Consider a structural system subjected to a set of forces ( … referred as P force) under stable equilibrium condition as shown in Figure 4.11(a). Further, consider a small element within the structural system and stresses on the surfaces caused by the P forces are shown in Figure 4.11(b) and referred as .

Let the body undergoes to a set of compatible virtual displacement . These displacements are imaginary and fictitious as shown by dotted line. While the body is displaced, the real forces acting on the body move through these displacements. These forces and virtual displacements must satisfy the principle of conservation of energy i.e.

This is the principle of virtual work.

If a system in equilibrium under a system of forces undergoes a deformation, the work done by the external forces ( P ) equals the work done by the internal stresses due to those forces, ( ).

In order to use the above principle for practical applications, we have to interchange the role of the forces and displacement. Let the structure acted upon by a virtual force is subjected to real displacements then the Eq. (4.9) can be written as

This is the principle of complimentary virtual work and used for computing displacements.

Consider a structure shown in Figure 4.12(a) and subjected to P force and it is required to find the displacement of point C in the direction specified. First apply a virtual force at C in the required direction. Next apply the external (real) loads acting on the structures as shown in Figure 4.12(a) with the virtual force remain in the position. The displacement of C in the required ditection be and the internal elements deform by an amount . Using Eq. (4.10)

The left hand side of Eq. (4.11) denotes the external work done by the virtual force moving through the real dispolacement . On the other hand, the right hand side of Eq. (4.11) represents the internal work done by the virtual internal element forces d f moving through the displacement .

where f denotes the internal force in the members due to virtual unit load.