Module 3 · Course notes

Stability: the movement a count can miss

Before a structure can carry anything, it must be unable to move as a rigid body. Deciding whether it can is the question of stability — and it is not answered by counting members and reactions alone. The reliable test is to look for a movement: can the joints shift while the members keep their length? If they can, it is a mechanism.

A mechanism is a permitted movement — a way the assembly can change shape (or move bodily) without stretching or shortening any member. A structure with a mechanism cannot carry the load that drives that movement; it simply moves.

A pin-jointed rectangle is the classic example. With four bars and pinned corners it can rack sideways into a parallelogram, its bars merely rotating. It is a mechanism.

The four bars keep their length while the joints move: the rectangle shears into a parallelogram. Because the joints can move without straining any bar, this is a mechanism — it cannot resist the sideways load.

Counting is only a screen. The rule reactions + members ≥ 2 × joints (for a plane pin-jointed frame) tells you whether there are enough restraints, but not whether they are arranged to work. Three reaction lines that are all parallel, or all pass through one point, leave a rigid-body movement free even though the count looks fine. Geometry decides stability; the count only flags a shortage.

Add one diagonal to the racking rectangle and it is triangulated: the joints can no longer move without changing a bar length. The mechanism is gone.

The diagonal ties opposite corners: racking would now have to lengthen or shorten it, which the bar resists. One diagonal turns the mechanism into a stable, triangulated frame.

Worked example

Worked example — is this arrangement stable?

A beam rests on three vertical rollers. There are three reactions — surely that is enough?

Step 1 — count, then look
Three reactions look sufficient for a plane body (which needs three restraints). But all three reactions are vertical — they are parallel. None restrains horizontal movement.
Step 2 — find the movement
Push the beam sideways: nothing resists it, so the whole beam slides horizontally. That is a rigid-body mechanism, hiding behind a count that looked adequate. Replace one roller with a pin and the horizontal movement is blocked.

In the exercises, decide for each assembly whether it is stable, a mechanism, or stable-but-redundant — by looking for a movement, not just counting.

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

Stability & mechanism exercises

5 exercises on deciding whether an assembly is stable, a mechanism, or stable-but-redundant by looking for a movement. 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.