Question

Flexibility coefficients are used in which of the following method?

• A. Force method
• B. Displacement method
• C. Both force and displacement method
• D. Virtual force method

Hint:

Remember: Force method → flexibility coefficients, Displacement method → stiffness coefficients.

Answer 1

The Correct Option is A

The force method treats certain support reactions or internal forces as redundant unknowns, and solves for them using compatibility conditions on displacement, and it is exactly displacement-due-to-unit-force values, i.e., flexibility coefficients, that build those compatibility equations.

1. Force method: redundant forces are solved using compatibility equations built from flexibility coefficients, so this matches directly.
2. Displacement method: unknowns are joint displacements, solved using stiffness coefficients (force needed per unit displacement), the opposite quantity.
3. Both: incorrect, since the two methods use two different, inverse-related coefficient sets.
4. Virtual force method: used for finding a single deflection via virtual work, not for a full system of redundant-force equations.

So flexibility coefficients belong to the force method.

Answer 2

A third way to see this clearly is to work through a concrete example. Consider a propped cantilever beam where the prop reaction \(R\) is chosen as the single redundant force.

Removing the prop leaves a simple cantilever, which deflects downward by some amount \(\delta_{L}\) at the prop location under the applied load alone. If a unit force is then applied at the same location (in place of the removed prop), the cantilever deflects by an amount \(f_{11}\) — this \(f_{11}\) is precisely a flexibility coefficient. The compatibility condition (that the actual deflection at the prop must be zero, since the prop is rigid) is written as \(\delta_{L} + R\,f_{11} = 0\), which is solved directly for the redundant reaction \(R\).

1. Force method: the worked example above shows the redundant force \(R\) being solved for using exactly a flexibility coefficient \(f_{11}\) inside a compatibility equation, confirming this is where flexibility coefficients are used.
2. Displacement method: here, instead, one would fix the joint displacement and compute the force needed to cause it, which is a stiffness coefficient, not the flexibility coefficient \(f_{11}\) used above.
3. Both: the worked example only required a flexibility coefficient, with no stiffness coefficient appearing anywhere in the redundant-force calculation, so this option overstates the case.
4. Virtual force method: that technique would be used separately to compute a single deflection like \(\delta_L\), but not to set up and solve the full compatibility equation for the redundant force as shown above.

The worked compatibility equation confirms flexibility coefficients are the tool of the force method.

Therefore, the correct answer is Force method.