Page 303 - Aircraft Stuctures for Engineering Student

284 Open and closed, thin-walled beams

or, when second-order terms are neglected

We may combine these results into a single expression
as. azMx
--w -A=- (9.11)
y- az a2
Similarly for loads in the xz plane
as, - azMy
-wx = - -- (9.12)
az az2

9.1.6 Deflections due to bending

We have noted that a beam bends about its neutral axis whose inclination relative to
arbitrary centroidal axes is determined from Eq. (9.10). Suppose that at some section
of an unsymmetrical beam the deflection normal to the neutral axis (and therefore an
absolute deflection) is C, as shown in Fig. 9.7. In other words the centroid C is
displaced from its initial position CI through an amount C to its final position CF.
Suppose also that the centre of curvature R of the beam at this particular section is
on the opposite side of the neutral axis to the direction of the displacement C and
that the radius of curvature is p. For this position of the centre of curvature and
from the usual approximate expression for curvature we have

(9.13)
The components u and 21 of C are in the negative directions of the x and y axes
respectively, so that
u = -Csina, 21 = -Ccosa (9.14)

R (centre of curvature)
t’ I 9

Loaded

Fig. 9.7 Determination of beam deflection due to bending.