Civil Engineering Reference
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which is generally close to the accurate solution, except when
N
y
,
r
≈
0.15
N
y
,
whenthemaximumerrorisoftheorderof5%.Thissmallerrorisquiteacceptable
since the accurate solutions ignore the strengthening effects of strain-hardening.
A similar analysis may be made of an I-section beam-column bent about its
minor axis. In this case, a satisfactory approximation for the load and moment at
full plasticity is given by
2
N
y
,
r
N
y
M
pl
,
r
,
z
M
pl
,
z
=
1.19
1
−
≤
1.0.
(7.18)
Beam-columns of more general cross-section may also be analysed to deter-
mine the axial load and moment at full plasticity. In general, these may be safely
approximated by the linear interaction equation
M
pl
,
r
M
pl
=
1
−
N
y
,
r
N
y
.
(7.19)
In a longer beam-column, instability effects become important, and failure
occursbeforeanysectionbecomesfullyplastic.Afurthercomplicationarisesfrom
the fact that as the beam-column deflects by
δ
, its maximum moment increases
fromthenominalvalue
M
to(
M
+
N
δ
).Thus,thevalueof
M
atwhichfullplasticity
occurs is given by
M
=
M
pl
,
r
−
N
δ
,
(7.20)
and so the value of
M
for full plasticity decreases from
M
pl
,
r
as the deflection
δ
increases,asshownbycurve7inFigure7.2.Thiscurverepresentsanupperbound
to the behaviour of beam-columns which is only approached after the maximum
strength is reached.
7.2.3 Ultimate resistance
7.2.3.1 General
Anisolatedbeam-columnreachesitsultimateresistanceataloadwhichisgreater
than that which causes first yield (see Section 7.2.1), but is less than that which
causes a cross-section to become fully plastic (see Section 7.2.2), as indicated
in Figure 7.2. These two bounds are often far apart, and when a more accurate
estimate of the resistance is required, an elastic-plastic analysis of the imperfect
beam-column must be made.Two different approximate analytical approaches to
beam-columnstrengthmaybeused,andthesearerelatedtotheinitialcrookedness
and residual stress methods discussed in Sections 3.2.2 and 3.3.4 of allowing for
theeffectsofimperfectionsontheresistancesofrealcompressionmembers.These
two approaches are discussed below.
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