Civil Engineering Reference
In-Depth Information
VIRTUAL WORK IN A DEFORMABLE BODY
In structural analysis we are not generally concerned with forces acting on a rigid body.
Structures and structural members deform under load, which means that if we assign a
virtual displacement to a particular point in a structure, not all points in the structure
will suffer the same virtual displacement as would be the case if the structure were
rigid. This means that the virtual work produced by the internal forces is not zero as it
is in the rigid body case, since the virtual work produced by the self-equilibrating forces
on adjacent particles does not cancel out. The total virtual work produced by applying
a virtual displacement to a deformable body acted upon by a system of external forces
is therefore given by Eq. (15.6).
If the body is in equilibrium under the action of the external force system then every
particle in the body is also in equilibrium. Therefore, from the principle of virtual
work, the virtual work done by the forces acting on the particle is zero irrespective
of whether the forces are external or internal. It follows that, since the virtual work
is zero for all particles in the body, it is zero for the complete body and Eq. (15.6)
becomes
W
e
+
W
i
=
0
(15.8)
Note that in the above argument only the conditions of equilibrium and the concept
of work are employed. Thus Eq. (15.8) does not require the deformable body to be
linearly elastic (i.e. it need not obey Hooke's law) so that the principle of virtual work
may be applied to any body or structure that is rigid, elastic or plastic. The principle
does require that displacements, whether real or imaginary, must be small, so that
we may assume that external and internal forces are unchanged in magnitude and
direction during the displacements. In addition the virtual displacements must be
compatible with the geometry of the structure and the constraints that are applied,
such as those at a support. The exception is the situation we have in Ex. 15.1 where
we apply a virtual displacement at a support. This approach is valid since we include
the work done by the support reactions in the total virtual work equation.
WORK DONE BY INTERNAL FORCE SYSTEMS
The calculation of the work done by an external force is straightforward in that it
is the product of the force and the displacement of its point of application in its
own line of action (Eqs (15.1), (15.2) or (15.3)) whereas the calculation of the work
done by an internal force system during a displacement is much more complicated. In
Chapter 3 we saw that no matter how complex a loading system is, it may be simplified
to a combination of up to four load types: axial load, shear force, bending moment
and torsion; these in turn produce corresponding internal force systems. We shall
now consider the work done by these internal force systems during arbitrary virtual
displacements.
