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.
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 prescribes a movement, which changes the redundant and can even reverse its sign.
Worked example
Worked example — a propped cantilever by the force method
Solve a propped cantilever by releasing the prop and restoring compatibility.
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.
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.
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
- 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
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.