Mat Chapter 8

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Mat Chapter 8
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    LOCAL BUCKLING AND SECTION CLASSIFICATION LOCAL BUCKLING AND SECTION CLASSIFICATION 1.0INTRODUCTION Sections normally used in steel structures are I-sections C!annels or an les etc# $!ic!are called o%en sections or rectan ular or circular tu&es $!ic! are called closed sections#T!ese sections can &e re arded as a com&ination o' indi(idual %late elements connectedto et!er to 'orm t!e re)uired s!a%e# T!e stren t! o' com%ression mem&ers made o' suc!sections de%ends on t!eir slenderness ratio# *i !er stren t!s can &e o&tained &y reducin t!e slenderness ratio i.e.  &y increasin t!e moment o' inertia o' t!e cross-section#Similarly t!e stren t!s o' &eams can &e increased &y increasin t!e moment o' inertia o' t!e cross-section# For a i(en cross-sectional area !i !er moment o' inertia can &eo&tained &y ma+in t!e sections t!in-$alled# As discussed earlier %late elements laterallysu%%orted alon ed es and su&,ected to mem&rane com%ression or s!ear may &uc+le %rematurely# T!ere'ore t!e &uc+lin o' t!e %late elements o' t!e cross section under com%ressions!ear may ta+e %lace &e'ore t!e o(erall column &uc+lin or o(erall &eam'ailure &y lateral &uc+lin or yieldin # T!is %!enomenon is called local buckling  # T!uslocal &uc+lin im%oses a limit to t!e e.tent to $!ic! sections can &e made t!in-$alled#Consider an I-section column su&,ected to uni'orm com%ression /Fi # 01 a 23# It $as %ointed out in t!e c!a%ter on 4Introduction to 5late Buc+lin 6 t!at %lates su%%orted ont!ree sides 1outstands2 !a(e a &uc+lin coe''icient k   rou !ly one-tent! t!at 'or %latessu%%orted on all 'our sides 1internal elements2# T!ere'ore in o%en sections suc! as I-sections t!e 'lan es $!ic! are outstands tend to &uc+le &e'ore t!e $e&s $!ic! aresu%%orted alon all ed es# Furt!er t!e entire len t! o' t!e 'lan es is li+ely to &uc+le int!e case o' t!e a.ially com%ressed mem&er under consideration in t!e 'orm o' $a(es# Ont!e ot!er !and in closed sections suc! as t!e !ollo$ rectan ular section &ot! 'lan es and$e&s &e!a(e as internal elements and t!e local &uc+lin o' t!e 'lan es and $e&s de%endson t!eir res%ecti(e $idt!-t!ic+ness ratios# In t!is case also local &uc+lin occurs alon t!e entire len t! o' t!e mem&er and t!e mem&er de(elo%s a 7c!e)uer &oard8 $a(e %attern/Fi # 01 b 23#In t!e case o' &eams t!e com%ression 'lan e &e!a(es as a %late element su&,ected touni'orm com%ression and de%endin on $!et!er it is an outstand or an internal elementunder oes local &uc+lin at t!e corres%ondin critical &uc+lin stress# *o$e(er t!e $e&is %artially under com%ression and %artially under tension# E(en t!e %art in com%ressionis not under uni'orm com%ression# T!ere'ore t!e $e& &uc+les as a %late su&,ected to in- %lane &endin com%ression#  Normally t!e &endin moment (aries o(er t!e len t! o' t!e &eam and so local &uc+lin may occur only in t!e re ion o' ma.imum &endin moment# Version II8-1   8    LOCAL BUCKLING AND SECTION CLASSIFICATION 9 Co%yri !t reser(edLocal &uc+lin !as t!e e''ect o' reducin t!e load carryin ca%acity o' columns and &eams due to t!e reduction in sti''ness and stren t! o' t!e locally &uc+led %late elements#T!ere'ore it is desira&le to a(oid local &uc+lin &e'ore yieldin o' t!e mem&er# :ost o' t!e !ot rolled steel sections !a(e enou ! $all t!ic+ness to eliminate local &uc+lin &e'oreyieldin # *o$e(er 'a&ricated sections and t!in-$alled cold-'ormed steel mem&ersusually e.%erience local &uc+lin o' %late elements &e'ore t!e yield stress is reac!ed#It is use'ul to classi'y sections &ased on t!eir tendency to &uc+le locally &e'ore o(erall'ailure o' t!e mem&er ta+es %lace# For t!ose cross-sections lia&le to &uc+le locallys%ecial %recautions need to &e ta+en in desi n# *o$e(er it s!ould &e remem&ered t!atlocal &uc+lin does not al$ays s%ell disaster# Local &uc+lin in(ol(es distortion o' t!ecross-section# T!ere is no s!i't in t!e %osition o' t!e cross-section as a $!ole as in lo&alor o(erall &uc+lin # In some cases local &uc+lin o' one o' t!e elements o' t!e cross-section may &e allo$ed since it does not ad(ersely a''ect t!e %er'ormance o' t!e mem&er as a $!ole# In t!e conte.t o' %late &uc+in  it $as %ointed out t!at su&stantial reser(estren t! e.ists in %lates &eyond t!e %oint o' elastic &uc+lin # Utili;ation o' t!is reser(eca%acity may also &e t!e o&,ecti(e o' desi n# T!ere'ore local &uc+lin may &e allo$ed insome cases %ro(ided due care is ta+en to estimate t!e reduction in t!e ca%acity o' t!esection due to it and t!e conse)uences are clearly understood# In $!at 'ollo$s 'irst t!e &asic conce%ts o' %lastic t!eory are introduced# T!en t!eclassi'ication o' cross-sections is descri&ed# T!e codal %ro(isions limitin t!e $idt!-t!ic+ness ratios o' %late elements in a cross-section are i(en# Finally t!e im%lications indesi n are discussed# Version II8-2 (b)(a) Fig. 1 Local buckling of Compression Members    LOCAL BUCKLING AND SECTION CLASSIFICATION 2.0BASIC CONCEPTS OF PLASTIC TEOR! Be'ore attem%tin t!e classi'ication o' sections t!e &asic conce%ts o' %lastic t!eory $ill &e introduced# :ore detailed descri%tions can &e 'ound in su&se)uent c!a%ters# Consider a &eam $it! &ot! ends 'i.ed and su&,ected to a uni'ormly distri&uted load o' w  %er meter len t! as s!o$n in Fi # <1 a 2# T!e elastic &endin moment at t!e ends is w  2  /12 and at mid-s%an is w  2  /24  $!ere   is t!e s%an# T!e stress distri&ution across any crosssection is linear /Fi # =1 a 23# As w  is increased radually t!e &endin moment at e(erysection increases and t!e stresses also increase# At a section close to t!e su%%ort $!eret!e &endin moment is ma.imum t!e stresses in t!e e.treme 'i&ers reac! t!e yield stress#T!e moment corres%ondin to t!is state is called t!e  first yield moment M   y  o' t!e crosssection# But t!is does not im%ly 'ailure as t!e &eam can continue to ta+e additional load#As t!e load continues to increase more and more 'i&ers reac! t!e yield stress and t!estress distri&ution is as s!o$n in Fi =1 b 2# E(entually t!e $!ole o' t!e cross sectionreac!es t!e yield stress and t!e corres%ondin stress distri&ution is as s!o$n in Fi # =1 c 2#T!e moment corres%ondin to t!is state is +no$n as t!e  plastic moment   o' t!e crosssection and is denoted &y  M   p # T!e ratio o' t!e %lastic moment to t!e yield moment is +no$n as t!e  shape factor   since itde%ends on t!e s!a%e o' t!e cross section# T!e cross section is not ca%a&le o' resistin any additional moment &ut may maintain t!is moment 'or some amount o' rotation in$!ic! case it acts li+e a  plastic hinge # I' t!is is so t!en 'or 'urt!er loadin  t!e &eam actsas i' it is sim%ly su%%orted $it! t$o additional moments  M   p  on eit!er side and continuesto carry additional loads until a t!ird %lastic !in e 'orms at mid-s%an $!en t!e &endin moment at t!at section reac!es  M   p # T!e &eam is t!en said to !a(e de(elo%ed a collapsemechanism  and $ill colla%se as s!o$n in Fi <1 b 2# I' t!e section is t!in-$alled due tolocal &uc+lin  it may not &e a&le to sustain t!e moment 'or additional rotations and may Version II8-   Bending Moment iagram !lastic hinges  M   p ollapse mechanism !lastic hinges M   p Fig. 2 Formation of a Collapse Mechanism in a Fixed Beam w  M   p    LOCAL BUCKLING AND SECTION CLASSIFICATION colla%se eit!er &e'ore or soon a'ter attainin t!e %lastic moment# It may &e noted t!at'ormation o' a sin le %lastic !in e i(es a colla%se mec!anism 'or a sim%ly su%%orted &eam# T!e ratio o' t!e ultimate rotation to t!e yield rotation is called t!e rotationcapacity  o' t!e section# T!e yield and t!e %lastic moments to et!er $it! t!e rotationca%acity o' t!e cross-section are used to classi'y t!e sections# .0SECTION CLASSIFICATION Sections are classi'ied de%endin on t!eir moment-rotation c!aracteristics 1Fi # >2# T!ecodes also s%eci'y t!e limitin $idt!-t!ic+ness ratios β   # b/t   'or com%onent %lates $!ic!ena&les t!e classi'ication to &e made# ã 5lastic cross-sections? 5lastic cross-sections are t!ose $!ic! can de(elo% t!eir 'ull- %lastic moment  M   p  and allo$ su''icient rotation at or a&o(e t!is moment so t!atredistri&ution o' &endin moments can ta+e %lace in t!e structure until com%lete'ailure mec!anism is 'ormed (b/t ≤    β  1  ) 1see Fi # @2# ã Com%act cross-sections? Com%act cross-sections are t!ose $!ic! can de(elo% t!eir 'ull-%lastic moment  M   p    &ut $!ere t!e local &uc+lin %re(ents t!e re)uired rotation att!is moment to ta+e %lace (  β  1 $ b/t $ β  2  ) # ã Semi-com%act cross-sections? Semi-com%act cross-sections are t!ose in $!ic! t!estress in t!e e.treme 'i&ers s!ould &e limited to yield stress &ecause local &uc+lin $ould %re(ent t!e de(elo%ment o' t!e 'ull-%lastic moment  M   p # Suc! sections cande(elo% only yield moment  M   y  (  β  2 $ b/t ≤    β  %  ). ã Slender cross-sections? Slender cross-sections are t!ose in $!ic! yield in t!e e.treme'i&ers cannot &e attained &ecause o' %remature local &uc+lin in t!e elastic ran e (  β  % $ b/t). Version II8-# (a) at M   y  (b) M   y  $ M$M   p (c) at M   p Fig. 3 Plastification of Crosssection under Bending 
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