Civil Engineering Reference
In-Depth Information
EXAMPLE 9.2
Determine the nodal force components representing the body force for the element of
Example 9.1, if the body force is gravitational attraction in the
y
direction, so that
F
BX
F
BY
0
−
386
.
4
in./sec
2
=
slug/in.
3
.
given the density of the element material is
=
7
.
3
×
10
−
4
■
Solution
As the
x
component of the body force is zero, the
x
components of the nodal force vector
will be, too, so we need not consider those components. The
y
components are computed
using the second of Equation 9.46:
=
t
A
f
(
b
)
iy
=
1, 3
N
i
F
BY
d
x
d
y
i
From the previous example, the interpolation functions are
1
2
[2
−
2
x
]
=
1
−
x
N
1
(
x
,
y
)
=
1
2
[2
x
−
y
]
N
2
(
x
,
y
)
=
1
2
y
N
3
(
x
,
y
)
=
We have, in this instance,
=
t
A
F
BY
N
1
d
x
d
y
=
t
A
f
(
b
)
1
y
F
BY
(1
−
x
)d
x
d
y
=
t
A
F
BY
N
2
d
x
d
y
=
t
A
F
BY
2
f
(
b
)
2
y
(2
x
−
y
)d
x
d
y
=
t
A
F
BY
N
3
d
x
d
y
=
t
A
F
BY
2
f
(
b
)
3
y
y
d
x
d
y
The limits of integration must be determined on the basis of the geometry of the area. In
this example, we utilize
x
as the basic integration variable and compute the
y
-integration
limits in terms of
x
. For the element under consideration, as
x
varies between zero and
one,
y
is the linear function
y
=
2
x
so the integrations become
=
tY
x
2
1
2
x
1
1
2
x
3
3
f
(
b
)
1
y
=
tF
BY
−
x
)d
y
d
x
=
tF
BY
−
x
)d
x
−
(1
2
x
(1
0
0
0
0
=
tF
BY
3
=−
0
.
0189 lb
