Mat Chapter 9

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Mat Chapter 9
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  LATERALLY RESTRAINED BEAMS LATERALLY RESTRAINED BEAMS 1.0INTRODUCTION Beams are structural members frequently used to carry loads that are transverse to ther lon!tudnal a s# They transfer loads $rmarly by bendn! and shear# In a rectan!ular  buldn! frame% beams that s$an bet&een ad'acent columns are called ( main or primarybeams/girders’  # Beams% &hch are used to transmt the floor loadn! to the man beams bet&een columns% are called ( secondary beams/joists’  # As far as the structural steelframn! n buldn!s s concerned% t s suffcent to consder only the bendn! effects for  beams% as torson s not !enerally $redomnant# )or a beam *loaded $redomnantly byfle ure+ t&o essental requrements must be met to develo$ ts full moment ca$acty,-# The elements of the beam *#e# flan!e and &eb+ should not buc.le locally and/# The beam as a &hole should not buc.le laterally#To ensure that the frst condton s met% the cross sectons of the flan!e and the &ebchosen must be 0  plastic 1 or 0 compact  1# *These defntons are e $laned n the cha$ter on(Local buc.ln!2 and also n later $art of ths cha$ter+# If the beam s requred to haves!nfcant ductlty% $lastc sectons must nvarably be used# To avod the lateral buc.ln! referred to under the second condton% restrants are $rovded to the beam n the $lane of the com$resson flan!e% hence such beams are called 0 laterally restrained beams 1# In many steel structures% es$ecally n buldn!s% beams carry floor dec.s on to$of them% and these floor dec.s $rovde restrant to the com$resson flan!e# In the absenceof any such restrants% and n case the lateral buc.ln! of beams s not accounted for ndes!n% the des!ner has to $rovde adequate lateral su$$orts to the com$resson flan!e#In ths cha$ter &e are concerned &th laterally restraned beams% n other &ords beams&hch have adequate lateral su$$ort to the com$resson flan!e# Beams% &hch buc.lelaterally% are covered n the ne t cha$ter# 2.0 BEHAVIOUR OF STEEL BEAMS Laterally stable steel beams can fal only by * a + fle ure * b + shear or * c + bearn!% assumn!that local buc.ln! of slender com$onents does not occur# These three condtons are thecrtera for Lmt State of colla$se for steel beams# Steel beams &ould also becomeunservceable due to e cessve deflecton and ths s classfed as a lmt state of servceablty# In the follo&n! sectons% &e reve& the fundamentals of these lmt states# 2.1 Flexural beha! ur # $%eel bea&$ It s m$ortant to reco!nse that only $lastc sectons can be used n “plastic design of  frames”  % &here moment redistribution is required throughout the frame # “Plasticanalysis”   of the cross secton s confned to the assessment of the behaviour of the crosssection at the instant of collapse # These t&o terms are not to be confused for each other#3 4o$yr!ht reserved Ver$! ' II( ) 1 I .e xeI .   e xe   (  LATERALLY RESTRAINED BEAMS If a fle ural member s $ro!ressvely loaded% t deflects and the curvature of such bendn!vares alon! ts len!th# Intally the beam s elastc throu!hout ts len!th# Let us consder a small $orton of the beam at a $ont  A  as sho&n n )!#- * a + &here the radus of curvature s # If &e consder a small se!ment of the beam at  A  5)!#- * b +6% then thevaraton of the stran across the de$th of the member could be found out !eometrcally as   z    *-+)rom Eq#-% the stran at any fbre s $ro$ortonal to ts dstance (  z  2 from the neutral a s#Ths s obtaned from the assum$ton that $lane sectons &hch are normal to thelon!tudnal a s before bendn!% remans $lane and normal even after bendn!# )or each Ver$! ' II( ) 2 I .e xe Fig.2 dealised elasto! plastic stress! strain curve for steel         S      t     r     e     s     s  strain 1 ε  2  y2  εε  = # 3 ε   $ 4 ε    f   y  Plastic range Elastic range Idealised stress strain curve ε   f  %   Radius of curvature=1  φ  Fig. &urvature of bending  !a ε  'a('b( A  )  %  ε   hc  #eflected s$a%e  *+ φ  d, d,   LATERALLY RESTRAINED BEAMS stran (   2 one can read off the corres$ondn! stress (  f  2 from the dealsed stress7strancurve for steel sho&n n )!# /# *The dealsed stress stran curve ne!lects the stran7hardenn! $orton for all $ractcal $ur$oses+# 8e choose four $onts -% /% 9% : on thestress7stran curve *)!# /+ for further dscusson and see ho& these four $onts are used&hen a sm$ly su$$orted beam s sub'ected to a md $ont load# 2.2 Ela$%!* #lexural beha! ur 4onsder the $ont *-+ n )!#/ n &hch the stran 1!a  ε ε   =  &hch s less than the yeldstran y  # At ths sta!e% as seen from )!ures / and 9% the stress s drectly $ro$ortonalto stran# ;ence from elementary Stren!th of Materals% the corres$ondn! moment of resstance *  &  c   + s !ven by c I   f   &   1 c  = */+&here (  f  1 2 s the e treme fbre stress% (  I  2 s the moment of nerta and ( c 2 s the e tremefbre dstance from the neutral a s# The term  ' = Ic s the elastc secton modulus&hch s a !eometrc $ro$erty of the secton# ;ence Eq#/ can be re&rtten n terms of elastc secton modulus as  '   f   &  1 c  = *9+ 2.+ Y!el, a', -la$%!* & &e'% *a-a*!%!e$  No& let us consder the $ont */+ n )!# /# The e treme fbre stran equals yeld stran#e#  y2!a  ε ε ε   ==  and also the stress  f  2  = f   y (  &here%  f   y s the yeld stress# <$ to ths sta!e%as sho&n n )!# 9% the stress and stran are $ro$ortonal to each other snce the e tremefbre of the beam s stressed &thn the elastc ran!e# The corres$ondn! moment%  )&   y  * % s 'ust suffcent to cause yeld n the e treme fbres and s !ven by Ver$! ' II( ) + I .e xe ε  1 + ε   y ε  2 = ε   y  f  2 =f   y  f  1 +f   y Fig.# -train and stress distributions in the elastic range2   LATERALLY RESTRAINED BEAMS  '   f   &   y y  = *:+&here  &   y s called the “yield moment”  % #e# the moment &hch 'ust causes the e tremefbres to yeld# It s evdent from )!# =* b + that once the e treme fbre stresses attan yeldstress they no lon!er ta.e any addtonal stresses#8hen the load and hence the moment s further ncreased% the outermost fbre stran ε ma near md s$an of the beam *#e# $ont of ma mum bendn! moment+ &ould attan a valuesay%  y 3   and ths s dentfed as $ont *9+ n )!# /# At ths sta!e the stran s n the $lastc sta!e% but e treme fbre stress stll equals yeld stress  f   y # 8e also note that thestresses have been redstrbuted to the nner fbres to&ards the neutral a s and thesefbres !radually attan a stress equal to  f   y  .  Ths s sho&n n )!#:# The remann! $ortonof the beam n the vcnty of the neutral a s s stll elastc# At ths sta!e the momentca$acty s calculated by consdern! both the $lastc $orton and the elastc core as% Ver$! ' II( )  I .e xe ε  3 > ε   y  f  3 =f   y  f  3 <f   y  Fig. $ -train and stress distributions in plastic range ε  4 >> ε   y  f  4 =f   y      + c  + t   z  c  z  t  B # f   y  , d  t  d  - d  -  z  c  z  t   -tress -train -train-tress
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