Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 320

whichconsiderableyieldingoccursbeforefailure.Sophisticatednumericalanaly-
seshavebeenmade[18,20-24]ofthebiaxialbendingofinelasticbeam-columns,
andgoodagreementwithtestresultshasbeenobtained.However,suchananalysis
requires a specialised computer program, and is only useful as a research tool.
A number of attempts have been made to develop approximate methods of
predictingtheresistancesofinelasticbeam-columns.Oneofthesimplestofthese
uses a linear extension
N
N b , Rd +

C my
( 1 − N / N cr , y )

M y
M b 0, y , Rd +

C mz
( 1 − N / N cr , z )

M z
M c 0, z , Rd ≤ 1

(7.62)

ofthelinearinteractionequationsforin-planebendingandflexural-torsionalbuck-
ling. In this extension, M b 0, y , Rd is the ultimate moment which the beam-column
cansupportwhen N = M z = 0forthecaseofequalendmoments( β m =− 1),while
M c 0, z , Rd is similarly defined. Thus equation 7.62 reduces to an equation similar
to equation 7.21 for in-plane behaviour when M y = 0, and to equation 7.57 for
flexural-torsional buckling when M z = 0. Equation 7.62 is used in conjunction
with

M y
M pl , r , y +

M z
M pl , r , z ≤ 1,

(7.63)

where M pl , r , y isgivenbyequation7.17and M pl , r , z byequation7.18.Equation7.63
representsalinearapproximationtothebiaxialbendingfullplasticitylimitforthe
cross-section of the beam-column, and reduces to equation 7.18 or 7.17 which
provides the uniaxial bending full plasticity limit when M y = 0or M z = 0.
A number of criticisms may be made of these extensions of the interaction
equations.Firstthereisnoallowanceinequation7.62fortheamplificationofthe
minoraxismoment M z bythemajoraxismoment M y (theterm ( 1 − N / N cr , z ) only
allowsfortheamplificationcausedbyaxialload N ).Asimplemethodofallowing
forthiswouldbetoreplacetheterm ( 1 − N / N cr , z ) byeither ( 1 − N / N cr , MN ) orby
( 1 − M y / M cr , MN ) wheretheflexural-torsionalbucklingload N cr , MN andmoment
M cr , MN are given by equation 7.45.
Second, studies [22-26] have shown that the linear additions of the moment
terms in equations 7.62 and 7.63 generally lead to predictions which are too con-
servative, as indicated in Figure 7.18. It has been proposed that for column type
hot-rolled I-sections, the section full plasticity limit of equation 7.63 should be
replaced by

α 0 α 0

M y
M pl , r , y

M z
M pl , r , z

+ ≤ 1, (7.64)

where M pl , r , y and M pl , r , z are given in equations 7.17 and 7.18 as before, and the
index α 0 by

N / N y
2ln ( N / N y ) .

α 0 = 1.60 − (7.65)