Fatigue cracks — count the inspections that land in the window

Drag upward to raise the stress range the joint sees; drag down to lower it. The arrow keys do the same, a step at a time. End goes to the heaviest duty; Home to the lightest. The marks along the bottom are inspections, one a year. The ones that fall inside the shaded band are the only chances anybody ever gets. Sound is off until you turn it on.

What you are looking at

A crack in a steel joint grows a little on every load cycle. How much depends on how hard the crack tip is being worked, which depends on the stress range and on how deep the crack already is. Both of those raise the rate, so growth accelerates: the crack spends a long time being small and a short time being large. That is the shape of the curve on the right, and it is why a crack that has been harmless for years is not reassuring.

Two sizes matter. There is the depth at which an inspector could actually find it — below that, it is there and nobody can see it. And there is the depth at which the remaining section can no longer carry the load. The span between them is the entire opportunity to do anything about it.

That span is measured in cycles, and it falls as the cube of the stress range. Doubling the stress range does not halve your chances; it divides them by eight.

The two things worth taking away

Total life is the wrong number to manage. A joint can have a long life and still be uninspectable, because almost all of that life is spent below the size anyone can find. What an inspection programme is buying is the window — and the window can be short while the life is long.

The fraction of life spent undetectable barely moves. Change the stress range and the total life changes enormously, but the proportion of it spent hidden stays about the same. So you cannot inspect your way out of a stress problem: lowering the stress range buys real time, while inspecting more often only helps if a window exists to inspect inside.

What is not verified

Stated plainly, because a piece whose value is trustworthiness should not overclaim. The first list is the part that would change how you read the result. The second is scope — true, worth saying, and not load-bearing.

• No engineer has reviewed this. If you are one and would check it, REVIEW.md says what would actually help and is meant to cost about fifteen minutes. The model is the Paris law with the Barsom mean-line constants for ferrite-pearlite steels, and the built-in checks confirm the integration reproduces the closed form exactly — but that check is the author's own, not a fracture-mechanics engineer's.
• A mean line is not a design line. Real crack growth data scatters by a factor of two or more about it, and assessment codes deliberately use an upper bound rather than the mean. Treated as design guidance, the numbers here are optimistic.
• One crack, one shape, one constant geometry factor. Real cracks change aspect ratio as they grow, interact with each other, and sit in stress fields that the single factor used here does not describe.
• No threshold and no closure. Below a certain stress intensity range cracks effectively stop growing, and this model keeps growing them, which is conservative. It also ignores retardation after overloads, which is not.

Scope, for completeness. True of the model, and none of it changes the answer above.

• Detection is drawn as a clean line. Real inspection has a probability of detection that rises gradually with size — an inspection inside the window can still miss, which makes the real case worse than the one drawn.
• The critical size is taken as the section depth over most of this range, with the fracture-limited size used when it is smaller. Real failure may come sooner, by plastic collapse of the remaining ligament.

Commissions

The subject changes; the method does not. If you have a failure mode, a mechanism, or a process that people keep getting wrong because the diagram cannot move, that is the shape of problem this suits. You get one file, the checks that hold it honest, and the notes explaining every decision including the ones that were reversed. Commissions start at £4,000 and take three to four weeks from an agreed brief. yinxueshi1017@gmail.com

Not commissioning anything? Still worth an email. If your version of this failed differently — other geometry, other numbers, a mechanism this piece leaves out — tell me in one line. Corrections are the main thing that decides what gets built next.