Page 384 - Aircraft Stuctures for Engineering Student

10.1 Tapered beams 365

2 mm and is fully effective in resisting direct stress. The beam tapers symmetrically
about its horizontal centroidal axis and the cross-sectional area of each flange is
400 mm2.

The internal bending moment and shear load at the section AA produced by the
externally applied load are, respectively
Mx = 20 x 1 = 20kNm, S, = -2OkN

The direct stresses parallel to the z axis in the flanges at this section are obtained either
from Eq. (9.6) or Eq. (9.7) in which M,, = 0 and Zx, = 0. Thus, from Eq. (9.6)

MXY
uz = -
In
in which
Ixx = 2 x 400 x 1502 + 2 x 3003/12
i.e.

Zxx = 22.5 x 106m4
Hence

The components parallel to the z axis of the axial loads in the flanges are therefore
PI = -Pz-2 = 133.3 x 400 = 53 320 N
1
-7
The shear load resisted by the beam web is then, from Eq. (10.5)
6Yl 6Y2
S,.,," = -20 x lo3 - 53 320- + 53 320-
6Z 6Z
in which, from Figs 10.1 and 10.2, we see that
6y1 - -100 - -0.05, 6Y2 - 100 - 0.05
- - ---
sz 2 x 103 6,. 2 x 103
Hence
S,.:w, = -20 x lo3 + 53 320 x 0.05 + 53 320 x 0.05 = -14668N
The shear flow distribution in the web follows either from Eq. (10.6) or Eq. (10.7) and
is (see Fig. 10.2(b))

412 = 22.5 x lo6 ([q150-s)ds+400 x 150
14'''
i.e.

412 = 6.52 x + 300s + 60000) (ii)
The maximum value of q12 occurs when s = 150mm and q12 (max) = 53.8 N/mm.
The values of shear flow at points 1 (s = 0) and 2 (s = 300mm) are q1 = 39.1 N/mm
and q2 = 39.1 N/mm; the complete distribution is shown in Fig. 10.3.