Page 157 - Aircraft Stuctures for Engineering Student

5.5 Thin plates having a small initial curvature 141

Therefore
1 . mm nry
sin
Ml = - sin - (ii)
2 2 2 a
I
I
( t2 i2) Nxm2]
.lr2Da2
Comparing Eq. (ii) with Eq. (i) of Example 5.1 we see that, as a physical inspection
would indicate, the presence of a tensile in-plane force decreases deflection. Con-
versely a compressive in-plane force would increase the deflection.

--e- ,.S.+.Y.T
aving

Suppose that a thin plate has an initial curvature so that the deflection of any point in
its middle plane is wo. We assume that wo is small compared with the thickness of the
plate. The application of transverse and in-plane loads will cause the plate to deflect a
further amount w1 so that the total deflection is then w = wo + wl. However, in the
derivation of Eq. (5.33) we note that the left-hand side was obtained from expressions
for bending moments which themselves depend on the change of curvature. We
therefore use the deflection w1 on the left-hand side, not w. The effect on bending
of the in-plane forces depends on the total deflection w so that we write Eq. (5.33)

The effect of an initial curvature on deflection is therefore equivalent to the applica-
tion of a transverse load of intensity

Thus, in-plane loads alone produce bending provided there is an initial curvature.
Assuming that the initial form of the deflected plate is

rnm . nry
wo = 2 2 A,, sin- sln b (5.35)
m=l n=l a
then by substitution in Eq. (5.34) we find that if Nx is compressive and N,. = N.yy = 0
mnx . nry
w1 = 2 2 Bmn sin- 3- (5.36)
m=l n=l a
Bmn=( , AmnNx
where

T~D a2)[m + (n2a2/mb2)12 - N,
We shall return to the consideration of initially curved plates in the discussion of the
experimental determination of buckling loads of flat plates in Chapter 6.