Four corners and a mechanism

The hole that costs nothing, and everything ends at a stress: the tee above an opening carries an axial force from the global moment and a local moment from the shear, the two add, and the sum is compared with the yield stress. On a 533 UB with a 400 × 300 hole at a sixth of the span, that sum is 238 N/mm² against 355 — 67 per cent utilised, and the check passes.

The check is answering the wrong question, and by a factor of two and a half.

First yield at one corner arrives at 1.49. The mechanism arrives at 3.89. Everything between those two numbers is a reserve that the elastic check cannot see, and knowing where it comes from also says where it disappears.

Which free body produced the number #

The free body is the length of beam containing the opening, cut vertically each side of it and horizontally along the mid-depth of the hole. That gives two tees, top and bottom, each spanning the length of the opening.

Crossing the cuts are three things, and separating them is the whole calculation.

A global moment, carried as a couple: compression in the top tee, tension in the bottom, each equal to $M/z$ where $z$ is the distance between the two tees’ centroids. On this beam that is 301 kN.

A global shear, shared between the two tees. For equal tees it is half each, and each half has to be carried across the opening by the tee bending in double curvature — which is exactly what a Vierendeel panel does, at the scale of a hole rather than a panel.

And the local moment that follows from that: each tee bends over half the opening’s length, so the moment at each of its ends is $V_{tee}a/2$ — 6.3 kNm here, at four places, with the sign reversing between the two ends of each tee.

So there are four locations where a moment and an axial force act together, and they are the four corners of the hole. The elastic check picks the one where the two stresses add most and compares that fibre with the yield stress. The mechanism asks a different question: at what load can all four form hinges, so that the two tees become a mechanism and the beam stops carrying the shear across the hole at all?

The tee’s own interaction, computed rather than quoted #

Answering it needs the tee’s plastic moment at the axial force it happens to be carrying, and that is a diagram rather than a number.

The construction is direct. Put the section fully plastic — every fibre at $\pm f_y$ — with the neutral axis at some height, and read off the resultant force and the resultant moment. Sweep the axis from one edge to the other and the pairs trace out the interaction. No formula is needed and none is used here, which matters because a tee is not a shape any of the standard formulae were written for.

Two properties of the diagram are worth reading off it.

It closes at both ends. A section fully in compression has its resultant at the area centroid, so its moment about that centroid is zero, and the same in tension. Getting that wrong is easy and instructive: take moments about the equal-area axis instead — the axis the neutral axis sits at when $N$ is zero — and the diagram does not close, because a uniform stress block has a moment about it.

And it is very flat near the top. The 301 kN in this tee is 20 per cent of its squash load and costs 13 per cent of its plastic moment: 28.1 kNm against 32.2. The reason is the flange. The axial force is accommodated by moving the plastic neutral axis into the flange, and the flange sits close to the plastic centroid, so the moment given up is a good deal less than the force would suggest. A tee is more tolerant of axial force than a proportional rule would allow, which is a good deal of why this reserve exists.

Where the reserve comes from, and where it goes #

The mechanism factor is not the yield factor times a shape factor, and the reason is that raising the load raises the axial force too — which eats the moment the hinges have to give. It is the root of an equation rather than a multiplication.

The coincidence at mid-span is the most useful line in the figure and it is exact rather than approximate.

At mid-span the shear is zero, so the Vierendeel moment is zero and each tee is in pure axial force. First yield of a uniformly stressed member is the same event as its full plasticity — there is no fibre left over, no shape factor, and no redistribution available. The two criteria are then the same criterion, and the reserve is exactly one.

Everything above one is therefore bought by the local bending, and the amount of it is the tee’s own shape factor discounted by the axial force it is carrying. That gives the practical rule: the elastic check on an opening is conservative in proportion to how much of the tee’s stress is Vierendeel bending rather than global axial force — which is largest at the supports and vanishes at mid-span, exactly where the elastic check said the worst position was least dangerous.

The two facts together are a genuine reversal. The elastic stress is worst near the support; the reserve is largest near the support; and the two nearly cancel, which is why openings can be placed over a much wider range of the span than either calculation alone suggests.

What a mechanism means for a beam that is not collapsing #

There is a reasonable objection to all of this and it deserves answering rather than deflecting.

A four-hinge mechanism at the opening does not drop the beam on the floor. The tees hinge, the shear stops crossing the hole in double curvature, and the load finds another route: the section either side of the opening is intact, and the beam can carry a good deal more by arching over the hole, dragging the flanges into tension, and deflecting a great deal to do it.

So the mechanism factor is not a collapse factor for the structure. It is the load at which the opening stops behaving in the way the analysis assumes, and past it the deflection grows fast and the beam’s behaviour depends on details nobody has computed. That is exactly the right thing for a limit state to be, and it is the same relationship a plastic hinge has to a real frame: the mechanism is where the model ends rather than where the steel parts.

What the number is good for is comparison and margin. An elastic utilisation of 0.67 says nothing about how close the detail is to behaving differently; a mechanism factor of 3.89 against a yield factor of 1.49 says the detail has a great deal of ductility available, which is what a designer wants to know about a hole that will be cut on site by somebody who has not read the drawing.

The post between two holes #

One opening has four corners. A cellular beam has an opening every 1.5 diameters along its whole length, and what separates them changes the problem.

Between two adjacent openings is a web post: a short piece of web, full depth of the hole, carrying the horizontal shear that transfers the change in chord force from one panel to the next. That shear is $\Delta N$ over the pitch, it acts on a post whose width is the pitch less the opening, and it puts the post into diagonal compression on one side and diagonal tension on the other.

The post has two ways to fail and neither is a mechanism of the tees. It can yield in horizontal shear across its narrowest section, and it can buckle, as a diagonal strut with one edge free at each hole. The second usually governs, it depends on the post’s width and the web’s thickness, and it is the reason cellular beams have a minimum spacing rule that has nothing to do with the openings themselves.

That is a different family of check from everything above — a stability one rather than a plastic one — and it means a cellular beam is governed by the material between its holes rather than by the holes. The hole is not the weak point; the gap between two holes is.

What it is worth on a real floor #

Numbers of this size change what a service opening costs, so it is worth putting them on a plan rather than on a beam.

A 9 m secondary beam at 3 m centres under 20 kN/m carries a shear of 63 kN at the sixth point where the first duct usually wants to go. The elastic check says the tee is 67 per cent utilised, which on most jobs is a pass with no comment. The mechanism says the same detail has a load factor of 3.9, which is a great deal more margin than the beam itself has anywhere else — its bending utilisation at mid-span is 0.7 or so, giving a factor of about 1.4.

So the opening is not the weakest thing about the beam, and the elastic check has been reporting it as the most heavily worked detail on the drawing. That is the practical consequence of a first-yield criterion applied to a detail with four hinges in it, and it is the reason openings get moved, shrunk and stiffened far more often than the mechanics require.

Where it inverts is the case nobody draws: a hole cut after the beam is designed, by a services contractor, at whatever position the duct wants. Move the same 400 × 300 opening from a sixth of the span to mid-span and the elastic factor improves — 2.58 rather than 1.49 — while the reserve disappears entirely. The detail becomes safer and much less ductile at the same time, which is exactly the trade a rule of thumb about “keep holes near mid-span” gets right for the wrong reason.

A note on how the interaction was built #

The diagram in the first two figures is worth one paragraph of method, because the same construction answers a great many questions on this site and it is three lines of arithmetic.

Put every fibre of the section at $\pm f_y$ with the boundary at some height $y_c$. Sum the forces to get $N$ and sum the moments about the section’s plastic centroid to get $M$. Sweep $y_c$ from one edge to the other and the pairs $(N, M)$ trace out the whole interaction — for any shape, with holes in it, made of two materials, with the reinforcement wherever it is.

No interaction formula appears anywhere in it. The familiar expressions — a parabola for a rectangle, the two-branch rule for an I-section — are what this sweep produces for those particular shapes, and they exist because the sweep was expensive before there were computers. For a tee, an angle or the section left beside a hole, there is no published formula to reach for, and the sweep is easier than looking one up.

The rule this replaces #

Design guides for openings state the check as a stress and add a table of permitted opening sizes, and it is worth saying what that table is standing in for.

The permitted sizes are a mechanism calculation done once, for a family of sections and a range of positions, and expressed as geometry so that nobody has to repeat it. That is a good way to publish an answer and a bad way to understand one, because the two variables it hides — the shear at the opening and the tee’s own shape factor — are exactly the ones that change when a beam is not the beam the table was written for.

A composite beam, a beam with a point load beside the hole, a beam with a hole in a hogging region, a beam that is not a rolled section at all: each of those breaks the table and none of them breaks the calculation. Which is the general shape of the argument running a plastic analysis rather than a stress check always makes — the mechanism is more work and it transfers.

Where the model stops #

Four hinges, and no shear in them. A plastic hinge in a tee whose web is also carrying shear has less moment than the interaction says, and near the support the tee’s web is carrying a good deal. The reduction is small on this section — the tees’ shear stress is well under half the shear yield — and it is not small on a deeper hole in a thinner web.

No local buckling. The compression tee above the hole is an outstand with a free edge along the opening, and if it is slender enough it buckles before it hinges. That is a class check on a section nobody rolled, and it is the usual reason a real opening needs a stiffener.

The tees are equal. Concentric openings give two identical tees; an eccentric one gives a large tee and a small one, the shear divides in proportion to their second moments rather than equally, and the small tee reaches its hinges first.

And nothing here is a composite beam. With a slab on top, most of the shear crosses the opening in the concrete rather than in the top tee, and the whole Vierendeel calculation applies to a fraction of the load it was given here.

What the pictures cannot show #

An interaction diagram is a surface of capacity and the path drawn on it is a proportional loading. Neither is a history: a beam propped during construction and unpropped afterwards arrives at the same point by a different route, and a section that has yielded once carries residual stresses into whatever comes next.

They also cannot show the deflection. Everything on this page is a strength argument, and an opening’s most common consequence is the deflection it adds — the tees’ local flexibility over the hole, which is a serviceability quantity that no amount of plastic reserve improves. The same is true of the fatigue of the corner, where the governing quantity is a stress range at a radius and the elastic concentration is felt in full. A detail with a mechanism factor of 3.9 and an unacceptable deflection is a perfectly ordinary outcome.

The assumption the figure rests on #

That the shear divides equally between the two tees and each bends over half the opening.

Both halves are the assumption that the tees are held in double curvature by the beam either side of the hole — that the sections at the ends of the opening stay vertical. They do not quite: the beam either side is itself flexible, so the contraflexure point is not exactly at the middle of each tee, and the four moments are not exactly equal.

The error is small for a short opening in a stiff beam and grows with the length of the hole, which is the same shape of assumption a Vierendeel girder makes about its panels and fails in the same way at the end panels. The honest version is a frame analysis with the opening as a panel, which returns four different moments and moves the first hinge to whichever corner has the largest.

The ladder from here #

Later rungs on this anchor: web-post buckling worked through properly, with the diagonal strut model and the spacing rule it produces. Stiffened openings, and the arithmetic of how much a horizontal rib restores. Openings in composite beams, where the slab carries the shear across the hole and the top tee’s job nearly disappears. Circular openings, where the tee’s depth varies along the hole and the governing section is not at the edge. Openings in concrete beams, where the model is strut-and-tie from the outset. And the fatigue of an opening, where the corner that governs is the one with the largest stress range rather than the largest stress, and a hole with a generous radius is worth several detail categories.