Restrained expansion — the length that does not matter, until it does

Both members are the same steel, held at both ends, and both have warmed by the same amount. One is short, one is long. Drag upward to allow them a little movement at one end; drag down to hold them rigidly. The arrow keys do the same. Home holds them solid; End gives them all the movement on offer. Sound is off until you turn it on.

Two members, the same steel, the same temperature rise. The bars on the right are the stress in each.

What you are looking at

Steel wants to get longer when it is heated, by a fixed fraction of its length per degree. Hold both ends and it cannot, so it is squeezed back to the length it started at — and the stress that takes is the stiffness times the strain you prevented.

The strain you prevented is a fraction. It is the same fraction whatever the length is. So the stress is the stiffness times the expansion coefficient times the temperature rise, and the length of the member is simply not in that expression. A short bar and a long pipeline, held rigidly and warmed by the same amount, are carrying exactly the same stress.

Now allow the end to move a few millimetres. What that buys is not a length of movement, it is a fraction of movement: the gap divided by the member. On a short member a few millimetres is a large fraction and the stress falls away. On a long one the same few millimetres is almost nothing, and the stress barely notices.

The two things worth taking away

Held rigidly, length is irrelevant. There is no such thing as a member that is "too long to restrain" — the short one is in just as much trouble. What decides whether it survives is the temperature rise and the steel, not the span.

Once you allow movement, length is the only thing that matters. An expansion allowance is not a quantity of millimetres, it is a quantity of millimetres per metre. The same detail that works on a short run does nothing on a long one, which is why the answer on long runs is more joints rather than bigger ones.

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 relations are the elementary ones and the built-in checks confirm them exactly — but that check is the author's own, not a structural or pipeline engineer's.
The restraints are perfectly rigid. Real abutments, anchors and supports deflect, and that flexibility relieves stress in exactly the way the gap does — so a real member is usually better off than this one, by an amount nobody knows precisely.
Nothing here buckles. A long member in compression will usually go sideways long before it reaches the stress shown — which is precisely how continuously welded rail fails on hot days. That makes the real case worse than the one drawn.

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

Friction is ignored. A buried pipeline or a rail is restrained by friction along its length rather than at its ends, which makes the stress build up over a transition length and then plateau. The plateau value is the one drawn here; the ends behave differently.
Elastic only, and one direction. Past yield the member relieves itself by deforming permanently, and then goes into tension when it cools back down. That cycle is the interesting part of the real problem and is out of scope here.

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.