Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 86

2.2.1 Principle of Virtual Work

The principle of virtual work for a solid can be derived from the equations of equilibrium
and vice versa. They are, in a sense, equivalent because the principle of virtual work is a
global (integral) form of the conditions of equilibrium and the static boundary conditions. To
look at this principle, consider a solid under prescribed body forces for which the conditions
of equilibrium hold at all points throughout the body, i.e., [Chapter 1, Eq. (1.53)]

σ + p Vi = 0in V

ij, j

or

D T σ + p V = 0 in V (2.42)

and for which the force (mechanical or static) boundary conditions are given on the surface,
i.e., [Chapter 1, Eq. (1.60)]

p i = p i or p i − p i = 0on S p

or

p = p or p − p = 0 on S p (2.43)

where p are tractions applied to the surface area S p . Recall that overbars signify applied
quantities. Equations (2.42) and (2.43) constitute the definition of a statically admissible
stress field.
The local conditions of equilibrium [Eq. (2.42)] and the static boundary conditions [Eq.
(2.43)] are now to be expressed in global (integral) form. To do so, multiply Eqs. (2.42) and
(2.43) by the virtual displacements u i , and integrate the first relation over V and the second

over S p . Take the sum of these two integrals, each of which is equal to zero, to form

δ

−

V (σ

+ p Vi )δ u i dV +

S p (

p i − p i )δ u i dS = 0

ij, j

or

u T D T σ + u T −

V δ

( p V ) dV +

S p δ

( p − p ) dS = 0 (2.44)

It is necessary to introduce the negative sign in order to obtain consistent relations later.
Note that according to the fundamental lemma of the calculus of variations (Appendix I),
the integral relations of Eq. (2.44) are equivalent to the local conditions of Eqs. (2.42) and
(2.43).
Now change the form of Eq. (2.44) such that the integrals can be identified as being
virtual work. This is accomplished using Gauss' integral theorem of Appendix II, Eq. (II.8)
[which was employed to de rive Eq. (2.40)]. Since S = S p + S u , where S u is the surface area

on which displacements u are applied, it is possible to write the integral S p p i δ u i dS of

Eq. (2.44) as

p i δ u i dS = p i δ u i dS − p i δ u i dS (2.45)

S p S S u