Beams and Frames by the Force Method

Analysis of beams and frames by the force method reduces an indeterminate problem to a series of analyses of one statically determinate primary structure. The main practical choices are the redundants and an efficient evaluation of compatibility displacements.

Primary structure

For a continuous beam, redundants may be support reactions or moments at intermediate supports. For frames, one may release support restraints, introduce hinges, or cut a closed loop and replace the released force connections by unknowns $X_i$.

Load state

Apply the prescribed external loading to the primary structure and construct the load diagrams $M_P$ and, where required, $N_P$ and $V_P$. Use one consistent sign convention in every state.

Unit states

For each redundant, set $X_i=1$ while the other redundants are zero and construct the unit diagram $M_i$. These unit states are used to evaluate $\delta_{ij}$ and $\Delta_{iP}$.

Superposition

After solving the compatibility equations, final bending moments are

$$M=M_P+X_1M_1+\cdots+X_nM_n,$$

with analogous expressions for other internal force effects included in the model.

Symmetric structures

For symmetric frames, separating symmetric and antisymmetric loading or unit states can make some coefficients vanish and substantially reduce the equation system.

Checks

Final reactions must satisfy global equilibrium, and internal-force diagrams must satisfy hinge and free-end conditions. The essential deformation check is that displacements in the restored restraint directions equal their prescribed values.