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CE434,Spring2010 AnalysisofCompressionMembers 1/7Thesenotespresenttheproceduresforanalyzingmemberssubjecttopurecompression

(axialcompressionthroughthecentroidalaxis,accordingtotheSpecifications,ChapterE

pg16.13243). Memberssubjecttocombinedcompressionandbendingwillbecovered

later(ChapterHintheSpecifications). Thesenoteswillfocusspecificallyontwopopular

typesofcompressiononlymembers: wideflangeandtubular(akaHollowStructural

Shapesor

HSS)

columns

of

braced

frames.

RelevantsectionsoftheSpecificationsinclude:

Section/Chapter TitleB3.12,16.112 DesignWallThicknessforHSS

B4.12,16.114>18 ClassificationofSectionsforLocalBuckling

ChapterC,16.119>31 StabilityAnalysis

ChapterE,16.132>43 DesignofMembersforCompression

CompressionMember

Failure

Modes

Therearetwoprincipalfailuremodesforcompressionmembers: yieldingandbuckling.

Bucklingrepresentsfailureduetoinstability,andstabilityisoneofthemorecomplicated

topicsinstructuralengineering. Longslendercolumnswillbuckleelastically,alsocalled

Eulerbuckling. Veryshortcolumnsorpedestalswillfailbyduetoyieldingoftheentire

crosssection. Columnsofintermediateslendernesswillfailduetoinelasticbucklingin

whichsomeofthecrosssectionhasyielded.

Figure1. Compressionfailuremodes.Crosssectionelementsofmemberscanalsobuckle. Anelementisconsideredslenderifit

cannotsupportthefullyieldstresswithoutbuckling. Crosssectionelementssubjectto

localbucklingaregenerallyplateshapeelements. Iftheelementissupportedalongboth

ElasticBuckling

(LongColumn)Inelastic

Buckling

(Interm. Column)

Yielding

(ShortColumn)

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CE434,Spring2010 AnalysisofCompressionMembers 2/7edges,itisconsideredstiffened. Forexample,intheWshapeinFigure2,theflangesare

consideredunstiffenedandthewebisconsideredstiffened. AllflangesofWshapesforA36

andGrade50steelarenonslender. AfewwebsforWshapesofGrade50steelare

consideredslender. ThewallsofmanyHSSshapesareconsideredslender.

Figure2. Examplesofstiffenedandunstiffenedcrosssectionelementssubjecttolocalbuckling.

ElasticBucklingAnaxialload(P)onacolumnthatisslightlydisplacedlaterallyadistancex

willcauseabendingmomentinthecolumn(M)

M=Px

Thedifferentialequationrelatingdisplacementtransversedisplacement(x)

tobendingmomentis:

02

2

2

2

=+

==

Px

dy

xd

PxMandEI

M

dy

xd

Solutionofthisequationyields:

+

= y

EI

PBy

EI

PAx cossin

Applyingboundaryconditions:

Outeredgeofhalfflange

isunsupported

Bothedgesofweb

aresupported

x

P

y

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CE434,Spring2010 AnalysisofCompressionMembers 3/7

===

===

LEI

PALyxii

Byxi

sin0,0)()(

0,0)0()(

Thenontrivialsolutionoftheequationaboveis

2

22

L

EInP

or

nLEI

P

=

=

whichisthefamiliarEulerbucklingequationwhenn=1.

InsteeldesignitisconvenienttoexpresstheEulerbucklingload,alsocalledtheelastic

bucklingload

(Pe),

in

terms

of

astress

2

2

2

2

22

2

2

,

=

=

=

r

L

EF

A

Irwhere

L

rEF

L

A

IE

A

P

e

e

e

Ferepresentstheaveragecompressivestressatwhichamemberwithpinnedendswill

buckleelastically. Theelasticbucklingstressformemberswithotherendconditionscanbe

calculatedbysubstitutingKLforLintheequationabove,whereKLiscalledthe

effectivelengthandKistheeffectivelengthfactor.

2

2

=

r

LK

EFe

Approximatevalues

of

Kfor

different

support

conditions

are

shown

in

Table

C

C2.2

on

pg.

16.1240oftheCommentary. Foreachsupportcondition,thetheoreticalKvalueinthe

tablerepresentsthemultipleofcolumnlengthnecessarytoachieveabuckledshapesimilar

tothepinnedpinnedconditionsassumedforthederivationofPeandFe. Thesupport

conditionsatthecolumnbaseareeitherpinnedorfixed. Fourpossiblesupportconditions

arepossibleatthecolumntop:

pinned(rotationfreeandtranslationfixed)

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CE434,Spring2010 AnalysisofCompressionMembers 5/7unbracedframes),pg.161240>241,basedontherelativerotationalstiffness(G)ofthe

columnstothegirdersconnectedtothejointinquestion.

=

g

gg

c

cc

LIE

L

IE

G

Themaximumeffectivelengthfactor(K)forabracedframeis1.0. Effectivelengthfactors

forcolumnsofbracedframescanconservativelybeestimatedas1.0. LowervaluesofKfor

bracedframescanbecalculatedforcolumnendswithrotationalrestraint. The

Commentary(pp.16.1241>242)listsadjustmentfactorsforpracticalconditionssuchas

connectionstofootings,inelasticbehaviorofcolumns,andgirderswithsignificantaxial

load.

InelasticBucklingInelastic

buckling

occurs

when

part

of

the

cross

section

yields,

resulting

in

adecrease

of

stiffnessforthecolumn. Residualcompressivestressesarecreatedinsteelmembersduring

themanufacturingprocess. Thesecompressivestressesaddtotheloadinduced

compressivestressesandcauseelementsofthecrosssectiontoyieldundersmallerloads.

Residualstressesariseasthemembercoolsafterbeingformed. Thecontractionof

membersthatcoollast,forexampletheintersectionoftheflangesandwebinanIbeam,is

resistedbyregionsthatcooledfirst,forexampletheendsoftheflangesandthemiddleof

thewebonanIbeam. Theendresultisresidualtensilestressesexistintheregionsthat

cooledlast,andresidualcompressivestressesexistintheregionsthatcooledfirst(see

Figure5below).

Figure5. Residualcompressivestresses()andtensilestresses(+)Residualcompressivestressesashighas20ksihavebeenmeasuredintheendsofthe

flangesofrolledshapes.ForA36steel,thisrepresentsaratioof56%

56.036

20=

= ksiy

ksi

F

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CE434,Spring2010 AnalysisofCompressionMembers 6/7Forthisinstance,anadditionalcompressivestress(duetoloading)equalto0.44Fywould

causetheflangetipstoyield. Iftheelasticbucklingstress(Fe)forthiscolumnwaslessthan

0.44Fyitwouldbuckleelastically,andiftheelasticbucklingstresswasgreaterthanorequal

to0.44Fythecolumnwouldbuckleinelastically.

ChapterEof

the

Specifications

(pp.

16.1

32

>43)

provide

two

equations

to

calculate

the

flexuralbucklingstress(Fcr),dependingonthevalueoftheelasticbucklingstress(Fe)relativetotheyieldstress(Fy).

WhenFe>=0.44Fy

y

F

F

cr FFe

y

= 658.0 (inelasticbuckling)

WhenFe

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CE434,Spring2010 AnalysisofCompressionMembers 7/7LocalBuckling(SlenderElements)Plateshapedelementsofcompressionmemberssuchasflangesandwebscanbuckle

locallybeforethememberitselfbuckles. Theequationsforflexuralbucklingstress(Fcr)

aremodifiedinSectionE7oftheSpecifications(pp.16.139 >43)byreplacingFywith

QFy,whereQisthereductionfactorforacompressionmemberwithslenderelements.

Anelementisclassifiedasslenderbasedonthewidthtothicknessratiooftheelement. An

elementwithawidthtothicknessratiogreaterthanrinTableB4.1(pg.16.116)isconsideredslender.

Adistinctionismadebetweenstiffenedandunstiffenedelements. Stiffenedelementsare

supportedalongtwoedgesparalleltothedirectionofthecompressionforce,while

unstiffenedelementsaresupportedalongonlyoneedge. Thereductionfactorfor

unstiffenedelements(Qs)isprovidedinSectionE7.1(pg.16.140 >42),andthereduction

factorforstiffenedelements(Qa)isprovidedinSectionE7.2(pg.16.142 >43). The

reduction

factor

(Q)

for

the

compression

member

is

the

product

of

Qs

and

Qa

Q=QsQa

Ifacompressionmemberhasnoslenderunstiffenedelements,Qs=1;andifacompression

memberhasnoslenderstiffenedelements,Qa=1.