Castigliano's theorem analyzes displacements and rotations in elastic structures. It relates the derivative of elastic strain energy to the applied forces or moments, allowing for the calculation of deformations. The theorem states that the partial derivative of the total strain energy of a system with respect to a specific load results in the displacement at the point where the load is applied. This principle applies to both forces and moments.

Strain energy, which measures the work done by external loads during elastic deformation, is crucial for these calculations. In conditions such as beams with non-uniform cross-sections or complex loading, the deflection at any point can be derived using Castigliano's theorem by differentiating the strain energy with respect to the load.

Similarly, for structural elements like shafts undergoing torsional deformation due to applied torque, the angle of twist can be determined using the same method. The strain energy due to torsion accounts for the material's properties and the geometrical attributes of the shaft.

Through Castigliano's theorem, the behavior of the structures under various loading conditions is predicted, aiding in the design of more efficient and safer systems. This theorem enhances understanding of structural responses and facilitates the optimization of structural designs.