Castigliano's 1st theorem

Castigliano's 2nd theorem

1.The first partial derivative of total internal energy (strain energy) in a structure with respect to any particular deflection component at a point is equal to the force applied at that point and in the direction corresponding to that deflection component.

1.The first partial derivative of total internal energy in a structure w.r.t the force applied at any point is equal to the deflection at the point of application of that force in the direction of its line of action.

2.Castigliano's 1st theorem is applicable to linearly or non-linearly elastic structures in which the temperature is constant and the supports are unyielding.

2.Castigliano's 2nd theorem is applicable to linearly elastic (Hookean material) structures with constant temperature & unyielding supports.