Frames - The Behaviour and Design of Steel Structures to EC3 - page 359

frame with negligible axial forces in the beams as

( N cr / L )

( N i / L )

α cr , s = (8.7)

in which N cr is the buckling load of a column in a storey obtained by using
equation 8.4, and the summations are made for each column in the storey. For a
storey in which all the columns are of the same length, this equation implies that
the approximate storey buckling load factor depends on the total stiffness of the
columnsandthetotalloadonthestorey,andisindependentofthedistributionsof
stiffnessandload.Theframebucklingloadfactor α cr maybeapproximatedbythe
lowestofthevaluesof α cr , s calculatedforallthestoriesoftheframe.Thismethod
gives close approximations when the buckling pattern is the same in each storey,
and is conservative when the horizontal members have zero axial forces and the
buckling pattern varies from storey to storey.
Aworked example of the application of this method is given in Section 8.5.3.
Alternatively, the storey buckling load factor α cr , s may be approximated by
using

α cr , s = H s
V s

h s
δ Hs

(8.8)

in which H s is the storey shear, h s is the storey height, V s is the vertical load
transmittedbythestorey,and δ Hs isthefirst-ordersheardisplacementofthestorey
caused by H s . This method is an adaptation of the sway buckling load solution
given in Section 3.10.6 for a column with an elastic brace (of stiffness H s /δ Hs ).
The approximate methods described above of analysing the elastic buckling
of unbraced frames are limited to rectangular frames. While the buckling of
non-rectangular unbraced frames may be analysed by using a suitable computer
program such as that described in [18], there are many published solutions for
specific frames [23].
Approximatesolutionsfortheelasticbucklingloadfactorsofsymmetricalportal
frames have been presented in [24-27]. The approximations of [27] for portal
frames with elastically restrained bases take the general form of

C − 1 / C

C N r

ρ r N cr , r

N c
ρ c N cr , c

α cr = + (8.9)

in which N c , N r are the column and rafter forces, N cr , c , N cr , r are the reference
buckling loads obtained from

N cr , c , r = π 2 EI c , r / L c , r (8.10)

and the values of C , ρ c , and ρ r depend on the base restraint stiffness and the
buckling mode.