Frames - The Behaviour and Design of Steel Structures to EC3 - page 362

of braced and unbraced frames are given in Figures 8.3 and 8.4. For the braced
frame of Figure 8.3, the second-order results are only slightly higher than the
first-order results. This is usually true for well-designed braced frames that have
substantialbendingeffectsandsmallaxialcompressions. Fortheunbracedframe
ofFigure8.4,thesecond-orderswaydeflectionsareabout18%largerthanthoseof
thefirst-orderanalysis,whilethemomentatthetopoftheright-handlower-storey
column is about 10% larger than that of the first-order analysis.
Second-order effects are often significant in unbraced frames.

8.3.5.6 Approximate second-order elastic analysis

Thereareanumberofmethodsofapproximatingsecond-ordereffectswhichallow
a general second-order analysis to be avoided. In many of these, the results of a
first-orderelasticanalysisareamplifiedbyusingtheresultsofanelasticbuckling
analysis (Sections 8.3.5.3 and 8.3.5.4).
Formemberswithouttransverseforcesinbracedframes,themaximummember
moment M max determined by first-order analysis may be used to approximate the
maximum second-order design moment M as

M = δ M max (8.19)

in which δ is an amplification factor given by

c m
1 − N / N cr , b ≥ 1,

δ = δ b = (8.20)

in which N cr , b is the elastic buckling load calculated for the braced member, and

c m = 0.6 − 0.4 β m ≤ 1.0 (8.21)

in which β m is the ratio of the smaller to the larger end moment (equations 8.20
and 8.21 are related to equations 7.7 and 7.8 used for isolated beam-columns).
A worked example of the application of this method is given in Section 8.5.6.
A procedure for approximating the value of β m to be used for members with
transverse forces is available [28], while more accurate solutions are obtained by
using equations 7.9-7.11 and Figures 7.5 and 7.6.
For unbraced frames, the amplification factor δ may be approximated by

1
1 − 1 /α cr

δ = δ s = (8.22)

in which α cr is the elastic sway buckling load factor calculated for the unbraced
frame (Section 8.3.5.4). A worked example of the application of this method is
given in Section 8.5.7.
For unbraced rectangular frames with negligible axial forces in the beams,
amoreaccurateapproximationcanbeobtainedbyamplifyingthecolumnmoments