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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.
  • Counting m+r against2j is necessary screening, not a rank/stability proof. Diagonal stability assumes a member capable of the required tension/compression and effective support restraint.

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.

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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.

For the ideal pin-jointed assemblies here, a mechanism is a permitted movement without first-order member extension or shortening. Rigid-jointed frames also resist movement through bending. 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.

a pin-jointed rectangle racks sideways
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.

a single diagonal makes it stable
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?

  1. 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.

  2. 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

  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

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.