468
12 Finite Element Method in Vibration Analysis
By performing the necessary differentiation in Eq. (12.124), the resulting equation
can be written as
{∈} = [B] {X } e
(12.127)
The ‘stress’ components of the plate are the bending moments and the twisting
moment, and they are related to the curvatures for an orthotropic plate as follows:
{σ } =
⎧
⎨
⎩
M x
M y
M xy
⎫
⎬
⎭
=
⎡
⎣
D x D 1 0
D 1 D y 0
0 0 D xy
⎤
⎦
⎧
⎪ ⎨
⎪ ⎩
−
∂
2 w
∂ x 2
−
∂
2 w
dy 2
2
∂
2 w
dx dy
⎫
⎪ ⎬
⎪ ⎭
(12.128)
or
{σ } = [D] (∈}
(12.129)
D x , D y , D 1 and D x y relate to the geometrical and material properties of the plate.
Combining Eqs. (12.127) and (12.128), we get
{σ } = [D] [B] {X } e
(12.130)
Following steps similar to that given in the previous sections of this chapter, the
stiffness matrix for the rectangular plate element is given by
[K ] e =
a
0
b
0
[B]
T
[D] [B] dx dy
(12.131)
The consistent mass matrix for the rectangular plate bending element is given by
[M] e = ρ t
A
[N ]
T
[I ] [N ] dA
(12.132)
Substituting [N] from Eq. (12.125) into Eq. (12.132), we get
[M] e = ρ t
a
0
b
0
([A]
− 1
)
T
[C]
T
[C] [A]
− 1 dA
(12.133)
Equation (12.133) is explicitly evaluated and it is given below