The theorem of least work states that redundant reaction forces in a linearly elastic structure with fixed supports minimize its strain energy, with the applied loads
held fixed. Redundant reactions are forces not determined by equilibrium equations
alone. Their values are fixed by displacement compatibility. Castigliano's
theorem expresses each displacement conjugate to a redundant reaction 
as 
. A support with zero displacement therefore
requires
 |
For independent redundant reactions, a stable linear elastic model gives
 |
where 
is a real symmetric matrix that is positive
definite and 
and 
are vectors. Thus 
attains its unique global
minimum at 
.
The minimum follows because 
is positive definite,
rather than a general rule that physical systems minimize every form of energy (Indian
Institute of Technology Kharagpur).
Nonzero prescribed support displacements change the compatibility conditions, so the zero-displacement formulation must then be modified.
