Civil Engineering Reference
In-Depth Information
The nodal displacements of a frame element as shown in Figure 3.8 are
=
[
]
T
u
v
u
v
(3.41)
a
e
φ
φ
1
1
1
2
2
2
The strains of a frame element contain the axial tension/compression strain
and bending strain as
du
dx
=
ε
ε
(3.42)
a
ε
e
=
2
y
d v
dx
b
−
2
where:
y
is the vertical distance of a fiber layer to the neutral axis of a cross section
u
and
v
are axial and vertical displacements, respectively
Their interpolation functions are
u
v
N
N
=
0
0
0
0
1
4
a
(3.43)
e
N
N
N
N
0
0
2
3
5
6
where:
(
)
2
2
N L N L
=
,
=
3 2
−
L
,
N L L l N L
=
,
=
,
1
1
2
1
1
3
1
2
4
2
(3.44)
x
l
x
l
(
)
2
2
N L
=
3 2
−
L
,
N
= −
L L l L
,
= −
1
,
L
=
5
2
2
6
1 2
1
2
Knowing φ =
dv dx
, it can be easily verified that the earlier functions satisfy
the conditions of a displacement function in Section 3.2.3.
The matrix
B
in Equation 3.28 is
1
1
−
0
0
l
l
B
=
x
x
12
6
6
4
−
y
−
−
y
−
0
0
l
3
l
2
l
2
l
(3.45)
0
0
x
x
12
6
6
2
y
−
−
y
−
3
2
2
l
l
l
l
When integrating over the entire element as in Equation 3.30, the beam-bending
assumption and a prismatic cross section can be taken into consideration.
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