The bolts that arrive one at a time

The hole that goes oval priced one bolt against one plate, and it called bearing the one connection failure that is reliably ductile. Everything in that essay treats the bolt as already touching the hole. A bearing bolt is not in that position. An M20 goes through a 22 mm hole, so it has two millimetres of nothing to cross, and the hole in the other plate was drilled in a different shop, or at least at a different moment, and may not be exactly opposite. When a lap of eight bolts is erected, some of them are drawn across to one side of their holes and some to the other, wherever the drift pin left them. The plates begin to slide under load, and for a while one or two bolts are carrying all of it.

The bolts that do not share followed a long lap and listed this at the end as something its model did not have: nothing is in contact until it moves. Two fasteners that never arrive together looked at the case where a weld and a bolt meet in one joint, and found that crossing a clearance is over in an instant compared with the range the bolt needs. Neither asks the plain question of a group of identical bolts in identical holes. Every design standard divides the shear by the number of bolts. Is that ever true, and when it is true, what makes it so?

One curve for every bolt, begun at a different place #

A bolt in bearing is not a spring. Fisher’s load–travel curve, from the tests at Lehigh that later went into the American treatment of eccentric groups, runs

$$R = R_u\left(1 - e^{-\mu\delta}\right)^{\lambda},$$

with $\mu = 10$ per inch, $\lambda = 0.55$, and the bolt failing when its own travel $\delta$ reaches about 0.34 inches, or 8.64 mm. The curve is steep at first, at 72 per cent by the time it has moved 2 mm, and it then takes a further 6 mm to find the last quarter. An M20 grade 8.8 bolt in single shear with $R_u = 94$ kN is the example here, though the shape is what matters.

Here the plates are taken as rigid, so every bolt sees the same travel $\Delta$ of one plate past the other. A bolt whose two holes are misaligned by $e$ along the load only starts when $\Delta$ has passed its own gap, and the origin of $\Delta$ is arbitrary, so only differences count. Bolt by bolt, $\delta_i = \Delta - s_i$, where $s_i$ is how much later than the first bolt this one arrives.

That single picture holds the whole of the argument, and its finding is already visible. The late bolt starts after the first has done most of its work, but the first bolt then flattens, and it stays nearly flat for several millimetres. The two curves come together, because a curve that is flat cannot stay ahead of one that is steep. By the time the first bolt is about to break, the late one has caught up to within a few per cent. A lag of 2 mm, a quarter of the bolt’s travel, has become a difference of 2 per cent in load.

The curve’s flat top is doing what a yield plateau does for a section and what the first plastic hinge does for a beam. Whatever got there first waits for the others, and strength is not lost by arriving in the wrong order so long as nothing breaks before the last one arrives.

Eight bolts, one draw of the holes #

The draw drawn here has its first arrivals at 0 and 0.17 mm and its last three bolts all near 3.85 mm, so it is a fairly bad case, with almost all of the tolerance used. The curve for the misaligned group runs well to the right of the fitted one. It reaches half of its nominal strength at 3.9 mm and three quarters at 5.0, where fitted bolts need 0.9 and 2.3. Then the two lines close, and at the moment the first bolt breaks they differ by 3.5 per cent, 95 against 98.

The fitted group’s 98 per cent deserves a remark of its own. Even perfectly fitted bolts do not reach 100, because the curve is still climbing at the point of failure, with $(1 - e^{-3.4})^{0.55} = 0.982$. A group in which every bolt arrives together gets its peak in the same instant that all its bolts break, and it has nothing beyond that. It has no warning either, and that is a point for later.

The hand check for the misaligned group is short. In the worst possible arrangement, half the bolts arrive 4 mm before the other half. When the early half reach 8.64 mm of travel, the late half have reached 4.64, where $(1 - e^{-0.394 \times 4.64})^{0.55} = 0.909$. The mean of 0.982 and 0.909 is 0.945. That is very nearly the figure’s 0.947 from a draw whose bolts are bunched in just that way.

So the rule that divides by the number of bolts holds, near enough, at failure, and it holds for an interesting reason. It is not that the bolts share, since until very late they do not. It is that sharing is the end state of a process that has room to finish.

What the journey looks like #

The end state is what a strength check reads, and the route to it is what a structure lives on. A bolted connection spends its service life well below its capacity.

At a quarter of the group’s nominal strength, four of the eight bolts have not touched anything. They are in their holes doing nothing, and two bolts carry about two thirds of what they will ever carry. At a half, with the connection roughly at its working load, every bolt is in contact, but the shares still run from less than a tenth of a bolt’s strength to 87 per cent. The two bolts with the highest loads are near their plateau, and three bolts are hardly loaded. Only at three quarters does the picture start to look like the one the calculation had in mind, and even then the spread is 55 to 87 kN.

Nothing in that figure is a safety problem for a static load. It does matter in three other places.

Stiffness. The misaligned group has moved 3.9 mm at half its strength where a fitted one has moved 0.9. That is 3 mm of joint slip, at each end of a member, that no stiffness calculation contains. The braces with slack in their bolts and stiffer than the model said both price a clearance as one gap, crossed and done with. A group with a spread of arrivals does not cross it once, it softens gradually, and the load–travel line has no straight part on which a single stiffness could be read.

Fatigue. A bearing bolt carries a share of a load cycle that depends on where it sits on its curve. Near the plateau, its share of a cycle is small and the bolt beside it, still on the steep part, takes more. Working out which bolt is doing what in a cycle needs the arrivals, and the arrivals are unknown. That is one reason why a fatigue-loaded joint is preloaded rather than calculated, which moves the problem to friction and away from the holes.

Eccentric groups. The instantaneous-centre method in the bolt that carries more than its share uses this same curve and assumes every bolt touches at zero travel. A group with a spread has its centre of rotation in the wrong place until the spread has been crossed. The peak barely changes, for the reason just given. What changes is the distance the group has turned on the way there.

How much the fit costs, over many draws #

One draw is one draw, so the next step is the population. Misalignments here are uniform and independent across the bolts, drawn repeatedly, and the figure gives the mean strength and the fractile that 5 per cent of connections fall below.

A standard 2 mm clearance allows up to ±2 mm of misalignment before the bolt will not enter, so the region up to there on the figure is what any bolt that goes in without reaming can produce. That whole range costs 3 per cent at the fractile. The cost is slow at first and picks up as the misalignment approaches half the bolt’s travel. Beyond ±2 mm the bolts will not go in, unless the holes are oversize, and an oversize hole of 26 mm for an M20 with ±4 mm of misalignment is where the 84 per cent lies. That is the price of arrivals alone, and it stands beside Eurocode 3’s 0.8 on an oversize hole, which the hole that goes oval traced to a shorter tear-out block rather than to any lag between bolts.

There is a further result that the figure does not show because it would be a flat line. The loss does not depend on the number of bolts. With two, four, sixteen and forty bolts at ±2 mm, the 5% fractile runs 96, 96, 95 and 95 per cent of nominal. A larger group draws a wider spread, but only slightly once it has already found its worst pair, and at failure each bolt carries a mean share rather than the extreme. The group’s strength is averaged over the bolts, and the average does not move much. Thus a long joint’s reduction, which grows with the length, has a different origin. In the bolts that do not share it comes from plate stretch building a difference along the joint that gets larger with every bolt added. Hole clearance gives a difference of bounded size whatever the count.

The property that pays for it #

All of those figures have used a travel at failure of 8.64 mm, as if that were a constant, and it is not. It belongs to a particular class of test: high-strength bolts of about 22 mm in double shear, in plates that hardly bear. Threads in the shear plane shorten it. A smaller bolt has less of it, as its deformation scales with its diameter. A brittle fastener has very little. A plate failing in bearing has much more, as the hole that goes oval showed, with a hole that can stretch by a bolt diameter or more before the load falls.

This is the surprising connection in the problem. The fitted line and the misaligned lines come together at high travel and separate sharply at low travel, and the variable along the bottom is the ductility of the fastener, a material property. The tolerance of the holes and the ductility of the bolts are the same quantity read from two ends. It is their ratio that decides whether a group behaves as the division by the number of bolts says it does.

It follows that the two bearing modes are very different in how they treat the fit. A connection governed by the plate, with bolts stronger than the holes, has the 25 mm line and does not care about fit at all. A connection governed by the bolts has a travel of about 8 mm and pays a few per cent. A bolt short of ductility, with threads in the shear plane of a thick lap or in a grade of 10.9 and above, may last only 4 or 5 mm, and there the fit costs up to a third of the strength. So the guidance that a plate should be the weak link of a bolted joint has a reason beyond the warning a stretching hole gives. A plate-governed joint is the one that does not need its holes to be accurate.

And the same reading explains an observation about a group that seems to be about something else. Standards put a limit on how far a connection may combine bolts of different grades, or bolts with welds. Two fasteners that never arrive together reached that limit from the direction of the weld, whose travel to failure is a fraction of a millimetre. It is this figure’s left-hand end. A weld in a group of bolts is a fastener with a travel well below the spread of arrivals, so it will always have left before the bolts have started.

The worst the holes can do #

A uniform random draw is a modelling choice. Real misalignment is partly random and partly a pattern, where a whole row is out by a pitch error, or one plate was drilled from a slightly different template. The pattern that matters most is the simplest one: half the bolts early by the whole tolerance and half late by it.

The line has a step in it. Above a travel of 4 mm the two sets overlap, and the group’s strength follows the mean of two curves. At 8.64 mm it is 94.5 per cent, and at 6 mm it is 83. Below that the early set has broken before the late set has done much, so the group reaches only what the early half could carry alone: 44 per cent at 4 mm, 36 at 2. Between 4 mm and 6 mm the strength nearly doubles. A joint whose bolt ductility sits close to the spread of its holes is near a cliff, in the sense that small changes in either value change which curve it is on.

This is an unzipping failure. The first half of the bolts break one after another in a few tenths of a millimetre, the load moves to the second half, and those then take a load they are nowhere near ready for and go in turn. The figure’s lower line assumes the second half has done nothing useful, which is close to the truth, and the load drops sharply at the moment the first half goes.

That is a contrast with the fitted group. There, every bolt reaches failure at the same moment, without warning, at 98 per cent. In the misaligned group of ordinary ductility, the first bolt to break is carrying very little more than its neighbours, and the load falls by one bolt’s share. Then the rest carry on at a slightly lower total, with some travel still in hand. Strictly, a fit that is a little loose gives a slightly weaker joint with a longer and more visible end. That is not a recommendation to loosen anything, but it does mean that the accuracy of a fit is not the same as its merit.

Reamed and fitted bolts, and how much they were worth #

A designer could remove the arrivals entirely. A fitted bolt in a reamed hole has no clearance, and nineteenth-century riveted work approached the same state by filling the hole with hot metal. Bridges were long built with turned bolts in reamed holes, and the reason given was fit. Their strength advantage, read from these figures, is 3 per cent, when the travel is long. What fitted bolts bought was the other side of the picture: at a quarter of nominal the fitted group has moved 0.2 mm where the misaligned one has moved 1.7, and it has no reversal slip at all under a load that changes sign. That is why they were used, and why a preloaded bolt, which gets the same stiffness by friction at far less expense on site, replaced them.

Put another way, every reason for a close fit is a serviceability or a fatigue reason. None of them is about the static strength of an ordinary bolt group, and the rule of dividing by the number of bolts survives, at bolt ductility of the usual size, because the arithmetic of the curve leaves a margin.

What the rigid plates hide #

The plates are rigid. In a real lap the plate stretches between bolts, and that adds a travel that grows along the joint. The combination has not been run here, and the bolts that do not share has the other half. Whether a long joint’s end bolts, already ahead by plate stretch, are helped or hurt when they are also the ones that happen to arrive early is a matter of chance, and the worst case compounds.

All the bolts have one curve. Fisher’s curve is a fit to a set of tests, with its own scatter, and real bolts have different strengths and ductility. A weak bolt that also arrives early breaks first, and that correlation is not here.

The bolts do not bend or tilt in their holes. A bolt with a long grip bends as it bears, particularly in single shear, where the lap’s own eccentricity rotates it. The same curve has been assumed for all of them.

The draws are uniform and independent. Drift pins, used to bring holes into line at erection, deform the plate and the hole, and leave a state of locked-in force that built to the wrong length priced for whole members. A bolt driven into a misaligned hole with a drift is touching on one side already. This model treats the gap as empty, and so overstates the lag on such a bolt.

The picture is a monotonic load. A load that reverses makes the group cross its gaps again in the other direction, with a different order of arrival, and each pass adds to the ovalling. How much of the reversal slip is worth carrying into a frame is a question about a frame.

Still open: the first bolt of a stair-step load #

Every case here reaches its peak by one push. A connection in a real structure sees small cycles first, from wind and traffic, with the early bolts ovalling their holes a little on each pass, and it reaches a large load only much later. If each small cycle deepens the seat of the bolts that are already carrying load, the spread of arrivals may close by itself over the service life, and then the group at the moment of the storm is better fitted than the group that was erected. Alternatively, the same ovalling may make the early bolts softer without making the late ones any busier. Which of those it is, and whether a joint is stiffer or looser after ten years of small loads, needs a contact law that remembers its own hole, and the curve used here has no memory at all.