**Introduction:
**

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.

**Description:**

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 K_{ij} 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,