MECHANICAL ENGINEERING DESIGN TUTORIAL 4 15: PRESSURE VESSEL DESIGN

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1 MECHANICAL ENGINEERING DESIGN TUTORIAL 4 5: PRESSURE VESSEL DESIGN PRESSURE VESSEL DESIGN MODELS FOR CYLINDERS:. Thck-walled Cylndes. Thn-walled Cylndes THICK-WALL THEORY Thck-wall hey s develped fm he They f Elascy whch yelds he sae f sess as a cnnuus funcn f adus ve he pessue vessel wall. The sae f sess s defned elave a cnvenen cylndcal cdnae sysem:. Tangenal Sess. Radal Sess 3. l Lngudnal Sess Sesses n a cylndcal pessue vessel depend upn he a f he nne adus he ue adus ( / ) ahe han he sze f he cylnde. Pncpal Sesses (,, 3). Deemned whu cmpuan f Mh s Ccle;. Equvalen cylndcal sesses (,, l) Applcable f any wall hckness--adus a. Cylnde unde Pessue Cnsde a cylnde, wh capped ends, subjeced an nenal pessue, p, and an exenal pessue, p, l p l p FIGURE T4-5- Tex Eq. efes Mechancal Engneeng Desgn, 7 h edn ex by Jseph Edwad Shgley, Chales R. Mschke and Rchad G. Budynas; equans and fgues wh he pefx T efe he pesen ual.

The sae f sess s defned elave a cnvenen cylndcal cdnae sysem:. Tangenal Sess. Radal Sess 3.

2 The cylnde gemey s defned by he nsde adus,, he usde adus,, and he cylnde lengh, l. In geneal, he sesses n he cylndcal pessue vessel (,, l) can be cmpued a any adal cdnae value,, whn he wall hckness bunded by and, and wll be chaacezed by he a f ad, ζ /. These cylndcal sesses epesen he pncpal sesses and can be cmpued decly usng Eq and 4-5. Thus we d n need use Mh s ccle assess he pncpal sesses. Tangenal Sess: p p ( p p ) / f (Tex Eq. 4-50) Radal Sess: p p + ( p p ) / f (Tex Eq. 4-50) Lngudnal Sess: Applcable cases whee he cylnde caes he lngudnal lad, such as capped ends. Only vald fa away fm end caps whee bendng, nnlneaes and sess cncenans ae n sgnfcan. Tw Mechancal Desgn Cases p p l f. Inenal Pessue Only ( p 0 ). Exenal Pessue Only ( p 0 ) Desgn Case : Inenal Pessue Only (Mdfed Tex Eq. 4-5) Only ne case cnsde he ccal secn whch exss a. Subsung p 0 n Eqs. (4-50) and ncpang ζ /, he lages value f each sess cmpnen s fund a he nne suface: + ζ + ( ),max p p pc ζ (T-) Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn /0

These cylndcal sesses epesen he pncpal sesses and can be cmpued decly usng Eq. 4-50 and 4-5. Thus we d n need use Mh s ccle assess he pncpal sesses. Tangenal Sess: p p ( p p ) / f (Tex Eq.

3 whee C ζ + ζ + s a funcn f cylnde gemey nly. ( p Naual Bunday Cndn (T-) ), max Lngudnal sess depends upn end cndns: l p C Capped Ends (T-3a) l 0 Uncapped Ends (T-3b) whee C l ζ. Desgn Case : Exenal Pessue Only The ccal secn s denfed by cnsdeng he sae f sess a w pns n he cylnde: and. Subsung p 0 n Tex Eqs. (4-50) f each case: ( ) 0 Naual Bunday Cndn (T-4a) ζ ( ),max p p p C ζ (T-4b) whee, C ζ ζ. ( p Naual Bunday Cndn (T-5a) ), max + ζ + ( ) p p p C ζ (T-5b) Lngudnal sess f a clsed cylnde nw depends upn exenal pessue and adus whle ha f an pen-ended cylnde emans ze: l p C Capped Ends (T-6a) l 0 Uncapped Ends (T-6b) Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 3/0

Desgn Case : Exenal Pessue Only The ccal secn s denfed by cnsdeng he sae f sess a w pns n he cylnde: and. Subsung p 0 n Tex Eqs.

4 whee C l ζ ζ. Example T4.5.: Thck-wall Cylnde Analyss Pblem Saemen: Cnsde a cylnde subjeced an exenal pessue f 50 MPa and an nenal pessue f ze. The cylnde has a 5 mm ID and a 50 mm OD, especvely. Assume he cylnde s capped. Fnd:. he sae f sess (,, l ) a he nne and ue cylnde sufaces;. he Mh s Ccle pl f he nsde and usde cylnde sufaces; 3. he ccal secn based upn he esmae f τ max. Slun Mehdlgy: Snce we have an exenal pessue case, we need cmpue he sae f sess (,, l ) a bh he nsde and usde adus n de deemne he ccal secn.. As he cylnde s clsed and expsed exenal pessue nly, Eq. (T-6a) may be appled calculae he lngudnal sess develped. Ths esul epesens he aveage sess acss he wall f he pessue vessel and hus may be used f bh he nne and ue ad analyses.. Assess he adal and angenal sesses usng Eqs. (T-4) and (T-5) f he nne and ue ad, especvely. 3. Assess he pncpal sesses f he nne and ue ad based upn he magnudes f (,, l ) a each adus. 4. Use he pncpal sesses calculae he maxmum shea sess a each adus. 5. Daw Mh s Ccle f bh saes f sess and deemne whch pvdes he ccal secn. Slun:. Lngudnal Sess Calculan: OD 50 mm ID 5 mm 5 mm ;.5 mm Cmpue he adus a, ζ 5 mm ζ.0.5 mm Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 4/0

Slun Mehdlgy: Snce we have an exenal pessue case, we need cmpue he sae f sess (,, l ) a bh he nsde and usde adus n de deemne he ccal secn.. As he cylnde s clsed and expsed exenal pessue nly, Eq.

5 Then, ζ () Cl.3333 mm ζ () ζ l( ) l( ) p p ( 50MPa)(.3333 mm ) Cl ζ 00 MPa l. Radal & Tangenal Sess Calculans: Inne Radus ( ) ζ () C.6667 ζ () ( ),max p p ( 50 MPa)(.6667) C ( ) 400 MPa Cmpessve ( ) 0 Naual Bunday Cndn f p 0 Oue Radus ( ) C ζ + () ζ () + ( ) ( 50 MPa)(.6667),mn p p C ( ) 50MPa Cmpessve ( ) p 50 MPa Naual Bunday Cndn 3. Defne Pncpal Sesses: Inne Radus ( ) Oue Radus ( ) 3 l 0 MPa 00 MPa 400 MPa 3 l 50 MPa 00 MPa 50 MPa 4. Maxmum Shea Sess Calculans: 3 0 ( 400) Inne Radus ( ) τ max ( ) 00 MPa Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 5/0

6667),mn p p C ( ) 50MPa Cmpessve ( ) p 50 MPa Naual Bunday Cndn 3.

6 3 ( 50) ( 50) Oue Radus ( ) τ max ( ) 50 MPa 5. Mh s Ccles: Inne Radus ( ) τ FIGURE T4-5- τ max 00 MPa MPa 0 MPa -00 MPa Oue Radus ( ) FIGURE T4-5-3 τ 3-50 MPa 50 MPa τ max 50 MPa -00 MPa Ccal Secn τ max ( ) 00 MPa Ccal Secn s a Insde Radus! Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 6/0

Oue Radus ( ) FIGURE T4-5-3 τ 3-50 MPa 50 MPa τ max 50 MPa -00 MPa Ccal Secn τ

7 THIN-WALL THEORY Thn-wall hey s develped fm a Sengh f Maeals slun whch yelds he sae f sess as an aveage ve he pessue vessel wall. Use esced by wall hckness--adus a: Accdng hey, Thn-wall They s jusfed f 0 In pacce, ypcally use a less cnsevave ule, 0 Sae f Sess Defnn:. Hp Sess,, assumed be unfm acss wall hckness.. Radal Sess s nsgnfcan cmpaed angenal sess, hus, Lngudnal Sess, l SExss f cylndes wh capped ends; SAssumed be unfmly dsbued acss wall hckness; SThs appxman f he lngudnal sess s nly vald fa away fm he end-caps. 4. These cylndcal sesses (,, l) ae pncpal sesses (,, l) whch can be deemned whu cmpuan f Mh s ccle pl. Analyss f Cylnde Secn FIGURE T4-5-4 d F V F Hp Pessue Acng ve Pjeced Vecal Aea F Hp Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 7/0

. Radal Sess s nsgnfcan cmpaed angenal sess, hus, 0. 3.

8 The nenal pessue exes a vecal fce, F V, n he cylnde wall whch s balanced by he angenal hp sess, F Hp. F F V Hp pa pj A p{( d sessed )()} pd {( )()} F y 0 F V F Hp pd Slvng f he angenal sess, pd Hp Sess (Tex Eq. 4-53) Cmpasn f sae f sess f cylnde unde nenal pessue veses exenal pessue: Inenal Pessue Only pd 0 pd l 4 Hp Sess By Defnn Capped Case (Tex Eq.4-55) Exenal Pessue Only pd 0 pd l 4 Hp Sess By Defnn Capped Case Example T4.5.: Thn-wall They Appled Cylnde Analyss Pblem Saemen: Repea Example T. usng he Thn-wall They (p 50 MPa, p 0, ID 5 mm, OD 50 mm). Fnd: The pecen dffeence f he maxmum shea sess esmaes fund usng he Thck-wall and Thn-wall Thees. Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 8/0

4-53) Cmpasn f sae f sess f cylnde unde nenal pessue veses exenal pessue: Inenal Pessue Only pd 0 pd l 4 Hp Sess By Defnn Capped Case (Tex Eq.

9 Slun Mehdlgy:. Check / a deemne f Thn-wall They s applcable.. Use he Thn-wall They cmpue he sae f sess 3. Idenfy he pncpal sesses based upn he sess magnudes. 4. Use he pncpal sesses assess he maxmum shea sess. 5. Calculae he pecen dffeence beween he maxmum shea sesses deved usng he Thck-wall and Thn-wall Thees. Slun:. Check / Ra:.5 mm 5 mm 0 0 The applcan f Thn-wall They esmae he sess sae f hs cylnde s hus n jusfed.. Cmpue sesses usng he Thn-wall They cmpae wh Thckwall hey esmaes. a. Hp Sess (aveage sess, unfm acss wall) p d (50 MPa)(50 mm) 300 MPa (.5 mm) b. Radal Sess 0 by defnn c. Lngudnal Sess (aveage sess, unfm acss wall) pd l 50 MPa 4 3. Idenfy Pncpal Sesses n ems f Aveage Sesses: 3 l 0 MPa 50 MPa 300 MPa 4. Maxmum Shea Sess Calculan: τ 3 0 ( 300 MPa) max + 50 MPa 5. Pecen Dffeence beween Thn- and Thck-wall Esmaes f he Ccal Secn: Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 9/0

5 mm 5 mm 0 0 The applcan f Thn-wall They esmae he sess sae f hs cylnde s hus n jusfed.. Cmpue sesses usng he Thn-wall They cmpae wh Thckwall hey esmaes. a. Hp Sess (aveage sess, unfm acss wall) p d (50 MPa)(50 mm) 300 MPa (.

10 τ τ τ max,thn max,thck % Dffeence 00% max,thck ( + 50) ( + 00) (00%) 5% ( + 00) Thn -wall esmae s 5% lw! Shgley, Mschke & Budynas Machne Desgn Tual 4 5: Pessue Vessel Desgn 0/0

Thick-Walled Cylinders and Press Fits. 2004 by W.H.Dornfeld PressCylinder:

Thick-Walled Cylinders and Press Fits. 2004 by W.H.Dornfeld PressCylinder: Thck-Walled Cylndes and Pess Fts 004 by W.H.Dnfeld PessCylnde: 1 Stesses n Thck-Walled Cylndes Thck-Walled cylndes have an aveage adus less than 0 tmes the wall thckness. σ σl They ae essuzed ntenally