Structural Engineering & Statics
where the load goes
Sections and stress

The hole that costs nothing, and everything

A service opening removes 30% of a beam's second moment and 0.7% of its deflection. What it costs is not that. Across the opening the shear has nowhere to go but through the two tees, and a tee carrying shear over a length bends.

Assumes The material far from the middle does nearly all the work, The shear nobody draws and Plane sections stay plane, and what the assumption costs.

19 min read 10 figures Which failure arrives firstThe load must go somewhere

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.

A hole in a web is a Vierendeel panelA 400 × 300 rectangular opening in a 533 deep beam, 15% along a 9 m span carrying 20 per metre — where the moment is 103 kNm and the shear 63 kN. The moment is a couple on the two tees, 301 kN on a lever arm of 343 mm, which is 70.3 N/mm² of uniform stress. The shear has nowhere to go but through the tees, so each carries 32 kN over the opening and bends in double curvature: a Vierendeel moment of 6.3 kNm and 167.6 N/mm² on top. So 70% of the stress at the corner exists because the hole has a LENGTH, and only 30% of the section's second moment has gone.contraflexure at the middle of the holefrom the moment: 70.3 N/mm²from the shear: 167.6 N/mm²total 237.9 of 355second moment lost: 30%deflection up by 0.7%the shear the tees must carry: 63 kN, half each, over 400 mm
Fig. 1 A 400 × 300 opening in a 533 mm beam, 15% along a 9 m span carrying 20 kN/m. The global moment of 103.3 kNm is a couple on the two tees — 301 kN on a lever arm of 343 mm, or 70.3 N/mm². The 63 kN of shear has nowhere to go but through the tees, so each bends in double curvature over the hole: 6.30 kNm, and 167.6 N/mm² on top. 70.5% of the corner stress is there because the hole has a LENGTH.

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=301N = 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.

Mvier=Vtee ao2=31.5×0.42=6.30 kNmM_{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 Vtee ao/2V_{tee}\,a_o/2. It is linear in the opening’s length and contains its depth only through the tee’s own properties.

aoa_o Vierendeel stress total
200 mm 83.8 154.1
300 125.7 196.0
400 167.6 237.9
600 251.4 321.6

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.

Every strip counts by the square of its distanceA rectangular section divided into equal strips, with each strip's contribution to the second moment of area drawn beside it. The strips are identical in size; only their distance from the neutral axis differs.neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work
Fig. 2 Why the loss of section is so small in the first place. A section’s second moment is dominated by the strips furthest from the neutral axis, and a web opening removes the strips nearest it. The 30.3% lost here is the material that was contributing least, which is exactly why the hole looks like a bargain.

The round hole, and why it is worth half

The worst place for a hole is where the bending is leastThe stress in the tee above a 400 × 300 opening, as the opening is moved along a 9 m span under a uniform load, split into the part that comes from the global moment and the part that comes from the local Vierendeel bending. The first rises toward mid-span and the second falls, and the second wins: the total is worst at 6% of the span, where the bending moment is only 46% of what it is at the middle. Services are routed near the supports because that is where the ceiling void is, which puts the holes exactly where the shear is — and the dashed curve is the same hole cut round instead of square, which halves the effect by shortening the span the tee has to bridge.0%10%20%30%40%50%050100150200250position of the opening along the spanstress in the tee (N/mm²)total, square holethe Vierendeel partthe global momentthe same hole, round
Fig. 3 The tee stress as the opening is moved along the span, split into the part from the global moment and the part from the local bending, with the same hole cut round instead of square shown dashed. The round hole halves the total, because the tee has a shorter distance to bridge and only reaches its full depth at one point.

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.

Three times the stress, and it does not matter how big the hole isThe hoop stress around a circular hole in a wide plate pulled at 100 N/mm², from Kirsch's exact solution. At the sides of the hole it is 3.0 times the applied stress — 300 N/mm² — and the factor is the same for a hole of any radius, because the radius cancels. At the top and bottom of the hole it is -1.0 times the applied stress, which is compression in a plate that nothing is pushing. The disturbance dies quickly: the stress is within 5% of the applied value by 3.5 hole radii, which is Saint-Venant's principle with a number on it.pulled at 100 N/mm², left and right300-100 — compressionhoop stress, tinted3.0× at the edgewithin 5% by 3.5 radiithe applied stressdistance from the centre, in hole radii12345
Fig. 4 The field round a hole, at the scale below the one this page works at. Every calculation here treats the tees as small beams with a section modulus, and within a few tens of millimetres of the corner that is a fiction — the real peak is a concentration factor times a nominal stress, and it decays over a distance of the order of the hole.

The worst place for a hole is where nothing is happening

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

position along span tee stress
5% 242 N/mm²
15% 238
25% 223
35% 197
50% 138

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.

Load, shear and moment — a simple spanThe applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear.4 per unit lengthshear16.0moment32.0 at x = 4.00the moment peaks exactly where the shear passes through zero
Fig. 5 The two diagrams the two terms follow. The Vierendeel moment tracks the upper one and the couple in the tees tracks the lower one, and the upper one is at its largest exactly where the lower one is at its smallest.

The deflection, which really is negligible

The deflected shape is the moment, integrated twiceA loaded beam with its deflected shape above and its bending-moment diagram below. The shape was obtained by integrating the moment twice and fitting the constants to the supports; the vertical scale is exaggerated enormously.the largest movement, at x = 4.00momentdrawn at roughly three hundred times the real deflection —a beam at its serviceability limit moves about a three-hundredth of its span
Fig. 6 The curve the hole barely alters. Virtual work over the opening gives the extra movement from the tees’ local double-curvature bending, and for a single 400 mm hole it is 0.75% of the plain beam’s deflection — small enough that the loss of section is not the reason to worry about openings.

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

Where a section exists, and where it does notThe same beam divided into the regions the two theories own. Within about one depth of a support, a concentrated load, a corner or an opening, the strain is not linear across the section and every calculation on this site that begins by choosing one is inapplicable — those are the D-regions, marked here. What is left between them is the B-region, where beam theory is exact enough to have been trusted for two centuries. On a beam this deep the D-regions are most of it, which is the practical reason the strut-and-tie model exists at all: 38% of this span is a region a section cannot describe.DBDBDshaded: one depth either side of every discontinuity, where no section describes the strain
Fig. 7 The general statement this is a case of. Within about a depth of any discontinuity the strain is not linear across the section, and an opening is a discontinuity. Everything on this page is a way of writing an equilibrium solution for a region where the ordinary section calculation is inapplicable.

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 shear goes round the corner instead of across itA 6-panel Vierendeel girder, 12 m by 1500 mm, under 100 kN at mid-span. There is no diagonal in it, so each panel's 50 kN of shear is carried as bending in the chords: the curves drawn along them are the chord moments, and every one passes through zero at the middle of its own panel. The local moment is the panel shear times the panel length over four, 25.0 kNm, and it adds to an axial force of 200 kN from the global moment at the same point. The girder deflects 6.20 mm against 3.18 mm for the same members triangulated — 1.95 times — and 68% of that movement is chord bending that a diagonal would have removed entirely.100 kNchord moment 25.0 kNm from the panel shear · axial 200 kN from the global moment6.20 mm against 3.18 mm triangulated — 68% of it is chord bending
Fig. 8 The full-size version: a girder with no diagonals at all, in which every panel is a hole and every chord is a tee. The curves drawn along the chords are their local moments, each passing through zero at the middle of its own panel — the same double curvature, the same contraflexure, and 68% of the girder’s deflection coming from it.

The correspondence is exact. A Vierendeel girder’s panel shear produces a chord moment of Vs/4Vs/4 — which is Vchord⋅s/2V_{chord} \cdot s/2 with the shear halved between two chords, and is this page’s Vteeao/2V_{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.

Moving the flanges apartThe second moment of area of an I-section against its depth, with the flange and web areas held constant. The growth is close to quadratic, because the parallel-axis term dominates everything the flanges contribute about their own centres.1001502002503000M10M20M30M40M50M60Moverall depth1.0×2.6×4.9×7.9×13.8×21.3×same steel, moved apart
Fig. 9 And the property that decides how bad it is. The tee’s section modulus is what the Vierendeel moment is divided by, and a tee is a shallow section with its material all at one end — so it has a small modulus for its area, and a hole that leaves deeper tees is very much kinder than the loss of second moment suggests.

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.

The same material, four waysFour cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 10 Why the tee is so weak: it is close to the first of these four shapes, with its material at one end and very little depth. The Vierendeel moment is divided by a section modulus, and a tee’s modulus for its area is among the worst on this chart.

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.

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.

Where this goes next

100 kNchord moment 25.0 kNm from the panel shear · axial 200 kN from the global moment6.20 mm against 3.18 mm triangulated — 68% of it is chord bending The truss with no diagonals A rectangle without a diagonal is a mechanism, so a Vierendeel girder makes its corners rigid instead. The shear a diagonal would have carried as axial force now travels as bending in the chords, and bending is a far more expensive way to move a force. neutral axispeak 0.4stressflow, q = VQ ÷ Imean stress 0.19 — the value a shear divided by an area would givepeak 2.15× that, and in the place bending ignoresthe flow is continuous; the stress jumps wherever the width does The shear nobody draws A stack of loose planks slides at its ends when it is loaded. Glue them and the sliding stops — and whatever the glue is now carrying is a stress that no bending calculation contains. 1200 kNtie 750 kNstrut 960 kN38.7°z = 1600strut 3.0 N/mm² over 812 mm · limit 15.8bursting across each strut 240 kN · tie steel 1724 mm² When there is no section to design Beam theory needs a section, and a section needs the strain to be linear across it. Within about a depth of a support, a load, a corner or a hole it is not — and those are the regions structures actually fail in. pulled at 100 N/mm², left and right300-100 — compressionhoop stress, tinted3.0× at the edgewithin 5% by 3.5 radiithe applied stressdistance from the centre, in hole radii12345 The hole that multiplies the stress by three The stress at the side of a hole is three times the applied stress whatever the hole's size, and at the top and bottom of the same hole it is minus one times it — compression in a plate that nothing is pushing.

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DeflectionDisturbed regionFlangeIdealisationLoad pathPlane sectionsPoint of contraflexureSecond moment of areaSecondary bendingSection modulusShear forceStress concentration