Page 85 - Aircraft Stuctures for Engineering Student

70 Energy methods of structural analysis

Fig. 4.2 Load-deflection curve for a linearly elastic member.

Hence

-=P, dP=n(h) 'I" =-y (4.3)
1
dU
P
1
dU
dY n
dC dC
dP
-- - y, - bny" = nP (4.4)
=
dY
When n = 1

and the strain and complementary energies are completely interchangeable. Such a
condition is found in a linearly elastic member; its related load-deflection curve
being that shown in Fig. 4.2. Clearly, area OBD(U) is equal to area OBA(C).
It will be observed that the latter of Eqs (4.5) is in the form of what is commonly
known as Castigliano's first theorem, in which the differential of the strain energy
U of a structure with respect to a load is equated to the deflection of the load. To
be mathematically correct, however, it is the differential of the complementary
energy C which should be equated to deflection (compare Eqs (4.3) and (4.4)).

In the spring-mass system shown in its unstrained position in Fig. 4.3(a) we normally
define the potential energy of the mass as the product of its weight, Mg, and its height,
h, above some arbitrarily fixed datum. In other words it possesses energy by virtue of
its position. After deflection to an equilibrium state (Fig. 4.3(b)), the mass has lost an
amount of potential energy equal to Mgy. Thus we may associate deflection with a
loss of potential energy. Alternatively, we may argue that the gravitational force
acting on the mass does work during its displacement, resulting in a loss of energy.
Applying this reasoning to the elastic system of Fig. 4.l(a) and assuming that the
potential energy of the system is zero in the unloaded state, then the loss of potential
energy of the load P as it produces a deflection y is Py. Thus, the potential energy V of