Page 145 - Aircraft Stuctures for Engineering Student

5.3 Distributed transverse load 129

whence from Eq. (5.12)

or

M -Gt3 6%
xy 6 axay
Replacing G by the expression E/2( 1 + v) established in Eq. (1.45) gives

Et3 @w
M,.. =
12( 1 + v) axay
Multiplying the numerator and denominator of this equation by the factor (1 - v)
yields

a2W
Mxy = D(1 - v)- (5.14)
axay
Equations (5.7), (5.8) and (5.14) relate the bending and twisting moments to the
plate deflection and are analogous to the bending moment-curvature relationship
for a simple beam.

The relationships between bending and twisting moments and plate deflection are
now employed in establishing the general differential equation for the solution of a
thin rectangular plate, supporting a distributed transverse load of intensity q per
unit area (see Fig. 5.8). The distributed load may, in general, vary over the surface
of the plate and is therefore a function of x and y. We assume, as in the preceding
analysis, that the middle plane of the plate is the neutral plane and that the plate
deforms such that plane sections remain plane after bending. This latter assumption
introduces an apparent inconsistency in the theory. For plane sections to remain

z
Fig. 5.8 Plate supporting a distributed transverse load.