# Castiglianos Theorems

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 Castiglianos first theorem.
If no displacement can occur at a support and Castiglianos theorem is applied to that support, Eq. (3.114) becomes

Equation (3.115) is commonly called the principle of least work, or Castiglianos 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. Castiglianos second theorem is restricted to linear elastic structures. On the other hand, Castiglianos 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