PREDICTION OF STRESS CONCENTRATION FACTORS IN UNLAPPED SQUARE HOLLOW "K" JOINTS BY THE FINITE ELEMENT METHOD

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1 Ngran Journal of chnology, Vol. 5, No., March 006 Jk 5 PREDICION OF SRESS CONCENRAION FACORS IN UNLAPPED SQUARE HOLLOW "K" JOINS BY HE FINIE ELEMEN MEHOD DR.P.N.JIKI Dpartmnt of Cvl Engnrng, Unvrsty of Agrcultur, P.M.B. 373 Makurd, Bnu Stat, Ngra. E-mal: ptrjk@yahoo.com ABSRAC h papr sts out to dscovr numrcally th ffct of brac spacng on strss concntratons n wldd squar hollow scton 'K' jonts. hn shll thory and soparamtrc formulaton ar mployd to obtan qulbrum quatons, whch wr solvd by th frontal soluton cod dvlopd for ths purpos. hraftr, th ffct of brac spacng was nvstgatd by varyng th spacng btwn th two bracs and makng computr runs. h rsults obtand for th prsnt study show a gradual dtroraton n strngth as th spacng btwn th two bracs was ncrasd. Howvr, du to gomtrcal dffrncs btwn th squar hollow scton and th tubular 'K' jonts, t was consdrd not ncssary to compar rsults of prsnt work drctly wth publshd ons on tubular 'K' jonts n th ltratur. It was, howvr, obsrvd that th trnd n bhavour btwn th tubular and th squar scton 'K' jonts s smlar. hus, th prsnt nvstgaton and othrs n th ltratur show that lappd tubular and hollow scton 'K' jonts ar strongr than unlappd jonts for th sam cross-sctons. NOAIONS U lmnt stran nrgy W xtrnal work don by th lmnt durng dformaton u dsplacmnt v dsplacmnt x global coordnat axs n u drcton y global coordnat axs n v drcton x local (lmnt) coordnat axs n u drcton y local (lmnt) coordnat axs n v drcton N lmnt shap functons [B] lmnt stran matrx [D] lmnt matral matrx [J] Jacoban matrx q dstrbutd prssur ovr lmnt [Q ] lmnt load vctor - [K ] assmbld systm stffnss matrx f strss from fnt lmnt analyss n calculatd strss from formula Q assmbld systm load vctor [] lmnt dsplacmnt vctor {} systm dsplacmnt vctor. non-dmnsonal coordnat non-dmnsonal coordnat s systm total potntal nrgy functonal Strss Stran.0 INRODUCION Lappd jonts gnrally, and lappd tubular jonts n partcular ar known to b strongr than un-lappd jonts followng a hurstc

2 Ngran Journal of chnology, Vol. 5, No., March 006 Jk 6 rasonng. Howvr, for th purpos of bracng, t s not always fasbl or vn possbl to brac a structural systm althrough wth lappd jonts. Hnc, a strong cas arss for usng un-lappd tubular or squar hollow 'K' jonts whch ar n many cass wldd on st du to as of constructon and transportaton. As usual, on of th problms wth wldd stl constructon s th dvlopmnt of strss concntratons or "hot spot" strsss. h "hotspots" affct th strngth (fatgu) of both lappd and un-iappd jonts n offshor structurs and stl brdgs [5, 9-]. Kuang t. al [] nvstgatd th dstrbuton of strss concntraton factors n tubular jonts usng th fnt lmnt mthod but dd not consdr th ffct of brac spacng on squar hollow jonts. Potun t. al [] carrd out a smlar study but thy too dd not consdr th ffct of brac spacng on th dstrbuton of strss concntratons. In 973, Gbstn [3] carrd out a paramtrc study of tubular jont, whch prcluds th study of brac spacng, and no squar hollow scton jonts wr usd n' th study. Wordsworth and Smdly [4] also nvstgatd th ffct of strss concntratons n unstffnd tubular jonts but dd not nvstgat th ffct of brac spacng on strss dstrbuton n th tubular K jont. Jk [8] also nvstgatd th ffct of brac spacng on strss concntratons n tubular K jonts, but dd not howvr, xamn th sam ffct on squar hollow K jonts. Som of th studs rvwd abov consdr th ffct of brac spacng on strngths of tubular K jonts wthout consdraton of sam ffct on squar scton K Jonts. hs lack of studs dsgn strngths of gappd squar scton K jonts maks avalablty of nformaton on strss concntraton factors n sam jonts scarc. Howvr, nformaton on 'Hot Spot' strsss s usful n th fatgu dsgn of such jonts. hus, th purpos of th prsnt study s to nvstgat th ffct of brac spacng on th dstrbuton of strsss' unlappd wldd squar hollow K jonts usng th Fnt Elmnt Mthod. h Fnt Elmnt dscrtzaton of th proposd squar 'K' jonts s shown n fg. h sktchs for tubular scton jon ar shown n rfrncs [,5,8]..0 EQUILIBRIUM EQUAIONS h usual curvd thn shll soparamtrc lmnt was usd for th prsnt nvstgaton [5-7]. h shap th lmnt s shown n Fg.. Usng potntal nrgy formulaton and th Raylgh-Rtz procss, th qulbrum quatons wr drvd as follows: h lmnt total potntal nrgy functonal s gvn as: U W () Whr U s th stran nrgy of th lmnt. W s th xtrnal work don by th lmnt durng dformaton. h stran nrgy of th 8-nod lmnt s gvn as: U j Ejdvol () v Whr j s drct strss j s drct stran. Usng so-paramtrc formulaton, th dsplacmnt ar ntrpolatd as u v 8 8 N, u (3) N, v (4) h 8 nod (curvd) soparamtrc lmnt s mappd nto th normalzd squar spac through th followng transformatons X y 8 8 N, x (5) N, y (6) Whr N ar th shap functons corrspondng to nod I wth Cartsan

3 Ngran Journal of chnology, Vol. 5, No., March 006 Jk 7 coordnats (x, y ) n X, y systm of axs and non-dmnsonal coordnats, coordnat systm whr, I =. For cornr nods and zro for md nods. hn th shap functons ar gvn as: N [ ] 4 (7) In trms of th shap functons N th stran matrx [B] s gvn as: [B] = [ ] (8) In whch N N x [ J ] (9) N N y And [J] s th Jacoban matrx gvn as x y [ J ] (0) x y In trms of th stran matrx of quaton (8) th lmnt stran vctor s gvn as: u [B] () v h lmnt stran nrgy of quaton () can also b wrttn n trms of matral matrx [D] as: U { j} [ D]{ j} dvol () v Substtuton of quaton () nto quaton () and assumng a unt thcknss for th lmnt gvs: U A [ B] u v [ D][ B] u v da (3) A [ B ] { } [ D][ B]{ } da (4) For th prsnt nvstgaton th lmnt shown n Fg. s loadd wth a unform prssur q ovr th lmnt boundary btwn nods. h xtrnal work don by th load durng dsplacmnt of th boundary s gvn as: W q{ } db (5) b Substtuton of quatons (4) and (5) nto quaton () gvs th total potntal nrgy of th lmnt as: [ B ] { } [ D][ B]{ } da q{ } db A (6) { } [ K ]{ } Q { } (7) In whch [K ] s th lmnt stffnss matrx Q s th lmnt boundary loads such that: [ K ] [ B] [ D][ B]dt[ J] d d (8) Q q (9) In practc th lmnt stffnss matrx of quaton (8) s obtand by prformng a numrcal ntgraton usng Gauss ntgraton rul [6, 7]. h Raylgh-Rtz procss gvs qulbrum quatons as: [ K { } ]{ } Q 0 b (0) Aftr assmbly of lmnt stffnss matrcs and appld load nto systm stffnss matrx and systm load vctor, th systm potntal nrgy functonal s gvn as: s s { } [ K ]{ } { } { Q} () In whch {A} s systm nod dsplacmnt vctor [K s ] s assmbld systm stffnss matrx

4 Ngran Journal of chnology, Vol. 5, No., March 006 Jk 8 {Q}s assmbld systm load vctor Agan applcaton of th Raylgh-Rtz procss lad to-th systm qulbrum quatons as: s K s Q () 3.0 SOLUION OF SYSEM EQUILIBRIUM EQUAIONS h systm qulbrum quaton for th prsnt problm wr solvd aftr assmbly and applcaton of bound condtons usng th Gaussan algorthm [5, 7]. h tchnqu of sub structurng was usd n th soluton procss so as to rduc th larg cor mmo rqurmnt n th bandd form of storag systm adoptd. h gan n usng ths form of soluton tchnqu s that lss cor mmory s usd at ach nstanc of th lmnaton and back substtuton procss. Aftr th soluton was carrd out th rsults wr prsntd lmnt by lmnt. 4.0 INVESIGAION OF SRESS CONCENRAIONS Du to symmtry, only hall th K jont was mployd to nvstgat strss concntraton n th prsnt study as shown n Fg.. h fnt lmnt dscrtzaton and loadng of th jont th purpos of strss analyss ar shown n Fg.. In ordr to us th Gauss lmnaton procss, lmnt nod numbrng systm was adoptd [5] hraftr, computr runs wr mad and both normal or drct strsss as wll as shar strsss wr obtand for lmnt aftr a computr run was mad Gauss ponts wth hghst lmnt drct and shar strsss from th fnt lmnt analyss wr markd as strss concntratons. Strss concntraton wr rcordd both n th chord and th two bracs of th 'K' jont. h rsults for th chord ar shown n Fgs. 3 and 4 for Gauss ponts typ for drct strsss.

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7 Ngran Journal of chnology, Vol. 5, No., March 006 Jk h gap btwn th two bracs was kpt constant at 50mm and Strss Concntraton factor (S CF ) was obtand as: F SCF (3) n n whch F s fnt lmnt strss n th longtudnal axs of th chord and N s avrag strss n th longtudnal axs gvn as: P M N (4) A Z In whch P and M ar th appld axal load and momnt rspctvly and A and Z ar th ara and lastc modulus of th scton rspctvly. Howvr, th rsult for th 50mm gap s not shown hr bcaus t s not crtcal. Crtcal cass ar thos for 60mm and 70mm gaps and ar shown hr n Fgs. 3 to GAP SUDY h ffct of gap btwn bracs (brac spacng) on th strss concntratons n th jont was nvstgatd by samplng or markng som lmnts and studyng th bhavour of th strss concntraton factors n th lmnts as gap btwn th two bracs was ncrasd from no gap to 70mm gap. h rsults of th ffct of brac spacng or gap on th chord ar shown n Fgs. 3 to 4 and thos of th ffct

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9 Ngran Journal of chnology, Vol. 5, No., March 006 Jk 3 of brac spacng on th two bracs ar shown n Fgs. 5 and 6. h ffct of shar strsss on th chord for no gap and 70mm gap ar compard n Fg DISCUSSION Gauss pont typ was usd n th sampl study. Plots of strss concntraton factors (S CF ) aganst th lmnt Gauss ponts for th cntr and cornr of th chord wr mad and crtcal valus ar shown n Fgs. 3 and 4. h rsults for no gap up to 30mm and vn 50mm gap follow a smlar pattrn and th strss concntratons ar not crtcal and ar thrfor not shown hr. Howvr, from th gap of 60mm up to 70mm, as shown n Fgs. 3 and 4, th pattrn has drastcally changd and at th mddl of th rang of th Gauss ponts, th strss concntraton factors, apart from havng hghr valus, hav ncrasd from ngatv to postv valus for cornr of th chord whl th cntr of th chord stll dsplayd ngatv valus. h chang from ngatv to postv strsss ndcats falur or ovrstrss at th ponts or th lmnts whr th chang occurs. h rsults for th bracs ar shown n Fgs. 5 and 6. h pattrn of strss concntraton factors rcordd for no gap up to 30mm gap s smlar to th on for th chord but startd to chang from 60mm up to 70mm gap. Hghr ngatv valus ar obsrvd n brac two from 60mm to 70mm gap. h rsults of brac two tnd to oscllat btwn ngatv and postv valus from 60mm to 70mm gap. h rsults of shar strsss for no gap and 70mm gap ar shown n Fg. 7 and thy ndcat hgh shar strss concntraton factors of about 45 whch occur n lmnt sampl pont 5 (Fg. ). hr s clarly a markd dffrnc as far as no gap and 70mm gap ar concrnd. h rsult obtand n th prsnt study usng rctangular hollow K jont s vry smlar to that rportd by Jk [8] that lappd unstffnd tubular K jonts ar strongr than unlappd un-stffnd tubular K jonts. 7.0 CONCLUSION From th abov rsults t can b sad that lappd unstffnd rctangular hollow K jonts ar strongr than un- lappd unstffnd rctangular hollow K jonts. h rsults obtand n th prsnt study usng rctangular hollow (un- stffnd) K jonts ar n agrmnt wth thos rportd arlr by Jk [8] for un- lappd unstffnd tubular (crcular) K jonts. In both studs, both hgh punchng shar strsss from bracs and bndng of th chord sm to b crtcal whn th spacng btwn bracs xcd 50mm. hrfor, th bracs n un-stffnd rctangular hollow scton K jonts should b spacd not mor than 60mm apart wth bracd chord rato of at last / for bttr rsults. 8.0 REFERENCES. Kuang, J.G., Potun, A.B. and Lck, R.D. Strss Concntraton Factors n ubular Jonts. Exxon Producton Rsrv Co: Offshor chnology Confrnc, 4th May, 97, pp Potun, G.J., Huang, J.G., Lck, R.D., and Hahlch, J.L. Strss Concntratons n ubular Jonts., S.P.E. Journal, August 97, pp Gbstn, M.S. Paramtrc Strss analyss of -Jonts. Offshor Stl Smnar, Cambrdg, papr 6, 978, pp Wordsworth, A.C. and Smdly, G.P. Strss Concntratons at Un-stffnd ubular Jonts. Offshor Stl Smnar, Cambrdg, papr 3, Jk,.N. An Invstgaton nto 'K' Jonts and th n th strss dstrbuton usng Rctangular/Fnt lmnt Mthod. hss submttd to Cranfld unvrsty, Cranfld, England, for th award of th dgr of Mastr of Scnc n Engnrng, Dsa, CS. and Abl, J.F Introducton to Fnt Elmnt - A Numrcal

10 Ngran Journal of chnology, Vol. 5, No., March 006 Jk 4 mthod for Engnrng Analyss. Van- Nostrand Rnhold, Nw York, 97, pp Cook, R.D. Concpts and applcatons of th Fnt Elmnt mthod. John Wlly and Sons., Nw York, 98, pp Jk, P.N., "Effct of Brac Spacng on Strngths of tubular 'K' Jonts." Ngran Journal of chncal Educaton, Vol., nos. and, 994, pp Soh, A.K. and Soh, C.K. 'Hot Spot' Strsss of ubular 'K' Jonts subjctd to combnd loadngs. Journal of Constructon Stl Rsarch. Vol. 6, 993, pp Paul, J.C, Makns, Y. and Kuroban, Y "Ultmat Rsstanc of Unstffnd Mult planar ubular I and KK Jonts". JOllmal Structural Engnrng, ASCE, Vol. 0 (no.0) 994, pp Parkr, J.A. "Concrt Flld HSS Connctons". Journal of Structural Engnrng, ASCE, Vol., (no.3), 995, pp Jk, P.N. "Strss Analyss of Squar Hollow Scton 'K' Jonts usng paramtrc Equatons". Accptd for publcaton n th Journal of Agrcultural Scnc and chnology (JAS), Unvrsty of Agrcultur, Makurd, 005.