Force Method

The force method is a classical procedure for analyzing linear-elastic statically indeterminate structures. Its unknowns are redundant reactions or internal force resultants that cannot be obtained from equilibrium alone.

Basic idea

Remove as many restraints as the degree of static indeterminacy. Replace the actions of those released restraints by unknown generalized forces $X_1,X_2,\ldots,X_n$. The resulting primary structure must be statically determinate and geometrically stable.

Compatibility

Displacements of the primary structure in the released directions, caused by external loading and the unknown redundants, must reproduce the deformation constraints of the original structure. When the corresponding original displacement is zero,

$$\sum_{j=1}^{n}\delta_{ij}X_j+\Delta_{iP}=0.$$

Here $\delta_{ij}$ is the displacement in direction $i$ caused by a unit value of redundant $X_j$, and $\Delta_{iP}$ is the displacement caused by the prescribed loading.

Coefficients

Flexibility coefficients and load terms are evaluated by displacement methods such as the Mohr integral or graphical diagram multiplication. For a linear-elastic reciprocal system, $\delta_{ij}=\delta_{ji}$.

Analysis sequence

1. Determine the degree of static indeterminacy.
2. Select redundants and a primary structure.
3. Construct the load state and unit states.
4. Evaluate the compatibility coefficients.
5. Solve for the redundants $X_j$.
6. Recover reactions and internal forces by superposition.
7. Check equilibrium and deformation compatibility.

Imposed deformations

Temperature changes, support settlements, and prescribed support movements enter the compatibility equations as additional kinematic terms, so the force method is not limited to mechanical force loading.