In any elastic structure having n independent displacements u1, u2, ....un corresponding to external forces P1, P2, ....Pn along their lines of action, if strain energy is expressed in terms of displacements.
n equilibrium equations may be written as follows.
This may be proved as follows. The strain energy of an elastic body may be written as
where Kij is the stiffness coefficient and is defined as the force at due to unit displacement applied at j. Hence, strain energy may be written as,
We know from reciprocal theorem . Hence, equation may be simplified as,
Now, differentiating the strain energy with respect to any displacement u1 gives the applied force p1 at that point, Hence,