Beam-columns - The Behaviour and Design of Steel Structures to EC3 - page 307

Figure 7.11b, and it can be seen that it is good except for high values of β m when
it tends to be oversafe, and for high values of M when it tends to overestimate
the analytical solutions for the ultimate resistance (although this overestimate is
reducedifstrain-hardeningisaccountedforintheanalyticalsolutions).Theover-
conservatismforhighvaluesof β m iscausedbytheuseofthe0.4cut-offinequation
7.22 for C m , but more accurate solutions can be obtained by using equation 7.8
for which the minimum value of C m is 0.2.
Thus, the interaction equations (equations 7.17 and 7.21) provide a reasonably
simple method of estimating the ultimate resistances of I-section beam-columns
bent about the major axis which fail in the plane of the applied moments. The
interaction equations have been successfully applied to a wide range of sections,
including solid and hollow circular sections as well as I-sections bent about the
minor axis. In the latter case, M pl , y , N b , y , Rd , and N cr , y must be replaced by their
minor axis equivalents, while equation 7.17 should be replaced by equation 7.18.
It should be noted that the interaction equations provide a convenient way of
estimating the ultimate resistances of beam-columns. An alternative method has
beensuggested[4]usingsimpledesignchartstoshowthevariationofthemoment
ratio M / M pl , r withtheendmomentratio β m andthelengthratio L / L c ,inwhich L c
isthelengthofacolumnwhichjustfailsundertheaxialload N alone(theeffects
of strain-hardening must be included in the calculation of L c ) .
7.2.3.3 First yield of crooked beam-columns
In the second approximate approach to the resistance of a real beam-column, the
first yield of an initially crooked beam-column without residual stresses is used.
For this, the magnitude of the initial crookedness is increased to allow approx-
imately for the effects of residual stresses. One logical way of doing this is to
use the same crookedness as is used in the design of the corresponding compres-
sion member, since this has already been increased so as to allow for residual
stresses.
The first yield of a crooked beam-column is analysed in Section 7.5.2, and
particularsolutionsareshowninFigure7.7bforabeam-columnwith N cr , y / N y =
1 / 1.5andwhoseinitialcrookednessisdefinedby η = 0.290 ( L / i − 0.152 ) . Itcan
be seen that as M decreases to zero, the axial load at first yield increases to the
column resistance N b , y , Rd which is reduced below the elastic buckling load N cr , y
by the effects of imperfections. As N decreases, the first yield loads for crooked
beam-columns approach the corresponding loads for straight beam-columns.
For beam-columns in uniform bending ( β m =− 1), the interaction between M
and N isnon-linearandconcave,asitwasforstraightbeam-columns(Figure7.7a).
If this interaction is plotted using the maximum moment M max instead of the end
moment M ,thentherelationshipsbetween M max and N becomesslightlyconvex,
since the non-linear effects of the amplification of M are then included in M max .
Thus a linear relationship between M max / M y and N / N b , y , Rd will provide a sim-
ple and conservative approximation for first yield which is of good accuracy.