Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 89

the equilibrium conditions are satisfied. This implies that the principle of virtual work
[Eq. (2.54)] is a necessary and sufficient condition for a structure to be in equilibrium.

Briefly, the term V δ T σ dV in Eq. (2.54) can, with the aid of Eq. (2.48) [a condition
underlying Eq. (2.54)], be transformed to

V σ ij δ V (σ ij δ

V σ ij δ ij dV

= u i,j dV = u i ) ,j dV −

V σ ij, j δ

u i dV (2.55)

With the help of the divergence theorem [Eq. (2.40)] the first integral on the right-hand
side can be transformed to a surface integral. Then

V σ ij, j δ

V σ ij δ ij dV

= p i δ u i dS + p i δ u i dS − u i dV (2.56)

S u S p

Invoking the remaining kinematic boundary condition in Eq. (2.54), i.e., 0on S u ,

the integral over S u in Eq. (2.56) is zero. Substitution of Eq. (2.56) into the first relation of
Eq. (2.54) gives Eq. (2.44), which can only be zero if the system is in equilibrium [Eq. (2.42)]
and if the static boundary conditions are satisfied. That is, according to the calculus of
variations, the principle of virtual work leads to the equilibrium conditions as Euler's
equations with the static (force) boundary conditions as the natural boundary conditions.
Thus, the principle of virtual work is basically a global formulation of the equilibrium
conditions and the static boundary conditions.

δ u i =

The principle of virtual work as expressed by Eq. (2.54) formally contains both stresses
and displacements, although only the displacements are the fundamental unknowns. This
equation can be written in a pure displacement form by replacing the stresses by displace-
ment gradients with the assistance of the stress-strain relations (a material law), which,
of course, makes the principle material dependent. Note that no use of the stress-strain
relations has been made in deriving the principle of virtual work of Eq. (2.54). Thus, this
form is valid for systems with nonlinear materials, as well as for linear materials. For the
case of linearly elastic material [Chapter 1, Eq. (1.34)], σ =

E . Using the kinematic relation

= Du , we find

σ = E = E ( Du ) = ED u u (2.57a)

Since T = ( Du )

T , we have

u T u D T (2.57b)
In Eq. (2.57a), the subscript u has been added to D to indicate clearly that the differential
operator matrix D operates on u . In Eq. (2.57b), however, as explained in Chapter 1, Section
1.8.3, the subscript index to the left of the operator matrix, u D T , signifies the application
of the operator to the preceding quantity which, in this case, is

δ T

T

= δ( Du ) = δ

u T . Substitution of Eqs. δ

(2.57a) and (2.57b) into the second relationship of Eq. (2.54) gives
V δ

u T u D T ED u u p V dV

u T p dS − −

S p δ

= 0 (2.58a)

Define

k D

u D T ED u = (2.58b)

where, due to the definition of u D and D u , the operator matrix k D is recognized as a sym-
metric operator matrix. Then the displacement form for the principle of virtual work relation
is

u T k D u u T p dS

− δ =

V δ

( −

p V )

−

S p δ

=

W dV 0 (2.58c)