Frames - The Behaviour and Design of Steel Structures to EC3 - page 363

ofeachstoreybyusingtheswaybucklingloadfactor α cr , s forthatstoreyobtained
using equation 8.7 in

1
1 − 1 /α cr , s

δ = δ s = (8.23)

in which α cr , s is the elastic sway buckling load factor calculated for the storey
(Section 8.3.5.4).Aworked example of the application of this method is given in
Section 8.5.8.
Alternatively, the first-order end moment

M f = M fb + M fs (8.24)

maybeseparatedintoabracedframecomponent M fb and acomplementarysway
component M fs [29].Thesecond-ordermaximummoment M maybeapproximated
by using these components in

M fs
( 1 − 1 /α cr )

M = M fb + (8.25)

in which α cr is the load factor at frame elastic sway buckling (Section 8.3.5.4).
Amoreaccuratesecond-orderanalysiscanbemadebyincludingtheP- effects
of sway displacement directly in the analysis. An approximate method of doing
this is to include fictitious horizontal forces equivalent to the P- effects in an
iterativeseriesoffirst-orderanalyses[30,31].Aseriesofanalysesmustbecarried
out because the fictitious horizontal forces increase with the deflections . If the
analysis series converges, then the frame is stable. An approximate method of
anticipating convergence is to examine the value of ( n + 1 − n )/( n − n − 1 )
computed from the values of at the steps ( n − 1), n , and ( n + 1). If this is
less than n /( n + 1 ) , then the analysis series probably converges. This method of
analysing frame buckling is slow and clumsy, but it does allow the use of the
widely available first-order elastic computer programs. The method ignores the
decreasesinthememberstiffnessescausedbyaxialcompressions(theP- δ effects),
and therefore tends to underestimate the second-order moments.

8.3.5.7 First-order plastic analysis

Inthefirst-ordermethodofplasticanalysis,allinstabilityeffectsareignored,and
thecollapsestrengthoftheframeisdeterminedbyusingtherigid-plasticassump-
tion (Section 5.5.2) and finding the plastic hinge locations which first convert the
frame to a collapse mechanism.
The methods used for the plastic analysis of frames are extensions of those
discussed in Section 5.5.5 which incorporate reductions in the plastic moment
capacity to account for the presence of axial force (Section 7.2.2). These meth-
ods are discussed in many texts, such as those [18-24: Chapter 5] referred to in