The load that never came near failing anything

Every capacity on this site is a comparison between one number and another: this stress against that strength, this load against that collapse load. Pass the comparison and the member is adequate, and adequacy is a property that does not expire.

Fatigue is not like that in any respect. There is no capacity to compare against, the quantity that governs is not a stress but a difference of stresses, the material’s strength does not enter, and adequacy is measured in cycles rather than in margin. A member that passes every static check by a factor of three can be decided entirely by a weld that appears nowhere in the strength calculation.

Why the range and not the stress #

A member cycled between 100 and 150 N/mm² and a member cycled between 0 and 50 have the same fatigue life. That is a startling claim on first hearing and it is very nearly true for a welded detail.

The reason is that a fatigue crack grows when it opens, and whether it opens depends on the change in stress rather than on its absolute level. And in a welded joint the absolute level is not what the applied loading says it is: the residual stress along a weld line is at yield, so the material at the toe is already sitting at the top of its stress range before anything is applied. Adding an applied tension does not move it up; it can only be moved down by an applied compression, and the crack tip experiences the full applied range as a range regardless of where the applied stresses nominally sit.

This is why fatigue design of welded structures uses the range and ignores the mean, and it is also why the same rule does not apply to a plain unwelded member or to a bolt, where the mean stress matters and has to be carried explicitly.

Why the material is nowhere on the plot #

The second startling claim is that a fatigue calculation contains no material property at all. The same detail in S275 and in S460 lies on the same line.

The reason is that fatigue life in a welded structure is almost entirely crack growth, and crack growth is governed by the stress intensity range $\Delta K = Y\Delta\sigma\sqrt{\pi a}$ — the same quantity as in the fracture essay, applied to a crack that is advancing a little on every cycle rather than running. The growth rate follows Paris’s law, $da/dN = C(\Delta K)^m$ with $m$ about 3, and the constants $C$ and $m$ are very nearly the same for all structural steels.

They are the same because crack growth is a process at the tip that does not much care how hard it is to move a dislocation somewhere else. Strengthening a steel changes its yield stress, which changes when it will yield, which changes nothing about how fast a crack advances per cycle at a given $\Delta K$.

And the slope of three on the S-N plot is not a fit. Integrating $da/dN = C(\Delta K)^m$ from an initial flaw to a critical one gives a life proportional to $\Delta\sigma^{-m}$, so the exponent on the S-N line is the exponent in Paris’s law. A log-log plot with slope $-1/3$ is a statement about crack growth wearing a design curve’s clothes.

What the categories are #

The category is set by a stress concentration nobody can compute. The toe of a fillet weld is a sharp re-entrant corner whose radius depends on the welder, the position, the electrode and the day; the same is true of the root, the end of a cover plate, and the profile of a butt weld’s cap.

So the profession stopped trying to compute the concentration and started classifying instead. A detail is assigned to a category by matching it against a catalogue of tested configurations, and the category carries the concentration, the local residual stress and the typical flaw size all folded together into one number that was obtained by testing that geometry to destruction many times.

That is a genuine retreat from calculation to classification, and it is worth noticing how unusual it is on this site. Almost everything else here is computed from stated principles. Fatigue design is done by looking a detail up in a table, and the reason is that the geometry that decides it is not knowable to the precision the calculation would need.

Which free body produced the number #

The free body for a fatigue crack is the same one as for a static fracture — the region ahead of the tip, with an energy balance across its boundary rather than a force balance. What is different is that the balance is not being tested against a critical value once; it is being evaluated on every cycle, and the crack advances by a tiny amount each time.

Integrating that gives the life, and the integration explains a feature of the curves that looks arbitrary. Most of the life is spent while the crack is small: $da/dN$ goes as $a^{3/2}$, so a crack takes far longer to grow from 0.1 mm to 1 mm than from 1 mm to 10. A member is therefore nearly all of the way through its fatigue life before its crack is large enough to find, which is the single most awkward fact about inspecting for fatigue.

The two slopes on each curve come from the same integration with a threshold in it. Below a stress intensity range of about 2 MPa√m a crack does not grow at all, so ranges below the corresponding stress do no damage on their own — hence the constant-amplitude limit, at 66.3 N/mm² for category 90. Under a variable spectrum that limit is not available, because a crack grown by the larger ranges is long enough that the small ones exceed the threshold too. That is why the slope changes to five past the knee rather than the line simply stopping, and it is a rare instance of a design curve whose kink is a mechanism rather than a compromise.

Miner’s rule, and the thing it assumes #

A real structure sees a spectrum of ranges, not one. The rule that reduces it to a number is Palmgren’s and Miner’s: each cycle at range $\Delta\sigma_i$ consumes a fraction $1/N_i$ of the life, damage adds, and failure is at $\sum n_i/N_i = 1$.

Worked on a spectrum of 5,000 cycles at 120 N/mm², 50,000 at 80, 500,000 at 50 and 5,000,000 at 30, on category 90: the damages are 0.0059, 0.0176, 0.0244 and zero, summing to 0.048. The structure could take about twenty-one such spectra, and the equivalent constant range doing the same damage is 34.7 N/mm².

Two things in that calculation deserve suspicion.

The five million cycles at 30 N/mm² contribute exactly nothing, because 30 is below the cut-off. That is the rule taking the threshold seriously, and it is the assumption most likely to be wrong: those cycles do no damage on their own and they do plenty once a crack exists.

And the sum contains no information about order. Miner’s rule says that a thousand large cycles followed by a million small ones does the same damage as the reverse, and it does not: a large cycle early grows a crack that the subsequent small ones can then extend, while the same large cycle late arrives at a member that has already spent its life. Measured sums at failure scatter between about 0.3 and 3, and the rule’s value is that it is simple and roughly unbiased rather than that it is right.

Where the model stops #

The curves are for welded details and much of the interesting behaviour is elsewhere. Bolts in tension, cables, reinforcement in concrete, and plain material all have their own treatments, and for all of them the mean stress matters in a way it does not here.

Low-cycle fatigue is a different phenomenon with the same name, and it is governed by the plastic strain range rather than by anything on these axes — the regime a section taken past yield on every cycle is in. Everything above concerns hundreds of thousands to hundreds of millions of cycles at stresses well inside the elastic range. A structure taken plastic on every cycle — by an earthquake, or by ratcheting — fails in tens or hundreds of cycles by a mechanism governed by plastic strain range rather than stress range, and none of the curves here applies to it.

Corrosion removes the endurance limit. In seawater or in any aggressive environment the threshold disappears, the curve continues downward without a knee, and there is no stress range that is safe indefinitely. Offshore structures are designed on curves with the limit deleted for exactly this reason.

And the whole apparatus is calibrated on details that were made properly. A detail’s category assumes the weld has the profile and the flaw population of the tested specimens. A weld with lack of penetration is not a category-56 detail with a slightly shorter life; it is a member with a crack in it, and the calculation that applies is the fracture one.

What the picture cannot show #

The plot has no scatter on it, and fatigue is the most scattered quantity in this field. Each category line is a mean-minus-two-standard-deviations fit to test data whose spread at a given stress range is typically a factor of two either side in life. A predicted life of ten million cycles means something closer to “somewhere between three and thirty million, probably”.

Nor does it have a crack size on it. The axis says “cycles to failure”, and what counts as failure in the underlying tests is usually a through-thickness crack in a small specimen. A real member with a through-thickness crack is not necessarily finished — it may leak, or be found, or redistribute to a member beside it — and damage-tolerant design is built on exactly that gap between a crack appearing and a structure failing.

And the working range drawn as a horizontal line is a fiction. A real structure sees a spectrum, and reducing it to one number requires either Miner’s rule or an equivalent range, both of which are on this page and neither of which is exact.

The generalisation #

The pattern is that the quantity that decides a structure is not always the quantity it was designed against, and fatigue is the case where the two share almost nothing.

A static design proceeds by sizing members: a bigger section is a stronger member, and the answer to a failed check is more material. A fatigue design proceeds by choosing details: the answer to a failed check is a better weld, a ground toe, a different attachment, a load path that avoids a transverse fillet. Adding material helps only by reducing the stress range, which enters at the cube — so a 25% reduction in range doubles the life, and a change of detail category from 56 to 90 multiplies it by four for no material at all.

That inverts the whole economics of the design. In a fatigue-governed structure the drawings matter more than the schedule, the fabricator’s practice matters more than the grade, and the most valuable thing a designer can do is remove an attachment rather than thicken a member.

It is worth noticing how differently the two calculations behave. The fracture length is a state: a member either has a crack that long or it does not, and the answer changes only if the crack changes. The fatigue life is a budget: it is consumed by every cycle and cannot be replenished, and a member half way through it looks exactly like a new one. Almost every failure mode on this site is of the first kind, and this is the only one of the second.

A surprising place this turns up #

The structures that fail in fatigue are almost never the ones carrying the largest loads. They are the ones carrying the most cycles, and the two are unrelated.

A bridge is designed for a load that moves, and the influence-line machinery for finding the worst position of that load is entirely a static argument — it locates the largest moment a vehicle can produce. The fatigue question is different: not how large the worst event is, but how many events there are and how big the swing is between a loaded and an unloaded deck. A short-span bridge is worse than a long one for fatigue while being better for strength, because every axle produces a full cycle on a short span and a long one sees one cycle per vehicle.

Cranes, crane runway girders, machine supports, wind-loaded masts, chimneys shedding vortices, offshore jackets under wave loading: the list of fatigue-critical structures is a list of things that move or are moved on, and it barely overlaps with the list of things carrying the largest loads. The floor of a building — the archetypal structural problem — is nearly never a fatigue problem, because the load arrives once and stays.

Where the ladder goes next #

Later rungs on this anchor: Paris’s law integrated properly, with an initial flaw size and a critical one, to produce a life rather than a category. Crack growth as the basis of inspection intervals, which is what makes damage-tolerant design possible. The detail categories derived, and the test programmes behind them. Mean stress effects and the Goodman diagram, for the unwelded cases where they matter. Rainflow counting, which is the algorithm that turns a measured stress history into a spectrum and is far less obvious than it looks. Low-cycle fatigue and the Coffin-Manson relation. Corrosion fatigue and the loss of the endurance limit. Improvement techniques — toe grinding, hammer peening, TIG dressing — and how much each is worth. And thickness effect, where a thicker plate of identical detail has a lower category, for the same reasons that made it a worse fracture prospect.

Historically the subject is the oldest of the failure modes in this field and was recognised before any of the others. Wöhler’s tests on railway axles in the 1860s established both the stress-range dependence and the existence of a limit, on a component that had been failing repeatedly at loads it obviously carried. The word fatigue comes from that period and carries an obsolete theory with it — the idea that the material became tired, its structure somehow deteriorating with use. Nothing deteriorates. A crack grows, and it grows by an amount that has been computable since the 1960s.