In-plane bending of beams - The Behaviour and Design of Steel Structures to EC3 - page 175

Thisapproachissimilartothatusedtoallowfortheredistributionofstresswhich
takesplaceinathincompressionflangeafterlocalbuckling(seeChapter4).How-
ever, the two effects of shear lag and local buckling are quite distinct, and should
not be confused.
5.5 Plastic analysis of beams
5.5.1 General
As the load on a ductile steel beam is increased, the stresses in the beam also
increase,untiltheyieldstressisreached.Withfurtherincreasesintheload,yielding
spreadsthroughthemosthighlystrainedcross-sectionofthebeamuntilitbecomes
fullyplasticatamoment M p .Atthisstagethesectionformsaplastichingewhich
allows the beam segments on either side to rotate freely under the moment M p .
If the beam was originally statically determinate, this plastic hinge reduces it to a
mechanism, and prevents it from supporting any additional load.
However, if the beam was statically indeterminate, the plastic hinge does not
reduce it to a mechanism, and it can support additional load.This additional load
causes a redistribution of the bending moment, during which the moment at the
plastic hinge remains fixed at M p , while the moment at another highly strained
cross-sectionincreasesuntilitformsaplastichinge.Thisprocessisrepeateduntil
enoughplastichingeshaveformedtoreducethebeamtoamechanism.Thebeam
is then unable to support any further increase in load, and its ultimate strength is
reached.
In the plastic analysis of beams, this mechanism condition is investigated to
determinetheultimatestrength.Theprinciplesandmethodsofplasticanalysisare
fully described in many textbooks [18-24], and so only a brief summary is given
in the following sub-sections.
5.5.2 The plastic hinge
Thebendingstressesinanelasticbeamaredistributedlinearlyacrossanysectionof
thebeam,asshowninFigure5.4,andthebendingmoment M isproportionaltothe
curvature − d 2 w / d x 2 (see equation 5.2). However, once the yield strain ε y = f y / E
(see Figure 5.24) of a steel beam is exceeded, the stress distribution is no longer
linear, as indicated in Figure 5.2c. Nevertheless, the strain distribution remains
linear, and so the inelastic bending stress distributions are similar to the basic
stress-strain relationship shown in Figure 5.24, provided the influence of shear
on yielding can be ignored (which is a reasonable assumption for many I-section
beams).Themomentresultant M ofthebendingstressesisnolongerproportional
to the curvature − d 2 w / d x 2 , but varies as shown in Figure 5.2d. Thus the section
becomes elastic-plastic when the yield moment M y = f y W el is exceeded, and the
curvature increases rapidly as yielding progresses through the section. At high
curvatures,thelimitingsituationisapproachedforwhichthesectioniscompletely