Civil Engineering Reference
In-Depth Information
body forces is
d
W
b
=
F
BX
ut
d
x
d
y
+
F
BY
vt
d
x
d
y
(9.42)
Considering the volume of interest to be a CST element in which the displace-
ments are expressed in terms of interpolation functions and nodal displace-
ments as
u
1
u
2
u
3
v
1
v
2
v
3
u
(
x
,
y
)
v
(
x
,
y
)
N
1
N
2
N
3
000
000
N
1
[
N
]
u
v
(9.43)
=
=
N
2
N
3
the total work done by the body forces acting on the element is expressed in
terms of nodal displacement as
t
W
b
=
F
BX
(
N
1
u
1
+
N
2
u
2
+
N
3
u
3
)d
x
d
y
A
t
+
F
BY
(
N
1
v
1
+
N
2
v
2
+
N
3
v
3
)d
x
d
y
(9.44)
A
As desired, Equation 9.44 is in the form
f
(
b
)
1
x
f
(
b
)
2
x
f
(
b
)
3
x
f
(
b
)
f
(
b
)
f
(
b
)
W
b
=
u
1
+
u
2
+
u
3
+
1
y
v
1
+
2
y
v
2
+
3
y
v
3
(9.45)
in terms of equivalent concentrated nodal forces. The superscript (
b
) is used to
indicate nodal-equivalent body force. Comparison of the last two equations
yields the nodal force components as
t
f
(
b
)
ix
=
N
i
F
BX
d
x
d
y
i
=
1, 3
A
(9.46)
t
f
(
b
)
iy
=
N
i
F
BY
d
x
d
y
i
=
1, 3
A
The nodal force components equivalent to the applied body forces can also be
written in the compact matrix form
[
N
]
T
F
BX
F
BY
d
x
d
y
t
f
(
b
)
=
(9.47)
A
While developed in the specific context of a constant strain triangular element in
plane stress, Equation 9.47 proves to be a general result for two-dimensional
elements. A quite similar expression holds for three-dimensional elements.
