Stress concentration in castellated Ibeams under transverse bending


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1 466 ISSN MECHANIKA olume 22(6): Stress onentrtion in stellte Iems uner trnsverse ening A. Pritykin Kliningr Stte Tehnil University (KGTU), Sovetsky v. 1, , Kliningr, Russi, Emil: Immnuel Knt Blti Feerl University, A. Nevskogo str. 14, , Kliningr, Russi 1. Introution Although stellte ems re lrey pplie out 100 yers, the theory of their stress stte still is not still finlly elorte. It n e onfirme, for exmple, with sene of reommentions on esigning of suh ems in Eurooe 3. This sitution n e expline, first of ll, with omplexity of prolem. To hoie the optiml imensions of stellte em it is nee to ppreite the imum level of stresses in it, euse this is one of importnt prmeters in struturl norms. Stress istriution in stellte ems ws investigte in ny works [113] minly using FEM n experiments. Anlytil reltions were nlyze in works [1415]. However relile formul for stress level ws not otine. ery suite instrument for ompre of stress stte of ems with ifferent weutting pttern is the stress onentrtion ftor α σ (SCF), representing y itself nonimension mgnitue. It is possile etermine SCF s rtio of imum equivlent stresses in zone of opening to imum stress in flnge of em with soli we uner given externl lo. In this work the etermintion of oeffiient α σ ws performe for se of trnsverse flexure. 2. Equivlent stresses in em For estimtion of the stress onentrtion level uner trnsverse ening it ws initilly onsiere simply supporte stellte Iem, performe on unwste tehnology from rolle profile #50 (GOST ), loe with onentrte fore, pplie in mile of spn. Depth of holes ws opte equl to h H, s more useful in struturl prtie. Wepost with ws equl to sie of hexgonl hole, i. e. lssi sheme of the em perfortion ws onsiere (Fig. 1). In this se imensions of ems were l сm In ommon se it n e interprete s l H t t m h / H /, i. e. length totl w f f height of em we thikness flnge with  thikness of flnge reltive epth of opening reltive with of wepost. Conentrte fore P = kn ws onstnt in ll ses of loing. Aury of lultions y FEM is minly etermine with the finite element sizes: the less FE, the more urte is lultion. But pplition of ll refine elements is not suitle euse of the restrite omputer memory n essentil inresing of ompute time. For exmple, solution of the eqution system with unknowns in omputer with 4 G RAМ emn more the one minute. Reuing the size of the eqution system n respetively time of lultion n e hieve using ifferent pprohes: pplition the super element metho; tking into ount the struture symmetry n onsiering only hlf of em; using no uniform mesh of finite elements n others. Lst two pprohes re simplest n rther effetive, ut pper question, wht size of elements will e suffiient for getting of emne ury of lultion? Theoretilly to etermine optimum sizes of FE is rther omplex, tht is why in most ses they re hosen on se of lultions with suessive reuing of element sizes, until ifferene in results will e negligile. Fig. 1 Clultion sheme of stellte em In this work the refine mesh of FE ws opte only in viinity of one opening, s it gives the lest system of equtions. Then this refine mesh ws isple in turn to eh opening. It is ler the size of element is to e onnete with rius of urvture of hexgonl opening. The smller fillet rius the lesser elements re to e, otherwise ontour of opening tkes form not smooth ut roken line, n ury of lultion will e reue. Although formlly openings in wes performe on unwste tehnology re not roune in relity they hve some fillets. In orne with reommentions of AISC (Amerin Institute of Steel Construtions) in lultions for strength of stellte ems the fillet rius of hexgonl openings is neessry opt equl to r 2t w, or to r 5/ 8", epening wht is igger. In lultions performing elow the rius of fillet ws tken r 0. 04h (h epth of opening). This orrespons pproximtely to onition r 2t. w Fulfille nlysis show the stisftory ury is rehe uner sizes of finite elements equl to 0.05r, tht is why in lultions sizes of FE ner the ontour of opening they were equl FE 1 mm, n in other prts of em their imensions were 20 mm. Depth of openings ws h = 500 mm600 mm. Uner trnsverse ening the importnt role in vlue of α σ ply s ening moment M so n sher fore. The first one etermines level of norml stress σ x, n seon is onnete with the sher stress τ xy in we. In tehnil
2 467 theory of flexure the stress stte of em with soli we is onsiering tking into ount only two stress omponents σ x n τ xy. In perforte ems ner openings the norml stress σ y is lso hieve ig vlues. Tht is why in stellte ems uner trnsverse ening tkes ple omplex stress stte, the integrl prmeter of whih for evlution of SCF uner joint tion of σ x, σ y n τ xy n e equivlent stress von Mises. In ommon form it n e expresse vi stress omponents s: tion of n M. For tht simply supporte stellte Iem of ritrry length, loe with onentrte fore in mile of spn ws onsiere. In this se ner ene opening uner ny length of em the sher fore will e onstnt n ening moment. Ner ny other opening the joint tion of n M tkes ple. M 0 P Const (1) x x y x xy Appreite vlue of with formul: /, (2) where is imum stress in flnge of em with soli we, etermine on tehnil theory of flexure s: М / W, (3) where W is moulus of inerti of em s ross setion with soli we: 2 f f w W t H H t / 6. (4) Of ourse suh pproh to lultion of SCF on Eq. (2) hs some peulirity, euse the level of imum stress is mesuring in one ross setion, n se vlue is tken in nother, ut it hs no importnt sense, euse α σ is nonimension mgnitue. Bsi vntge here is ommoity of lultion, n orresponene of otine vlue α σ to physil piture of stress stte of we in the onentrtion zone. With the im to istinguish influene of n M on vlue α σ it ws performe lultions uner onstnt vlue n prtilly sent moment М n uner joint Fig. 2 e versus 0 of ene wepost:   ; с  e  с 0. 4Н с0 0. 2Н 0 с Н ;  с Н с Н First of ll efine originl lotion of ene opening, i.e. istne 0 from the opening ege to the support setion. Otine y FEM lultions of versus with of ene wepost re shown in Fig. 2, from whih it n e seen the pik of stress tkes ple in zone ( )Н, fter tht level of stress is stilizing, lthough flexure moment is growing. Differene in vlues (Fig. 2) oes not exee 3%, i. е. this mgnitue is rther stle uner hnging of with of ene wepost. All this llow in further lultions opt с H. Evlute now influene of em length l t imum level of equivlent stress ner first opening lote ner support. As show lultions y FEM (Fig. 3), uner onstnt trnsverse fore n sent flexure moment М imum equivlent stress ner ontour of ene hole will e prtilly onstnt. Moreover relte length of em ose not ply ny role (ompre Fig. 3, а n 3, ). From this n e onlue the stress stte in viinity of ene hole of simply supporte em ompletely etermine with vlue of trnsverse fore. ; ; Fig. 3 Stress stte ner ene hole uner onstnt fore in ems l см of ifferent length:  l = 15H;  l 20 Н lue of stress, proue with tion fore, n e represente with reltion:, (5) Ht w where α is numeril oeffiient, etermine from FE nlysis. Clultion with FEM shows the oeffiient 41. For exmple, for stellte Iem with imensions m uner tion of onentrte fore Р = kn level of imum equivlent stress in viinity of 1st hole oring to Eq. (5) will e: / MP. Now perform with help of lultions y FEM nlysis the influene on mgnitue of flexure moment in the sme em with imensions сm , ut in zone of seon n following openings (Fig. 4). As it n e seen, level of imum stress grows proportionlly to moment М. On se of these results the stress in viinity of ny opening n e represente
3 468 s sum of two items: one use y sher fore in orne with Eq. (5) n seon use y moment М: М, (6) W М Htw where α M is numeril oeffiient, etermine y FEM nlysis. Flexure moment М for nth opening n e pproximte s: М x n 1 s, (7) where s is step of openings; n is orinl numer of opening, in viinity of whih the stress vlue is etermining. In ommon se the step of openings uner ny perfortion n e written s: 2 2 3, (8) s а H / where is reltive with of wepost; reltive epth of opening. с / h / H is Sustitution of Eq. (4), Eq. (7) n Eq. (8) in Eq. (6) ring to expression: М n t / Ht 1 Ht f f w w, n 2 In Eq. (9) vlue (n  2) is written inste of (n  1), euse proportionl growth of stresses is ppering only eginning from 2n opening. Tking into ount the oeffiient is onstnt for ll lulte elow ems Eq. (9) n e rewritten s: М экв 6.4 n t / Ht 1 Ht f f w w (9). (10) erify Eq. (10) for stellte Iem сm loe with onentrte fore Р kn, pplie in mispn. For this vrint it ws opte vlue. Then for 6th hole uner n in orne with Eq. (10) it n e otine МP Fig. 4 Stress stte of Iem with imensions m uner tion of onentrte fore Р = kn:  2n;  3r;  4th;  6th holes Aoring to result of FEM, s shown in Fig. 4, for 6th opening vlue 433 МP. It inites on prtilly full oiniene with result of Eq. (11). For other openings of this em the vlues otine nlytilly y Eq. (10) re shown in Tle 1. It is nee to note tht Eq. (10) remins pplile for ny length of em uner unhngele prmeters of perfortion. So for rtio of em length l / Н 20 in orne with Eq. (10) vlue 495 МP, n lultion y FEM gives МPа, tht inite on ivergene pproximtely in 1%. For other em with imensions сm uner the sme lo the stress stte of em otine y FEM is represente in Fig. 5. / Fig. 5 Stress stte of Iem with imensions сm uner trnsverse ening:  2n;  4th;  5th;  6th holes
4 469 Clultion oring to Eq. (10) uner ξ = 1, β = n α = 41 for 6th opening gives МP n oring to FEM (Fig. 5, ) result. is МP (the ivergene oes not exee 2.6%). Suh result is stisfying to engineering ury of lultions. But it is possile to otin prtilly full oiniene of results y FEM n y Eq. (10) for this em, if perform orretion of oeffiient α, reuing it from 41 to 39.5 n remining seon oeffiient α M unhngele n equl to 6.4. In this se results otine y FEM n y Eq. (10) prtilly oinie. lues of stresses for other openings of this em n for ems of other sizes re shown in Tle 1. In Tle 1 it were onsiere stellte ems frite on unwste tehnology from rolle profiles #45, 50, 55 n 60. They re quite similr in proportions of height n we thikness ut min ifferene is in reltions of were to flngere. This ifferene n e seen in vrition of oeffiient α whih is hnging in very nrrow rnge: from 38.8 to 41. Stress Tle 1 in zone of hexgonl openings with rius of fillet in Iems of ifferent profile uner onentrte lo Р = kn pplie in mispn r 0. 04h Numer of hole Prmeters of em ; ; сm Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; ; 15Н m Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; ; 15Н m Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; М 6. 4 ; 15Н m Stress y FEM Stress y Eq. (10), MP Divergene, % М М М Influene of epth of opening t the stress level Evlute now influene of epth of opening t the stress. Consier ems with reltive epth of opening 07. n Results of lultion of em with imensions сm0.71, performe y Eq. (10) with vlue of α = 46.5 n ompute y FEM re shown in Tle 2. Stress stte of we ner the ifferent openings of this em uner onentrte lo Р = kn is lso shown in Fig. 6. It n e seen from Tle 2 n Tle 1 the inreseing of the epth of opening les to growth of oeffiient of fore α, i. e. role of sher fore in vlue of equivlent stresses is inresing. Depenene α on mgnitue β is proportionl n n e pproximte with expression: (11) Similr linel epenene there is n for ems with other imensions. Stresses экв in viinity of hexgonl opening of ifferent epth in Iems uner onentrte lo Р = kn Numer of hole Prmeters of em ; M 6. 4 ; сm Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; 6. 4 ; сm Stress y FEM Stress y Eq. (10), MP Divergene, % M Tle 2
5 470 e Fig. 6 Stresses in simply supporte Iem сm uner trnsverse ening:  1st;  2n;  3r;  4th; e  5th; f  6th holes f Fig. 7 Stresses in simply supporte Iem сm uner trnsverse ening:  2n;  3r;  4th;  6th holes For relte epth of openings β = 0.73 oeffiient α tkes vlue 55.3 ut α M remins unhngele. lues of for Iem with imensions сm re shown in Tle 2, from whih it is seen the ivergene in ifferent lultions oes not exee 2.5%. Results of lultion y FEM re shown in Fig. 7. Clultions performe for ems m n m show tht if opt oeffiients α = 47.3 n α = 56.2 respetively ivergene oes not exee 1% for thir n following openings (see Tle 3). It is importnt the high ury tkes ple ner the most loe openings, lote in the mile prt of em. Stress istriution in em m is shown in Fig. 8. Stresses ner hexgonl openings of ifferent epth in Iems Tle 3 Numer of hole Prmeters of em ; 6. 4 ; m Stress y FEM Stress y Eq. (10), MP Divergene, % Prmeters of em ; 6. 4 ; m Stress y FEM Stress y Eq. (10), MP Divergene, % M M Fig. 8 Stresses in simply supporte Iem m uner trnsverse ening:  2n;  6th holes
6 Influene of reltive with of wepost t stress level As it is known, eveloping of perforte ems is irete on lightening of we with ifferent wys: inresing the epth of openings; inresing the length of opening, y performing them with elongte form suh s ovl, retngulr or sinusoil; reuing the with of weposts. Author propose tehnology of frition of stellte ems with regulr hexgonl openings uner ny with of weposts [16]. Tht is why influene of reltive with of wepost t stress is onsiering elow. Results of lultion y FEM on progrm ANSYS of Iem сm with reltive with of wepost β = 0.5 with sequent isplement of smll mesh re shown in Fig. 9. Due to reuing of with of weposts the numer of openings t hlf of the em length inrese to 7. Anlytil lultion of equivlent stress llows onfirm, tht oring to Eq. (10) reuing of β le to less level of. In Eq. (10), s in previous vrints with β = 1 ftor of influene of moment remins the sme α M = 6.4, ut ftor of sher fore tkes vlue α = Results of lultion of inite em y Eq. (10) re shown in Tle 4. Reue the reltive wepost with till β = 0.3 n lulte gin the sme simply supporte em with Н = 90 m uner tion of onentrte lo Р = kn. The vlues of stresses otine y FEM re shown in Fig 10. Due to reuing the wepost with the numer of openings t hlf of em length inrese to 8. Clultions in oring to Eq. (10) re performe uner the sme vlues of α Q = 37.6 n α M = 6.4. e f Fig. 9 Stress stte of Iem with imensions сm uner trnsverse ening:  2n;  3r;  4th ;  5th ; e  6th ; f  7th holes Stress ner hexgonl openings with rius of fillet r 0. 04h n ifferent with of weposts in simply supporte Iem uner trnsverse ening Numer of hole Bem s prmeters α = 37.6; α M = 6.4; сm Stress y FEM, MP Stress y Eq. (10), MP Divergene, % Bem s prmeters α = 37.6; α M = 6.4; сm Stress y FEM, MP Stress y Eq. (10), MP Divergene, % Tle 4 Fig. 10 Stress stte of Iem with imensions сm uner trnsverse ening:  2n;  5th;  6th;  7th holes
7 Experimentl investigtion In orer to verify Eq. (10) it ws put n experiment on steel moel in form of oule ntilevere Iem with imensions m , loe y two onentrte fores = 10 kn pplie t the ene setions vi ynmometers DR20 (Fig. 11). Mteril of em ws steel S345. Instlltion h two rigi posts lote t istne 1 m from eh other. Length of eh ntilever ws 1.5 m. During loing the level of stresses in viinity of openings ws mesure y strin guges with se 1mm lote on we in form of strin rosettes. Reings of guges were registere with Dt Aquisition Controller of English firm Shlumerger. Guges were glue ner the fillet orner openings in ples, etermine with lultion y FEM. Bem ws simply supporte n loe symmetrilly t ens. Fig. 11 Test setup with stellte Iem moel m equivlent stress Results of tests show the mesure imum ner ontour of 2n opening ws equl to 181 MP n in viinity of 3r opening it ws 190 MP. Clultion of em with finite element metho (Fig. 12, n 11, ) inite vlues of in the sme lotions equl to 184 MP n 195 MP respetively. Differene in vlues reh 2.5%. Determintion of equivlent stress ppering in viinity of thir opening in orne with Eq. (10) for teste em gives: / МP (12) Otine results inite the stresses lulte y Eq. (12), y FEM n registere in experiment uner trnsverse ening re in goo orreltion. As it n e seen oeffiient α M = 6.4 is onstnt for ll imensions of ems n ifferent perfortion. Fig. 12 Stress istriution in oule ntilevere simply supporte Iem moel m uner trnsverse ening:  1st;  2n;  3r openings 6. Stress onentrtion ftor Evlute now stress onentrtion ppering in we uner tion of flexure moment M n trnsverse fore. For this purpose it will e using Eq. (2) in whih we sustitute vlue of imum equivlent stress in ritrry se ТТ tion Eq. (10) n stress, etermine on tehnil theory of flexure s: ТТ l 2 t H H t / 6 2 f f w. (13) Sustituting Eq. (10) n Eq. (13) in Eq. (2), the stress onentrtion oeffiient α σ is etermine s follows * , (14). n / where ω * = 6 f t f / Ht w + 1 n η = l / H. As it n e seen from Eq. (14) SCF oes not epen on lo ftors ut is etermine only with geometry of em, reltive length η = l / H n prmeters of perfortion in nonimension form ξ n β. Clulte y Eq. (14) oeffiients α σ for ems with imensions сm n сm will e equl α σ = 433 / 110 = 3.93 n α σ = 298 / 81.4 = 3.67 respetively. The less vlue α σ for em with height Н = 90 сm ompre with em with Н = 75 сm n e expline y reution of relte re of flnge: if in em with Н = 75 сm
8 473 vlue ω f / ω w = 0.345, then in em with Н = 90 сm it will e ω f / ω w = The flnge effet n e ompre with inresing the plte imensions uner evlution of stress onentrtion in viinity of lone opening uner plne stress stte. It is nee to rememer the otine results re pplile to stellte ems with fillet rius of opening r = 0.04h. 7. Conlusions 1. Anlytil expression for SCF for se of trnsverse ening is otine s sum of two omponents refleting influene of sher fore n flexure moment М respetively. 2. Otine reltions for α σ n for equivlent stresses re pplile for reltive epth of openings in ipson n for reltive with of weposts in ipson uner fillet rius r 0. 04h 3. Ftor of influene of moment α M = 6.4 oes not epen on the reltive vlues ξ n β. 4. Ftor of influene of sher fore α grows with inresing of epth openings n is lmost proportionl to vlue β. 5. Stress onentrtion ftor ner hexgonl openings uner trnsverse ening n reh vlue α σ = 4. Referenes 1. esrghvhry, K Stress istriution in stellte em, Pro of ASCE, Strut Div 95(2): Cheng, W.K.; Hosin, M.U.; Neis, Anlysis of stellte steel ems y the finite elements metho, Pro of Speil Conf on FEM in Civil Eng, Moutree, Cn, Liu, T.C.H.; Chung, K.F Steel em with lrge we openings of vrious shpes n sizes: finite element investigtion, J Constr Steel Res 59(9): erissimo, G.S.; Fkury, R.H.; Riero, J.C Design is for unreinfore we openings in steel n omposite ems with Wshpes, Eng J. 20(3): Dionisio, M. t l Determintion of ritil lotion for servie lo ening stresses in nonomposite ellulr ems, Reserh Report 8. illnov University. 6. Lgros, N.D. t l Optimum esign of steel strutures with we opening, J of Eng Strut 30(4): Devinis, B.; Kvers A.K Investigtion of rtionl epth of stellte steel Iem, J of Civil Eng. n Mngement 149(3): Hoffmn, R. t l Anlysis of stress istriution n filure ehvior of ellulr ems, Reserh Report 7. illnov University. 9. Tsvriis, K.D Filure moes of omposite n nonomposite perforte ems setions with vrious shpes n sizes of we openings, PhD thesis, City University, Lonon. 10. Сhhpkhne, N.K.; Sshiknt, R.K Anlysis of stress istriution in stellte em using finite element metho n experimentl tehniques, Int. J. of Meh Eng Appl Res 3(3): Wkhure, M.R.; Sge, A Finite element. nlysis of stellte steel em, Int. J. of Eng. n Innovtive Tehnology (IJEIT), 2(1): Wng, P.; Wng, X.; M, N ertil sher ukling pity of weposts in stellte steel ems with fillet orner hexgonl we openings, Engineering Strutures. 75: Durif, S.; Bouhir, A.; ssrt, O Experimentl n numeril investigtion on wepost memer from ellulr ems with sinusoil openings, Eng. Strut. 59: Dorhev,.М.; Litvinov, Е Anlytil etermintion of stressstrin stte of wepost of perforte em, Izvesti vuzov, Constrution 5: (in Russin). 15. Pritykin, A Stress onentrtion in ems with one row of hexgonl openings, estnik of Mosow Stte Strut. University 1: (in Russin). 16. Ptent Russin Feertion Perforte metlli em. Pulishe Bulletin 30(3). А. Pritykin STRESS CONCENTRATION IN CASTELLATED IBEAMS UNDER TRANSERSE BENDING S u m m r y In the work on se of lultions y FEM of stellte Iems the pproximte reltions for evlution of stress level n stress onentrtion ftor in viinity of hexgonl fillet openings uner trnsverse ening re erive. Clultion of simply supporte stellte Iems uner tion of one onentrte fore pplie in mispn n two symmetrilly pplie fores ws performe. Propose reltion for equivlent stress ner openings ifferentite role of eh fore ftor n М n llow etermine level of stresses in stellte ems in wie ipson of the opening prmeters uner ifferent length rtio with engineering ury. Otine results were verifie with experiment test on steel stellte em with 4 m length. Keywors: stress onentrtion ftor, stellte Iems, hexgonl openings, von Mises stress, FEM, experiment. Reeive Novemer 04, 2015 Aepte Novemer 25, 2016
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