Page 87 - Aircraft Stuctures for Engineering Student

72 Energy methods of structural analysis

Actual displaced

(a) (b)
Fig. 4.4 (a) Principle of virtual displacements; (b) principle of virtual forces.

For the case where the particle is in equilibrium the resultant PR of the forces must
be zero and Eq. (4.7) reduces to
PIS1 + P2S2 + . + P,S, = 0
or
n
c PrSr = 0
r=l
The principle of virtual work may therefore be stated as:
A particle is in equilibrium under the action of a force system if the total virtual work
done by the force system is zero for a small virtual displacement.
This statement is often termed the principle of virtual displacements.
An alternative formulation of the principle of virtual work forms the basis of the
application of total complementary energy (Section 4.5) to the determination of
deflections of structures. In this alternative approach, small virtual forces are applied
to a system in the direction of real displacements.
Consider the elastic body shown in Fig. 4.4(b) subjected to a system of real loa&
which may be represented by P. Due to P the body will be displaced such that
points 1,2,. . . ,n move through displacements Al, A2,. . . ,A, to It, 2', . . . ,n'. Now
suppose that small imaginary loads SPI , 6P2, . . . , SP, were in position and acting in
the directions of All A2,. . . ,A, before P was applied; since SP1, SP2,. . . , SP, are
imaginary they will not affect the real displacements. The total imaginary, or virtual,
work SW* done by these loads is then given by

n
SW* =AlSPl + A2SP2+...+AnSP, = CAJP,.
r= 1
which, by the law of conservation of energy, is equal to the imaginary, or virtual,
strain energy stored SU*. This is due to small imaginary internal forces SP, produced
by the external imaginary loads, moving through real internal displacements y and