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This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

Checks needed for project use

  • An independent structural reviewer must check the illustrated models, prediction feedback, geometry and full interactive input ranges.
  • Hinge-count shortcut applies to the stated beam flexural mechanisms, not every frame or local/partial mechanism. Check rotation capacity, buckling, axial/shear interaction, hardening and geometric effects before applying a plastic model.
  • The shortcut r+1 is not valid for every frame or partial mechanism; first hinge may trigger a local mechanism despite overall indeterminacy.

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Module 14 · Course notes

Plastic behaviour: hinges, redistribution and collapse

An elastic analysis does not describe the spread of yielding beyond first yield. In the ideal ductile beam models used here, yielding can spread across a section and moment can redistribute before a collapse mechanism forms. This requires sufficient rotation capacity and restraint against other failure modes.

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.

bending moment (tension side)

where the plastic hinges form
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.

For the beam flexural mechanisms illustrated here, the number of hinges is one more than the flexural 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.

a beam collapse mechanism
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?

  1. 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.

    bending moment (tension side)

  2. 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.

    bending moment (tension side)

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

  1. 1Points of certainty. Match the displacement and rotation that each support prescribes, including any settlement. Infer the displacement direction from the whole structure and load case; it need not be downward at every loaded point.
  2. 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).
  3. 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.Moment is zero at a genuine moment release and at an unloaded simple end. A continuous member can carry moment over a translational support.
  • 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.For an initially straight elastic beam without imposed curvature, zero curvature implies zero bending moment.
  • 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.