A hole takes a tenth of every perimeter

The studs that send the check outward found that shear reinforcement in a flat slab turns one punching check into three: the control perimeter, which the studs carry; the outer perimeter beyond them, where the concrete has to carry the shear alone and which stands further from the column the larger the load; and the column face, which nothing reinforces. Every perimeter in that essay was a closed loop round the column. Its last paragraph named the commonest way a real slab breaks the loop — “a service riser, a drainage outlet or a duct cut through the slab within a few depths of the column” — and asked whether such an opening takes more from the outer check than from the inner.

The answer turns out to depend on what “more” means, and the part of the check that pays for the hole is not the one the question was about.

The shadow from the column’s centre #

EN 1992-1-1 deals with openings in a single geometric instruction. If an opening is closer to the column face than six effective depths, draw two straight lines from the centre of the column to the opening’s outline, tangent to it on either side. The part of any control perimeter lying between those two lines is ineffective, and the check is made on what is left.

It is a rule about free bodies. A check made on a perimeter showed that the punching check is a free body of the whole bay outside a perimeter, with the shear leaving across that perimeter’s length. An opening is a place where no shear can cross. The shadow is EN 1992-1-1’s estimate of how much of the perimeter is cut off from the column by it: shear heading straight out from the column towards the hole has nowhere to go.

Take the slab from the essays on punching — 260 mm deep with an effective depth $d$ of 225 mm, carrying 12 kN/m² on 400 mm square internal columns, with a moment from the frame putting the shear 300 mm off centre — on a 10 m bay, where each column delivers 1,182 kN. Put a 300 × 300 mm riser one effective depth from the face of the column, square to its side.

The two tangents enclose 38.9 degrees. The control perimeter, two depths out, is 4,427 mm long and loses 458 mm to the shadow. The outer perimeter — wherever it ends up — is much longer and loses much more: 1,093 mm. Measured in millimetres, the riser takes more than twice as much from the outer check as from the inner one.

Measured as a share, the same #

Measured as a share of each perimeter, the answer is different, and it is the share that sets each check.

The control perimeter loses 0.10 of its length, the outer perimeter 0.11. The reason is that the shadow is an angle. A punching perimeter far from a column is close to a circle centred on the column, and a sector of a circle takes its angle’s share of the circumference however large the circle is. Both perimeters lengthen as they move out, and the sector widens in proportion, so the share hardly moves: $38.9/360 = 0.108$.

Near the column it does move, and which way depends on where the opening is. Close in, a perimeter is mostly the column’s own straight sides, joined by quarter circles at the corners. A straight side is nearer the column’s centre than a corner is, so a sector aimed at the middle of a side cuts less length than the same sector aimed at a corner. At the column face itself, the side opening’s shadow takes 0.088 of the square and the corner opening’s 0.135. Two depths out the two have drawn together, 0.10 and 0.12; by the outer perimeter both are within a hundredth of the sector’s own share. An opening costs its angle, and the column’s shape only modifies that close to the face.

So the answer to the question as asked is yes in length and no in proportion. The outer check is not hit harder than the inner. Both lose a tenth.

A tenth of the perimeter is not a tenth of the steel #

If both checks lose the same share, the obvious estimate is that the slab needs about a tenth more of everything. It needs more than that, and the reason lies in how the outer check is met.

The control perimeter is carried by studs. Its demand, the shear stress $\beta V_{Ed}/(u_1 d)$, rises by $1/(1 - 0.103)$, from 2.07 times the concrete’s resistance to 2.31. But the studs on each ring only have to close the gap between that demand and three-quarters of the concrete’s resistance, and the concrete’s three-quarters is lost only on the missing 10 per cent of the perimeter. The steel on each ring rises from 1,415 mm² to 1,498 mm², 6 per cent.

The outer perimeter is carried by concrete alone, and it is met by distance. It has to be long enough that the concrete’s resistance over its effective length carries the whole shear. Without the opening that happens 5.36 depths from the face. With it the perimeter has to make up 11 per cent of a perimeter that is itself growing by $2\pi$ per unit of distance, and it does so at 6.13 depths. The last ring of studs has to be within 1.5 depths of the outer perimeter, so the studs run from 3.86 depths to 4.63, and on a spacing of 0.75 depths that is a seventh ring.

Six rings of 1,415 mm² become seven of 1,498: 10,490 mm² against 8,490, 24 per cent more. Most of the increase is the extra ring rather than heavier rings. Across the range of bays the same thing happens in steps — every time the outer perimeter’s extra distance crosses another 0.75 depths, a ring is added — so the gap between the two curves is ragged rather than proportional. And the bay at which studs are first needed moves from 7.00 m to 6.64 m: a slab that needed no studs on a 6.8 m grid needs them once a riser is cut beside its column.

This is the same lever that made studs grow as the shear to the power 3.5: reach. The steel on a ring scales with the demand, but the number of rings scales with how far out the concrete becomes enough, and that distance grows faster than the demand. An opening raises both, and it is the reach that costs.

Studs that cannot be placed #

The extra reach has a geometric awkwardness that the arithmetic does not show.

The riser occupies the slab from one depth to 2.3 depths from the face. At every bay worth studding, the outer perimeter is well beyond it, so the stud zone has to reach past the hole. On the eleven rails that are not aimed at it, that is just a longer rail. On the twelfth it is impossible: the rail runs into the opening at its second stud, and nothing beyond the hole on that line can be connected to the column. The shadow rule has already written off that part of every perimeter, so the check is consistent. But it means the stud layout round a column with a riser is not a rotated copy of the one without it. It has a gap in it, and the rails either side of the gap carry the reach the missing rail would have carried.

Off the corner, the 45-degree opening is closer to the column’s corner and its shadow catches two rails rather than one. It takes 516 mm of the control perimeter and 1,180 mm of the outer, puts the outer perimeter at 6.19 depths, and also needs seven rings: on this slab, the position of the opening round the column changes the details and not the outcome.

The closer, the worse — and then nothing #

The distance of the opening from the face changes the outcome a great deal.

Against the column face the opening’s shadow is 74 degrees wide, a fifth of a full turn. The control perimeter’s demand rises to 2.60 times the concrete’s resistance, the outer perimeter moves out to seven depths, and the slab needs eight rings of studs. Moved out to a depth from the face, the shadow narrows to 39 degrees; at three depths, 20; at five, 13. The opening’s effect falls away roughly as the inverse of its distance from the column’s centre, because its angle does.

Then, at six depths, it stops. The rule’s limit is a cliff: an opening at 5.9 depths takes 3 per cent of every perimeter, and the same opening at 6.1 depths takes nothing. On this slab the cliff is small — 2.13 against 2.07, and the same number of rings — because by six depths the shadow is only 11 degrees. That is the honest reading of the limit. It is not a claim that openings beyond six depths have no effect; it is a judgement that the effect has by then fallen below what the check is precise enough to see.

Wider is worse, and it is still the angle #

The steel climbs in the same steps as the opening widens, each step a ring. A metre-wide service duct one depth from the face takes 0.28 of every perimeter and needs 14,700 mm² of studs, three-quarters more than none. But throughout, the column face check sits at 0.71 of its limit, exactly where it was. The shadow is drawn from the centre of the column outward, and the face check is made on the column’s own outline, inside the shadow’s apex. An opening beyond the face cannot shorten it.

That makes the face the one check that tells a designer whether an opening can be paid for at all. Studs can make up any amount of lost control perimeter, and distance can make up any amount of lost outer perimeter, at a price. Neither can do anything at the face, so an opening that pushes the demand towards the face’s limit — by sitting against the column, or by being so wide that the stud zone has to be enormous — is a question for the architect rather than for the reinforcement.

Half an edge column #

There is a scale against which to read these shares, and the slab already has it. A column at the edge of a slab has no slab on one side, and the perimeter whose middle is not the column’s worked through what that does: the control perimeter becomes the column’s three inner sides and half the rounding, $c_2 + 2c_1 + \pi a$. For the 400 mm column at two depths that is 2,614 mm against the internal column’s 4,427 — a loss of 41 per cent. A corner column keeps two sides and a quarter of the rounding, 1,507 mm, and has lost 66 per cent.

An opening against the column face, with its 74-degree shadow, takes 20 per cent of the perimeter: it has made the internal column about half of an edge column. The 300 mm riser a depth out, at 10 per cent, has made it a quarter of one. That is the useful way to think about where a riser is placed. Nobody would put an edge column into a flat slab without a separate look at its punching. A riser against an internal column does half as much damage, and it is usually drawn by a building-services engineer who will never see the punching check.

The comparison also shows why the edge column was worse than its lost length suggests. An edge column’s missing perimeter is all on one side, so the shear has to run round the remaining three sides unevenly, and the eccentricity factor for an edge column is large even with no moment from the frame. An opening on one side of an internal column does the same thing in miniature, which is the first of the omissions listed below. The figures hold that factor fixed; the edge column showed what releasing it costs.

What the opening does not change #

The concrete’s own resistance per unit of perimeter is not touched by the opening, and it is worth saying what sets it, because it bounds what a designer can do. The resistance falls as the slab gets deeper — the bigger one is the weaker one traced that size effect through the factor $k = 1 + \sqrt{200/d}$ — and rises only as the cube root of the reinforcement ratio. Neither lever is large. So an opening’s lost tenth has to be made up by length, which means distance, which means rings of studs, or by a column that presents more perimeter to the slab. Turn the column, and the slab passes found that a rectangular column can be oriented to help its own punching check; with an opening on one side the same reasoning says to turn the column so the hole sits off one of its ends: the end of a long column is further from its centre than its side is, and the same hole seen from further away subtends a narrower angle.

The load does not change either. The column still collects one bay of slab, by the same tributary argument that gives every column its load; the riser’s own area is a few hundredths of a square metre out of a hundred. The opening is entirely a matter of how the load leaves, never of how much arrives.

The rule is a free body, not a flow #

The shadow rule is a convention, and it is worth being exact about what kind.

It is not a calculation of where the shear goes. Round a real opening, the slab carries load past the hole, and the shear in the slab beside the opening is larger than it would be with no hole, not merely redirected. The rule makes no attempt to find that. It removes a sector and treats the rest of the perimeter as if it were the perimeter of a slab with no opening, uniformly stressed. The strength with no mechanism in it made the point about the concrete’s own shear resistance: an empirical number that works because it was fitted to tests, not because it describes a mechanism. The shadow is the same kind of object, a geometric fit to tests on slabs with holes in them, and it should be read as one.

What the rule does preserve is the free body. Choosing the free body is the whole of the method, and the punching free body is the bay outside a perimeter with the shear leaving across it. An opening does not change the load on that free body. It changes the length the load can leave across, and the shadow is a statement of how much length to deduct. That is why the deduction lands identically on every perimeter, and why the share of each is the angle: the free body does not care how far out it is drawn.

There is a nice contrast with the hole that costs nothing, and everything, where an opening in a steel beam’s web costs almost nothing in bending, because the flanges carry the moment, and almost everything in shear, because the web carries the shear and the hole is in the web. A hole in a flat slab beside a column is the same kind of hole in the same kind of place. Punching is the slab’s shear, and the slab has no flange to fall back on.

What the figures leave out #

The eccentricity factor is held at its value for the whole perimeter. EN 1992-1-1’s $\beta$ depends on the perimeter’s own first moment, and a hole on one side moves the perimeter’s centroid away from it. When the moment the frame puts into the column acts towards the hole, $\beta$ rises and the opening costs more than drawn; when it acts away, less. The figures keep $\beta$ fixed, which isolates the length.

The opening’s aspect is not refined. The rule has a second clause for a long opening running along the perimeter, which replaces its length by the root of its two dimensions before drawing the tangents. The openings here are square or widened across the ray, and the clause is not applied.

The outer perimeter’s own rule for widely spaced rails is not applied. Where the last studs on adjacent rails are more than two depths apart, EN 1992-1-1 draws the outer perimeter inward between them, and a rail missing at the hole makes that gap wider. The figures draw the outer perimeter as a smooth loop with the shadow removed.

The flexural reinforcement is the same. The concrete’s resistance depends on the reinforcement ratio through the slab, and a hole interrupts bars, which are usually trimmed round it and gathered beside it. A slab that lost 10 per cent of its bars across the control perimeter would have a lower resistance on what is left.

Still open: the hole that is cut after the slab is built #

Every opening here is on the drawing before the studs are laid out, so the rails can be arranged round it. Most risers are not. A building in use acquires new services, and a core drill through an existing slab beside a column is a routine instruction that nobody associates with punching. The studs are already cast, at a spacing chosen for a slab without the hole, and the drill finds whatever is there — a stud, a rail, or the bars that carried the moment. Whether a slab designed to pass with six rings still passes once a 150 mm core is taken one depth from the face, which rail it is acceptable to cut through, and whether the reach the slab already has covers an outer perimeter that has just moved out by half a depth, are questions about a check that was right on the day it was made and is now being made on a different slab.