LATERAL BUCKLING STABILITY OF TRUSSED BEAM WITH UNDER UNIFORMLY DISTRIBUTED LOADS

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1 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) TER BKING STBIITY OF TRSSED BE WITH NDER NIFORY DISTRIBTED ODS RI FTIH Fut o ii Enginring, nirsit o Sin n Thnoog Houri Bouin, BP, E i,bb Eour girs, gri ri_t@hoo.r DN REDONE Fut o ii Enginring, nirsit o Sin n Thnoog Houri Bouin, BP, E i,bb Eour girs, gri r_n@hoo.r BOKHF DI Fut o ii Enginring, nirsit o Sin n Thnoog Houri Bouin, BP, E i,bb Eour girs, gri.boukh@hoo.r bstrt: Th prsnt ppr instigts th tr stbiit o unrstrin tti girr. oring to th Europn st o, Euroo, th unrstrin b buking rsistn is untion o th thorti tr buking ont o th b. Th prssion o th tr buking ont is propos b th o or soi wb girr; howr, no inortion is gin or th s o truss b. tti girr, whih n proi rti ight struturs with high rigiit, r otn us s bs, n whn suh unrstrin nts bn bout thir strong inrti is, th hibit tr buking phnonon. Th ppr prsnts o tht ws op or th nsis o th tr buking stbiit o truss bs with onstnt ross stion. Bs on this o, n prssion o th tr buking ont is stbish in th s o unior istribut os. Kwors: Euroo ; tr buking; tti girr; thin w; truss b; wrping. ISSN : Vo. 5 No. Fbrur 77

2 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) Nontur w,, E G I, I I J r q n W u,, w u,, w,,, n ross stion r r o quint wb o truss b oiints o thorti tr buking ont Young s ouus o otion in rtion to shr ntr Shr ouus Prinip ont o inrti bout n is Wrping onstnt Torsion onstnt B ngth Thorti tr buking ont Bning ont bout is Distribut o in th irtion Strin nrg inr prt o th strin nrg Son orr prt o th strin nrg Work o th ppi os Dispnt oponnts o th shr ntr in th, n is Dispnt oponnts o point in th, n is Prinip o-orint o point in th gob G rrn Shr o-orint in th G rrn Wgnr s oiint i strin oponnt inr prt o th i strin Son orr prt o th i strin Tot potnti Torsion ng Torsion pitu o Stori or wrping o-orint. Introution Ents us in st struturs b onsir s n ssb o nubr o thin t ws. For p, in th s o n I-stion, h w is onnt to nothr w prpniur to it. Suh nts r "thin-w nts with opn stion" or n "thin-w nts with opn proi". Whn n unrstrin thin-w nt bns bout its strong inrti is, it hibit tr buking phnonon. s rsut, prt o th stion is in oprssion n th othr is in tnsion. rs in oprssion tn to buk tr in th irtion prpniur to th pn o bning, whrs rs in tnsion tn to oppos th tr ispnt, thror, tr buking is opni b rottion o th stions roun th ongituin is o th b. This bhiour o thin-w bs is or op sin th phnonon onsists o ouping btwn bning n torsion n so th torsion is opni with wrping. Vso s o [], op or non-unior torsion is oon opt or th ution o thin-w struturs. tr buking is prnt in nts whr th strutur rrngnts h bn opt to prnt th ont o th oprssion on out o pn bning. Whn prt b bns bout its strong inrti is, tr buking tks p or riti u o th iu bning ont «tr buking ont». This u is untion o th bning istribution, th n onitions, th o prtr n th gr o ono-str o th stion. In th Europn st o, Euroo [], n prssion o th tr buking ont is propos or th s soi wb girr; howr, no inortion is gin or th s o tti girr. Truss b r otn us s bs bning bout thir strong is o inrti n thror pron to th tr buking phnon whn th r unrstrin. In th prsnt work, truss b is ssiit to iti soi wbb b with r wb quint to truss brs. Bhiour o Suh soi wb girr is t b shr ortion. Thror, introuing shr ortion t in quiibriu qution, thorti o is op or th tr buking o truss bs. n prssion o th tr buking ont is propos or sip support b subjt to unior istribut os. ISSN : Vo. 5 No. Fbrur 78

3 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST). Equint Wb o Truss B Equint wb o truss b w is th r wb o soi wb girr whih hs th s shr ortion s th truss b. Th u o w is trint suh s trnsrs ortion o truss b pn o ngth is: V V ) G w V Z V Z Z h ; ; Y X B uting V oring to Fig. : Fig.. Trnsrs ortion o truss b pn. V V ) Eh sing Eq. ) n Eq. ), oowing prssion o quint wb o truss b w is obtin: w G G Eh ot ot E gθ gθ ) sin Θ sin Θ. Thorti tr buking nsis.. Strin nrg o thin-w nt with opn stion In th s o thin-w nt s shown in Fig., n ssuing n sti bhiour, th strin nrg o this nt inuing th torsion is gin b [], []: E GJ ') ) ) ISSN : Vo. 5 No. Fbrur 79

4 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) Fig. : Shti o thin-w nt in rtsin o-orint rrn [] In Eq. ) bo, is th i oponnt o strin tnsor whos u n b pproit b th prssion: u' ' ) w' )) 5) whr u, t w r th ritis with rspt to th rib o th thr oponnts o th ispnt o point on th ontour o th ross stion Fig.) in b its o-orints,, ) ; whr is th stori o-orint o th point introu in Vso s o u to non-unior torsion. B opting th ssuption tht th ontour o th ross stion is rigi in its pn, th thr ispnt oponnts o point n b ri ro thos o th shr ntr, in b its oorints, ) in G rtsin rrn. Th oowing rtionships r oon us: u u ' w' ' 6) ) 7) w w ) 8) whr u, n w r th ispnt oponnts o th shr ntr rspti in th, n irtions n is th twist ng ng o rottion) roun. onsiring tht th ispnt onsists o two prts, th inr prt n son orr prt n suh tht: 9) n whr, u' ) n ' ) w' )) ) sing Eq. 6), Eq. 7) n Eq. 8), th oowing rtionships b writtn: u' '' w'' " ) n ' w' R ') ) w' ' ) ' ' ) in Eq. ), th prssion o R is : R ) ) ) Substituting in Eq. ), th strin nrg bos: ' E n n ) GJ ) 5) ISSN : Vo. 5 No. Fbrur 8

5 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) In th s o inr stbiit, th ontribution o n is oitt, n th tot strin nrg n b writtn s th su o inr prt n son orr prt: whr, ' E GJ ) E GJ ' ) n thror: n Knowing tht: E n 7) E 8) n ; I ; I ; I n tht th i or N whih is qu to: n 6) ; 9) N E Eu' ) is ro in th s o th tr buking o bs, oowing prssion or th inr prt o th strin nrg is obtin: ' '' ") w") )) GJ ) ) In th s nnr, th son orr prt o strin nrg n b orut s untion o th initi os n s to: N) ) ) B ) ) n n n n n in whih, N nots th i or, n rprsnt th bning onts n B is th bi-ont introu in Vso s o. In th s o tr buking o bs o initi bout th strong inrti is is), initi os r ru to. Introuing t o shr ortion, th son orr prt o strin nrg is thn: V n n ) ' " ) Gw ) is Wgnr s oiint in s : ) ) I Th prssion o th strin nrg is thn ru to: w") ") GJ ') " ) q ) ' " ) 5) Whr: G w t q V.. Forution o th quiibriu qution onsir stright b with I ross stion, unr unior istribut os q ppi ong points P ot t th hight ro th shr ntr, s shown in Fig.. ISSN : Vo. 5 No. Fbrur 8

6 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) q q P w P w p Fig. : tr buking o b with I ross stion Th rti ispnt w p o points P, tking into ount th tr buking ortion n b prss s untion o th ispnt w o th shr ntr. It n b writtn s: w w os 6) p nr th ssuption o s ortions, th untions os is pproit b: os 7) Th prssion o th ispnt w p is thn: w w 8) p Th trn o work W, oring to th ispnt w p Eq. 8), is in b th rtionship: W qwp qw q 9) sing Eq. 5) n Eq. 9), th tot potnti nrg o th b in tr buking bhiour is: W w") ") GJ ') " ) q ) ' " ) q w q Whn th tot potnti nrg, whih is untion o th irtu ispnts n thir ritis, is iu, th oowing rtionships n b writtn: ' '' w w w w ' '' q ) " q ) q ) ' '' ) ) ) q ) " ) Z GJ '' q ) ')' ) ISSN : Vo. 5 No. Fbrur 8

7 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) Eq. ) rrs on to ispnt w n orrspons to th ssi quiibriu qution o th b bor buking. It hs no t on th tr buking ont. Howr, in th s o sip support b, th quiibriu qution Eq. ) n b trnsor to: ' ' q ) ) Z Th tr ispnt n b iint b soing Eq. ) or n substituting this rsut into Eq. ). Th gorning tr buking qution is thn: )' ' q ) ) GJ q ) ) " q q ) 5) For unior istribut o q, th prssion o th bning ont ) is: ) q q 6) whih ipis tht th o q is positi i ting ownwr... Dopnt o tr buking ont prssion sing Grkin s tho [5], th quiibriu qution Eq. 5) n b trnsor to sip or b utiping b n intgrting ong th b ngth. t irst, in optibiit with th bounr onitions o th b whih r ssu to b: t n 7) t n ng o twist ) is pproit b th untion : ) sin 9) Whn Grkin s tho is ppi to quiibriu qution Eq. 5), th oowing rtionship is obtin: q sin sin sin sin q q GJ) sin sin q ' )sin sin os sin sin sin To sipi th bo prssion, hng o ribs [6] is so tht th intgrtion oin [, ] is in: ; ; ) ition, th bning ont is nori to unit b iiing b th iu u o th bning ont: ) ) ) ) V ) ' 8) ) ) ISSN : Vo. 5 No. Fbrur 8

8 " ) q 5) B king ths hngs in qution ), th oowing is thn obtin: "sin "sin sin " ) sin ) sin os ' "sin sin sin sin GJ 6) t us tk: sin sin. b sin os '. ) " sin )sin " )sin " Intgrting th irst two trs in Eq. 6), th tr buking ont is gin b th soutions o qurti qution writtn in th or: GJ b 7) Th positi root o th bo qution is: ω b b r GJ β β 8) To obtin n prssion o r siir to tht opt in th Europn st o, Euroo [], th oiints, n r in s: 9) 5) ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) ISSN : Vo. 5 No. Fbrur 8

9 ri Ftih t. / Intrntion Journ o Enginring Sin n Thnoog IJEST) b 5) In this s, Eq. 8) bos: r I GJ ) ) I oiints, n r obtin tr intgrtion ong th b. Th r thn:,77 5),898 6,9 Gw Gw,866 5),898 6,9 G,898 6,9 G w,8 G w G G w w 55). onusion In this ppr, thorti o hs bn op or th ution o th tr buking ont r or sip support truss b unr unior istribut os. Eprssion o r siir to tht opt in th Europn st o, Euroo is propos. Shr ortion t ppr through th oiints, n. On n obsr tht, i shr ortion t is ngigib, th bo oiints r th s s Euroo s oiints in this o s =,; =,59; =,55). Rrns [] B.Z. Vsso Piès ongus n ois ins Eros 96 [] Euroo - u s struturs n ir t ount ppition ntion. Prti. : Règs générs t règs pour bâtint - Eros 997. [] S.P. Tioshnko, J.. Gr Thor o sti stbiit Grw-Hi 96. [] F. ohri,. Brouki, J.. Roth Thorti n nuri stbiit nss o unrstrin, ono-stri thin-w bs Journ o onstrution st rsrh. [5] H. Dj Théori u son orr stbiité éstiqu s brrs à prois ins t proi ourt t son ppition nns ITBTP. Suppént u n Sptbr 97. [6] S.. Wikrson Ipro oiints or sti tr-torsion buking 6 th I/SE/SE/HS/S Struturs, Strutur Dnis n tris onrn, ustin, Ts 8- pri 5. 5) ISSN : Vo. 5 No. Fbrur 85