Castigliano's method

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Castigliano's method, named for Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the strain energy. In the mathematics of structures, he is known for his two theorems:

Castigliano's first theorem – for forces in an elastic structure
Castigliano's second theorem – for displacements in a linearly elastic structure

Castigliano's method for calculating forces is an application of his first theorem, which states:

If the strain energy of an elastic structure can be expressed as a function of generalised displacement q_i; then the partial derivative of the strain energy with respect to generalised displacement gives the generalised force Q_i.

Castigliano's method for calculating displacements is an application of his second theorem, which states:

If the strain energy of a linearly elastic structure can be expressed as a function of generalised force Q_i; then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q_i in the direction of Q_i.