Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 110

Essentially, this combination, i.e., A B which we will denote as AB, along with the material

law is a global form of all the basic equations of Table 2.1. The combined form AB can
be thought of as being derived from a general variational form (functional) by employing
variations of the displacements and stresses simultaneously. Note that for the displacements
and stresses, derivatives of at least the first order must exist.
Combinations, such as AB, of the variational principles of Table 2.2 are referred to as
generalized variational principles. They can be considered to be extensions of classical vari-
ational principles to which additional terms corresponding to governing equations not
yet considered have been added with the aid of Lagrange multipliers (Appendix I). Thus,
the functional corresponding to A

+

B can be considered to have been constructed from
the functional corresponding to B which incorporates the kinematical equations and dis-
placement boundary conditions to which the constraints for the force boundary conditions
and the equilibrium equations are appended with the displacements u (or

+

δ u ) as Lagrange

multipliers.
Suppose the expression AB is represented by AB is the corresponding

functional. Then Eq. (2.86) implies that the relationships that render

δ AB , where

AB stationary are the

complete governing equations of elasticity. Note that 0 leads to only a station-

ary value of the functional. It cannot be proven that the stationary value is a minimum.
Normally this same situation occurs with other generalized variational principles.
Table 2.3 shows three other possible combinations: C

δ AB =

+ D = CD, C + B = CB, and

A + AD [Wunderlich, 1973].

The four combinations may be summarized as follows:

D =

A + B = AB Global form of the fundamental equations

C + D = CD Combination of the principle of virtual work and the

principle of complementary virtual work

C + B = CB Extended principle of virtual work

A + D = AD Extended principle of complementary virtual work

2.3.1 Matrix Form
The generalized variational principles are displayed with index notation in Table 2.3. In
order to utilize these principles for numerical solutions, it is useful to write them in matrix
notation [Wunderlich, 1972]. Previously, the combination A

+ B = AB was treated in this

way; now the case of C + B = CB will be discussed. From C and B of Table 2.2,

δ

T σ

σ T dV σ T Du

u T p V dV

−

V δ

+ + δ( ) −

V δ

Du dV

V

= V δ(σ ij u i, j ) dV in index notation

) dS
This term vanishes if the
displacement boundary
conditions are satisfied.

S u δ p T

u T p dS −

S p δ

− ( u − u = 0 (2.88)

In order to convert this relationship to a more useful matrix form, introduce the material
law

E − 1 σ , and make use of the notation

=

σ T Du σ T D u u ,

T σ

u T D T u T u D T σ δ = δ δ( Du ) = δ( ) σ = δ

(2.89)

σ T A
where D u has been used specifically to indicate that D operates on u , and u D T

A T σ , p T

p = =

signifies the

application of the operator D T to the preceding variable u T . Then Eq. (2.88) can be rewritten