Civil Engineering Reference
In-Depth Information
the natural coordinates. From Equation 6.87, we have
I
11
∂
I
11
∂
∂
N
i
∂
∂
N
j
∂
N
i
∂
I
12
∂
N
i
∂
N
j
∂
I
12
∂
N
j
∂
x
=
r
+
r
+
(6.88)
x
s
s
so the integrand is transformed using the terms of [J]
−
1
. As shown in advanced
calculus [9], the differential area relationship is
d
A
=
d
x
d
y
= |
J
|
d
r
d
s
(6.89)
so integrals of the form described previously become
∂
d
A
I
11
∂
I
11
∂
1
1
N
i
∂
∂
N
j
∂
N
i
∂
I
12
∂
N
i
∂
N
j
∂
I
12
∂
N
j
∂
=
r
+
r
+
|
J
|
d
r
d
s
x
x
s
s
(6.90)
Such integrals are discussed in greater detail in later chapters in problem-specific
contexts. The intent of this discussion is to emphasize the importance of the
Jacobian matrix in development of isoparametric elements.
Rather than work with individual interpolation functions, it is convenient to
combine Equations 6.84 and 6.85 into matrix form as
A
−
1
−
1
∂
[
N
]
∂
∂
x
∂
y
∂
[
N
]
∂
x
∂
r
∂
r
r
(6.91)
=
∂
[
N
]
∂
∂
x
∂
y
∂
[
N
]
∂
y
∂
s
∂
s
s
where [
N
] is the
1
×
4
row matrix
(6.92)
[
N
]
=
[
N
1
N
2
N
3
N
4
]
and Equation 6.91 in matrix notation is the same as
∂
∂
∂
∂
∂
x
∂
x
x
∂
∂
r
∂
∂
∂
r
∂
s
(6.93)
[
N
]
=
[
N
]
∂
y
∂
y
y
s
∂
r
∂
s
We use this matrix notation to advantage in later chapters, when we examine
specific applications.
While the isoparametric formulation just described is mathematically
straightforward, the algebraic complexity is significant, as illustrated in the
following example.
EXAMPLE 6.4
Determine the Jacobian matrix for a four-node, two-dimensional quadrilateral element
having the parent element whose interpolation functions are given by Equation 6.56.
