Page 59 - Aircraft Stuctures for Engineering Student

44 Two-dimensional problems in elasticity

Fig. 2.5 Bending of an end-loaded cantilever.

the cantilever beam shown in Fig. 2.5 is

Bxy3
f$ = Axy+-
6
where A and B are unknown constants. Hence

I (ii)
BY2 J
b
$4
Txy=--=-A--
axay 2
Substitution for f$ in the biharmonic equation shows that the form of the stress
function satisfies compatibility for all values of the constants A and B. The actual
values of A and B are chosen to satisfy the boundary condition, viz. T,~ = 0 along
the upper and lower edges of the beam, and the resultant shear load over the free
end is equal to P.
From the first of these
BY2 b
T~,, = -A --= 0 at y = f-
2 2
giving
Bb2
A=---
8
From the second
bI2
T,~ dy = P (see sign convention for 7-yy)
- J -b/2
or

- dy = P
-b/2