The screw that is pulled instead of bent

Johansen’s model treats a fastener across a timber joint as a beam on a foundation of wood: it resists the joint’s slip by crushing the wood it bears on and by bending into one or two plastic hinges, and the joint’s capacity is the cheapest of six ways that can happen. The essay before this one added the force Johansen left out — the fastener, tilted by the slip, is pulled along its own axis, and if its thread or washer holds, that pull clamps the members together. For a screw the addition is allowed to equal the whole Johansen capacity.

All of that is about a fastener square to the joint, which is pulled only after it has tilted. The question left was what happens when a screw starts tilted — driven at forty-five degrees across the shear plane, as the long fully threaded screws in modern timber construction usually are. Then the slip pulls it from the first millimetre, and the question is whether Johansen’s model still decides anything.

Two ways to resist one slip #

Take an 8 mm fully threaded screw through two softwood members, each 100 mm thick, the timber of density 350 kg/m³. The screw’s yield moment is 20 N·m and its tensile capacity 20 kN, which is the order of a commercial screw of that size.

Drive it at an angle $\alpha$ to the joint, and let the upper member slide over the lower by a slip $u$. The two ends of the screw move apart by $u$, and that movement has two components. Along the screw’s own axis it is $u\cos\alpha$, which pulls the screw against its thread’s grip in the wood. Across the screw it is $u\sin\alpha$, which bends the screw and crushes the wood around it, exactly as in Johansen’s model.

The two resistances are very different in size. The thread’s grip, over the 141 mm of screw in each member at 45 degrees, is worth 13.7 kN by the European withdrawal expression. The screw’s bending and embedment, by Johansen’s model on the same lengths, are worth 3.3 kN — the screw bends into two hinges whatever the members’ thickness, so its lateral capacity does not grow with the inclined length at all.

Resolved along the joint, the pull contributes $T\cos\alpha$ and the push $V\sin\alpha$. The pull also clamps the two members together with $T\sin\alpha$, and the push prises them apart with $V\cos\alpha$, so friction adds $\mu$ times whatever net clamping is left — a quarter, the coefficient the design rules use. At 45 degrees, with a pull at its peak and a push at its, that resolution adds to 13.9 kN.

Peaks that arrive at different slips #

Adding two peaks assumes they arrive together. They do not, and the joint’s real capacity is the largest total the two give at any one slip.

The thread grips stiffly and lets go early. A screw being withdrawn reaches its greatest resistance after an axial movement of a millimetre or two and then softens as the wood fibres around the thread shear. The bending is the opposite: a screw being pushed sideways through wood rises slowly toward its Johansen capacity, which it approaches only after several millimetres of crushing, and goes on rising slowly after that as the hinges harden.

To add them slip by slip, each needs a curve, and neither is measured here. The thread is taken to peak at 1.5 mm of axial movement and soften after it; the bending follows Foschi’s exponential curve, starting at the slip modulus EN 1995-1-1 gives a screw and rising to nine tenths of Johansen’s capacity before hardening gently. Those are shapes of the kind tests report, stated so that every number below can be traced to them.

At 45 degrees the thread reaches its peak at a slip of 2.1 mm — its 1.5 mm of axial movement divided by $\cos 45°$ — and the joint peaks a moment later, at 13.4 kN after 2.2 mm of slip. By then the bending has reached 73 per cent of its own capacity. Added peak to peak the two would have made 13.9 kN, so the shortfall is 3.5 per cent.

That is a small number, and the reason it is small is the finding. The bending is a small share of the joint — 1.3 kN of 13.4 — so the fact that it arrives late costs almost nothing. At 45 degrees the screw is a tension member with a little bending attached, and Johansen’s model, which was the whole of the square screw’s design, has become a correction of about a tenth.

Stronger, and finished sooner #

Across the angles the curves change shape completely. The square screw, bending alone, rises slowly and keeps rising: at 12 mm of slip it is at 3.6 kN and still climbing, which is the ductile plateau that made Johansen’s model and its rope effect a comfortable basis for design. Lean the screw and the curve gets taller, steeper and shorter. At 60 degrees it peaks at 10.4 kN after 3.2 mm; at 45, 13.4 after 2.2; at 30, 19.1 after 1.8. Past the peak each falls: at 45 degrees the joint is down to four fifths of its peak by 4.1 mm, where the square screw has barely started.

A leaning screw trades ductility for strength, and the trade is steep. The strength more than doubles at 45 degrees and triples at 30; the slip at which the strength is spent falls from more than twelve millimetres to two or three. The same screw in the same wood has become a different kind of joint — stiff, strong and brittle where it was soft, modest and forgiving — because the slip now works its thread instead of its shank.

That is the same distinction a bolted steel joint makes between a bearing connection and a slip-critical one, and it has the same consequence for a structure with many fasteners. Fasteners that peak early and soften do not share load by yielding: the first to reach its peak starts to let go while its neighbours are still climbing, and the bolts that do not share are the rule rather than the exception.

And several times as stiff #

The steepness of those curves at their foot is a stiffness, and for a connection in service it matters more than the peak. A floor whose timber joists are joined to a concrete topping by screws, or a wall whose panels are screwed to each other, is designed for deflection and vibration long before it is designed for collapse — stiffness, not strength, usually governs — and both depend on how much the screws let the joint slip under everyday load.

The square screw’s stiffness is the slip modulus the design rules give it, about 3.0 kN per millimetre for this screw in this wood. Leaned to 45 degrees, the same screw starts at 16.7 kN per millimetre — five and a half times as stiff — because the thread’s grip is much stiffer than the wood’s resistance to being crushed sideways, and at 45 degrees half of every millimetre of slip goes into pulling the thread. At 60 degrees the stiffness is 9.4, at 30 degrees 29.6. Over the first 0.4 mm of slip, which is the range a service load works in, the secant stiffnesses are 13.9 at 45 degrees against 2.5 square.

Those numbers inherit the assumed withdrawal curve — the thread’s stiffness is its peak over its peak slip, so a thread that peaked at 3 mm rather than 1.5 would be half as stiff — and the ratio is the claim, not the kilonewtons. It is why leaning screws are so widely used in timber–concrete composite floors, where the connection’s stiffness sets how much of the concrete the timber can use: a joint that slips five times less makes a beam that is much nearer the bonded one than the loose one.

Where bending still decides #

Plotted against the angle, the joint’s peak follows the thread’s line almost exactly. The thread’s contribution rises steeply as the screw leans — partly because the pull is resolved more directly along the joint, partly because the screw’s length in each member grows as $t/\sin\alpha$ and its grip with it — until, at shallow angles, the screw’s own tensile capacity stops it. The bending’s contribution falls as the screw leans, because less of the slip pushes it sideways.

The two lines cross at 85.7 degrees. Only within about four degrees of square is the bending the larger share of the joint, and the square screw itself, at 90 degrees, carries 5.9 kN: its 3.3 kN of bending plus the rope effect, which is the inclined resolution’s friction term at the one angle where the screw is not pulled until it tilts. Johansen’s model is not wrong for an inclined screw. It is about a mechanism that has stopped carrying the joint.

And the crossing depends on the members. The thread’s grip grows with the length it has to grip, and the bending does not, so in thin members the bending holds its share further: in 40 mm members it is the larger share steeper than 69 degrees, in 60 mm steeper than 77, and in 200 mm members it never is.

The thickness sweep makes the same point as a design choice. In 40 mm members the leaning screw is worth half as much again as the square one, 6.6 kN against 4.4; in 100 mm members, more than twice; in 200 mm members nearly three times, 19.0 against 6.7. The square screw’s capacity stops growing almost at once — its bending is set by the screw and its rope effect is capped at the bending — while the leaning screw’s goes on growing until the screw breaks before the wood lets go of it, at about 155 mm here. The deeper the timber, the more the lean is worth, which is why it is in the heavy glulam and cross-laminated panels that inclined screws are used most.

One way better than the other #

The friction term has a sign, and that makes the joint directional.

When the slip pulls the screw, its axial force draws the two members together and the friction it creates adds to the joint. Reverse the slip and the same screw is pushed. A fully threaded screw grips the wood about as well in compression as in tension, so its axial resistance is still there — but now its axial force pushes the members apart, the clamping comes off, and the friction that the lateral push was already prising open is not replaced.

At 45 degrees the joint carries 13.4 kN one way and 11.4 kN the other, 85 per cent. The loss grows as the screw stands up, because a steeper screw’s clamping is a larger share of what it gives: at 70 degrees the two directions are 8.9 and 6.6 kN, 75 per cent. At 30 degrees, where the screw pulls almost along the joint and clamps very little, the two are within ten per cent.

And the pushed direction has a failure the pulled one does not. A long slender screw in compression is a column embedded in wood, and if the wood around it is soft or the screw is unsupported near the joint, it buckles before its thread lets go. None of that is in the dashed line, which is therefore the more optimistic of the two.

A wall or a floor whose screws all lean one way is stronger against wind from one side than from the other. The usual answer is to cross them — pairs at plus and minus 45 degrees, so that whichever way the load comes, one screw of each pair is pulled. A crossed pair carries about the sum of one pulled and one pushed screw, some 25 kN here, in both directions: not twice the pulled screw, and not a coincidence that the design of crossed-screw joints is written for the pair rather than the screw.

The screw cut at the joint #

The resolution behind every curve here is a free body of the upper member’s share of the screw. Cut the screw at the joint. What crosses the cut is the screw’s axial force $T$ along its own line and its shear $V$ across it, and the wood on either side of the cut is held to the other by whatever normal force those two leave pressing across the plane.

Resolve along the joint and the screw resists slip with $T\cos\alpha + V\sin\alpha$. Resolve across it and the screw presses the members together with $T\sin\alpha - V\cos\alpha$ when it is pulled and pulls them apart when that is negative; friction is $\mu$ times the pressing force and only when there is one, whatever it needs to be up to that limit. That is the whole model. Every number above is these two lines evaluated with $T$ and $V$ read off their curves at the same slip — and the square screw’s rope effect is the friction line with $\alpha = 90°$, where $\mu T$ with $\mu = 1/4$ is exactly Eurocode 5’s quarter of the axial capacity.

The 45-degree screw by hand #

The withdrawal capacity of an 8 mm screw over a length $\ell$ in softwood of density 350 is, by the European expression, $0.52 \, d^{-0.5} \ell^{-0.1} \rho^{0.8} \cdot d\ell$. At 45 degrees, $\ell = 100/\sin 45° = 141$ mm, and that is $0.52 \times 0.354 \times 0.609 \times 108 \times 8 \times 141 = 13.7$ kN.

Resolved along the joint with its friction, the pull at its peak gives $13.7 \times (\cos 45° + 0.25 \sin 45°) = 13.7 \times 0.884 = 12.1$ kN. The bending at its full capacity would give $3.3 \times (\sin 45° - 0.25 \cos 45°) = 3.3 \times 0.530 = 1.8$ kN; at the slip where the thread peaks it is about three quarters of that, 1.3. The joint is 13.4 kN.

The square screw is $3.3 + 0.25 \times 10.1 = 5.9$ kN, the 10.1 being the thread’s grip over only 100 mm. The whole of the factor of 2.3 is the thread, put to work by the angle.

What the curves assume #

The two load–slip shapes. Where the thread peaks, how fast it softens, and how quickly the bending rises are taken, not measured; a thread that peaks later would let the bending catch up, and a softer wood would delay both. The peak-to-peak shortfall of a few per cent depends on them. The factor of two between the leaning and the square screw does not: it is in the capacities, not the shapes.

The withdrawal expression’s angle. A screw’s grip depends on its angle to the grain as well as its length, and the expression is used here as though the screw met the grain at the angle it was calibrated for. A screw driven along the grain grips much less, and the angles that matter in a real connection are a joint’s angle to the grain, which is a separate decision from its angle to the joint.

And one screw. A row of leaning screws shares the load only as well as their peaks coincide, and early-peaking fasteners in a long row share it badly. Spacing, edge distances and the effective number of screws all enter a real design, and none of them is here.

What the figures cannot show #

They cannot show the wood splitting. A screw that grips hard and leans puts a wedging force into the wood along its thread, and the closer it is to an end or an edge the more likely the member splits before the thread lets go — a failure with no curve on this page.

They cannot show what happens on reloading. A joint past its peak has a thread that has partly sheared the wood around it, and a second load cycle finds a weaker grip. Leaning screws under cyclic load are therefore a different design problem from leaning screws under one load, and the softening that makes the first peak brittle makes the second one lower.

And they cannot show the screw breaking. At shallow angles in deep members the thread outgrips the steel, and the joint fails when the screw snaps in tension. That is the flat top of the thickness sweep, and it is the most brittle failure of all: no slip, no warning, and a capacity set by the steel rather than the wood.

What it comes to #

Leaning a screw puts its thread to work. At 45 degrees an 8 mm screw through two 100 mm members carries 13.4 kN against 5.9 square — 2.3 times, all of it from the thread’s withdrawal.

It peaks early and softens. 2.2 mm of slip to the peak and 4.1 to four fifths of it, where the square screw is still rising past 12 mm.

Johansen’s bending becomes a correction. It is the larger share only steeper than 86 degrees in these members, and steeper than 69 in 40 mm ones.

And the joint has a direction. Pushed instead of pulled, the screws carry 85 per cent at 45 degrees and 75 at 70, before buckling.

Still open: a row of screws that peak at different slips #

Every number here is for one screw. A connection is twenty of them in a row, each at its own distance from the end of the member, each loaded through wood that stretches between it and the next. A row of square screws shares its load because each one keeps rising while its neighbours catch up; a row of leaning screws reaches its first peak at two millimetres, and the screw nearest the load starts to let go before the one furthest from it has done much. Whether the effective number of leaning screws in a row is close to the actual number, as it nearly is for square ones, or falls away with the row’s length as it does for a long bolted joint, is the question a timber connection made of inclined screws has to answer before its capacity is twenty times anything on this page.