Energy theorems

When a structure deforms under load, the work done by the loads is stored as strain energy in its members. Energy theorems turn that bookkeeping into a way of finding deflections and redundant forces. They are especially handy for frames, curved members and trusses, where drawing deflected shapes by geometry is awkward.

Castigliano's second theorem says that the partial derivative of the total strain energy with respect to a load gives the deflection at that load, in its direction. To find a deflection where there is no load, a dummy load is added, the derivative taken, and the dummy then set to zero. The unit load method, also called virtual work, reaches the same result more directly: apply a unit load at the point and in the direction of the wanted deflection, find the moment m it causes, and integrate M times m over EI along the structure.

For indeterminate structures, the theorem of least work, a corollary of Castigliano, states that the redundant forces take values that make the strain energy a minimum, so the derivative of energy with respect to each redundant is zero. The Maxwell-Betti reciprocal theorem says the deflection at A due to a unit load at B equals the deflection at B due to a unit load at A; it underlies flexibility matrices and influence line methods.

In bending-dominated members the bending energy is usually all that matters; axial and shear energy become important for trusses, short deep members and arches. These theorems assume linear elastic behaviour and small deflections, so they connect naturally to the matrix stiffness method used in software.