Page 150 - Aircraft Stuctures for Engineering Student

134 Bending of thin plates

Mxy + (dMx,,/dy)Sy. At the common surface of the two adjacent elements there is
now a resultant force (aMx,,/ay)by or a vertical force per unit length of aMxy/ay.
For the sign convention for Q, shown in Fig. 5.9 we have a statically equivalent
vertical force per unit length of (Q, - 8Mx,,/ay). The separate conditions for a free
edge of (Mxy)x=o = 0 and (Qx)x=o = 0 are therefore replaced by the equivalent
condition

(ex-%) x=o =O

or in terms of deflection

(5.25)

Also, for the bending moment along the free edge to be zero

(5.26)

The replacement of the twisting moment Mxy along the edges x = 0 and x = a of a
thin plate by a vertical force distribution results in leftover concentrated forces at the
corners of Mxy as shown in Fig. 5.11. By the same argument there are concentrated
forces Myx produced by the replacement of the twisting moment Myx. Since
Mxy = -My,, then resultant forces 2Mxy act at each corner as shown and must be
provided by external supports if the corners of the plate are not to move. The
directions of these forces are easily obtained if the deflected shape of the plate is
known. For example, a thin plate simply supported along all four edges and uni-
formly loaded has awlax positive and numerically increasing, with increasing y
near the corner x = 0, y = 0. Hence #w/axay is positive at this point and from
Eq. (5.14) we see that Mxy is positive and Myx negative; the resultant force 2Mv is
therefore downwards. From symmetry the force at each remaining corner is also
2Mxy downwards so that the tendency is for the corners of the plate to rise.
Having discussed various types of boundary conditions we shall proceed to obtain
the solution for the relatively simple case of a thin rectangular plate of dimensions
a x b, simply supported along each of its four edges and carrying a distributed load
q(x, y). We have shown that the deflected form of the plate must satisfy the differential
equation

a4w a4w a4w q(x,y)
-+2-+-=-
a# ax~ay2 ay4 D
with the boundary conditions