Civil Engineering Reference
In-Depth Information
Position and torsional
end restraints
L
2
L
1
L
1
L
2
(a) Beam supported by cross-beams
(b) Symmetrical portal frame
Figure 6.26
Simple rigid-jointed structures.
Byanalysingtheminoraxisanddifferentialflangebendingofthecross-beams,
it can be shown that
R
2
=
R
4
=
1
/(
1
+
L
1
I
z
2
/
6
L
2
I
z
1
)
.
(6.70)
This result is based on the assumption that there is no likelihood of the cross-
beamsthemselvesbuckling,sothattheireffectiveminoraxisrigiditycanbetaken
as
EI
z
1
. With these values for the restraint parameters, the elastic buckling load
can be determined from the tabulations in [33, 35].
The buckling loads of the symmetrical portal frame shown in Figure 6.26b can
bedeterminedinasimilarmannerprovidedtherearesufficientexternalrestraints
to position-fix the joints of the frame. The columns of portal frames of this type
are usually placed so that the plane of greatest bending stiffness is that of the
frame. In this case, the minor axis stiffness of each column provides only a small
resistance against end twisting of the beam, and this resistance is reduced by the
axial force transmitted by the column, so that it may be necessary to provide
additional torsional end restraints to the beam. If these additional restraints are
stiffenough(andtheinformationgivenin[16,34]willgivesomeguidance),then
it may be assumed without serious error that
R
3
=
0. The major axis stiffness of
a pinned base column is 3
EI
y
1
/
L
1
and of a fixed base column is 4
EI
y
1
/
L
1
, and it
can be shown by analysing the in-plane flexure of a portal frame that for pinned
base portals
R
1
=
1
/(
1
+
2
L
1
I
y
2
/
3
L
2
I
y
1
)
,
(6.71)
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