Civil Engineering Reference
In-Depth Information
First, the distributed loads are converted to equivalent loadings in the global
coordinate directions, as in Figure 9.3b, via
p
x
=
p
n
n
x
−
p
t
n
y
(9.33)
p
y
=
p
n
n
y
+
p
t
n
x
with
n
x
and
n
y
corresponding to the components of the unit outward normal vec-
tor to edge 2-3. Here, we use the notation
p
for such loadings, as the units are
those of pressure. The mechanical work done by the distributed loads is
3
3
W
p
=
t
p
x
u
(
x
,
y
)d
S
+
t
p
y
v
(
x
,
y
)d
S
(9.34)
2
2
where the integrations are performed along the edge defined by nodes 2 and 3.
Recalling that interpolation function
N
1
(
x
,
y
)
is zero along edge 2-3, the finite
element representations of the displacements along the edge are
u
(
x
,
y
)
=
N
2
(
x
,
y
)
u
2
+
N
3
(
x
,
y
)
u
3
(9.35)
v
(
x
,
y
)
=
N
2
(
x
,
y
)
v
2
+
N
3
(
x
,
y
)
v
3
The work expression becomes
3
W
p
=
t
p
x
[
N
2
(
x
,
y
)
u
2
+
N
3
(
x
,
y
)
u
3
]d
S
2
3
+
t
p
y
[
N
2
(
x
,
y
)
v
2
+
N
3
(
x
,
y
)
v
3
]d
S
(9.36)
2
and is of the form
f
(
p
)
2
x
f
(
p
)
3
x
f
(
p
)
2
y
f
(
p
)
3
y
v
3
(9.37)
Comparison of the last two equations yields the equivalent nodal forces as
W
p
=
u
2
+
u
3
+
v
2
+
3
f
(
p
)
2
x
=
t
p
x
N
2
(
x
,
y
)d
S
2
3
f
(
p
)
3
x
=
t
p
x
N
3
(
x
,
y
)d
S
2
(9.38)
3
f
(
p
)
2
y
=
t
p
y
N
2
(
x
,
y
)d
S
2
3
f
(
p
)
3
x
=
t
p
y
N
3
(
x
,
y
)d
S
2
