Canonical Equations of the Force Method

The canonical equations of the force method express displacement compatibility in the directions of the released redundant restraints. A structure with degree of static indeterminacy $n$ produces $n$ linear equations.

General form

If each released direction has zero displacement in the original structure,

$$\delta_{i1}X_1+\delta_{i2}X_2+\cdots+\delta_{in}X_n+\Delta_{iP}=0,\qquad i=1,\ldots,n.$$

In matrix form,

$$[\delta]\{X\}=-\{\Delta_P\}.$$

Meaning of the coefficients

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

Evaluation

For bending-dominated structures with constant or piecewise-constant stiffness,

$$\delta_{ij}=\sum\int\frac{M_iM_j}{EI}\,dx,$$

and

$$\Delta_{iP}=\sum\int\frac{M_iM_P}{EI}\,dx.$$

Axial, torsional, and shear contributions are added when required by the model.

Matrix properties

For a linear-elastic conservative reciprocal system, $\delta_{ij}=\delta_{ji}$. Diagonal coefficients $\delta_{ii}$ are positive for independent nonzero unit states.

Recovery

After solving for the redundants, final internal forces follow by superposition, for example $M=M_P+\sum X_jM_j$. Check both equilibrium and the prescribed compatibility conditions.