Civil Engineering Reference
In-Depth Information
the discretized representation of the displacement field can be written in matrix
form as
u
v
w
[
N
]
[0]
[0]
{}=
=
[0]
[
N
]
[0]
[
N
3
]
{}
(9.108)
[0]
[0]
[
N
]
In the last equation, each submatrix
[
N
]
is the
1
×
M
row matrix of interpolation
functions
[
N
]
=
[
N
1
N
2
···
N
M
]
(9.109)
so the matrix we have chosen to denote as
[
N
3
]
is a
3
×
3
M
matrix composed of
the interpolation functions and many zero values. (Before proceeding, we
emphasize that the order of nodal displacements in Equation 9.107 is convenient
for purposes of development but
not
efficient for computational purposes. Much
higher computational efficiency is obtained in the model solution phase if the
displacement vector is defined as
{}=
[
u
1
v
1
w
1
u
2
v
2
w
2
u
M
v
M
w
M
]
T
.)
Recalling Equations 9.10 and 9.19, total potential energy of an element can
be expressed as
···
1
2
T
[
D
]
T
=
U
e
−
W
=
{
ε
}
{
ε
}
d
V
−{}
{
f
}
(9.110)
V
The element nodal force vector is defined in the column matrix
{
f
Mz
]
T
(9.111)
and may include the effects of concentrated forces applied at the nodes, nodal
equivalents to body forces, and nodal equivalents to applied pressure loadings.
Considering the foregoing developments, Equation 9.110 can be expressed
(using Equations 9.104, 9.105, and 9.108), as
f
}=
[
f
1
x
f
2
x
···
f
Mx
f
1
y
f
2
y
···
f
My
f
1
z
f
2
z
···
1
2
T
[
L
]
T
[
N
3
]
T
[
D
][
L
][
N
3
]
T
=
U
e
−
W
=
{}
d
V
−{}
{
f
}
(9.112)
V
As the nodal displacement components are independent of the integration over
the volume, Equation 9.112 can be written as
T
1
2
{}
[
L
]
T
[
N
3
]
T
[
D
][
L
][
N
3
]d
V
T
=
U
e
−
W
=
{}−{}
{
f
}
(9.113)
V
which is in the form
T
1
2
{}
[
B
]
T
[
D
][
B
]d
V
T
=
U
e
−
W
=
{}−{}
{
f
}
(9.114)
V
