ABSTRACT
In Chapter 17, we established the governing partial differential equation for plate deformation due to bending as a result of loading normal to the plate (Equation 17.107):
r4w ¼ p=D (18:1)
where w is the displacement normal to the plate p is the loading function D is (Equation 17.91)
D ¼ Eh3=12(1 n2) (18:2)
where h is the plate thickness E and n are the elastic modulus and Poisson’s ratio
In Cartesian coordinates the r4 operator has the form
r4( ) ¼ @ 4( )
@x4 þ 2 @
4( )
@x2@y2 þ @
4( )
@y4 (18:3)
In cylindrical coordinates the r2 operator has the form [1]
r2( ) ¼ @ 2( )
@r2 þ 1
r
@( )
@r þ 1 r2
@2( )
@u2
¼ 1 r
@
@r r @( )
@r
þ 1
r2 @2( )
@u2 (18:4)
so that r4( ) is then
r4( ) ¼ r2r2( ) (18:5)
In Cartesian coordinates, p is a function of x and y. In cylindrical coordinates, p is a function of r and u, although for most circular plate problems of practical importance the loading is axisymmetric, that is p¼ p(r).