Page 166 - Aircraft Stuctures for Engineering Student

150 Bending of thin plates

P.5.4 A simply supported square plate a x a carries a distributed load according
to the formula

where qo is its intensity at the edge x = a. Determine the deflected shape of the
plate.
w- *4oa4 2 2 (-I)~+’ mrx nry
Ans. sin - -
sin
r6D m=1,2,3 n=1.3,5 mn(m2+n2)2 a U
P.5.5 An elliptic plate of major and minor axes 2a and 2b and of small thickness t
is clamped along its boundary and is subjected to a uniform pressure difference p
between the two faces. Show that the usual differential equation for normal displace-
ments of a thin flat plate subject to lateral loading is satisfied by the solution

where wo is the deflection at the centre which is taken as the origin.
Determine wo in terms of p and the relevant material properties of the plate and
hence expressions for the greatest stresses due to bending at the centre and at the
ends of the minor axis.
3PU - 3)
Am. wO=
3
2
2~t3 -+-+-
(d a2b2 b4
f3pa2b2(b2 + v2) k3pa2b2(d + vb2)
Centre, ~;r,,.,= =
9(3b4 + 2a2b2 + 3d) ’ uyulmax = t2(3b4 + 2a2b2 + 3d)
Ends of minor axis
*6pa4b2 &6pb4d
’
ux,max = 9(3b4 + 2a2b2 + 3d) ’ = t-(3b4 + 2a2b2 + 3a4)
P.5.6 Use the energy method to determine the deflected shape of a rectangular
plate a x b, simply supported along each edge and carrying a concentrated load W
at a position ((’17) referred to axes through a comer of the plate. The deflected
shape of the plate can be represented by the series
mm nry
w= 2 2Amnsin- U sin
m=l n=l
m..5 nrrr;l
4Wsin- sin
Ans. A,,,,, = U
r4Dab[(m2/d) + (n2/b2)I2
P.5.7 If, in addition to the point load W, the plate of problem P.5.6 supports an
in-plane compressive load of Nx per unit length on the edges x = 0 and x = a,
calculate the resulting deflected shape.