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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.
  • Primary structure must be stable and solvable. Maxwell reciprocity needs linear conservative elastic response and identical restraint conditions. Loss of support contact needs a changed model.
  • Any accepted answer saying a settled prop reaction reverses requires a hold-down-capable support; an ordinary bearing loses contact.

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.

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Module 11 · Course notes

The flexibility method: release, restore, superpose

When equilibrium alone cannot solve a structure — because it has redundant restraints — the flexibility method supplies the missing equations from geometry. The idea is to remove a redundant, see the incompatibility that appears, and then find the force needed to put it right. It is the force method, and its heart is a compatibility condition, not an equilibrium one.

An indeterminate structure has more unknowns than equilibrium equations. The flexibility method turns that into a recipe:

1. Choose a redundant and release it, leaving a stable, determinate primary structure. 2. Analyse the primary structure under the real load. Where the redundant used to be, there is now an incompatibility — a gap, an overlap, or a movement that should not be there. 3. Analyse the primary structure under the redundant alone, as an unknown. 4. Write the compatibility condition: the two effects must cancel. Solve for the redundant. 5. Superpose to get everything else.

deflected shape

step 1–2: release the prop, and the primary structure deflects
Take a propped cantilever and remove the prop. What remains is a plain cantilever — determinate. Under the real load its free end drops. That drop is the incompatibility: in the real structure the prop holds that point at zero.

The redundant — here the prop force — is then whatever it takes to push the tip back up to zero. That single sentence is the whole analysis. The equation that fixes it is compatibility: the deflection at the prop is zero. It says nothing about forces balancing; it is a statement about geometry, and it is exactly the equation equilibrium could not provide.

Two cautions. The release must leave a stable, determinate primary structure — release the wrong restraint and you get a mechanism, and the method collapses. And the compatibility target is not always zero: a settling or flexible support changes the displacement target and therefore the redundant. A reversal requires a support that can carry tension; an ordinary bearing instead loses contact.

bending moment (tension side)

step 5: superpose to recover the propped cantilever
Adding the redundant prop force back restores the real structure: hogging at the fixing, sagging in the span, and a point of contraflexure. The prop reaction and the fixing moment both follow from the one compatibility equation.

Worked example

Worked example — a propped cantilever by the force method

Solve a propped cantilever by releasing the prop and restoring compatibility.

  1. Step 1 — release and see the gap

    Remove the prop. The primary structure is a cantilever; under the load its tip drops. In the real structure the prop holds that point at zero, so this drop is the incompatibility to be removed.

    deflected shape

  2. Step 2 — restore with the redundant

    Apply an upward prop force to the cantilever tip and choose it so the tip returns exactly to zero — that is the compatibility condition. Superposing the two cases gives the real propped cantilever: hogging at the wall, sagging in the span, one contraflexure. Every reaction follows from that one geometric equation.

    bending moment (tension side)

In the exercises, choose good releases, identify the compatibility condition, and reason about what settlement does to a redundant.

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

The flexibility method exercises

5 exercises on choosing a redundant, the compatibility condition, settlement and reciprocity. 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.