Page 90 - Aircraft Stuctures for Engineering Student

4.4 Stationary value of the total potential energy 75

single particle for which the strain energy is zero. Clearly at each position the first
order variation, a( U + V)/au, is zero (indicating equilibrium), but only at B where
the total potential energy is a minimum is the equilibrium stable. At A and C we
have unstable and neutral equilibrium respectively.
To summarize, the principle of the stationary value of the total potential energy may
be stated as:
The total potential energy of an elastic system has a stationary value for all smull
displacements when the system is in equilibrium; further, the equilibrium is stable if
the stationary value is a minimum.
This principle may often be used in the approximate analysis of structures where an
exact analysis does not exist. We shall illustrate the application of the principle in
Example 4.1 below, where we shall suppose that the displaced form of the beam is
unknown and must be assumed; this approach is called the Rayleigh-Ritz method
(see also Sections 5.6 and 6.5).

Example 4. I
Determine the deflection of the mid-span point of the linearly elastic, simply sup-
ported beam shown in Fig. 4.6; the flexural rigidity of the beam is EI.

The assumed displaced shape of the beam must satisfy the boundary conditions for
the beam. Generally, trigonometric or polynomial functions have been found to be
the most convenient where, however, the simpler the function the less accurate the
solution. Let us suppose that the displaced shape of the beam is given by
TZ
w = wB sin-
L
in which Q is the displacement at the mid-span point. From Eq. (i) we see that w = 0
when z = 0 and z = L and that v = WB when z = L/2. Also dvldz = 0 when z = L/2
so that the displacement function satisfies the boundary conditions of the beam.
The strain energy, U, due to bending of the beam, is given by (see Ref. 3)

(ii)

wl

A B C

//////

Fig. 4.6 Approximate determination of beam deflection using total potential energy.