Question

To generate the $j^{\text{th}}$ column of the flexibility matrix

• A. A unit force is applied at coordinate $j$ and the displacements are calculated at all coordinates
• B. A unit displacement is applied at co-ordinate $j$ and the forces are calculated at all coordinates
• C. A unit force is applied at coordinate $j$ and the forces are calculated at all coordinates
• D. A unit displacement is applied at co-ordinate $j$ and the displacements are calculated at all co-ordinates

Hint:

Flexibility matrix → apply unit {force}, find {displacements}. Stiffness matrix → apply unit {displacement}, find {forces}.

Answer 1

The Correct Option is A

Think of building the flexibility matrix experimentally, one column at a time: to get column j, you apply a unit load only at coordinate j and measure how every coordinate on the structure moves.

1. Unit force at j, displacements at all coordinates: matches this experimental procedure directly and is how each column of the flexibility matrix is built.
2. Unit displacement at j, forces at all coordinates: this instead is how a column of the stiffness matrix would be built, not the flexibility matrix.
3. Unit force at j, forces at all coordinates: forces cannot be the output when a force is already the applied input in this matrix relation.
4. Unit displacement at j, displacements at all coordinates: mismatched input and output types that don't correspond to either standard matrix.

So the flexibility matrix's \(j^{\text{th}}\) column comes from applying a unit force at coordinate j and reading displacements everywhere.

Answer 2

A third way to settle this is purely from the units of a flexibility coefficient. Each entry \(f_{ij}\) of the flexibility matrix has units of displacement per unit force (for example, mm/kN for a translational coordinate), since it physically represents a displacement produced by a unit applied force.

1. Unit force at j, displacements at all coordinates: Here the applied quantity has units of force, and the measured output quantity has units of displacement, giving a ratio with units of displacement per force — exactly matching the units of a flexibility coefficient.
2. Unit displacement at j, forces at all coordinates: Here the applied quantity is a displacement and the output is a force, giving units of force per displacement, which is the unit of a stiffness coefficient, not a flexibility coefficient.
3. Unit force at j, forces at all coordinates: Both quantities here would have units of force, giving a dimensionless ratio, which does not match the displacement-per-force units of a flexibility coefficient.
4. Unit displacement at j, displacements at all coordinates: Both quantities here have units of displacement, again giving a dimensionless ratio, not matching flexibility coefficient units either.

Only applying a unit force and measuring displacement gives a ratio with the correct displacement-per-force units of a flexibility coefficient.

Therefore, the correct answer is a unit force is applied at coordinate j and the displacements are calculated at all coordinates.