1. Castigliano’s theorems
Castigliano’s first theorem:
For linearly elastic structure, the Castigliano’s first theorem may be defined as the first partial derivative of the strain energy of the structure with respect to any particular force gives the displacement of the point of application of that force in the direction of its line of action.

2. Castigliano’s First theorem derivation
Consider an elastic beam AB subjected to loads W1 and W2, acting at points 1 and 2 respectively

3. Castigliano’s First theorem derivationIf
where ∆ 11 = deflection at 1 due to a unit load at 1 and
with ∆ 21 = deflection at 2 due to a unit load at 1

4. Castigliano’s First theorem derivation
, with ∆ 22 = deflection at 2 due to a unit load at 2 &
, with ∆ 12 = deflection at 1 due to a unit load at 2.
Then
Similarly,
Considering the work done = Ui

5. Castigliano’s First theorem derivation
Now applying W2 at Point 2 first and then applying W1 at Point 1,
Similarly,
Strain energy, Ui

6. Castigliano’s First theorem derivation
Considering equation (III) and (IV), and equating them, it can be shown that
This is called Betti – Maxwell’s reciprocal theorem
Deflection at point 2 due to a unit load at point 1 is equal to the deflection at point 1 due to a unit load at point 2. 

7. Castigliano’s First theorem derivation
From Eqn. (III),
From Eqn. (IV),
This is Castigliano’s first theorem. ∴

8. Castigliano’s second theorem
Similarly the energy Ui can be express in terms of spring stiffnesses k11, k12 (or k21), & k22 and deflections δ1 and δ2; then it can be shown that
This is Castigliano’s second theorem.
When rotations are to be determined,