Castigliano's theorem

Italian engineer Alberto Castigliano developed a method of determining the deflection of structures by strain energy method.

He is known for his two theorems :

1. Castigliano's first theorem - for forces in an elastic structure
2. Castigliano's second theorem - for displacements in a linearly elastic structure.

We can discuss the second theorem below :

Castigliano's theorem statement : 

It is mainly used in machine design while designing shafts, keys and coupling.

• When a body is elastically deflected by any combination of forces or moments, the deflection at any point and any direction are equal to the partial derivative of total strain energy of the body with respect to the force located at that point and acting in that direction.

Castigliano's theorem is one of the important techniques for determining the deflection of a complex structure.

Castigliano's theorem is applicable only in the elastic range of the materials.

Procedure :

Steps :

1. Write an expression for each of the internal actions (axial force, bending moment, shear force, and torque) in each member of the structure in terms of external loads. 
2. Take derivatives of strain energy to get deflections and/or rotations.

For example :

Consider an elastic body subjected to a system of forces P₁, P₂, P₃ etc and U is total strain energy of the body. ∂1,∂2,∂3 are deflection at point of application and in direction of P₁, P₂, P₃ etc. then according to the theorem,

∂₁ = ∂U / ∂P₁

∂₂ = ∂U / ∂P₂

∂₃ = ∂U / ∂P₃

∂_I = ∂U / ∂P_i