Flexibility — exercises | Queensferry

Module 11Level 3stable or mechanism

Choosing a redundant to release

In the flexibility method you release a redundant to form the primary structure. What must be true of the structure that remains?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AIt must be stable and determinate — you must be able to analyse it. Releasing a restraint the structure genuinely needs leaves a mechanism, and the method fails.BIt must be exactly the same as the original structure.CAny release will do, because you add the redundant back later.

Module 11Level 3reactions and load path

What fixes the redundant?

For a propped cantilever, the prop reaction is taken as the redundant. Which condition determines its value?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

ACompatibility: the deflection at the prop must be zero. That geometric condition — not any equilibrium equation — sets the prop force.BA further equilibrium equation, taken about a different point.CThe prop simply carries half the load, by symmetry.

Module 11Level 4change and sensitivity

What if the prop settles?

The prop of the propped cantilever settles downward by a small amount instead of holding its point exactly. What happens to the redundant prop force?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AIt falls — the compatibility target is no longer zero but the settlement, so less force is needed; settle enough and the prop force can even reverse.BNothing changes — the prop force depends only on the load.CIt increases, because settlement adds load.

Module 11Level 4reactions and load path

How many compatibility equations?

A continuous beam is indeterminate to the second degree (it has two redundants). How many compatibility equations does the flexibility method need?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

ATwo — one compatibility condition for each redundant, solved together, because each redundant generally affects both released coordinates.BOne — a single equation always suffices.CThree — one for each equilibrium equation as well.

Module 11Level 4change and sensitivity

Cross-flexibility and reciprocity

In a flexibility analysis, f₁₂ is the movement at coordinate 1 due to a unit redundant at coordinate 2, and f₂₁ the reverse. For a linear elastic structure, how do they compare?

Predict first, then reveal

Reason from the supports, the movement and equilibrium — not from what looks familiar.

AThey are equal: f₁₂ = f₂₁. This is Maxwell's reciprocal theorem, a property of linear elastic structures.BThey are generally different and unrelated.Cf₁₂ is always larger, because coordinate 1 comes first.