The row that remembers its overload

Twenty screws that peak together followed a row of leaning screws slip by slip and found the fear of it misplaced. A screw driven at 45° across a timber joint peaks at about 2.2 mm of slip and then softens as its thread lets go of the wood, and in a row of twenty the end screws reach that peak before the middle ones because the members stretch between them. But the hump of a screw’s curve is broad, and twenty screws in members of ordinary depth spread their slips across it: at the row’s greatest load the end screws are a millimetre past their peak, the middle ones a little short of it, and the row carries 98% of twenty times one screw. One ratio decided it — the slip the members’ stretch spreads along the row against the width of the screw’s hump — and every row of every length fell on one curve in it.

Every row there was loaded once. Timber connections in a roof, a grandstand or a bridge are loaded by wind, crowds and traffic that come and go. A screw past its peak does not return to its curve when it is unloaded and loaded again. Its thread has sheared the wood around it, and the next loading finds a softer path to a lower peak. After an overload that took the end screws past their peak and left the middle ones short of it, the row holds two populations of screw. The question is what the next load does with them.

What a screw keeps #

The screw’s curve on first loading is the one from the earlier essays: an assumed shape, not a test curve, with the thread peaking at 1.5 mm of axial movement and the screw as a whole at 2.22 mm of slip. What happens when it is unloaded and loaded again has to be assumed too. The rule used here is the simplest one that matches what cyclic tests of screwed timber joints report, and its one free number is varied later.

Unloaded, a screw comes back along its initial stiffness. It drops its load almost elastically. Because the wood around its thread has been crushed and sheared, it does not return to zero slip. It stops at a permanent set, and below that set it is slack: the thread has pushed the wood away, and the screw carries nothing until the joint slips back to where it was.

Reloaded, it follows a straight line from its permanent set towards the slip it reached before. It arrives there carrying only a share ρ of the load it had — here 0.85, the kind of loss between the first and third cycle at one amplitude that the cyclic test protocols for timber joints measure and call impairment of strength. Past that slip it climbs steeply until it meets its own curve again. A screw that has been to its peak loses the whole 15%; one that stopped part-way up its curve loses that part of it.

A screw loses again only when it is pushed back onto its curve. One that went a little past its old slip on the stiff reload line without regaining its curve keeps what it carried.

That last rule is the one that decides everything below, and it is a statement about damage: the thread shears new wood only when the screw is loaded as hard as the wood around it allows.

The slip it reached #

The first consequence is the plainest. Take the row past its peak, to an end slip of 4 mm, where it has fallen to 95% of its first peak, unload it, and load it again.

The reloaded row starts slack, which is the first surprise. Every screw has a permanent set, and the sets are not equal: the end screws, which went furthest, set at 3.3 mm, and the middle ones at 1.8. On reloading the joint must slip back to 1.8 mm before any screw carries anything. Then the middle screws take up first, and the end screws join only when the slip reaches theirs.

Then it climbs, and it does not go past where it was. It reaches 0.942 of its first peak at an end slip of 4.10 mm, just beyond the 4.00 it was taken to, and then turns down along the same falling path it was on when it was stopped. A row loaded again carries what it carried at the deepest slip it reached, a little less, and no more. The peak it once had is gone, because the screws that made it are past theirs.

An overload that stops short of the row’s peak is a different thing, and the difference is the whole of what the falling path means.

Stopped at 2.40 mm, the row carries 94% of its first peak, and its end screws are just past their own peak while the middle ones are well short of theirs. Unloaded and loaded again, it comes back slack as before, takes up, reaches the slip it had been to with its damaged screws carrying a little less than they did, and keeps climbing, because the middle screws still have most of their curve to go. It reaches its first peak again, at the same end slip as from new.

So the rule is not that an overload costs a row its peak. An overload costs a row whatever lay behind it on the row’s own path. Short of the peak, the screws that went past theirs lose a share of their load, and the screws that did not have curve enough to make it up. Past the peak, the row has used up that curve, and what it was carrying when it stopped is the most it will carry again. Inspecting a joint after an event, the question that decides its capacity is not whether its screws were damaged but whether the joint as a whole was taken past its peak.

Reloaded, the middle goes first #

The slack start reverses something the earlier essay found about service loads. A row loaded from new is uneven at low load: the end screws, where the two members strain most differently, carry 1.74 times the average at a fifth of the peak, as a long bolted joint’s end bolts do. After an overload the end screws have the larger permanent set, so at low load they are slack and the middle screws carry everything. The row’s service stiffness after an overload belongs to its middle, and the load path inside it has moved by half a row.

That matters for whatever the joint’s service checks were. A check of the cycles a repeated load puts on the most loaded fastener chose the end screws on the virgin row’s evidence. After one overload the most loaded screws at service are the middle ones, and the end ones are not working until the load is high.

The load that comes back #

Now the question the row was left with. Load it from new, again and again, to one level below its first peak, and follow the end screw’s slip at the top of each cycle.

Both answers the question proposed are true, at different loads. At 0.94 of the first peak the row settles almost at once: the first cycle damages the end screws a little, the second finds them on their reload lines, and nothing moves again. At 0.96 it takes seven cycles. The end slip creeps from 2.56 mm to 2.83 and stops, because each cycle the end screws are pushed onto their curve, lose a little, and pass load to the screws behind them, and the screws behind them have room to take it.

At 0.97 it never stops. The end slip grows every cycle, slowly at first and faster as the end screws work down their falling curve, until on the 21st cycle the row cannot reach the load at all. At 0.99 it fails on the 8th. A row of leaning screws can fail under a load it has already carried, by doing nothing but carry it again.

Between the two lies a limit: the largest load the row carries for ever when it comes back. For this row it is 0.965 of the first peak. Below it the row evens itself out; above it the row ratchets down. The question was evening out or ratcheting, and the answer is a line between them three and a half per cent below the peak.

How the middle catches up #

The settling cycles are worth looking inside, because they are the evening-out the question hoped for, and it comes with a price.

On the first cycle at the limit the row looks as it did on its first loading: the end screws near their peak, the middle ones well short of theirs, at 0.87. By the seventh cycle the ends have dropped to 0.96 and the middle has risen to 0.92. The row carries the same load with a flatter distribution. It has evened out by giving up some of what its end screws could do and spending some of the slip the middle ones had in hand. The end screw has slipped 0.29 mm further to get there.

The profile is not smooth any more. The fourth screw from each end dips below its neighbours, because in one of the intermediate cycles it was pushed back onto its curve and took a fresh loss while the screws either side of it did not. A row that has been cycled carries the history of which screws were damaged when, and a screw’s position along the row no longer says everything about its load.

A row that peaks together loses least #

The limit depends on the one number that had to be assumed: how much load a screw carries back to the slip it reached.

With no loss on reloading the limit is the first peak: every curve meets at 1, within the precision of the search. With a loss, the row of ten screws barely notices. It keeps 98% of its peak even if each screw regains only 60% of its load on reloading, because its screws all peak at nearly the same slip, so none of them is pushed far past its peak while the others are still climbing. The rows of twenty and thirty lose much more — to 0.91 and 0.90 at a share of 0.6 — because they are the rows that the earlier essay found spread across the hump, with end screws past their peak while the middle ones approach theirs.

The same ratio decides both. The spread is a load handed across an interface along a length, and a row whose spread of slip is small beside the width of its screws’ hump reaches its sum once and keeps nearly all of it under repeated load. A row whose spread fills the hump reaches its sum once, comfortably, by redistributing from damaged ends to intact middles — and that redistribution is exactly what repeated loading eats. The earlier essay found that the 98% the row of twenty reached was bought with a hump that let the ends go past their peak without losing much. Under a load that comes back, going past the peak is what costs.

One large overload against many small ones #

The rule that a screw loses only when pushed back onto its curve has a consequence that looks backwards.

A row that has been taken once to its own peak, to an end slip of 3.25 mm, and then loaded again and again, settles at every level drawn up to 0.99 of its first peak. The same row loaded again and again from new fails above 0.963. The row that was overloaded is the one that survives the repeated load.

The reason is in when the damage happens. The single overload took every screw as far as the peak took it and damaged each of them once, by the full 15% for the end screws. Every later load below 0.99 finds the end screws on their reload lines, short of the slip they reached, and pushes none of them back onto its curve. The repeated loads from new take each screw only as far as each cycle needs, and a cycle near the peak needs a little more slip every time, so the end screws are pushed back onto their curves cycle after cycle and lose 15% of their load on each. The row is damaged by being worked near the edge, not by having been over it once.

It would be wrong to read this as advice. The overloaded row’s settled slip is half a millimetre larger than the cycled row’s at every level, its end screws carry 15% less than they could, and the overload itself was a load the joint was never meant to see. What it says is that a joint’s history matters in a particular way. The loads to worry about are the ones that bring the screws back to their curves repeatedly, not the single large one — the same lesson a frame walked by the wind teaches in steel, where a load applied once is carried and the same load with a reversing companion is not.

What it means for the rule #

A designer does not meet any of this through the effective number. EN 1995-1-1 and the product approvals for screws in rows use $n_\text{ef} = n^{0.9}$, which for twenty screws is 14.8 — three quarters of the 19.6 the row reaches once, and well under the 0.965 × 19.6 = 18.9 it carries for ever under a load that comes back. The rule covers the repeated-load loss many times over. Design loads at service are a fraction of this again: a row checked to its effective number at the ultimate limit state is carrying under 40% of its own peak at service, where nothing here is near happening.

The loss matters in the cases that bring a joint near its peak more than once. A timber frame in a storm that loads it to near its capacity in gust after gust, or a structure damaged in one event and loaded again in the next, before anyone has looked at its joints. For those the row’s capacity is not its first peak but the lower of two things: the load at the deepest slip it has reached, and the repeatable limit below its peak.

Twenty screws, by hand #

The single screw peaks at 13.4 kN at 2.22 mm of slip, with an initial stiffness of 16.3 kN/mm. The row of twenty peaks at 263 kN with its end screw at 3.25 mm.

Taken to an end slip of 4.00 mm, the end screw has fallen down its curve to 11.0 kN. Unloaded at 16.3 kN/mm it sets at $4.00 - 11.0/16.3 = 3.32$ mm, and reloaded it will reach 4.00 mm carrying $0.85 \times 11.0 = 9.4$ kN. The middle screw, at 2.58 mm, was just past its peak at 13.3 kN; it sets at 1.76 mm and will be back at 2.58 mm carrying 11.3 kN. The row’s slack start runs from zero to 1.76 mm, and the end screws join at 3.32.

At a repeated load of 0.97 of the peak, 255 kN, the row needs an average of 12.8 kN a screw. With its end screws on reload lines topping out at 0.85 of what they last carried, that average needs either more from the middle or a push of the ends back onto their curves. The first cycle supplies both. Each later cycle supplies a little more of the second, because the middle screws are themselves approaching their peak, and the ends, pushed onto falling curves, carry less every time. At 0.96 the middle screws’ remaining climb is enough and the sequence converges; at 0.97 it is not.

A rule assumed and a direction fixed #

The reload rule is assumed. The softer path to a lower peak is what cyclic tests on screwed joints show. The share regained, 0.85, is typical of the strength impairment the cyclic test protocols record, and the limit is drawn against the share precisely because it is not known for any particular screw. The rule that a screw loses only when it is pushed back onto its curve is the one that produces the ratchet. A rule under which a screw lost a little on every cycle, wherever it was, would ratchet at lower loads still; one under which damage depended only on the largest slip ever reached would never ratchet.

The load never reverses. Every cycle goes from nothing to a load in one direction and back. A screw driven square to the joint, bending and crushing the wood rather than pulled along its thread, has a curve with no peak worth the name, and a row of those would neither lose its peak to an overload nor ratchet below it. A row of leaning screws loaded the other way is a row of screws pushed rather than pulled, much weaker, and an earthquake or a reversing wind that loads the joint both ways would mix the two.

Time does not enter. Timber under sustained load creeps, and a screw’s withdrawal resistance falls with moisture and duration; a load that comes back for years is not a load that comes back forty times in an afternoon.

The members stay elastic and do not split. As in the earlier essay, a row whose damage concentrates at its ends is a row more likely to split the timber there, which is one of the six failures a dowel joint is checked for and the one a ratcheting end screw invites.

Every screw is the same. In a real row the screws scatter, and the weakest of the end screws is the first to be pushed back onto its curve, so a row of scattered screws begins to ratchet sooner and less evenly than the identical row here.

The joint as a record of its loads #

The row of screws joins a short list of things in this collection whose state is a record of their history rather than a function of their load. The order the loads arrived in decided the force in a pier under a sliding bearing, and the shakedown of a frame decided whether a load it carried once could be carried for ever. Here the record is in every screw: its permanent set, the slip it reached, the share it lost. Two rows of identical screws, one overloaded once and one worked repeatedly near its peak, carry the same load with different distributions and fail at different repeated loads.

That is the honest content of the effective number. A single exponent cannot know a joint’s history, so it is set low enough that history rarely matters. A row loaded within it is far from everything here. A row loaded beyond it, more than once, carries the history of each loading in where its screws have set.

Still open: the row whose load reverses #

The loads here come and go in one direction. A screwed joint in a shear wall or a moment frame under wind or earthquake is loaded one way and then the other, and a leaning screw resists the two directions differently: pulled along its axis one way, pushed the other, strong in the first and much weaker in the second. A row reversed after an overload has end screws slack in one direction and damaged in the other, and middle screws that took up first in one direction and last in 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 from new, is the question the screws’ own asymmetry puts to any joint that has to carry a load both ways.