Page 86 - Aircraft Stuctures for Engineering Student

4.3 Principle of virtual work 71

Mass M
t

1” t

(a) (b)
Fig. 4.3 (a) Potential energy of a spring-mass system; (b) loss in potential energy due to change in position.

P in the deflected equilibrium state is given by
v = -Py
We now define the totalpotential energy (TPE) of a system in its deflected equilibrium
state as the sum of its internal or strain energy and the potential energy of the applied
s:
external forces. Hence, for the single member-force configuration of Fig. 4.l(a)
TPE=U+V= Pdy-Py

For a general system consisting of loads PI, P2, . . . , Pn producing corresponding
displacements (i.e. displacements in the directions of the loads: see Section 4.10)
A,, A2,. . . , A, the potential energy of all the loads is

and the total potential energy of the system is given by

Suppose that a particle (Fig. 4.4(a)) is subjected to a system of loads PI, P2, . . . , P,
and that their resultant is PR. If we now impose a small and imaginary displacement,
i.e. a virtual displacement, 6R, on the particle in the direction of PR, then by the law of
conservation of energy the imaginary or virtual work done by PR must be equal to the
sum of the virtual work done by the loads PI, P2,. . . , P,. Thus
PR6R = PI61 + P262 + ‘ ’ ’ + Pn6n (4.7)
where SI, S2, . . . , 6, are the virtual displacements in the directions of PI, P2,. . . , P,
produced by SR. The argument is valid for small displacements only since a significant
change in the geometry of the system would induce changes in the loads themselves.