Page 499 - Aircraft Stuctures for Engineering Student

480 Structural constraint

displacements of its cross-section. In the analysis we assume that the cross-section of
the beam is undistorted by the loading and that displacements corresponding to the
shear strains are negligible. In Fig. 11.39 the tangential displacement ut is given by
Eq. (9.27), i.e.

ut = pR8 + ucos$ + usin$ (1 1.65)
Also, since shear strains are assumed to be negligible, Eq. (9.26) becomes

(1 1.66)

Substituting for vt in Eq. (1 1.66) from Eq. (1 1.65) and integrating from the origin for s
to any point s around the cross-section, we have
d8 du dv
W, - WO = --2AR,o - -(x - XO) - - (y -yo) (11.67)
dz dz dz
where 240 = p~ ds. The direct stress at any point in the wall of the beam is given
by

Thus, from Eq. (1 1.67)

Now AR,O = Ak + AR (Fig. 11.39) so that Eq. (1 1.68) may be rewritten

d28 d2u d2v
uZ = fi (z) - E - ~AR E - - E - (1 1.69)
x
-
dz2 dz2 dz2
in which

The axial load P on the section is given by

where Jc denotes integration taken completely around the section. From Eq. (11.55)
we see that sc 2A~t dr = 0. Also, if the origin of axes coincides with the centroid of the
section Jc txds = Jc tyds = 0 and J tyds = 0 so that
I
P = a2tds =fi(z)A (1 1.70)

in which A is the cross-sectional area of the material in the wall of the beam.
The component of bending moment, M,, about the x axis is given by