Structural Engineering & Statics
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Six ways for one dowel to fail, and the capacity is the smallest

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing 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.

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.

7 essays call dowel-modes. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

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. Connections

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.

 The load path has a kink in it, and the kink is a plate thick. Two 10 mm plates lapped over 60 mm and pulled with 60 kN. The two load paths are offset by the thickness of a plate, so the joint carries a moment nobody applied: P × 10.0 mm / 2. Taken at face value that gives a peak stress 4.00 times the mean. The joint rotates under load and the moment falls to 86 per cent of it, leaving 3.57 times — a saving of 11 per cent and not, on a plate this thick, a rescue. The bolt is bent as well as sheared: 382 N/mm² of bending against 191 of shear. Connections

The joint that is crooked by construction

Lap two plates and fasten them and the two load paths are offset by the thickness of a plate. The joint carries a moment nobody applied, the peak stress is four times the mean, and the rotation that is supposed to straighten it out saves eleven per cent — because the rescue works for thin sheet with a long lap and a bolted structural joint is neither.

 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. Connections

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.

 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. Connections

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.

 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. Connections

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.

 Below a limit the row settles; above it, it ratchets. The end screw's slip at the top of each cycle, against the number of cycles, for 20 leaning screws 56 mm apart with 40 mm of member depth per row loaded again and again from new to four levels, each screw, reloaded, reaching the slip it had been to with 85% of the load it had there. At 0.94 it settles at 2.50 mm after 3 cycles; at 0.96 it settles at 2.83 mm after 7 cycles; at 0.97 of the peak the end slip grows every cycle until the row cannot reach the load, at cycle 21; at 0.99 of the peak the end slip grows every cycle until the row cannot reach the load, at cycle 8. The row's first peak is 263 kN; every level drawn is below it. Connections

The row that remembers its overload

A row of twenty leaning screws reaches 98% of twenty screws' peak, once. Loaded again, a screw that has been past its peak comes back along a softer path and reaches the slip it had been to with less load than it carried there. So a row loaded again and again neither simply evens out nor simply ratchets down: below 0.965 of its first peak it settles after a few cycles with the middle screws catching up, and above it the end screws are pushed back onto their curve every cycle, lose a little more each time, and the row fails at a load it carried the first time. One large overload costs it less than many smaller ones near the peak.

 The more of the load that comes back, the less of it the row can carry for ever. The largest load 20 screws leaning at 45° in a row 56 mm apart carries for ever, as a share of its forward peak, against the share of it that comes back the other way each cycle (0 the same load repeated, 1 fully reversed), for each screw carrying none, half and all of its thread's first loss into the other way. Carrying 0 per cent: 0.96 repeated, 0.96 half reversed, 0.82 fully reversed; carrying 50 per cent: 0.96 repeated, 0.94 half reversed, 0.79 fully reversed; carrying 100 per cent: 0.96 repeated, 0.92 half reversed, 0.76 fully reversed. Connections

The row that is pushed as often as it is pulled

A row of leaning screws in a shear wall is loaded one way by the wind and then the other, and a screw that leans is pulled along its axis one way and pushed the other. Pulled, its axial force clamps the joint and friction helps it; pushed, it prises the joint open and the friction goes. So the row has a weaker side, and under a reversing load it settles exactly as it settles under a repeated one — the reversal does not start the ratchet again — but at a ceiling the pushed side sets, 0.79 of the pulled peak against 0.97. And it settles with a gap in the middle that every reversal crosses carrying nothing.

The library, page 1 of 7 — where dowel-modes sits