Page 99 - Aircraft Stuctures for Engineering Student

4.6 Application to deflection problems 83

w/un i t length Pf .
-1

AT
t
L I

Fig. 4.10 Deflection of a uniformly loaded cantilever by the method of complementary energy.

length (see Fig. 4.10). First we apply a fictitious load Pf at the point where the deflec-
tion is required. The total complementary energy of the system is
c
C = IL 1; dt'dM - ATP, - Awdz

where the symbols take their previous meanings and A is the vertical deflection of any
point on the beam. Then

(4.20)

As before
M
dt' = - dz
EI
but
W2
M = PfZ + - (Pf = 0)
2
Hence

Substituting in Eq. (4.20) for de, M and dM/dPf, and remembering that Pf = 0, we
have

giving
wL4
AT =-
8 EI
It will be noted that here, unlike the method for the solution of the pin-jointed
framework, the fictitious load is applied to the loaded beam. There is, however, no
arithmetical advantage to be gained by the former approach although the result
would obviously be the same since M would equal w2/2 and aM/aPf would have
the value z.