Development of a Simplified Theoretical Model for Dynamic Burst Time And Pressure of a Cylindrical Shell

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1 Te Open Ocean Engineeing Jounal 9 6 Open Access Developmen of a Simplified Teoeical Model fo Dynamic Bus Time And essue of a Cylindical Sell Cunjiang Ceng and G E Oo Widea Bjoksen Reseac Lab BIT 7 INC Madison Wisconsin Cene fo Joining and Manufacuing Assembly Mauee Univesiy Milwaukee Wisconsin Absac: Te objec of is sudy is o deemine e soem bus pessue and ime of meal cylindes unde soem dynamic loading condiions A simplified eoeical model o calculae ese dynamic bus ime and pessue of cylindical sells as been developed and e esuls ae compaed wi finie elemen analysis FEA esuls via e use of e LSDYNA code [] Based on e ageemen beween e wo esuls i can be concluded a a popely fomulaed simplified eoeical model can be employed wi sufficien accuacy o deemine e soem dynamic bus pessues of meal cylindes Keywods: Cylindical sell ipeline Dynamic bus pessue Dynamic bus ime INTRODUCTION Te cylindical sell is a basic sucue a is used all ove e wold in e sipping peocemical powe and aeospace indusies as well as in oes Envionmenal polluion popey damage pesonal injuy and even dea can occu wen a pessuied cylindical sell buss Teefoe an accuae deeminaion of e bus pessue of cylindical sells is a ciical paamee in e design pocess Consideable wok as been caied ou o deemine e saic bus pessue using eie analyical soluions expeimenal sudies o finie elemen analysis Dynamic loading suc as wae o seam amme penomena is also vey common in a pipeline o piping In e 95 s Coope [] developed an analyical euaion o pedic e saic bus pessue fo cylindes made of an isoopic ducile maeial Tis euaion povided e desied elaionsip beween e bus pessue maeial caaceisics oiginal dimensions and ulimae ensile seng of e maeial A e same ime Svensson [] deived a soluion of e bus pessue fo an abiay ick endcapped pipe based on e vonmises yield cieion Tadmo e al [4] developed an analyical expession of e bus pessue of mulilayeed cylindes Tey pefomed a lage sain analysis aking ino consideaion e elasicplasic defomaion wi e Hill yield funcion and abiay adening An oveall effecive modulus was used o deemine e onse of busing and ey en deived e elaions fo inwalled cylindes wi e neglec of e elasic sains Kleve [5] pesened an analyical model o deemine e bus seng of e inwall uncooded and cooded pipelines Te model esuls compaed well wi ose of an uncooded pipe es Sewa e al [6] eexamined e fundamenal elaionsips a goven e euilibium and sabiliy Addess coespondence o is auo a e Bjoksen Reseac Lab BIT 7 INC Madison Wisconsin; Tel: 68477; Fax: ; cengcunjiang@gmailcom geowidea@muedu 87485X9 of a pipe unde inceasing inenal pessue Bo Tesca s and vonmises yield cieia wee consideed Te auos concluded a ei analyical model is suiable fo bo in and ickwall cases Adaci e al [7] poposed a maemaical soluion fo pipe elasic sess based on e assumpion of onedimensional popagaion of e pessue and e uasisaic defomaion of e pipe o calculae e elasic sesses in a ube due o wae amme Leisea e al [8] pesened bo an analyical soluion and a finie elemen analysis o deemine e dynamic pipe sess due o e vibaions ceaed by wae amme wi e assumpions a e maeial is linealy elasic and e wae amme load as been limied o a sep pessue incease insead of a amped pessue incease He concluded a e maximum dynamic sess is wice a of e saic sess wen damping is ignoed Neie Adaci no Leisea consideed dynamic plasic defomaion Only a limied amoun wok as been publised o deemine e dynamic elasic sess in pipes bu no lieaue is available on e dynamic bus pessue of cylindical sells SIMLIFIED THEORETICAL MODEL To make easily undesand e effec of facos suc as e pessue vs ime funcion and e diamee vs ickness aio on e bus pessue as well as examine e consisency of e FEA esuls a eoeical analysis meod based on a simplified inwalled cylinde geomey is developed Conside a inwalled axisymmeic cylinde unde unifom dynamic inenal pessue Te geomey and beavio of is cylinde can be descibed by use of cylindical coodinaes Assume plane sain beavio Assume fue a e oop sess is e dominan sess and a i can be aken o be consan ougou e ickness of e cylinde Wi efeence o Fig e summaion of foces in e adial diecion fo an elemen of volume 9 Benam Open d dl is

2 Te Open Ocean Engineeing Jounal 9 Volume Ceng and Widea dθ dl 8 o σ dθ 9 Fig Simplified Geomey and Loading Condiion of Cylinde d dl d dl sin F Assume a e mass of is small volume of e cylinde will emain consan; i can eefoe be expessed as m d dl In e above is e insananeous wall ickness e iniial wall ickness dl e incemen of cylinde leng e densiy e insananeous inside adius e iniial inside adius and d e small incemen in ac angle Define e adial acceleaion in a d d a as Wen d is vey small e euaion 4 can be esablised d d sin Because e cylinde is assumed o be inwalled e acceleaion can be aken o be consan ougou e ickness Te euaion of moion in e adical diecion fo e volume elemen can us be wien as d d dl dl d dl 5 Te assumpion of volume consancy duing e defomaion pocess saisfies e following euaion: wic yields dl Euaion 5 can now be wien as 4 dl 6 7 Fo a vey small incemen of e adial sain one can wie d d One can inegae bo side of e euaion and obain d ln ln wic yields e Wi a consideaion of e exisence of sain ae effecs e following euaion can be developed o fi e maeial daa elaing sess sain and sain ae cuves by using MaCAD [] sofwae: n k D Hee is e effecive vonmises sess and e effecive vonmises sain Fue k and n ae consans fo a paicula maeial Te assumpion of volume consancy duing e defomaion also leads o e following euaion: 4 Fo plane sain and us 5 Te effecive vonmises sain is defined by e following euaion: 6 Fo plane sain i becomes 7

3 Developmen of a Simplified Teoeical Model Te Open Ocean Engineeing Jounal 9 Volume Fo a inwalled cylinde adius is muc lage an e ickness and e adial sess is small compaed o e oe wo sesses Teefoe one can assume 8 Also 9 Te effecive vonmises sess is defined by e following euaion: Wi e assumpions made is expession can be solved fo and becomes 4 k n D Euaing e euaion o e euaion 9 one obains Subsiuing euaions 7 7 and 9 ino e euaion esuls in: Hee ln ln Te mos geneal appoac fo e soluion of e dynamic esponse of sucual sysems is e diec numeical inegaion of e dynamic euilibium euaions Euaion is a vey complicaed diffeenial euaion based on e vaiable I canno be solved diecly by using e egula cenal diffeence meod o e Newmak meod [] Hee a simila explici numeical meod is developed based on e geneal cenal diffeence meod Tus 4 Combine e expessions of 4 wi e euaion o obain e euaion 5: eoeically one can solve e euaion 5 using e iniial condiions of and Hee is e known iniial inside adius Radius can be solved fom following euaions: D k n 4 4 k n D

4 4 Te Open Ocean Engineeing Jounal 9 Volume Ceng and Widea p 4 k ln n D ln p Combining euaions 6 and 7 wi e euaion 5 one obains euaion 8: Table ASTM A6B Maeial Descipion Cemical Analysis C Mn S Si Ω Condiion Seamless oolled In e sample calculaion e cylindical sell as an ouside diamee of 4 inces and a wall ickness of 65 4 k ln n D ln 8 In euaion 8 e only unknown vaiable is Teefoe eoeically i is solvable Howeve due o e ig nonlineaiy i is difficul o solve is euaion manually Euaion 8 and en euaion 5 can be solved using a commecial maemaics sofwae suc as MaCAD [] To make sue a e euaion 5 is always solvable e incemenal ime sep sould be vey small Howeve wen e ime sep is smalle an e minimum pecision of e MaCAD sofwae e soluion will be unsable A sample calculaion is pefomed by using MaCAD Te maeial is ASTM6 seel [] Te maeial descipion is lised in Table Te sess sain and sain ae cuves ae sown in Fig By fiing e sess sain and sain ae cuves one can ge e maeial coefficiens of k n and Te pessue vs ime cuve is aken as e amped cuve sown in Fig Hee e ime d is pessue pulse duaion Sess E9 a Sess Sain Cuves a Diffeen Sain Raes Sain mmmm Saic Es Sain Rae 4Es Sain Rae 48Es Sain Rae Fig Sesssain Cuve of A6B Seel a Diffeen Sain Raes

Developmen of a Simplified Teoeical Model Te Open Ocean Engineeing Jounal 9 Volume 5 d d Fig essue psi vs Time second Cuve Fixed Symmeic Dynamic essue Inside Fig 4 FEA Model Loading and Bounday

5 Developmen of a Simplified Teoeical Model Te Open Ocean Engineeing Jounal 9 Volume 5 d d Fig essue psi vs Time second Cuve Fixed Symmeic Dynamic essue Inside Fig 4 FEA Model Loading and Bounday Condiions inces Te esuling bus ime is abou milliseconds wen e peak pessue is psi To confim ese calculaions a finie elemen analysis FEA is caied ou using e dynamic compue code LS DYNA [] In developing e FEA mes e eig node solid elemen is employed and e guidelines saed in efeence [4] ae followed In paicula e mes as ee elemens acoss e ickness see Fig 5 and e elemen sie is 97 in Te bounday condiions and loading see Fig 4 ae as follows: One end is fixed e oe end is fee Dynamic Foce Inenal pessue is a funcion of ime see Fig wi e maximum ampliude aken as 5 8 Fig 5 FEA Mes 76 and 7 psi A dynamic foce eual o e pessue muliplied by e inside aea of e cylinde is added a e fee end Gaviy acs along e Y axis In e FEA simulaion e bus cieion is based on e maximum plasic sain a diffeen sain aes Once e plasic sain of an elemen eaces e maximum plasic sain fo a ceain sain ae e elemen is deleed fom e calculaion Te bus pessue and ime can be ecoded fom e oupu of e FEA model Moe deailed seup mes esuls and configuaions can be found in efeence [5] Table clealy sows a e bus pessue pediced by FEA is 8 psi wile a pediced by eoeical meod is 76 psi fo e cylinde unde consideaion Te bus imes ae close as well SUMMARY A simplified explici numeical meod o calculae e dynamic bus pessue and ime of a inwalled cylindical sell is developed A sample esul fom e use of is meod is compaed wi FEA esuls and i is sown a e pesen simplified eoeical meod can be used o accuaely deemine e bus pessue and ime of a inwalled cylindical sell Table Compaison of Euaion and FEA Resuls OD inc Tickness inc d ms eak essue psi FEA Bus Time ms Euaion Bus Time ms No bus No bus No bus

6 6 Te Open Ocean Engineeing Jounal 9 Volume Ceng and Widea REFERENCES [] LSDYNA Teoeical manual Livemoe Sofwae Tecnology Copoaion 998 [] WE Coope Te significance of e ensile es o pessue vessel design Welding Reseac Supplemen pp 4956 Jan 957 [] NL Svensson Bus pessue of cylindical and speical vessels Jounal of Applied Mecanics vol 5 No pp [4] EB Tadmo and D Duban lasic defomaion and bus of pessuied mulilayeed cylindes Jounal of essue Vessel Tecnology vol 7 pp 859 Feb 995 [5] FJ Kleve Bus seng of cooded pipe: flow sess evisied Twenyfou annual Offsoe Tecnology Confeence May 47 Houson Texas 99 [6] G Sewa and FJ Kleve An analyical model o pedic e bus capabiliy of pipelines Inenaional Confeence on Offsoe Mecanics and Acic Engineeing vol 5 Houson Texas ipeline Tecnology 994 [7] T Adaci S Ujiasi and H Masumoo Impulsive esponses of a cicula cylinde sell subjec o wae amme waves Jounal of essue Vessel Tecnology vol pp 575 Nov 99 [8] RA Leisea CA Young and E M Alfod Dynamic pipe sesses duing wae amme I finie elemen appoac Design and Analysis of iping Vessels and Componens ASME essue Vessels and iping Confeence Vancouve: Biis Columbia Canada Augus 59 [9] RA Leisea Dynamic pipe sesses duing wae amme II a vibaion analysis Design and analysis of piping vessels and componens ASME essue Vessels and iping Confeence Vancouve: Biis Columbia Canada Augus 59 [] RA Leisea Dynamic pipe sesses duing wae amme III complex sesses Design and analysis of piping vessels and componens ASME essue Vessels and iping Confeence Vancouve: Biis Columbia Canada Augus 59 [] MaCAD Use s Guide Masof Engineeing Educaion Inc Cambidge MA USA 5 [] NM Newmak A meod of compuaion fo sucual dynamics Jounal of Engineeing Mecanics Division ASCE EM vol 85 pp [] D eeson ipe wip dynamics: an expeimenal and analyical invesigaion D Disseaion Univesiy of Akon 98 [4] GEO Widea and L Xue Guidelines fo FEA fodeling of cylindeocylinde Inesecions WRC Bullein No 49 New Yok: Welding Reseac Council 4 [5] C Ceng and GEO Widea Dynamic bus pessue simulaion of cylindical sells in oceedings of e 8 ASME essue Vessels iping Confeence Cicago IL USA July 7 8 Received: June 8 Revised: Januay 9 Acceped: Januay 9 Ceng and Widea; Licensee Benam Open Tis is an open access aicle licensed unde e ems of e Ceaive Commons Aibuion NonCommecial License p:ceaivecommonsoglicensesbync wic pemis unesiced noncommecial use disibuion and epoducion in any medium povided e wok is popely cied

Heat Conduction Problem in a Thick Circular Plate and its Thermal Stresses due to Ramp Type Heating

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