Define force method and primary structure

Force method

The force method (flexibility or compatibility method) is a method of analysing statically indeterminate structures in which the redundant forces are taken as the unknowns. The redundants are removed to get a determinate structure, the loads and the unit redundants are applied to it, and the redundants are found from compatibility equations that restore the real displacement conditions at the releases:

$$\Delta_{i0} + \sum_j f_{ij}X_j = \delta_i$$

Here $\Delta_{i0}$ is the displacement due to the loads, $f_{ij}$ the flexibility coefficient and $\delta_i$ the actual displacement of the support (usually zero). After finding the $X_j$, the final forces follow by superposition, $M = M_0 + \sum X_jm_j$. The number of unknowns equals the degree of static indeterminacy $D_s$.

Primary structure

The primary structure is the statically determinate, stable structure obtained by removing the redundants (supports, members or moment restraints) from the indeterminate structure. It is not unique, but it must stay stable.

Example: a propped cantilever (fixed at A, roller at B) has $D_s = 1$. If the roller reaction $R_B$ is removed, the primary structure is a cantilever; if the fixed-end moment $M_A$ is released, it is a simply supported beam.