The shear that decides a timber beam

Horizontal shear is the shear nobody draws, and on a steel beam that is a fair description of how much attention it gets. Jourawski derived the expression in 1855 while investigating why timber railway bridges were splitting along their length, and the material he was looking at is the one where the omission is not survivable.

Why the crossover contains nothing but the material #

Write the two utilisations for a rectangular beam under a uniform load.

Bending: the moment is $wL^2/8$ and the modulus is $bd^2/6$, so the stress is $3wL^2/4bd^2$ and the utilisation is that over $f_m$.

Shear: the shear force is $wL/2$ and the peak stress is 1.5 times the mean, so it is $3wL/4bd$ and the utilisation is that over $f_v$.

Divide one by the other and everything cancels but two lengths and two strengths:

$$\frac{\text{bending}}{\text{shear}} = \frac{L}{d}\cdot\frac{f_v}{f_m}$$

The load has gone, the width has gone, and the units have gone. The two checks are equal when $L/d = f_m/f_v$, and that number is a property of the material and of nothing else.

For structural steel, $f_m$ is $f_y$ and $f_v$ is $f_y/\sqrt3$, so the ratio is $\sqrt3 = 1.73$. A steel beam would have to be less than twice its own depth long for shear to govern, and an object of those proportions is not a beam. That is why shear is a formality in steel design and why a generation of engineers trained on steel find the timber result surprising.

For a softwood at $f_m = 28$ and $f_v = 4.2$ the ratio is 6.7, and beams at span-to-depth ratios below that are ordinary: a lintel, a trimmer, a header over an opening, a beam carrying a heavy point load close to a support.

Which free body produced the number #

The free body is the top half of a length of beam, cut horizontally at the neutral axis and vertically at each end.

Crossing the two vertical cuts is the compression in that half, and it differs between them because the moment differs. Crossing the horizontal cut is the shear traction that makes up the difference — which is the whole of Jourawski’s argument and gives $q = VQ/I$.

Two properties of that distribution matter here and are easy to lose.

The peak is 1.5 times the mean, exactly, for any rectangle. A section has two areas and the tables give one of them; the shear check on a timber beam uses the peak, so the 1.5 is not a refinement but a factor on the answer.

And the peak is at the neutral axis, which is the plane the grain runs along and where the bending stress is nothing. A timber beam therefore fails in shear at the one place where a bending check reports zero, along the one plane the material is weakest on.

What the crossover means for a real member #

The ratio is clean and the design consequence is worth spelling out, because it inverts the order in which checks are usually made.

On a beam of span-to-depth 20 — a joist, a rafter, a floor beam — bending governs by a factor of three and the shear check is a formality, exactly as in steel.

On a beam of span-to-depth 8 the two are within 20 per cent of each other, and the section that satisfies one barely satisfies the other. This is the range of a trimmer round a stair opening, a header over a wide door, or a beam supporting a wall.

On a beam of span-to-depth 4 — a deep lintel, a short transfer member, a beam cut down to fit under a ceiling — the shear utilisation is 1.7 times the bending one, and deepening the member makes it worse: shear stress goes as $wL/bd$, so adding depth reduces it in proportion, while adding depth reduces bending as the square. A member in the shear-governed range gets less benefit from depth than one in the bending-governed range, which is the opposite of the habit every designer brings to a timber beam.

The remedy for a shear-governed timber beam is width, not depth, and it is the only place in this collection where that sentence is true.

Three shear strengths, not one #

Timber’s shear strength depends on the direction of the shear relative to the fibres, and there are three cases rather than two.

Parallel to the grain, with the shear tending to slide one layer of fibres along the next: about 4.2 N/mm² for softwood. This is the one in the beam above.

Perpendicular to the grain, cutting the fibres across: several times stronger, and it never governs because the parallel plane is always available.

Rolling shear, with the shear acting across the fibres in the plane perpendicular to them, so the fibres roll over each other like a bundle of straws: about 1.2 N/mm², a third of the parallel value and the weakest strength timber has.

That figure is the whole case for caring about which shear plane a member has. The same beam, the same load, the same section, and the governing check has changed because the fibres in the critical layer happen to run the other way.

It is worth being clear about why the three differ so much, because the mechanism is geometric rather than chemical.

Timber is a bundle of hollow tubes glued together — cells running along the grain, with much weaker material between them. Shear parallel to the grain slides one bundle of tubes along the next, and what resists it is the lignin between the cell walls: weak, but acting over a very large surface.

Shear perpendicular to the grain, cutting across the tubes, has to sever the cell walls themselves, which is why it is several times stronger. It never governs because a plane parallel to the grain is always available a fraction of a millimetre away, and a material fails on the weakest plane it has rather than on the one the load suggests.

Rolling shear acts in the plane at right angles to the fibres, where the cells are free to roll over one another like logs. Nothing has to be sheared or severed — the tubes simply rotate — and the resistance is the smallest of the three by a wide margin.

That ordering is a property of the microstructure and it does not vary between species by anything like the amounts the absolute values do. The ratios are more reliable than the numbers, which is a useful thing to know when working with a timber whose test data is thin.

Where a rolling plane comes from #

Nobody designs a member to fail in rolling shear; it arrives with a manufactured product.

Cross-laminated timber is layers of boards glued at right angles. The layers whose grain runs across the span carry the beam’s shear flow across their fibres — a rolling plane, by construction, through a third of the panel’s thickness.

Plywood webs in an I-joist have plies in alternating directions for the same reason, and the transverse plies are rolling-shear planes in the web where the shear is largest.

So both of the engineered products designed to remove timber’s anisotropy problem introduce, in doing so, its weakest strength into the plane where the shear is greatest. It is a good trade — the cross layers buy two-way spanning and dimensional stability, which are worth more — and it is a trade rather than a solution, and the design consequence is that a CLT floor panel is very often governed by rolling shear rather than by bending or deflection.

The other anisotropy, which decides the details #

The shear planes are one instance of a general problem, and the general problem is the reason timber design looks the way it does.

Two things follow for the details rather than for the members.

Bearing across the grain governs more often than anything. In the first figure it sits at 0.40 and does not move with the span, because it depends on the reaction and the bearing length and on nothing about the beam’s proportions. A member that is comfortable in bending and shear at every span can still be limited by 100 mm of bearing at its end.

And a skewed detail is a real loss. Hankinson’s curve falls away in the first twenty degrees, so a load applied 30° off the grain — a birdsmouth, a notched joint, a skewed bearing — has lost most of the way to the transverse strength already.

Why the check is on a peak and not on an average #

There is one more factor of 1.5 in the timber case that has no counterpart in steel, and it is worth separating from the rest.

A steel I-section’s shear check divides the shear force by the web area and compares the result with $f_y/\sqrt3$. That is a mean over the web, and it is legitimate because the web’s shear stress is nearly uniform — the flanges carry almost none of the shear, so the web carries almost all of it at almost the same stress everywhere.

A rectangle has no web. Its shear stress is a parabola, its peak is 1.5 times its mean, and the peak is at the neutral axis. A timber check that used the mean would be 33 per cent unconservative, so the 1.5 appears explicitly in every timber code’s shear expression.

Two materials, two shapes, and the difference in the check is a consequence of the second rather than the first. A steel bar of solid rectangular section has the same parabola and the same 1.5; a timber member built as an I-joist with a plywood web has a nearly uniform web stress and does not. The factor belongs to the shape, and it has ended up attached to the material because each material is usually made into one shape.

Steel’s own third direction #

The habit is worth generalising, because steel has an anisotropy too and it is invisible until it matters.

The direction a plate was never tested in is steel’s version of the same thing: a rolled product with a manufacturing direction, three different properties, and a design that quietly assumes one number. The difference in degree is enormous — a factor of 4 in ductility for steel against a factor of 29 in modulus for timber — and the difference in kind is nil.

The general habit is to ask, of any strength being used, which direction it was measured in and which direction the member needs it in. For an isotropic material the question is empty; for every real structural material it is not, and the two that are usually treated as isotropic are the two where it has caused most trouble.

What happens at the glue line #

A laminated member has interfaces as well as fibres, and the shear flow crosses both.

For a glued lamination the interface is at least as strong as the timber, so it does not govern and the member behaves as one piece. For a mechanically laminated member — nailed, screwed or bolted — it does not, and the member is only partly composite: the interface slips, the two halves each bend about their own axis, and the second moment falls toward the sum of the parts.

That is the partial-interaction problem with a much softer connector than a stud, and it is why a nailed built-up timber beam is analysed with a slip modulus and a reduction factor while a glulam is not.

The point load close to a support #

There is one loading arrangement that concentrates everything above, and it is common enough to be worth naming.

A load applied within a beam depth or so of a support delivers almost all of its force straight down into the support through a compression strut, and very little of it bends the beam. The moment is small, the shear is nearly the whole load, and the beam is in the shear-governed regime whatever its overall span-to-depth ratio is.

Timber codes handle it by permitting the shear force to be reduced for loads within a distance of the support — the same allowance concrete codes make for a load applied close to a support, and for the same reason. The load has found a shorter route than the beam, and a shear check on the full reaction is checking a mechanism that is not happening.

The allowance is generous and it is conditional, and the condition is the part that gets lost: it applies to a load on the top of the beam, where the strut can form. A load hung from the soffit within the same distance has to be lifted back into the section before it can go anywhere, and no reduction applies at all — the same distinction, in a material with no reinforcement to hang it with.

What to carry away #

Shear governs a timber beam at proportions people build. The crossover is $L/d = f_m/f_v$, it contains nothing but the material, and it is 6.7 for softwood against 1.73 for steel.

Ask which shear plane the member has. Parallel to the grain gives 4.2 N/mm²; a rolling plane gives 1.2, and moves the crossover to 23.3, which is past the end of the practical range.

And check the bearing before anything else. It does not scale with the span, it depends on a detail rather than on a member, and on a beam of ordinary proportions it is often the highest utilisation on the page.

Where the model stops #

The strengths are characteristic values with wide scatter. Timber’s coefficient of variation is far larger than steel’s, and a shear strength quoted to two figures is known to about one.

The section is assumed uncracked and free of defects. A knot, a shake or a check at the neutral axis is a pre-existing shear plane, and timber shear failures start at defects rather than at the calculated peak.

Duration is left out. A load held for fifty years leaves 59 per cent of the short-term strength, and the duration factor applies to shear as it does to everything else — so the crossover moves with the load’s duration as well as with the material.

Nothing here is a notch calculation. A notched end concentrates the shear stress at the re-entrant corner, and the check there is a fracture mechanics one rather than a stress one.

And the shear stress is taken as VQ/It. That is a beam-theory result and it is poor near supports and point loads, which is exactly where the shear is largest — so the quantity being compared against a shear strength is least reliable at the section that governs.

The same shear flow decides two other things elsewhere in this collection. The point that is not in the section is where the flow’s resultant passes, which is what makes an open section twist; and the connection that is busiest where the beam is not is the same integral read as a force per metre along an interface.

The ladder from here #

Later rungs on this anchor: rolling shear in cross-laminated panels worked properly, with the layer build-up and the effective section. Notched ends and the fracture-based check that replaces a stress one there. Shear in glued laminated members with the glue line’s own strength. Diagonal tension and link design in concrete, which is the same shear on a material with no grain. Interface shear between precast and in-situ concrete, where the plane is a construction joint rather than a fibre direction. And shear deformation, which is the movement this stress produces and which beam theory omits entirely — a one per cent correction in steel and a six per cent one here.

Jourawski’s investigation of 1855 was into timber beams splitting horizontally on the St Petersburg to Moscow railway. The theory of the day had nothing to say about it, the failures were not bending failures, and the expression he produced is the one used unchanged today. That the material which forced the derivation is also the one where the result governs is not a coincidence: nobody would have looked for the effect in a material where it never decides anything.