The row that is pushed as often as it is pulled

The row that remembers its overload loaded a row of twenty leaning screws, unloaded it, and loaded it again. A screw past its peak comes back along a softer path and reaches the slip it had been to with less load than it carried there, and the row loaded again and again neither simply evens out nor simply ratchets down: up to 0.965 of its first peak it settles after a few cycles, the middle screws catching up with the end ones, and above that its end screws are pushed back on to their curve every cycle and the row fails at a load it once carried.

Every load in that essay came and went one way. A screwed joint in a shear wall carries wind from one side and then the other, and an earthquake reverses it every second. A leaning screw resists the two directions differently — the screw that is pulled instead of bent found it strong when it is pulled along its axis and weaker when it is pushed — and a row reversed after an overload has end screws worn one way and middle screws worn the other. Whether a row of leaning screws under a reversing load settles as it does under a repeated one, or whether each reversal resets the slack and the ratchet begins again, is the question.

A screw has a strong side #

A screw driven at 45° across a timber lap joint resists the joint’s slip in two ways at once: its thread is pulled out of the wood along its axis, and its shank bends and bears sideways. Loaded the way it leans, its axial force pulls the two members together, and the clamping makes friction between them that adds to both. Loaded the other way, its axial force pushes the members apart; there is no clamping, and so no friction. The screw’s thread grips the wood equally either way — a fully threaded screw is withdrawn and pushed through by the same thread — so the difference is the friction, and at 45° with a friction coefficient of a quarter it makes the screw pushed 0.85 as strong as the screw pulled.

The row inherits that difference. Twenty screws 56 mm apart in members of ordinary depth peak at 263 kN pulled — 19.6 times one screw, the earlier essays’ row — and at 226 kN pushed, 0.857 of it.

Two memories, and a gap between them #

Each screw in the row keeps the earlier essay’s memory, once for each direction: the furthest slip it has been to that way, the permanent set it left behind when it was unloaded, and the load a reload reaches at that slip — 0.85 of what it carried there, if it had been to its peak. Between its two sets it is slack. The thread has pushed the wood back on one side of the screw when it was loaded one way and on the other side when it was loaded the other, and the screw moves through the hole it has made carrying nothing.

The one thing the earlier essay could not have needed is a link between the two memories. The thread and the wood round it are one interface whichever way the screw is loaded, so some of what the thread loses one way it has lost the other way too. How much is a property of screws in timber that has not been measured as such, and it is carried here as a share: none, half or all of a thread’s loss on its first excursion one way is taken off its curve the other way. Half is the default, and the figures say what the other two do.

The first loading in each direction is the row’s own curve. Every later one starts at nought load with the joint able to move a little each way carrying nothing — 0.16 mm towards the pulled side and 0.40 mm towards the pushed — and then picks up its load along a reload path stiffer and lower than the first. The loops are pinched: thin through the middle, where the joint crosses its gap, and wide only near the peaks.

The free play is the least of the row’s sets, not the end screw’s. At nought load the members carry no force and do not stretch, so every screw has the same slip, and the row starts carrying as soon as any screw does. The middle screws have yielded least and pick up first; the end screw, which has been furthest, is slack over 2.08 mm and only joins in near the peak.

Reversed, the row settles — at a lower ceiling #

At 0.70 of the forward peak, reversed every cycle, the row settles in four cycles: the end screw’s slip at each peak stops growing on both sides. The reversal has not reset anything. Each direction’s memory carries on from where the last loading that way left it, and a screw that has been to a slip one way reloads to it the next time along its reload path, whatever was done to it the other way in between. A reversal does not start the ratchet again; it adds a second one, on the other side.

Why it does not is a matter of where each memory lives. A screw’s set one way is the wood its thread has crushed on that side; loading it the other way crushes wood on the other side and leaves the first alone. For a reversal to reset the ratchet it would have to put the crushed wood back, and nothing does. The row’s two sides fill with the record of their own loads separately, as what is left when the load comes off found of any material that yields — and they meet only through the share of the thread’s loss that is carried across, which the earlier sections made explicit.

At 0.80 the pulled side settles in three cycles and the pushed side does not. Its end screw is driven back on to its weakened curve every cycle and its slip grows by a few hundredths of a millimetre each time — 2.5 mm after the first cycle, 3.3 mm after twenty — which is the earlier essay’s ratchet, on the weak side only. The row fails pushed.

The share of the load that comes back #

The row’s ceiling under a load that partly comes back falls with the share that does. With none of a thread’s loss carried across, a load half reversed changes nothing — the pushed side, at half the load, is well inside its own ceiling — and a fully reversed one brings the ceiling down to 0.82 of the forward peak, which is the pushed row’s own repeated-load ceiling: 0.965 of its 0.857. The fully reversed ceiling is the weaker direction’s repeated ceiling, and nothing worse, when the two directions do not share their damage.

The share carried across decides the rest. Half of it brings the fully reversed ceiling to 0.79 and the half-reversed one to 0.94; all of it, to 0.76 and 0.92. The spread between the three curves, 0.82 to 0.76 at full reversal, is the cost of not knowing a property no monotonic test of a screw measures — which is how much a thread that has been pulled loses when it is pushed. It is a cyclic test with a reversal in it, and it would turn the band into a line.

A wall designed for wind is designed for loads that reverse fully: a storm from the west and then one from the east. The ceiling that applies to it is 0.76 to 0.82 of the row’s pulled peak, not the 0.965 of the earlier essay and not the peak itself.

The gap the row keeps #

A row that has settled has not recovered. Under a reversed load of half its peak it keeps 0.20 mm of free play in a swing of 2.05 mm, and at 0.70 it keeps 0.56 mm in 3.40. The end screw’s own slack is four times the joint’s, because the end screws have been furthest each way, and they are idle for most of each cycle.

Free play is what makes a timber wall’s hysteresis pinched, and it matters for what the wall does to an earthquake rather than for whether it stands. A joint that crosses a gap carrying nothing dissipates nothing there; the loops are thin, the energy each cycle absorbs is less than a full loop’s, and the wall’s effective damping falls as the gap grows — the only thing that stops it found how much a structure’s response depends on how much each cycle takes out. A row of leaning screws that settles under reversal is safe against the load and has become a worse damper than it was.

What a test of three cycles can see #

Seismic rules for timber ask a dissipative connection to show, in a test, that it keeps its strength through a few fully reversed cycles at a stated ductility: in the European rule, three cycles, losing no more than a fifth of its resistance. The row here would pass that at almost any load below its peak. Its pushed side, the weak one, settles at 0.70 within four cycles and creeps at 0.80 by a few hundredths of a millimetre a cycle; neither loses a fifth of anything in three.

What a three-cycle test does not measure is the ceiling. The row’s strength in the test is its first-cycle peak, and the load it carries for ever reversed is 0.79 of that — a number that only many cycles reveal, and that a wind storm, which reverses the load a few hundred times, will find. The rule is a test of ductility and the ceiling is a question of shakedown, and a row can pass one and fall short of the other by a fifth.

Crossing the screws #

The obvious answer to a joint with a strong side and a weak one is to give it two strong sides, by crossing the screws in pairs, half leaning each way. Then whichever way the joint is loaded, half its screws are pulled and half pushed, and the row is the same both ways. It pays for that with its forward strength: 245 kN against the leaning row’s 263, 7 per cent less, because half its screws are always pushed.

Under a fully reversed load the crossed row is the stronger: 220 kN carried for ever against 208, with half the threads’ loss carried across, and 233 against 215 with none. Its ceiling as a share of its own peak is 0.90, against the leaning row’s 0.79. Crossing the screws gives up a little strength one way to gain more both ways, and the trade is right for a wall and wrong for a beam whose load always comes from above.

It also changes the gap. A crossed pair picks up its load when either of its halves does, and the half that is pulled in a given direction is always the one whose memory that way is its own; the joint’s free play is set by the less worn half of each pair rather than by a single screw worn both ways. Settled at 0.70 of its own peak the crossed row keeps 0.32 mm of free play against the leaning row’s 0.56, and at 0.50, 0.10 mm against 0.20. So the crossed row’s loops are fuller as well as higher, and for a wall whose job in an earthquake includes absorbing energy every cycle, that is the larger gain of the two.

The angle a screw leans at #

A screw that leans steeply — at 30° to the shear plane, nearly along the joint — is the strongest both ways, and loses least to a reversal, 13 per cent: its thread carries most of the load whichever way it is loaded, and the friction that comes and goes is a small part of a large force. A screw at 90° does not lean at all; it is the same pushed and pulled, and its reversal costs only the damage each way does to the other, 7 per cent. Between them, a screw that leans a little — 60° or 75° — is the worst under reversal: its friction is a large part of a small force, and pushing it takes away 28 per cent of its strength. At 75° a fully reversed load costs the row 30 per cent of what it would carry one way.

The angle is usually chosen for the forward strength, which falls steadily from 30° to 90°. The reversal makes the choice sharper: a screw that leans steeply is better one way and better both ways, and the angles in between are the ones that suffer most.

The pushed screw by hand #

At 45° a screw’s thread can be withdrawn by 13,749 N and its shank, bent and bearing, carries 3,343 N across. Resolved on the joint, with the axial force clamping the members and a friction coefficient of a quarter, the pulled screw’s peaks add to

$$R_{ax}(\cos\alpha + \mu\sin\alpha) + R_{la}(\sin\alpha - \mu\cos\alpha) = 13{,}749 \times 0.884 + 3{,}343 \times 0.530 = 13{,}926\ \text{N}.$$

Pushed, the axial force prises the members apart, the clamping force is negative and the friction is nothing, so the two resolve at $\cos 45° = \sin 45° = 0.707$ each: $(13{,}749 + 3{,}343) \times 0.707 = 12{,}086$ N, 0.868 of the pulled sum. The peaks of the two curves are reached at slightly different slips and the ratio of the curves’ own peaks is 0.852.

The row follows. Pushed from new it peaks at 226 kN, and if its pushed side behaved as its pulled side does under repeated load it would settle up to 0.965 of that, 218 kN. Reversed with no loss carried across, the row settles up to 0.82 of its 263 kN forward peak, 216 kN — within a per cent of the pushed side’s own repeated ceiling. With half the loss carried across, 208 kN, 0.92 of the pushed peak: the forward excursions have taken 8 per cent off the pushed side before it is first pushed.

What the memory assumes #

The thread’s loss carries across once. A screw’s first excursion each way crushes the wood round its thread, and the share of that loss carried into the other direction is taken once. Carried again on every return to the curve it would compound without limit, at loads far below the peak, which no test shows; but a screw cycled to its peak many times may well lose more than once.

One screw’s curve, measured one way. The screw’s forward and reverse curves are the screw that is pulled instead of bent’s resolution of a thread and a shank whose own strengths come from the smallest of six failures; neither was measured on a screw that had already been loaded the other way.

The wood does not split. A row of screws loaded both ways puts tension across the grain on both sides of each screw, and a row close to the end of a member can split along it, which ends the joint at a load nothing here sees.

The reload rule is the same both ways. The 0.85 of the load at the slip a screw had reached is the earlier essay’s, from tests loaded one way; a screw reloaded after being pushed may come back more softly or less.

The members stay elastic and the row straight. A shear wall’s sheathing and studs move as well, and the screws at a panel’s corners see a reversal that is part rotation.

Still open: the wall whose screws see different loads #

Every screw in this row carried the same load history, scaled by its place along the row. The screws round a shear wall’s panel do not: the ones at the panel’s corners are loaded diagonally and reversed through a full circle, while the ones along its middle edges move mainly along the edge. A wall’s free play is then a sum of rows with different histories, and its hysteresis loop a sum of rows with different gaps. Whether a wall’s pinching is set by its corner screws, which are worn most and in every direction, and whether the gap they keep after a storm is what decides the wall’s stiffness under the next one, is what the screws round a panel ask of the bolts that do not share a load evenly in the first place.