Page 141 - Aircraft Stuctures for Engineering Student

5.2 Plates subjected to bending and twisting 125

Fig. 5.3 Anticlastic bending

Equations (5.7) and (5.8) define the deflected shape of the plate provided that M, and
My are known. If either M, or My is zero then
d2W - a2W d2W - d2W
- -v- or - -v-
ax2 - 8Y2 ay2 - dX2
and the plate has curvatures of opposite signs. The case of My = 0 is illustrated in
Fig. 5.3. A surface possessing two curvatures of opposite sign is known as an
anticlastic surface, as opposed to a synclastic surface which has curvatures of the
same sign. Further, if M, = My = M then from Eqs (5.5) and (5.6)
1
----_ 1 1
--
PX Py P
Therefore, the deformed shape of the plate is spherical and of curvature

-_
--d*-v-7. =
g
In general, the bending moments applied to the plate will not be in planes
perpendicular to its edges. Such bending moments, however, may be resolved in
the normal manner into tangential and perpendicular components, as shown in
Fig. 5.4. The perpendicular components are seen to be M, and My as before, while

Fig. 5.4 Plate subjected to bending and twisting.