Frames - The Behaviour and Design of Steel Structures to EC3 - page 357

effective length factor so determined is used with equations 8.1 and 8.2 to obtain
an estimate of the frame buckling load factor as

α cr , i = π 2 EI i /( k cr , i L i ) 2
N i

. (8.6)

The lowest of these provides a conservative approximation of the actual frame
buckling load factor α cr .
The accuracy of this method of calculating effective lengths using the stiffness
approximationsofFigure3.19isindicatedinFigure8.6cfortherectangularframe
shown in Figure 8.6a and b. For the buckling mode shown in Figure 8.6a, the
buckling of the vertical members is restrained by the horizontal members. These
horizontal members are braced restraining members which bend in symmetrical
single curvature, so that their stiffnesses are ( 2 EI 1 / L 1 )( 1 − N 1 / N cr 1 ) in which
N cr1 = π 2 EI 1 / L 1 ,asindicatedinFigure3.19.ItcanbeseenfromFigure8.6cthat
theapproximatebucklingloadsareinverycloseagreementwiththeaccurateval-
ues.WorkedexamplesoftheapplicationofthismethodaregiveninSections8.5.1
and 8.5.2.
Amore accurate iterative procedure [22] may also be used, in which the accu-
racies of the approximations for the member end stiffness factors k increase with
each iteration.

8.3.5.4 Elastic buckling of unbraced frames

Thedeterminationoftheframebucklingloadfactor α cr ofarigid-jointedunbraced
frame may also be carried out using a suitable computer program such as that
described in [16, 17].

I/L 2

2
I/L 1

---- ---- ---- ----------

=1.0

2.0

1

Mode 1

Exact

1.5

N 2 N 2 N 2 N 2

N 1 N 1

I , L 1 I,L 1

1 1

1.0

Mode 2

I , L 2 I,L 2

2 2

0.5

Approximate

N 1 N 1

0

0 0.5 1.0 1.5 2.0

I , L 1 I,L 1

1
(a) Buckling mode1

1
(b) Buckling mode 2

N/N

1 cr 1

(c) Buckling loads

Figure 8.6 Buckling of a braced frame.