Properties of stiffness and flexibility matrices

A system is described by $n$ coordinates (points and directions of force or displacement). The flexibility matrix $[F]$ gives displacements from forces, $\{\Delta\} = [F]\{P\}$, and the stiffness matrix $[K]$ gives forces from displacements, $\{P\} = [K]\{\Delta\}$.

Properties of the flexibility matrix $[F]$

1. It is a square matrix of order $n$ (number of coordinates).
2. It is symmetric, $f_{ij} = f_{ji}$ (Maxwell's reciprocal theorem).
3. The diagonal terms $f_{ii}$ are always positive (a force does positive work through its own displacement).
4. Off-diagonal terms may be positive, negative or zero.
5. It exists only for a stable, adequately supported system; a mechanism has no flexibility matrix.
6. It is positive definite, and $[F] = [K]^{-1}$.
7. Column $j$ is the deformed shape caused by a unit force at coordinate $j$.

Properties of the stiffness matrix $[K]$

1. It is a square, symmetric matrix of order $n$: $k_{ij} = k_{ji}$.
2. The diagonal terms $k_{ii}$ are always positive.
3. Off-diagonal terms may be positive, negative or zero.
4. It is positive definite when the rigid-body motion is prevented. If the system is free, $[K]$ is singular (it has zero determinant, because rigid-body motions need no force).
5. Column $j$ is the set of forces needed to hold the system in the shape of a unit displacement at $j$ (all others zero).
6. $[K] = [F]^{-1}$, and for any coordinate $k_{ii} \ge 1/f_{ii}$.

Illustration: cantilever of length $L$ with two coordinates

Coordinate 1 is the vertical deflection at the free end and coordinate 2 is the rotation at the free end.

  ||==========================o  -> (1) deflection
  fixed                       |  -> (2) rotation

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

Both are symmetric with positive diagonals. Also $k_{11} = 12EI/L^3 > 1/f_{11} = 3EI/L^3$, because $k_{11}$ holds the rotation at zero while $f_{11}$ leaves it free.