12.6 Inplane Vibration of Plates
465
N 1 =
1 −
x
a
1 −
y
b
N 2 =
1 −
x
a
y
b
N 3 =
x
a
1 −
y
b
N 4 =
xy
ab
⎫
⎪ ⎪ ⎬
⎪ ⎪ ⎭
(12.111)
Strain components for a plane problem is given by
{∈} =
⎧
⎨
⎩
∈ x
∈ y
γ xy
⎫
⎬
⎭
=
⎡
⎢
⎣
∂
∂ x
0
0
∂
∂ y
∂
∂ y
∂
∂ x
⎤
⎥
⎦ =
u
v
(12.112)
Combining Eqs. (12.109) and (12.112), yields
{∈} = [B] {X } e
(12.113)
The relationship of stress with strain is given by
{σ } = [D] {∈} = [D] [B] {X } e
(12.114)
[D] is same as Eq. (12.90) for an isotropic plate.
As already shown, the element stiffness matrix is given by
[K ] e = t
A
[B]
T
[D] [B] dx dy
(12.115)
[K ] e in explicit form for an isotropic plate is given by Eq. (12.116).
The consistent mass matrix is given by Eq. (12.99).
Substituting Eqs. (12.110) in (12.99), yields
(12.116)