Civil Engineering Reference
In-Depth Information
per node. The stiffness matrix of an element in a grid model with warping
partially restrained is as follows:
GK
l
−
GK
l
te
te
0
0
0
0
EI
l
−
EI
l
EI
l
EI
l
4
6
2
6
y
y
y
y
0
0
2
2
−
EI
l
EI
−
EI
l
−
EI
6
12
6
12
y
y
y
y
0
0
2
3
2
3
l
l
[
]
=
(7.4)
K
e
−
GK
l
GK
l
te
te
0
0
0
0
EI
l
−
EI
l
EI
l
EI
l
2
6
4
6
y
y
y
y
0
0
2
2
EI
l
−
EI
EI
l
EI
l
6
12
6
12
y
y
y
y
0
0
2
l
3
2
3
A similar study was done years later by the NCHRP Project 12-79 Report
725 (White et al. 2012) with two equivalent equations with warping fix-
ity at each end of a given unbraced length
L
b
(Equation 7.5a) and warping
fixity at one end and warping free boundary conditions (Equation 7.5b),
where
J
eq
is equivalent to
K
te
in Equation 7.3.
−
1
[
1
2
]
cosh(
pL
)
−
sinh(
pL
pL
)
b
b
J
=
J
1
−
+
(7.5a)
eq fx fx
(
−
)
pL
sinh
(
pL
)
b
b
b
−
1
sinh(
pL
)
(7.5b)
b
J
=
J
1
−
eq s fx
(
−
)
pL
cosh
(
pL
)
b
b
A cross frame between girders for a grid analysis can be formed by steel
beam, X-type, and K-type cross frames. For the 3D-modeling purpose, at
least four or five nodes are needed for the definition of a cross frame as seen
on the right side of Figure 7.2. For a 2D grid model, idealization in beam
solutions is used to simulate the
exact
equivalent beam stiffness of this cross
frame. In the NCHRP Project 12-79 Report 725 (White et al. 2012), it is
called
Timoshenko beam element
.
This approach simply involves the calculation of an equivalent moment of
inertia,
I
eq
, as well as an equivalent shear area
As
eq
(as shown in Equations 7.6
and 7.7) for a shear-deformable (Timoshenko) beam element representation
of the cross frame.
1. The equivalent moment of inertia is determined first based on pure
flexural deformation of the cross frame (zero shear). The cross frame
is supported as a cantilever at one end and is subjected to a force
couple applied at the corner joints at the other end (Figure 7.5).
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