ANALYSIS OF RING BEAMS FOR DISCRETELY SUPPORTED CYLINDRICAL SILOS

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1 ANALYSIS OF RING BEAS FOR DISCRETELY SUPPORTED CYLINDRICAL SILOS Öz Zybk Rsach Assistat Dpatmt of Ciil Eii, il East Tchical Uisity, Akaa, Tuky Cm Topkaya Pofsso Dpatmt of Ciil Eii, il East Tchical Uisity, Akaa, Tuky J. ichal Rott Pofsso Istitut fo Ifastuctu a Eiomt, Uisity of Eibuh, Scotla, UK m.ott@.ac.uk. ABSTRACT I cyliical silos, th psc of isct suppots sults i a o-uifomity of miioal stsss aou th cicumfc. I la silos, a sizabl i bam is omally us, sti o isct colum suppots bath th cyliical shll to istibut th majoity of th isct focs fom th colum suppots ito a mo uifom stss stat i th cyliical wall. Althouh th is a compl itactio btw cylis a suppoti i bam, EN pois th bi a tosioal momt quatios pouc i th isolat i bam u uifom tass loai by assumi that th miioal stss istibutio i th shll is cicumftially uifom. I this stuy, ths quatios i EN99-4- fo th si of th i bam a -i usi Vlaso s cu bam thoy. Th, th sha foc quatio i th tass ictio fo th i bam is i usi th quilibium quatios. oo, th tass isplacmt of th i bam is obtai fom th lobal focfomatio latioships. Ths clos-fom quatios a ifi by complmtay fiit lmt aalyss fo pactical si applicatios.. INTRODUCTION Th cyliical mtal shll is th commost fom fo mtal silos a taks. Ths stuctus ca b suppot ith o th ou o o a fw colum suppots, pi o th quimts of th ischa systm. Colums at qual cicumftial spaci a ally mploy to lat th silo stuctu Fi.. Th psc of th isct suppots sults i a o-uifomity of miioal stsss aou th cicumfc. I

2 liht silos, th isct local colums a commoly attach to th si of th wall usi ith a umb of backts o a colums []. I la silos, a sizabl i bam is omally us, sti o isct colum suppots bath th cyliical shll. Th fuctio of th i bam is twofol. Fist, it is qui to cay cicumftial focs to maitai suppot fo th aial compot of th miioal tsio i th hopp []. Sco, th i bam plays a impotat ol i istibuti majoity of th isct local focs fom th colum suppots ito a mo uifom stss stat i th cyliical wall []. Coical oof Cyliical shll Ri bam Hopp Colum Fi. : Typical cicula plafom silo Ealy stuis of isctly suppot cylis [-5] mostat th at complity of this bhaiou, a attmpts to al with th itactio of cylis a suppoti i bams ha ot l to simpl si pocus [6, 7]. Althouh th is a compl itactio btw th cyliical shll a th i bam, si is typically atly simplifi by assumi that th miioal stss istibutio i th shll is cicumftially uifom. As illustat i Fi. a, th taitio is fo ach compot to b tat spaatly u th actio of uifom loai aou th cicumfc [, 8- ]. Fo this loai to b ali, th i bam must fuctio poply i istibuti th isct suppot loas ito a mo uifom stat of stss. Th tt to which this istibutio of th suppot focs ca b achi is ictly lat to th stiffss of th i bam lati to th stiffss of th cyliical shll. I oth wos, fomatios of th cyliical shll a th i bam must b compatibl as show i Fi. b fo th aliity of th ulyi assumptio. Cosii that th cyliical shll is ath stiff i its ow pla, th i bam shoul b y stiff, a much stiff tha th shll, to b abl to satisfactoily istibut th suppot loas. A appoimat citio to tmi th appopiat i bam stiffss was fist itifi by Rott [8] a was futh lop a ifi by Topkaya a Rott []. Th citio lop by ths authos is y mai a usually las to y bi i bams fo typical omtis. EN [9] oly pois si quatios fo stss sultats pouc i th isolat i bam u uifom tass loai. I this pap, ths stss sultats pouc i th i bam w -i usi Vlaso s cu bam ifftial quatios [, ]. Th, th sha foc quatio i th tass ictio fo th i bam is i usi th quilibium quatios. Th aata of usi Vlaso s quatios is that th tass isplacmts ca also b obtai fom th ifftial latioships. Th afomtio si quatios fo cass wh th i bam

3 itacts with th silo shll a ifi by complmtay fiit lmt aalyss fo pactical si applicatios. Cyliical shll Aisymmtic wall loai a bottom pssus Ri i aious s-sctio omtis Uifom suppot to cyli fom i i Uifom loai of i i by cyli Disct local suppots Disct suppot Shll wall Disct suppot I-pla tical flctios Ri i flct shap a Taitioal si mol fo isctly suppot silos b Dfomatio quimt o cyli impos by compatibility with bam fomatio Fi. : Aial fomatio compatibility btw i i a shll [, 8, 9]. ALGEBRAIC CLOSED-FOR SOLUTION OF STRESS RESULTANTS IN THE RING BEA USING VLASOV S CURVED BEA THEORY Th si basic quilibium quatios fo th cu bam lmt show i Fi. ca b pss usi th Vlaso s ifftial quatios as follows [-]: Q Q Q Q q q Q q T T Q m m Q m wh = bi momt i th i about a aial ais; = bi momt i th i about a tass ais; T = tosioal momt i th i; = aius of th i bam ctoi; = cicumftial cooiat; q, q, q = istibut li loas p uit lth i th tass, cicumftial a aial ictios spctily; m, m, m = istibut appli toqus p uit cicumfc about th tass, cicumftial a aial ictios spctily; Q = cicumftial foc i th i; Q, Q = sha focs i th i i tass a aial ictios spctily. Th f boy iaam of a clos sctio i bam s-sctio is iicat i Fi. 4. By amiatio of this f boy iaam, th focs acti i th two picipal ictios, th uifomly istibut tical loa a th aial iwa loa, ca b pss as follows: c h hsi wh c = th si alu of compssi mmba stss sultat at th bas of th cyli; h = th si alu of tsil mmba stss sultat at th top of th hopp; β = th hopp half al.

4 Q T +T Q T + Q +Q Q +Q u u u q m m m q Q Q +Q q Loai a isplacmts + Focs Fi. : Difftial cu bam lmt a si cotios As show i Fi. 4 th focs i th two picipal ictios act cctically with spct to th i bam ctoi G. Ths focs ca b compos ito th basic loas o th i bam show i Fi. 4 wh th tical a iwa loas act thouh th ctoi of th s-sctio. Th th basic loas a th tass istibut loa q, th cicumftially istibut toqu m, a th coctat toqu at th suppots ms which ca b pss by th followi latioships: q mθ ms Qs - s 4 wh = th aial ccticity of th cyli fom th i bam ctoi; s = th aial ccticity of th suppot fom th i bam ctoi; = th tical ccticity of th joit ct fom th i bam ctoi; = th cicumftial al i aias subt by th half spa of th i bam = π/; = th umb of qually spac isct suppots; Q = suppot foc. Th i bam is aalyz fo ths th ifft loa cass i tu. c uifom loai q Rsultat Focs Cyli/co tasitio juctio β G Disct local suppots + uifom toqu m θ ϕh Ri bam ffcti sctio ctoi, G G Disct local suppots + coctat toqu at th suppot m s s Q Disct local suppots Ais Fi. 4: Simplifi loa-cayi mchaism mol fo th i bam []. Diatio of stss sultats i u tass istibut loa Fo th cas wh oly a tass istibut loa q acts o th i bam i.. q = q = m = m = m =, th si basic quilibium quatios ca b uc to th ifftial latioships. O ifftial quatio cocs bi of th i i its ow pla a

5 ca b ucoupl fom th oth two. Th two coupl ifftial quatios of quilibium ca b pss as : T q T 5 Simultaous solutio of q. 5 usi th appopiat bouay coitios =θ, T=T θ = als th followi latioships: cot si 6 si cot T 7. Diatio of stss sultats i u cicumftially istibut toqu I th sam ma, th followi two coupl ifftial quatios a obtai fo th cas wh oly a cicumftially istibut toqu mθ is appli to th i i.. q = q = q = m = m = : m T T 8 Simultaous solutio of q. 8 usi th appopiat bouay coitios =θ a Tθ=Tθθ= als th followi latioships: T 9. Diatio of stss sultats i u coctat toqu at th suppots Th followi coupl ifftial quatios a obtai fo th cas wh th coctat toqus ms a appli at th suppots of th i i.. q = q = qθ = m = m = mθ = : T T Simultaous solutio of q. usi th appopiat symmty bouay coitios that i symmty =θ, Tθ= a T=ms/ als th followi latioships: cot si s si cot s T Eqs 6, 9 a ca b suppos to obtai th bi stss sultat a similaly Eqs 7, 9 a ca b suppos to obtai th tosioal stss sultat as follows : cot si s si cot s T 4 Eqs a 4 a itical to thos os lop by Rott [] a poi i EN [9]. Sha foc i tass ictio ca b obtai usi q. as follows: Q 5 O aata of usi th Vlaso s cu bam ifftial quatios is that th tass isplacmts of th i ca b obtai. Th lobal foc-fomatio latioships fo th tass isplacmts ca b pss as follows []: u EI u EI u GK T w T 6

6 wh u = tass tical isplacmt omal to th pla of th i, as show i Fi. ; = otatio; I = bi momt of itia of th i bam about aial ais; Iw = wapi tat of th i bam; KT = uifom tosioal tat of th i bam; A = s sctioal aa of th i bam; E = moulus of lasticity a G = sha moulus. Th fist tm i tosioal momt pssio i q. 6 psts th spos u St. Vat tosio, whil th sco tm psts wapi cotibutio. Th pst stuy focuss o clos sctios wh St. Vat s tm omiats o th wapi tm. Wapi is lct i ths calculatios. Th sulti bi momt a tosioal momt aiatios which a i i qs a 4 ca b ictly ist q. 6 to obtai th followi ifftial latioship: u u T 7 GKT EI Eq. 7 ca b sol usi th appopiat bouay coitios u = uθ =, u = u θ to fi th tass isplacmts as: EI GK EI T s 8 u 4EI GK T si si si si EI GK s T s 4. COPUTATIONAL VERIFICATION OF THE CLOSED FOR SOLUTIONS Th commcial fiit lmt poam, ANSYS. [], was us to ify th accuacy of ths quatios fo th stss sultats a isplacmts i th tass ictio. Th isolat i bam aalyz h sts o = 4 suppots a has a cyli aius of = mm. A tat uifom tass loai of.5 kn/mm was appli to th i bam. All ccticity tms w lct fo simplicity. Th i bam was slct a squa hollow sctio a moll usi two-o bam lmts bam4 as show Fi. 5. Th moulus of lasticity was tak as GPa a Poisso s atio as.. 5 mm appli loai 5 mm 5 mm 5 mm symmty bouay coitios symmty bouay coitios Ri bam mol with bam 4 lmts Ri bam mol with bam 4 lmts colum actio u = Fi. 5: Fiit-lmt molli fo th i bam a cyliical shll Th calculat aiatios of, Tθ, Q a u i th cicumftial ictio a show i Fi. 6, toth with th pictios of th clos fom solutios qs, 4, 5 a

7 8. Wh a isolat i bam is i, th compaisos show that th abo quatios poi y accuat solutios, with th last iffcs bi.8%,.6%, 4.8% a.% fo, Tθ, Q a u spctily. To istiat th bhaiou of a i that itacts with th silo shll, silo stuctus with a i bam w t i with ifft shll thicksss t = 4 a mm a a tat hiht H = mm. Th sam suppot coitios a i bam poptis w us. A uifom aial loa =.5 kn/mm was appli to th top of th cyliical shll aou th full cicumfc. Fo a cyliical shll sti o qually spac isct suppots, th a plas of symmty. Th computatioal tim was uc by molli oly a smt coi a al of /, as show i Fi. 5. Fou-o shll lmts shll6, with a siz of mm i both th aial a cicumftial ictios, w mploy to mol th cyliical shll. Ri Bi omt kn-m Ri Sha Foc Q kn Numical Solutio-Isolat i bam Clos Fom Solutio a Euoco Appoach Numical Solutio-shll&i bam itactio with t=4mm Numical Solutio-shll&i bam itactio with t=mm Cicumftial Cooiat Numical Solutio-Isolat i bam Clos Fom Solutio Numical Solutio-shll&i bam itactio with t=4mm Numical Solutio-shll&i bam itactio with t=mm Cicumftial Cooiat Ri Tosioal omt T ϴ kn-m Ri Tass Displacmt u mm Numical Solutio-Isolat i bam Clos Fom Solutio a Euoco Appoach Numical Solutio-shll&i bam itactio with t=4mm Numical Solutio-shll&i bam itactio with t=mm Cicumftial Cooiat Numical Solutio-Isolat i bam Clos Fom Solutio Numical Solutio-shll&i bam itactio with t=4mm Numical Solutio-shll&i bam itactio with t=mm Cicumftial Cooiat Fi. 6: Compaiso of clos fom solutio with umical solutios fo th i that itacts with th silo shll Th aiatio of th stss sultats a isplacmt a show i Fi. 6. Wh th 4 mm thick shll was i, th compaisos show that th abo quatios poi ati solutios fo th i, with th last iffcs bi 5.54%,.%,.8% a.% fo, Tθ, Q a u spctily. Wh th mm thick shll was i, th compaisos show that th abo quatios poi mo ati solutios fo th i, with th last iffcs bi 44.5%, 6.8%, 58.49% a 4.8% fo, Tθ, Q a u spctily.

8 5. SUARY AND CONCLUSIONS This pap has both lop a -i si quatios fo i bams us to suppot cyliical shlls. Th stuy coctat o clos sctio i bams wh th wapi tm is lct i th calculatios. Clos fom si quatios obtai fom Vlaso s cu bam thoy w compa with umical sults. Th compaisos show that ths quatios poi accptably accuat solutios wh th isolat i bam is i. O th oth ha, wh th i bam a shll itactio is i, th alus obtai fom fiit lmt calculatios i fom thos of th clos fom solutios. As th thickss of th shll ts thick, th iffc btw th clos fom solutio a th umical calculatios icass. Th si of a i bam as a isolat cu bam is ati, sic it lcts th cotibutio of th attach shll a hopp. 6. REFERENCES [] TENG, J.G. a ROTTER, J.. Th Stth of Silo Tasitio Ris a Hopps, Tasactios of chaical Eii, Istitutio of Eis, Austalia, Vol. E4, No., 989, pp [] ROTTER, J.. Gui fo th Ecoomic Dsi of Cicula tal Silos, Spo, Loo,. [] GOULD, P. L., SEN, S. K., WANG, R. S. C., SURYOUTOO, H., LOWREY, R. D. Colum Suppot Cyliical-Coical Taks, Joual of Stuct. Di., ASCE, ST, 976, pp [4] GUGGENBERGER, W., GREINER, R., ROTTER, J.. Th Bhaiou of Locally-Suppot Cyliical Shlls: Ustiff Shlls, J. Cost. Stl Rs., 56,, pp [5] GUGGENBERGER, W., GREINER, R., ROTTER, J.. Cyliical shlls abo local suppots, i Buckli of Thi tal Shlls, s J.G. T & J.. Rott, Spo, Loo, 4, pp [6] GREINER, R. Zu Laskaftilitu i sth zyliisch Bhalt aus Stahl., Stahlbau, 5 7, 984, pp. 5 i Gma. [7] ÖRY, H. a REIERDES, H.G. Stsss i a Stability of Thi Wall Shlls u No-ial Loa Distibutio, Poc. It. Colloq. Stability Plat & Shll Stucts, Gt, ECCS, 987, pp [8] ROTTER, J.. Aalysis a Dsi of Ribams, i Dsi of Stl Bis fo Stoa of Bulk Solis, J.. Rott, Uisity of Syy, 985, pp [9] EN Euoco : Dsi of stl stuctus, Pat 4.: Silos, Euoco Pat 4., CEN, Bussls, 7. [] TOPKAYA, C. a ROTTER, J.. Ri bam stiffss citio fo colum suppot mtal silos, ASCE, Joual of Eii chaics, 7,, pp [] Vlaso, V.Z. Thi-wall lastic bams, Natioal Scic Fouatio, Washito, D.C, 96. [] HEINS, C.P. Bi a tosioal si i stuctual mmbs, Lito Books, Lito, assachustts, 975. [] ANSYS, Vsio. O-Li Us s maual,.