Civil Engineering Reference
In-Depth Information
The [
B
] matrix contains derivatives of the shape functions with respect to global coor-
dinates, but first these are computed in the local coordinate system as
∂
fun
T
∂ξ
∂
fun
T
∂η
der
=
(3.43)
or
−
(
1
1
4
−
η)
−
(
1
+
η) (
1
+
η)
(
1
−
η)
der
=
(3.44)
−
(
1
−
ξ)
(
1
−
ξ) (
1
+
ξ)
−
(
1
+
ξ)
The information in (3.42) and (3.44) for a 4-node quadrilateral (
nod=4
) is formed by
the subroutines,
shape_fun
and
shape_der
for the specific Gaussian integration points
(ξ, η)
i
held in the array
points
where
i
runs from 1 to
nip
, the total number of sampling
points specified in each element. Figure 3.9(a) shows the node numbering and the order in
which the sampling (Gauss) points are scanned in a “2-point” scheme. Since there are two
integrating points in each coordinate direction
nip=4
in this example. In all cases,
points
and their corresponding
weights
(Table 3.1) are found by the subroutine
sample
,where
nip
can take the values 1, 4 or 9 for quadrilaterals.
The derivatives
der
must then be converted into their counterparts in the
(x, y)
coor-
dinate system,
deriv
, by means of the Jacobian matrix transformation (3.4). From the
isoparametric property (3.2),
1
4
(
1
1
4
(
1
x
=
−
ξ)(
1
−
η)x
1
+
−
ξ)(
1
+
η)x
2
1
4
(
1
1
4
(
1
+
+
ξ)(
1
+
η)x
3
+
+
ξ)(
1
−
η)x
4
(3.45)
1
4
(
1
1
4
(
1
y
=
−
ξ)(
1
−
η)y
1
+
−
ξ)(
1
+
η)y
2
1
4
(
1
1
4
(
1
+
+
ξ)(
1
+
η)y
3
+
+
ξ)(
1
−
η)y
4
L
2
h
2
3
i
=
1
i
=
2
3
x
=
=
i
3
i
4
=
i
1
1
4
L
1
1
2
Figure 3.9 Integration schemes for (a) quadrilateral element with
nip=4
, and (b) trian-
gular element with
nip=1
