Page 144 - Aircraft Stuctures for Engineering Student

128 Bending of thin plates

Fig. 5.7 Determination of shear strain 5.

and on the face ADFE
I,
tl2
MxySx = - rxySxzdz

giving

or in terms of the shear strain yxy and modulus of rigidity G

(5.12)

Referring to Eqs (1.20), the shear strain yxy is given by
av au
yxy = - + -
ax ay
We require, of course, to express -yxy in terms of the deflection w of the plate; this may
be accomplished as follows. An element taken through the thickness of the plate will
suffer rotations equal to dw/dx and aw/ay in the xz and yz planes respectively.
Considering the rotation of such an element in the xz plane, as shown in Fig. 5.7,
we see that the displacement u in the x direction of a point a distance z below the
neutral plane is

Similarly, the displacement in the y direction is
v= --z dW
aY
Hence, substituting for u and v in the expression for -yxy we have
a2W
-yxy = -22- (5.13)
axay