Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 102

in V and the static (mechanical) boundary conditions [Chapter 1, Eq. (1.60)] on S p . Then,
for the first variation of the stresses, it follows that

δσ = 0in V (2.71)

ij, j

δ p i = 0on S p (2.72)

In Eq. (2.71), no p V term appears, since p V is prescribed and its variation would be zero.
Equation (2.72) follows from [Chapter 1, Eq. (1.58)] σ ij a i = p j or δσ ij a i = δ p j .If δσ ij = 0

on S p , then 0on S p .

The conditions of Eqs. (2.71) and (2.72) require that two of the integrals in the Gauss
integral expression of Eq. (2.41) be zero, leaving

δ p j =

u i δ p i dS = u i,j δσ ij dV (2.73)

S u V

Substitution of Eq. (2.73) into Eq. (2.70) gives

V

δσ

ij dV

− u i δ p i dS = 0 (2.74)

ij

S u

or

V δ σ T dV

p T u dS

−

S u δ

= 0

The integrals in Eq. (2.74) can be interpreted as virtual work expressions. The first integral
corresponds to the definition of complementary work of internal forces [Eq. (2.37)]

V δ σ T dV

W i = − δ

V ij δσ ij dV

= (2.75)

The external complementary virtual work [Eq. (2.38)], in terms of prescribed displacements,
is

W e = p T u dS

δ u i δ p i dS =

S u δ

(2.76)

S u

Thus, Eq. (2.74) can be written as

W i − δ W e =− δ W ∗ W i + W e ) =

− δ or − δ( 0 (2.77)

with W ∗ = W i + W e

.

Equation (2.74) or (2.77) and the requirement of statical admissibility
constitute the principle of complementary virtual work . This is also known as the principle of
virtual stresses and as the principle of virtual forces.
In summary,

V ij δσ ij dV

− u i δ p i dS = 0

S u

or

σ T dV p T u dS

V δ

−

S u δ

= 0

W i W e =

− δ − δ 0 ( 2 . 78 )

with statically admissible δσ ij , i.e.,

D T σ δσ = 0 or = 0 in V

ij,j

δ p i = 0or p = p on S p