Civil Engineering Reference
In-Depth Information
Figure 5.12
First-order and second-order bending moments in a
column length
The greatest first-order bending moment is multiplied by a factor
k
end
given by
k
=
β
/[1
−
(
N
Ed
/
N
cr,eff
) ]
(5.28)
where
β
=
0.66
+
0.44(
M
2
/
M
1
)
≥
0.44
(5.29)
with
N
cr,eff
as above. The coefficient
allows for the more adverse effect
of single-curvature bending than of double-curvature bending, for which
M
2
/
M
1
is negative.
In EN 1994-1-1, Equation 5.28 appears with the further condition
β
k
1.0. It is over-conservative to apply this when combining two sets of
second-order effects, and
≥
1. This value need
not be increased to 1.0. The first-order end moments, for example AB in
Fig. 5.12(c), are replaced by an equivalent uniform moment
β
end
is often such that
k
end
<
β
end
M
1,Ed
,
which is increased to
k
end
M
1,Ed
at mid-length to allow for second-order
effects, as shown. This always exceeds
β
end
M
1,Ed
, from Equation 5.28.
For the bending moment from the member imperfection, EN 1994-1-1
specifies
β
imp
=
1.0 so, from Equation 5.28,
k
imp
always exceeds 1.0,
Fig. 5.12(d).
The design bending moment for the column length is usually
M
Ed
=
k
end
M
1,Ed
+
k
imp
N
Ed
e
0
(5.30)
but is
M
1,Ed
, if greater.
