Castigliano's theorem relates the strain energy of an elastic structure to its applied forces and corresponding displacements through partial
derivatives. At constant temperature with fixed supports, its displacement formulation
states that if 
is the strain energy, the force conjugate to 
is
 |
For a linearly elastic structure, the load formulation instead gives
 |
where 
is expressed as a function of the applied forces. A
force and its conjugate displacement may also be a moment and an angular displacement
(Purdue University, p. 1).
For example, a stable linear model has a real symmetric matrix 
that is positive definite,
with 
and 
.
Expressing the same energy as 
verifies both formulations by differentiation.
The load formulation leads to the theorem of least
work for redundant support forces.
