Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 85

This represents the actual external loads moving through the corresponding virtual dis-
placements.
Also useful is the concept of complementary virtual work in which the variation applies to
stress rather than displacement. The internal complementary virtual work is defined as

W i =− δ

V ij δσ ij dV

(2.37)

while the external complementary virtual work would be

W e =

δ u i δ p i dS (2.38)

S u

which corresponds to the complementary work of Eqs. (2.17) and (2.29), respectively.
Surface integrals can be transformed into volume integrals and vice versa using the
Gauss (divergence) integral theorem described in Appendix II, Section II.3. Typically, the
transformation of a surface integral of the sort

p i δ u i dS (2.39)

S

into a volume integral is desired. Use Chapter 1, Eq. (1.58), p i = σ

ij a j , and the Gauss

integral theorem of Appendix II, Eq. (II.8) to obtain

V (σ V σ

p i δ u i dS = δ u i ) , j dV = δ u i dV +

V σ

δ u i,j dV (2.40)

ij ij, j ij

S

If the variation in stress is considered, Gauss' theorem takes the form

δ = δσ =

V (

δσ )

u i p i dS u i

ij a i dS

u i

,j dV ij

S S

= u i δσ ij, j dV + u i,j δσ ij dV (2.41)

V V

2.2 Classical Variational Principles of Elasticity

In this section, the classical variational principles which are essential for pursuing varia-
tional formulations of structural mechanics are presented. At first glance, such principles
tend to appear very comprehensive due, in part, to the general nomenclature and terminol-
ogy that are used to express them. As will be seen, this generality is necessary to develop a
proper foundation. In order to understand the basic concepts better, several simple exam-
ples are given.
Before considering the variational principles, it is worthwhile to mention the principle
of conservation of energy, which is a familiar energy theorem. For solids of interest to us,
this principle is the first law of thermodynamics for adiabatic processes. As an example,
consider an elastic solid with static loading for which there is no loss of energy through
the conversion of mechanical work into heat or through friction or other dissipative forces.
Assume a potential exists for the internal forces. For this conservative system, the principle
of conservation of energy is as follows: work done by the applied forces is equal to the
strain energy stored in the solid. It is important to understand that this principle deals with
changes in energy because the work being done corresponds to a change in energy.