Page 382 - Aircraft Stuctures for Engineering Student

10.1 Tapered beams 363

calculated at the particular section being considered; Eqs (9.6)-(9.10) may therefore
be used with reasonable accuracy. On the other hand, the calculation of shear stresses
in beam webs can be significantly affected by taper.

Consider first the simple case of a beam positioned in the yz plane and comprising two
flanges and a web; an elemental length Sz of the beam is shown in Fig. 10.1. At the
section z the beam is subjected to a positive bending moment My and a positive
shear force Sy. The bending moment resultants Pz,l and P3:2 are parallel to the z
axis of the beam. For a beam in which the flanges are assumed to resist all the
direct stresses, Pz,l = Mx/h and Pz,2 = -Mx/h. In the case where the web is assumed
to be fully effective in resisting direct stress, PZ;~ and PQ are determined by multiply-
ing the direct stresses oZ,] and found using Eq. (9.6) or Eq. (9.7) by the flange areas
B1 and B2. PZ,] and Pz,2 are the components in the z direction of the axial loads PI and
P2 in the flanges. These have components Py,l and Py,2 parallel to the y axis given by

(10.1)

in which, for the direction of taper shown, Sy2 is negative. The axial load in flange 0
is given by

Substituting for P,,l from Eqs (10.1) we have

(10.2)

Fig. 10.1 Effect of taper on beam analysis.