The detail decides and the steel does not

Take a bridge detail — a transverse stiffener welded to a girder flange — and look up its fatigue strength. It is a number like 80, meaning 80 N/mm² of stress range survives two million cycles.

Now ask what steel it is in. The question has no answer, because the number does not depend on it. S235, S355 and S690 all give 80.

That is a genuinely odd property for a strength to have, and it is the whole of this essay.

Which free body produced the number #

None, and the reason is that fatigue is not a strength problem at all. It is a crack growth problem.

A weld contains, from the moment it is made, small defects at its toe — undercuts, slag inclusions, lack-of-fusion, and the sharp geometric notch where the weld meets the parent metal. Those are the initial cracks. Each cycle of stress extends them a little, and the rate at which they extend is governed by the stress intensity at the crack tip, which depends on the stress range and the crack’s own size.

Nothing in that description mentions a yield stress. What matters is:

• the stress range $\Delta\sigma$, because the crack opens and closes through it;
• the geometry of the notch, because it sets the initial stress intensity;
• the size of the initial defect, which the welding process fixes;
• and $E$, which is the same for every steel.

Integrate the crack growth law from the initial defect to failure and the result is $N\Delta\sigma^m =$ constant with $m = 3$ — the slope on the plot, which is a property of the growth law and not of the material.

The hole that multiplies the stress is the static version: a stress concentration that a ductile material redistributes around and largely ignores. Fatigue is where the same concentration matters, because a crack does not redistribute.

Why a stronger steel is worse #

Follow the consequence and it goes somewhere uncomfortable.

Design a member in S355 for a static load and it comes out at some size. Design the same member in S690 and it comes out roughly half the area, because the strength has doubled. Same load, half the section, so twice the stress range.

Twice the range is eight times the damage, so the member in the stronger steel has an eighth of the life.

Specifying a higher grade for a fatigue-loaded member makes it worse, and the mechanism is that the grade changes the size and the size changes the range while leaving the fatigue curve exactly where it was. That is why high-strength steels are used freely in buildings and cautiously in bridges and cranes, and why the members that are fatigue-governed in a structure are frequently the ones somebody optimised.

The general form of it is worth stating: a fatigue check is a check on a stress range, and any decision that reduces a section increases a stress range. Strength, optimisation, refinement of the analysis, a more accurate load model — all of them make a member smaller, and all of them shorten its fatigue life.

The detail category is a shape #

If the material does not decide the number, what does?

The detail category is a classification of the joint’s geometry, and it runs over about a factor of four in stress range and therefore a factor of sixty in life. A plain rolled section with no attachments is category 160 — the case the weld that is stronger across never has to consider, because there is no weld. A transverse butt weld ground flush is 112. A transverse stiffener fillet-welded to a flange is 80. A cover plate ending on a flange is 50 or below. A cruciform joint with a partial penetration weld is 36.

Every step in that list is a change of shape — how abruptly the section changes, how sharp the notch at the weld toe is, whether the weld is ground, whether the load passes through the weld or beside it.

That is a very different kind of design variable from the ones this collection usually deals with. A designer choosing a section chooses a number from a table; a designer choosing a detail category chooses a drawing. The variable is the shape of the joint, and improving it means grinding the toe, changing the termination, moving the attachment, or eliminating the weld entirely — none of which is a quantity in a calculation.

There is a second way to see why the material drops out, and it is worth having because it explains the one exception.

A fatigue life has two stages: initiation, during which a crack forms where none existed, and propagation, during which it grows. Initiation depends strongly on the material — a harder, stronger, cleaner steel takes very much longer to nucleate a crack — and propagation barely does.

A welded detail has no initiation phase. The crack is already there when the weld cools, in the form of the defects at its toe, so the entire life is propagation and the material’s contribution has been skipped. A plain machined component with no welds does have an initiation phase, and for those the material does matter — which is why an S–N curve for a polished specimen orders steels by strength and one for a welded detail does not.

That is the exception, and its practical form is that the highest detail categories, where there is no weld, are the only ones with any material dependence in them at all. Everything below about 125 is a weld, and below a weld the steel is a spectator.

Two per cent of the traffic, most of the damage #

The cube law has a second consequence, and it is the one that decides how a fatigue calculation is actually done.

A structure’s load history is a spectrum: many small cycles, some medium ones, a few large. Miner’s rule sums the damage as $\sum n_i/N_i$, and $N_i$ goes as $\Delta\sigma_i^{-3}$ — so the damage is a cube-weighted sum.

The heaviest band — a full vehicle, two per cent of the crossings — carries most of the damage. The lightest — cars, a third of the crossings — carries none at all, because its range is below the cut-off.

Which means a fatigue assessment is dominated by the tail of the traffic distribution, and the tail is the part that is measured worst. The train that is worse than its heaviest axle makes the neighbouring point about static effects: what matters about a load spectrum is rarely the largest member of it, and here it is not the largest but the largest-cubed weighted by how often it comes. A weigh-in-motion survey characterises the average vehicle well and the rare overloaded one badly, and the rare one is the answer.

It also explains the equivalent constant range. Reduce a spectrum to a single number that does the same damage in the same number of cycles and the answer is

$$\Delta\sigma_e = \left(\frac{\sum n_i\Delta\sigma_i^3}{\sum n_i}\right)^{1/3}$$

a cube-weighted average, which sits far closer to the largest band than to the mean. A designer handed that number and told it is “the equivalent range” may reasonably assume it is an average of some kind; it is, and it is an average that puts eight times the weight on a cycle twice the size.

What a category is worth, in years #

It is worth converting the classification into the units the decision is taken in, because the factor of four in stress range is not the number anybody feels.

Two details on the same girder, carrying the same stress range of 62 N/mm² under the same traffic. One is a plain rolled flange with a bolted splice, category 112. The other has a cover plate welded across it, category 50.

The lives differ by $(112/50)^3 = 11$. A detail with a design life of 120 years becomes one with a design life of 11, on the same member, in the same steel, under the same load — because somebody welded a plate to a flange.

That is the real content of a detail category and it is why fatigue design is so unlike the rest of structural engineering. There is no gradual trade here: the categories are steps, the law is cubic, and moving between two adjacent entries in the table is worth a factor of two or three in life. A change that a static calculation would not notice at all — a stiffener 20 mm longer, a weld returned round a corner instead of stopping short, an attachment welded rather than bolted — changes the answer by an order of magnitude.

The design response is a hierarchy that has nothing to do with sizing. Avoid the attachment; if it must be there, move it away from the peak stress; if it must be at the peak, improve its termination; and only then make the member bigger. The last of the four is the only one a calculation naturally produces, and it is the least effective.

Where the model stops #

The cut-off is not a cliff. Below the constant-amplitude limit a cycle does no damage on its own; in a variable spectrum it does, because larger cycles have already grown the crack and small ones then extend it. The rules handle this with a second, shallower slope below the knee, which is a fitted compromise rather than a mechanism.

Residual stresses were ignored and they are the reason the mean stress does not matter. A welded detail has a residual tension at yield along the weld, so the whole applied cycle happens in tension whatever the applied mean is — the stress that was there before the load — and that is why a fatigue rule for welded details uses the range alone and one for plain material does not.

Nothing here is about inspection. A fatigue crack is detectable for some fraction of its life, and the design decision that follows — whether a detail is inspectable, and whether the structure survives its failure — belongs to the structure that survives losing a member rather than to the S–N curve.

Nothing here is about temperature or environment. A detail in seawater has a shorter life than the same detail in air by a factor of two or more, because the corrosion opens the crack and removes the cut-off entirely. A detail at low temperature can fail by brittle fracture from a crack that fatigue grew, which is a different failure with a different criterion — the flaw that sets the strength is where the crack becomes a fracture problem rather than a growth one.

And the categories are a classification, not a measurement of the joint that will be built. A detail is assigned to a category by matching it to a picture in a table, and two joints in the same category can differ by a factor of two in life according to how they were welded.

Where the check belongs in a design #

One last practical point, and it is about sequence rather than about mechanics.

A fatigue check is made on a detail, and details are drawn late. The member sizes are fixed in the scheme, the analysis is run, the connections are designed, and the fabrication drawings — which is where a stiffener’s termination and a weld’s return are actually decided — are produced by somebody else, months later, from a specification.

So the design variable that dominates the answer is set after the answer has been checked. The profession’s response is to specify the category rather than the geometry: the drawings say “category 80 minimum at this location”, and the fabricator’s detail has to satisfy it. That works, and it depends entirely on the requirement having been written down, which requires the designer to have known at scheme stage which locations would be fatigue-governed.

The alternative is worse and is common: the fatigue check is made on an assumed category, the assumption is not recorded, and the joint that gets built is category 50 where 80 was assumed. Nothing in the built structure looks wrong, no inspection finds it, and the life is an eighth of what was calculated. It is the same shape of problem as the dimension nobody can measure: a quantity that decides the answer, is set by somebody outside the calculation, and leaves no trace in the finished work.

The generalisation #

The habit worth taking away is to notice when a check has no material property in it, and to ask what that implies about which decisions matter.

A fatigue check depends on a geometry and a load history and on nothing else. So every decision that improves it is a geometric decision: a detail change, an attachment moved, a weld ground, a termination radiused. And every decision about the material — a higher grade, a tougher steel, a better certificate — is worth exactly nothing, except insofar as it changes the size of the member, which makes things worse.

There is a companion habit for reading load spectra. Whenever damage accumulates as a power of a load, the design question is about the tail of the distribution and not about its middle, and the higher the power the further out the answer lives. A cube law puts it at the heaviest two per cent; the twenty-first power in the load that was left on too long puts it at a single load state; a linear law would put it at the mean. The exponent says which part of the data collection actually matters, and it is usually the part nobody funded.

That inverts the usual hierarchy. In a static design the material is a lever and the detail is a consequence. In a fatigue design the detail is the lever and the material is not one at all.

Geometry beats material is the theme, and this is its purest instance in the collection: a design problem in which the material appears in the answer only through $E$, which is the same for every grade, and in which the whole of the available design freedom is in the shape of a weld that will be made by somebody who has never seen the calculation.