The hole that costs nothing, and everything

Somebody has to get a 300 mm duct through a 533 mm beam, and the beam has 200 mm of web that appears to be doing nothing — a web whose shear it is carrying almost all of. Cutting a hole in it removes 30% of the section’s second moment and adds three quarters of one per cent to the deflection, which sounds like an excellent trade. It is not, and the reason has nothing to do with the section that was removed.

The hole has turned one member into two, over a short distance, and the two have to carry the shear between them by bending. That is a Vierendeel panel, and everything about a Vierendeel girder applies to it at the scale of one hole.

Which free body produced the number #

Cut the beam vertically at the middle of the opening. What crosses the cut is: the direct force in the top tee, the direct force in the bottom tee, and a shear in each.

The global moment is carried as a couple between the two direct forces, on the lever arm between the tees’ centroids — 343 mm here. So $N = M/z = 103.3/0.343 = 301$ kN in each, tension below and compression above, giving 70.3 N/mm² of uniform stress over each tee’s 4,279 mm².

The shear is the interesting one. It cannot cross the opening in the web because there is no web, so it is shared between the two tees — half each for equal tees, 31.5 kN. Now cut again at the end of the opening and take the piece of top tee between the two cuts. It carries 31.5 kN of shear over 400 mm with no vertical load on it, so it must be in double curvature: equal and opposite moments at the two ends and a point of contraflexure at the middle.

$$M_{vier} = \frac{V_{tee}\,a_o}{2} = \frac{31.5 \times 0.4}{2} = 6.30\ \text{kNm}$$

on a tee whose section modulus is only 37.6 × 10³ mm³ — a small tee, not a beam — which gives 167.6 N/mm².

Add them: 237.9 N/mm² at the corner, of which 70% is the term the section calculation does not contain.

The length matters more than the depth #

The Vierendeel moment is $V_{tee}\,a_o/2$. It is linear in the opening’s length and contains its depth only through the tee’s own properties.

Tripling the length triples the local bending and doubles the total. A long shallow hole is worse than a short deep one, which is precisely backwards from the intuition that a hole’s harm is the material it removes — and it is the reason a slot for a flat duct is more dangerous than a round hole for a fatter one, and why the effective width of a wide flange is a different question with a similar flavour.

The round hole, and why it is worth half #

A circular opening of the same depth is credited with an effective length of about 0.45 of its diameter rather than the whole of it, and an effective depth of about 0.9. Both credits point the same way, and the arithmetic is decisive: 44.3 N/mm² of Vierendeel stress against 167.6, and 117.1 total against 237.9.

A round hole costs about half of what a square one of the same depth costs. The reason is geometric rather than a fudge: the tee above a round hole is only cut to its minimum depth at one station, and either side of that the section is deepening — so the length over which it has to span as a small beam is genuinely shorter than the hole’s overall dimension.

The same argument explains why the corners of a rectangular opening are always specified with a radius. A sharp re-entrant corner is a stress concentration sitting exactly where the Vierendeel moment is largest, and multiplying an already-governing stress by two or three is not a margin anybody has.

The worst place for a hole is where nothing is happening #

The position sweep is the finding worth carrying away from the page.

The global-moment term rises toward mid-span and the Vierendeel term falls, and the second is much the larger, so the total falls monotonically toward the middle. The hole is worst near the support.

That is precisely where services are routed, because that is where the ceiling void is deepest, where the duct is running along the corridor, and where the beam looks least busy on a bending diagram. It is also the region a designer glances at least: the moment diagram is small there, the section is nowhere near its limit in bending, and every instinct says the material is spare.

The deflection, which really is negligible #

Computed by virtual work over the opening — the real shear against the unit-load shear, on the tees’ local bending — the extra deflection is 0.0075 of the plain beam’s. That is a genuine result and it is worth stating because it is so often used as a defence: an opening does not make a beam floppy.

The number grows with the cube of the opening length, so four openings of 400 mm add 3%, and one of 800 mm adds 6%. A cellular beam, which is a rolled section cut into a zigzag and rewelded so that it is riddled with holes, is a different object again: its deflection is dominated by the Vierendeel flexibility of dozens of openings, and it is analysed as a Vierendeel girder from the start rather than as a beam with corrections.

The region has no section in it #

The general statement this is a case of is that within about a depth of any discontinuity the strain stops being linear across the section, and an opening is a discontinuity: the region round it wants a strut-and-tie treatment rather than a section calculation.

Both terms above are equilibrium statements rather than section calculations, and that is not an accident. An opening is a D-region: within roughly one beam depth either side of it, plane sections do not stay plane, and the strain distribution across a cut has a discontinuity where the web is missing.

So the analysis on this page is a lower-bound argument wearing a beam’s clothes — an assumed load path, in equilibrium with the applied actions, checked against the strength of the pieces it uses. Which is why the answers are conservative and why the assumption about how the shear divides between the tees can be varied: taking it in proportion to the tees’ plastic moments rather than half each is an equally admissible model and gives different numbers.

The same beam, without the hole, doing the same thing #

The mechanism is not peculiar to openings, and seeing it elsewhere is the fastest way to trust the arithmetic.

The correspondence is exact. A Vierendeel girder’s panel shear produces a chord moment of $Vs/4$ — which is $V_{chord} \cdot s/2$ with the shear halved between two chords, and is this page’s $V_{tee}a_o/2$ with different letters. A beam with a web opening is a one-panel Vierendeel girder embedded in a beam, and a beam with a row of them is a Vierendeel girder outright.

The property that decides how bad it is is the tee’s section modulus, which is what the Vierendeel moment is divided by — and a tee is a shallow section with its material at one end, so that modulus is small however much steel is in it.

So the design lever is the tee depth rather than the hole depth, and the two are related by subtraction rather than by proportion. Reducing a 300 mm hole in a 533 mm beam to 250 mm takes the tee from 117 mm to 142 mm and its modulus from 37.6 to 52.8 × 10³ mm³ — a 17% smaller hole for a 40% smaller stress, which is the most useful sentence on this page for anybody actually holding a pen.

The three ways to make it acceptable #

The arithmetic points at three levers and they are worth ranking, because two of them are cheap and one is not.

That is why the tee is so weak: it is close to a flat plate, with its material at one end and very little depth, and the Vierendeel moment is divided by a modulus that has almost no lever arm in it.

Shorten the hole. The Vierendeel moment is linear in the length, so this is the most direct lever there is — and a round hole is the shortened version of a square one for free, since a circular opening is credited with 0.45 of its diameter.

Move it toward mid-span. The position sweep says 138 N/mm² at the middle against 242 near the support: a factor of 1.75 for a change that costs nothing structurally and everything in coordination.

Stiffen it. Welding a plate above and below the opening deepens the tee and raises its modulus, and because the modulus of a tee goes roughly as the square of its depth, a small rib is worth a great deal. It is also the expensive option — a fabricated detail on every hole — and it is the one that gets specified when the first two conversations have failed.

The order matters because the first two are decisions and the third is a cost. A hole in a beam is a coordination problem that becomes a structural one only when the coordination fails, which is why it is one of the few subjects in this collection whose answer is usually a drawing rather than a calculation.

The simpler check that comes first #

All of the arithmetic above is about the Vierendeel moment, which is the interesting failure. There is a duller one that has to be made first and that governs a hole placed where holes are usually placed.

The shear capacity of the section at the opening is what is left of its web. A 533 mm deep section with a 10.1 mm web has a shear area of about 5,380 mm²; cut a 300 mm hole through it and 233 mm of web depth survives, so the shear area is 2,350 mm² — 44 per cent of what it was. Nothing subtle has happened: the material that carries shear has been removed, and the capacity has fallen with it.

Near mid-span that costs nothing, because there is no shear there. Near a support it is decisive, and it is decisive before the Vierendeel check is, because the Vierendeel moment is a consequence of the shear crossing the opening and the shear capacity is a limit on the same quantity. So the sequence is: check that the remaining web can carry the shear at all, then check what that shear does as a Vierendeel moment.

The reduced web also has a second, less obvious problem. The two tee webs are now short unstiffened plates with a free edge along the hole, and a plate with a free edge buckles in shear at a fraction of the stress a plate held on both edges does. So the effective shear capacity of the tees is below the yield-based figure by an amount that depends on the opening’s length as well as its depth — which is the same length that drives the Vierendeel moment, pushing the same way.

Between them these two effects are the reason the practical limits on opening depth are stated as a fraction of the section depth rather than derived: a hole deeper than about half the section has taken most of the shear capacity and left two slender tees to carry the rest.

How the hole was made #

The analysis treats the opening as a shape. The fabricator treats it as an operation, and the operation leaves something behind.

A hole flame-cut through a web releases part of the section’s locked-in residual stress field, so the piece distorts as it is cut — and it leaves a new field of its own, with a strip of yield-level tension along the cut edge where the metal cooled against the restraint of the plate around it.

Two consequences follow, and they point in different directions.

For a statically loaded member it barely matters. The tension at the edge yields, redistributes, and by the time the section is anywhere near its capacity the residual field has been erased — as it always is by yielding.

For a member under repeated load it matters a great deal. A flame-cut edge carries yield-level residual tension, a rough surface, and a heat-affected microstructure, at the exact corner where the Vierendeel stress peaks. That combination is a low fatigue category, and the response is procedural rather than structural: drill or saw rather than flame-cut, grind the edge, and provide a generous corner radius.

The corner radius is worth naming because it appears in the rules as a bare number — at least twice the web thickness, or fifteen millimetres, whichever is greater — and it is a stress-concentration requirement standing in for a calculation nobody makes. A square corner has an elastic concentration factor that grows without limit as the radius falls, which is the same divergence that made fracture mechanics necessary; a radius turns it into a finite number that the ductility can absorb.

Where the model stops #

The shear is assumed to divide equally. That is right for equal tees and wrong for the common case of an opening placed off centre, where the larger tee attracts more — in proportion to stiffness elastically, and in proportion to plastic capacity at collapse, which are two different splits.

Local buckling of the tees is ignored. The compression tee is a slender outstand carrying axial force and bending across an unrestrained length, and for a large opening in a slender section it can go locally before it yields — which the stress check above cannot see.

The interaction of the four corner stresses is not a check. Each corner has an axial stress and a bending stress, and the resulting state is biaxial with shear present, so the honest criterion is a combination rather than the sum of two numbers.

Two openings close together are one opening. If the web post between them is shorter than the openings themselves, the post is a Vierendeel post rather than a piece of web, and it has its own buckling and bending checks.

Nothing here is a fatigue check. The corner of an opening is a stress concentration in a member under repeated load, and a detail category rather than a stress governs whenever the beam is a crane girder or a bridge.

And the effective-length credits for a circular hole are empirical. The 0.45 and 0.9 used here are representative values from the literature; they are not derived from anything on this page, and different sources give different numbers.

What the pictures cannot show #

The hero figure draws the Vierendeel moment as a small double-curvature line inside each tee, at a scale chosen so that it is visible. There is no such curve in the beam; the tee bends by a fraction of a millimetre, and the drawn shape is a moment diagram laid over a member rather than a deflection.

The point of contraflexure is drawn as a dot at the exact middle of the opening, and it is only there if the two tees are identical and the global moment does not vary over the opening’s length. Both of those are approximations, and the second is worse for a long opening: the moment does change across 400 mm, and the contraflexure point shifts accordingly.

And no figure shows the thing a fabricator would ask about first, which is the stiffening. A large opening is almost always given horizontal stiffeners above and below, sometimes a full ring, and every number on this page is for an unstiffened hole — which is the case that is easy to compute and rare to build.

The ladder from here #

Later rungs on this anchor: the plastic analysis of an opening, where all four corners reach their interaction limit together and the capacity is a mechanism rather than a stress. Stiffened openings, and the arithmetic of how much a horizontal rib restores. Cellular and castellated beams, which are the same argument at every 400 mm and are designed as Vierendeel girders. Web post buckling between adjacent openings, which is a diagonal compression member with no diagonal drawn. Openings in composite beams, where the concrete slab carries a large part of the shear across the hole and the tee’s job is much reduced. Openings in concrete beams, where the model is strut-and-tie from the outset and the reinforcement round the hole is the answer. And the retrofit case — a hole cut into an existing beam, where the loss of section is real, the Vierendeel moment is real, and neither was in the original calculation.

The rule of thumb that survives all of it is short: the hole’s length is the number to argue about, and its position should be as near mid-span as the services will allow — which is the opposite of what the building wants, and is a conversation rather than a calculation.