Displacement Method

The displacement method analyzes linear-elastic statically indeterminate framed structures using independent joint translations and rotations as the primary unknowns.

Basic idea

Unlike the force method, which uses redundant force quantities, the displacement method temporarily restrains all independent joint movements by additional constraints. The resulting structure is the primary system of the displacement method.

Primary unknowns

The unknowns $Z_1,Z_2,\ldots,Z_n$ represent independent rotations of rigid joints and independent translations of joints or joint groups. Their number is the degree of kinematic indeterminacy of the adopted structural model.

Canonical equations

The artificial restraints do not exist in the real structure, so their total reactions must vanish after superposition:

$$\sum_{j=1}^{n}r_{ij}Z_j+R_{iP}=0,\qquad i=1,\ldots,n.$$

Here $r_{ij}$ is the reaction in artificial restraint $i$ caused by unit displacement $Z_j=1$, and $R_{iP}$ is the reaction in the same restraint caused by the prescribed loading while all primary displacements are restrained.

Analysis states

The primary system is analyzed under the prescribed loading and under each unit displacement separately. Once $Z_j$ are known, member end moments, shears, axial forces, and support reactions are recovered by superposition.

Applications

The method is especially effective for continuous beams and frames when the number of independent joint displacements is smaller than the number of force redundants. Its stiffness-based logic leads directly to matrix structural analysis.