Structural Engineering & Statics
where the load goes
Concept

Johansen model — where it appears

The capacity of a laterally loaded dowel in timber, taken as the least of its collapse mechanisms. The timber crushes, the dowel rotates, or it forms one or two plastic hinges; the calculation is an upper bound, and it leaves out the clamping a fastener adds when pulled along its axis.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Six ways for one dowel to fail, and the capacity is the smallest. Johansen's single-shear mechanisms for a 12 mm dowel through 40 and 40 mm members, each drawn as the shape it is: the dowel straight and the timber crushing, the dowel rotating rigidly, one plastic hinge, then two. The capacity under each is that mechanism's own, and the joint's strength is the smallest — 5.02 kN by mode c, which is the dowel rotates rigidly and both members crush. That is the kinematic theorem of plasticity: every mechanism gives an upper bound and the true collapse is the lowest of them. The crushed timber is shaded, and the circles are plastic hinges in the steel.

The smallest of six failures

Everything else in this collection that fails does so in one way at a time. A dowel through timber does not — the wood can crush while the steel stays straight, or one plastic hinge can form in it, or two — and the capacity is the smallest of the six, which is the kinematic theorem of plasticity applied to a joint rather than to a frame.

connections · Dowel yield The modes that tilt, and the ones that do not. Johansen's mechanisms for a 12 mm bolt with washers in single shear through 40 and 40 mm of timber of density 350 kg/m³, each with the rope term the fastener's axial resistance of 6.64 kN adds to it. Modes a and b, where the fastener only translates, gain nothing; the modes in which it tilts gain a quarter of its axial resistance, capped at a quarter of their own Johansen value. The joint's capacity rises from 5.02 kN (mode c) to 6.28 kN (mode c), 25 per cent.

The pull that Johansen left out

Johansen's mechanisms treat a timber fastener as a beam in a bed of crushing wood, bent and pushed sideways and nothing else. A real fastener that tilts across the joint is also pulled along its own axis, and if a washer or a thread resists the pull, it clamps the two members together and adds to what the joint can carry. The addition is a quarter of the axial resistance, it goes only to the modes in which the fastener tilts, and it is capped by fastener type — nothing for a dowel, a quarter for a bolt, all of it for a screw — which is how a screw keeps gaining strength past the thickness at which Johansen's capacity stops.

connections · Dowel yield Lean a screw and its bending stops mattering. The peak load on an 8 mm fully threaded screw (yield moment 20 N·m, tensile capacity 20 kN) through two 100 mm softwood members of density 350 kg/m³, against the angle it is driven at to the joint: the joint, its two resistances added slip by slip (solid); the thread's alone, resolved along the joint with its friction (dashed); the bending and embedment's alone (dotted); and the square screw with the rope effect, 5.9 kN (thin). At 45 degrees the joint carries 13.4 kN, 2.3 times the square screw, of which the bending gives 1.8; at 30, 19.1. The bending is the larger share only steeper than 85.7 degrees, within 4.3 of square. Withdrawal is taken to peak at 1.5 mm of axial movement and soften after it; bending follows Foschi's curve from EN 1995-1-1's slip modulus — shapes, not measurements.

The screw that is pulled instead of bent

A screw square to a timber joint resists slip by bending, and its thread is engaged only when it has tilted far enough to be pulled. Lean the same screw to 45 degrees and the slip pulls it from the first millimetre. An 8 mm screw through two 100 mm members goes from 5.9 kN to 13.4, reached after 2.2 mm of slip instead of more than twelve, and the bending that Johansen's model is about becomes an eighth of the joint. The price is a joint that holds one way better than the other.

connections · Dowel yield One ratio decides how much of a row works. The share of n single screws' peaks a row reaches, for rows of 10, 20 and 30 8 mm screws at 45° to the joint, 56 mm apart along the grain, in two 100 mm softwood members of modulus 11,500 N/mm² with 40, 80 and 160 mm of member depth each, and threads peaking at 0.5, 1, 1.5 and 2.5 mm, against one ratio: the slip difference the members' stretch puts between the end screw and the middle one at the row's peak, PL(1/EA₁ + 1/EA₂)/8, over the width of a single screw's curve above nine tenths of its peak. Thirty-six rows fall on one curve: every row with the ratio below 0.3 reaches at least 1.00 of its screws' sum; at a ratio of 1 they reach about 0.96; and the most stretched, at 3.5, 0.72. The colour is the thread's peak slip; a brittler thread has a narrower plateau and lands further right for the same row.

Twenty screws that peak together

A leaning screw reaches its peak at two millimetres of slip and then lets go, so a row of twenty along the grain looks like the long bolted joint again — the end screws past their peak and falling while the middle ones have barely started. Follow the row slip by slip and it is not: in a member of ordinary depth twenty leaning screws carry 98 per cent of twenty times one, against the 74 per cent the effective-number rule allows. What decides it is one ratio — the slip the timber's stretch spreads along the row against the width of a screw's peak — and the rule is right only for a brittle thread in a thin member.

connections · Dowel yield

Named alongside it

The objects these essays reach for when they reach for this one.

Embedment strengthTimberBearingDuctilityFrictionPlastic hingeClamping forceAnisotropyBound theoremsCollapse loadConnection designDowel yield

All concepts