Flexibility, stiffness coefficient; suitable method

Flexibility coefficient

The flexibility coefficient $f_{ij}$ is the displacement at coordinate $i$ due to a unit force applied at coordinate $j$ (all other coordinates unloaded). Its unit is m/kN (or rad/kNm). The flexibility matrix $[F]$ relates displacements and forces: $\{\Delta\} = [F]\{P\}$.

Stiffness coefficient

The stiffness coefficient $k_{ij}$ is the force required at coordinate $i$ to produce a unit displacement at coordinate $j$ while all other coordinates are held fixed. The stiffness matrix gives $\{P\} = [K]\{\Delta\}$, and $[K] = [F]^{-1}$.

Suitable method

Read the figure as a single-storey, two-bay frame with fixed outer bases and a hinged base under the middle column: $j = 6,\ m = 5,\ r = 3 + 2 + 3 = 8$.

$D_s = 3(5) + 8 - 3(6) = 5$

$D_k$ (inextensible) $= 3(6) - 8 - 5 = 5$ (three top-joint rotations, one base-hinge rotation, one sway).

Because the base of the middle column is hinged, its far-end rotation can be removed with the modified stiffness $3EI/L$. The effective unknowns of the displacement method fall to 4, compared with 5 redundants in the force method.

Displacement method is more suitable because

1. it has fewer unknowns (4 against 5);
2. the frame is a regular framed system with several rotating joints and sway, which suits slope-deflection or moment distribution well;
3. the force method needs a stable released structure and many moment diagrams for 5 redundants, which is long hand work.