If strain energy U, as defined in Art. 3.23, is expressed as a function of external forces, the partial derivative of the strain energy with respect to one of the external forces Pi gives the displacement i corresponding to that force:
This is known as Castigliano’s first theorem.
If no displacement can occur at a support and Castigliano’s theorem is applied to that support, Eq. (3.114) becomes
Equation (3.115) is commonly called the principle of least work, or Castigliano’s second theorem. It implies that any reaction components in a structure will take on loads that will result in a minimum strain energy for the system. Castigliano’s second theorem is restricted to linear elastic structures. On the other hand, Castigliano’s first theorem is only limited to elastic structures and hence can be applied to nonlinear structures.
As an example, the principle of least work will be applied to determine the force in
the vertical member of the truss shown in Fig. 3.61. If Sa denotes the force in the vertical bar, then vertical equilibrium requires the force in each of the inclined bars to be (P – Sa) (2 cos alpha). According to Eq. (3.103), the total strain energy in the system is