Civil Engineering Reference
In-Depth Information
3.0
Beam with equal and
opposite end moments
2.5
Second mode
buckling
2.0
2
1.5
x
Axis of buckled beam
Braced first mode
buckling
f
L/2
1.0
M
cr
L/2
0.5
t
(
-
z
t
f
)
L/2
0
-z
t
0 5 10 15 20 25
r
f
L/2
(
)
M
cr
L
3
2
t
z
2
z
t
Translational restraint stiffness
1
+
/
d
d
1
6
EI
y
z
(
)
3
Rotational restraint stiffness
2
z
2
z
L
r
t
/
1
-
z
1
6
EI
d
d
w
(a) Beam
(b) Effective length factors
Figure 6.15
Beam with elastic intermediate restraints.
These relationships are shown graphically in Figure 6.15b, and are similar to
thatshowninFigure3.16cforcompressionmemberswithintermediaterestraints.
Itcanbeseenthattheeffectivelengthfactor
k
cr
variesfrom1whentherestraints
are of zero stiffness to 0.5 when
1
−
2
z
t
d
f
α
t
L
3
16
EI
z
=
α
r
L
3
16
EI
w
1
+
2
z
t
/
d
f
2
z
c
/
d
f
2
z
c
d
f
=
π
2
.
(6.43)
If the restraint stiffnesses exceed these values, the beam buckles in the second
mode with zero central deflection and twist at a moment which corresponds to
k
cr
=
0.5. When the height
−
z
t
of the translational restraint above the centroid
is equal to
−
d
f
/
4
z
c
, the required rotational stiffness
α
r
given by equation 6.43
is zero. Since
z
c
is never less than
d
f
/
2 (see equation 6.42), it follows that a top
flange translational restraint of stiffness
α
L
=
16
π
2
EI
z
/
L
3
1
+
d
f
/
2
z
c
(6.44)
isalwayssufficienttobracethebeamintothesecondmode.Thisminimumstiffness
can be expressed as
α
L
=
8
M
cr
Ld
f
1
{
1
+
√
(
1
+
4
K
2
)
}
,
(6.45)
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