Page 156 - Aircraft Stuctures for Engineering Student

140 Bending of thin plates

in which Nyf is equal to and is replaced by N,,.. Using Eqs (5.31) and (5.32) we reduce
this expression to

(N,$ + NJ $ + 2Nx, -
axay
Since the in-plane forces do not produce moments along the edges of the element
then Eqs (5.17) and (5.18) remain unaffected. Further, Eq. (5.16) may be modified
simply by the addition of the above vertical component of the in-plane loads to
qbxby. Therefore, the governing differential equation for a thin plate supporting
transverse and in-plane loads is, from Eq. (5.20)

Example 5.2
Determine the deflected form of the thin rectangular plate of Example 5.1
if, in addition to a uniformly distributed transverse load of intensity qo, it supports
an in-plane tensile force N, per unit length.

The uniform transverse load may be expressed as a Fourier series (see Eq. (5.28)
and Example 5.1), viz.
w co
16qo 1 . mrx . nry
sin
sin - -
q=7C n=1,3:5 Cz a b
m=1,3,5
Equation (5.33) then becomes, on substituting for q
a4w
-sin - -
a4W
-+2-+--2-=- @w N d2w 16q0 2 2 1 mrx sin nry (i)
ax4 ax2dy2 ay4 D ax’ $0 mn a b
m=1,3,5 n=1,3,5
The appropriate boundary conditions are
a2W
w=-=O at x=O and a
ax2
a2
w=-=O w at y=O and b
aY2
These conditions may be satisfied by the assumption of a deflected form of the plate
given by

w = 2 2Am,sin-sin- mrx nry
n h
Substituting this expression into Eq. (i) gives
1670
Am, = for odd rn and n

A,,, = 0 for even m and n