What is strain energy and how is it used in analysis?

Strain energy is the energy stored in a deformed elastic body, equal to work done by external forces. For axial loading U = P²L/(2AE), for bending U = ∫M²dx/(2EI), for torsion U = T²L/(2GJ). Castigliano's theorem uses ∂U/∂P = δ to find deflections. Strain energy methods are powerful for complex and indeterminate structures.