Canonical Equations of the Displacement Method

The canonical equations of the displacement method express the condition that the artificial restraints introduced in the primary system carry no final reaction after the structure acquires its actual joint displacements.

General form

For $n$ independent displacements,

$$r_{i1}Z_1+r_{i2}Z_2+\cdots+r_{in}Z_n+R_{iP}=0,\qquad i=1,\ldots,n.$$

In matrix form,

$$[r]\{Z\}=-\{R_P\}.$$

Stiffness coefficients

$r_{ij}$ is the reaction in artificial restraint $i$ caused by unit displacement $Z_j=1$ while all other primary displacements are zero. The diagonal coefficient $r_{ii}$ measures the structural resistance associated with coordinate $Z_i$.

Load terms

$R_{iP}$ is the reaction in restraint $i$ caused by the prescribed external loading when all primary displacements are blocked. It is obtained from member end forces and equilibrium of the corresponding joint or structural part.

Reciprocity

For a linear-elastic conservative system, the coefficient matrix is symmetric: $r_{ij}=r_{ji}$. This provides a useful check on the unit states and equation assembly.

Force recovery

After solving for $Z_j$, any internal-force quantity can be recovered by superposition, for example

$$M=M_P+Z_1M_1+\cdots+Z_nM_n.$$

Joint equilibrium, support reactions, and boundary conditions should then be verified.