Twenty screws that peak together

A screw driven at an angle across a timber joint resists a slip by being pulled along its axis as well as bent across it, and the pull is the stronger of the two. It is also the less forgiving. The thread’s grip on the wood rises to its peak at an axial movement of a millimetre or two and then falls away as the wood around the thread shears, so a single leaning screw reaches its best at a slip of about 2.2 mm and has lost a tenth of it by 3.5 mm. A square screw, bending and crushing the wood, goes on rising for ten times that slip.

A real connection is not one screw. It is a row of them along the grain, each at its own distance from the end of the member, each loaded through wood that stretches between it and the next. A long row of bolts does not share its load evenly, because the two members strain differently along the lap: the end bolts slip most and carry most, and a joint long enough reaches its first failure at the ends while the middle bolts are still lightly loaded. For bolts that is survivable, because a bolt in bearing goes on carrying load as it slips. A leaning screw does not. The fear the single screw left was that in a row of leaning screws the end screws reach their peak and let go before the middle ones have done much, and the row’s capacity falls well short of the sum.

A row followed slip by slip #

The row is twenty 8 mm fully threaded screws at 45° to the joint, 56 mm apart along the grain — seven diameters — through two softwood members 100 mm thick, with a modulus of 11,500 N/mm² along the grain. A connection usually has several rows side by side sharing one member, so each row has some depth of member to itself; at five diameters between rows that is 40 mm, and the member area each row stretches is 100 × 40 mm. Each screw follows the single screw’s curve from the essay before: a withdrawal resistance that peaks at 1.5 mm of axial movement and softens after it, and a bending resistance on Foschi’s curve, resolved along the joint with friction. Those curves are assumed shapes, not test results, and the argument below is built so that its conclusion depends on one property of them that can be measured.

The calculation follows the row as a chain. Choose the slip of the end screw; guess the load on the joint; walk along the row screw by screw, each screw’s slip differing from the last by the difference in the two members’ strains over one pitch, and each screw’s load read off its curve at its own slip; adjust the guess until the screws’ loads add up to it. Repeating for increasing end slips traces the whole joint, past its peak and down the other side.

The fear is half right. At the row’s greatest load the end screws have slipped 3.25 mm, a millimetre past their peak, and are letting go; the middle ones have slipped 1.80 mm and are still climbing. Ten of the twenty are on each side of the peak. But none of them is far from it. The end screw carries 92 per cent of its peak and the middle one 98, and between them the row carries 98 per cent of twenty times one screw.

The reason is in the shape of the curve, not the row. A screw’s curve near its peak is a broad hump: between 1.35 and 3.45 mm of slip it stays above nine tenths of its best. The row spreads its screws across 1.8 to 3.25 mm, inside that hump. The ends have passed the top and the middle has not reached it, and both are on the flat part.

How long a row has to be #

That conclusion depends on how far apart the end and middle screws’ slips are, and that grows with the row.

The slip spread along a row is the members’ stretch: the load passes from one member to the other along the row, so one member is carrying more of it than the other everywhere except the middle, and the two strain differently. For a row that hands its load across uniformly, the end screw slips more than the middle one by PL(1/EA₁ + 1/EA₂)/8, where P is the row’s load, L its length, and EA₁ and EA₂ the two members’ axial stiffnesses. Both P and L grow with the number of screws, so the spread grows as the square of the row’s length. Twenty screws spread their slips by 1.45 mm; forty by 5.5; sixty by 8.6.

Past twenty the spread outgrows the hump. At forty screws the end screws are at 6.65 mm when the row peaks, far down the softening side of their curve, while the middle ones have not reached theirs, and the row carries 83 per cent of its screws’ sum. At sixty, 63 per cent. Long enough, a row of leaning screws does behave as the fear predicted, and its capacity nearly stops growing with the number of screws: sixty carry 506 kN, less than twice the 263 kN of twenty, and the end screws of the sixty-screw row are down to 29 per cent of their peak when it peaks.

Uneven in service, even at the peak #

The row that shares its peak so well does not share its service load at all, and the difference is worth seeing because it is the opposite of what the elastic picture of a joint suggests.

At a fifth of its peak load the twenty-screw row is nearly elastic. Every screw is low on the rising part of its curve, where its resistance grows almost in proportion to its slip, and the row behaves as the long bolted joint does: the end screws, where the two members strain most differently, carry 1.74 times the average and the middle ones 0.60. At two fifths of the peak the ends carry 1.59 times the average; at three fifths, 1.43; at four fifths, 1.24. By the peak the ratio between the end and middle screws is 0.93 to 0.98, the other way round — the ends, past their peak, carrying slightly less than the middle.

The row evens itself out because its screws soften. The end screw, carrying most, reaches the flat top of its curve first and stops gaining; every further increment of load then goes to the screws behind it, which are still climbing. That is the same redistribution a ductile bolt makes when it yields in bearing, and it is the whole reason a long bolted joint reaches its plastic capacity. The leaning screw does it with a hump rather than a plateau, so it has a limited range of slip in which to do it — the width of the hump — and that range is exactly what the ratio measures. A row whose spread fits inside the hump redistributes completely; a row whose spread is wider runs out of hump at its ends before its middle arrives.

So there are two efficiencies and they answer different questions. At service the end screws carry 60 per cent more than the average and the row’s stiffness is set by them; a check of a joint’s slip under service load, or of the cycles a repeated load puts on its most loaded fastener, must count the ends at their elastic share. At the peak the row is even, and its strength is close to the sum. The effective-number rule is a statement about the second, and it reads the first’s unevenness into it.

The effective number #

Design does not follow a row slip by slip. It multiplies one screw’s capacity by an effective number, and for screws loaded along their axes in a row EN 1995-1-1 and the product approvals for self-tapping screws use $n_\text{ef} = n^{0.9}$.

For rows of up to thirty screws in members of ordinary depth, the effective number is the actual number within a few per cent. The rule’s $n^{0.9}$ sits far below every curve: at twenty screws it allows 14.8 against the row’s 19.6 in the thinnest member drawn, three quarters of what the row delivers. Only the thinnest member at thirty screws begins to bend away from the actual number, and even it, at 27.7, is well above the rule’s 21.4.

The effective number of leaning screws in a row is close to the actual number, as it nearly is for square ones, for the rows a timber connection actually uses. The rule is not wrong; it is conservative here by a quarter, and the next question is what it is conservative against.

One ratio for every row #

The three things that set the spread — the row’s length, the members’ stiffness, the load — and the one thing that sets the hump’s width — how quickly the screw’s thread lets go — can be put into one number.

The ratio χ is the slip spread over the plateau width. Thirty-six rows of every length, member depth and thread fall on one curve in it. Below 0.3 the row is the sum of its screws. At 1 — the ends at the far edge of the hump while the middle is at the near edge — it reaches 96 per cent. Only at a ratio of 2 or more, when the spread is twice the hump, does the row lose a tenth or more.

That collapse is the useful result, because the ratio separates what the designer controls from what the screw supplies. The spread is a calculation from the joint’s geometry and the timber’s modulus; a load handed across an interface along a length spreads its slip in the same way whether the interface is a bar’s bond in concrete, a row of bolts or a row of screws. The plateau is a property of the screw in that timber, and it is exactly what a withdrawal test measures: the slip over which the force stays within a tenth of its peak. A connection designer with the test curve could check a row’s efficiency without following it.

A brittle thread in a thin member #

The plateau is where the rule’s conservatism goes. A thread that grips harder and lets go sooner — a smaller screw, a denser timber, a thread form that cuts rather than displaces — has a narrower hump, and the same row is then further along the curve.

With a thread peaking at half a millimetre the hump is a third as wide, and in the thinnest member the row falls away much sooner: 17.8 of 20, and 21.7 of 30 — the rule’s own 21.4 to within one and a half per cent. In deeper members the same brittle screws still share almost perfectly. $n^{0.9}$ is what a brittle thread in a thin member gets, and a generous allowance for anything else. That reads like a rule calibrated on its worst case, which is what a single exponent applied to every screw and every member has to be.

The brittle row is the one the fear described. Its end screws have slipped more than twice the slip of their peak and carry less than two thirds of it, while the middle ones are only a little over half way up. Here the row is losing what the ends drop faster than the middle gains, and it peaks at 89 per cent of its sum, at an end slip at which the single screw has lost a third of its strength.

Two ways out follow from the ratio, and neither is a smaller number of screws. A deeper member per row — fewer rows in the same timber, or a thicker timber — cuts the spread in proportion to its area. And a row cut into two shorter rows, each anchored separately, cuts the spread to a quarter, since it goes as the square of the length. The square screw, whose curve has no peak worth the name, needs neither: its row reaches its sum whatever its length, at a third of the leaning screw’s load per screw.

The numbers for twenty screws, by hand #

The ratio can be checked for the twenty-screw row without the chain. The row’s peak load is very nearly twenty times one screw’s, 20 × 13.4 = 268 kN. Its length, end screw to end screw, is 19 × 56 = 1,064 mm. Each member’s axial stiffness over the depth a row has is 11,500 × 100 × 40 = 46 MN. The spread is then 268,000 × 1,064 × (2/46,000,000)/8 = 1.55 mm, against the 1.45 the chain finds at its slightly lower actual peak.

One screw’s curve stays above nine tenths of its peak between 1.35 and 3.45 mm of slip, a plateau of 2.10 mm. The ratio is 1.55/2.10 = 0.74, and the collapse figure puts a row at that ratio at about 0.98 of its sum, which is what the chain found. Doubling the row to forty screws doubles both the load and the length and quadruples the spread, to about 5.5 mm and a ratio of 2.5; the collapse puts that a little above 0.8, against the chain’s 0.83.

The same arithmetic run backwards gives a rule a designer can use. For the row to reach 95 per cent of its sum the ratio must stay below about 1.2, so the spread must be less than 1.2 times the plateau. With the plateau from a withdrawal test and the spread from the joint’s geometry, that fixes the longest row a member of given depth can take without losing more than a twentieth — about 25 screws at seven diameters in members with 40 mm per row, and about 50 in members with 160 mm.

What the row leaves out #

Assumed curves. Every screw follows a curve whose shape is stated rather than measured, and the conclusions are drawn in the one ratio that does not need the shape: if a test gives a narrower plateau, the row lands further along the same curve.

Identical screws. Real screws in timber scatter in their peak loads and slips, and a row of scattered screws shares differently: the weakest screw’s peak, not the mean, can trigger the row’s. A scatter of a tenth in the peak slip is a fraction of the plateau and would change little; a scatter in the peak load would.

Elastic members, no splitting. The timber stretches elastically and nothing else happens to it. A row of screws close to the end of a member, or at a close spacing across the grain, can split the timber along the row, which is one of the failures a dowel joint must be checked for, and the row’s capacity is then the timber’s.

One row. Rows side by side in one member share the member’s stiffness; the depth per row here is that share. They are springs in parallel, where the parts of a joint made of springs in series add their flexibilities instead. If the rows are of different lengths or carry different loads, they do not stretch the member alike, and each row’s spread depends on its neighbours’.

Still open: the row loaded and unloaded #

Every row here is loaded once, to its peak and past it. A timber connection under wind or a crowd is loaded and unloaded many times, and a leaning screw that has gone past its peak does not return to its curve on reloading: its thread has damaged the wood, and it comes back along a softer path to a lower peak. After one overload that took the end screws past their peak and the middle ones not, the row has two populations — damaged ends and intact middles — and the next load finds the ends softer than before, so it sends more to the middle. Whether repeated loading evens a row out, by softening the overloaded ends until the middle catches up, or ratchets it down, by taking the ends further down their softening branch each time, is the question a row of leaning screws asks under any load that comes back.