Castigliano's method

Remarks

The first theorem follows from the work–energy balance: the change of $U$ when $q_i$ alone changes by $dq_i$ is the work $F_i \, dq_i$. For the second theorem, linearity gives $q = A F$ with a symmetric flexibility matrix $A$, so $U = \tfrac12 F \cdot A F$ and $\partial U / \partial F_i = (A F)_i = q_i$. For nonlinear elastic structures the second theorem holds with the complementary energy in place of $U$. Example: a cantilever of length $L$ and bending stiffness $EI$, loaded by $P$ at its tip, has $U = \int_0^L \frac{(P x)^2}{2EI} \, dx = \frac{P^2 L^3}{6 EI}$, so the tip deflection is $\partial U / \partial P = \frac{P L^3}{3 EI}$. To find a displacement where no force acts, one adds a fictitious force there, differentiates, and sets it to $0$.

Going deeper

For a student

The method is standard in the analysis of beams, frames, trusses and springs. For statically indeterminate structures, the second theorem applied to a redundant reaction whose support does not move gives $\partial U / \partial R = 0$, an extra equation that determines $R$. It is closely related to Betti's reciprocal theorem, which gives the symmetry of the flexibility matrix, and to the principle of minimum complementary energy.