Mat Chapter 6

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Mat Chapter 6
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  INTRODUCTION TO COLUMN BUCKLING INTRODUCTION TO COLUMN BUCKLING 1.0INTRODUCTION AND BASIC CONCEPTS There are many types of compression members, the colmn bein! the best no#n$ Topchor%s of trsses, bracin! members an% compression flan!es of bilt p beams an% rolle% beams are all e&amples of compression elements$ Colmns are sally tho!ht of asstrai!ht 'ertical members #hose len!ths are consi%erably !reater than their cross(sectional %imensions$ )n initially strai!ht strt or colmn, compresse% by !ra%allyincreasin! e*al an% opposite a&ial forces at the en%s is consi%ere% first$ Colmns an%strts are terme% + long   or +  short   %epen%in! on their proneness to bc lin!$ If the strtis +short, the applie% forces #ill case a compressi'e strain, #hich reslts in theshortenin! of the strt in the %irection of the applie% forces$ Un%er incremental loa%in!,this shortenin! contines ntil the colmn -s*ashes-$ .o#e'er, if the strt is +lon!,similar a&ial shortenin! is obser'e% only at the initial sta!es of incremental loa%in!$Thereafter, as the applie% forces are increase% in ma!nit%e, the strt becomes + unstable an% %e'elops a %eformation in a %irection normal to the loa%in! a&is$ /0ee 1i!$23$ Thestrt is in a + buckled   state$    Buckling behaviour is thus characterized by deformations developed in a direction (or  plane) normal to that of the loading that produces it  $ 4hen the applie% loa%in! isincrease%, the bc lin! %eformation also increases$ Bc lin! occrs mainly in memberssb5ecte% to compressi'e forces$ If the member has hi!h ben%in! stiffness, its bc lin!resistance is hi!h$ )lso, #hen the member len!th is increase%, the bc lin! resistance is%ecrease%$ Ths the bc lin! resistance is hi!h #hen the member is +  stocky  /i$e$ themember has a hi!h ben%in! stiffness an% is short3 con'ersely, the bc lin! resistance islo# #hen the member is +  slender  $0trctral steel has hi!h yiel% stren!th an% ltimate stren!th compare% #ith other constrction materials$ .ence compression members ma%e of steel ten% to be slen%er$Bc lin! is of particlar interest #hile employin! steel members, #hich ten% to beslen%er, compare% #ith reinforce% concrete or prestresse% concrete compressionmembers$ Members fabricate% from steel platin! or sheetin! an% sb5ecte% tocompressi'e stresses also e&perience local bc lin! of the plate elements$ This chapter intro%ces bc lin! in the conte&t of a&ially compresse% strts an% i%entifies the factors!o'ernin! the bc lin! beha'ior$ The local bc lin! of thin flan!es6#ebs is notconsi%ere% at this sta!e$ These concepts are %e'elope% frther in a sbse*ent chapter$7 Copyri!ht reser'e% Version II6-1 6  INTRODUCTION TO COLUMN BUCKLING Version II6-2  A “short” column fails by compression yield  Buckled shape A “long” column failsby predominant buckling  Fig 1: “short” vs “long” columns δ   INTRODUCTION TO COLUMN BUCKLING 2.0ELASTIC BUCKLING O AN IDEAL COLUMN OR STRUT !IT PINNED END To be!in #ith, #e #ill consi%er the elastic beha'ior of an i%eali8e%, pin(en%e%, niformstrt$ The classical 9ler analysis of this problem ma es the follo#in! assmptions$ ã the material of #hich the strt is ma%e is homo!eneos an% linearly elastic /i$e$ itobeys .oo e:s La#3$ ã the strt is perfectly strai!ht an% there are no imperfections$ ã the loa%in! is applie% at the centroi% of the cross section at the en%s$4e #ill assme that the member is able to ben% abot one of the principal a&es$ /0ee 1i!$;3$ Initially, the strt #ill remain strai!ht for all 'ales of  P  , bt at a particlar 'ale  P = P  cr  , it bc les$ Let the bc lin! %eformation at a section %istant  x  from the en%  B  be  y $ The ben%in! moment at this section <  P  cr  .y The %ifferential e*ation !o'ernin! the small bc lin! %eformation is !i'en by The !eneral soltion for this %ifferential e*ation is  EI  P  x B EI  P  x A y  cr cr  sincos 22  += #here  A   an%  A !  are constants$0ince  y =  #hen  x = # A   = $ #hen  x   =   # y = $   Version II6-#  y P dx yd  EI  cr  $ ;; =−  x y B P  cr   P  cr  Fig ! olumn Buckling    INTRODUCTION TO COLUMN BUCKLING .ence 9ither  B   =  or  B   <  means  y =  for all 'ales of  x  /i$e$ the colmn remains strai!ht3$ )lternati'ely  EI  P  sin  cr  =  This e*ation is satisfie% only #hen !!!!!!!cr cr   EI n..... EI % # EI  P ........ #! # #  EI  P   π π π π π  == #here n  is any inte!er$ 4hile there are se'eral bc lin! mo%es correspon%in! to n   = # !# &#  =, the lo#est  stable  bc lin! mo%e correspon%s to n = .  /0ee 1i!$ >3$ The lo#est 'ale of the critical loa% /i$e$ the loa% casin! bc lin!3 is !i'en by Version II6-$  EI  P  sin B  cr   =  ? =  EI  P  cr   sin Fig # Buckling load $s %ateral deflection &elationship '% (nstable buckling modes ;; @   EI  π  ;; A   EI  π  ;;   EI  π        ;;   EI  P  π δ   All )alues abo)e are unstable
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