Page 56 - Aircraft Stuctures for Engineering Student

2.3 Inverse and semi-inverse methods 41

(a) (b)
Fig. 2.2 (a) Required loading conditions on rectangular sheet in Example 2.2 for A = B = C = 0; (b) as in (a)
butA = C = D= 0.

the coefficients are related in a certain way. Thus, for a stress function in the form of a
polynomial of the fourth degree

Ax4 Bx3y Cx2g Dxy3 Ey4
$=-+- +-+-
12 6 2 6 '12
and

Substituting these values in Eq. (2.9) we have
E = -(2C + A)
The stress components are then
a24
ox = - Cx2 + Dxy - (2C + A)y2
=
aY2
84
g --=A x + Bxy + Cy2
E'- ax2
#q5 - B2 DY2
5=------ 2cxy - -
2
axay
2
The coefficients A, B, C and D are arbitrary and may be chosen to produce various
loading conditions as in the previous examples.
The obvious disadvantage of the inverse method is that we are determining
problems to fit assumed solutions, whereas in structural analysis the reverse is the
case. However, in some problems the shape of the body and the applied loading
allow simplifying assumptions to be made, thereby enabling a solution to be obtained.
St. Venant suggested a semi-inverse method for the solution of this type of problem in
which assumptions are made as to stress or displacement components. These assump-
tions may be based on experimental evidence or intuition. St. Venant first applied the