Civil Engineering Reference
In-Depth Information
USE OF VIRTUAL FORCE SYSTEMS
So far, in all the structural systems we have considered, virtual work has been pro-
duced by actual forces moving through imposed virtual displacements. However, the
actual forces are not related to the virtual displacements in any way since, as we have
seen, the magnitudes and directions of the actual forces are unchanged by the virtual
displacements so long as the displacements are small. Thus the principle of virtual
work applies for
any
set of forces in equilibrium and
any
set of displacements. Equally,
therefore, we could specify that the forces are a set of virtual forces
in equilibrium
and
that the displacements are actual displacements. Therefore, instead of relating actual
external and internal force systems through virtual displacements, we can relate actual
external and internal displacements through virtual forces.
If we apply a virtual force system to a deformable body it will induce an internal virtual
force system which will move through the actual displacements; thus, internal virtual
work will be produced. In this case, for example, Eq. (15.10) becomes
w
i,
N
=
N
v
ε
A
d
x
L
in which
N
v
is the internal virtual normal force and
ε
A
is the actual strain. Then, for
a linear system, in which the actual internal normal force is
N
A
,
ε
A
=
N
A
/
EA
, so that
for a structure comprising a number of members the total internal virtual work due to
a virtual normal force is
N
v
N
A
EA
W
i,
N
=
d
x
L
which is identical to Eq. (15.12). Equations (15.17), (15.21) and (15.22) may be shown
to apply to virtual force systems in a similar manner.
APPLICATIONS OF THE PRINCIPLE OF VIRTUAL WORK
We have now seen that the principle of virtual work may be used either in the form of
imposed virtual displacements or in the form of imposed virtual forces. Generally the
former approach, as we saw in Ex. 15.1, is used to determine forces, while the latter is
used to obtain displacements.
For statically determinate structures the use of virtual displacements to determine
force systems is a relatively trivial use of the principle although problems of this type
provide a useful illustration of the method. The real power of this approach lies in
its application to the solution of statically indeterminate structures, as we shall see in
Chapter 16. However, the use of virtual forces is particularly useful in determining
actual displacements of structures. We shall illustrate both approaches by examples.
E
XAMPLE
15.2
Determine the bending moment at the point B in the simply
supported beam ABC shown in Fig. 15.8(a).
