Junal Meanial Decembe 007, No. 4, - 4 OTIMUM AUTOFRETTAGE RESSURE IN THICK CLINDERS Aman Ayob *, M. Kabashi Elbashee Depatment of Applied Mechanics, Faculty of Mechanical Engineeing, Univesiti Tenologi Malaysia, 830 UTM Sudai, Joho, Malaysia. ABSTRACT In optimal design of pessuised thic-walled cylindes, an incease in the allowable intenal pessue can be achieved by an autofettage pocess. An analysis is caied out on plain cylindes by using the von Mises and Tesca yield citeia to develop a pocedue in which the autofettage pessue is detemined analytically, esulting in a educed stess distibution. A validation by a numeical simulation shows that the analytical and numeical simulations coelate well in tems of tend and magnitude of stesses. Keywods: Autofettage, pessue vessels, esidual stess, plastic collapse, finite element analysis.0 INTRODUCTION Autofettage is a common pocess of poducing esidual stesses in the wall of a usually thic-walled cylinde pio to use. An appopiate pessue, lage enough to cause yielding within the wall, is applied to the inne wall of the cylinde and then emoved. Lage scale yielding occus in the autofettaged thic-walled cylinde wall, []. Upon the elease of this pessue, a compessive esidual cicumfeential stess is developed to a cetain adial depth at the boe. These esidual stesses seve to educe the tensile stesses developed as a esult of subsequent application of an opeating pessue, thus inceasing the load beaing capacity, [], [3]. Due to the eve-inceasing industial demand fo axisymmetic pessue vessels which have wide applications in chemical, nuclea, fluid tansmitting plants, powe plants and militay equipment, the attention of designes has been concentated on this paticula banch of engineeing. The inceasing scacity and high cost of mateials have led eseaches not to confine themselves to the customay elastic egime but attacted thei attention to the elastic-plastic appoach which offes moe efficient use of mateials, [4]. * Coesponding autho: E-mail: aman@fm.utm.my
Junal Meanial, Decembe 007.0 CLINDER SUBJECTED TO INTERNAL RESSURE Fo a cylinde subjected to an intenal pessue, i, the adial stess,, and cicumfeential stess, θ, distibutions ae given by Lame s fomulation: and i o = () θ i o = + () θ Fo a cylinde with end caps and fee to change in length, the axial stess is given by [5]: i z = (3) 3.0 IELD CRITERIA Accoding to the Tesca yield theoy, yielding occus when the Tesca equivalent stess is [5]: = ( ) = (4) T θ Based on the von Mises yield theoy, yielding occus when the von Mises equivalent stess is [6]: {( ) ( ) ( ) } vm = θ + z + z θ = (5) Two impotant pessue limits,,i and,o, ae consideed to be of impotance in the study of pessuised cylindes.,i coesponds to the intenal pessue equied at the onset of yielding at the inne suface of the cylinde, and,o is the intenal pessue equied to cause the wall thicness of cylinde to yield completely. The magnitudes of,i and, o, accoding to Tesca yield citeion ae, [], [6], [7]: T,,i ( - ) = (6) T,,o ( - ) = (7)
Junal Meanial, Decembe 007 and based on the von Mises yield citeion, the magnitudes of, i and, o ae, []: ( - ) vm,,i = (8) 3 vm,,o ( - ) = (9) 3 Equations (4) and (5) give the elation between the von Mises and Tesca equivalent stesses fo the state of stess in a pessuised thic-walled cylinde: 3 3 vm = ( θ ) = T (0) and shows that the Tesca citeion is moe consevative than the von Mises citeion by 5.5 %. 4.0 RESIDUAL STRESSES If the intenal pessue is emoved afte pat of the cylinde thicness has become plastic, a esidual stess is set up in the wall. Assuming that duing unloading the mateial follows Hooe s Law, the esidual stesses can be obtained fom the equations below. Fo the plastic egion, i a, the espective esidual stesses in the adial, hoop and axial diections ae [8]: m m o, p, R = ln + ln ( m ) + a m m o θ, p, R = + ln + ln ( m ) + + a m m z, p, R = + ln + ln ( m + ) a (a) (b) (c) Fo the elastic egion, a o, the espective esidual stesses in the adial, hoop and axial diections ae: o m m = ln ( m ) +, e, R (a) 3
Junal Meanial, Decembe 007 o m m θ, e, R = ln ( m ) + + m m z, e, R = ln + ( m ) (b) (c) whee m i Equations (a-c), the esidual stesses at the junction adius a ae obtained: a = and a is the autofettage adius. By substituting = a in m m, R = ln ( m ) m + m m = ln ( m ) + m + θ,r m m z, R = ln ( m ) + (3a) (3b) (3c) The esidual stess distibutions ae shown in Figues and..0e+08.8 0 E+0 8.4 0 E+0 8.0 0 E+0 8 θ θ,t Stess [Ma] 6.00E+07.00E+07 -.00E+07 0. 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0. -6.00E+07 θ,r -.00E+08 -.40E+08 -.80E+08 lastic Region Elastic Region -.0E+08 i Radius in cylinde wall [m] o Figue : Hoop stesses ( θ ) due to opeating pessue, esidual autofettage pessue ( θ,r ) and total ( θ,t ) 4
Junal Meanial, Decembe 007.0E+08.80E+08.40E+08 a,opt = 0 Ma a,opt = 49. mm Stess [Ma].00E+08 6.00E+07.00E+07 -.00E+07 0. 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0. -6.00E+07,R θ.r -.00E+08 -.40E+08 -.80E+08 z,r lastic Region Elastic Region -.0E+08 i Radius in cylinde wall [m] o Figue : Residual stess distibutions in cylinde wall On application of the opeating pessue the total stess of the patially autofettaged cylinde is the summation of the esidual stess and the stess due to the opeating pessue, i.e.: = + (4a), T,R,op = + (4b) θ, T θ,r θ,op = + (4c) z, T z,r z,op The above total stesses ae shown in Figue 3..0E+08.80E+08.40E+08 θ,t Stess [Ma].00E+08 z,t 6.00E+07.00E+07 -.00E+07 0. 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0. -6.00E+07 -.00E+08 -.40E+08 -.80E+08,T lastic Region Elastic Region -.0E+08 i Radius in cylinde wall [m] o Figue 3: Total hoop, adial and axial stess distibutions in cylinde wall when subjected to opeating pessue, afte autofettage 5
Junal Meanial, Decembe 007 Hence at = a, when the cylinde is subjected to an intenal opeating pessue, afte being teated by autofettage, the Tesca equivalent stess at the elasticplastic junction is: m m op T = ( ln ( m )) m + + m Diffeentiating T with espect to m and equating the diffeential to zeo: dt 4 = ln ( m) 4op 0 3 3 3 3 dm + = m m m m to obtain op m = exp( ) (6) d T Since 0, the a (= a,opt ) obtained is the optimum and minimum da op autofettage adius. Letting n =, theefoe, and mt mvm (5) = exp( n) - Tesca (7) 3 = exp n - von Mises (8) Figue 4 shows how the optimum autofettage adius is influenced by the opeating pessue. It shows that fo a cetain opeating pessue, the optimum autofettage adius obtained using Tesca yield citeia is moe than that using von Mises citeion. 3 Optimum autofettage / i (mopt).5.5 0.5 m T m vm 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9.0 Opeating pessue / ield stess (n) Figue 4: Effect of yield citeia on optimum autofettage adius 6
Junal Meanial, Decembe 007 5.0 MAXIMUM INTERNAL RESSURE OF AUTOFRETTAGED CLINDER Equations (6) and (7) ae used to obtain the (Tesca) autofettage pessue to cause diffeent stages of yielding in a vigin cylinde. Fo a cylinde teated with patial autofettage, the intenal pessue to cause the inne suface to yield again can be obtained. Substituting Equations (), () and () into Equation (4), and using Tesca yield citeion, when = i, the intenal pessue to cause yielding at the inne suface is, m, i = ln ( m) + (9) and when = o, by substituting Equations (), () and () into Equation (4), and using Tesca yield citeion, the intenal pessue to cause the whole wall thicness to yield is,, o = ln ( m) + m (0) Figues 5 and 6 espectively show the maximum intenal pessue to cause yielding at the inne suface and to cause the whole cylinde to yield. These pessues ae influenced by diffeent optimum autofettage pessue levels which wee obtained when an opeating pessue was initially nown. The intenal pessue to cause yielding at the inne suface of a cylinde which is teated with optimum autofettage pessue, is geate than that fo a non-teated cylinde (Figue 5). On the othe hand, the intenal pessue to cause full yielding in a cylinde which has been teated with optimum autofettage, is lowe than that which is non-autofettaged (Figue 6)..5 0.5 0..4 3.5 4.7 5.8 Maximum intenal pessue / ield stess 7.0 8. 9.3 0.4.6.7 3.9 5.0 Optimum Autofettage Non-Autofettage 0. -- 0.5 0.3 -- 0.35 0.4 -- 0.45 0.5 -- 0.55 0.6 -- 0.65 0.7 -- 0.75 0.8 -- 0.85 0.9 -- 0.95 Opeating pessue / ield stess [n] Radius atio [] Figue 5: Maximum intenal pessue to cause inne suface to yield, with diffeent optimum autofettage levels- Tesca 7
Junal Meanial, Decembe 007 Maximum opeating pessue / ield stess 6 5 4 3 Non-Autofettage Optimum Autofettage.0 0.9 0.8 0.7 0.6 0.5 0.45 0.4 0.35 0.3 0.5 Opeating pessue / ield stess [n] 0..5.8..4.7 3 3.3 3.6 Radius atio [] Figue 6: Maximum intenal pessue to cause the whole wall to yield, with diffeent optimum autofettage levels-tesca 6.0 FULL AUTOFRETTAGED CLINDER A special case is when the cylinde is fully autofettaged, i.e. a = o. Theefoe m = and the Tesca equivalent stess at any adius can be obtained fom Equation (5): ( ) ln o op o = + T () Theefoe the intenal pessue to cause the intenal suface and whole wall to yield is obtained by substituting = i, = o and m = in Equations (9) and (0). The compaison of allowable intenal pessues of a cylinde teated with full and non-autofettage, ae shown in Table and in Figues 7 and 8. Figue 7 shows that full autofettage is beneficial if yielding of the inne suface is equied, in which case the cylinde can sustain the highest intenal pessue. To cause the whole wall to yield, the cylinde should not be autofettaged, in which case the cylinde can sustain the highest intenal pessue, as shown in Figue 8. 8
Junal Meanial, Decembe 007 Autofettage Level Table : Compaison between allowable intenal pessues on non-autofettaged, optimumly autofettaged and fully autofettaged thic-walled cylinde Intenal pessue to cause the inne suface to yield i / Intenal pessue to cause the whole cylinde wall to yield i / Non autofettage Optimum autofettage Full autofettage ( ) m ln ( m) + 0.375 ( ) 0.6 ln ( ).5 m + m.87 ln 0.693 ln 0.693 Intenal essue/ield stess [n].8.6.4. 0.8 0.6 0.4 0. 0.96 0.40 Fully- Autofettaged Non-Autofettaged 0 Radius Ratio [].3.7..5.9 3.3 3.7 4. 4.5 4.9 Radius Ratio [] Figue 7: Maximum intenal pessue to cause intenal suface to yield Intenal essue/ield stess [n] 5 Non-Autofettaged 4.5 4 3.5 3.65.5.5 Fully- Autofettaged 0.96 0.5 0.3.6.9..5.8 3. 3.4 Radius Ratio [] Figue 8: Maximum intenal pessue to cause whole thicness to yield 9
Junal Meanial, Decembe 007 7.0 THEORETICAL OTIMUM AUTOFRETTAGE RESSURE The autofettage pessue a is a sufficiently high intenal pessue applied befoe a cylinde is put into use by applying an opeating pessue. The adius of the elastic-plastic junction line is called the autofettage adius a. The objective is to design fo a total minimum equivalent stess at the junction line. The value of autofettage pessue which satisfies this condition is called the Optimum Autofettage essue, a, opt and the adius of elastic-plastic junction line is called the Optimum Autofettage Radius, a, opt. The intenal pessue to cause (Tesca) yielding to a depth of is: = + ln o i Fom Equations (7) and (8) the optimum autofettage adius is deduced as, e n a, opt i 3 n a, opt i e = - Tesca = - von Mises Theefoe the optimum autofettage pessue is: () n e = n + a, opt, T - Tesca (3) 3n e = + 3n a, opt, vm - von Mises (4) The above optimum autofettage pessues esult in the minimum equivalent stess and occus on the elastic-plastic junction line as shown in Figue 9. Equivalent stess at junction line/ield stess a, opt a, opt eq 49.8 mm 0 Ma 39 Ma i o 00 mm 00 mm o 30 Ma 35 Ma n 0.4 Autofettage pessue/ield stess 0 Figue 9: Optimum autofettage pessue and adius
Junal Meanial, Decembe 007 8.0 OTIMUM AUTOFRETTAGE AND MAXIMUM OERATING RESSURES The elation between the optimum autofettage pessue and opeating pessue of thic-walled pessuised cylindes can be obtained: a,opt,t op,t = + e n n - Tesca (5) Figue 0 shows the optimum autofettage pessue/opeating pessue atio vaying with the adius atio, using Tesca yield citeion. Fo thic walled cylindes, inceasing the opeating pessue leads to an incease in the optimum autofettage pessue, which in tun leads to an incease in autofettage adius. a,opt,vm op,vm 3 e = + n 3n - von Mises (6) Figue0: Optimum autofettage fo diffeent values of opeating pessue and adius ation
Junal Meanial, Decembe 007 9.0 FINITE ELEMENT ANALSIS The autofettage pocess may be simulated by Finite Element Method, maing use of elastic-plastic analysis. It is possible to model the autofettage pocess by applying pessue to the inne suface of the model, emoving it and then calculating the esidual stess field, followed by eloading with an opeating pessue. Using a D axisymmetic element available in ABAQUS v6.5 [9] a finite element mesh of a cylinde with an inside adius 00 mm and outside adius of 00 mm was geneated. An autofettage pessue of 0 Ma was applied, and then emoved. The esidual stess distibutions wee evaluated in the thicwalled cylinde. The opeating pessue of 30 Ma was then applied. The von Mises equivalent stess was used in the subsequent analysis. The mateial used was steel and this mateial has the following popeties: E = 03 Ga = 35 Ma ν = 0.33 The mateial is assumed to be isotopic, linealy elastic and has bilinea inematic hadening using von Mises plasticity esponse. 9. FEM Results Using the above FE cylinde model and compaing between Tesca and von Mises citeia, the autofettaged thic-walled cylinde give the following esults on the diffeence in the optimum autofettage adius and pessue. Tesca citeion von Mises citeion m.49.44 a, opt 49.0 mm 4.4 mm a, opt 0 MN/m 94 MN/m Table shows the influence of autofettage level on the allowable intenal pessue, using Tesca yield citeion, the diffeent levels being cylindes not teated with autofettage, teatment with optimum autofettage and autofettaged until the whole cylinde has yielded. 0.0 CONCLUSIONS The following conclusions ae thus dawn: ) The autofettage pocess inceases the maximum allowable intenal pessue. ) The autofettage pocess cannot incease the maximum intenal pessue to cause the whole thicness of the cylinde to yield. 3) If the opeating pessue, op, is lage, the optimum bounday adius ( a, opt ) is also lage.
Junal Meanial, Decembe 007 4) If the yield stess,, is lage, the optimum bounday adius ( a,opt ) is small. 5) The optimum autofettage pessue causes the lowest equivalent stess duing application of opeating pessue, and this occus at the elasticplastic junction line. 6) The optimum autofettage adius, a,opt depends on the opeating pessue op, and the inne adius of the thic-wall cylinde i, apat fom the mateial popety. NOMENCLATURE t m n τ pessue adius thicness oute:inne adius atio autofettage:inne adius atio opeating pessue:yield stess atio nomal stess shea stess Subscipts i o a θ z p e opt op max min T vm R T inne oute autofettage adial hoop axial yield plastic elastic optimum opeating maximum minimum Tesca von Mises esidual total REFERENCES. Thumse, R., Begmann, J.W., Vomwald, M., 00. Residual stess fields and fatigue analysis of autofettage pats, Int Jnl essue Vessels and iping, 79(), 3-7.. ey, J., Aboudi, J., 003. Elastic-plastic stesses in thic-walled cylindes, ASME Jnl essue Vessel Technology, 5, 48-5. 3
Junal Meanial, Decembe 007 3. Zhao, W., Seshadi, R., Dubey, R.N., 003. On thic-walled cylinde unde intenal pessue, ASME Jnl essue Vessel Technology, 5, 67-73. 4. Wang Zhiqun, 995. Elastic-plastic factue analysis of a thic-walled cylinde, Int Jnl essue Vessels and iping, 63, 65-68. 5. Mott, R.L., 00. Applied Stength of Mateials, New Jesey, Ohio: entice Hall. 6. Hill, R., 967. The mathematical theoy of plasticity, Oxfod Univesity ess. 7. Chaaban, A., Baae, N., 993. Elasto-plastic analysis of high pessue vessels with adial coss-boes, High essue Technology, ASME essue Vessel and iping; 63, 67-3. 8. Wuxue, Z., Zichu, Z., 985. An elastic-plastic analysis of autofettage thic-walled cylindes, oc Int Conf on Non-Linea Mechanics, Shanghai, 663-667. 9. ABAQUS v6.5., 005. Getting stated/standad Use s Manual. 4