BEHAVIOR OF THICK SPHERICAL VESSELS UNDER HIGH PRESSURE

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1 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 BEHAVIOR OF HICK SPHERICAL VESSELS UNDER HIGH PRESSURE Mse Fsel Milk College of engineeing, Univesity of Qdisiy Astct he im of this study ws to investigte elsto-plstic theml stesses in themoplstic of hick Spheicl vessels unde high pessue he pesent study dels with spheicl shells nlysis, he elstic nd plstic theoy of spheicl shell is conside in this sech, with thick sphee sujected to diffeent types of loding such s intenl pessue, extenl pessue nd theml loding hve een studies. he esc yield condition used in this study. When the pplied pessue exceeds the minimum pessue equied to initite the yielding t the inne dius, plstic zone stts to e fomed. he esidul stess components lso wee clculted using elstic nd elsto-plstic solution esult سلوك الخزانات السميكة الكروية تحت ألضغوط العالية ما سي فيصل ملك آلية الهندسة جامعة القادسية ألخلاصة: اله دف م ن ه ذا البح ث آ ان تح ري الاجه ادات ألحراري ة ف ي ط ور اللدون ة-المرون ة للخزان ات الكروي ة ال سميكة وتح ت ال ضغوط ألعالية.اعتمدت نظريات اللدونة والمرونة للق شريات ألكروي ة م ع ت سليط أن واع مختلف ة م ن الا حم ال آال ضغوط الداخلي ة وال ضغوط الخارجية والا حمال الحرارية جميعه ا أخ ذت بنظ ر الاعتب ار.ت م اس تخدام نظري ة ترس كا للخ ضوع ف ي ه ذه الدراس ة.عن دما ال ضغط المسلط يتجاوز الحد الا دنى للضغط المطلوب لبدء الخضوع في نصف القطر الداخلي ينتج عنه تشكل منطقة التشوه اللدن.آذلك ت م دراسة الاجهادات المخزونة أو المتبقية وتا ثيرها على التشوه اللدن والمرن Nomencltue y: Yield stength of mteil of unixil tension test : Rdil Stess : Hoop stess A nd B: Constnt : Rdius of thick sphee : inne Rdius : oute Rdius p e :pessue t plstic Zone c ( : he dil stess t =c C: Rdius of sphee t plstic zone P i : inne dius P o : oute dius nd : tempetues t the oute nd inne dius espectively α : line coefficient of expnsion. β: Dimensionless quntity ٣٦٨

2 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 Intoduction Cylindicl o spheicl pessue vessels (e.g., hydulic cylindes, sewge tetment plnts, gun els, pipes, oiles nd tnks e commonly used in industy to cy oth liquid nd gses unde pessue. When the pessue vessel is exposed to this pessue, the mteil compising the vessel is sujected to pessue loding, nd hence stesses, fom ll diections. he noml stesses esulting fom this pessue e functions of the dius of the element unde considetion; the shpe of the pessue vessel (i.e., open ended cylinde, closed end cylinde, o sphee s well s the pplied pessue. A detiled study of stess nlysis of spheicl pessue vessel with intenl pessue nd extenl pessue e to e consideed. Spheicl pessue vessels e clssified into thick sphee if the oute dius is lge then the inne dius y 0 %, othewise the sphee cn e consideed s thin sphee. in the thin sphees the dil nd tngentil stesses e ssumed to e constnt long the thickness of the sphee ut in thick sphees the dil nd tngentil stesses vey coss the thickness of the sphee ccoding to the type of loding such s intenl pessue nd extenl pessue. An effective numeicl itetive method is poposed fo the pessue loding nlysis. Elstic stess ehvio is consideed fist, followed y some cses of elsto-plstic stess distiutions, nd finlly plstic defomtion. fo the spheicl pessue vessel, the hoop nd xil stesses e equl nd e one hlf of the hoop stess in the cylindicl pessue vessel s shown in Figue (. his mkes the spheicl pessue vessel moe efficient pessue vessel geomety [- 4] Filue Citei he pupose of filue citei is to pedict o estimte the filue/yield of stuctul memes sujected ixil o tixil sttes of stess. hee e moe one theoy fo Filue Citei, dependent on the ntue of the mteil (i.e. ittle o ductile, s shown elow: - Mximum she stess citeion, Von Mises citeion fo Ductile mteil. - Mximum She Stess Citeion, esc's citeion fo Ductile mteil. - Mximum noml stess citeion, Moh s theoy fo Bittle mteil. Whethe mteil is ittle o ductile could e sujective guess, nd often depends on tempetue, stin levels, nd othe envionmentl conditions. Howeve, 5% elongtion citeion t ek is esonle dividing line. Mteils with lge elongtion cn e consideed ductile nd those with lowe vlue ittle. Anothe distinction is ittle mteil's compession stength is usully significntly lge thn its tensile stength. All popul filue citei ely on only hndful of sic tests (such s unixil tensile nd/o compession stength, even though most mchine pts nd stuctul memes e typiclly sujected to multi-xil loding. his dispity is usully diven y cost, since complete multi-xil filue testing equies extensive, complicted, nd expensive tests. he success of ll mchine pts nd stuctul memes e not necessily detemined y thei stength. Whethe pt succeeds o fils my depend on othe fctos, such s stiffness, vitionl chcteistics, ftigue esistnce, nd/o ceep esistnce. he mximum she stess citeion, lso known s esc's o Guest's citeion [4], is often used to pedict the yielding of ductile mteils. Yield in ductile mteils is usully cused y the slippge of cystl plnes long the mximum she stess sufce. heefoe, given point in the ody is consideed sfe s long s the mximum she stess t tht point is unde the yield she stess otined fom unixil tensile test. With espect to plne stess, the mximum she stess is elted to the diffeence in the two pincipl stesses. heefoe, the citeion equies the pincipl stess diffeence, long with the pincipl stesses themselves, to e ٣٦٩

3 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 less thn the yield she stess, this citeion hs good geement with expeimented esults otined fo ductile mteil nd due to simplicity of this citei will e used in this study. < y, < y, - < y Whee, y = Yield stength of mteil of unixil tension test As shown elow, the mximum she stess citeion equies tht the two pincipl stesses. Mthemticl Fomultion he condition of spheicl symmety is ssumed fo the thick sphee with stess-stin eltions nd incompessiility conditions fo plstic defomtion. he eqution is solved in the spheicl Coodintes. he govening eqution (equiliium eqution fo the thick spheicl vessel is: ( d ( d ( d ( d ( ( d d ed = 0 simplified nd dividingy( d d = 0 d d d ( = d Fomultion Of esc's ( d d d = 0 4 dd sin( d / = 0 his theoy pedicts tht yielding will stt when the mximum she stess in the mteil ecomes equl to the mximum she stess t yielding in simple tension test: = Y = ±..( Y Y Fo thick sphees the pinciple stess e nd,whee is compessive nd is tension so tht: = Y.( ٣٧٠

4 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 Cse : Intenl Pessue Only Elstic Anlysis: In the nlysis of elstic nd elstic-plstic cse we must use some fundmentl equtions, such s equiliium nd comptiility equtions. Fom Equiliium eqution fo spheicl coodintes hs een deived in lst section, nd it is given s follows: d d ( =.( Comptiility eqution in tems of stesses is given y: d ( = 0 d.(4 In the elstic solution fo this cse nd fo othe, Lmes eqution hs een used to find the solution, fom eqution (4, it cn e seen tht the mount ( is independent of the dius, thee foe: ( =A Whee A is constnt. By sustitute Eq. (4 into the Eq. ( nd integting, will e otined (5 B = A B = A..(6 t = = p = 0 t = By sustituting the ove oundy condition in Eq.(5 nd (6, we otin: A = B = P P hus Eq. (5 nd (6 cn e witten s: p( =.(7 ( ٣٧١

5 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 p( = (.(8 hese Equtions expess the dil nd tngentil elstic stess distiution coss the thickness of the wll nd it is noted tht the edil stess is compessive while the tngentil stess is tensile. Plstic Anlysis If the intenl pessue is incesed to citicl vlue (p e, plstic yielding egins t the dius, whee the yield citeion (esc is fist stisfied. By sustituting Eq. (7 nd (8 into Eq. (: p p = Y his eqution educes to P e = Y It hs note tht hs minimum vlue t = (i.e the diffeence ( ( hs the getest vlue t =, this mens tht the initition of the plstic zone will egin-fom the inne dius when p=p e P e = Y ( With futhe incese in the intenl pessue, the plstic zone speds out wds nd the elsticplstic oundy is spheicl tce t ech stge. o find the stess distiution t the plstic zone (t ech point in this zone the yield citeion will e stisfied s will s the equiliium eqution theefoe y sustituting eqution. Fom Eq ( nd ( we get: d = Y then integting d = Y ln A Whee A is constnt to e evluted fom the oundy condition, At =, = p then A= (-P-Yln, And = Y ln P (9 o otin the expession fo Eq. (9 is sustituted into the esc's citeion Eq. ( we get. ٣٧٢

6 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 = Y ( ln( P (0 o find the stesses in the elstic zone, Lmes eqution e lso pplied s fo the elstic cse, ut the constnt will e function of the plstic dius (c, thee fo the sphee is ssumed to e consisted fom two sphees,the inne sphee is completely plstic nd the oute sphee is completely elstic, thus the oundy condition ecome: = 0 t = = ( c t =c Whee ( is the dil stess t =c, nd it is equl to: c -( ( = ( c c Y fom the fist oundy condition: A=-B/ C And fom second oundy condition, B= Y ( c hen Lmes equtions ecome C = Y (. ( nd C = Y ( ( At the elstic-plstic oundy the stesses must stisfy the continuity condition theefoe equlizing Eqs. ( nd ( t = c, will yield c ( P c = ln Y his eqution is used to detemine the dius of plstic phse. he solution is ccomplished t ny vlue of pessue (P, y solving it y using suitle numeicl method Residul Stess Anlysis he esidul stesses cn e found y sutcting Eqs. (7 nd (8 fom Eqs (9 nd (0.ut in the elstic zone the esidul stess cn e found y sutcting Eqs. (7 nd (8 fom Eqs ( nd (. Y C P = ( (,c P Y C = ( e P ( P e, c Y p = ( ( ln, c p e ٣٧٣

7 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 Y = ( p p e ( ln, c Cse : Extenl Pessue Only hee e some cse in which sphee is sujected to extenl pessue, the nlysis fo such cse is simil to fo the intenl pessue cse, ut oundy condition e vied: Elstic Anlysis he Lmes equtions fte pplying the oundy condition ecomes : P O P = ( nd O = ( ( ( It is noted tht the dil stess nd s well s the tngentil stess e oth compessive while they e tensile fo the cse of intenl pessue only Plstic Anlysis o find the stess distiution t the plstic zone (<c esc yield citei should e stisfied s well s the equiliium eqution. hus the stesses will e: c = P Y ( R c = P Y (,Since these stesses must e continuous coss the elstic-plstic R oundy: = then So R=C,[ ] elstic [ ] plstic c Y ( c = Y ln -P- c C So P = P = Y ln Y ( By solving the ove eqution numeiclly using Newton Rphson method the vlue of the dius C cn e otin fo ny vlue of pplied pessue. Cse : Intenl nd Extenl Pessue he sphee in this cse is sujected to intenl nd extenl pessue pplied simultneously. By supeposition of the two elstic stesses fo the intenl nd extenl pessue cse nd y pplying the following oundy condition in to lmes eqution: = t = P i =-P O t = hen ٣٧٤

8 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 ٣٧٥ ( ( P P O i = nd ( ( P P O i = o otin the citicl of the diffeence in pessues tht will cse the initition of plstic zone t the inne dius (=,su, nd in esc citeion( Y = ( Cse 4: heml Loding he theml loding, hs significnt effect on the vlue of stesses due to tempetue diffeence tht my occus etween the inne nd the oute dius of the sphee, this will cuse diffeence in the expnsion o contction etween the lyes of the thick sphee, thee fo this will give ise-to the dil nd tngentil stesses. he stedy stte tempetue distiution in the cse of spheicl symmety is give y: ( = Whee & e tempetues t the oute nd inne dius espectively. he stin in the sphee is the supeposition of tht due to pessue loding nd theml loding, then the elstic stess-stin eqution is: E E α ν ν ε α ν ε = = (( ( whee α is the line coefficient of expnsion. Afte Appling comptiility eqution nd sustitution with integting we cn get the dimensionless quntity given s: ( ( ν α β = he dil stess nd s well s the tngentil stess e given y the Equtions elow: = βe = βe Result And Discussion he Clcultion show tht the intenl pessue equied to initite the plstic zone fom the inne dius is P i =4 Mp nd the stte of totl plsticity is eched when P i =.Mp, ecuse of incesing the loding of vessel, the volume incese, which leds to plstic defomtions tke

9 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 plce. At intenl pessue P i =8Mp, the whole sphee is in fully elstic stte. his pessue is lowe thn the pessue needed to initite the plstic defomtion t the inne sufce, Figue( shows the non-dimensionl dil nd tngentil stess distiution long the thickness of the sphee, we conclude fom tht when the extenl pessue incesed eyond the citicl vlue of the pessue the plstic zone will popgte outwd tht s leds to incesing the dil stess. Figue (4 shows the vition of the dimensionl esidul dil nd tngentil stesses coss the thickness of the sphee when it is loded with pessue P=0 Mp. we see tht the hoop stess incese duing the thickness of the sphee ecuse of the Residul stess distiution incese due to ovelods. Figue (5 show the non-dimensionl dil nd tngentil stess distiutions due to theml loding,we see tht the hoop stess is lge then dil stess ecuse of the Coefficient of theml expnsion mismtch etween diffeent phses. Figue (6 show the elsto-plstic stte of stesses fo extenl pessues of 0Mp this figues lso show the popgtion of plstic zone s the pessue inceses. due to the plstic e incese leds to yielding tke plce. Conclusions Fo the intenl pessue cse it cn e oseved tht the plstic zone stts t the inne dius nd speds outwd s the pessue incese, lso thee is diect popotionlity etween elstic oundy dius nd intenl pessue. he esidul stesses inceses s the mount of plstic defomtion inceses. hee is diect popotionlity etween the intenl pessue nd the vlue of the esidul tngentil. Fo theml loding it is noted tht the dil stess is negtive nd eches mximum vlue etween the inne nd the oute dius. his mximum stess will e t =0.64.it lso oseved those vlues of stesses inceses s tempetue diffeence inceses. Effects of esidul stess my e eithe eneficil o detimentl, depending upon the mgnitude, sign, nd distiution of the stess with espect to the lod-induced stesses. Vey commonly, the esidul stesses e detimentl, nd thee e mny documented cses in which these stesses wee the pedominnt fcto contiuting to ftigue nd othe stuctul filues when the sevice stesses wee supeimposed on the ledy pesent esidul stesses. Refeences [] Pvel Elesztos - Ldislv Ecsi stess test of spheicl pessue-etining vessels with hydocon Gses [] Dechnt,K.E.: Effects of high pessue testing technique on pipelines. 996 OMAE - Volume V, Pipeline echnology, ASME 996. [] Hossein Mhdi- Mohmd R. Eslmi Elstic-plstic-ceep Cyclic Loding of hick Pessue Vessels sed on the Fedeick-Amstong Kinemtic Hdening Model Ppe # F05-7 [4] Fuk Sen nd Metin Sye Elsto-Plstic theml stess nlysis in themoplstic composite Disc unde unifom tempetue using FEM Mthemticl nd Computtionl Applictions, Vol., No., pp. -9, 006. ٣٧٦

Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 Figue (: Pessue nd Intenl Hoop nd Axil Stesses. 0.4 0.40 0. /Y 0 /Y 0.0 0.00-0. -0.0-0.4 Figue (: Elstic stess distiution fo intenl pessue(p i =8 Mp 0.

10 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye 009 Figue (: Pessue nd Intenl Hoop nd Axil Stesses /Y 0 /Y Figue (: Elstic stess distiution fo intenl pessue(p i =8 Mp Figue (: Elstic stess distiution fo intenl pessue(p i = 8 Mp /Y 0.00 Plstic zone Elstic zone /Y Figue (:Elstic-Plstic stess Distiution fo( p i =5 Mp Figue (4:Residul stess distiution fo intenl pessue ( p i =0 Mp ٣٧٧

11 Al-Qdisiy Jounl Fo Engineeing Sciences Vol. No. ٢ Ye /Y /Y Figue (5:Rdil nd tngentil stess Distiution fo theml loding ( =500, = Figue (6:Elstic plstic stess distiution fo extenl pessue(p o =0 Mp ٣٧٨

Available online at ScienceDirect. Procedia Engineering 91 (2014 ) 32 36

Available online at ScienceDirect. Procedia Engineering 91 (2014 ) 32 36 Aville online t wwwsciencediectcom ScienceDiect Pocedi Engineeing 91 (014 ) 3 36 XXIII R-S-P semin Theoeticl Foundtion of Civil Engineeing (3RSP) (TFoCE 014) Stess Stte of Rdil Inhomogeneous Semi Sphee