Civil Engineering Reference
In-Depth Information
Use the integration formula of Equation 6.49 to confirm the result of Example 6.6.
6.25
Use the integration formula for area coordinates to show that
6.26
d
x
d
y
=
1
1
−
L
2
A
=
d
A
=
d
L
1
d
L
2
A
0
0
Consider the isoparametric quadrilateral element in Figure P6.27. Map the point
r
=
0
.
5,
s
=
0
in the parent element to the corresponding physical point in the
quadrilateral element.
6.27
3 (2.4, 2.6)
4 (2, 2.5)
y
2 (2.5, 2.1)
x
1 (2, 2)
Figure P6.27
Again referring to the element in Figure P6.27, map the line
r
=
0
in the parent
element to the physical element. Plot the mapping on a scaled drawing of the
quadrilateral element.
6.28
Repeat Problem 6.28 for the line
s
=
0
in the parent element.
6.29
Consider the two-node line element in Figure P6.30 with interpolation functions
6.30
N
1
(
r
)
=
1
−
r
2
(
r
)
=
r
Using this as the parent element, examine the isoparametric mapping
x
=
N
1
(
r
)
x
1
+
N
2
(
r
)
x
2
for arbitrary values
x
1
and
x
2
such that
x
1
<
x
2
.
a.
What has been accomplished by the mapping?
b.
Determine the Jacobian matrix for the transformation.
r
1
2
r
1
0
r
2
1
Figure P6.30
Consider the three-node line element in Figure P6.31 with interpolation functions
N
1
(
r
)
=
(2
r
−
1)(
r
−
1)
N
2
(
r
)
=
4
r
(1
−
r
)
N
3
(
r
)
=
r
(2
r
−
1)
6.31
