Page 60 - Aircraft Stuctures for Engineering Student

2.6 Bending of an end-loaded cantilever 45

from which

The stresses follow from Eqs (ii)
l2Pxy Px
a,=----- -
b3 IY
u,, = 0 (iii)

where I = b3/12 the second moment of area of the beam cross-section.
We note from the discussion of Section 2.4 that Eqs (iii) represent an exact solution
subject to the following conditions.
(1) That the shear force Pis distributed over the free end in the same manner as the
shear stress T~~ given by Eqs (iii).
(2) That the distribution of shear and direct stresses at the built-in end is the same
as those given by Eqs (iii).
(3) That all sections of the beam, including the built-in end, are free to distort.

In practical cases none of these conditions is satisfied, but by virtue of St. Venant's
principle we may assume that the solution is exact for regions of the beam away from
the built-in end and the applied load. For many solid sections the inaccuracies in these
regions are small. However, for thin-walled structures, with which we are primarily
concerned, significant changes occur and we shall consider the effects of axial
constraint on this type of structure in Chapter 11.
We now proceed to determine the displacements corresponding to the stress system
of Eqs (iii). Applying the strain-displacement and stress-strain relationships, Eqs
(1.27), (1.28) and (1.47), we have
du ux Pxy
E, = - = _- --
-
dx E EI
dv vux vPxy
E =-=--=- EI (v)
dy
E
du dv ryy P
=-+-=-=--(b'-4 8 IG Y2) (vi)
xy dy dx G
Integrating Eqs (iv) and (v) and noting that E, and sY are partial derivatives of the
displacements, we find
Px2y uPxy2
u=-- +fib), v=- 2EI +f2(x) (vii)
2EI
where fi (y) andfi(x) are unknown functions of x and y. Substituting these values of
u and Y in Eq. (vi)

px2 'fi(Y) vpY2 ah(x) -
-- +-+- - P (b2 - 4y2)
2EI 8y 2EI ax 81G