Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 103

This principle can be stated as the following: a deformable system satisfies all kinematical
requirements if the sum of the external complementary virtual work and the internal complementary
virtual work is zero for all statically admissible virtual stresses

ij .
The fundamental unknowns for the principle of complementary virtual work are stresses (forces).
The variations are always taken on the stresses or forces.

δσ

As indicated previously, Eq. (2.78) holds if the displacement field of a structural system
is kinematically admissible. Conversely, it can be shown that the conditions of Eq. (2.78)
lead to kinematically admissible displacements. To do so, transform the first integral in
Eq. (2.74) as follows:
V ij δσ ij dV

=

V ( ij −

u i,j )δσ ij dV + u i, j δσ ij dV

V

=

V ( ij −

u i,j )δσ ij dV +

V (

u i δσ ij ) ,j dV − u i δσ ij, j dV (2.79)

V

Apply Gauss' theorem to the second integral on the right-hand side giving
V (

u i δσ ij ) ,j dV = u i δ p i dS = u i δ p i dS + u i δ p i dS (2.80)

S S p S u

With the aid of Eqs. (2.79) and (2.80) and the conditions of Eqs. (2.71) and (2.72), Eq. (2.70)
can be recovered from Eq. (2.74). Thus, the Euler equations of the variational princi-
ple (2.78) are the strain-displacement relations [Chapter 1, Eqs. (1.19) or (1.21)] and the
displacement (geometrical) boundary conditions [Chapter 1, Eq. (1.61)] are the natural
boundary conditions.
Thus, the fulfillment of the principle of complementary virtual work is an alternative
statement of the conditions for kinematic admissibility of the displacement field.

Finally, a material law can be introduced to express the strains in Eq. (2.78) in terms of the
stresses, and the surface tractions p can be written as functions of the stresses. The stresses
or forces and not the displacements are the fundamental unknowns for the principle of
complementary virtual work. Thus, setting = E − 1 σ [Chapter 1, Eq. (1.32)] and p A T σ =

[Eq. (1.57)], the second relation of Eq. (2.78) becomes

σ T E − 1 σ dV σ T Au dS

V δ

−

S u δ

= 0 (2.81)

This expression can be used as the basis for the force method of structural analysis.

EXAMPLE 2.9 Torsion, An Example of Field Theory Equations
Derive the force form of the governing equations for Saint Venant torsion. See Chapter
1, Section 1.9, and Example 2.8 for related material, including notation and definitions.
Consider a shaft of length L with the x

=

a end fixed and with an angle of twist of

magnitude b end.

The principle of complementary virtual work [Eq. (2.78)] takes the form

φ L imposed at the x =

V

δσ

ij dV

− u i δ p i dS

ij

S u

S u ( − φ Lz

δτ xy + φ Ly =

V (γ xy δτ xy + γ xz δτ xz )

dV − δτ xz ) dS = 0 (1)

= v =− φ Lz and u 3 = w = φ Ly . Also, from Chapter where, from Chapter 1, Eq. (1.139), u 2

1, Eq. (1.146), p y = τ

xy and p z

= τ

xz on the x

= b end. Introduce the Prandtl stress function

ψ

of Chapter 1, Eq. (1.155). This implies that the conditions of equilibrium are satisfied as