Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 96

In order to extract the corresponding Euler's equations, the first integral of (13) should
be rewritten in a form such that the variations are taken on the displacements and not on
the displacements and their derivatives, as is the case with (13). This will permit (13) to
be expressed such that each term contains the factor u T . The desired transformation is

accomplished by integration by parts, which, in serving to switch the derivatives from one
variable to another, is the one-dimensional equivalent of the multi-dimensional divergence
theorem (Appendix II). This changes the initial integral to

δ

δ

M L

u 0 N + δw V − δθ M ) u 0 N + δw V + δθ

0 −

x (δ

V + δθ dx (14)

The complete expression for the virtual work is now

N + V + M + − δ W =− [ ( p x )δ u 0 + ( p z )δw + ( − V + m )δθ ] dx

x

+ ( )δθ L

0

N − N )δ u 0 + ( V − V )δw + ( M − M (15)

− δ =

Since the variations are arbitrary, W 0 gives the equilibrium equations

N =− V =− M =

p x , p z , V − m 0 ≤ x ≤ L (16)

and the boundary terms

N = N, V = V, M = M at x = 0 ,L (17)

Thus, the principle of virtual work has led to the differential equations of equilibrium and
the static boundary conditions for a beam.
In matrix notation, (15) could be expressed as

u T D s s p dx + δ ) L

0

u T − δ W =−

x δ

+ ( s − s (18)

where from Chapter 1, Eq. (1.116)

d x 00

D s =

0 d x 0 (19)

0 − 1 d x

The equivalent local differ e ntial equilibrium equations would be D s s + p = 0 with the force

boundary conditions s = s . Of course, these are the same relations obtained in Chapter 1,

Section 1.8.
These manipulations are typical of those employed in using the principle of virtual work
to establish local governing differential equations. The divergence theorem is used to trans-
form the integrals, so that the fundamental unknowns (the displacements) on which the
variations are taken can be factored out, thus permitting Euler's equations to be extracted.
This, of course, is the same procedure used to show the equivalence of the global form of
the equilibrium equations and the principle of virtual work.

EXAMPLE 2.7 The Principle of Virtual Work in Terms of Displacements for Beams
In the principle of virtual work, displacements are treated as the unknowns. If = D u u

(δ T = δ( )

T

= δ

u T u D T

) =

E

=

D u u and s ED u u are substituted in Eq. (13) of Example 2.6,