The clamp the pair holds longer

The flanges that lean on each other solved an end plate and a column flange on one bolt as a pair: two strips clamped at their webs, the bolt between them, their tips pressed together. The component method’s two tee stubs on rigid bases, each with half the bolt, turned out to be the exact halves of a matched pair, because the rigid base is the matched pair’s plane of symmetry. A real end plate is not the column flange’s mirror image, and the pair came out a few per cent softer than its stubs in series, twelve to fifteen per cent when the end plate’s bolt was twice as far from its web as the column flange’s.

Its bolts carried no preload. Most bolts in an end-plate joint that matters do: a preloaded bolt clamps the plates together over a zone around it, and the joint is stiffer until the applied tension relieves the clamp. That essay ended on whether a mismatched pair decompresses earlier than its stubs say — its end plate prying the column flange’s tip away and opening the clamped zone from its edge — and what preload buys a joint whose two flanges are not mirror images.

A clamp at the bolt line #

The joint is the earlier one: an end plate 18 mm thick with its bolt 45 mm from its web, on a column flange 14 mm thick with its bolt 30 mm from its web, the tips 35 mm beyond the bolt, one bolt’s strip 90 mm wide. The bolt is an M20, 50 mm long, now preloaded to 110 kN. Under the washer the preload compresses a zone of plate that spreads through the grip — a cylinder about 48 mm across, less the hole — and that zone is a compression spring of 9,617 kN/mm between the two plates, against the bolt’s 1,029.

At rest, the bolt pulls with 110 kN and the clamped zone pushes back with the same. Pull the webs apart with a tension per bolt, and the plates begin to part at the bolt line: the bolt stretches, gaining force, and the clamped zone relaxes, losing it, in the ratio of their stiffnesses. The bolt sees about a tenth of each kilonewton; the clamp gives up the rest. Meanwhile each flange still bends from its web to its bolt line, and its tip still presses on the other flange’s tip. The joint stays closed at the bolt until the clamp’s force reaches nothing, and then it is the unpreloaded pair again, with its bolt starting from 110 kN.

Below the clamp’s limit, the preload does what the bolt that was already stretched found it does to any bolt: it protects the bolt from the load. The bolt’s force rises from 110 to 122 kN while the tension rises from nothing to 88. Past the clamp’s limit the bolt takes everything, and its line joins the unpreloaded pair’s.

Why the clamp lets go at 88 and not at 122 #

The joint diagram every preloaded joint is taught with — a bolt pulling straight through two plates — says the plates part when the applied tension reaches the preload over one minus the bolt’s share: $110/(1 - 0.097) = 121.8$ kN. That is the bolt force at which this pair’s clamp lets go. The applied tension at that moment is only 88.2.

The difference is prying. While the clamp holds, the flanges still bend, and their tips still press on each other with a contact force that has to be carried back through the bolt line. The bolt line therefore feels the tension and the prying together, and the clamp lets go when they reach 121.8 kN between them.

In the pair the tips carry 0.38 of the tension while the clamp holds, so the clamp lets go at $121.8/1.38 = 88.2$ kN. The end plate as a stub on a rigid base pries harder, 0.50 of the tension, and lets go at $121.8/1.50 = 81.1$ kN. The column flange as a stub pries less, 0.30, and lets go at 93.4. Prying divides the decompression load by one plus the prying ratio, and every comparison below follows from that one fraction. The ratios are higher than the unpreloaded pair’s — 0.38 against 0.32 — because while the clamp holds, the plates cannot part at the bolt line, and a flange that cannot open at its bolt tips its load further onto its tip.

A clamp is a stiff bolt #

The higher prying ratio has a simple reading. Prying in a tee stub grows with the stiffness of whatever holds the flange at its bolt line: a flange held by a soft spring opens at the bolt and has no reason to press on its tip, and one held rigidly turns about the bolt line and presses its tip down hard. In the unpreloaded pair the bolt line is held by the bolt alone, 1,029 kN/mm. Lengthen the bolt to 100 mm and the pair’s tips carry 0.26 of the tension; to half a metre, nothing at all. Shorten it to 10 mm and they carry 0.37; to 5 mm, 0.38; and a bolt that does not stretch at all gives 0.39.

While the clamp holds, the bolt line is held by the bolt and the clamped zone together, 10,646 kN/mm — the stiffness of an M20 about 5 mm long. A preloaded bolt, until its clamp lets go, is a very short bolt, and the flanges pry as hard as they would against an almost rigid one. That is why preload stiffens the joint, and also why it raises the prying: both are what a stiffer bolt line does. Once the clamp lets go, the bolt line falls back to the bolt’s own stiffness and the prying falls with it towards the unpreloaded pair’s 0.32.

Stiffer, and still softer than its stubs #

The preload stiffens the pair by more than half, from 235 kN/mm a bolt to 375 — not to anything like rigid. The clamp stops the plates parting at the bolt line, which removes the bolt’s stretch from the joint’s flexibility, but the two flanges still bend from their webs to their bolts, and that bending was always the larger part. A preloaded end plate is a stiffer spring, not a solid block.

And the pair is still softer than its stubs in series, at 0.93 of their stiffness, for the reason the earlier essay found without preload: each flange’s tip leans on a tip that moves, so the end plate’s prying is less effective at stiffening it than a rigid base would be. Preload does not remove the mismatch. It moves the whole curve and leaves the ratio about where it was.

The pair holds its clamp longer #

The earlier essay’s question was whether the mismatch makes the pair decompress early. The component method checks each flange as a stub, and a joint whose stubs are in series lets go when the first of them does — here the end plate, at 81.1 kN.

The pair lets go later, not earlier. Where the lever arms match the pair and its stubs agree, as the symmetry requires. As the end plate’s bolt moves away from its web, its stub’s decompression falls steeply — a longer lever pries harder — while the column flange’s stays where it is. The pair falls between them, much nearer the column flange than the end plate: at a 45 mm lever it holds its clamp to 88.2 kN, 9 per cent past the end plate’s stub; at 60 mm, to 82.0 against 71.2, 15 per cent past.

The mechanism the question proposed is real but acts the other way. The end plate does pry the column flange’s tip, and the column flange’s tip does give way. But a tip that gives way is a weaker prop, and a weaker prop draws less contact force: the pair’s tips carry 0.38 of the tension where the end plate’s stub, propped on a rigid base, would carry 0.50. Less contact means less pull at the bolt line, and the clamp lasts longer. The softness that made the pair less stiff than its stubs is exactly what makes its clamp outlast the weaker of them.

Every mismatch drawn delays it #

Across the joints drawn, from 10 mm end plates to 30 mm and from 10 mm column flanges to 20, the pair never lets go before its weaker stub. The ratio is smallest where the two flanges are nearly alike in stiffness — a thin end plate on a thick column flange, which pries similarly despite the lever arms — and largest, about 1.12, where a thick end plate with a long lever bears on a thin column flange that cannot prop it well.

The pair also never lets go after its stronger stub. Across 150 combinations of end plate thickness, column flange thickness and lever arm, its decompression load lies between its two stubs’ every time, which is what a shared contact makes inevitable: the pair’s prying is a compromise between the two flanges’, and so is the bolt-line force that relieves its clamp. The two stubs therefore bracket the joint. The weaker one gives a safe decompression load; the stronger one gives a limit the joint cannot reach; and a joint whose stubs are close together has its decompression load fixed to within their difference without any further analysis.

For a designer that is the comfortable direction. The component method’s decompression load, read from the weaker stub, is safe for the pair. It errs by up to an eighth in the joints drawn, always low.

Where the difference is worth a bolt #

The decompression load matters most to fatigue. A preloaded bolt whose joint stays closed sees a tenth of each load cycle; a bolt whose joint opens sees all of it, and its prying besides. The detail decides and the steel does not, and for a bolt in tension the detail is whether the joint opens.

Below 81 kN every model agrees: the joint stays closed and the bolt’s range is a tenth of the tension’s and a little more. Above 88 they agree again: the joint opens and the bolt’s range rises steeply in all of them. Between the two the component model has the joint open when it is not. For a cycle peaking at 84.7 kN the stub gives the bolt a range of 16.8 kN, the pair 11.3. In a fatigue check where the range enters cubed, half as much again is more than three times the damage.

The band is narrow — 81 to 88 kN of tension a bolt — and a joint whose repeated load peaks inside it is a joint designed with little margin against opening; the usual advice is to keep the peak well below the decompression load, which both models then agree about. Where the band matters is in assessment. A connection under a crane girder or a vibrating machine, checked by the component method and found to open on every cycle, may in fact stay closed, and a bolt that was condemned on a range of 16.8 kN may be carrying 11.3. The difference is not in the bolt; it is in which flange the model let the joint’s prying be decided by. The force the bolt never saw applied found prying on the stub’s side of the truth for strength; here it is on the same side for fatigue.

The decompression load by hand #

The joint diagram’s limit is the bolt force at which a straight bolt’s clamp lets go, $P_0/(1 - \varphi)$ with $\varphi = k_b/(k_b + k_c)$. With $k_b = 1{,}029$ and $k_c = 9{,}617$ kN/mm, $\varphi = 0.097$, and $110/0.903 = 121.8$ kN.

A prying flange feels that force at its bolt line from the tension $T$ and the tip contact $qT$ together. So it lets go at $T = 121.8/(1 + q)$. The pair’s $q$ while clamped is 0.38: 88.2 kN. The end plate’s stub’s is 0.50: 81.1 kN. The column flange’s stub’s is 0.30: 93.4 kN. A designer who has the stub’s prying ratio already, from the strength check, has its decompression load too. What the stub cannot give is the pair’s ratio, which needs the two flanges solved together, and it is the stub’s that is the larger.

The preload enters only through $P_0$, so every decompression load here scales with it: the pair lets go at 0.80 of its preload, at 55 kN or 140.

The row in a joint #

Translated into a joint, the per-bolt numbers become a moment. Take a beam end with two such bolts in the row beside its tension flange and its compression centre 400 mm below that row. Ignoring the rows below, the row keeps its clamp up to a tension of $2 \times 88.2 = 176$ kN, a moment of 71 kN·m; checked by the end plate’s stub, $2 \times 81.1 = 162$ kN, 65 kN·m. Up to that moment the row is half as stiff again as the unpreloaded pair and its bolts see a tenth of the load; past it, the row softens to the unpreloaded pair’s stiffness and its bolts begin to carry the moment’s tension and its prying whole.

The joint therefore has two stiffnesses, and the moment at which it passes from one to the other is the row’s decompression moment. A frame analysed with the preloaded stiffness under its service loads and the unpreloaded one under its ultimate loads uses the first only as far as that moment. The component method puts the change at 65 kN·m; the pair puts it at 71. For a joint whose service moment sits between the two, the difference decides which stiffness its frame actually has in service, and the joint is the stiffer of the two for longer than its components say.

A clamped zone that is one spring #

The clamped zone is a single spring at the bolt line. A real clamped zone is a cone of pressure spreading from the washer, and a flange bending under it shifts the pressure towards its tip-ward edge before it lets go. The cone’s edge opens first, and the joint’s stiffness falls gradually towards the unpreloaded pair rather than with a corner. The decompression loads here are where the clamp’s centre lets go.

The strips are elastic and stay so. Past decompression the flanges approach the mechanisms the hinge that forms in the other flange found, and the elastic pair is no longer the joint.

The preload is the preload. A bolt tightened by torque carries a preload uncertain by a quarter either way, as the torque that goes into the thread found, and every decompression load here moves with it. The ratio between the pair and its stub does not.

The tips touch, and the strip width is fixed. As before, the effective width of each flange at a bolt row stands in for how much of the plate is bending, and the comparison is made with the same widths on both sides.

What a joint’s preload buys #

A preload is sometimes described as making a joint rigid until it opens. For a prying end plate it makes the joint about half as stiff again and keeps its bolt nearly unloaded, until the tension and the prying it causes together reach the clamp’s limit — a limit a quarter lower than the joint diagram says. Between those two statements sits the whole of what preload does in a tee stub: it takes the bolt’s stretch out of the joint and leaves the flanges’ bending in.

The pair’s own contribution is a small, safe-sided one. A real end plate leaning on a real column flange pries less than the end plate would against a rigid base, so the pair is a little softer than its stubs and holds its clamp a little longer than the weaker of them — two effects of one cause. For a joint’s stiffness the mismatch is a small penalty; for its decompression and its bolts’ fatigue it is a small gift, and the component method takes neither.

Still open: the joint whose rows share a plate #

Every number here is for one bolt’s strip. An end plate carries several bolt rows, and the rows nearest the beam’s tension flange carry most of the moment. They decompress first, and once a row has opened, its share of the moment moves to the rows still clamped, which are further from the flange and pry differently. Whether a multi-row end plate lets go row by row from the flange outwards, whether the rows still clamped can take what the opened ones shed without opening in turn, and at what fraction of the joint’s moment the last clamp gives way, is the question a single strip cannot answer about the joint that was chosen for its stiffness.