CRITICAL VELOCITY OF FLUID CONVEYING PIPES RESTING ON TWO-PARAMETER FOUNDATION

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1 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam CRITICA EOCITY OF FUID CONEYING PIPES RESTING ON TWO-PARAMETER FOUNDATION Kamesaa Rao Chellapilla Pofesso Mechaical Egieeig Depame Muffaham Jah College of Egieeig & Techology Hydeabad Idia Phoe: cao_5@yahoo.com H. Simha Scieis Desig & Egieeig Divisio Idia Isiue of Chemical Techology Hydeabad Idia Phoe: simha@iice.og Absac Pipelies ae used eesively fo aspoaio of fluids. The velociy of he fluid i he pipelie impas eegy o he pipelie maig i o vibae. I is ell esablished fom published lieaue ha hee eiss a ciical velociy of he fluid ea hich he aual fequecy of he pipelie eds o zeo. This is he equied codiio fo buclig of he pipelie. ieaue abouds ih aalyses hich give ifomaio o he ifluece of bouday codiios o he sabiliy of fluid coveyig pipes. Hoeve much of hese sudies have bee caied ou fo pipelies esig o Wile ype foudaios. This has povided he moivaio fo sudyig he ifluece of o-wile ype foudaio o he ciical velociies of a fluid-coveyig pipe. The foudaio cosideed i his sudy is a o-paamee foudaio model such as he Pasea foudaio. Epessios ae deived fo he ciical flo velociy by uilizig Fouie seies ad Galei mehod fo hee simple bouday codiios amely: Pied-Pied Pied-Fied ad Fied-Fied. Resuls ae peseed fo vayig values of he foudaio siffess paamee ad ieesig coclusios ae da o he effec of he foudaio paamees o he ciical flo velociy of he pipelie. Ioducio The echology of aspoig fluids especially peoleum liquids hough log pipelies hich cove diffee ypes of eai has evolved ove he yeas. Iees i sudyig he dyamic behaviou of such fluid coveyig pipes as simulaed he ecessive asvese vibaios ee obseved ad subsequely aalyzed fis by Ashley ad Havilad i 95 [] ad lae by House i 95 []. House cosideed a simply suppoed beam model fo he pipelie ad aalyzed i usig a seies soluio appoach ad shoed he eisece of a ciical flo velociy fo a pipelie hich could cause buclig. I 955 og [3] sudied he ifluece of fied-fied ad fied-pied bouday codiios o he ciical velociy. I 966 Gegoy & Paidoussis [] peseed esuls o he dyamic behaviou of a caileveed pipe coveyig fluid. All he above sudies did o coside elasic suppo codiios. Whe a pipelie ess o a elasic medium such as a soil a model of he soil medium mus be icluded i he goveig diffeeial equaio. A vey commo sucual model of he soil medium is he Wile model i hich soil is epeseed by a seies of cosa siffess closely spaced liea spigs. I 97 Sei & Tobie [5] sudied he vibaios of a fluid coveyig pipe esig o a elasic foudaio. oai ad Koeci i 986 [6] sudied he ifluece of he elasic foudaio o he sabiliy of he pipelie. ae i 99 Demedia-Ivaova [7] ivesigaed he behaviou of a fluid coveyig pipe esig o a elasic foudaio ad obaied he ciical fluid velociy. I 993 Chay [8] peseed a deailed aalysis of fluid coveyig pipes esig o elasic foudaio. He also cosideed he ieia of he foudaio ad alalyzed he seismic espose of such pipelies. All hese sudies modeled he elasic foudaio as a Wile model. I his pape he o of Chay has bee eeded o sudy he ifluece of a o-paamee foudaio model o he ciical flo velociy. Resuls ae peseed fo vaious values of he foudaio siffess paamees.

2 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam Equaio of moio The diffeeial equaio of moio fo laeal displaceme of a uifom fluid coveyig pipe esig o a Wile ype elasic foudaio is give by [8]: I he above equaio E deoes he modulus of elasiciy I is he mome of ieia of he he pipe secio MmA is he oal mass of he pipe pe ui legh m is he mass of he pipe aloe pe ui legh A is he mass of he fluid pe ui legh v is he seady flo velociy ad is he siffess of he elasic medium pe ui legh of he pipe. I his equaio he elasic medium is modeled o he Wile ype foudaio. Folloig he mehod give by Paelides [9] The equaio of moio fo a fluid coveyig pipe esig o a o-paamee foudaio becomes: I equaio above epeses he addiioal paamee defiig he foudaio usually emed as he shea cosa of he foudaio. The model is sho i figue. Equaio is o solved fo hee simple bouday codiios. Pied-pied pipe The bouday codiios fo a pied-pied pipe ae Taig he soluio of equaio hich saisfies he bouday codiios 3 as Whee ω epeses he aual fequecy of he h mode of vibaio. Subsiuio of equaio i ad epadig i a Fouie seies e have a equaio of he fom: hee K is he siffess mai hose elemes ae eumeaed i [8] ad ill o be epeaed hee I is he ideiy mai ad a T {a a..a }. Reaiig he fis o ems of he above equaio ad seig he deemia equal o zeo e ge Av Av M - Av Av M cos si si si a a ω ω [ ]{ } 5 a MI K? Ω Ω β

3 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam 3 I equaio 6 he folloig o-dimesioal paamees have bee used: Whe he fluid velociy eaches a ceai value c he fudameal aual fequecy becomes zeo. Hece seig Ω i equaio 8 e obai: Solvig equaio 7 fo e obai he ciical flo velociy fo he pied-pied case. Pied-fied ad fied-fied pipe The bouday codiios fo a pied-fied pipe ae Ad hose fo a fied-fied pipe ae We assume he deflecio of he pipe o be of he fom I equaio R deoes he eal pa φ is a seies of beam eige-fucios ψ give by: I he above equaio is he fequecy paamee of he pipe ihou fluid flo hich is cosideed as a beam ad i s values ae [] ad fo he pied-fied case ad.73 ad fo he fied-fied case A v M M A ; ; ; 3... ; Ω ω β R i e ω φ σ σ ψ si sih cos cosh ; 3... si sih cos cosh

4 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam Subsiuig equaio i he equaio of moio gives φ Av iωav Mω φ φ Folloig he mehod give i [8] usig Galei s mehod ad miimizig he mea squae of he esidual ove he legh of he pipe e have he folloig equaios i. Fo he pied-fied case: CC CC [ C C ] [ ] 3 Fo he fied-fied case: CC [ C C ] [ ] I equaios 3 ad he cosas C ec. ae iegal values hich ae eumeaed i [8]. Resuls Table gives he umeical values of he ciical velociy paamee. The o foudaio paamees ae vaied fom.e-6 o.e. The fis value is equivale o o foudaio hile he las value epeses a vey siff foudaio. The esuls ae abulaed fo all he hee bouday codiios. I figues 3 & he ifluece of o he ciical velociy paamee of he pipe fo a pied-pied bouday codiio is sho fo vaious values of. The esuls fo he ohe o cases of piedfied ad fied-fied follo a simila ed. Figue 5 shos he vaiaio of he ciical velociy paamee ih fo vaious values of. Fially a compaiso of he idividual effecs of each of he o foudaio paamees he he ohe is equivale o zeo o he ciical velociy paamee is sho i figue 6. Coclusios I is see fom figue ha hee is o ay pecepible chage i he dyamic behaviou of he pipe uil he shea cosa of he o-paamee foudaio aes a value of.. The ciical velociy iceases slighly fo he value of of.. Fo a value of of. hee is a shap ump i he value of he ciical velociy paamee ad his ed coiues fo iceasig values of as sho i figues 3 ad. Aohe obsevaio fom hese plos is ha fo loe values of hee is a shap icease i he value of ciical velociy fo he Wile foudaio cosa values geae ha.. The ciical velociy does o seem o be effeced by he value of he Wile cosa fo highe values of. This is sho i aohe ay i figue 5. I ca be see ha he Wile cosa has lile effec o he ciical velociy especially fo values of geae ha abou..

5 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam I figue 6 a compaiso beee he elaive effecs of each of he o foudaio paamees is sho. The op cuve shos ha hee is a shap icease i he ciical velociy he hee is a pogessive icease i he value of beyod.. This cuve epeses he case hee is ea zeo. The boom cuve shos he vaiaio of ciical velociy ih he is ea zeo. I ca be obseved ha he ifluece of he shea cosa of he o-paamee foudaio is moe ha ha of he Wile cosa o he ciical velociy. Refeeces. Ashley H. ad Havilad G. 95 Bedig ibaios of a Pipe ie Coaiig Floig Fluid J. Appl. Mech. Tas. ASME Sepembe pp House G. W. 95 Bedig ibaios of a Pipe ie Coaiig Floig Fluid J. Appl. Mech. Tas. ASME Jue pp og J. R. H. 955 Epeimeal ad Theoeical Sudy of Tasvese ibaio of a Tube Coaiig Floig Fluid J. Appl. Mech. Tas. ASME Mach pp Gegoy R. W. ad Paidoussis M. P. 966 Usable Oscillaios of Tubula Caileves Coveyig Fluid Poc. Roy. Soc odo 93A pp Sei R. A. ad Tobie M. W. 97 ibaio of Pipes Coaiig Floig Fluids J. Appl. Mech. Tas. ASME Decembe pp oai I. Ad Koeci A. 986 The effec of a Elasic Foudaio ad of Dissipaive Foces o he Sabiliy of Fluid-Coveyig Pipes J. Soud ibaio 9 pp Demedia-Ivaova D. S. 99 Ciical Flo elociies of a Simply Suppoed Pipelie o a Elasic Foudaio J. Soud ibaio 57 pp Chay S. R. 993 ibaios of Pipes Resig o Soil Medium M.S. Thesis Idia Isiue of Sciece Bagaloe. 9. Paelides C. P. 99 Sabiliy of Colums o Bipaameic Foudaios Comp. S pp Rao S. S. 986 Mechaical ibaios Addiso Wesley Publishig Compay Massachuses p

6 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam Table alues of he ciical velociy paamee fo vaious values of g ad g fo he hee bouday codiios Pied-pied pipe Pied-fied Pipe Fied-fied pipe C C C.E-6.E E-6.E E-6.E E-5.E E-5.E E-5.E E.E E.E E.E E3.E E3.E E3.E E.E E.E E.E E-6.E E-6.E E-6.E E-5.E E-5.E E-5.E E.E E.E E.E E3.E E3.E E3.E E.E E.E E.E E-6.E.5753.E-6.E E-6.E E-5.E.5753.E-5.E E-5.E E.E E.E E.E 6.73.E3.E 8.69.E3.E 9.93.E3.E 9.96.E.E 7.6.E.E 7.53.E.E 6.35.E-6.E.887.E-6.E E-6.E.38.E-5.E.887.E-5.E E-5.E.38.E.E.9553.E.E.339.E.E.6375.E3.E E3.E 3.3.E3.E E.E E.E E.E E-6.E E-6.E E-6.E E-5.E E-5.E E-5.E E.E E.E E.E E3.E E3.E E3.E E.E E.E E.E E-6 5.E E-6 5.E E-6 5.E E-5 5.E E-5 5.E E-5 5.E E 5.E E 5.E E 5.E E3 5.E E3 5.E E3 5.E E 5.E E 5.E E 5.E E-6 8.E E-6 8.E E-6 8.E E-5 8.E E-5 8.E E-5 8.E E 8.E E 8.E E 8.E E3 8.E E3 8.E E3 8.E E 8.E E 8.E E 8.E E-6.E.93.E-6.E..E-6.E.76.E-5.E.93.E-5.E..E-5.E.76.E.E..E.E.9.E.E.77.E3.E.335.E3.E.376.E3.E.7.E.E.53.E.E.3.E.E.7 6

7 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam Figue Model of a pipe fluid coveyig pipe esig o a o-paamee foudaio 8 C 6.Ε6.Ε5.Ε.Ε3.Ε.Ε 8 6.Ε.Ε.E-6.E-5.E-.E-3.E-.E-.E.E.E.E3.E.E5 Fig. - Pied-pied pipe: aiaio of C ih fo vaious values of Figue Pied-pied pipe: aiaio of C ih g fo vaious values of g 7

8 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam 35 3 C 5.Ε.Ε3 5.E-.E-.E.E.E.E3.E.E5 Fig. 3 - Pied-pied pipe: aiaio of C ih fo vaious values of - Cod. Figue 3 Pied-pied pipe: aiaio of C ih g fo vaious values of g cod C Ε3 8.Ε3.Ε 75 7.E-.E-.E.E.E.E3.E.E5 g Fig. - Pied-pied pipe: aiaio of C g fo vaious values of g - Cod. Figue Pied-pied pipe: aiaio of C ih g fo vaious values of g cod. 8

9 Ieaioal Cofeece o Advaces i Sucual Dyamics ad is Applicaios ICASDA-5 Decembe isahapaam C 8 6.Ε6.Ε.Ε3.Ε.E-6.E-5.E-.E-3.E-.E-.E.E.E.E3.E.E5 g Figue 5 Pied-pied Fig. 5 - pipe: Pied-pied Ifluece pipe: Ifluece of g of g o he C fo vaious vaious values of values g of g 8.Ε6.Ε6 C 6.E-6.E-5.E-.E-3.E-.E-.E.E.E.E3.E.E5 g g Fig. 6 - Pied-pied pipe: Compaiso of he effec of g he g C Figue 6 Pied-pied pipe: Compaiso of he effec of g ad g o C 9