Castigliano's method

Castigliano's method calculates forces or displacements in elastic structures by differentiating strain energy with respect to loads or imposed displacements.

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Strain energy: Energy stored in a material as it deforms elastically. This is the energy function differentiated to obtain structural forces and displacements.
Bending moment: The internal moment that resists bending in a structural member. Its square contributes to beam bending strain energy.
Beam deflection: The transverse displacement of a beam caused by loading. Differentiating beam strain energy yields deflection at a chosen load location.
Principle of minimum potential energy: An equilibrium principle stating that a stable conservative system minimizes total potential energy. It finds equilibrium through energy stationarity rather than response derivatives.
Castigliano's first theorem: A theorem relating displacement to the derivative of strain energy with respect to an applied force. It gives displacements by differentiating energy with respect to force.
Torsion: The twisting of a body under an applied torque. Torsional energy must be included when a structure carries torque.
Statically indeterminate structure: A structure whose support reactions or internal forces cannot be found from equilibrium alone. Compatibility conditions and energy derivatives can determine its redundant forces.
Maxwell–Betti reciprocal theorem: A theorem stating reciprocal equality between paired force-displacement effects in linear elastic systems. It connects the reciprocal displacements that energy-based structural analysis predicts.
Castigliano's second theorem: A theorem relating force to the derivative of complementary strain energy with respect to an imposed displacement. It gives forces when the corresponding displacement is prescribed.
Axial force: An internal force acting along a member's longitudinal axis. Axial deformation adds an energy term alongside bending and torsion.