Analysis of Cylindrical Shells Using Mixed Formulation of Curved Finite Strip Element

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1 IJIET - Intentionl Jounl of Innovtive cience, Engineeing & Technology, Vol. 4 Iue 5, My 7 IN (Online) Ipct Fcto (6) Anlyi of ylindicl hell Uing Mixed Foultion of uved Finite tip Eleent lh Bony, Ezzedin Glut nd Ahed Tuhi ivil Engineeing Deptent, Acdey of Gdute tudie, Tipoli, Liby ivil Engineeing Deptent Univeity of Tipoli, Tipoli, Liby Abtct In thi ppe, the ixed foultion of cuved finite tip eleent fo cylindicl hell i developed. In the ixed foultion, tee nd oent nol to the nodl line e included piy vible in ddition to the diplceent coponent. The cuved hell eleent i divided into nube of longitudinl tip. Ech tip eleent h two nodl line with fou degee of feedo pe nodl line. The nodl degee of feedo e conited of the diplceent u, v, nd w in x, y nd z diection epectively, well the oent M bout y-xi. A nueicl exple i peented which clely deontte the vlidity of the ixed foultion to nlyze diffeent type of cylindicl hell. The ipotnt deign quntitie uch veticl deflection nd ltel oent e found to be in good geeent, when coped with n ccute olution. Keywod: Mixed foultion, cuved finite tip, cylindicl hell nd finite tip ethod.. Intoduction The fit known utiliztion of Reine vitionl pinciple in finite eleent ethod w popoed by Henn []. The foultion include the effect of tnvee he tin, whee the conideed field vible of the poble e the tnvee diplceent nd te couple. He ued line intepoltion function fo tnvee deflection nd contnt te couple within ech eleent. Lte contibution to the ue of ixed foultion in plte wee given by onno [], in which good dicuion of Reine vitionl pinciple nd it ppliction in finite eleent nlyi of plte w given. Pto [] intoduced odified Reine pinciple fo hllow hell. He ued cuved tingul eleent with line intepoltion function fo diplceent nd te couple. Fo pecil ppliction of ixed foultion to hell of evolution, Eli [4] ued cuved ing eleent fo the xiyetic nlyi of hell of evolution. Gould nd en [5] ued peviouly teted eleent nd foultion in tiffne ethod with ixed ode of nodl unknown fo the yeticl nlyi of ottionl hell. A finite eleent foultion fo the yeticl nlyi of ottionl hell with geoetic non-lineity w developed by Bony nd Tottenh [6] uing ixed finite eleent odel of cuved ottionl hell type. The pecibed geoety t evey nodl cicle copie the co-odinte nd eidionl lope ngle nd cuvtue. Newton-Rphon itetion i ued in olving the non-line yte of eqution. icul plte nd pheicl cp e ued exple to tet the foultion; good eult wee chieved in copion to the nlyticl olution. The conventionl finite tip ethod oiginlly developed by heung [7], ue continuouly diffeentible tigonoetic function eie in the long diection. Aftewd Abdunne [8] intoduced the ixed finite tip foultion nd peented coptive tudy between ixed nd tiffne foultion fo plte poble. In thi ppe the ixed foultion of two-nodl line cuved finite tip will be peented. In the ixed foultion, oent e included piy vible in ddition to diplceent coponent. The behvio of the tip i divided into two ction which e in-plne nd out-of-plne ction. The cuved hell i dicetized into nube of longitudinl tip eleent. The deivtion pocedue i iil to tht of the ixed finite eleent given by [6].. Mixed Foultion of uved Finite tip Eleent The cuved hell eleent i dicetized into et of finite tip eleent nd ech tip eleent conit of two nodl line hown in Figue. Ech nodl line h fou degee of feedo with totl of eight degee of feedo pe ech eleent. The nodl degee of feedo e coniting of the diplceent u, v, nd w in the x, y nd z diection epectively, well the oent M bout y-xi. Detil of the eleent unknown on the nodl line e hown in Figue. The geoety of cuved eleent tken fo the cuved ing eleent which w peented by [6] i hown in Figue. 84

2 z IJIET - Intentionl Jounl of Innovtive cience, Engineeing & Technology, Vol. 4 Iue 5, My IN (Online) Ipct Fcto (6) 5.64 y cuved tip with diplceent u, v nd w, nd nol oent M nodl pete, i hown in Figue 4. w M u w u x Figue : Dicetized hell into et of eleent. u M w v v Figue 4: Nodl unknown of the eleent A uitble hpe function fo the cuved eleent cn be witten : nd l l Figue : Nodl unknown of the eleent. Fo the pecil ce of both end e iply uppoted, the function (hpe ode) nd it deivtive tht tify the end condition in the y diection e given by [7] : in( k y) ' k co( k y) '' k in( k y) π k ;,,,..., Whee i the tip length. (). tin-diplceent Reltionhip. Model Function Figue : Geoety of the cuved eleent. The ixed foultion of two-nodl line cuved finite tip will be peented heefte. A typicl The tin fo plne te poble e given : u w ε x ε v y u v γ + () xy 85

3 IJIET - Intentionl Jounl of Innovtive cience, Engineeing & Technology, Vol. 4 Iue 5, My 7. Tnvee he-moent Reltionhip The tnvee he-oent eltionhip e given : M x M xy Q x M y M xy Q y ().4 he tin-diplceent Reltionhip Once the diplceent function i vilble it i eltively iple tte to obtin the cuvtue by pefoing the ppopite diffeentition. u w χ x + w χ y u w χ xy + (4).5 Vitionl Pinciple The functionl ued by Eli [4] nd extended by Bony nd Tottenh [6] fo the hell of evolution which w pplied to line nlyi of xiyetic hell of evolution i given by Eq. (5) : π ( ωuε ωσ ωσ + ωγ + ωu ) d dy + ψ y (5) Whee ω uε : i the tin enegy of the in-plne tee. ω σ : i the copleenty enegy of the te couple. ω σ : i the copleenty enegy of the tnvee he tin. ω γ : i the wok done by the tnvee he. ω u : i the wok done by the pplied lod. ψ : ψ(δ, M) i function of the vible δ nd M giving the wok done t the boundie whee δ nd/o M e pecified. The enegie in the integnd of Eq. (5) e elted to the tin, diplceent, intenl foce, nd pplied lod by: IN (Online) Ipct Fcto (6) T ωuε ε knε T ωσ M f xm T ωσ Q fγ Q M T ωγ Q θ T ωu p δ (6) In thi foultion the fee nd independent vible e δ nd M, i.e. π(δ, M) π(u, v, w, M x, M y nd M z ) With the fee nd independent vible efeed to thei nodl vlue u nd hown in Figue 4, nd uing line expnion in te of the eidionl cuve, Eq. (5) develop into: ( ) ( ) u T F T π δ, M π u, knf ddy u T T N T H T f xn f y H T H T + I ddy ddy ddy T T q ( N N ) ddy u +ψy (7) Whee the vlue of the pete in Eq. (7) e given by [9] : F ð N, H ð N nd I ð N (8) Whee; N. J. Ө u 86

4 IJIET - Intentionl Jounl of Innovtive cience, Engineeing & Technology, Vol. 4 Iue 5, My 7 IN (Online) Ipct Fcto (6) π u T T ku k T b k + l T kcu q T k pu + A + B (9) Whee l T k F k nf d dy l k T b N f xn d dy l T k H fvh d dy l T kc H I d dy l k T p N N d dy () To iniize the functionl, it ppe in Eq. (9), fit vition i conideed, with epect to the two yte of nodl vlue u nd, which poduce et of line eqution which cn be olved fo the nodl unknown (δ). π k T k T u k p q u c π k k + k b c u () () Eqution () nd () cn be witten in tix fo : The function,,, Ỳ, nd k e defined peviouly in ection.. Afte ubtituting the vlue of Ỳ, J nd Ө into N : N k k Detil of the tice F, H, nd I in Eq. (8) cn be found in Ref. [9[. The tice inide ech integnd e ll independent of the nodl vible, nd dependent only on the geoeticl nd eltic popetie of ech eleent. Eq. (7) cn be witten in oe concie fo : k δ F Detil of the k tix, δ nd F vecto e defined in Ref. [9]. In ode to clculte the integl of Eq. (), the hell i eplced by et of cuved tip eleent hown peviouly in the Figue. The ctul geoety t ech nodl cicle (co-odinte, lope, nd eidionl cuvtue) i ued to evlute the contnt of the fifth ode polynoil igned fo the ubtitute cuve, which i epeented by the non-dienionl locl co-odinte of the eleent (η, ζ). The polynoil i given by Eq. () while the locl co-odinte of the cuved eleent e hown in Figue 5. η ζ + ζ + ζ + 4ζ + 5ζ () 87

5 η IJIET - Intentionl Jounl of Innovtive cience, Engineeing & Technology, Vol. 4 Iue 5, My 7 IN (Online) Ipct Fcto (6) Reult nd Dicuion ζ ζ L Figue 5: Locl nolized coodinte of n eleent The geoeticl vible of the eleent e deteined polynoil in te of thee nolized co-odinte nd thei deivtive (η nd η ). Refeing to the Figue, the geoeticl vible e; i the c length of the eleent, x nd z epeent the globl coodinte of the tip, R i the diu of cuvtue of the cuved tip, R i the cicufeentil diu of cuvtue, φ i the lope ngle of the cuved tip, nd β i the lope ngle of the ubtitute cuve with epect to it locl coodinte. Fo Figue : tn co + + β β η η + η 4... (4) 8 L d dζ L η + dζ co β L η + η 4... dζ (5) 8 Whee L i the chod length of the eleent. ζ The vlidity of the Mixed foultion of uved Finite tip eleent () nd the ccucy of eult e deontted by one nueicl exple of cylindicl hell with iply uppoted t the cuved edge (longitudinl diection) nd fee long the tight edge (tnvee diection) nd it i ubjected to loding of q 9 Ib/ft. The thickne of the hell i inche nd the diu of cuvtue i 5 feet while the length of the tight edge i 5 feet. Detil of the hell geoety nd teil popetie e given below in Figue 6. The eult obtined fo the () wee coped with the hllow hell Theoy () olution peented by codeli []. E x 6 pi, ν., h in., R 5 ft., 5 ft., ϕ i - 4 o nd ϕ f 4 o ( L in θ ) ζ ( coθ ) η x x + (6) + L Whee x i the ditnce co-odinte of node (), θ i the ngle between the eleent chod nd the longitudinl xi. x coφ in( θ + β ) co β inθ + coθ η x { ( ) } { inθ + ( coθ ) η } η + η 4... (7) 8 { coθ ( θ ) η } inφ co β in { coθ ( inθ ) η } η + η 4... (8) 8 The c length i deteined by integting (d) nd the c length of the eleent (l) i evluted follow: l + L 4 η η... dζ d 8 (9) Figue 6: Detil of the hell geoety nd teil popetie. onvegence i invetigted by chnging the nube of tip eleent (NE). The cylindicl hell i dicetized into ix, eight nd ten longitudinl tip eleent while the hpe ode ein equl to one. Due to yety, only the eult of the hlf hell e peented. The eult of the veticl diplceent (w) nd the ltel oent (M x ) obtined by () uing diffeent tip eleent e coped with the () olution nd uized in Tble, nd. The diffeence in eult between () nd () olution e deontted by in ll the Tble. 88

6 IJIET - Intentionl Jounl of Innovtive cience, Engineeing & Technology, Vol. 4 Iue 5, My 7 Tble : Veticl diplceent nd ltel oent (NE 6). Veticl Ltel Moent Diplceent, w φ M x x -5 (in) (Ib.ft/ft) (7%) (7%) (49%) (4%) φ IN (Online) Ipct Fcto (6) (9%) Tble : Veticl diplceent nd ltel oent (NE 8). Veticl Ltel Moent Diplceent, w M x x -5 (in) (Ib.ft/ft) (.%) (.7%).547 (.%) (5%) 9. (5%).6 (9%) 4. (8%) Tble : Veticl diplceent nd ltel oent (NE ). Veticl Ltel Moent Diplceent, w φ M x x -5 (in) (Ib.ft/ft.) (.5%).69 (.5%).49 (.4%).45 (.%) -.8 (.7%) (.7%) (5%).8. (7%).4 (5%).8 (5%) A we cn ee fo the eult in Tble, nd, when the hell i odeled by 6 longitudinl tip eleent (i.e. NE 6), the veticl diplceent obtined by () e highe thn thoe clculted by () olution. The xiu diffeence in the eult of veticl diplceent i 49%. The eult convege quickly when oe tip eleent (i.e. NE 8 nd ) e ued, whee the xiu diffeence in the eult of veticl diplceent becoe.7%. A light ipoveent in the eult of the veticl diplceent cn be obeved when the nube of the longitudinl tip eleent e 8 nd. The xiu diffeence in the eult of the ltel oent i high fo the eh with 6 tip eleent, i.e. 4%. The oent convege lightly to 5% nd % when the nube of tip eleent i inceed to 8 nd epectively. The convegence of the xiu oent t the cente of the hell i ipoved fo 9% to 5%. Howeve, the vlue of the oent inceed to 5% highe thn the () olution t the tip ne the longitudinl fee edge. iil obevtion cn be de fo the eh with tip eleent, whee the diffeence in thi eh cn be high 5% in the tip ne the longitudinl fee edge (i.e. ϕ o ). Fo thi pticul exple, the eult of the oent e conveged they oved wy fo the fee edge. 4. oncluion The nueicl exple how, in genel, the vlidity of the ixed finite tip foultion to nlyze cylindicl hell. The eult of the veticl diplceent nd oent e found to be in geeent when coped with the hllow hell theoy olution. Refeence [] Henn, L. R., Finite eleent bending nlyi of plte, Poc. AE, J. Eng. Mech. Div., vol. 9, EM5, pp. -6, 967. [] onno, J., Mixed odel fo hell, Finite eleent technique in tuctul echnic, Tottenh nd Bebbi. Univeity of outhpton, U. K., 97. [] Pto.., hell finite eleent ethod vi Reine' pinciple, Int. J. olid tuctue, vol.5, pp. 9-, 969. [4] Eli, Z., Mixed finite eleent ethod fo xiyetic hell, Int. J. Nu. Method Eng., vol. 4, pp. 6-77, 97. [5] Gould, P. nd en,.k., Refined ixed finite eleent fo hell of evolution, Poc. Thid onf. On tix ethod in tuctul echnic, Wight-Ptteon. AFB, Ohio, 97. [6] Bony. nd Tottenh H., The nlyi of ottionl hell uing cuved ing eleent nd ixed vitionl 89

7 IJIET - Intentionl Jounl of Innovtive cience, Engineeing & Technology, Vol. 4 Iue 5, My 7 foultion, Int. J. Nu. Meth. Eng., vol., 978, pp [7] heung. K., Finite tip Method in tuctul Anlyi, t ed., Pegu Pe, 976. [8] Abdunne M. oune, Finite tip Method A coptive tudy between the tiffne nd the ixed ppoche, M.c. Thei, Al-Fth Univeity, Liby, ping 996. [9] Tuhi A., Anlyi of ylindicl hell Uing Mixed Foultion of uved Finite tip Eleent, M. c. Thei, Univeity of Tipoli, 7. []codeli, A. nd Lo, K., opute Anlyi of ylindicl hell, J. A. onc., Int., 6, No. 5, My 964. IN (Online) Ipct Fcto (6)