Stress concentration — the hole's shape, not its size

The plate is being pulled. Drag upward to flatten the hole into a slit; drag down to round it back out. The arrow keys do the same. End sharpens it as far as it goes; Home makes it round again. The hole keeps the same width across the pull the whole time, so the amount of material removed hardly changes. Sound is off until you turn it on.

What you are looking at

Pull a plate and the stress is spread evenly across it. Put a hole in it and two things happen. The obvious one: there is less material left, so the average stress across what remains goes up a little. The other one is much larger and much less obvious: the lines of force have to go round the hole, they crowd together as they pass it, and right beside the hole the stress is several times the average.

For a round hole in a wide plate that crowding factor is exactly three. Not about three — three. And it does not depend on how big the hole is. A pinhole and an opening you could put your arm through concentrate the stress beside them by the same amount.

What it does depend on is the shape. The factor is set by how sharply the hole turns at its ends, and as you flatten a round hole into a slit that turn gets tighter and the multiplier climbs without any limit. This is why a crack is in a different category from a hole: a crack is the same feature with its ends sharpened to nothing.

The two things worth taking away

Counting the lost area is not an assessment. The lost area is the small term. If a check on a perforated member only accounts for the net section, it has accounted for the part that barely matters and missed the multiplier entirely.

The fix is a radius. Because the multiplier depends on sharpness and not size, you cannot help a sharp feature by making it smaller — but you can transform it by rounding it. That is the whole reason for drilling a hole at the end of a crack, for generous fillets on re-entrant corners, and for why a scratch across the pull is worse than a much bigger smooth recess.

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 relation is the classical elliptical-hole result and the built-in checks confirm it exactly — but that check is the author's own, not a stress engineer's.
• The plate is infinitely wide. The factor of three is the wide-plate answer; as the hole becomes a significant fraction of the width the real factor changes and the net-section term stops being small.
• The multiplier runs to infinity here and does not in reality. Once the tip radius approaches the scale of the material's own structure the continuum answer stops meaning anything, and the right tool is fracture mechanics rather than a concentration factor.

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

• Elastic, isotropic, two-dimensional. Real materials yield at the hot spot and shed the peak, which is why a ductile part survives a concentration that this arithmetic says should fail it. A brittle one does not get that mercy, and neither does anything under fatigue loading.
• The colour around the hole is indicative, not a solved field. The peak value at the edge is exact; the way the drawing fades it away from the hole is a rendering choice, and its only job is to show where the crowding is.

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.