PVP Plastic Instability Pressure of Conical Shells. PVP2011 July 17-21, 2011, Baltimore, Maryland, USA. R. Adibi-Asl *

Transcription

1 Proeedigs Proeedigs of he ASME of 11 he Pressure ASME 11 Vessels Pressure & Pipig Vessels Divisio & Pipig Coferee Divisio PVP11 July 17-1, 11, Balimore, Marylad, USA Plasi Isabiliy Pressure of Coial Shells PVP PVP Adibi-Asl * AMEC NSS Ld., 9 Uiversiy Ave., Toroo, ON, Caada Absra There several failure modes ha are osidered i he available odes ad sadards i field of pressure vessel ad pipig. Oe of hese failure modes is plasi isabiliy. This failure mode is defied as he pressure for whih he ompoes/sruures approah dimesioal isabiliy (large deformaio), i.e. ubouded displaeme for a small ireme i he applied load. I order o fid his pressure boh large deformaio ad srai hardeig urve are osidered. Whe he slope of umerially geeraed load-deformaio urve approahig zero he orrespodig applied load is osidered as plasi isabiliy load. Ulike ylidrial ad spherial pressure vessels, available heoreial soluio i rage of plasi isabiliy load for oial shells are very limied. Hee, i would be very useful o predi he behavior of hese ompoes wih aepable auray for desig purposes. Aalyial expressios are derived o deermied plasi isabiliy load of s oial shell subjeed o ieral pressure ad whe i is subjeed o hydrosai pressure (i.e., sorage aks). The geomerial hages a be esimaed usig he proposed soluios whe osiderig he maerial srai hardeig urve. 1 INTODUCTION Esimaig he ulimae load (i.e., ieral pressure) ha he ompoe a wihsad is very useful for desig ad fiess for servie assessme. The ulimae load is defied as he load or load ombiaio for whih he ompoe approahes dimesioal isabiliy (plasi isabiliy), i.e. ubouded displaeme for a small ireme i he applied load. The plasi isabiliy is oe of he failure modes ha is required o be evaluaed by he available desig odes ad sadards (i.e., VIII of ASME Code [1]), ad fiess for servie guidelies (i.e., API 579-1/ASME FFS-1 Code []) of pressure vessels ad pipig. For isae, he desig-by-aalysis opio available i Se. VIII of ASME Code [1] oais elasiplasi sress aalysis opio ha requires esimaio of plasi isabiliy load. As disussed i [], he ose of esile plasi isabiliy ours whe sregheig due o srai hardeig of a maerial ad various effes due o geomery hages are i balaed. Whe esimaig he ulimae load or load ombiaio boh realisi sress-srai urve ad geomerial olieariy have o be osidered i he aalysis. The isabiliy ours a he ulimae load. I umerial (i.e., fiie eleme aalysis) ad experimeal approahes, whe he slope of geeraed loaddeformaio urve approahig zero he orrespodig applied load is osidered as plasi isabiliy load. This a be easily observed whe performig esio es. As soo as he load reahes is ulimae value i he esio es, he esio speime begis ekig ad deformaio sars o loalize (plasi isabiliy). The early work o plasi isabiliy daes bak o he work doe by Hill [4] o isabiliy aalysis of a meal diaphragm. The esile plasi isabiliies of he ylidrial ad spherial * reza.adibiasl@ame.om 1 Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms

2 pressurized shells were ivesigaed i -[7]. The esile plasi isabiliy of a hi-walled ylider subjeed o ombied ieral pressure ad axial load was proposed by Hillier [8]. Sorakers [9] proposed he plasi isabiliy soluio of a ylider subjeed o ombied ieral pressure, axial load ad orsio. Ulike ylidrial ad spherial pressure vessels, available heoreial soluio i rage of plasi isabiliy load for oher shell ofiguraios are very limied. Hee, i would be very useful o predi he behavior of differe ompoes for desig ad fiess-for-servie evaluaio purposes. eely, Vu ad Blahu [1] sudied he plasi isabiliy pressure i a oroidal shell uder ieral uiform pressure. I was deermied i [1] ha he plasi isabiliy soluio for a oroidal shell is he geeral ase for ylidrial ad spherial shells. Nomelaure A, B, Parameers o defie sress-srai urve g Graviy aeleraio H Fluid Heigh P Ieral pressure Shell radius s ij x z ij Deviaor sress ompoe Shell hikess Sress raio Criial Sub-age Kroeker dela Equivale srai Equivale sress i Priipal srai ompoes (i=1,, ) ρ Fluid Desiy i Priipal sress ompoes (i=1,, ) H Cirumfereial (hoop) sress L Logiudial (meridioal) sress Half oe agle Subsrip L Logiudial adial θ Cirumfereial Criial loaio Origial odiios BASIC FOMULATION Based o he heories of iremeal plasiiy, he followig expressios are valid for he sress ad srai i a body wih srai hardeig maerial: 1. The vo-mises yield odiio, whih will be used i he prese paper, is defied by Where as s ij s ij s ij (1) s ij is deviaor sress whih is defied. ij 1 ij kk. The vo-mises equivale srai is also defied as d d () ijd. The relaioship bewee he priipal plasi srai ompoes o he priipal sress ompoes is defied by Levy-Mises expressio as, d1 d 1.5( ).5( 1) d.5( ) 4. The ose of esile plasi isabiliy for muliaxial sress sae is defied as ij 1 () d 1 (4) d z where z is riial sub-age, he graphial represeaio of he riial sub-age is preseed i Figure 1. A rue sress-srai urve is required o perform a ulimae load aalysis. The ommoly used urve is suggesed by Swif [11] as, A ( B ) (5) where A is rae hardeig parameer, is ireme of his rae ad B iiial sage parameer. These parameers are exraed from experimeal daa. Differeiaig he above equaio respe o srai leads o, Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms

3 d A B 1 ( ) (6) d B z Therefore, he rue equivale sress ad srai a be deermied as, ad A ( z) (7) z B (8) The vo-mise yield rierio is used i his paper, ad he elasi srai assumed o egligible. The appropriae hoie for he yield fuio depeds o he maerial. For isae, for mealli maerials vo-mises ad Tresa yield fuios are beig used. Ulike Tresa, he vo-mises yield fuio does o ilude ay sigulariies i he formulaio; herefore, i is more suiable i umerial aalysis. a 1 d ( ) d Figure 1: Coep of Maerial Isabiliy o a True Sress-Srai Diagram CONICAL SHELL SUBJECTED TO UNIFOM INTENAL PESSUE.1 Criial Sub-Tage z A The sress ompoes i a hi walled oial shell subjeed i ieral pressure (see Figure ) are similar o hose i ylidrial shell. The irumfereial (hoop direio) sress is give as, P r P (9) os( ) Similarly, he meridioal sress a be wrie as P r P L (1) os( ) A ay poi i he oial shell, he raio of meridioal ad irumfereial sress a be deermied as x L /. 5. Therefore, he vo-mises equivale sress a be wrie as, (11) The srai relaioships are defied as follows, d d d d (1) ds d L s Usig he Levy-Mises flow rule, he srai ireme ompoes a be wrie as, d d (1) Please oe ha he meridioal srai is assumed o be zero (i.e., d L ). Therefore, he equivale srai a be wrie as d d (14) Usig Eq. (9), he ieral pressure a be wrie as, Differeiaig Eq. (15) resuls, P (15) dp d d d (16) Applyig he isabiliy odiio ( dp ) i Eq. (16) resuls i he followig relaioship, d d d (17) Makig use of Eqs. (11), (1) ad (17), he followig equaio a be obaied, Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms

4 d d (18) Compariso of Eqs. (4) ad (18) he riial sub-age a be deermied usig he followig equaio, 1 z.5774 (19) Equaio (19) is ideial o a lose-eded ylidrial shell. r r Shell Thikess= P Iegraio of Eq. () gives, d d () d l () Combiig Eq. (8) wih Eq. (), he hikess a he plasi isabiliy a be deermied as, ( Bz) / e (4) I a similar way, iegraig Eq. (14) gives, d l (5) Combiig Eq. (8) wih Eq. (5), he radius a he plasi isabiliy a be deermied as he followig expressio, s r 1 e (6) ( Bz) / Subsiuig Eqs. (5) ad (6) io (1) he plasi isabiliy is deermied as, P (7) ( Bz) A( z) e Isabiliy ours a he riial loaio where he equivale sress is maximum (i.e., r r ) ; hus, r (8) os( ) Figure : Coial Shell Subjeed o Ieral Pressure. Plasi Isabiliy Pressure The isabiliy pressure for oial shell subjeed o he ieral pressure is ivesigaed ex. Makig use of Eqs. (7) ad (11), he followig expressio a be obaied, Thus, P A( z) () P A( z) (1) Makig use of Eqs. (1)-(14), we have 4 CONICAL SHELL SUBJECTED TO INTENAL HYDOSTATIC PESSUE 4.1 Criial Sub-Tage For a hi-walled oial shell subjeed o ieral hydrosai pressure (see Figure ) he irumfereial ad meridioal sresses are give as [1]. ad a( ) P gy( H y) (9) os( ) a( ) L gy( H y) () os( ) The sress raio a be deermied as, 4 Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms

5 L x H y / ( H y) Therefore, he vo Mises equivale sress a be wrie as, (1) 1 x x () Usig he Levy-Mises flow rule, he ompoes of he srai ireme a be wrie as d L d d ( x.5) (1.5x) (1 x) The equivale srai a be wrie as, 1 x x d d x Shell Thikess= () (4) L dp d d d (6) The plasi isabiliy odiio is dp ; herefore, Eq. (6) a be re-wrie as, d d Equaio (7) a also be re-wrie i erms of srais, i.e., d d (7) L d d (8) L Makig use of Eq. (4), Eq. (8) a be re-wrie a riial loaio as, d d (9) 1 x x I is assumed ha isabiliy begis a he riial loaio (i.e., x x, i whih x will be deermied laer); herefore, differeiaig Eq. () yields, d 1 x x d (4) Combiig Eqs. (9) ad (4) resuls i he followig expressio, H y s r d d 1 x x (41) The verial loaio of maximum ad miimum equivale sress a be aalyially deermied by differeiaig Eq. () wih respe o y (see Figure 4), i.e., mi max a y H a y H 56 (4) The sress raio of sress a he maximum equivale sress loaio a be deermied as, Figure : Coial Shell uder Ieral Hydrosai Pressure Equaio (9) a be rewrie i followig form, P (5) Differeiaio Eq. (5) resuls i he followig equaio, (1 51) x (4) Subsiuig Eq. (4) io (41) yields, d 1 51 d 4 (44) Comparig Eqs. (4) ad (41), he riial sub-age a be deermied as, 5 Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms

6 y/h z (45) σ θ σ L σ eq Usig he Levy-Mises flow rule, he ompoes of he srai ireme a be wrie as, d L d d ( x.5) (1.5x) (1 x) The equivale srai a be wrie as 1 x x d d x Shell Thikess= (5) (51) σ/σ θ,max Figure 4: Variaio of Sresses wih Verial Posiio for Coial Shell uder Ieral Hydrosai Pressure Also plasi isabiliy of a ylidrial shell subjeed o ieral hydrosai pressure is also ivesigaed i here. For a hi-walled ylidrial shell subjeed o a ieral hydrosai pressure (see Figure 5) he irumfereial ad meridioal elasi sresses are give as, ad where P g( H y). gy P ( H y) (46) gy P y L y (47) H y Variaios of hoop, meridioal ad equivale sresses are ploed i Figure 5 versus y/h. The sress raio a be deermied as, L x y ( H y) Therefore, he vo-mises equivale sress a be wrie as, (48) 1 x x (49) H y Figure 5: Cylidrial Shell uder Ieral Hydrosai Pressure I a similar way as i oial shell uder ieral hydrosai pressure, he followig expressio is valid for he ylidrial shell, d d 1 x x (5) Assumig he isabiliy ours a he riial loaio (i.e., y= where x =), Eq. (5) a be rewrie as follows, d d (5) Compariso of Eqs. (4) ad (5), parameer z a be deermied as, 6 Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms

7 z.6667 (54) Equaio (54) is ideial as i a ope-eded ylidrial shell uder uiform ieral pressure. y/h σ θ σ L σ eq σ/σ θ,max Figure 6: Variaio of Sresses wih Verial Posiio for Cylidrial Shell uder Ieral Hydrosai Pressure 4. Plasi Isabiliy Pressure The isabiliy pressure for oial shell subjeed o he hydrosai pressure is ivesigaed i here. Makig use of Eqs. (7) ad (46), he followig expressio a be obaied, Thus, P A 1 x x ( z) A P ( z) 1 x x Makig use of Eqs. (1), (5) ad (51), we have Iegraig Eq. (57) gives, (55) (56) 1 d d (57) 1 x 1 d 1 l (58) 1 1 Combiig Eq. (8) wih Eq. (58), he hikess a he plasi isabiliy a be deermied usig he followig expressio, e 1 ( Bz) 1 x x I a similar way, iegraig Eq. (51) yields, (59) 1 d 1 l (6) Combiig Eq. (8) wih Eq. (6), he radius a he plasi isabiliy a be deermied as follows, e ( Bz) 1 (61) Subsiuig Eqs. (59) ad (61) io (56) he plasi isabiliy is deermied as, ( Bz) 1 x x A P ( z) e (6) 1 x x Equaio (6) is isabiliy hydrosai pressure aig o riial loaio. This equaio is valid boh for oial ad ylidrial shell uder hydrosai pressure wih he osideraios lised i Table 1. Table 1: Parameers for Evaluaio of Plasi Isabiliy defied i Eq. (6) for Coial ad Cylidrial Shell subjeed o Ieral Hydrosai Pressure Criial Loaio ( y ) Coial H 56 Criial Sress (1 51) aio ( x ) Iiial ieral radius a Criial Loaio ( ) a( ) H 56 os( ) Cylidrial Criial Subage ( z ) Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms

8 5 CONCLUSION Aalyial expressios are developed for he oial shells o esimae he riial sub-age (z) i esile plasi isabiliy aalysis. The proposed soluios are give for wo ypes of oial shell ofiguraios: oial shell subjeed o a uiform ieral pressure, ad oial shell subjeed o a ieral hydrosai pressure. The proposed soluios are he used o deermie he ulimae pressure i he oial shells for a speifi sress-sai urve proposed by Swif [11]. The soluios a be used o ivesigae he plasi isabiliy of a ompoe of a differe sress-srai relaioship or realisi sress-srai urve. 6 EFEENCES [1] ASME Boiler ad Pressure Vessel Code, 7, Seio VIII, Divisio, Ameria Soiey of Mehaial Egieers, New York. [] API, (7), Fiess-for-Servie Sadard API 579-1/ASME FFS-1, The Ameria Peroleum Isiue, Washigo DC 5-47 U.S.A. [] Updike, D. P., ad Kalis, A., 1998, Tesile Plasi Isabiliy of Axisymmeri Pressure Vessels, ASME J. Pressure Vessel Tehol., Vol. 1, pp [4] Hill,., 195, A Theory of he Plasi Bulgig of a Meal Diaphragm by Laeral Pressure, Philos. Mag., Vol. 7, pp [5] Johso, W., ad Mellor, P. B., 198, Egieerig Plasiiy, Ellis Horwood Ld., Chiheser. [6] Mellor, P. B., 196, The Ulimae Sregh of Thi- Walled Shells ad Cirular Diaphragms Subjeed o Hydrosai Pressure, I. J. Meh. Si., Vol. 1, pp [7] Mellor, P. B., 196, Tesile Isabiliy i Thi- Walled Tubes, J. Meh. Eg. Si., Vol. 4, pp [8] Hillier, M. J., 1965, Tesile Plasi Isabiliy of Thi Tubes, I. J. Meh. Si., Vol. 7, pp [9] Sorakers, B., 1968, Plasi ad Viso-Plasi Isabiliy of a Thi Tube Uder Ieral Pressure, Torsio ad Axial Tesio, I. J. Meh. Si., Vol. 1, pp [1] Vu, T. V., ad Blahu, J., 9, Plasi Isabiliy Pressure of Toroidal Shells, ASME J. Pressure Vessel Tehol., Vol. 11, Paper # 51. [11] Swif, H., 195, "Plasi Isabiliy udre Plae Sress," J. Meh. Phys. Solids, Vol. 1, pp [1] agab, A.., Bayoumi, S.E., 1998, Egieerig Solid Mehais: Fudameals ad Appliaios, CC Press. 8 Copyrigh 11 by ASME Dowloaded From: hp://proeedigs.asmedigialolleio.asme.org/ o 11/5/14 Terms of Use: hp://asme.org/erms