Compression members - The Behaviour and Design of Steel Structures to EC3 - page 78

1000

N cr,y

N cr,z

750

75

6

500

N cr,T

N cr

250

100

y

z

0

0 1 2 3 4 5

Length L (m)

(a) Cross-section

(b) Elastic buckling loads

Figure 3.25 Elastic buckling of a simply supported angle section column.

coincidewiththeloadingaxisthroughthecentroid,andanytwistingwhichoccurs
causesthecentroidalaxistodeflect.Forsimplysupportedmembers,itisshownin
[5,24,25]thatthe(lowest)elasticbucklingload N cr isthelowestrootofthecubic
equation
N cr i 0 − y 0 − z 0 − N cr ( N cr , y + N cr , z + N cr , T ) i 0 − N cr , z y 0 − N cr , y z 0
+ N cr i 0 { N cr , y N cr , z + N cr , z N cr , T + N cr , T N cr , y }− N cr , y N cr , z N cr , T i 0 = 0.
(3.57)

Forexample,theelasticbucklingload N cr forapin-endedunequalangleisshown
in Figure 3.25, where it can be seen that N cr is less than any of N cr , y , N cr , z ,or
N cr , T .
These and other cases of flexural-torsional buckling (referred to as torsional-
flexural buckling in EC3) are treated in a number of textbooks and papers [4-6,
24-26],andtabulationsofsolutionsarealsoavailable[16,27].Oncethebuckling
load N cr hasbeendetermined,thecompressionresistance N b , Rd canbefoundfrom
Figure 3.23.

3.8 Appendix - elastic compression members

3.8.1 Buckling of straight members

Theelasticbucklingload N cr ofthecompressionmembershowninFigure3.3can
be determined by finding a deflected position which is one of equilibrium. The
differential equilibrium equation of bending of the member is

EI d 2 v

d x 2 =− N cr v .

(3.58)