Page 151 - Aircraft Stuctures for Engineering Student

5.3 Distributed transverse load 135

Navier (1820) showed that these conditions are satisfied by representing the deflection
w as an infinite trigonometrical or Fourier series
m w mrx nry
w= EA,,,,sin- a sin (5.27)
m=l n=l
in which m represents the number of half waves in the x direction and n the
corresponding number in the y direction. Further, A,,, are unknown coefficients
which must satisfy the above differential equation and may be determined as follows.
We may also represent the load q(x, y) by a Fourier series, thus
m w mrx nry
q(x,y) = Camnsin-sinb (5.28)
m=l n=l a
A particular coefficient amtnt is calculated by first multiplying both sides of Eq. (5.28)
by sin(m'rx/a) sin(n'ry/b) and integrating with respect to x from 0 to a and with
respect to y from 0 to b. Thus

m'rx . n'ry
q(x,y)sin-sin- dxdy
a b
. mrx m'rx nry n'ry
= m=l 2 n=l T/:jIamsin-sin- a a sin - sin - dx dy
b
b
ab
-
-
4
since

- a
_- when m=m'
2
and
so" sin-sin- b dy = 0 when n # n'
. nry . n'ry
b
b
_- when n=n'
-
2
It follows that
4 a b m'rx n'ry
am!,# = - 1 1 q(x, y) sin - sin- dx dy (5.29)
ab o o a b
Substituting now for w and q(x, y) from Eqs (5.27) and (5.28) into the differential
equation for w we have