Plates: Fundamental Bending Configurations and Applications

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).