Energy methods

When a structure deforms elastically, the work done by the loads is stored as strain energy, like a compressed spring. Energy methods use this to find deflections and to solve indeterminate structures without writing differential equations. They are general, work for trusses, beams and frames alike, and form a bridge to the matrix and finite element methods used in software.

Strain energy for axial load is P² L / (2 A E); for bending, ∫ M² / (2 E I) dx; for torsion, ∫ T² / (2 G J) dx. Castigliano's first theorem gives load from the derivative of energy with respect to displacement, and his second theorem gives the deflection at a load as the partial derivative of total strain energy with respect to that load: δ = ∂U/∂P. If you need deflection where there is no load, add a dummy load Q, differentiate, then set Q = 0. This is equivalent to the unit load method.

For indeterminate structures, the theorem of least work says that redundant forces take the values that minimise strain energy, so ∂U/∂R = 0 for each redundant. Maxwell's reciprocal theorem states that the deflection at A due to a load at B equals the deflection at B due to the same load at A, which makes flexibility matrices symmetric. Impact problems use energy too: a weight dropped onto a bar produces much larger stress than the same weight applied slowly.