Page 153 - Aircraft Stuctures for Engineering Student

5.4 Combined bending and in-plane loading 137

Substitution for MI from Eq. (i) into the expressions for bending moment, Eqs (5.7)
and (5.8), yields

[(m2/a2) + v(n2/h2)] . m7rx n7ry
sin
mn[(m2/a2) + (n2/b2)12 sin - (iii)
a
[v(m2/a2) + (n2/h2)] . m7rx . n7ry
sin
M,. = - sin - - (iv)
mn[(m2/a2) + (n2/h2)12 a h
Maximum values occur at the centre of the plate. For a square plate a = h and the first
five terms give
M,,,,, = M,,,,, = 0.0479qoa2

Comparing Eqs (5.3) with Eqs (5.5) and (5.6) we observe that
12M,z 12M"Z
, ay=-
a,y = -
t3 t3
Again the maximum values of these stresses occur at the centre of the plate at
z = ft/2 so that
6MX 6M,
ax,rnax = - ay,rnax = -
t2
>
t2
For the square plate
a2
ax,rnax = gypax = 0.287q0 -
t2
The twisting moment and shear stress distributions follow in a similar manner.

So far our discussion has been limited to small deflections of thin plates produced by
different forms of transverse loading. In these cases we assumed that the middle or
neutral plane of the plate remained unstressed. Additional in-plane tensile, compres-
sive or shear loads will produce stresses in the middle plane, and these, if of sufficient
magnitude, will affect the bending of the plate. Where the in-plane stresses are small
compared with the critical buckling stresses it is sufficient to consider the two systems
separately; the total stresses are then obtained by superposition. On the other hand, if
the in-plane stresses are not small then their effect on the bending of the plate must be
considered.
The elevation and plan of a small element SxSy of the middle plane of a thin
deflected plate are shown in Fig. 5.12. Direct and shear forces per unit length pro-
duced by the in-plane loads are given the notation Nx, Ny and Nxy and are assumed
to be acting in positive senses in the directions shown. Since there are no resultant
forces in the x or y directions from the transverse loads (see Fig. 5.9) we need only
include the in-plane loads shown in Fig. 5.12 when considering the equilibrium of