Civil Engineering Reference
In-Depth Information
4.2.4.2
Buckling moment
The value
M
cr
is relevant only to an initially perfect member that remains
elastic. Evidence is limited on the influence of initial imperfections,
residual stresses and yielding of steel on this type of buckling; but the
Perry-Robertson formulation and the strut curves developed for overall
buckling of steel columns provide a suitable basis. The method of EN
1994-1-1 is therefore as follows.
The slenderness l
LT
is given for a Class 1 or 2 section by
l
LT
=
(
M
Rk
/
M
cr
)
1/2
(4.23)
where
M
Rk
is the value that would be obtained for
M
Rd
in hogging bending
if the partial factors
γ
S
were taken as 1.0. This is because these
factors do not occur in the calculation of
M
cr
. For a Class 3 section,
M
Rk
is
the characteristic yield moment.
The buckling moment is given by
γ
A
and
M
b
,
Rd
=
χ
LT
M
Rd
(4.24)
where
χ
LT
is a function of l
LT
that in practice is taken from the relevant
strut curve in Eurocode 3: Part 1.1. For rolled steel sections this curve is
given by
χ
LT
=
[
Φ
LT
+
(
Φ
LT
−
l
LT
)
1/2
]
−1
but
χ
LT
≤
1
(4.25)
where
Φ
LT
=
0.5[1
+
α
LT
(l
LT
−
l
LT
,0
)
+
β
l
LT
]
(4.26)
and
α
LT
is an imperfection factor. For rolled sections,
α
LT
=
0.21 where
h
a
/
b
f
0.34 otherwise.
For rolled or equivalent welded steel sections, national annexes may
give values for l
LT
,0
that are
≤
2 and
α
LT
=
0.75. EN 1993-1-
1 recommends the use of these limiting values. Their effect is that lateral
buckling does not reduce
M
Rd
until l
LT
≤
0.4, and for
β
that are
≥
>
0.4.
Simplified expression for
l
LT
For cross-sections in Class 1 or 2, and with some loss of economy, Equa-
tion 4.23 can be replaced by
025
.
2
3
⎡
⎤
⎡
⎢
th
bt
⎤
⎥
⎛
⎜
f
EC
⎞
⎟
⎛
⎜
h
t
⎞
⎟
⎛
⎜
t
b
⎞
⎟
y
ws
s
f
.
=
50 1
+
⎢
⎢
⎥
⎥
l
LT
4
⎣
⎦
ff
a
4
w
f
