Information Technology Reference
In-Depth Information
where
J
is the
Jacobian
.
7
From Eq. (6.130), the Jacobian can be expressed as
∂
N
i
∂ξ
x
i
∂
N
i
∂ξ
J
11
y
i
J
12
J
=
=
(6.135b)
∂
N
i
∂η
x
i
∂
N
i
∂η
J
21
J
22
y
i
A necessary and sufficient condition for Eq. (6.135a) to be invertible is that det
J
.
This same condition must be satisfied if the coordinate relations of Eq. (6.132) are invertible.
The desired derivatives
=|
J
|=
0
∂
x
and
∂
y
can be expressed as
∂
x
∂
∂
J
−
1
∂ξ
∂
∂η
=
(6.136a)
y
where
J
−
1
is the inverse of the Jacobian,
∂ξ
∂
J
22
J
11
∂η
∂
J
12
1
−
J
12
J
−
1
x
x
=
=
=
(6.136b)
∂ξ
∂
∂η
∂
|
J
|
J
21
J
22
−
J
21
J
11
y
y
A comparison of Eq. (6.24) and the above relation shows that
J
11
=
1
a
,J
22
=
1
b
,
and
J
12
=
J
21
=
.
The determinant of the Jacobian relates the differential change in the two coordinate
0
systems in the sense that, from the calculus,
S
=
S
g
Here,
f
and
g
are equivalent expressions for a function in the two coordinate systems. The differential
area in our quadrilateral element would be
f
(
x, y
)
dx dy
(ξ
,
η)
|
J
|
d
ξ
d
η.
dA
=|
J
|
d
ξ
d
η
(6.137)
In order to establish the stiffness matrix and the loading vector, return to the principle of
virtual work. The principle for a thin plate can be represented by the formula developed in
Section 6.4.1. The element stiffness matrix and loading vector are
k
i
T
ED
u
N
dV
N
T
u
D
T
=
V
(
D
u
N
)
=
V
(
)
E
(
D
u
N
)
dV
t
N
T
u
D
T
=
A
(
)
E
(
D
u
N
)
dA
(6.138)
p
i
N
T
p
dS
=
S
p
where
t
is the thickness of the plate, and
N
1
N
2
N
3
N
4
0
0
N
1
N
2
N
3
N
4
N
=
The material law matrix
E
is defined in Chapter 1, Eq. (1.39). In the isoparametric sense, i.e.,
N
is used for both displacements and coordinates,
D
can be altered such that the integration
7
Carl Gustav Jacob Jacobi (1804-1851). As the son of a German Jewish banker, he was raised in a wealthy, cultured
atmosphere. Jacobi was forced to privately study the works of mathematicians as the leading mathematicians then
were in Paris. An exception was Gauss in G ottingen. In 1826, Jacobi left Berlin for the University of K onigsberg
where he joined the physicists Franz Neumann and Heinrich Dove and astronomer Friedrich Bessel. There he
attacked many applied problems. His mathematical accomplishments are often compared with such predecessors
as Euler. His interests were varied: he once assisted Alexander von Humboldt, who was preparing his topic
Kosmos,
by proving theorems from ancient Greek mathematics.
Search WWH ::
Custom Search