Page 84 - Aircraft Stuctures for Engineering Student

4.1 Strain energy and complementary energy 69

Complementary energy C

(a) (b)

Fig. 4.1 (a) Strain energy of a member subjected to simple tension; (b) load-deflection curve for a non-
linearly elastic member.

this work is stored in the member as strain energy. A typical load-deflection curve for
a member possessing non-linear elastic characteristics is shown in Fig. 4.l(b). The
strain energy U produced by a load P and corresponding extension y is then
1
U= Pdy

and is clearly represented by the area OBD under the load-deflection curve. Engesser
(1889) called the area OBA above the curve the complementary energy C, and from
Fig. 4.l(b)
I
C= ydP (4.2)

Complementary energy, as opposed to strain energy, has no physical meaning, being
purely a convenient mathematical quantity. However, it is possible to show that
complementary energy obeys the law of conservation of energy in the type of situation
usually arising in engineering structures, so that its use as an energy method is valid.
Differentiation of Eqs (4.1) and (4.2) with respect to y and P respectively gives
dU dC
-= P, --
dY dP-'
Bearing these relationships in mind we can now consider the interchangeability of
strain and complementary energy. Suppose that the curve of Fig. 4.l(b) is represented
by the function
P = by"
where the coefficient b and exponent n are constants. Then
(x-'"dP
U = Pdy = tJo P
P

C= JIydP=n[by"dy