Page 164 - Aircraft Stuctures for Engineering Student

148 Bending of thin plates

The above solution is exact since we know the true deflected shape of the plate in
the form of an infinite series for w. Frequently, the appropriate infinite series is not
known so that only an approximate solution may be obtained. The method of
solution, known as the Rayleigh-Rifz method, involves the selection of a series for
w containing a finite number of functions of x and y. These functions are chosen to
satisfy the boundary conditions of the problem as far as possible and also to give
the type of deflection pattern expected. Naturally, the more representative the
‘guessed’ functions are the more accurate the solution becomes.
Suppose that the ‘guessed’ series for w in a particular problem contains three
different functions of x and y. Thus

w = Alfi(X,Y) + A2f2(X,Y) + A3h(X,Y)
where Al, A2 and A3 are unknown coefficients. We now substitute for w in the
appropriate expression for the total potential energy of the system and assign station-
ary values with respect to AI, A2 and A3 in turn. Thus
8(U+ V) a( u + V) 8(U+ V)
= 0, = 0, =O
8A 1 8A2 8-43
giving three equations which are solved for Al, A2 and A3.
To illustrate the method we return to the rectangular plate a x by simply supported
along each edge and carrying a uniformly distributed load of intensity qo. Let us
assume a shape given by
7rx 7ry
w = All sin-sin-
a b
This expression satisfies the boundary conditions of zero deflection and zero curva-
ture (Le. zero bending moment) along each edge of the plate. Substituting for w in
Eq. (5.46) we have
4 7rx 7ry
(2 + b2)2 sin2- sin2 - - 2( 1 - v)
a b
7r4 sin 27rx -sin 27ry - - -cosz~cos2~]}
7r4
[&? a b a2b2 a b

whence

so that
~
8(~+ V) ~ ~ (a + b2)’ - qo- 4ab - 0 r ~
7
~
=-
aAll 4a3b3 9-
and
1 6qoa4b4
All =
7r6D(a2 + b2)’