The castellated beams deflections calculated with theory of composed bars

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1 67 ISSN MECHANIA. 1 Volue 1(): 6771 The stellted es defletions lulted ith theory of oposed rs A. Pritykin liningrd Stte Tehnil University (GTU), Sovetsky v. 1, 6, liningrd, Russi, E-il: prit_lex@il.ru Inuel nt Blti Federl University, A. Nevskogo str. 1, 61, liningrd, Russi 1. Introdution In Russin struturl Nors SN&R [1], s in Euroode [], one of the si dend to stellted es is seuring of neessry rigidity, i.e. restrition of reltive defletion f / l. For different strutures dissile vlue of this gnitude is different, ut ore often for es it hd to stisfy to ondition f / l 1/. In SN&R [1] the reltive height of openings in stellted es is restrited y vlue h / H. 667, s ore useful in struturl prtie. Nely es ith suh openings ill e onsidered in the rtile. For tody there re three different ethods of lultion of the stellted es defletions: ethod sed on the theory of oposed rs (TCB) []; ethod using the theory of Vierendeel truss [] nd the finite eleent ethod (FEM) []. First to ethods re pproxite ones ut FEM is n ext ethod. It s sees the est of ll to use FEM for lultion of defletions, ut pplition of FEM dends of existene of rther expensive progr oplex for odeling of lulted struture (for exple, ANSYS) nd eside of this reserher is to e qulified in using ANSYS, so s even sll devitions in dopted oundry onditions or desription of odel n ring to signifint distortions in results. Tht is hy different nlytil pproxite ethods re elorted for predition of the stellted es defletions. In Russin prtie the ethod of the theory of oposed rs, elorted y А.R. Rznizyn [] is ore populr nd in rod Vierendeel ethod is idespred, pplition of hih to lultion of stellted es s even inluded in one of the previous vrints of Euroode []. In the ork the tsk of otining relile reltion for predition of defletions of the stellted es s put. A riterion of the lultion ury is the finite eleent ethod results. Clultion of defortions of orted es defletions ith the theory of oposed rs s odified y uthor ith integrtion of differentil eqution in Fourier series [6].. Testing proedures Perforing of openings in es of stellted es led to reduing of their sher rigidity nd hene to inresing of its defletions. For siply supported es its groth n reh 6%, nd for lped es even -. ties opre to defletions of the se diensions es ith solid e. In ork it s put prole of oprison of different pprohes for lultions of defletions of siply supported stellted e, loded ith distriuted lod (Fig. 1) under different reltive length l/h. Fig. 1 Shee of loding of siply supported e ith uniforly distriuted lod In the theory of oposed rs orted e is onsidering s to ering T-rs, loted ove nd elo of openings, nd elsti ly, fored ith e-posts eteen openings, involving rs in joint ork (Fig. ). Fig. Geoetry preters nd lultion shee of stellted e ording to TCB Belo it is onsidering vrint of lultion of defletions ith TCB sed on integrtion of differentil eqution in Fourier series. In ork [] differentil eqution of the flexure xis of oposed e s otined in for: I M M, (1) " "" " Efi Ei E fi

2 68 here E is Young odulus of teril; f is ross re of every T-elt; is oeffiient of rigidity of elsti lyer, fored ith e-posts; I is oent of inerti of e lulted for ekened y opening setion; i is proper oent of inerti of T-elt loted ove opening; M is flexure oent. Fro Eq. (1) s privte ses it n e otined eqution of flexure of pket fored ith to rs E i ties ( EI M IV M ", not jointed eteen theselves y sher ), or eqution of flexure of onolith e under. For otining siple for of solution Eq. (1) s solved in Fourier series for se of tion of uniforly distriuted lod q. Then y deoposition of funtion of flexure oent M in series y sinus: ql 1 M sin n, () n 8 n 1, solution of Eq. (1) n e otined in for of series: ql I / i / n sin n, () EI n 1, n 1 / n is diension- here l is length of e, x / l, nd less oeffiient, deterined on reltion: l I / Efi. () Anlysis of results of lultions y FEM nd TCB of stellted es sho tht for siply supported e (Fig. 1), loded ith uniforly distriuted lod of intensity q, good results n e otined reining only one ter of series in deoposition Eq. (), hnging oeffiient / t /8. Then insted of Eq. () e get: ql I / i. () 8EI 1 First ultiple in Eq. () represents the defletion of siply supported e, lulted on tehnil theory of flexure: ql. 8EI (6) Moent of inerti I lulted for setion ith opening hve vie: I Isol h t / 1. (7) The inerti oent I sol of e ith solid e n e lulted pproxitely s: I t H t / t H t / 1. (8) sol f f f f Expression (8) is pproxite so s it is not tken into ount proper oent inerti of shelves, influene of hih for I-es usully not exeed 1%. If tke into ount tht in ny ses 1 it is possile to neglet ith unity in denointor of Eq. () nd rite it in ore opt for, onvenient for prtil lultions: 1 I / i. (9) Seond dditive in rkets in Eq. (9) is refleting the sher oponent of e defletion of orted e. Although the expression for defletion Eq. (9) hs opt for it n e siplified ore if sustitute in it Eq. (). Then reltion Eq. (9) n e ritten s: 1. Ef / l. (1) For getting of relile result on theory of oposed rs it is need orretly deterine oeffiient, hih is funtion of height of opening h, reltive idth of e-post с / а, thikness t, teril of e E nd for of e-post. Inorret finding of vlue n led to essentil loss of ury in lultions of the e defletions. Unknon oeffiient n e deterined fro experient on stti testing of orted e or using results of nueril lultion of e y FEM. In the ork it s used seond pproh. Coeffiient of rigidity n e lulted s: Gt h / 1 here G E / 1, (11) is sher odulus;. is Poisson ftor; t is thikness of e; h is height of openings; с / is reltive idth of e-post; is iniu idth of e-posts; is side of hexgonl opening; is nueril oeffiient depending on y of fixing e (siply supported or lped) nd on reltive idth of e-post. Tody in struturl prtie ellulr es ith nrro e-posts (Fig., ) re pplied idely. Tehnology of the stellted es oring suggested y uthor [7] llo to get ny idth of e-post independently of side of opening (Fig., ). Tht is hy oeffiient s hosen s funtion of. For s iply supported e ith height of openings h =.667Н oeffiient hve vie: (1) Using of reltion Eq. (1) led to good results for ny idth of e-post in rnge. 1. Sustitution of Eq. (11) into Eq. (1) led to: 11. h f 1 / / t l. (1)

3 69 Fig. Perforted es ith nrro e-posts: - irulr; - hexgonl openings So, defletion of orted e ording to theory of oposed rs n e esy lulted on reltion Eq. (1), if it is knon lod, preters of ortion nd diensions of e. All lultions of defletions on TCB re onvenient to or ith help of eletroni tles Exel. Are f of T-elt n e lulted s: f f f f t t. H h t. (1) Preliinry lultions y FEM shoed tht etter results n e otined if insted of I in reltion Eq. (6) use the inerti oent I deterined s verge ritheti of the inerti oents lulted in to ross setions: in ross setion ithout opening I sol Eq. (8) nd in ross setion ith opening I Eq. (7): I t H t / f f f t H t / 1 t h /. (1) f Nely reltion Eq. (1) ill e used elo for lultion of defortions of siply supported es nd only in expression for Eq. (6) it is need to sustitute the inerti oent I.. Nueril lultion of defletions of stellted es Estition of ury of otined ove nlytil reltion is possile y oprison of nueril results ith lultions ored y FEM, hih re ost relile. For tht it s onsidered siply supported stellted es ith different vlues of the e-post slenderness H / t. It s lulted three es ith 6 / ; 7 /. 6 1 nd 1 /. 6, euse ith inresing of the e slenderness the influene of ortion t defletions under se reltive height of openings is gro. It n e onluded fro the seond ter in Eq. (1), hih is proportionl to vlue h / t. As it is knon the orted es ith hexgonl openings re eing extensively used in lightly loded nd long-spn oposite floor onstrutions, for exple, in ulty-level prking grges. Cstellted es re eing pushed to spn gret lengths hih n pproh 16-8 under height of e Н 8, i. e. reltive length of suh es n vried in ig dipson. Tht is hy in lultion elo vrints of es ith reltive length 1 l / H ere onsidered. First e hd diensions: l с hih n e interpreted s l H t t h / H / i. e. length height f f e thikness of e idth of shelves thikness of shelves reltive height of opening reltive idth of e-post. Lod on e s uniforly distriuted ith intensity q 1 kn/. Clultions ere ored for es ith reltive length hnging in dipson 1 h / H. Preters of first lulted e ere next: q 1kN/; l 9 ; E. E. 11 G MP : MP The inerti oent of e Eq. (1) is:. I ording to I / / / / 98. Defletion of e (6) ill e: ql EI Are of T-elt is deterining s Eq. (1): f t. H h t t 181. f f f Then lultion of defletion Eq. (1) gives: / / The finite eleent lultions ring to vlue 9.. As it n e seen divergene ith the FEM FEM is 1.%. Results of lultions of types of es for different reltive length of e ith diensions: l с ; l с nd l 1. 6 с re shon in T-

4 7 le 1. As it n e seen fro Tle 1, ury of lultion on TCB is rther high; divergene ith FEM does not exeed.% even for shot es. As it is knon, iggest effet of sher on defletions is registered for shot nd high es, i.e. es ith sll rtio l / H. Clultion of defletions ith FEM s ored ith progr oplex ANSYS using qudrngulr eleents Shell6 ith 6 degrees of freedo in every node. Chrteristis of teril ere: odulus of elstiity Е. 11 MP nd Poisson oeffiient.. Mesh of eleents s unifor ith size FE. Suh pproh joins siple odeling ith good results. Using of eleents Shell6 the thikness of hih is ttriute it is need height of e dopt equl H t, then rel height of odel ill e equl full height of e H. Due to syetry it is possile onsider only hlf e tht suffiiently redue tie of lultions. Perfored ith FEM lultions of es l ith different length re shon in Fig.. f Method of lultion Defletions () of the siply supported stellted es, uniforly loded q = 1 kn/ Tle 1 Reltive length of e, l / H l TCB FEM Divergene, % l TCB FEM Divergene, % l TCB FEM Divergene, % d Fig. Defletions of siply supported es l с under distriuted lod q = 1 kn/: - l 6; - l 7. ; - l 9; d - l 1 ; e - l = 18 e In Tle results of lultion y FEM nd TCB of stellted es ith nrro e-posts re shon. Width of e-post in first se s equl to с. а of horizontl side of opening nd in seond se с. а. As it n e seen ury of lultion on TCB is even etter thn for es ith ide e-posts. Influene of idth of e-posts on defletions is not ig; for es ith length l H it does not exeed %. Results of lultion of defletions of stellted es ith nrro e-posts y FEM re shon in Fig.. As it n e seen divergene eteen defletions of es ith reltive e-posts. nd e-posts. is less of % for es ith reltive length l / H 1 (see Tle ). If opre defletions of orted e nd e ith solid e ( l 9 ) it n e seen (Fig., ) the openings redue rigidity of e pproxitely t % for lssi ortion ith 1 nd t 1% for ortion ith.. For short es ( l/н 1 ) the influene of ortion is inresing up to 6%.

5 71 Tle Defletions () of the siply supported stellted es ith nrro e-posts, lod q = 1 kn/ Method of lultion Reltive length of e, l / H l TCB FEM Divergene, % l TCB FEM Divergene, % Fig. Defletions of siply supported es с. 667 ith nrro eposts: -. ; -. ; - ith solid e. Conlusions 1. Theory of oposed rs is quite pplile to predition of defletions of orted es.. Otined results sho the defletions of stellted es ith height of openings h. 667H n e deterined ith good ury on reltion Eq. (1) for ny idth of e-posts in liits. 1. Divergene ith FEM does not exeed %.. For reltive length l / H it is not need to use reltion (1) euse good results give lultion of defletion ith the tehnil theory of flexure Eq. (6) using n verge inerti oent I. Divergene ith FEM results is less 1%.. The otined reltion Eq. (1) n e reoended for inluding in Struturl Nors & Rules for predition of the stellted es defletions. Referenes 1. SN&R II--81 Steel Strutures. Nors of Design p. (in Russin).. Euroode. EN :. Design of steel strutures Prt 1.. Brussel: Europen Coittee on Stndrtiztion. p.. Rznizyn, А Coposite Brs nd Pltes. М., Strojizdt. 16p (in Russin).. Euroode. ENV :199. Design of steel strutures. Annex N: Openings in es. Europen Coittee for Stndrdiztion.. Hrok, M.; Hosin, M Cstellted es defletions using sustruturing, J. of the Struturl Division Proeedings of the ASCE 1(1): Pritykin, A. 1. Influene of sher on defortions of orted es ith hexgonl openings, Izvesti vuzov, Constrution : (in Russin). A. Pritykin THE CASTELLATED BEAMS DEFLECTIONS CALCULATED WITH THEORY OF COMPOSED BARS S u r y In the rtile it s investigted defletions of the stellted es using the theory of oposite rs. Differentil eqution of the flexure xis of oposed e s integrted ith Fourier series. Relile expression for defletion s otined due to good hoie of the rigidity oeffiient of elsti ly fored ith e-posts. Anlytil solution for defletion is pplile for ide rnge of ortions: l/н 1 ;. с / 1; h. 667Н. An ury of otined reltion s estited ith lultions y the finite eleent ethod. Divergene of results does not exeed %. eyords: defletion, stellted es, hexgonl openings, theory of oposite rs, oeffiient of rigidity of elstilly, FEM. Reeived My 1, 1 Aepted June, 1