The bolt that was already stretched

Every bolt in the essays below has been a bar that the load stretches. The tee stub’s bolt carries the applied tension plus a prying force; the thickness of the flange decides which of them gives way; the rows of an end plate fold together rather than one at a time. In all three the bolt is slack until something pulls on it.

A preloaded bolt is stretched before the joint has been asked to do anything, and it changes the arithmetic at the first line rather than at the last.

Two springs, and the load goes mostly into the wrong one #

Tightening a bolt stretches it and squashes the plates under the head by the same amount. That is one displacement shared by two elastic objects, which is the definition of springs in parallel — not in series, which is the arrangement a joint’s other components are in and which behaves the opposite way.

The two stiffnesses are wildly unequal. An M20 bolt in a 40 mm grip has a stress area of 245 mm² over an elastic length of about 60 mm, giving 857 kN/mm. The clamped plates are not a cylinder but a cone spreading from each bearing face at roughly thirty degrees, with the hole taken out of it, and integrating that cone gives 3,781 kN/mm — four and a half times stiffer, because it is short and wide where the bolt is long and thin.

Apply a tension to the joint. It lengthens by some amount, the bolt takes that extension and stiffens, and the clamped cone takes the same extension and unloads. The bolt’s share is

$$\Phi = \frac{k_b}{k_b + k_c} = \frac{857}{857 + 3781} = 0.185$$

so eighteen per cent of every kilonewton applied reaches the bolt and eighty-two per cent is spent undoing the clamping. Nothing about that is a safety factor or an allowance; it is a stiffness ratio and it is available from two dimensions and a modulus.

Which free body produced the number #

The stiffness ratio is easy to write down and easy to lose confidence in, so it is worth taking the free body that produces it.

Cut the joint on a plane through the clamped plates, between the two bearing faces. Three forces cross that plane: the applied tension $T$ pulling the halves apart, the bolt’s force $B$ holding them together, and the contact pressure $C$ between the plates, compressive, distributed over whatever area is still touching. Equilibrium gives $B = T + C$ with $C$ negative in the sense that it closes the joint, so

$$B - C = T$$

with two unknowns and one equation. The second equation is not equilibrium at all — it is compatibility. The bolt and the clamped stack share one displacement, so $\delta = (B - F_p)/k_b = (F_p - (-C))/k_c$, and eliminating $\delta$ gives the split.

So a preloaded joint is statically indeterminate to the first degree, which is why nothing about it can be read off a force diagram and why the answer contains stiffnesses at all. It is about the smallest indeterminate structure there is — two springs, one release — and it has every feature of a large one: the stiffer path takes the load, the self-equilibrating preload vanishes at collapse, and the answer is sensitive to a stiffness nobody measures.

That last point is the one to hold. $k_c$ is not measured on any site and is not stated on any drawing. It comes out of an assumed cone, and every number in this essay is proportional to how good that assumption is.

The lines meet, which is the whole of the strength answer #

The figure contains the result that matters most and it is a negative one. Past the crossing the two lines are the same line. A preloaded bolt and an untightened one in the same tee stub carry identical forces at every load above 138 kN, fail at the same load, and are governed by the same mode.

That is not an approximation. Once the plates have parted there is no clamped cone left to share anything, so the joint is exactly the joint the earlier essays analysed, and the preload has been entirely consumed in getting there. A preload is a displacement imposed on a self-equilibrating pair, and a self-equilibrating pair contributes nothing to a failure load — the same reason a lack-of-fit force disappears at collapse.

So every strength check on a tee stub stands unchanged. The tee’s mode 1, mode 2 and mode 3 are what they were, the group rules are what they were, and the preload appears in none of them.

Where the crossing is, and why it is earlier than it looks #

The obvious place to expect the plates to part is where the applied load has undone the preload, at $F_p/(1-\Phi)$, which for these numbers is 210 kN. The crossing is at 138.

The difference is prying. The un-preloaded bolt’s line is not $B = T$; it is $B = T + Q$, and on a flange that pries, $Q$ grows at $m/n$ per unit of applied load — here 45/40, so the line climbs at 2.125 rather than 1. A steeper line meets the flat one sooner. Prying brings the separation forward by a third, on a joint whose whole purpose in being preloaded was to postpone it.

That is worth carrying because it inverts the intuition. Prying is usually described as something that happens to a bolt after the flange starts to bend; here it is deciding when the flange starts to bend at all, through a line it has made steeper.

Below the crossing the bolt barely notices the load #

The reason any of this is worth the cost of a preloaded assembly is not visible in a strength calculation at all.

A fatigue calculation reads a range, and the range the bolt sees is the applied range multiplied by the slope of the line it is sitting on. Below the crossing that slope is 0.185; above it, 2.125. The ratio is 11.5, and any endurance curve cubes it.

A preloaded bolt below its separation load is, for fatigue purposes, not a fatigue detail. It is carrying a range a tenth of what the joint is carrying, at a stress amplitude that is likely below the constant-amplitude endurance limit entirely — which is why a detail category for a preloaded bolt in tension is a different object from one for an ordinary bolt, and why the two cannot be compared as though they described the same fastener at different stresses.

And the same arithmetic run backwards is a design rule with one number in it. The separation load has to sit above the highest load the joint ever sees, not above the load it is designed for — because the benefit does not degrade gracefully as the crossing is approached. It is a step.

The curve comes back down, and that is not good news #

The range curve does not keep rising, which is worth explaining because the shape looks like a reprieve.

Past the crossing the prying force climbs until the flange has formed its own collapse mechanism, at which point $Q$ is capped at $M_p/n$ and stops growing. The bolt’s line then climbs at 1.0 again, so the range it sees falls back toward the applied range. A joint whose mean load is well past separation has a smaller stress range than one just past it — because the flange has already failed, and a failed flange transmits load without amplifying its variation.

That is the whole of what the falling limb means, and it is a good example of a curve whose worst point is in the middle. Reading the right-hand end as safe would be reading the strength of a plastic mechanism as a serviceability result.

A longer bolt is a better bolt, which nothing else in this subject says #

The load factor is a ratio of two stiffnesses and both of them fall as the grip lengthens, but not at the same rate, and the direction that produces is the one useful design lever in the whole calculation.

The bolt’s stiffness falls as the inverse of its length — 1,285 kN/mm at a 20 mm grip, 857 at 40, 514 at 80, 367 at 120. The clamped cone’s stiffness also falls, because the cone is longer, but far more slowly: 4,984, 3,781, 3,153, 2,936. The cone is spreading outward as it lengthens, so the extra area partly compensates for the extra length, and by a 120 mm grip it has lost only two fifths of its stiffness where the bolt has lost seven tenths.

So the ratio improves with grip, and Φ falls with it: 0.205 at a 20 mm grip, 0.185 at 40, 0.140 at 80 and 0.111 at 120. A long bolt through a thick stack takes eleven per cent of the applied load where a short one takes twenty.

That is the reverse of nearly every other length effect in structures, where a longer member is a worse member. It is the reason a bolt in a fatigue-loaded tension joint is sometimes deliberately given a reduced shank, or a long spacer under the nut, or is simply specified longer than the grip requires — all three of which lengthen the bolt without lengthening the clamped stack, and all three of which are cheap.

Bolt diameter moves it the other way and hardly at all. Φ runs 0.175, 0.185, 0.191 and 0.199 for M16, M20, M24 and M30 at the same 40 mm grip, because both stiffnesses scale with the square of the diameter and only the cone geometry breaks the tie. The load factor is a property of the grip and not of the bolt, which is a more useful sentence than it looks: it means the fatigue behaviour of a tension joint is decided by how thick the plies are, a dimension chosen for entirely unrelated reasons.

What does scale with the bolt is the separation load, in direct proportion to the preload: 133 kN for an M16, 170 for an M20, 210 for an M24 and 292 for an M30. So a larger bolt buys a higher crossing and very slightly worse protection below it, and the first effect is much the larger.

The preload has to survive being installed, and has to survive the year after #

Every number above rests on 171 kN actually being in the bolt, which is the least certain quantity in the calculation.

What a torque wrench delivers is a preload with a scatter of roughly ±25 per cent, because most of the torque goes into friction under the nut and in the thread rather than into stretching the bolt. The separation load is exactly proportional to the preload, so the crossing moves with it: at 0.7 of the bolt’s tensile strength it is 170 kN on a 32 mm flange, at 0.5 it is 145, and at 0.3 it is 90.

Relaxation takes more of it. Embedment of the surface roughness under the bearing faces is worth a few per cent in the first hours; coating creep, if the plies are painted or galvanised, can be worth much more; and a joint that is unbolted and re-tightened has a different preload again. None of it changes any strength check and all of it moves the crossing, which is the recurring shape of this essay: everything uncertain about a preload affects only the quantity nothing is checked against.

A thin flange never gets there #

The benefit has a condition attached that is easy to miss, and it is the dimension the whole anchor turns on.

For the preload to be worth anything, the plates have to part while the flange is still elastic enough to matter. On these proportions that needs about 25 mm of flange against an M20 at full preload, and below it the flange’s own collapse arrives first.

So the thickness that decides who fails decides this too, and in the same direction: a thin flange gets no fatigue benefit from preload, gets the most prying, and is the case where a designer is most tempted to reach for a preloaded assembly because the bolt looks overloaded. The flange has to be designed first and the bolt afterwards, which is the same conclusion the essay on flange thickness reached from the strength side.

What a strength check and a fatigue check are each reading #

The two answers this joint gives are so different that it is worth naming what each check actually looks at on the same curve, because they are not two accuracies of one calculation.

A strength check reads the curve’s end. It asks where the bolt reaches its tensile resistance and what load that corresponds to, and the preload has moved neither, because the curve’s end is on the line an untightened bolt was on all along. Every term in the check belongs to the ultimate state, and the ultimate state has forgotten the preload.

A fatigue check reads the curve’s slope, at one point, and the point is decided by the mean load rather than by anything the strength check computed. It does not care where the curve ends. It cares only whether the working load sits on the flat part or the steep one, and the answer is binary in everything but a few tens of kilonewtons either side of the crossing.

Two checks on one drawing that share no quantity at all is unusual, and it explains why the two get confused. A designer’s habit is that a conservative assumption is conservative everywhere; here, assuming no preload is conservative for fatigue and exactly neutral for strength, while assuming the full preload is unconservative for fatigue and, again, exactly neutral for strength. There is no direction in which being wrong about the preload costs anything in strength, and every direction in which it costs a factor of a thousand in life.

The other thing a preload is for, and why it is not this #

Preload is far more often installed for shear than for tension, and the two mechanisms have almost nothing in common beyond the bolt.

A slip-critical joint uses the preload as a normal force, so its resistance is $\mu F_p$ and is directly proportional to what was installed. Lose a third of the preload and lose a third of the resistance. In tension, lose a third of the preload and lose nothing except the position of a crossing.

One is a strength that is proportional to the preload and the other is a stiffness effect that is invisible in strength, and they are routinely described with the same sentence — the bolt is tightened so the joint works better. A designer who carries the shear intuition into a tension joint expects a stronger connection and gets a quieter one; one who carries the tension intuition into a shear joint treats the preload as a refinement and loses the mechanism.

What the two springs cannot be asked #

The clamped stiffness is a cone, and a cone is a model. Rötscher’s construction is a closed-form integral over an assumed geometry with an assumed half-angle, and measured values scatter around it. The half-angle depends on the grip-to-diameter ratio and on whether the plies are the same material as the bolt.

The load is assumed to arrive at the bearing faces. It does not: an end plate is pulled at its own surface, somewhere inside the clamped stack, which reduces the effective load factor further. The correction is a factor between zero and one on Φ, it always helps, and it is omitted here.

Nothing above treats the contact as a distributed thing. Separation is drawn as an event, and a real interface opens progressively from its outer edge inward, so the transition between the two lines is a curve a few tens of kilonewtons wide rather than a corner.

And the prying force after separation is an ultimate-limit quantity while everything before it is elastic. The two halves of the curve are computed by different kinds of theory, joined at a point where neither is exactly right, which is the least satisfactory feature of the figure and is stated rather than hidden.

Bolt relaxation and flange creep are absent. Both reduce the preload over time and neither is in any of the arithmetic, so the crossing drawn is the crossing on the day of installation.

Still open: the column flange on the other side of the same bolt #

Every tee stub so far has been one flange with a bolt through it and something rigid behind. A real bolted moment connection has two flanges — the beam’s end plate and the column’s flange — and the same bolt passes through both, with a different thickness, a different bolt-to-web distance and possibly a different governing mode on each side.

The two are components in series, so the joint’s resistance is the lesser of the two and its stiffness is the harmonic sum. That much follows from the component method. What does not follow is what the clamped cone does when the stack it is spreading through is two plates of different thicknesses with a gap in the middle, or which side’s yield-line groups govern when the two sides group differently — and whether the prying lever on one side can be generated by a flange that has already folded on the other.