Civil Engineering Reference
In-Depth Information
y
d
y
d
z
d
x
x
z
Figure 10.11
Differential element of a general
three-dimensional body.
For the general case, we consider the three-dimensional body depicted in
Figure 10.11 and examine a differential mass
d
m
=
d
x
d
y
d
z
located at arbi-
trary position
(
x
,
y
,
z
)
.
Displacement of the differential mass in the coordinate
directions are
(
u
,
v
,
w
)
and the velocity components are
(
u
,
v
,
w
)
, respectively.
As we previously examined the potential energy, we now focus on kinetic energy
of the differential mass given by
˙
1
2
(
1
2
(
u
2
v
2
w
2
)d
m
u
2
v
2
w
2
)
d
T
=
˙
+˙
+˙
=
˙
+˙
+˙
d
x
d
y
d
z
(10.85)
Total kinetic energy of the body is then
(
(
1
2
1
2
u
2
v
2
w
2
)d
m
u
2
v
2
w
2
)
T
=
˙
+˙
+˙
=
˙
+˙
+˙
d
x
d
y
d
z
(10.86)
and the integration is performed over the entire mass (volume) of the body.
Considering the body to be a finite element with the displacement field
discretized as
M
u
(
x
,
y
,
z
,
t
)
=
N
i
(
x
,
y
,
z
)
u
i
(
t
)
=
[
N
]
{
u
}
i
=
1
M
v
(
x
,
y
,
z
,
t
)
=
N
i
(
x
,
y
,
z
)
v
i
(
t
)
=
[
N
]
{
v
}
(10.87)
i
=
1
M
w
(
x
,
y
,
z
,
t
)
=
N
i
(
x
,
y
,
z
)
w
i
(
t
)
=
[
N
]
{
w
}
i
=
1
(where
M
is the number of element nodes), the velocity components can be
expressed as
∂
u
u
˙
=
t
=
[
N
]
{˙
u
}
∂
∂
v
(10.88)
v
˙
=
t
=
[
N
]
{˙
v
}
∂
∂
w
∂
w
˙
=
t
=
[
N
]
{˙
w
}
