Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 326

process can be repeated until the values are found which satisfy equation 7.80.
Solutions obtained in this way are plotted in Figure 7.7b.

7.6 Appendix - flexural-torsional buckling of
elastic beam-columns

7.6.1 Simply supported beam-columns

Theelasticbeam-columnshowninFigure7.1bissimplysupportedandprevented
from twisting at its ends so that

( v ) 0, L = (φ) 0, L = 0, (7.82)

and is free to warp (see Section 10.8.3) so that

d 2 φ
d x 2

= 0. (7.83)

0, L

The combination at elastic buckling of the axial load N cr , MN and equal and
opposite end moments M cr , MN (i.e. β m =− 1) can be determined by finding
a deflected and twisted position which is one of equilibrium. The differential
equilibrium equations for such a position are

EI z d 2 v

d x 2 + N cr , MN v =− M cr , MN φ

(7.84)

for minor axis bending, and

d x − EI w d 3 φ

GJ − N cr , MN i p d φ

d x 3 = M cr , MN d v

(7.85)

d x

fortorsion,where i p = √ [ ( I y + I z )/ A ] isthepolarradiusofgyration.Equation7.84
reduces to equation 3.59 for compression member buckling when M cr , MN = 0,
and to equation 6.82 for beam buckling when N cr , MN = 0. Equation 7.85 can
be used to derive equation 3.73 for torsional buckling of a compression mem-
ber when M cr , MN = 0, and reduces to equation 6.82 for beam buckling when
N cr , MN = 0.
The buckled shape of the beam-column is given by

N cr , z − N cr , MN φ = δ sin π x

M cr , MN

v =

L ,

(7.34)

where N cr , z = π 2 EI z / L 2 is the minor axis elastic buckling load of an axially
loaded column, and δ the undetermined magnitude of the central deflection. The