The notch that costs more than its depth

The shear that decides a timber beam found that a timber beam is governed by shear at proportions people build every day, because its bending strength is only six or seven times its shear strength along the grain. It ended on a qualification: nothing in it was a notch calculation. A notched end concentrates the shear at the re-entrant corner, it said, and the check there is a fracture one rather than a stress one.

Notches are not exotic. A beam is notched at its bearing to bring its top flush with a joist it meets, to drop it onto a wall plate, to clear a service, or to correct a support that was built a little high. The cut is quick and the remaining depth looks ample. The question this essay answers is how much of the beam’s end survives the cut, and the answer is that it depends on the beam’s size in a way no stress check can express.

A crack, not a stress #

The beam is glued-laminated softwood of an ordinary grade: shear strength 3.5 N/mm², modulus along the grain 11,500 N/mm², shear modulus 650 N/mm². It is 400 mm deep and 140 mm wide, and at its support the bottom half is cut away, so the reaction is carried by the 200 mm that remain. The corner of the notch stands 50 mm from the line of the reaction, which is where the edge of a 100 mm bearing would put it.

A stress check would reason like this. The shear stress in a rectangular section peaks at 1.5 times the average, so the end carries $V = f_v b h / 1.5$, which is 130.7 kN at full depth; halve the depth and it carries half, 65.3 kN. That is a reasonable calculation for a material that yields, and timber in shear along the grain does not yield. It splits. The corner of the notch is a re-entrant angle — a sharper version of the hole that multiplies the stress — where the stress in an elastic solid is unbounded in principle, and in timber a crack opens there at a modest load and runs along the grain, between the fibres, towards mid-span. Once it has run, the bottom of the beam beyond the notch is hanging from the top by nothing, and the end is carrying its reaction through the 200 mm strip above the crack as a cantilever.

What decides whether the crack runs is energy. As it grows by a small length, the part of the beam the reaction has to pass through is shallower over that length, the beam is more flexible, and the work the reaction has done is partly released. If what is released per unit of new crack surface exceeds what the timber needs to create that surface, its fracture energy, the crack runs. Eurocode 5 builds this in through a factor $k_v$ that multiplies the shear strength on the reduced section:

$$k_v = \min\left\{1,\ \frac{k_n\left(1 + \dfrac{1.1\, i^{1.5}}{\sqrt h}\right)}{\sqrt h\left(\sqrt{\alpha(1-\alpha)} + 0.8\,\dfrac{x}{h}\sqrt{\dfrac{1}{\alpha} - \alpha^2}\right)}\right\},$$

with $h$ in millimetres, $\alpha$ the share of the depth left over the support, $x$ the distance from the reaction to the corner, $i$ the slope of a sloped notch and $k_n = 5$ for glulam. For this end it is 0.40, and the end carries 25.8 kN: a fifth of the un-notched beam, and two fifths of what the stress check promised.

The beam keeps a quarter #

To see the depth’s own effect, put the corner over the reaction for the moment, so that $x = 0$, and vary only how much of the depth is left.

Every curve lies below the diagonal, which is the stress check’s answer, and the deeper the beam the further below it lies. Notched to half depth, the 400 mm beam keeps a quarter of its shear resistance, not a half. The 800 mm beam keeps 18 per cent. Only the 200 mm beam keeps more than a third.

The curves climb steeply towards the right, and a shallow notch is nearly harmless: the factor reaches one, and the stress check takes over, once the notch removes less than a seventh of a 200 mm beam’s depth, less than a sixteenth of a 400 mm one’s and less than a thirtieth of an 800 mm one’s. The rule of thumb that a notch of a tenth of the depth does not matter is true of small beams and false of large ones. A 50 mm adjustment notch in an 800 mm glulam beam — a correction for a bearing built a little high, a sixteenth of the depth — takes a third of its end’s shear resistance.

A size the stress check does not have #

The size effect is the whole difference between this check and a stress check, so it is worth seeing directly.

The factor falls steadily with depth for any notch, and it never levels off. Between a 200 mm joist and a 1,400 mm glulam girder, the same proportion of notch costs the larger member two and a half times as much of its shear strength. The curves for half and seven tenths are close together, because $\alpha(1-\alpha)$ is nearly the same for both — 0.25 and 0.21 — and a notch of seven tenths’ depth left is not much better than one of a half. A notch is costly at any depth beyond a shallow one, and costliest in the members where the cut is most tempting, the deep ones with depth to spare.

On logarithmic axes the dependence is a straight line.

A slope of one half is the signature of linear elastic fracture, and the bigger one is the weaker one explained where it comes from. The energy stored in a beam’s end, per unit width, grows as the square of its size: the stress times the strain, over an area. The energy needed to grow a crack, per unit width, grows only with its length, which scales with the size once. At a fixed stress, a larger beam has more energy to release per unit of crack than a smaller one, so the stress at which the release reaches the fracture energy falls with size, as one over its square root. A stress check has no length in it and cannot see any of this; a fracture check has nothing else.

Where the formula comes from #

The factor in Eurocode 5 is not empirical in form. It comes from a calculation that can be done on the back of the drawing.

When the crack has grown by a length $\mathrm{d}a$, the reaction’s path through that length has gone from the full depth $h$ to the strip $\alpha h$ above the crack. The strip is more flexible in shear, by $1.2/(G b)$ times $(1/\alpha h - 1/h)$ per unit length, and it is more flexible in bending, because the reaction’s moment, $V(x + a)$ at the crack tip, now bends the strip alone rather than the whole section. The extra compliance per unit of crack, times $V^2/2b$, is the energy release rate; setting it equal to the fracture energy $G_c$ gives

$$\frac{V_f}{b\,\alpha h} = \frac{\sqrt{G_c/h}}{\sqrt{\dfrac{0.6(\alpha - \alpha^2)}{G} + \dfrac{6\beta^2(1/\alpha - \alpha^2)}{E}}}, \qquad \beta = \frac{x}{h}.$$

The $\sqrt{G_c/h}$ is the size effect, out in front where it cannot be missed. The first term in the root is the shear flexibility, which is all there is when the corner is over the reaction; the second is the bending flexibility, which grows with the square of the corner’s distance from it.

Gustafsson’s formula, from which the Eurocode factor is derived, has the same two terms but adds their square roots instead of taking the root of their sum. The plain balance treats the strip below the crack tip as a cantilever fixed at the tip, and it is not fixed: the uncracked timber beyond the tip rotates with it, and that rotation couples the two terms. The sum of the roots is never less than the root of the sum, so his form is the more cautious, and the Eurocode factor follows it closely: its 0.8 in front of the bending term is $\sqrt{10G/E}$ for a timber whose stiffness along the grain is sixteen times its shear modulus. With the corner over the reaction the two forms are identical, and Eurocode 5’s $k_n = 5$ for glulam corresponds, through them, to a fracture energy of 0.13 N/mm with this shear strength and modulus.

The corner’s distance from the support #

The second term is the one a designer controls without changing the beam: where the corner stands.

The Eurocode curve and Gustafsson’s lie on top of each other, and both fall fast: every 50 mm the corner moves away from the reaction, on this beam, costs about a fifth of what is left at first. A corner half the beam’s depth from the reaction leaves the end a quarter of the material’s strength on its reduced section, an eighth of the un-notched beam. The plain energy balance falls more slowly at first, because without the root’s rotation its bending term enters as a square, and it ends up a third to a half higher. The difference is a reminder that Eurocode 5’s version is chosen to be safe as well as to be right.

The practical instruction is in the shape. A notch should end at the bearing, not beyond it. A notch cut long — to clear a bearing plate with room to spare, or to let a beam sit on a wall plate whose edge is set back — puts the corner out in the span, where the reaction has a lever arm on the strip above the crack and the crack runs at a fraction of the load. The 50 mm in the first figure, half a bearing’s length, already cost a fifth.

A slope buys it back #

The other control is the shape of the corner itself. A notch cut at a slope instead of square spreads the change of depth over a length, and the crack, which needs a sharp change of compliance to run, has less to work with.

The taper term in Eurocode 5 grows as the slope’s 1.5 power over the root of the depth, so it helps small beams most and large beams least, and it helps everyone more than any reasonable amount of extra depth would. A 1 : 10 slope restores the 400 mm beam’s notched end to its full shear strength on the reduced depth — doubling it — and takes the 800 mm beam from 0.35 to 0.79. A 1 : 2 chamfer, the sort a carpenter adds out of habit, buys a sixth on the 400 mm beam and a tenth on the 800 mm one. The slope has to be long to be worth having, and a long slope is a longer cut and a longer bearing; that is the trade Eurocode 5 is pricing.

The alternative that every timber design guide prefers is not to notch at all, and the second is to reinforce the end across the potential crack with a fully threaded screw or a glued-in rod driven vertically through the depth near the corner — the screw pulled instead of bent used as a tie across the grain. A screw does not stop the crack starting; it holds the two sides together once it has, so that the strip above and the beam below go on working as one, and the check becomes a check of the screw in tension instead of the timber in fracture.

Two beams drawn to the same proportions #

The size effect is easiest to underestimate when a detail is scaled. Take a joist 200 mm deep and 70 mm wide, notched to half its depth with the corner over the reaction, and a glulam girder exactly five times larger in every dimension: 1,000 mm deep, 350 mm wide, notched to 500 mm. Drawn at the same size on two sheets, they are the same picture.

Un-notched, the joist’s end carries $3.5 \times 70 \times 200/1.5 = 32.7$ kN and the girder’s $3.5 \times 350 \times 1000/1.5 = 816.7$ kN — twenty-five times as much, as the areas say. Notched, the joist keeps $k_v = 5/(\sqrt{200} \times 0.5) = 0.71$ on its 100 mm and carries 11.5 kN, 35 per cent of its full end. The girder keeps $k_v = 5/(\sqrt{1000} \times 0.5) = 0.32$ on its 500 mm and carries 129 kN, 16 per cent. The picture is the same; the girder’s notched end is worth less than half as much of its own beam as the joist’s is.

This is how a detail that has worked for years in small members fails the first time it is used in a large one. A carpenter’s habit of notching a joist onto a ledger is sound because joists are shallow, and a 200 mm member sits only a factor of two above the depth at which fracture stops governing. The same habit applied to a deep glulam beam, a factor of ten above it, is not.

It is also why a test on a model says less than it seems to. A notched specimen at a fifth of full size, loaded to failure, cracks at a shear stress $\sqrt 5 = 2.24$ times higher than the full-size member will. The test is correct about the model; scaled up by its area, as a stress would be, it overstates the full-size end’s capacity by more than double. The fracture energy, not the strength, is the property a notch test measures — the same trap as the strength no specimen had, read off the wrong quantity — and Eurocode 5’s $k_n$ is fitted to tests of several sizes for exactly that reason.

The 400 mm beam by hand #

The figures can be reproduced in a few lines. For the 400 mm beam with half its depth left and the corner over the reaction, $\sqrt{\alpha(1-\alpha)} = 0.5$ and the bending term vanishes, so

$$k_v = \frac{5}{\sqrt{400} \times 0.5} = \frac{5}{10} = 0.50,$$

and the end carries $0.50 \times 3.5 \times 140 \times 200 / 1.5 = 32{,}667$ N, 32.7 kN: a quarter of the 130.7 kN the beam carries un-notched. At 800 mm deep, $\sqrt{800} = 28.3$, $k_v = 0.354$, and the share is $0.354 \times 0.5 = 0.177$.

With the corner 50 mm from the reaction, $\beta = 0.125$ and the bending term is $0.8 \times 0.125 \times \sqrt{2 - 0.25} = 0.132$, so the bracket becomes 0.632 and $k_v = 5/(20 \times 0.632) = 0.395$: the first figure’s 25.8 kN. And the fracture energy behind $k_n$: equating the two forms at $x = 0$ gives $G_c = 0.6\,(k_n f_v/1.5)^2/G = 0.6 \times (5 \times 3.5/1.5)^2/650 = 0.126$ N/mm.

What the factor leaves out #

The fracture model treats the timber as a linear elastic solid with one crack path, along the grain at the level of the corner, and one fracture energy. Real timber has a fracture energy that differs between opening and sliding, varies by species and density, and falls with moisture cycling; Eurocode 5’s single $k_n$ per product is calibrated to tests rather than derived, which is why it is quoted as a product property and not a material one. The model has no fatigue in it, and notched ends in service often fail after years, as seasonal drying opens small checks at the corner that the next wet season’s load drives further. A notch on the compression face at the support — the top cut away, the reaction under the full depth — is not covered at all, because the crack there is pressed shut, and Eurocode 5 takes $k_v$ as one. And the model is for a notch at the support; a hole or a notch out in the span meets a combination of shear and moment that the factor is not written for.

The crack between the two checks found the same structure in steel: a stress check that a ductile material passes and a fracture check that a flaw fails, with size deciding which governs. Timber’s notch is the version of that where the flaw is cut deliberately, at a known place, by the person who will later sign the calculation, and the flaw that sets the strength is a corner the drawing shows.

Still open: the hole through the web of a timber I-joist #

Engineered timber I-joists have thin webs of oriented strand board or plywood, and services are passed through them in holes that the manufacturer’s tables allow by size and position. A hole near the support of an I-joist is a pair of notches — one above and one below it — in a web whose fracture energy and stiffness are not those of solid timber, joined to flanges that carry the moment. Whether a hole’s corners crack by the same size law, and whether the tables, which are given in absolute sizes rather than proportions, are quietly a size-effect correction of the kind this essay has drawn, is the question the notched end leaves for the material that has a direction when it is cut into a web.