Page 140 - Aircraft Stuctures for Engineering Student

124 Bending of thin plates

As would be expected from our assumption of plane sections remaining plane the
direct stresses vary linearly across the thickness of the plate, their magnitudes depend-
ing on the curvatures (i.e. bending moments) of the plate. The internal direct stress
distribution on each vertical surface of the element must be in equilibrium with the
applied bending moments. Thus

and

Substituting for ux and cy from Eqs (5.3) gives
Mx= Jli2 --(‘+;)dz
Ez2
-t/2 1 - lJ2 px

Let
Ez2
D=J r/2 - Et3 (5.4)
dz=
-t/2 1 - 3 12(1 - 3)
Then

My = D(; + ;)

in which D is known as theflexural rigidity of the plate.
If w is the deflection of any point on the plate in the z direction, then we may relate
w to the curvature of the plate in the same manner as the well-known expression for
beam curvature. Hence

1 - a2w 1 a2w
Px a$’ P,=-- ay”
the negative signs resulting from the fact that the centres of curvature occur above the
plate in which region z is negative. Equations (5.5) and (5.6) then become

Mx=-D(S+~*) dY2 (5.7)