Civil Engineering Reference
In-Depth Information
x
x
x
mz
M
z
N
N
N
m
M
L
m
M
L
my
M
y
Lateral
restraints
z
z
z
M
z
M
M
N
N
N
M
y
y
y
y
(a) In-plane behaviour.
Member deflects
w
in
zx
plane only
(b) Flexural-torsional
buckling. Member deflects
w
in
zx
plane, then buckles
by deflecting
v
in
yx
plane
and twisting
(c) Biaxial bending.
Member deflects
v
,
w
and twists
Figure 7.1
Beam-column behaviour.
7.2 In-plane behaviour of isolated beam-columns
When the deformations of an isolated beam-column are confined to the plane
of bending (see Figure 7.1a), its behaviour shows an interaction between beam
bendingandcompressionmemberbuckling,asindicatedinFigure7.2.Curve1of
thisfigureshowsthelinearbehaviourofanelasticbeam,whilecurve6showsthe
limitingbehaviourofarigid-plasticbeamatthefullplasticmoment
M
pl
.Curve2
shows the transition of a real elastic - plastic beam from curve 1 to curve 6.
The elastic buckling ofa concentrically loaded compression memberat its elastic
buckling load
N
cr
,
y
is shown by curve 4. Curve 3 shows the interaction between
bendingandbucklinginanelasticmember,andallowsfortheadditionalmoment
N
δ
exerted by the axial load. Curve 7 shows the interaction between the bending
moment and axial force which causes the member to become fully plastic. This
curveallowsforreductionfromthefullplasticmoment
M
pl
to
M
pl
,
r
causedbythe
axial load, and for the additional moment
N
δ
. The actual behaviour of a beam-
columnisshownbycurve5whichprovidesatransitionfromcurve3foranelastic
member to curve 7 for full plasticity.
7.2.1 Elastic beam-columns
A beam-column is shown in Figures 7.1a and 7.3a which is acted on by axial
forces
N
and end moments
M
and
β
m
M
, where
β
m
can have any value between
−
1 (single curvature bending) and
+
1 (double curvature bending). For isolated
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