Page 88 - Aircraft Stuctures for Engineering Student

4.4 Stationary value of the total potential energy 73

is given by

Therefore, since S W* = SU*

(4.9)

Equation (4.9) is known as the principle of virtual forces. Comparison of the right-
hand side of Eq. (4.9) with Eq. (4.2) shows that SU* represents an increment in
complementary energy; by the same argument the left-hand side may be regarded
as virtual complementary work.
Although we are not concerned with the direct application of the principle of
virtual work to the solution of structural problems it is instructive to examine possible
uses of Eqs (4.8) and (4.9). The virtual displacements of Eq. (4.8) must obey the
requirements of compatibility for a particular structural system so that their relation-
ship is unique. Substitution of this relationship in Eq. (4.8) results in equations of
statical equilibrium. Conversely, the known relationship between forces may be
substituted in Eq. (4.9) to form equations of geometrical compatibility. Note that
the former approach producing equations of equilibrium is a displacement method,
the latter giving equations of compatibility of displacement, a force method.

4.4 The principle of the stationary value of the
total potential energy

In the previous section we derived the principle of virtual work by considering virtual
displacements (or virtual forces) applied to a particle or body in equilibrium. Clearly,
for the principle to be of any value and for our present purpose of establishing the
principle of the stationary value of the total potential energy, we need to justify its
application to elastic bodies generally.
An elastic body in equilibrium under externally applied loads may be considered to
consist of a system of particles on each of which acts a system of forces in equilibrium.
Thus, for any virtual displacement the virtual work done by the forces on any particle
is, from the previous discussion, zero. It follows that the total virtual work done by all
the forces on the system vanishes. However, in prescribing virtual displacements for
an elastic body we must ensure that the condition of compatibility of displacement
within the body is satisfied and also that the virtual displacements are consistent
with the known physical restraints of the system. The former condition is satisfied
if, as we saw in Chapter 1, the virtual displacements can be expressed in terms of
single valued functions; the latter condition may be met by specifying zero virtual
displacements at support points. This means of course that reactive forces at supports
do no work and therefore, conveniently, do not enter the analysis.
Let us now consider an elastic body in equilibrium under a series of external
loads, PI, Pz, . . . , P,, and suppose that we impose small virtual displacements
SAl, SA2,. . . , SA, in the directions of the loads. The virtual work done by the loads