CASTIGLIANO THEOREM

• Castigliano's first theorem – for forces in an 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.

In equation form,

where U is the strain energy.

• Castigliano's second theorem – for displacements in a linearly elastic structure.

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.

As above this can also be expressed as: