Page 148 - Aircraft Stuctures for Engineering Student

132 Bending of thin plates

substitution for M,, My and Mxy in terms of the deflection w of the plate; thus
aMxy - -D - (- + e)
aMx a a2w (5.21)
Q ax ax2
x-
ax ay ayz
Q aM, dMXY (5.22)
- ay ax

Direct and shear stresses are then calculated from the relevant expressions relating
them to M,, My, Mxy, Q, and Qy.
Before discussing the solution of Eq. (5.20) for particular cases we shall establish
boundary conditions for various types of edge support.

5.3.1 The simply supported edge
I
Let us suppose that the edge x = 0 of the thin plate shown in Fig. 5.10 is free to rotate
but not to deflect. The edge is then said to be simply supported. The bending moment
along this edge must be zero and also the deflection w = 0. Thus

The condition that w = 0 along the edge x = 0 also means that
aw a2w
-
ay ay2 -O
along this edge. The above boundary conditions therefore reduce to

(5.23)

Fig. 5.10 Plate of dimensions a x b.