For a linearly elastic, conservative structure with well-defined strain energy U as a function of applied generalized loads, the partial derivative of U with respect to an applied force equals the displacement at the point and in the direction of that force; similarly, the partial derivative of U with respect to an applied moment equals the rotation at that location. The theorem requires differentiable strain energy and small deformations so that virtual work and superposition hold.

Principle

Energy–kinematic duality: incremental change in stored strain energy produced by an infinitesimal change in a generalized load equals the conjugate generalized displacement produced by that load.

Demonstration

Illustrative scenario (linear elastic beam): Situation — a simply supported beam carries a concentrated vertical load P at midspan. Recognition — write internal bending moment M(x) as function of P and compute strain energy U = ∫ (M(x)^2 / 2EI) dx. Action — form ∂U/∂P. Consequence — ∂U/∂P yields the vertical deflection at the load point, reproducing the same displacement found by direct deflection formula, demonstrating the theorem in operation.

Misapplication

Treating the theorem as valid for structures with nonconservative (follower) loads, time-dependent materials (viscoelasticity), large deformations, or when strain energy is not uniquely defined; the error is assuming differentiability and energy-conservation where they do not hold.

Consequence

Enables displacement or rotation calculation from internal force expressions without solving equilibrium equations for displacements directly; facilitates analysis of statically indeterminate systems and derivation of influence coefficients when the elastic constitutive relation and geometry are known.

Reversal

Fails or requires modification when materials are nonlinear (plasticity), loading includes follower forces or nonconservative work, deformations are large enough to invalidate linearization, or when path-dependent dissipative processes alter stored energy.

Boundary

Clearly within — linear-elastic prismatic beam under conservative static loads. Boundary case — structure with small plastic zones where elastic–plastic partitioning may permit local use but global differentiability is lost. Clearly outside — rigid-body mechanisms (no strain energy) or structures dominated by viscous dissipation.

Semantic Tension

Energy methods (Castigliano) ⇄ equilibrium/stiffness methods: energy-based formulations produce displacements via differentiating scalar energy while stiffness/matrix methods produce displacements via solving equilibrium; each approach trades scalar integrals for system solves and may be preferred depending on convenience and available information.

Synthesis

Castigliano expresses a direct duality: once the strain energy functional is known, kinematic responses are available by simple differentiation—this converts a field of internal forces into measurable displacements, but only within the linear, conservative regime where a single-valued energy exists.