Physical significance of flexibility and stiffness matrices

Flexibility matrix $[F]$

Each element $f_{ij}$ is the displacement at coordinate $i$ caused by a unit force at coordinate $j$. The relation is $\{\Delta\} = [F]\{P\}$. It measures how soft the structure is: a large $f_{ij}$ means a small force gives a large displacement. Column $j$ is the deflected shape produced by a unit load at $j$. Diagonal terms are always positive, and the matrix is symmetric ($f_{ij} = f_{ji}$, Maxwell's theorem). It exists only for a stable, restrained structure and is the matrix used in the force method.

Stiffness matrix $[K]$

Each element $k_{ij}$ is the force needed at coordinate $i$ to give a unit displacement at coordinate $j$ while all other coordinates are held fixed. The relation is $\{P\} = [K]\{\Delta\}$. It measures how stiff the structure is: column $j$ is the set of restraint forces that holds the structure in the shape of a unit displacement at $j$. Diagonal terms are positive, and the matrix is symmetric. It is used in the displacement method.

The two are inverses: $[K] = [F]^{-1}$.

Example: cantilever of length $L$

Coordinates: 1 = vertical deflection at the free end, 2 = rotation at the free end.

$$[F] = \begin{bmatrix} \dfrac{L^3}{3EI} & \dfrac{L^2}{2EI} \\[2mm] \dfrac{L^2}{2EI} & \dfrac{L}{EI} \end{bmatrix}, \qquad [K] = [F]^{-1} = \begin{bmatrix} \dfrac{12EI}{L^3} & -\dfrac{6EI}{L^2} \\[2mm] -\dfrac{6EI}{L^2} & \dfrac{4EI}{L} \end{bmatrix}$$

• $f_{11} = L^3/3EI$ is the tip deflection due to a unit tip load.
• $k_{22} = 4EI/L$ is the moment needed at the tip to give a unit rotation when the tip deflection is held zero.
• Note $k_{11} = 12EI/L^3 \neq 1/f_{11} = 3EI/L^3$. The stiffness is not the reciprocal of the flexibility for a matrix, because the stiffness holds the other coordinate fixed while the flexibility leaves it free.