Page 179 - Aircraft Stuctures for Engineering Student

6.4 Beams under transverse and axial loads 163

where A and B are unknown constants and A' = P/EI. Substituting the boundary
conditions v = 0 at z = 0 and 1 gives
W W
A=-, B= (1 - cos XI)
X2P X2P sin XI
so that the deflection is determinate for any value of w and P and is given by

v=- X2 P [ cosXz+ ('~i-~~)sinXz] +&(z2-lz-$-) (6.30)

In beam-columns, as in beams, we are primarily interested in maximum values of
stress and deflection. For this particular case the maximum deflection occurs at the
centre of the beam and is, after some transformation of Eq. (6.30)
vma=~(sec2-i) -8p w12
A1
(6.31)
The corresponding maximum bending moment is
w12
M,,, = -Puma - -
8
or, from Eq. (6.31)

(6.32)

We may rewrite Eq. (6.32) in terms of the Euler buckling load PCR = dEI/12 for a
pin-ended column. Hence

(6.33)

As P approaches PCR the bending moment (and deflection) becomes infinite.
However, the above theory is based on the assumption of small deflections (otherwise
d2v/d2 would not be a close approximation for curvature) so that such a deduction is
invalid. The indication is, though, that large deflections will be produced by the
presence of a compressive axial load no matter how small the transverse load
might be.
Let us consider now the beam-column of Fig. 6.10 with hinged ends carrying a
concentrated load W at a distance u from the right-hand support. For

(6.34)

and for
d2v W
z 2 1 - a, EI - - -M = -Pv - - (I - u> (1 - z) (6.35)
d.9 - I
Writing

X2 = P/EI