Civil Engineering Reference
In-Depth Information
and as before the isoparametric property gives,
x
=
N
1
x
1
+
N
2
x
2
+
N
3
x
3
=
[
N
]
{
x
}
y
=
N
1
y
1
+
N
2
y
2
+
N
3
y
3
(3.11)
=
[
N
]
{
y
}
Equations (3.3) and (3.4) from the previous paragraph still apply regarding the Jacobian
matrix but equation (3.5) must be modified for triangles to give,
d
x
d
y
=
1
1
−
L
1
|
|
det
J
d
L
2
d
L
1
(3.12)
0
0
3.3.1 Numerical integration for triangles
Numerical integration over triangular regions is similar to that for quadrilaterals, and takes
the general form
1
1
−
L
1
nip
f(L
1
,L
2
)
d
L
2
d
L
1
≈
W
i
f(L
1
,L
2
)
i
(3.13)
0
0
i
=
1
where
W
i
is the weighting coefficient corresponding to the sampling point
(L
1
,L
2
)
i
and
nip
represents the number of integrating points. Typical values of the weights and sampling
points are given in Table 3.2.
As with quadrilaterals, numerical integration can be exact for certain polynomials. For
example, in Table 3.2, the 1-point rule is exact for integration of first degree polynomials
and the 3-point rule is exact for polynomials of second degree. Reduced integration can
again be beneficial in some instances.
Computer formulations involving local coordinates, transformations of coordinates and
numerical integration are described in subsequent paragraphs.
Table 3.2 Coordinates and weights
in triangular integration formulae
nip
(L
1
,L
2
)
i
W
i
3
,
3
1
2
1
2
,
2
1
6
3
2
,
0
1
6
0
,
2
1
6
