Module 14 · Course notes

Plastic behaviour: hinges, redistribution and collapse

Elastic analysis stops at first yield, but a ductile structure carries far more before it falls. As sections yield fully they form plastic hinges, the structure sheds moment to where there is spare capacity, and it keeps standing until enough hinges turn it into a mechanism. Reading that sequence — first hinge, redistribution, collapse — is what plastic thinking adds.

A ductile section does not fail at first yield. As the moment rises past first yield the section yields further until it is fully plastic, reaching its plastic moment Mp. From then on it can rotate at essentially constant moment — a plastic hinge.

A plastic hinge is not a release of moment like a real pin: it holds a constant moment (Mp) while rotating. But its effect on the structure — allowing relative rotation — is the same, and each hinge that forms removes one degree of indeterminacy.

In a fixed-ended beam the moments are largest at the two fixings and at the load point. As the load rises, hinges form there first — and once enough have formed, the beam becomes a mechanism.

In an indeterminate structure, the first hinge does not mean collapse. The structure simply redistributes: it carries on taking load through its remaining restraints, forming more hinges where the moment now peaks, until there are enough hinges to make a mechanism — a way to move without further increase of load. That is collapse.

The number of hinges needed is one more than the degree of indeterminacy: a simply supported beam (determinate) collapses at its first hinge; a propped cantilever (one redundant) needs two; a fixed-ended beam (two redundants) needs three.

When enough plastic hinges have formed, the beam folds at those hinges as a mechanism — it can move without carrying more load. The collapse load is the load at which this mechanism becomes possible.

Worked example

Worked example — counting the hinges

How many plastic hinges does each of these beams need to collapse?

Step 1 — the determinate beam
A simply supported beam is determinate (no redundants). The first plastic hinge — at mid-span under the load — turns it straight into a mechanism. One hinge is enough to collapse it.
Step 2 — the indeterminate beams
A propped cantilever has one redundant, so it needs two hinges (at the fixing and in the span). A fixed-ended beam has two redundants, so it needs three (both ends and mid-span). The rule is hinges = indeterminacy + 1.

In the exercises, locate where hinges form, follow the redistribution, and count the hinges to collapse.

How to read these problems

The three-step method

1Points of certainty. The deflected curve must pass through every support and deflect downward under the load. Mark what each support prevents before drawing anything.
2Deflected shape and reaction directions. Sketch the compatible deflected shape. To find a reaction's direction, imagine removing that support: the direction that pushes the structure back to its place is the reaction's sense (it may be a hold-down).
3Bending moment and contraflexure. Draw the bending-moment diagram on the tension side and check it against the shape: hogging where the curve is convex-up, sagging where convex-down, zero at pins and at every contraflexure.

Rules that must always hold

1.The bending moment is zero at a simple support and at an internal pin or hinge.
2.A bending-moment diagram crosses the baseline exactly at a point of contraflexure.
3.Under a distributed load the bending-moment diagram is curved; under point loads alone it is straight lines.
4.At a fully fixed support the deflected shape leaves the support with zero rotation (tangent along the member).
5.If a part of the structure stays straight after loading, it carries no bending moment there.
6.The moment is drawn on the tension side: sagging below the member, hogging above it.

Now predict for yourself

Plastic behaviour and collapse exercises

5 exercises on plastic hinges, redistribution, counting hinges to a mechanism and collapse above first yield. Predict, then reveal an explanation.

Start the exercises →

This lesson is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions.