Building Design and Construction Handbook - Merritt - Ventech!

5.68 SECTION FIVE
FIGURE 5.49 Principle of virtual work applied
to determination of a simple-beam reaction
(a) and (b) and to the reaction of a beam with
a suspended span (c) and (d).
As an example of how the principle
may be used to find a reaction of a statically
determinate beam, consider the
simple beam in Fig. 5.49a, for which the
reaction R is to be determined. First, replace
the support by an unknown force
R. Next, move that end of the beam upward
a small amount dy as in Fig. 5.49b.
The displacement under the load P will
be xdy/L, upward. Then, by the principle
of virtual work, Rdy� Px dy/L �
0, from which R � Px/L.
The principle may also be used to
find the reaction R of the more complex
beam in Fig. 5.49c. The first step again
is to replace the support by an unknown
force R. Next, apply a virtual downward
displacement dy at hinge A (Fig. 5.49d
). Displacement under load P is xdy/c,
and at the reaction R, ady/(a � b). According
to the principle of virtual work,
�Ra dy/(a � b) � Px dy/c � 0, from
which reaction R � Px(a � b)/ac. In
this type of problem, the method has the
advantage that only one reaction need
be considered at a time and internal
forces are not involved.
5.10.2 Strain Energy
When an elastic body is deformed, the
virtual work done by the internal forces
is equal to the corresponding increment
of the strain energy dU, in accordance with the principle of virtual work.
Assume a constrained elastic body acted upon by forces P 1, P 2,...,forwhich
the corresponding deformations are e 1, e 2 ....Then, �P n de n � dU. The increment
of the strain energy due to the increments of the deformations is given by
�U �U
dU � de1 � de2 � ���
�e �e
1 2
In solving a specific problem, a virtual displacement that is not convenient in simplifying
the solution should be chosen. Suppose, for example, a virtual displacement
is selected that affects only the deformation e n corresponding to the load P n, other
deformations being unchanged. Then, the principle of virtual work requires that
This is equivalent to
�U
P de � de
n n n
�en