Civil Engineering Reference
In-Depth Information
m
n
s
Beam-column
First-order moment (
N
= 0)
k
M
max
,0
cr
2
qL
/8
q
qL
2
---------
1.0
1.0
1.0 -0.03
+
8
q
qL
2
-
2
qL
---------
2.0
1.0
1.0 0.4
/2
2
q
2
2
qL
qL
2
qL
9
/128
9
16
+
------
---------
---------
1.0
1.0 0.29
-
8
8
2
qL
/8
q
2
qL
qL
2
9
/128
+
---------
0.7
1.0
1.0 0.36
-
8
2
qL
/8
q
2
qL
qL
2
9
/128
---------
+
+
1.0
1.0
0.49 0.18
8
-
L
L
2
qL
/8
2
qL
q
2
2
/24
qL
2
qL
qL
+
---------
---------
---------
1.0
0.5
1.0 0.4
-
-
12
12
12
2
qL
/12
2
qL
qL
2
q
/24
+
0.5
1.0
1.0
0.37
---------
-
-
12
2
qL
/12
Figure 7.6
Approximations for beam-columns with distributed loads.
in which
σ
ac
=
N
/
A
and
σ
bcy
=
M
/
W
el
,
y
. If the member has no residual stresses,
then it will remain elastic while
σ
max
is less than the yield stress
f
y
, and so the
above results are valid while
N
y
+
M
N
M
max
M
≤
1.0
(7.13)
M
y
in which
N
y
=
Af
y
is the squash load and
M
y
=
f
y
W
el
the nominal first yield
moment.Atypicalelasticlimitofthistypeisshownbythefirstyieldpointmarked
on curve 3 of Figure 7.2. It can be seen that this limit provides a lower bound
estimate of the resistance of a straight beam-column, while the elastic buckling
load
N
cr
,
y
provides an upper bound.
Variationsofthefirstyieldlimitsof
N
/
N
y
determinedfromequation7.13with
M
/
M
y
and
β
m
are shown in Figure 7.7a for the case where
N
cr
,
y
=
N
y
/
1.5. For a
beam-column in double curvature bending (
β
m
=
1),
M
max
=
M
, and so
N
varies
linearlywith
M
untiltheelasticbucklingload
N
cr
,
y
isreached.Forabeam-column
inuniformbending(
β
m
=−
1),therelationshipbetween
N
and
M
isnon-linearand
concave due to the amplification of the bending moment from
M
to
M
max
by the
axial load.Also shown in Figure 7.7a are solutions of equation 7.13 based on the
use of the approximations of equations 7.7 and 7.8. A comparison of these with
the accurate solutions also shown in Figure 7.7a again demonstrates the
comparative accuracy of the approximate equations 7.7 and 7.8.
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