FEM Analysis of Welded Spherical Joints Stiffness Fan WANG a, Qin-Kai CHEN b, Qun WANG b, Ke-Wei ZHU b, Xing WANG a

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1 Intrnational Confrnc on Mchanics and Civil Enginring (ICMCE 014) FEM Analysis of Wldd phrical Joints tiffnss Fan WANG a, Qin-Kai CHEN b, Qun WANG b, K-Wi ZHU b, Xing WANG a chool of Architctur and Civil Enginring, Ningbo Univrsity of Tchnology, Ningbo , China a wyuqi45@163.com, b @qq.com Kywords: Wldd phrical Joints, FEM Analysis, Joints tiffnss, Non-linar tability Analysis. Abstract. Wldd sphrical joints stiffnss has important influncs on th strngth and stability baring capacity of rticulatd shlls. For complication of thortical analysis and difficulty on xprimnts, th stiffnss of wldd sphrical joints which is applid widly in th larg-span rticulatd shlls, is analyzd and computd undr th diffrnt conditions by finit lmnt mthod with th structural analysis univrsal program AP5 in this papr, th xprssions of axial flxibility function f and flxural stiffnss k of wldd sphrical joints with paramtrs: joints diamtr and thicknss, mmbrs diamtr ar attaind by rgrssion analysis for lots of computing rsults. Hr, orthogonal dsign and orthogonal multi-itm formula ar accptd. Ths xprssions can b adoptd by th non-linar stability analysis of th larg-span spac rticulatd shlls with th wldd sphrical joints considring th influnc of joints stiffnss. Introduction Wldd sphrical joints stiffnss has important influncs on th stability baring capacity of larg-span spatial rticulatd shlls. According to th Currnt <sign and Construction Rgulation of Latticd hll>, th influnc of joints stiffnss can b ignord for th analysis of latticd shll structur, supposing th joints ar pinnd and th mmbr only bars axial forc. It is gnrally accptd that th joints ar compltly pinnd or rigidly connctd in th stability analysis of latticd shll. In fact, th joints of spac trusss or rticulatd shll hav crtain axial and flxural rigidity, which ar flxibl or smi-rigid. Joints stiffnss of rticulatd shll is an important factor of shll s stability and its fixing ability is of vital significanc to maintaining th stability of rticulatd shll. Howvr, joints of rticulatd shlls constitut th som spatial load systm, with th complicatd loading situation. Morovr, th gomtric dimnsion, matrial, tchnology, prcision, wldd quality of joints and mmbrs all hav a dirct ffct on th joints rigidity. Thrfor, it is difficult to offr analytical solutions to strss and dformation. At prsnt, th dtrmination of joints stiffnss domstically and abroad has bn obtaind from xprimnts and is only to boltd sphrical joint. Howvr, in trms of larg spac rticulatd structur, thr xist many joints, and th dimnsion of which is th sam as th connction of mmbrs. If dtrminations of joints stiffnss ar mad on aftr anothr, it will turn out to b tim-consuming and unpractical. In this papr, th structural gnral program AP5 is adoptd to conduct finit lmnt analysis and calculation of wldd sphrical joints widly applid in th larg spac rticulatd structur. It focuss on th rsarch of dformation of th boundary btwn hollow sphr and stl tub; finally th formulas of th axial flxibility cofficint f and flxural stiffnss k ar gottn by rgrssion analysis of orthogonal polynomials. FEM Analysis Th Mchanical Modl According to th trial rsult of th whol sphr, whn th joints hav a singl load in on dirction, th displacmnt in th othr dirction is on ordr of magnitud gratr than that. Hnc, it can b ovrlookd. Whn it coms to calculation of joints of hollow sphr, simplifying th 014. Th authors - Publishd by Atlantis Prss 773

2 multidirctional load to unidirctional on in turn simplifis gratly th analysis. Manwhil, it satisfis th prcision rquird by nginring. inc th joints of hollow sphr tak on symmtry in unidirctional load, th hmisphr is chosn as th mchanical modl. Th us of slid support and th fixd support rspctivly as boundary condition of symmtric plan xrts littl influnc on th calculation rsult, which conforms to limitations of strss concntration. Th trial rsult of th whol sphr shows that both th circumfrntial and radial displacmnts of th boundary ar small. Thrfor, it is appropriat to mak fixd sid th boundary condition of symmtric plan. Elmnt Msh ubdivision Th calculation in this papr adopts th structural gnral procdur AP5. As th comparison of hollow sphr s thicknss and diamtr is within th rang of thin-shll, plat shll lmnt of thr-dimnsional problm in th program is usd. Outsid th sphrical top lmnt msh subdivision utilizs th constant-strain triangular lmnt whil othr parts mploy conforming quadrilatral lmnt. On th boundary of sphr joint and stl tub, msh subdivision is dnsr. In th calculation of axial flxibility cofficint, axial load has an vnnss ffct on sphr joint and stl tub. Accordingly, it is a mattr of axial symmtry. Th papr dirctly sks th answr through axially symmtrical problm. In th calculation of flxural rigidity cofficint, th calculatd modl is dividd into 160 lmnts and 161 joints, that is 0 qual divisions horizontally and 8 vrtically. Calculation of Nodal Load In th calculation of flxural rigidity cofficint, bnding momnt is transformd into quivalnt load; th dirction of nodal load is dtrmind by th dirction of bnding momnt; siz changs with qual proportion from XOZ plan. Th forc momnt producd quals to th known bnding momnt. Aftr calculation of joints vrtical displacmnt on th boundary of sphr joint and stl tub, th conclusion is obtaind about wldd sphr s axial flxibility and flxural stiffnss rigidity f, Mthod of Analysis In th calculation of axial flxibility cofficint f and flxural rigidity cofficint considring that f and k ar rlatd to many factors such as sphr diamtr, pip diamtr d, sphr thicknss, tc, orthogonal dsign mthod is adoptd to calculat f and k undr various conditions. Furthrmor, orthogonal polynomial is followd to mak rgrssion analysis about th calculation rsult, with a satisfying rgrssiv formula subsquntly. Bcaus of AP5 s wors pr-procssing function, th workload of filling in th data fil is havy for sphrical shll lmnt with mor joints. It only nds to calculat 5 cass by using orthogonal tabl 5 (56), rducing th workload gratly. Th calculation finds that sphr diamtr has an inconspicuous ffct on axial flxibility cofficint f and flxural rigidity cofficint k, howvr, th ratio of sphr diamtr and pip diamtr d constitut th main factor. Th othr important factor is sphr thicknss. Thrfor, in this papr th factors of d/ and rgrss to axial flxibility cofficint f and flxural rigidity cofficint k. Quadratic orthogonal polynomial is applid to th rgrssion of calculation rsult. Th rsult indicats that prcision mts th rquirmnts. Basd on th rgulations in Currnt sign and Construction Rgulation of Latticd hll, d = /7. Thus, d/ is within th rang of 0.3 to 0.5. In trms of larg span rticulatd structur, sphr diamtr is usually abov 400 mm and sphr thicknss ovr 1 mm. To improv th prcision of rgrssiv formula, taks 5 lvls of 1 mm, 14 mm, 16 mm, 18 mm, 0 mm. k. k, 774

3 Rgrssion Analysis Rgrssion of Axial Flxibility Cofficint Th valu of axial flxibility cofficint f is shown in Tabl 1. Tab. 1 ats of f (10 6 mm/kg) d/ /mm Th rgrssion formula of axial flxibility cofficint f can b concludd. f ( d /, ) = f + ( d / ) + ( ) + ( d /, ) (1) Whr ( d / ) and ( ) ar th ffct functions of factors d / and rspctivly. ( d /, ) is th ffct function of d / and intractions. f is th avrag valu of axial flxibility cofficint. In th abov formula, ( d / ) = P( d / ) k1 ko ( ) = ( ) t1 P ot t ( d /, ) = P ( d / ) P ( ) Whr u1 v1 uv u v d d d d d d P / 0.05 (5 1) / 0 8 P (0 8) (5 1) /1 (0 8), P1 ( ) ( 1) / (5 1) / 8, P ( ) ( 8) ( 8) 1 Put th data in Tabl of data f into th abov formula and it can b obtaind that d d 1 f ( d /, ) (0 8) 0.04[(0 8) ] 0.8( 8) d d 1 d 1 d 1 ( 8) 0.001[(0 8) ][( 8) ] Th significanc tst is followd. Rgrssion squar sum can b infrrd from calculation: [( 8) ] 0.06(0 8) 0.009(0 8)[( 8) ] 0.009[(0 8) ] U Whr α is th variation of rgrssion cofficint α. Th squar sum l 00 of f total variation is l 00 fij N f i j

4 o surplus sum of squars Q is Q l00 U Bcaus 9.646/ 8 F 1.64 F0.01(8,16) /(58 1) Rgrssion formula is highly significant. According to th rsult of significanc tst, in th obtaind f rgrssion formula, ar liminatd. Aftr simplification, P d / ) P ( ), P d / ) P ( ), P d / ) P ( ) 1( ( 1 ( 6 f ( d /, ) d / 16( d / ) ( d / ) (10 mm/ kg) Th Rgrssion of Flxural Rigidity Cofficint Th valus of flxural rigidity cofficint ar shown in Tabl. k imilarly, th rgrssion formula of flxural rigidity cofficint is k () k ( d / ) k ( d / ) ( ) ( d /, ) Th rgrssion cofficints and thir variations can b calculatd through th sam mthod. d d d k ( ) (4) Th rgrssion formula is also highly significant. (3) Tab. ats of 8 k ( 10 kg mm/rad) /mm d/ Conclusion Th wldd hollow sphrical joint is a spatial load systm. Its rigidity is rlatd to many factors such as load condition, th dimnsions of gomtry and mmbr, matrial, tchnology, prcision and so on. Obviously, without considring th quality of wld sam and wld, th most important factors that dtrmin th rigidity of joints li in th sphr diamtr, pip diamtr d and sphr thicknss. Th calculation rsult in this papr suggsts that axial th flxibility cofficint f and flxural rigidity cofficint k tak on a rgular chang in accordanc with th sphr diamtr, pip diamtr d and sphr thicknss. Th givn rgrssion formulas boast high significanc. Thy can b applid to rigidity and stability analyss of larg span rticulatd structur of th wldd hollow sphrical joints which ar influncd by joints rigidity. Rfrncs [1]hi, H.Z. and Hu, X.R., Calculat Mthod of ingl-layr Rticulatd hlls of Boltd phrical 776

5 Joint. Collction of pac Lattic tructurs, hanghai: Tongji Univrsity Prss, []Wang, X., ong,.l.and Wan, H.Y., Th Influnc of tiffnss of Wldd phrical Joints on Intrnal Forc and isplacmnt of pac Trusss. patial tructurs, 1996, (4), [3]Wu, T. and Yan, H., Analysis of Limit Bar Forc of pac Trusss on Joints Rstriction. Collction of pac Lattic tructurs, hanghai: Tongji Univrsity Prss, [4]Xu, A.L., Th Influnc of tiffnss of Axial Joints on Intrnal Forc and isplacmnt of pac Trusss. Collction of patial tructurs, Bijing: cinc Prss, [5]Yin,.Y., Liu,.W. and Qian, R.J., tructural sign of Rticulatd hll, Bijing: China Architctur and Building Prss,