Civil Engineering Reference
In-Depth Information
becomes simply
(
e
)
2
(
e
)
T
[
B
]
T
[
D
][
B
]
1
U
(
e
)
e
d
V
(
e
)
=
V
(
e
)
2
(
e
)
T
V
(
e
)
[
B
]
T
[
D
][
B
]
(
e
)
1
=
(9.16)
where
V
(
e
)
is the total volume of the element.
Considering the element forces to be as in Figure 9.2b (for this element for-
mulation, we require that forces be applied only at nodes; distributed loads are
considered subsequently), the work done by the applied forces can be expressed
as
f
3
y
v
3
(9.17)
and we note that the subscript notation becomes unwieldy rather quickly in the
case of 2-D stress analysis. To simplify the notation, we use the force notation
W
=
f
1
x
u
1
+
f
2
x
u
2
+
f
3
x
u
3
+
f
1
y
v
1
+
f
2
y
v
2
+
f
1
x
f
2
x
f
3
x
f
1
y
f
2
y
f
3
y
{
f
} =
(9.18)
so that we can express the work of the external forces (using Equation 9.13) as
W
T
={}
{
f
}
(9.19)
Per Equation 2.53, the total potential energy for an element is then
V
e
2
{}
T
[
B
]
T
[
D
][
B
]
T
=
U
e
−
W
=
{}−{}
{
f
}
(9.20)
If the element is a portion of a larger structure that is in equilibrium, then the
element must be in equilibrium. Consequently, the total potential energy of the
element must be minimum (we consider only stable equilibrium), and for this
minimum, we must have mathematically
∂
∂
i
=
0
i
=
1, 6
(9.21)
If the indicated mathematical operations of Equation 9.21 are carried out on
Equation 9.20, the result is the matrix relation
V
e
[
B
]
T
[
D
][
B
]
{}={
f
}
(9.22)
and this matrix equation is of the form
[
k
]
{}={
f
}
(9.23)
