Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 312

supportedandfreetowarpbutendtwistrotationsareprevented.Itisalsoassumed
that the cross-section of the beam-column has two axes of symmetry so that the
shear centre and centroid coincide.
When the applied load and moments reach the elastic buckling values N cr , MN ,
M cr , MN , a deflected and twisted equilibrium position is possible. It is shown in
Section 7.6.1 that this position is given by

N cr , z − N cr , MN φ = δ sin π x

M cr , MN

v = (7.34)

L

in which δ is the undetermined magnitude of the central deflection, and that the
elastic buckling combination N cr , MN , M cr , MN is given by

1 − N cr , MN
N cr , T

M cr , MN
i p N cr , z N cr , T =

1 − N cr , MN
N cr , z

(7.35)

in which i p = √ [ ( I y + I z )/ A ] is the polar radius of gyration, and

N cr , z = π 2 EI z / L 2 ,

(7.36)

1 + π 2 EI w
GI t L 2

N cr , T = GI t

(7.37)

i p

aretheminoraxisandtorsionalbucklingloadsofanelasticaxiallyloadedcolumn
(see Sections 3.2.1 and 3.7.5). It can be seen that for the limiting case when
M cr , MN = 0, the beam-column buckles as a compression member at the lower of
N cr , z and N cr , T ,andwhen N cr , MN = 0,itbucklesasabeamattheelasticbuckling
moment (see equation 6.3)

π 2 EI z
L 2

GI t + π 2 EI w = i p M cr = M zx = N cr , z N cr , T . (7.38)

L 2

Some more general solutions of equation 7.35 are shown in Figure 7.14.
The derivation of equation 7.35 given in Section 7.6.1 neglects the approx-
imate amplification by the axial load of the in-plane moments to M cr , MN /
( 1 − N cr , MN / N cr , y ) (seeSection7.2.1).Whenthisisaccountedfor, equation7.35
for the elastic buckling combination of N cr , MN and M cr , MN changes to

1 − N cr , MN
N cr , z

1 − N cr , MN
N cr , T

M cr , MN
i p N cr , z N cr , T =

1 − N cr , MN
N cr , y

(7.39)

The maximum possible value of N cr , MN is the lowest value of N cr , y , N cr , z , and
N cr , T .Formostsectionsthisismuchlessthan N cr , y andsoequation7.39isusually
very close to equation 7.35. For most hot-rolled sections, N cr , z is less than N cr , T ,