Civil Engineering Reference
In-Depth Information
The required partial derivatives are
∂
N
1
∂
r
1
4
(
s
−
∂
N
1
∂
s
1
4
(
r
−
=
1)
=
1)
∂
N
2
∂
r
1
4
(1
∂
N
2
∂
s
1
4
(1
=
−
s
)
=−
+
r
)
∂
N
3
∂
r
1
4
(1
∂
N
3
∂
s
1
4
(1
=
+
s
)
=
+
r
)
∂
N
4
∂
r
1
4
(1
∂
N
4
∂
s
1
4
(1
=−
+
s
)
=
−
r
)
Substituting numerical values (noting that
a
=
b
), we obtain, for example,
1
16
(
s
−
1)
2
0
.
5
12
d
r
d
s
1
1
1
16
(
r
−
1)
2
k
11
=
20
+
−
1
−
1
−
s
)
2
0
.
5
12
2
1
1
1
16
−
r
)
2
(
1
+
2(50)
(
1
d
r
d
s
−
1
−
1
Integrating first on
r
,
1
1
1
)
2
r
(
r
−
1)
3
3
20(0
.
5)
16(12)
1
−
1
+
k
11
=
(
s
−
d
s
−
1
−
1
0
.
5
12
2
1
1
−
s
)
2
(1
−
r
)
3
3
100
16
−
(
1
d
s
−
1
−
1
or
(
s
−
1)
2
(2)
+
d
s
+
0
2
1
1
20(0
5)
16(12)
.
8
3
100
16
5
12
.
(1
−
s
)
2
8
3
k
11
=
d
s
−
1
−
1
Then, integrating on
s
, we obtain
3
s
2(
s
0
2
8
3
(1
1
1
20(0
5)
16(12)
.
−
1)
3
8
100
16
5
12
.
−
s
)
3
k
11
=
+
1
−
3
3
−
−
1
or
16
3
+
0
2
8
3
8
3
20(0
5)
16(12)
.
16
3
100
16
5
12
.
=
0
.
6327
Btu/(hr-
◦
F
)
k
11
=
+
The analytical integration procedure just used to determine
k
11
is
not
the method used by
finite element software packages; instead, numerical methods are used, primarily the
Gauss quadrature procedure discussed in Chapter 6. If we examine the terms in the inte-
grands of the equation defining
k
ij
, we find that the integrands are quadratic functions
