Civil Engineering Reference
In-Depth Information
Table 3.15.
(
Continued
)
x
y
weight
f(x,y)
weight*f(x,y)
2
8
4
1
4
4
0
2
0
0
4
2
8
0.5
4
4
4
4
1
4
4
6
8
1.5
12
4
8
2
2
4
6
0
4
0
0
6
2
16
0.75
12
6
4
8
1.5
12
6
6
16
2.25
36
6
8
4
3
12
8
0
1
0
0
8
2
4
1
4
8
4
2
2
4
8
6
4
3
12
8
8
1
4
4
144
S
3.6
DOubLE intEgRAtiOn bY gAuSSiAn
QuADRAtuRE
Double integration by Gaussian quadrature is very similar to the single
integration process. In this case, the Gauss equation is applied in both
directions and then multiplied by the weighting factors.
bb
,
n
(
)
xy
(
)
∑
=
∫∫
V
=
f xydxdy ss fs sx s
,
=
±
,
±
s yww
xy
i
x
1
x
i y xi xi
1
yi
ab
,
xx
1
s
ba
−
ba
+
=
x
x
and
s
=
x
x
x
x
1
2
2
s
ba
s
ba
−
−
ba
ba
+
+
y
y
y
y
=
and
s
=
y
y
1
2
2
The number of points used should closely match the degree of the equa-
tion to integrate. The same Gauss points and weights from Section 3.4 are
used in each direction.
