Flexibility, stiffness; properties of flexibility matrix

Flexibility

The flexibility $f_{ij}$ is the displacement at coordinate i caused by a unit force acting at coordinate j (all other coordinates carry no force). For a beam fixed at one end, the tip deflection due to a unit tip load is $f=L^3/3EI$. Its unit is m/kN (displacement per unit force). The flexibility matrix $[F]$ collects all $f_{ij}$, so that

$$\{\Delta\}=[F]\{P\}$$

Stiffness

The stiffness $k_{ij}$ is the force required at coordinate i to produce a unit displacement at coordinate j while all other coordinates are held fixed (zero displacement). For the same cantilever the tip stiffness is $k=3EI/L^3$. Its unit is kN/m. The stiffness matrix $[K]$ gives

$$\{P\}=[K]\{\Delta\}$$

For a stable structure with the same coordinates, $[K]=[F]^{-1}$.

Properties of the flexibility matrix

1. Square: its order equals the number of coordinates (redundants) chosen.
2. Symmetric: $f_{ij}=f_{ji}$ (Maxwell's reciprocal theorem).
3. Diagonal terms are positive: $f_{ii}>0$, because a force always moves its own point in its own direction (work done is positive).
4. Positive definite: the strain energy $U=\tfrac12\{P\}^T[F]\{P\}$ is always positive, so the determinant is positive.
5. Inverse is the stiffness matrix: $[F]=[K]^{-1}$.
6. Depends on the chosen released structure and coordinates, but the final answer does not.
7. Depends only on geometry and material (L, E, I, A), not on the loads. Off-diagonal terms may be positive, negative or zero.