470
12 Finite Element Method in Vibration Analysis
Fig. 12.11 Deformation of
plate in XZ plane
A general formulation is presented so that the derivation is valid for all types
of isoparametric elements—linear, quadratic, cubic etc. Only the appropriate shape
function is to be substituted.
The coordinates within an element are
x
y
=
n
r =1
1 0
0 1
N r
x r
y r
(12.137)
x r , y r are nodal coordinates and Nr is the shape function.
The shape functions for the linear element of Fig. 12.12 are given by
N r =
1
4
(1 + ξ ξ r ) (1 + η η r )
(12.138)
The shape functions for the quadratic element of Fig. 12.13.
Fig. 12.12 Isoparametric linear element