In a linear, elastic, and statically admissible structure under small displacements with conservative loading, the work done by one system of external forces acting through the displacements produced by a second system equals the work done by the second system acting through the displacements produced by the first. Equivalently, influence (flexibility) coefficients are symmetric.

Principle

Reciprocity: pairwise influence is symmetric — the displacement at point i due to a unit load at j equals the displacement at point j due to the same unit load at i, provided linearity and conservative conditions hold.

Demonstration

Illustrative scenario (two-point loading): Situation — apply load A at point 1 and load B at point 2 on a linear elastic frame. Recognition — measure work WA = ∑ Fi·ui(B) where ui(B) are displacements under load B, and WB = ∑ Fi·ui(A) similarly. Action — compute WA and WB. Consequence — WA = WB; experimentally measuring displacement at one point under load at the other yields the reciprocal value.

Misapplication

Assuming reciprocity holds under geometric nonlinearity, when loads are follower-type, when material behaviour is path-dependent, or with active/anisotropic elements; the mistake is treating linear superposition and work symmetry as universally valid.

Consequence

Provides practical shortcuts: experimental measurement or computation of one influence coefficient yields its reciprocal without re-testing; justifies symmetric stiffness and flexibility matrices in linear structural analysis and reduces required calculations or tests.

Reversal

Reciprocity does not hold if linearity is violated (large deformations, geometric nonlinearity), loading is nonconservative (follower forces), the material exhibits nonelastic path dependence, or active elements impose nonreciprocal interactions.

Boundary

Clearly within — small-strain linear-elastic truss or frame under conservative static loads. Boundary case — preloaded or prestrained members where initial stress alters linear response but reciprocity may still hold for incremental linearization if assumptions are met. Clearly outside — systems with nonconservative actuators or strong geometric nonlinearity.

Semantic Tension

Reciprocity (symmetry of influence) ⇄ causality/ordering in load application: reciprocity is a symmetric mathematical property under linear assumptions, while physical experiments and dynamic responses may show directional or time-ordered effects that do not contradict reciprocity but constrain its direct experimental use.

Synthesis

Maxwell–Betti reveals that, in the linear conservative regime, structural response coefficients are symmetric—this reduces testing and computation by allowing one measured or computed influence to stand for its reciprocal, but the symmetry depends strictly on linearity and energy-conservation assumptions.