The pull that Johansen left out

The smallest of six failures worked out the capacity of a dowel through timber as the least of Johansen’s mechanisms: the wood crushing along the dowel’s length, the dowel rotating rigidly, or the dowel bending into one or two plastic hinges. It named one thing the six free bodies leave out. “A dowel with a nut and washer on it, or a screw with a head, resists being pulled through as it bends — so as the joint deforms, the dowel goes into tension and its axial force clamps the two members together.” It called the effect genuine, not small, and entirely absent from its calculation.

This essay puts it back. The addition has a name, the rope effect, and a form in the European timber code that is short enough to state in a line: a quarter of the fastener’s axial resistance, added only to the modes in which the fastener tilts, and capped at a fraction of the Johansen value that depends on what kind of fastener it is. Each part of that line has a reason, and the reasons change which fastener a designer should choose.

The modes that tilt, and the ones that do not #

A fastener in a joint that slips is pushed sideways by the wood on each side. In Johansen’s two crushing modes, a and b, it stays straight and moves bodily: the wood crushes along its whole length in one member, and nothing about the fastener’s own axis changes. In every other mode it tilts — rotating rigidly in mode c, bending into hinges in d, e and f — and a fastener that crosses the joint at an angle is longer, between its ends, than one that crosses square. If both ends are held, by a washer and nut, or by a thread in each member, the tilt stretches it, and the stretch is a tension.

The tension does two things, and both resist the slip. It pulls the two members against each other, and the clamping force makes friction on the shear plane between them. And because the fastener crosses that plane at an angle, its tension has a component along the plane, pulling the joint back. Eurocode 5 folds both into one term, a quarter of the fastener’s axial resistance, which is what friction at a coefficient of 0.25 would give from a clamp equal to that resistance. It is a simplification, but the kinematics behind it are exact: no tilt, no pull. That is why modes a and b get nothing in the figure, and why the joint’s governing mode can change once the rope term is added. A mode that tilts can climb above one that does not.

Three fasteners with one Johansen capacity #

Johansen’s calculation sees only the fastener’s diameter and bending strength and the wood’s crushing strength. A smooth dowel, a bolt and a screw of the same diameter and steel are the same fastener to it. The rope effect separates them, because they differ in what holds them along their axis.

The smooth dowel has nothing holding it. It slides along its hole as the joint tilts it, its axial force is zero, and it is exactly Johansen’s fastener. The bolt is held by its head and nut, through washers bearing on the timber faces. The screw is held along its whole threaded length by the wood itself, and the harder the wood grips the thread, the more it resists.

At these thicknesses the three are ranked by their axial resistance alone: 9.61 kN for the dowel, 11.27 for the bolt, 12.13 for the screw. The difference between the dowel and the screw is not in the steel, which is the same, or the diameter, which is the same, but in the thread — a detail of the fastener’s surface that Johansen’s calculation cannot see.

The plateau a screw does not have #

The central result of Johansen’s model was a plateau. Mode f, with two plastic hinges, has no member thickness in its expression, so past a few diameters of embedment the joint’s capacity stops rising and a thicker member buys nothing. That result holds for the fastener Johansen modelled. It does not hold for one held along its axis by the wood.

A screw’s withdrawal resistance grows with the length of thread in the wood, nearly in proportion. The timber code’s expression for the withdrawal strength per unit of thread falls only as the length to the minus one-tenth, so a thread twice as long holds about 1.9 times as much. So the rope term grows with the thickness of the members, while Johansen’s term has stopped growing, and the screw’s capacity keeps rising past the plateau. At ten diameters the 8 mm screw carries 44 per cent more than Johansen’s plateau, and the cap — all of the Johansen value, for a screw — is still far away.

That is the reason for a practice that looks, from Johansen’s model alone, like a waste of steel. Long, fully threaded screws in thick timber are not buying bending strength past the plateau; they are buying withdrawal resistance, which becomes lateral resistance as soon as the joint slips enough to tilt them. Screws inclined across the joint take this further: the thread’s withdrawal resistance is then engaged directly by the slip, and the fastener acts more as a tie than as a dowel. That arrangement is a different mechanism from any of Johansen’s, and the one-quarter rule does not describe it.

A bolt is held by its washer #

The bolt does not share the screw’s advantage, because its axial resistance does not grow with the thickness. It is held at its two faces, and what resists the pull is the timber under the washers. The bolt’s capacity follows Johansen’s curve up by a fixed step wherever the step is below the cap, and by a quarter of Johansen’s value where it is not, and it reaches the same plateau as the dowel, raised by 1.66 kN.

What decides that step is the washer and the wood, not the bolt. A washer three diameters across, less its hole, is 885 mm² of steel pressing on the face of the timber across the grain, and the timber code allows three times the timber’s crushing strength across the grain there — 2.5 N/mm² for this timber — for 6.64 kN. The bolt itself could carry 47.6 kN in tension. The pull is limited by the wood under the washer by a factor of seven, and this is the anisotropy the material that has a direction describes arriving at a place nobody draws: the washer bears across the grain, where timber is weakest.

The washer, not the bolt #

Since the washer decides, the washer is the thing to change. Its bearing area grows as the square of its diameter, so a modestly larger washer raises the axial resistance quickly, and the joint’s capacity with it, until the rope term meets the bolt’s cap of a quarter of the Johansen value. For this joint that happens at a washer about 3.6 diameters across; beyond it, a larger washer buys nothing, because the cap, not the washer, now binds.

It is the same conclusion three times as strong under a smaller pad reached about concrete, from the opposite direction: a bearing’s strength depends on what surrounds the loaded area, and the numbers that come out are properties of the geometry as much as of the material. Here the loaded area is the washer, and the question is how much of the timber’s weak direction it can mobilise before the fastener’s own rule stops counting.

The cap is a decision about slip #

The cap is the part of the rule that looks arbitrary, and it is the part that carries a physical argument. The rope effect needs the fastener to tilt, and a fastener tilts by slipping. Johansen’s mechanisms reach their capacity at a slip of a few millimetres, when the wood has crushed and the hinges have formed; the rope effect grows with the tilt, and reaches its full value only at slips several times larger. A joint designed on the sum is designed on a load it reaches only after it has moved a long way.

Read that way, the cap says how much slip the design rule is willing to credit. For a bolt in an oversized hole, whose washers must be drawn into the wood before they bear, it credits a quarter. For a screw, whose thread grips from the first movement and does not need to seat, it credits as much again as Johansen’s value. For a smooth dowel, which has nothing to hold, it credits nothing. The caps are not a ranking of fasteners by quality; they are a statement about how early in the slip each one’s axial resistance arrives, and a joint that must stay stiff — a joint made of springs in series whose softest spring is its slip — cannot use what arrives late.

Friction that the clamp makes #

The clamping half of the rope effect is the mechanism of the joint that carries nothing until it slips in steel, arriving in timber by a different route. A preloaded steel joint clamps its plates before any load arrives, and friction carries the load until it is exceeded. A timber joint cannot be preloaded that way, because the wood creeps and shrinks and the preload would be gone in a season. Its clamp is made by the load itself: the slip tilts the fastener, the tilt stretches it, and the tension presses the members together. The friction is therefore not there at the start. It grows with the slip it resists, which is why it cannot contribute to stiffness and why the design rule counts it only at a fraction.

Which free body produced the number #

Take the fastener out of the joint in a tilting mode and draw it. Along its length the wood pushes on it sideways at the embedment strength, and at its hinges it carries its plastic moment; that is Johansen’s free body, and its equilibrium gives his capacity. Add the two end forces: at each end, the washer or the thread pulls it along its axis with a force up to its axial resistance. The shear plane now sees the members pressed together by that force, and the fastener crossing it at an angle carries a share of that force along it. The design rule takes a quarter of the axial resistance as the sum of both effects.

The upper-bound character of Johansen’s model survives the addition. Each mode’s capacity, with the rope term, is still the load at which one assumed mechanism can move, and the joint’s capacity is still the least of them. What changes is which mechanism is least, because the rope term favours the tilting ones. That is the general property of the reasoning the ground’s bearing capacity is built from: add a resisting term to some mechanisms and not others, and the answer can move to a different mechanism, not merely to a larger number.

The 12 mm bolt by hand #

For the bolt in the first figure, the embedment strength of 350 kg/m³ timber along the grain is $0.082(1 - 0.01 \times 12) \times 350 = 25.3$ N/mm², and the bolt’s yield moment, from Eurocode 5’s expression $0.3 f_u d^{2.6}$, is 115 kN·mm. Johansen’s rigid-rotation mode for two equal 40 mm members then gives 5.02 kN, the least of the six. The washer, 36 mm across with a 13 mm hole, bears on 885 mm² at $3 \times 2.5 = 7.5$ N/mm², so the bolt is held along its axis by 6.64 kN, and a quarter of that is 1.66 kN. The cap for a bolt is a quarter of the mode’s own Johansen value, $0.25 \times 5.02 = 1.26$ kN, which is less, so the joint carries $5.02 + 1.26 = 6.28$ kN.

At 80 mm the governing mode is the two-hinge one, 9.61 kN, whose quarter is 2.40 kN; the washer’s 1.66 kN is now the smaller, and the joint carries 11.27 kN. The same bolt and washer are capped by the rule in one joint and by the washer in the other, and the thickness decides which.

The hinges it adds to, and the bearing it relies on #

The rope term is added on top of mechanisms that are themselves plastic. Modes d, e and f are plastic hinges in a short steel beam, forming where the wood’s resistance and the fastener’s moment balance, and they are what lets the joint slip far enough for the rope effect to develop at all. A fastener too stiff to hinge, in wood too thin to crush much, never tilts far, and its axial resistance is never called on.

And the wood on which both depend is loaded in bearing. The hole that goes oval is the steel version of the same process, a plate yielding around a bolt until the hole elongates, and it is the connection failure that reliably gives warning. The timber equivalent is the embedment crushing that every one of Johansen’s modes contains, and it is the reason a doweled timber joint slips a long way before it fails — long enough, in the tilting modes, for the rope effect to be worth something.

A bolt can be tightened, and the torque that goes into the thread sets a clamp before any load arrives. In timber that clamp does not last: the wood shrinks across the grain as it dries and creeps under the washer, and a bolt tightened at erection is often loose a year later. The rope effect does not depend on it. It is the clamp the joint makes for itself as it slips, which is why a design rule can count it at all in a material that will not hold a preload.

Where the model stops #

The quarter is a rule, not a derivation. Friction coefficients between timber faces run from about 0.2 to over 0.5 with moisture and surface, and the component of the fastener’s tension along the shear plane depends on the tilt, which depends on the slip. The rule’s single fraction stands in for both, and the caps exist partly because the fraction is not safe at large axial resistances.

The axial resistance is taken as fully mobilised. A washer must be drawn tight against the wood before it bears, and a bolt in a hole a millimetre oversize, in timber that has shrunk since the nut was tightened, may be slack. A slack bolt is a dowel until the slack is taken up, which is another reason its cap is low.

Withdrawal is the thread’s only. A screw’s axial resistance may be limited by its head pulling through the member it holds, or by the steel of its shank, and either can be less than the thread’s withdrawal in a thick member. The figures here assume the thread governs, which is the case for which a long screw is chosen.

The joint is in single shear, loaded along the grain. Across the grain the embedment strength falls and Johansen’s capacity with it, while the rope term does not, so the rope effect is a larger share of a joint loaded across the grain.

What the pictures cannot show #

That the rope effect is a property of the joint’s history. A joint that has slipped once has already tilted its fasteners, stretched its bolts and crushed wood under its washers; loaded again, it may carry its rope term from the start, or may have lost it to a washer that has sunk into the wood and a bolt that has stretched. The rule is written for a joint loaded once to failure, and a joint loaded back and forth by wind has neither of the histories it assumes.

Nor can they show the ductility. A joint whose capacity comes from Johansen’s hinges yields and holds its load over a long slip; the rope term adds to it, but the axial resistance it draws on can end abruptly — a thread stripping from the wood, a washer punching through. A joint whose capacity depends heavily on the rope term is stronger than Johansen predicts and may be less forgiving at the end.

Still open: the screw put in at an angle #

Every figure here had the fastener square to the joint, tilting only as the joint slipped. A screw driven at forty-five degrees across the shear plane starts tilted, and the slip pulls it along its axis from the first millimetre: its withdrawal resistance is engaged directly rather than through a tilt, and the joint behaves as a truss of screws in tension rather than as a row of dowels in bending. Whether that makes Johansen’s model and the rope term irrelevant for such joints — or whether the lateral bending of an inclined screw still decides the joint at some angle and thickness, so that the two mechanisms trade places as the angle changes — is the question a square fastener cannot answer, and it is the one the long inclined screws in modern timber construction are relying on.