Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 88

The integrals in Eq. (2.51) can be identified as work expressions. The first integral corre-
sponds to the internal virtual work of Eq. (2.35), and the final two integrals are [Eq. (2.36)]
expressions for the work of the actual external forces moving through the corresponding
virtual displacements δ u i . Hence,

− δ W i − δ W e = 0 (2.52)

This can also be expressed as

δ( + ) = δ =

W i W e 0or W 0 (2.53)

with

= +

W W i W e

Equations (2.51), (2.52), or (2.53) are expressions for the principle of virtual work , if the addi-
tional requirement of kinematically admissible virtual displacements is satisfied.
In summary, the equations representing the principle of virtual work appear as

V σ ij δ ij dV

− p Vi δ u i dV − p i δ u i dS = 0

V S p

or

V δ

T σ dV u T p V dV u T p dS −

V δ

−

S p δ

= 0

− δ W i − δ W e = 0 ( 2 . 54 )

with kinematically admissible δ u i , i.e.,

1
2 δ(

1
2 (

δ ij = u i,j + u j,i ) or ij = u i,j + u j,i ) in V

δ u i = 0or u i = u i on S u

or

δ = D δ u or = D u in V

δ u = 0 or u = u on S u

This principle can be stated as the following: A deformable system is in equilibrium if the
sum of the external virtual work and the internal virtual work is zero for virtual displacements
δ u i that satisfy the kinematic equations and kinematic boundary conditions, i.e., for δ u i that are

kinematically admissible.
This principle is independent of the material properties of the solid. Moreover, it has not
been necessary to assume that a potential function exists for the internal or external virtual
work.
The fundamental unknowns for the principle of virtual work are displacements. Although
stresses or forces often appear in the equations representing the principle, these variables
should be considered as being expressed as functions of the displacements. Also, the vari-
ations are always taken on the displacements in the principle of virtual work. In fact, this
principle is also known as the principle of virtual displacements.

It has been shown in this section that for any structural system in equilibrium, Eq. (2.54)
holds. This derivation can be reversed by beginning with Eq. (2.54) and then proving that