Define flexibility coefficient

The flexibility coefficient $f_{ij}$ is the displacement at coordinate $i$ (in the direction of the redundant $X_i$) caused by a unit force acting at coordinate $j$, with all other redundants and loads absent.

$$f_{ij} = \int \frac{m_i\,m_j}{EI}\,dx$$

for flexural members (for trusses, $f_{ij} = \sum n_in_jL/AE$). Its unit is m/kN for a force coordinate or rad/kNm for a moment coordinate. The coefficients form the flexibility matrix $[F]$, which is symmetric ($f_{ij} = f_{ji}$, Maxwell). Example: for a cantilever of length $L$, the tip deflection due to a unit tip load is $f_{11} = L^3/3EI$.