distance between the two is an eccentricity of 0.00 m before the 5% that has to be assumed anyway. The table below the plan splits each wall's force into its direct share and its torsional one. Torsion relieves the walls near the centre of rigidity and loads the far ones, so the wall in trouble is not the wall carrying the most: west wall is asked for 8% more than its direct share, and the walls at right angles to the push carry 19 kN each with nothing applied along them at all. The corner that moves most A lateral force is shared out in proportion to stiffness only if it passes through the centre of rigidity, which is not the centre of the plan and not the centre of mass. The distance between the two is a torque, and the wall that pays for it is the one furthest away and carrying least. Take that one away and the load finds another route. A 6-panel pratt truss under 20 kN at each top node, before and after member 2 is removed. The load redistributes. The worst-affected survivor now carries 2.03 times what it did, and four members that carried nothing before are now working. Whether that is survival depends on how much spare capacity was there, which is a different question from whether the frame was strong enough. The structure that survives losing a member Every check in this collection asks what a structure carries. None of them asks what is left when part of it is gone — and two frames with the same members, the same weight and the same factor of safety can answer that question completely differently.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

What links here

Every essay whose body links to this one.

The objects this essay names

Each one links to every other essay that touches it.

Brittle failureControl perimeterEccentricityEffective depthFlat slabLoad pathProgressive collapsePunching shearReinforcement ratioRobustnessShear stressSize effect