The wrench that reads the rust

A slip-resistant joint is designed on a force nobody can see. The four or sixteen bolts in a splice are tightened until each carries a preload of 171 kN or more, the plates are clamped together by the sum, and the joint carries its shear by friction on the faying surfaces for as long as that clamping force is there. The essay on how a preload gets into a bolt showed that the commonest method of putting it in, a torque, is the least reliable, because nine-tenths of the torque on a nut is spent on friction in the thread and under the nut, and the friction is a surface condition nobody measures.

That essay ended on a question it could not answer. Every method it ranked is verified by watching the force go in. Once the joint is closed and the steelwork has been painted, clad and loaded for twenty years, the tension in a bolt is invisible, and somebody — a surveyor before a change of use, an engineer after a fire or a fatigue crack nearby, an owner whose bridge has reached its first major inspection — needs to know whether it is still there. The obvious instrument is the one everybody has: a wrench on the nut. This essay is about what that wrench measures, and why on an old joint the answer is mostly the rust.

What holds a nut still #

Turn a nut in the tightening direction on a bolt that is already clamping something, and it does not move until the wrench exceeds the torque the nut is being held by. That holding torque is the same three-term bracket that governs tightening, because the nut cannot advance without doing the same three kinds of work — stretching the bolt by one pitch a turn, sliding the thread flanks and sliding the nut face:

$$T_b = F\left[\frac{p}{2\pi} + \frac{\mu_s d_2}{2\cos 30^\circ} + \frac{\mu_s D_{km}}{2}\right] + T_r$$

Two things differ from the tightening relation. The coefficient is the static one, $\mu_s$, because the nut has been at rest for years and has to be started rather than kept moving; static friction on a lubricated thread is typically a tenth or two above the sliding value. And there is a term that does not multiply the preload at all: $T_r$, the torque with which corrosion product, paint and grit in the thread hold the nut regardless of what the bolt carries. A bolt with no tension in it at all can have a nut that will not turn by hand.

The inspection practice built on this is simple. A small number of bolts from the same lot are tightened in a calibration device to the minimum preload, the torque that does it is recorded, and that torque — or a small multiple of it — is applied to the installed bolts in the tightening direction. A nut that does not move passes. Here the inspection torque is set at the minimum preload of an M20 grade 10.9 bolt, 171 kN, and the nominal sliding coefficient of 0.14: a bracket of 3.77 mm, and 647 N·m.

Read across at the dashed line. On a new joint the inspection torque is reached by bolts between 121 and 188 kN, which is the scatter of a static coefficient with a spread of fifteen per cent: a wide band, but one centred close to the preload it was calibrated at. The same check on the same joint twenty years later is reached by bolts between 39 and 110 kN. Nothing about the bolts has changed in that comparison. The line has rotated because the bracket has grown — a static coefficient of 0.30 gives a bracket of 7.63 mm against 4.28 mm new — and it has lifted because the rust holds the nut by itself.

The steepening alone does the damage. The torque is a product of two unknowns, and a wrench can only ever report the product. When the friction is known to within fifteen per cent, the product says something about the force. When the friction has doubled by an amount nobody measured, the product says almost nothing about either.

Pass and fail, as a curve #

An inspection does not read the breakaway torque; it applies one torque and records a binary result. The useful way to describe such a test is the chance that it passes, against the thing it is supposed to be testing — the operating characteristic that an acceptance plan for any manufactured product is judged by.

With the static coefficient lognormal about its median, the probability that a bolt carrying $F$ holds the inspection torque $T_i$ is the probability that its bracket is big enough:

$$P(\text{pass}\mid F) = P\!\left(\mu_s \ge \mu^*\right),\qquad \mu^* = \frac{(T_i - T_r)/F - p/2\pi}{d_2/(2\cos 30^\circ) + D_{km}/2}$$

which is one minus a normal distribution in $\ln\mu^*$. A perfect test would be a step: zero below the minimum preload, one above it.

The new joint’s curve is a reasonable test. It is not a step: a bolt exactly at the minimum passes only 83 per cent of the time, because a fifth of the static coefficients sit below the nominal sliding one, and a bolt at 188 kN — the 1.1 times the minimum a torque method aims at — passes 95 per cent. But it separates a tight bolt from a slack one cleanly. Nothing at half the minimum preload passes.

Then the joint ages, and the curve walks left. At one year the half-preload bolt still almost never passes. At five years it passes half the time. At twenty it passes four times in five, and the preload at which a bolt is equally likely to pass or fail is 66 kN — 38 per cent of the minimum preload the test was set to find. A joint whose bolts have lost most of their clamping force is accepted by the instrument that exists to reject it, and the instrument gives no sign that anything has changed: the wrench clicks at the same 647 N·m it always did.

The direction of the error is the part to dwell on. A test that drifts towards failing good bolts is a nuisance — it costs re-tightening and it is noticed. A test that drifts towards passing slack ones is silent. Every incentive on site points the same way, because a bolt that passes is a bolt nobody has to deal with.

How the threshold moves with the years #

The ageing in those curves is an assumption, and it is the weakest number in the essay. The static coefficient is taken to drift from 0.16 on a new lubricated thread towards 0.30, the rust torque from nothing towards 150 N·m, both with a time constant of five years, and the coefficient’s scatter to widen from fifteen to thirty-five per cent as corrosion makes each thread its own. No standard publishes the friction in a twenty-year-old thread. That absence is the argument: the inspection torque is calibrated against a quantity that exists only on the day of calibration.

Two things in this picture survive any reasonable choice of the ageing values. The first is that every line falls, because both the friction and the rust can only make a nut harder to start; there is no ageing mechanism that makes an inspection stricter. The second is that the fall happens early. Most of it is over by year ten, and by year five the line a bolt must clear to pass half the time has dropped from 151 to 86 kN. An inspection carried out at the first major survey of a structure is already reading a different instrument from the one calibrated at handover.

The values can be pulled apart to see which of them matters. Take the rust torque away altogether and leave only the friction’s drift: at twenty years the half-pass preload is 85 kN instead of 66, and the half-preload bolt passes half the time instead of four times in five. Take the friction’s drift down to 0.20, a thread that has merely dried out rather than corroded: the half-pass preload is 96 kN and the half-preload bolt passes 27 per cent of the time. No combination of plausible values restores the new joint’s curve, because the curve was set by the friction on the day it was measured, and every year after that day moves the friction one way.

The hand check, once #

The number at the centre of the second figure can be reproduced on the back of an envelope. The M20 has a pitch of 2.5 mm, a pitch diameter of 18.38 mm and a nut-face mean diameter of 27 mm, so the bracket is

$$b(\mu) = 0.398 + \mu\left(\frac{18.38}{2 \times 0.866} + \frac{27}{2}\right) = 0.398 + 24.1\,\mu \;\text{mm}.$$

At $\mu = 0.14$ that is 3.77 mm, and 171.4 kN times 3.77 mm is the inspection torque of 647 N·m. On the twenty-year joint the median static coefficient is 0.297 and the rust holds 147 N·m. A bolt at half the minimum, 85.7 kN, needs a bracket of $(647 - 147)/85.7 = 5.83$ mm to hold, which is a coefficient of $(5.83 - 0.40)/24.1 = 0.225$. That sits 0.82 standard deviations below the median on a logarithmic scale with $\sigma = 0.34$, and the normal distribution above $-0.82$ holds 80 per cent of its area. A bolt carrying half the clamping force the joint was designed on holds its nut four times in five because its thread holds nearly twice as hard as it did.

The same envelope shows why the converted reading is worse than the pass–fail result. An inspector who turns the wrench up until the nut moves and divides by the new bracket gets a number, and for the 66 kN bolt with median friction that number is $647/3.77 = 171$ kN — exactly the minimum preload. The bolt reports the right answer for the wrong bolt.

The washer that remembers the wrong day #

One of the four methods of installation leaves a witness behind. A load-indicating washer carries small protrusions on one face that are flattened as the bolt is tightened, so that the gap under the nut, measured with a feeler gauge, says how much force crushed them. It is the method the earlier essay ranked as the only one that measures the clamping force itself, and its scatter is the washer’s calibration rather than anybody’s friction. It is the natural thing to read twenty years later.

It reads a peak. The protrusions deform plastically, and plastic deformation does not recover when the force falls. Meanwhile the force does fall: the interfaces between the plates, the washer and the nut bed in under the first loading, and a coating on the faying surfaces creeps under the clamping pressure for years. Each of those is a shortening of the clamped pack, and the bolt and the pack are springs in series, so a shortening costs preload at the series stiffness.

The losses drawn are modest — a bedding-in of 0.015 mm and a coating creep of 0.008 mm a decade of time — and they still take 31 kN out of the bolt, because 688 kN/mm turns a few hundredths of a millimetre into a large fraction of a preload. After the first day the bolt has lost 12 kN; by the first year 24; by twenty years 31, and it has fallen below the 171 kN minimum within the first fortnight. Thicker coatings, more plies and softer packs make each of those numbers larger. Throughout, the washer under the nut reports 188 kN.

This is not a defect of the washer. It was designed to answer the question asked at installation — has this bolt been taken to its preload? — and it answers it permanently, which is useful: a washer with an open gap twenty years later shows a bolt that was never tightened, and that is a real finding. What the gap cannot show is relaxation, and relaxation is the thing a long-service inspection is usually looking for. A peak-hold gauge is a witness to the installation and to nothing after it. The column base whose bolts were retightened after each storm is the case where that matters most, because there the bolts stretch permanently in the first storm and the washers under them would say they were never anything but tight.

The direct measurement, and what it needs #

There is one way to read the tension in an installed bolt that bypasses friction altogether. An ultrasonic pulse sent down the bolt from its head and reflected from its far end returns after a time set by the bolt’s length and the speed of sound in steel, and both change with tension: the bolt is longer, and the stressed steel carries sound slightly more slowly. Gauges built on the principle resolve the change in length to about a hundredth of a millimetre, and the change is proportional to the force:

$$\delta = \frac{F\,L_b}{E\,A_s}$$

where $L_b$ is the stretching length, the grip plus roughly 0.8 of a diameter for the parts of the head and the nut that stretch. For the M20 at its minimum preload over a 40 mm grip, $L_b = 56$ mm and $\delta = 171{,}400 \times 56 / (210{,}000 \times 244.8) = 0.19$ mm — nineteen times the gauge’s resolution. As a measurement it is excellent.

It is excellent only relative to a reference. A pulse measures the bolt’s present length, and the preload is the difference between that and the length the bolt had before it was tightened. If that length was recorded — every bolt measured, numbered and logged at installation, as is done on wind turbine flanges and some bridge splices — the gauge reads the force directly, years later, through paint, with no wrench and no friction. If it was not, the reference has to be the bolt’s nominal length, and the doubt about a manufactured bolt’s actual length is of the order of half a millimetre: more than twice the stretch the whole preload produces on a short grip. The stretch only clears that doubt on grips over 130 mm, which is not where most splices are.

There are research methods that try to escape the need for a reference — measuring two kinds of wave whose speeds change differently with stress, so that their ratio reports the stress alone — and they show the shape of the problem rather than solving it. The measurement that works needs something written down on the day of installation. That is a statement about records, not about steel, and it puts the decisive step of a long-service inspection twenty years before the inspection.

A joint, not a bolt #

An inspection does not turn every bolt. A splice of sixteen bolts may be accepted on two, and the question becomes a sampling one: given that some bolts in the joint have lost their preload, what is the chance that the bolts chosen include one whose nut turns? The arithmetic is the hypergeometric draw that any acceptance sample uses, with the twist that a slack bolt drawn is itself only found with the probability the second figure gave it.

On a new joint the sampling is the only limitation, and it behaves as sampling does: two bolts in sixteen find a joint with four slack bolts a little under half the time, eight bolts find it nine times in ten, and turning every bolt finds it with certainty. On the twenty-year joint the curve for the same defect never gets there. Turning all sixteen bolts finds the joint three times in five, because each slack bolt drawn is itself passed four times in five. Sampling harder cannot fix a test that cannot see what it is sampling. Only a bolt that has lost three-quarters of its preload is still caught reliably on an old joint — and a quarter of the clamping force is a joint that slipped long ago.

The new joint’s curve carries a cost the old joint’s does not. Its good bolts, at 188 kN, fail the inspection torque one time in twenty, because the scatter of the static coefficient on a new thread reaches low values. Turning all sixteen bolts of a new joint therefore flags at least one good bolt 45 per cent of the time — a false alarm that costs a re-tightening and a report. On the old joint the same figure is under half a per cent. The ageing that blinds the test to slack bolts makes it quiet about good ones too, and a quiet inspection reads as a reassuring one.

Which free body the reading belongs to #

Cut the nut free with the wrench on it. On the cut through the thread act the flank forces, whose axial components sum to the bolt’s tension $F$ and whose tangential components are friction; on the cut through the washer act the bearing pressure, which also sums to $F$, and its friction; and in the thread, independent of $F$, whatever the corrosion product transmits by adhesion. Moments about the bolt’s axis are the whole of the breakaway equation, and the wrench is the only external torque.

Everything the inspection learns, it learns from that one equilibrium equation in one unknown torque. The unknowns inside it are three — the tension, the static coefficient and the rust torque — and the measurement fixes one combination of them. One equation cannot be solved for three unknowns, and the inspection procedure resolves it by assuming two of them have the values they had at calibration. On the day of calibration they do. The free body is exact; the assumption goes stale.

Cut the bolt’s shank instead and the ultrasonic reading’s free body appears: an axial force and an extension, related by the shank’s stiffness, with nothing else in the equation. That is why it works and why it needs the reference length — the extension is a difference of two lengths, and one of them belongs to a day in the past.

Where this sits in a design #

The slip check counts the preload at the minimum value and nothing else, and the essay that set out what a preloaded joint does showed why: below its slip load a friction joint carries shear without any movement, and once it slips it becomes a bearing joint with a different load path. A joint whose bolts have relaxed slips at a lower load, and the slip is permanent rather than catastrophic. That is why the consequences of an undetected loss of preload are usually serviceability — a splice that moves, a bridge expansion that jumps, a plate whose holes were made larger so the steel would fit and that now takes the whole clearance at once.

Fatigue is the exception, and it is the case where the invisible loss is expensive. A preloaded bolt in a joint carrying fluctuating tension sees only a small part of the tension’s range until the plates separate, because the clamped pack carries the rest; the prying essay found how strongly the bolt force depends on the flange’s flexibility, and preload is the other half of that shielding. Lose the preload and the bolt sees the whole range, and the detail category it was given assumed it would not. An inspection that passes a relaxed bolt in a fatigue-loaded joint has passed a bolt whose life is now a different number.

What an assumed twenty years of rust leaves open #

The ageing is assumed, and the assumption is labelled as such. The drift of the static coefficient to 0.30, the rust torque of 150 N·m and the five-year time constant are representative of a painted, weathered joint rather than measured on one. The direction of every result is independent of them, and the third figure is drawn so that the effect of each can be read off; the size is not.

The friction is one coefficient for two surfaces. The thread and the nut face age differently — the nut face is under the paint, the thread is open to water at the end of the bolt — and a two-coefficient bracket stays linear in each, so the argument survives with different numbers.

The relaxation is a two-term model. A bedding-in at the first loading and a coating creep that grows with the logarithm of time is the shape the measurements on coated faying surfaces show; the magnitudes depend on coating thickness, number of plies and the clamping pressure, and are inputs here.

The scatter is independent from bolt to bolt. In a real joint the bolts share a coating, a paint system and a weather, so their frictions are correlated, which makes a sample less informative than independent draws — the sampling figure is an upper bound on what a sample can find.

Still open: whether a joint can be re-qualified by its own slip #

Every method here tries to read the tension in a bolt. The quantity the design actually relies on is the joint’s slip resistance — the clamping force times the coefficient on the faying surfaces — and that can in principle be tested directly, by loading the joint in shear and watching for the first slip. A proof load to the design slip load on a whole joint is rarely possible in a standing structure, but a small fraction of it, with a displacement gauge across the joint, can show whether the joint is still on the stiff, pre-slip branch of the load–slip curve. Whether a partial proof load — a passing lorry of known weight, a jacked load across a splice — can bound the clamping force from below without ever reaching it, by how stiff the joint is rather than by whether it slips, and whether the stiffness of a friction joint below slip depends on the preload strongly enough to be read at all, is the question that would turn a test of each bolt into a test of what the bolts are for.