Elastic deformation stores strain energy. Once that stored energy has been expressed in terms of the applied loads, Castigliano's theorem can extract selected displacements and rotations without solving the complete deflected shape.

For a linearly elastic structure with total strain energy $U$, the displacement in the direction of an applied generalized force $P$ is

$$\boxed{\delta_P=\frac{\partial U}{\partial P}}.$$

Similarly, if an applied generalized moment $M_0$ is used as a load parameter, the corresponding rotation is

$$\boxed{\theta=\frac{\partial U}{\partial M_0}}.$$

Strain energy in Euler-Bernoulli bending

For a slender Euler-Bernoulli beam, the bending strain energy is

$$\boxed{U=\int\frac{M^2(x)}{2EI},dx},$$

where $M(x)$ is the internal bending moment, $E$ Young's modulus, and $I$ the second moment of area.

Regions with large bending moment or low flexural rigidity $EI$ contribute most strongly to the stored energy.

If axial, torsional, shear, or other deformation modes are also important, their strain-energy contributions must be included as appropriate. Castigliano's theorem acts on the total elastic strain energy of the model.

Worked example: cantilever with an end load

Consider a cantilever beam of length $L$ and constant flexural rigidity $EI$, carrying a downward end load $P$.

Measure $x$ from the fixed end. The bending-moment magnitude is

$$M(x)=P(L-x).$$

The bending strain energy is therefore

$$U=\int_0^L\frac{P^2(L-x)^2}{2EI},dx.$$

Since

$$\int_0^L(L-x)^2,dx=\frac{L^3}{3},$$

we obtain

$$U=\boxed{\frac{P^2L^3}{6EI}}.$$

Castigliano's theorem gives the end displacement in the direction of $P$:

$$\delta_P=\frac{\partial U}{\partial P} =\boxed{\frac{PL^3}{3EI}}.$$

This reproduces the familiar cantilever-tip deflection without integrating the beam-curvature equation twice.

For example, if

$$P=1000,\mathrm N,\qquad L=2.0,\mathrm m,$$

and

$$EI=2.0\times10^6,\mathrm{N,m^2},$$

then

$$\delta_P =\frac{(1000)(2.0)^3}{3(2.0\times10^6)} \approx1.33\times10^{-3},\mathrm m.$$

Thus

$$\boxed{\delta_P\approx1.33,\mathrm{mm}}.$$

Why the method works well

A direct beam solution determines a displacement field first and then evaluates the point of interest. An energy method can instead target one displacement directly after the internal-force distribution is known.

This is particularly useful for structures where equilibrium gives internal forces readily but a full displacement solution would be cumbersome.

Castigliano's theorem is therefore a structural application of the broader relation between mechanical work and stored elastic strain energy.