The torque that goes into the thread

A slip-resistant joint of four M20 grade 10.9 bolts is designed on a preload of 171 kN a bolt, and every number downstream of that — the friction plateau, the margin to bearing, the cost of an oversize hole — is that preload multiplied by something. On the drawing the preload is a force. On site it is a number stamped on a torque wrench.

The essay on what preloading does lists the four ways the force gets into a bolt and ranks them, torque last, with a scatter of about thirty per cent. That is the right ranking. This essay is about why, because the reason is a single piece of statics and it decides which of the four a designer can believe.

Three things a torque does #

Turn a nut on a bolt that is already clamping something and three kinds of work are done at once.

The nut advances along the thread by one pitch a turn, and that advance, against the tension in the bolt, is the work that stretches it. The flanks of the nut’s thread slide over the flanks of the bolt’s, pressed together by the same tension, and that is friction. And the face of the nut slides round on the washer or the plate beneath it, pressed down by the same tension again, and that is friction too.

Each of the three is a force times a lever, and each force is proportional to the preload $F$, so the torque needed is the preload times a bracket of three lengths:

$$T = F\left[\frac{p}{2\pi} + \frac{\mu_t\,d_2}{2\cos 30°} + \frac{\mu_h\,D_{km}}{2}\right]$$

The first length is the thread’s lead divided by a full circle: for an M20 with a 2.5 mm pitch it is 0.40 mm. The second is the thread’s friction at the pitch diameter $d_2$ of 18.4 mm, enlarged by the 60 degree flank that presses the surfaces together harder than a flat thread would: at a coefficient of 0.14 it is 1.49 mm. The third is friction under the nut at the mean diameter of its bearing face, 27 mm: 1.89 mm.

Add them and the bracket is 3.77 mm, so a preload of 188 kN needs 711 N·m. Of that bracket, 0.40 mm is doing what the torque is for. The other 3.38 mm is spent sliding metal on metal, 39 per cent of the torque in the thread and 50 per cent under the nut.

That is the figure’s whole content, and its shape carries the argument. The lead’s share is not a fixed tenth. It is the lead divided by a bracket whose other two terms are proportional to a coefficient nobody measured on these particular bolts: 22 per cent if the thread happens to be at 0.06, 6 per cent if it is at 0.24. Whatever the coefficient turns out to be, the same torque is divided up differently, and only one of the pieces is preload.

A bolt has to be a bad machine #

It is tempting to treat the nine-tenths as waste — something better lubrication could remove. Some of it can be removed, and not most of it, and the reason is worth knowing because it is the same result that governs a screw jack.

A bolt carrying a preload is a screw carrying a load. Take the wrench away and the tension in the bolt pushes on the flanks of the thread, and the flanks are inclined, so the load is trying to turn the nut back off. It does not, because the thread’s helix angle — 2.5 degrees on an M20 — is far inside the friction angle of the flanks, about nine degrees at a coefficient of 0.14. The bolt is self-locking, and it must be: a preloaded bolt that backed itself off would unload its joint the first time the structure vibrated.

A self-locking screw cannot be more than half efficient, and the argument that proves it is the argument here. The friction that makes a screw hold its load when the effort is removed is the same friction the effort has to overcome to raise it. There is no coefficient that keeps the first and removes the second. So the wasted torque is not a defect of the bolt; it is the price of the bolt staying tight, and a bolt that converted most of its torque into preload would be one that undid itself.

That settles what lubrication can and cannot do. It can lower both friction terms and so raise the share of the torque that becomes preload, and bolt makers apply it for exactly that reason. What it cannot do is make the answer independent of the coefficient, because the coefficient is two-thirds of the bracket at any lubricated value a nut will stay on with.

One torque, and a range of preloads #

The specification therefore fixes a torque, and the torque fixes the product of the preload and the bracket. The preload is what is left when the bracket turns out to be whatever it is.

The curve is a hyperbola in the coefficient, and its steepness is the point. The torque was deliberately set ten per cent above the minimum preload, which is the ordinary way to leave a margin. At the dry end of the range that margin is consumed and overdrawn: from a coefficient of 0.156 upward the bolt carries less than the 171 kN the slip calculation credited it with, and at 0.18 it carries 150. At the lubricated end the same torque would put 256 kN into the bolt if the bolt could hold it, and it cannot — it yields at 220 kN while the wrench is still on it.

So one torque on one bolt specification delivers anything from 12 per cent below the minimum to the bolt’s own yield force, and nothing about the installed bolt looks different. The joint’s friction plateau follows the preload down.

This is where the scatter of a friction coefficient enters a calculation twice without anybody noticing. The faying surface’s coefficient multiplies the preload to give the slip resistance, and the thread’s coefficient divides the torque to give the preload, and the two are different surfaces with different scatter, both absent from the finished joint.

Why the procedures look the way they do #

None of this is news to the people who write tightening procedures, and the shape of those procedures is a direct response to the bracket.

The torque is not taken from a table for a real job. A torque procedure worth the name establishes the relation between torque and tension on bolts from the actual delivery batch, in a tension-measuring device, under the lubrication condition they will be installed in. That calibration is measuring the bracket, and it is only valid for as long as the bracket stays what it was on the day — the same lubricant, the same surface finish, the bolts kept dry and clean between the box and the joint.

A torque procedure also targets above the minimum, by an amount related to the scatter the calibration found, because a characteristic value of a preload is a fractile of a distribution like any other and the target has to sit high enough that the low tail clears it.

And a torque procedure is where most inspections stop. An inspector with a wrench checks that a nut does not turn at the specified torque — which measures the torque needed to overcome the friction the nut now has, and says very little about the tension in the bolt behind it.

The turn, and why yielding is the point #

Turn-of-nut replaces the torque with an angle. Bring the joint to snug, so the plates are in contact, then turn the nut a specified further fraction of a revolution.

An angle is a length. Each degree advances the nut by a 360th of the pitch, and that advance is shared between stretching the bolt and squeezing the clamped plates, two springs in series. With a 40 mm grip and plates three times as stiff as the bolt, every degree adds 4.78 kN of preload. Friction does not appear anywhere in that, which is the method’s whole advantage over torque.

Its disadvantage is the word snug. Snug is a condition, not a position — the nut run up by a spanner until the plates are in firm contact — and a snug nut is somewhere within a range of angles of where another person’s snug nut would be. Twenty degrees of doubt about snug is ninety-six kilonewtons of doubt about preload, if the bolt stays elastic.

The specified turn does not keep the bolt elastic. It is chosen to take the bolt well past yield, onto the nearly flat part of its force–elongation curve, and there the twenty degrees of doubt about snug move the preload by five kilonewtons rather than ninety-six. The method is accurate because it deliberately puts the bolt where extra rotation produces almost no extra force.

That is only possible because the bolt steel is ductile enough to be taken past yield in service and stay there, which is the property no design equation contains doing essential work in a component whose drawings never mention it. It is also why a bolt tightened this way is not reused: it has spent part of its elongation capacity being installed, and the next installation would spend more.

Four methods on one axis #

The four intervals are four answers to one question — what fixes the preload — and the answers are different kinds of thing.

For the torque, the preload is fixed by friction in the thread and under the nut, two coefficients of a surface condition. For the turn, it is fixed by the bolt’s own yield strength, a material property with a narrow and well-documented scatter. For the washer, it is fixed by the calibration of a small part designed to flatten at a known force, which measures the clamping force itself and can be checked afterwards with a feeler gauge. Only one of the four is controlled by something that varies from joint to joint on site, and it is the one most often used.

The ranking is not the whole story, because the turn’s narrow interval sits above the yield force of the bolt. A designer reading the chart as “turn-of-nut delivers more preload” has read it correctly and should not credit the extra: the slip check uses the minimum preload whatever the method, and what the method changes is how far above that minimum the real bolts are. For the torque method the answer is sometimes below it.

The preload above the minimum is real and not credited #

The slip check is written on the minimum preload whatever method delivers it, and the chart above says what that leaves out. Take the four-bolt joint from the start of this essay and install its bolts by turning the nuts rather than by torque.

On paper the joint slips at 342 kN, using 171 kN a bolt, which is ninety-five per cent of its bearing capacity: a joint whose serviceability limit is one small step below its collapse load. The bolts actually in it put the friction plateau above the bearing capacity, so there is no serviceability limit state left to reach — the joint cannot move before it has to fail in bearing. The same drawing describes two joints, and which one was built depends on a word in the tightening specification.

That asymmetry is the practical content of the four intervals. A method whose preloads lie entirely above the minimum delivers a joint better than its calculation. A method whose preloads straddle the minimum delivers some joints worse than their calculation, in an unknown proportion, with nothing to tell them apart. Only the second kind of error is a problem, and it is the one torque control has.

It also matters away from slip. A preloaded joint carrying a fluctuating tension shields its bolts from the range of that tension until the plates decompress, and the fatigue category a detail is given assumes the shielding is there. A bolt installed at the dry end of the torque’s range decompresses at a lower applied load than one installed by turning, and past decompression the bolt sees the whole of the range. The preload’s scatter is therefore a scatter in fatigue life as well as in slip resistance, and the fatigue version is invisible for decades rather than for a load event.

The bolt that is a wrench #

A tension-control bolt looks like a fifth method. It has a splined extension beyond its thread, the installation tool holds the spline and turns the nut, and when the torque reaches a value set by a groove machined into the spline, the spline shears off. No wrench is calibrated and the sheared spline is visible evidence that the bolt was installed.

It is evidence that a torque was reached. The bolt is the torque method with the wrench built in, and it carries exactly the same bracket.

What a tension-control bolt improves is the coefficient, not the physics. The bolts come from the factory with a controlled lubricant on the thread and under the nut, so the bracket is narrower than on bolts lubricated by whoever opened the box. The advantage lasts exactly as long as the lubricant does. A box left open in the rain, bolts left in the steelwork overnight before installation, and a thread that has picked up grit are all a return to the site range, with the spline still shearing at the same torque and the same visible evidence that everything was done correctly.

The free body a torque is applied to #

Cut the nut free. Across the cut into the bolt’s thread pass the flank forces, whose axial components add up to the preload and whose tangential components are the thread friction. Across the cut into the washer pass the nut’s bearing pressure, adding up to the preload again, and the friction round the nut face. The wrench applies the torque. Moments about the bolt’s axis give the bracket.

Cut the bolt’s shank and something else is visible: while the wrench is on, the shank carries not only the preload but the torsion from the thread friction, which has to pass down the shank to the head. A shank in combined tension and torsion yields by a criterion that combines the two, the same combination a von Mises check makes, so a bolt being tightened reaches yield at a lower tension than a bolt being pulled. When the wrench comes off, most of that torsion relaxes and the tension it was costing is recovered. This is the reason the torque curve above caps at the bolt’s yield force as an idealisation rather than a precise limit: the real cap during tightening is somewhat lower, and it moves with the thread friction too.

What the bracket leaves out #

The two coefficients are taken as equal. They are separate surfaces — thread flanks and nut face — and on a real bolt they differ. The bracket stays linear in each, so the argument survives; the numbers move.

The thread is an idealised 60 degree form. The standard relation uses the pitch diameter and ignores the detail of the thread profile and the load distribution along the engaged turns. It is accurate to a few per cent, which is small beside the effect it is being used to show.

The joint stiffness is a single ratio. Clamped plates three times as stiff as the bolt is a representative figure for a short steel grip. Thin plates, packs and coated faying surfaces are softer, which lowers the preload per degree and moves the angle at which the bolt yields.

The plastic branch is a straight line. The bolt’s hardening past yield is taken as a small constant slope, capped at its tensile strength. Real grade 10.9 bolts harden very little and have limited elongation, which is precisely why the turn is limited and why the method is not used on long grips without testing.

Nothing happens after tightening. Embedment, relaxation and creep in coatings take preload away over hours and years, and the essay on preloading follows them; a preload that was delivered accurately is only the starting point for how much of it is still there.

Still open: whether a preload can be checked once the joint is closed #

Every method above is a way of putting a force into a bolt, and every one of them except the washer is verified by watching it go in. Once the joint is closed, the tension in the bolt is invisible, and the obvious check — a wrench on the nut — measures the friction the nut now has rather than the force behind it, which is the very quantity this essay has spent itself showing to be unreliable. A washer’s gap can be read afterwards. A bolt’s elongation can be measured ultrasonically, from the time a pulse takes to travel its length and back, which is a direct measurement of the thing wanted and needs the bolt’s length to have been recorded before it was tightened. Whether any practical inspection can establish the preload in a joint that has been in service for twenty years, without that record and without taking the joint apart, is a question about what the bolt does after the load arrives as much as about how it was installed.