Analytical Approach in Autofrettaged Spherical Pressure Vessels Considering the Bauschinger Effect

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1 R. Adibi-Asl 1 Deatment of Mechanical ngineeing, Univesity of Tehan, Tehan, Ian adibi@eng.mun.ca P. Liviei 2 Deatment of ngineeing, Univesity of Feaa, Feaa, Italy liviei@ing.unife.it Analytical Aoach in Autofettaged Sheical Pessue Vessels Consideing the Bauschinge ffect This ae esents an analytical study of sheical autofettage-teated essue vessels, consideing the Bauschinge effect. A geneal analytical solution fo stess and stain distibutions is oosed fo both loading and unloading hases. Diffeent mateial models incooating the Bauschinge effect deending on the loading hase ae consideed in the esent study. Some actical analytical exessions in exlicit fom ae oosed fo a bilinea mateial model and the modified Rambeg Osgood model. DOI: / Keywods: autofettage, esidual stess, sheical essue vessels, Bauschinge effect, bilinea mateial, Rambeg Osgood law 1 Intoduction Residual stesses ae setu on unloading intenally essuized comonents subsequent to the alication of high essues such that the stess distibution beyond yield is obseved in the comonent 1. A favoable distibution of esidual stesses inceases the stength of the essue comonents. The autofettage ocess adoted by the essue vessel industy enhances the static limit essue of the comonents. In addition, a significant incease in the cyclically loaded autofettage comonents is also obseved due to the inhibition of cack initiation and oagation 2. The alication of autofettage teated vessels can be extended to the owe geneation industy fossil and nuclea, the etochemical industy, the food industy bacteial eadication containe, and automotive alications injection um, among many othes. In aticula, sheical essue vessels, due to thei inheent stess and stain distibutions equie thinne walls comaed to cylindical vessels; theefoe, they ae extensively used in gas-cooled nuclea eactos, gas o liquid containes athe than heads of close-ended cylindical vessels. Rees in 3 intoduced the geneal exessions fo the stessstain field fo close-ended cylindical essue vessels, which wee suitable fo alying numeical methods. Lazzain and Liviei 4,5 extended study by Rees to vaious mateial models such as bilinea mateial and modified Rambeg Osgood law. In the autofettage ocess elastic unloading occus only if the essues do not imose the eyielding of the mateial at the boe. In the case of eyielding, fom a theoetical oint of view, excet in some aticula cases, the esidual stesses cannot be obtained by sueimosition of two indeendent analyses the fist fo loading and the second fo unloading, as citically discussed in 5. The kinematic and the isotoic hadening ules ae only two efeence schemes that give an idea of the value of the evese yielding. The manufactuing tubs made with high stength medium alloy steel shows a stong Bauschinge effect 2; theefoe, the eduction of the evese yielding may be consideed An 1 Pesent addess: Faculty of ngineeing and Alied Science, Memoial Univesity of Newfoundland, Canada 2 Coesonding autho. Contibuted by the Pessue Vessel and Piing Division of ASM fo ublishing in the JOURNAL OF PRSSUR VSSL TCHNOLOGY. Manuscit eceived Decembe 19, 2005; final manuscit eceived Novembe 21, Review conducted by Modechai Pel. analytical fame of esidual stesses with the Bauschinge effect was given in 5 fo selected - law duing loading and unloading ocess such as bilinea mateial and the modified Rambe Osgood law simila to that oosed by Wang 7. Howeve, as ecently shown in 12, fo steel 30CNiMo8 a bilinea behavio gives a good inteolation of the exeimental - law data, so that evious analytical equations oosed in 5 may be again used. Fo the geneal Bauschinge effect a numeical ocedue is necessay as oosed in 4 o as indicated by Jahed and Dubey 10 whee the linea elastic solution of axisymmetic bounday value oblem is used as a basis to geneate its inelastic solution with a vaiable Bauschinge effect facto. As indicated above, many studies have been conducted to evaluate stess and stain in autofettaged oen and close ended cylindical essue vessels. These investigations addessed vaious issues: such as, the influence of the yield citeion Tesca, von Mises, mateial models elastic-efectly lastic, bilinea mateials, and Rambeg Osgood, and hadening ule. Howeve, only few such studies exist fo sheical vessels see fo examle Ref. 13. In the aeosace industy, tyically, the shae of the tank is sheical in ode to effectively maintain the intenal hydostatic gas essue 14. The inceasing alication of sheical vessels fo high essue alications motivates the use of autofettage techniques fo an efficient and economic design. In this ae analytical exessions have been deived fo stess and stain duing loading and fo autofettage ocess of sheical vessels with diffeent mateial models. These fomulas can be used to evaluate the esidual stesses, useful fo designes that would otimize the weight of comonents. The aims of this ae can be summaized as follows: a b c To futhe develo the Lazzain and Liviei 4,5 woks, which wee fo cylindical autofettage teated essue vessels, to sheical autofettage teated essue vessels. To obtain, within the lastic zone, some geneal analytical exessions fo stess and stain fo both loading and unloading hases. In aticula such as bilinea mateial and modified Rambeg Osgood law, the stess and stain will be given by closed-fom exessions. To analyze the influence of the Bauschinge effect on the esidual stesses. Jounal of Pessue Vessel Technology Coyight 2007 by ASM AUGUST 2007, Vol. 129 / 411 Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

2 e = 1 2 f lastic stains e = 1 1 = eq = 1 2 eq 5 6 Adoting the analytical ocedue suggested by Rees 3 in the case of cylindical vessels, the combination of qs. 1 6 esults fo sheical vessels in the following diffeential equation: whee d = 1A d eq d eq 3 eq A eq d eq A = 21 Integating q. 7 and alying the bounday condition eq =0 at =R yields 7 R d = 0 eq 1A d eq d eq 3 eq A eq d eq 8 Fig. 1 Sheical essue vessels: a sheical coodinate system and b elastic-lastic egion loading and unloading 2 Geneal quation fo lastic zone ÏR The following exessions ae valid fo a sheical vessel subjected to a unifom intenal essue, P, with isotoic elastic and lastic zones as esented in Fig. 1. Unde these conditions the stesses and the stains deend only on the distance fom the cente of the shee. So that, any sheical coodinate system with the oigin in the cente of the shee is a incial coodinate system see Fig. 1a. The incial assumtions ae: a equilibium b c d e comatibility d e d Pandtl Reuss equation equivalent stess elastic stains d d 2 =0 1 e e d eq =2d eq = = Since eq eq R= y is valid at =R, the following geneal equation can be obtained: = eq A eq A y Radial stess is evaluated using the following elationshi obtained using qs. 1 and 4 togethe eq =2R d C eq 1 y =2R d 2 3 b 1 10 whee C 1 is the constant of integation detemined by imosing the condition that at lastic adius =R, the stess is equal to the stess evaluated in the elastic egion. 3 Analytical Integation fo Diffeent Mateial Model in the Loading Phase In this section, vaious mateial models, such as the bilinea model with both kinematic and isotoic hadening in unloading hase and the Rambeg Osgood law will be studied. By using the elationshi between equivalent stess and equivalent lastic stain eithe eq = f eq law o its invese eq =f 1 eq, and by using the equations intoduced in the evious section, it is ossible to obtain closed fom exessions fo the stess and stain comonents. 3.1 Bilinea Mateial. Fo the bilinea mateial model as esented in Fig. 2, the elation between equivalent stess and equivalent lastic stain can be stated as below 4 eq = y P eq 11 whee = t t Now this mateial model is alied to deive exessions fo stesses and stains in sheical essue vessels. Substituting eq / Vol. 129, AUGUST 2007 Tansactions of the ASM Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

3 and 17 with the afoementioned assumtion, the equivalent stess and stain within the lastic zone can be obtained as eq = y /n eq = y R3 3 R 18 3/n 19 fom q. 11 into q. 9, the equivalent lastic stain can be exessed as eq = A y 1A 1 12 Using qs. 11 and 12 the equivalent stess is obtained as y eq = A 1A R By substituting q. 13 into q. 10, the exession fo adial stess fo bilinea mateial is obtained as 3ln R A 1 R = 2 y 3 b 1 1A Using eq and with q. 4, an exession fo hoo stess is easily deived 3ln R A R3 3 = 2 y 3 b 2 1 1A 15 The lastic adius R may be calculated by imosing in q. 14: = P fo =a. On the othe hand, by intoducing the bounday condition = P at =a in qs. 9 and 15, it is ossible to obtain the lastic adius R in elation to the autofetage essue P as R = a 3 Lambetw A Aa 3 a 3 Ab3 Aa 3 a 3 ex 3 2 A 3 P y 2 P y 1/3 16 LambetwX, also called the omega function, evaluates Lambet s W function at X. This funcation solves the equation We W =X fo W as a function of X 15 and it has many alications in mathematics and engineeing oblems. 3.2 Modified Rambeg Osgood Law. A modified fom of the Rambeg Osgood equation is used instead of the oiginal Rambeg Osgood law to ovecome the difficulty in obtaining the yield stess exlicitly. The - cuve using the modified Rambeg Osgood law is descibed by eq = eq Fig. 2 eq = y eq Bilinea mateial model n y when eq y 17 eq when eq y An assumtion A1/ has been taken 4 which coesonds to the mateial incomessibility condition, =0.5. Using qs. 9 = 2 yn 1 3 R n b3 3/n R 20 = 2 3 yn 1 R b3 4 Unloading Phase 3 2 3/n nr Geneal quations. The alication of autofettage essue duing the loading hase esults in a lastic egion of adius R, Fig. 1. Residual stesses ae intoduced in the intenal essue bounday of sheical essue vessels, in the unloading hase. Due to the high value of the autofettage essue and also the Bauschinge effect, a new lastic adius R would aea, such that arr. Fo the unloading hase, the stesses i i =,, due to a negative vaiation of the intenal essue P, may be obtained by eaanging qs. 1 6 as suggested by Bland fo cylindical vessels 16. Hence the esidual stess i is given by i = i i whee i =,, 22 In analogy with cylindical vessels see Liviei and Lazzain 4,5 the unloading diffeential equation tuns out to be 1A d eq d = d eq 3 eq A eq d eq 23 Fo wok-hadening mateials, consideing the Bauschinge coefficient eq, the unloading yield stess assumes the fom 5 y =2 eq 24 As in the case of cylindical vessels 5, by taking into account q. 24, a good aoximation of the solution of q. 23 can be obtained as = eq A eq 2A eq 25 quation 25 may be solved as well as the unloading - law is known and the hadening ule is assumed. The hadening ule deends on the mateial behavio and only afte a test loading and unloading cuve is it ossible to choose the coect model. On the othe hand, kinematic and isotoic hadening give two easonable bounday limits. Hence, in this ae, an analytical equation fo esidual stess will be develoed fo both kinematic and isotoic hadening ules. Futhemoe, we have consideed a bilinea - law and a owe hadening mateial namely modified Rambeg Osgood whee the load histoy duing unloading hase is elated to the evious loading hase. 4.2 Bilinea Mateial. Fo bilinea mateial, in the unloading hase, the equivalent stess is exessed as eq =2 eq u eq 26 Fom the geneal unloading solution in q. 25, qs. 12 and 13 can be ewitten fo unloading hase as 1 27 eq = 2A eq 1A u R 3 Jounal of Pessue Vessel Technology AUGUST 2007, Vol. 129 / 413 Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

4 Fig. 3 Bilinea mateial unde unloading: kinematic hadening Fig. 4 Bilinea mateial unde unloading: constant isotoic hadening with =1 eq = 2 eq 1A u1a u R 3 28 Also fo obtaining unloading stess, q. 10 can be modified as = 2R eq d C 1 = 2R 2 eq 1A u 1A u R 3d 4 R eq R R b In the esent section, to deive exlicit foms of and, the value of the Bauschinge coefficient is studied. Bilinea mateial exhibits two main hadening behavios: kinematic hadening and isotoic hadening duing the unloading hase Kinematic Hadening. In the case of kinematic hadening, if the tensile stess eaches the value of eq duing lastic defomations, then on unloading the yielding stess comes to a constant value of 2 y Fig. 3. Theefoe, the lastic defomations do not affect the elastic ange and only influence the yield stess in tension and comession. The Bauschinge coefficient intoduced in q. 24 can be witten as q. 30 fo this case = y 30 eq Now, to obtain the stess and stain comonents using equations eviously obtained duing the unloading hase, the substitution of y with 2 y and with u, should be done. Hence eq = 2A y 1A u y eq = 1A u1a u R = 4 A 3 y1 R3 u R3 3 = 4 3 y R 3ln A 2 u A u 3 32 R 1 3ln 1A u 33 R3 34 The esidual stesses can be evaluated using the following equations: R 6ln 2A u 1 R3 3 = 2 3 y = 2 3 y 1A u A 1 3ln R3 1A R3 2R R 6ln A u 1 R A R3 1A u 3 1 3ln R 1A R Isotoic Hadening. In the case of the bilinea isotoic hadening mateial model, duing the unloading hase, the yield stess assumes the value 2 eq, as shown in Fig. 4. Theefoe, the magnitude of the yield stess in tensile and comession ae the same as that obtained fom neglecting the Bauschinge effect. The isotoic hadening is obtained by assuming the value of to be unity eq 1 37 In geneal, by assuming =constant, the exessions of stess and stain comonents ae obtained as = eq = eq = 2A y 1A 1A u 1A 2 y 1A 1A u 1A 2 y 31A 1A u 6ln R 2A u R A u R A2 u 1 R3 3 C / Vol. 129, AUGUST 2007 Tansactions of the ASM Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

5 2 y = 31A 1A u 6ln R 3 A2 u 1 2 R6 6 2A u R 1 3 R3 2 3 C 2 41 whee 4 y R3 C 2 = 31A 1A R31 The esidual stess distibutions ae A 1 R3 = 4 3 y1 = 1A 2 y 31A 1A u 6ln R A 2 u R A u R 3 R y 1 y ln 3 R3 A R3 y R3 1A 4 A y1 1 R3 31A 6ln R A 2 u R6 6 R y R A y1 3 1 A u 3 1A 1A u y 31A 1A u 2A u ln 3 R3 1 A 2 R3 3 1A Modified Rambeg Osgood Law. The loading and unloading - cuves incooating the Bauschinge effect fo the modified Rambeg Osgood law ae shown in Fig. 5, whee n and n ae hadening exonents fo the loading and unloading hase. In geneal, n and n may assume diffeent values; futhemoe, they ae also deendent on the equivalent lastic stain eq. As esented in Fig. 6, thee diffeent cuves ae consideed by assuming n=n=constant duing the unloading hase as suggested by Liviei and Lazzain 5 Case a: Case b: eq eq = y eq eq = y eq Case c: eq 2 1 eq y = 2 eq n eq 2 1 eq 44 n 2 eq y y eq 2 eq y 45 n eq 2 eq eq 46 Fig. 5 - cuve fo the Rambeg Osgood law By means of qs. 25 and qs the eq and eq can be obtained as a function of the adius. Afte some calculations, the eq and eq may be witten as follows: Case a: eq = y 2 /n R 3/n /n 2 R 3/n 31/n R 47 2 R eq = y2 1 3/n /n 2 /n 31/n R 48 Fig. 6 Thee diffeent cases in unloading hase in - cuve of the Rambeg Osgood law Jounal of Pessue Vessel Technology AUGUST 2007, Vol. 129 / 415 Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

6 Case b: Case c: eq = y 12 R3 3 2 R3 3 3/n 1R 3/n 1R eq = y2 12 R 3/n R3 3 eq = 2 y 3/n R R3 3 11/n 49 3/n 1R R 11/n 50 3/n 51 eq =2 y 3/n R /n 52 Hee, as fo the bilinea mateial, exessions will be ewitten including the value of the Bauschinge coefficient Kinematic Hadening. Fo kinematic hadening mateial, keeing in mind that = y / eq, qs can be witten as follows: Case a: Case b: Case c: eq = y 2 R3 3 1 /n 2 R3 3 eq = y2 2 R 3/n R3 3 eq = y 2 R R3 3 1 eq = y12 R3 3 eq = 2 y R3 R 3 eq =2 y /n 1 31/n R /n R /n 55 1/n 56 3/n In ode to detemine and, it is equied to use the integated q. 29. Howeve, fo cases a and b, the wites could not deive the exlicit fom of and ; theefoe, fo the uose of this ae, only case c is studied = 4 3 yn 1 R = 4 3 yn 1 b 3 b n R 3 2 nr 3/n 59 3/n 60 And also fo case c the esidual stess is given by = 2 3 y1 n 2 R b3 n 2n R /n /n 61 = 2 3 y1 n R b /n nr R 2 R 3 3 2n b 3/n Isotoic Hadening. By assuming =1 o constant in qs , valid exessions fo the isotoic hadening mateial can be obtained. As discussed ealie fo kinematic hadening mateial, the exlicit foms of and ae only ossible fo case c = 2 3 yn/n /n 1 6/n 1 R 6/n 2 R R R = 2 3 y R R3/n2 R 3/n1 R3 1 R3 n 3 R 6/n Also the esidual stesses fo case c ae as follows: = 2 3 yn/n /n 1 6/n 1 R 6/n 2 R R R n 3/n1 R n 1 R b3 n /n 65 = y 2 3 R3/n2 R R n 1 R b /n nr 5 Veification 1 R3 n 3 R 6/n 66 n In ode to validate the exessions of esidual stess, the finite element method 17 has been used. The quadatic axisymmetic 8-node elements have been used fo inelastic finite element analysis. A tyical finite element mesh with bounday conditions is esented in Fig. 7. The vaiation of stess comonents fo tyical geomety and mateial oety ae esented in Figs. 8 and 9. In Fig. 8, bilinea kinematic hadening BKH with the mateial oeties of =206 GPa, t = tu =10 GPa, y =850 MPa, and =0.3, has been used. Fo bilinea isotoic hadening BIH, Fig. 9, the same mateial oety is used, assuming ==1. As can be seen fom Figs. 8 and 9, a good ageement between numeical simulation and the esent study is obtained. Fom the above figues, the loading lastic adius, R, aoaches the analytical value of m. Duing the unloading hase, the value of the lastic adius, R, obtained fom the finite element solution fo BKH and BIH models ae about R= and R=0.064, esectively. The detail analytical stess distibution along the thickness of the sheical vessel in thee conditions, loading, unloading and esidual, ae illustated in Figs fo each mateial model. As can be obseved fom the figues, all cuves ae continuous. It imlies that the exessions alicable fo elastic and lastic egion ae continuous. In all cases the same geomety and loading condition ae used: a=0.06 m, b=0.240 m, and P=3 y y =850 MPa. The lot of stess fields fo the BKH mateial model with =206 GPa and t = tu =10 GPa is shown in Fig. 10. Fom the figue, it can be seen that the analytical lastic adius afte loading is R= and afte unloading it is R= m. Figue 11 shows the stess fields fo BIH mateial model with =206 GPa, t = tu =10 GPa, and ==1. Refeing to the figue, the lastic adius afte loading is R= and afte unloading is R=0.064 m. 416 / Vol. 129, AUGUST 2007 Tansactions of the ASM Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

7 Fig. 7 Tyical finite element mesh Fig. 8 The vaiation of esidual stess comonents in the thick wall of a shee fo a=0.06 m, b=0.240 m, P=3 y, =206 GPa, and t = tu =10 GPa with the BKH mateial model Fig. 10 The stess distibution along the wall thickness fo BKH a=0.06 m, b=0.240 m, P=3 y, y =850 MPa, =206 GPa, and t = tu =10 GPa, a loading, b unloading, c esidual Fig. 9 The vaiation of esidual stess comonents in the thick wall of a shee fo a=0.06 m, b=0.240 m, P=3 y, =206 GPa, and t = tu =10 GPa with the BIH mateial model Figue 12 esents the stess fields fo the Rambeg Osgood kinematic hadening mateial model with =206 GPa and n=5. Fom the figue it can be seen that the analytical lastic adius afte loading is R= and afte unloading is R= m. Figue 13 shows the stess fields fo the Rambeg Osgood isotoic hadening mateial model with =206 GPa, ==1, and n=5. The analytical lastic adius afte loading is R= m and afte unloading is R= m. As in the case of cylindical vessels, the Bauschinge effect oduces less comessive esidual stess comaed to ideal autofettage solutions, theefoe, in ode to obtain an accuate esidual stess ediction, the Bauschinge effect has to be taken into account. Jounal of Pessue Vessel Technology AUGUST 2007, Vol. 129 / 417 Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

8 Fig. 11 The stess distibution along the wall thickness fo BIH mateial model a=0.06 m, b=0.240 m, P=3 y, y =850 MPa, =206 GPa, and t = tu =10 GPa, a loading, b unloading, c esidual Fig. 12 The stess distibution along the wall thickness fo the Rambeg Osgood kinematic hadening mateial model a =0.06 m, b=0.240 m, P=3 y, y =850 MPa, =206 GPa, and n =5, a loading, b unloading, c esidual 6 Conclusion The main conclusions of this study on autofettage oblems in sheical vessels may be summaized as follows: 1 As in the case of cylindical essue vessels, the esidual stess in sheical vessels due to the autofettage ocess is vey sensitive to both the Bauschinge effect and the tensile law. 2 Fo two classes of mateials bilinea and owe hadening mateial, analytical exessions have been obtained both fo the loading and the unloading hases with kinematic and isotoic hadening. 3 As in the case of a cylinde, comaing the induced esidual stess in the hollow shee fo kinematic and isotoic mateial models, it can be concluded that in the intenal adius the magnitudes of tangential stesses ae almost the same; howeve, the lastic adius, afte unloading fo the kinematic hadening is consideably lage than the isotoic hadening mateial model. 4 The knowledge of the analytical esidual stesses would be useful fo the otimization of autofettage essue fo a given geomety and mateial oeties. In some essue vessel comonents involving comound shaes, e.g., combination of cylindical and sheical comonents, such as sheical closed end vessels, the stesses of these comonents ae needed. 418 / Vol. 129, AUGUST 2007 Tansactions of the ASM Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://

9 Fig. 13 The stess distibution along the wall thickness fo the Rambeg Osgood isotoic hadening mateial model a =0.06 m, b=0.240 m, P=3 y, y =850 MPa, =206 GPa, and n =5, a loading, b unloading, c esidual Nomenclatue a inne adius b oute adius d, d stain and stess incement elastic modulus t tangent modulus loading hase tu tangent modulus unloading hase lastic modulus loading hase u lastic modulus unloading hase n stain-hadening coefficient loading hase n stain-hadening coefficient unloading hase P intenal essue adius R lastic adius afte loading R lastic adius afte unloading Bauschinge coefficient,, total stain comonents e,, elastic stain comonents,, lastic stain comonents eq equivalent stain loading hase eq equivalent stain unloading hase e eq equivalent elastic stain equivalent lastic stain loading hase eq eq equivalent lastic stain unloading hase Poisson s atio eq Von Mises equivalent stess,, stess comonents y yield stess The subscit,, indicate the comonent in a sheical coodinate system. Subscit adial diection tangential diection ola diection y yield Suescit e elastic lastic Abbeviation BKH bilinea kinematic hadening BIH bilinea isotoic hadening FM finite element method Refeences 1 Megahed, M. M., and Abbas, A. T., 1991, Influence of Revese Yielding on the Residual Stesses Induced by Autofettage, Int. J. Mech. Sci., 332, Pay, J. S. C., 1965, Fatigue of Thick Cylindes: Futhe Pactical Infomation, Poc. Inst. Mech. ng., 180, Rees, D. W. A., 1987, A Theoy of Autofettage With Alications to Cee and Fatigue, Int. J. Pessue Vessels Piing, 30, Lazzain, P., and Liviei, P., 1997, Diffeent Solution fo Stess and Stain Fields in Autofettaged Thick-Walled Cylindes, Int. J. Pessue Vessels Piing, 71, Liviei, P., and Lazzain, P., 2002, Autofettage Cylindical Vessels and Bauschinge ffect: An Analytical Fame fo valuating Residual Stess Distibutions, ASM J. Pessue Vessel Technol., 124, Chen, P. C. T., 1986, Bauschinge and Hadening ffect on Residual Stesses in an Autofettaged Thick Walled Cylinde, ASM J. Pessue Vessel Technol., 108, Wang, G. S., 1988, An lastic Plastic Solution fo Nomally Loaded Cente Hole in a Finite Cicula Body, Int. J. Pessue Vessels Piing, 33, Pake, A. P., and Undewood, J. H., 1999, Detemination of Residual Stess Distibutions in Autofettaged Thick Walled Cylindes, Fatigue and Factue Mechanics: 29th Vol. ASTM STP, 1332, Pake, A. P., 2001, Autofettage of Oen nd Tubes Pessues, Stesses, Stains and Code Comaisons, ASM J. Pessue Vessel Technol., 123, Jahed, H., and Dubey, R. N., 1997, An Axisymmetic Method of lastic- Plastic Analysis Caable of Pedicting Residual Stess Field, ASM J. Pessue Vessel Technol., 119, Pake, A. P., Undewood, J. H., and Kendall, D. P., 1999, Bauschinge ffect Design Pocedues fo Autofettaged Tubes Including Mateial Removal and Sachs Method, ASM J. Pessue Vessel Technol., 1214, Huang, X. P., 2005, A Geneal Autofettage Model of a Thick-Walled Cylinde Based on Tensile-Comessive Stess-Stain Cuve of a Mateial, J. Stain Anal. ng. Des., 406, Nayebi, A., and l Abdi, R., 2002, Cyclic Plastic and Cee Behaviou of Pessue Vessels Unde Themomechanical Loading, Comut. Mate. Sci., 253, Lee, H.-S., Yoon, J.-H., Pak, J.-S., and Yi, Y.-M., 2005, A Study on Failue Chaacteistic of Sheical Pessue Vessel, J. Mate. Pocess. Technol., , Coless, R. M., Gonnet, G. H., Hae, D.. G., Jeffey, D. J., and Knuth, D.., 1996, On the Lambet W Function, Adv. Comut. Math., 5, Bland, D. R., 1956, lastolastic Thick-Walled Tubes of Wok-Hadening Mateial Subject to Intenal and xtenal Pessues and to Temeatue Gadients, J. Mech. Phys. Solids, 4, ANSYS, 2004, Univesity Reseach Vesion, 7.1, SAS IP, Inc. Jounal of Pessue Vessel Technology AUGUST 2007, Vol. 129 / 419 Downloaded 11 Nov 2008 to Redistibution subject to ASM license o coyight; see htt://