The Deformation of Cylindrical Shells Subjected to Radial Loads Using Mixed Formulation and Analytic Solutions

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1 Uvesal Joual of Mechacal Egeeg (4): 8-3, 03 DOI: 0.389/ujme The Defomato of Cyldcal Shells Subjected to Radal Loads Usg Mxed Fomulato ad Aalytc Solutos Lusa R. Maduea,*, Elza M. M. Foseca, Facsco Q. Melo 3 Uvesty of Poto, Faculty of Egeeg, Depatmet of Mechacal Egeeg, Potugal Polytechc Isttute of Bagaça, Depatmet of Appled Mechacs, Potugal 3 Uvesty of Aveo, Depatmet of Mechacal Egeeg, Potugal *Coespodg Autho: lusa.maduea@fe.up.pt Copyght 03 Hozo Reseach Publshg All ghts eseved. Abstact The objectve of ths wok s to cotbute wth a smple ad elable umecal tool fo the stess aalyss of cyldcal vessels subjected to geealzed foces usg a mxed fomulato. Vaatoal techques coupled wth fuctoal aalyss ae used to obta a optmzed soluto fo the shell dsplacemet ad futhe stess feld evaluato usg a combato of ukow aalytc fuctos wth Foue expasos. A lage cyldcal shell subjected to pchg loads s cosdeed. These elemets ae teded to povde accuate modellg of the tally ccula ppes espose. Because of ths behavou, the bed s coss-secto abados ts ogal oudess, tug to a oval o occula cofguato. I addto, the tally plae coss-secto, teds to defom out of ts ow plae. These two defomato pattes ae temed ovalzato ad wapg, espectvely. I ths wok the esults fo the adal dsplacemet ad secto ovalzato ae aalysed whee the soluto has sx tems fo a acceptable accuacy. The tasvese dsplacemet pesets mpotat depedece o the shell thckess vs adus, whee the case of th shells the ovalzato s estcted to a local aea ad ths s the case aalysed. The poposed method leads to accuate esults wth low complex put data. The coclusos of ths wok ae that the defto of the load system ad bouday codtos ae easly pocessed as the method has pe-defed possbltes fo each load case o edge bouday codtos. A aalytc soluto s obtaed ad a low umbe of tems the Foue sees show good accuacy as ca be see by a compaso wth fte elemet methods. Keywods Ppg Egeeg; Foue Sees; System Of Dffeetal Equatos; Bouday Codtos. Itoducto The soluto hee poposed deals wth the combato of ukow aalytc fuctos wth Foue expasos, whee the fome deped o the axal shell coodate ad the tgoometc tems ae depedet upo the cylde ccumfeetal pola agle []. Wth ths fomulato ad the evaluato of the total eegy a system of oday dffeetal equatos s obtaed ad solved whee the soluto s aalytc afte evaluato of egevalues ad egevectos. The bouday codtos ae the used to acheve the fal soluto. The cyldcal shell defomatos clude axal, ccumfeetal ad shea membae tems ε, ε θθ, ad γ. I ths fomulato the shell s assumed to be ccumfeetally extesble, so ε θθ =0. Due to the shell elastcty the defto of the dsplacemets cyldcal th ppes leads to a defomato feld ogatg teal foces, whch fo the geomety cosdeed, clude the membae axal foce N, the ccumfeetal foce N θθ ad the shea foce N. ad also the bedg momets N, N θθ ad the twst momet N.[].They ae elated to the defomatos by the elastcty matx as show by Eq.(): A N ε 0 S N γ 0 0 D 0 0 Mθθ = kθθ D 0 M k ν M D k The costats A, S, ad D Eq.() ae gve by: 3 Eh Eh Eh A= S = D= ν ( ν) ( ν ) whee E ad ν ae, espectvely, the mateal Youg s modulus ad Posso s ato whle h s the ppe thckess (assumed ufom). The tems peseted Eq.() togethe wth teal foces complete the soluto ukows. The dsplacemet feld cossts of the axal, the ccumfeetal ad the tasvese shell dsplacemets u, v ad w espectvely. These dsplacemets esult fom the supeposto of the so-called beam type wth the oes assged to the tasvese secto dstoto. The fst goup esults fom the bedg of a ppe, () ()

2 Uvesal Joual of Mechacal Egeeg (4): 8-3, 03 9 modelled ude cuet assumptos of a geometcally uchaged secto, oly udegog gd body dsplacemet o otatos. Howeve they ae able to geeate axal defomatos alog the ppe axs. The secod goup of defomatos s the most mpotat whe adal cocetate o fte aea dstbuted loads ae cosdeed. Ths last type of dsplacemets cosdes that the tasvese secto ovalzes ad waps. Fo the same esultg load, the magtude of the adal dsplacemets compaed wth the beam type oe s much hghe; beg possble to eglect these last oes f compaed wth the local shell defomato as metoed. The shell dsplacemets fom adal loads ae appoached by tgoometc expasos [,3,4] ad ae defed Eq.(3): ux (, ) = θ θ b( x) cos( θ) θ = = a ( x) vx (, θ) = s( θ ) (3) θ wx (, θ) = a ( x)cos( θ) = The fuctos a (x) ad b (x) ae the ampltudes fo the ovalzato ad wapg dsplacemets [4]. All expasos stat wth dex = oce the ode oe tem efes to the udefomed beam type dsplacemet [,]. These tems ae cluded the Foue θ-expasos, but they ae oly x-depedet fuctos. θ s equal to 6 the studed example. The defomato eegy U s a fuctoal U(u, v, w, N N, M θθ M, M ), fo whch the mmum s obtaed pefomg a weak fomulato wth Foue sees ampltude paamete vaatos. I ths poblem the tems used fo evaluato of the total eegy ae peseted Eq.(4) as suggested by [4]. = D( ν ) k Dk dω U = Aε Sγ Dkθθ 7 ( ) [ ( ( ) cos ) ( ) x Ab x S a b ( s = θ x θ) ( D ) 4 ( a ( ) cos x θ ) D( ν ) ( a ( ) s ) ( ( )) cos ] x θ Da x θ dω whee d Ω= dθdx ad (the shell axal cuvatue) k = a ( x) cos θ (the shell suface twst) (4) (5) k = a ( x) s x θ θ (the ccumfeetal cuvatue) k = (( ) ) a ( x) cos θ (6) θθ (the axal membae sta) ε (the membae shea sta) γ = b ( x) cos θ a ( x) = b ( ) s x θ Coespodg to a mmum of the fuctoal U [5], sepaate vaatos fo each of the ukows a (x), b (x) gve the followg equatos: U b x U a x U δb ( x) δb ( x) = 0 ( ) b ( x) U δa ( x) δa ( x) = 0 ( ) a ( x) The soluto s ow wtte wth totally assumed dsplacemets ad ts devatves. Followg ths pocedue, a system of dffeetal equatos s obtaed. The dffeetal equatos fo the staght ppe, whee the defomato s caused by a foce F appled o the suface of the ppe ae descbed Eq.(8) ad solved by evaluato of egevalues ad egevectos [6]: π a v ( ) π D π 3 D ( x) = a ( x) Sb ( x) π π ν. Example S D( ) a ( x) F πs π Ab ( x) = b ( x) Sa ( x) π (7) (8.) (8.) A steel cylde wth gd ed daphagms s aalysed. The cylde s subjected to opposte loads (pchg mode) appled the adal decto as show Fg.. Ths s a bechmak case study fequetly aalysed by seveal authos [7,8]. The example of a pched cylde wth the peset methodology s compaed wth the wok poposed by [7,9]. Smultaeously, umecal smulato usg ANSYS fte elemet pogam wll be peseted. I ths example legths ae ches ad load pouds as

3 30 The Defomato of Cyldcal Shells Subjected to Radal Loads Usg Mxed Fomulato ad Aalytc Solutos epoted [7,9]. Fgue shows the case study ad all cosdeed paametes. The mateal popetes ad the geometc paametes ae the followg (uts ae accodg to the efeeces fo compaso): 6 E = 3 0 ps ν = 0. 3 =300ch L=600ch h=3 ch Supeposg the sx solutos a (x),,a 7 (x) (Fg.3) obtaed fom solvg Eq.(8) effectve soluto fo the defomato s gve. The vetcal dsplacemet alog the legth of the cylde obtaed wth ths mxed fomulato s show Fg.4. The ext table (Table ) shows that these esults obtaed wth the mxed fomulato match the oes obtaed wth fte elemet methods by [7,0]. Foseca et al [7] Table. Vetcal dsplacemet (ch) Dvok ad Bathe [0] Mxed fomulato -.836E E E-5.. The fte Elemet Method Applcato Fgue. Geomety of a two-adal loads pched cylde.. The Vaatoal Method Applcato Equato 9 epesets the load F submtted fo ths type of aalyss ad toduced the dffeetal system gve by Eq.(8). A Foue s expaso s cosdeed to smulate the pot load. It cotas odd tems oly ad fo a estmated quadatc eo [6] of 6%, 59 tems ae used. F(x) = 59 F 86π 34π πx cos cos s = π 600 (9) Fgue shows the load cosdeed o the suface as a Foue expaso. The pched cylde poblem s useful to assess the accuacy of a stuctual shell model ude bedg ad membae states. Ths study s a bechmak test the eseach of the pefomace of cuved beam o shell elemets [0]. The fte elemet method was used fo ths example. Results ae compaed wth the pevous soluto obtaed wth the fuctoal aalyss. Dsplacemets obtaed usg ANSYS pogam, ae show the ext fgues. A fte Shell93 elemet was used, wth 8 odes ad 6 degees of feedom pe each ode: taslatos the odal x, y, ad z dectos ad otatos about the odal x, y, ad z-axes. Shell93 s patculaly well suted to model cuved shells. The defomato shapes ae quadatc both -plae dectos. Due the geomety ad load symmety a oe-eghth of the cylde s modelled ad all complete cylde, beg the fte elemet mesh used Fg.5. Fo the oe-eghth symmety model, oe-fouth of the load s appled. Fgue. Appled load Fgue 3. Sepaate solutos coespodg to each ovalzato mode a ( x ),a ( x ),...,a ( x ) 3 7 Fgue 4. a( x) a3( x)... a7( x) fo ppe defomato

4 Uvesal Joual of Mechacal Egeeg (4): 8-3, 03 3 (a) The efeece dsplacemet, as epoted by [0,] ad the exact deflecto at the load pot s equal to 0.848E-4ch, usg the elato:. The same value was obtaed wth ANSYS pogam ad the vaatoal method, both peseted hee. Ths s kow as a bechmak esult whee thee s a domat bedg behavou the th shell lmt. The vetcal dsplacemet alog the legth of the cylde s show Fg.5 whee two dffeet fte elemet appoaches (ths oe wth ANSYS ad as publshed by [7]) ad the mxed fomulato ae compaed. Results ae smla to the oes obtaed by [9]. Fgue 5. Fte elemet mesh used fo a eghth (a) ad total cylde legth (b) Fgue 6 shows the postpocesso obtaed fo the vetcal dsplacemet esults usg ANSYS pogam. (a) (b) Fgue 7. Vetcal dsplacemet alog the cylde legth (as [7], peseted solutos ANSYS ad mxed fomulato) 3. Coclusos The peseted umecal tools have show a good ageemet altogethe whe aalysg a ecuet poblem ppg egeeg, the stuctual behavou of th walled cyldcal ppes subjected to adal loads. The vaatoal soluto based o fuctoal aalyss, though tally demadg some wok wth closed fom aalytc methods, has the advatage of leadg to a qute smple computatoal soluto by compute mplemetato ad s ftted to othe bouday codtos poblems. As the poposed soluto deals wth aalytcal fomulatos hghe ode devatves ae possble gvg cotuous ad ealstc esults whle fte elemets ths s ot possble. Moeove, the esults match cosstetly wth coespodg wok the lteatue usg fte elemet techques ad commecal codes. The soluto accuacy ca be mpoved by easly expadg the umbe of tgoometc elemets cosdeed the Foue expasos cluded the peseted soluto. Ackowledgemets The authos wsh to thak Fudação paa a Cêca e Tecologa fo the facal suppot leadg to ths publcato. (b) Fgue 6. Vetcal dsplacemet: a) oe eghth; b) total cylde legth REFERENCES

5 3 The Defomato of Cyldcal Shells Subjected to Radal Loads Usg Mxed Fomulato ad Aalytc Solutos [] O. C. Zekewcz, The Fte Elemet Method, 3d Ed McGaw-Hll Co. NY, 979 [] E. Oñate, Calculo de Estuctuas po el Metodo de Elemetos Ftos CIMNE Baeceloa, Spa [3] F. J. M. Q. Melo, P. M. S. T. Casto, The lea elastc stess aalyss of cuved ppes ude geealzed loads usg a educed tegato fte g elemet, Joual of Sta Aalyss, Vol.3, Nº, 47-59, 997 [4] A. Mllad, R. Roche, Elemetay Solutos fo the Popagato of Ovalzato alog Staght Ppes ad Elbows. It J Pessue Vessels Ppg, Vol.6, No., 0-9, 984 [5] T. H. Rchads, Eegy Methods Sta Aalyss, Ells-Howood, 98 [6] C. Wyle, L. Baet. Advaced Egeeg Mathematcs, 6th ed McGaw-Hll, 995 [7] E. Foseca, F. Melo, C. A. Olvea, Tgoometc fucto used to fomulate a mult-odal fte tubula elemet, Mech Res Comm, Vol.34, No., 54-6, 007 [8] R. H. Mac Neal, R.L. Hade, A poposed stadad set of poblems to test fte elemet accuacy, Joual of Fte Elemets Aalyss ad Desg, Vol., 3-0, 985 [9] F. Cak, M. Otz, P. Schode, Subdvso sufaces: a ew paadgm fo th shell fte elemet aalyss, It J Nume Meth Eg- Vol.47, No., , 000 [0] E. Dvok, K. J. Bathe, A cotuum mechacs based fou-ode shell elemet fo geeal o-lea aalyss, Eg Comput, Vol., 77-88, 984 [] R. Kouha, R. Stebeg, A smple lea ocofomg shell elemet, IASS-IACM 000 Fouth Iteatoal Colloquum o Computatoal of Shell Spatal Stuctues, Geece, 000.