Civil Engineering Reference
In-Depth Information
Asthedeformationsincreasewiththeappliedmoments
M
,sodothestresses.It
is shown in Section 6.12.2 that the limiting moment
M
L
at which a beam without
residual stresses first yields is given by
M
L
M
y
=
1
(6.18)
Φ
2
−
λ
2
Φ
+
in which
Φ
=
(
1
+
η
+
λ
2
)/
2,
(6.19)
λ
=
√
(
M
y
/
M
zx
)
(6.20)
is a generalised slenderness,
M
y
=
W
el
,
y
f
y
is the nominal first yield moment, and
η
is a factor defining the imperfection magnitudes, when the central crookedness
δ
0
is given by
δ
0
N
cr
,
z
M
zx
W
el
,
z
/
W
el
,
y
1
+
(
d
f
/
2
)(
N
cr
,
z
/
M
zx
)
=
θ
0
=
η
,
(6.21)
in which
N
cr
,
z
is given by equation 6.12. Equations 6.18-6.20 are similar to
equations 3.11, 3.12, and 3.5 for the limiting axial force at first yield of a
1.2
Yielding due to major
axis bending alone
1.0
0.8
Elastic buckling
M
zx
/
M
y
0.6
First yield
M
L
/
M
y
due to initial
crookedness and twist
0.4
N
W
W
/
0
cr,z
el
,
z
el
,
y
=
=
0
(
)
M
N
0.2
d
zx
f
cr
,
z
-
1
2
M
zx
0
0
0.5
1.0
1.5
2.0
2.5
Generalised slenderness
=
(
)
M
y
/
M
cr
Figure 6.9
Buckling and yielding of beams.
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