Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 318

Onemethodofpredictingthesereducedresistancesistomodifytheapproximate
interaction equation for in-plane bending (equation 7.21) to

N
N b , z , Rd +

C my
( 1 − N / N cr , y )

M y
M b 0, y , Rd ≤ 1,

(7.57)

where N b , z , Rd is the minor axis strength of a concentrically loaded column
( M y = 0), and M b 0, y , Rd the resistance of a beam ( N = 0) with equal and oppo-
site end moments ( β m =− 1) which fails either by in-plane plasticity at M pl , y ,
or by flexural-torsional buckling. The basis for this modification is the remark-
able similarity between equation 7.57 and the simple linear approximation of
equation 7.40 for the elastic flexural-torsional buckling of beam-columns with
equal and opposite end moments ( β m =− 1).The application of equation 7.57 to
beam-columnswithunequalendmomentsissimplifiedbythesimilarityshownin

Figure 7.15 between the values of C m for in-plane bending and 1 / √ F and 1 /α m
for flexural-torsional buckling.
However, the value of this simple modification is lost if the in-plane strength
of a beam-column is to be predicted more accurately than by equation 7.21. If,
for example, equation 7.23 is to be used for the in-plane strength as suggested in
Section7.2.3.3,thenitwouldbemoreappropriatetomodifytheflexural-torsional
bucklingequations7.45and7.53toobtainout-of-planestrengthpredictions.Ithas
been suggested [15] that the modified equations should take the form

2

1 − 1 −

M
α bcu M b 0, y , Rd

N
N b , z , Rd

N
N cr , T

= (7.58)

in which

1 − β m
2

1 + β m
2

3

1
α bcu =

N
N b , z , Rd

+ 0.40 − 0.23 . (7.59)

7.3.4 Design rules

The design of beam-columns against out-of-plane buckling will generally be
governed by

N b , z , Rd + k zy M y , Ed

N Ed

M b , Rd ≤ 1

(7.60)

in which k zy is an interaction factor that may be determined from Annex A or
Annex B of EC3. Equation 7.60 is similar to equation 7.32 for in-plane major
axisbucklingresistance, exceptfortheallowanceforlateraltorsionalbucklingin
the second term and the use of the minor axis column buckling resistance in the
first term.