472
12 Finite Element Method in Vibration Analysis
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−
∂ θ x
∂ x
−
∂ θ y
∂ y
−
∂ θ x
∂ y
+
∂ θ y
∂ x
− φ x
− φ y
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
=
n
r =1
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0 −
∂ N r
∂ x
0
0
0 −
∂ N r
∂ y
0 −
∂ N r
∂ y
−
∂ N r
∂ x
∂ N r
∂ x
− N r
0
∂ N r
∂ y
0
− N r
⎤
⎥
⎥
⎥
⎥
⎥
⎦
⎧
⎨
⎩
w r
θ x r
θ y r
⎫
⎬
⎭
(12.145)
while deriving Eq. (12.144), use has been made of Eq. (12.135).
Equation (12.144) can be written in compact form as given below
{∈} =
n
r =1
[B] r {X r } e = [B] {X } e
(12.146)
The quantities ∂ N r /∂ x and ∂ N r /∂ y are to be computed to evaluate [B] matrix.
∂ N r
∂ ξ
and
∂ N r
∂ η
are related to
∂ N r
∂ x
and
∂ N r
∂ y
as follows:
∂ N r
∂ ξ
∂ N r
∂ η
=
∂ x
∂ ξ
∂ y
∂ ξ
∂ x
∂ η
∂ y
∂ η
∂ N r
∂ x
∂ N r
∂ y
(12.147)
Equation (12.147) can now be written as
∂ N r
∂ x
∂ N r
∂ y
= [J ]
− 1
∂ N r
∂ ξ
∂ N r
∂ η
(12.148)
where
[J ]
− 1
=
∂ ξ
∂ x
∂ η
∂ x
∂ ξ
∂ y
∂ η
∂ y
(12.149)
The elements of [J] are calculated by
∂ x
∂ ξ
=
n
r =1
∂ N r
∂ ξ
x r
∂ y
∂ ξ
=
n
r =1
∂ N r
∂ ξ
y r
∂ x
∂ η
=
n
r =1
∂ N r
∂ η
x r
∂ y
∂ η
=
n
r =1
∂ N r
∂ η
y r
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(12.150)
The stress components to be considered in this is
{σ }
T
=
M x M y M x y V x V y
(12.151)