##  [Castigliano's Theorem](/castiglianos-theorem-0) 

 Definition

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

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

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

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

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

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

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

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

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.