Page 299 - Aircraft Stuctures for Engineering Student

280 Open and closed, thin-walled beams

i.e. the first moment of area of the cross-section of the beam about the neutral axis is
zero. It follows that the neutral axis passes through the centroid of the cross-section
as shown in Fig. 9.4(b).
Suppose that the inclination of the neutral axis to Cx is a (measured clockwise from
Cx), then
<=xsina+ycosa (9.3)
and from Eq. (9.2)
E
az= -(xsina+ycosa) (9-4)
P
The moment resultants of the internal direct stress distribution have the same sense as
the applied moments Mx and My. Thus
r r

Substituting for a= from Eq. (9.4) in Eqs (9.5) and defining the second moments of
area of the section about the axes Cx, Cy as
1
Zxx = y2dA,

gives
E sin a E cos a E sina E cos a
Mx =- zxy + p My =- ZYY +p
zxx,
ZXY
P P
or, in matrix form

from which

i.e.
1

so that, from Eq. (9.4)

Alternatively, Eq. (9.6) may be rearranged in the form
+
Mx(ZyyY - Ixyx) My (ZXXX - ZxyY)
a, = (9.7)
IxxZyy - Zx.xZyy - cy
z:y
From Eq. (9.7) it can be seen that if, say, My = 0 the moment Mx produces a stress
which varies with both x and y; similarly for My if Mx = 0.