Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 114

Further transformation of Eq. (2.95) leads to a generalized form of the principle of sta-
tionary potential energy. Add

V ij σ ij dV V ij σ ij dV

δ − (2.97)

to Eq. (2.95) and use Eq. (2.14) to obtain

V σ

U 0 (σ ) − dV +

ij dV

= U 0 () dV (2.98)

ij

V V

Then Eq. (2.95) appears as

u T p V dV u T p dS δ U 0 () dV − −

V V S p

Principle of virtual work C

σ T p T

− ( − Du ) dV − ( u − u ) dS = 0 (2.99)

V S u

Kinematics B

The underlined terms can be dropped for a kinematically admissible u . The potential func-
tion implied by Eq. (2.99) is

= U 0 () dV − p Vi u i dV − p i u i dS

V V S p

+

V (

− )σ −

S u (

− )

u i, j

ij dV

u i u i p i dS (2.100)

ij

which, by comparison with Eq. (2.64), is an expanded form of the principle of stationary
potential energy.
In a similar fashion, AD = A + D leads to an expanded form of the functional of the

complementary energy:

V (σ ij, j +

∗ =− U 0 (σ ) dV + u i p i dS − p Vi ) u i dV +

S p (

p i − p i ) u i dS (2.101)

V S u

Another important form of the generalized principles leads to the so-called hybrid func-
tionals which are the basis of hybrid methods of analysis . Suppose, for example, that the
displacements u that satisfy the kinematical relations in the body V are introduced, i.e.,
=

Du . Then Eq. (2.99) reduces to

0

u T p V dV u T p dS p T

δ U 0 () dV − − − ( u − u ) dS = (2.102)

V V S p S u

Additional term for the
boundary

Principle of virtual work C

δ u = 0on S u

The corresponding functional is the hybrid functional

u T p V dV u T p dS p T

= () − − − ( − )

U 0 dV u u dS (2.103)

H

V V S p S u