same

analnls,ander,pples chislopessateauanaatre29

b appropriate subdivision of the regionc implementation of physically realistic boundary

conditions

Accordingly, the results from an FE model of a thick-walledcylinder are here compared to results from an analytical model 3and the Hencky programme 4. Specifically, the autofrettage and

DownlAll of these requirements are met in the work reported herein aspart of the initial development and validation of a finite elementFE procedure.

The Bauschinger effect 1 manifests itself as a reduction ofcompressive yield stress as a result of prior plastic tensile strain2. The Bauschinger effect factor is the normalized magnitudeof compressive stress during unloading at which behavior deviatesfrom linear elastic. In the development phase reported here, thestrain hardening during loading and unloading and Bauschingereffect during unloading are assumed to be bilinear. A value of isselected to ensure reyielding near the bore as autofrettage pressureis removed.

Figure 1 shows the inner bore and outer radii ra and rb, aswell as the primary and secondary yield radii rp and rs, whichdesignate the extent of plastic deformation during the loading andunloading processes, respectively.

This paper summarizes the development of a family of finiteelement models aimed at creating an accurate representation ofthe different end conditions used when modeling hydraulic autof-rettage, rather than a precise material model of a gun steel. Instead

residual stresses are scrutinized: the former referring to stressesdeveloped when the tube is subject to the peak autofrettage pres-sure; the latter to stresses that remain when the autofrettage pres-sure is removed.

2 Compared MethodsThis paper compares the current FE model with other methods,

for a number of different end conditions, which are describedbelow. All three methods use the von Mises yield criterion, gen-erally more suitable than even a well-developed Tresca solution,such as that formulated by Liu 5, to predict behavior within theplastic region. All configurations are pressurized to cause initialplastic deformation throughout an equal proportion of the tubewall thickness. This proportion is termed overstrain and is oftendefined as a percentage of the wall thickness.

2.1 End Conditions

Open-ended: Hydraulic autofrettage in which pressure ismaintained by two frictionless pistons, with zero netaxial force in the tube itself. A gun barrel may be con-sidered as an open-ended tube as one end is left free, andthere is zero net axial force in the tube itself.

Closed-ended: Hydraulic autofrettage in which pressureis maintained by a cap at each end, creating a net axialforce in the tube itself. Provided calculations are per-formed at least one St. Venant distance from each cap,

Contributed by the Pressure Vessel and Piping Division of ASME for publicationin the JOURNAL OF PRESSURE VESSEL TECHNOLOGY. Manuscript received December 5,2005; final manuscript received December 19, 2005. Review conducted by John H.Underwood. Paper presented at the Gun Tubes Conference 2005, April 1014, 2005,Keble College Oxford, hosted by Cranfield University, RMCS, Shrivenham, PaperNo. S3P11.

Journal of Pressure Vessel Technology MAY 2006, Vol. 128 / 217Copyright 2006 by ASMEMichael C. Gibsone-mail: [email protected]

Amer Hameed

Anthony P. Parker

John G. Hetherington

Defence College of Management and Technology,Engineering Systems Department,

Cranfield University at the Defence Academy,Swindon, SN6 8LA, UK

A CompPredictiStrain-HThick CyBauschiHigh-pressure vesseoperating pressuresection of the cylindation. Subsequent abefore tensile stressworking pressure. Tthat have been deveconditions: plane strplane strain). The mhardening and the Bmodel and a recentThis demonstrates thsimulate high-pressuDOI: 10.1115/1.217

1 IntroductionNumerical stress analysis of the relatively complex behavior of

a tube during autofrettage pressurization and depressurization re-quires:

a implementation of an appropriate yield criterionoaded 11 Nov 2008 to 129.5.224.57. Redistribution subject to ASMErison of Methods forg Residual Stresses inrdening, Autofrettagedinders, Including theger Effectsuch as gun barrels, are autofrettaged in order to increase their

fatigue life. Autofrettage causes plastic expansion of the innersetting up residual compressive stresses at the bore after relax-ication of pressure has to overcome these compressive stressesan be developed, thereby increasing its fatigue lifetime and safepaper presents the results from a series of finite element modelsed to predict the magnitude of these stresses for a range of endand several plane-strain states (open and closed ended, plus true

rial model is currently bilinear and allows consideration of strainschinger effect. Results are compared to an alternative numericallytical model (developed by Huang), and show close agreement.general purpose finite element analysis software may be used tovessels, justifying further refining of the models.64

a bilinear material, incorporating the Bauschinger effect reverseyield stress set by was selected corresponding to the - plotshown below Fig. 2.

To confirm the validity of a new model, comparison to resultsobtained using alternative models is essential. Only then may thenew model be applied to new configurations. license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Table 1 Huang loading parameters

K=2.0 K=2.5

Downlthe tube is in a plane state with z constant, but nonzerothroughout the tube wall.

Plane strain: More formally termed true plane strain, thiscondition sets axial strain to zero throughout the tubewall. This represents the midlength state of a long tubeconstrained to its original length, such as a built-in pres-sure vessel.

Plane stress: In this condition axial stress is set to zero.This may be used to represent a relatively thin sheet.Accordingly, this condition is not particularly useful interms of pressure vessels and gun tubes, which tend to belong. It may be used to model configurations such asholes in aircraft skins subjected to mandrel enlargement.Also, analytical plane stress solutions are frequently usedto normalize results for other end conditions.

The numerical modeling of each of the above end conditions isdescribed in later sections.

2.2 Huangs Model. Huang has developed a von Mises solu-tion of an elastic-plastic tube. The solution is made possible by thefollowing simplifications:

Incompressibility r + + z = 0 1

Plane strain z = 0 2Coincidentally, for this case the incompressibility condition cre-ates stresses that are identical to the closed-end condition. Themodel considers the tube material to behave linearly in the elasticphase both loading and unloading, and either linearly or accord-ing to a power law in the plastic phase loading and unloading

Fig. 1 Tube geometry and yield diagram

Fig. 2 Material stress-strain diagram

218 / Vol. 128, MAY 2006oaded 11 Nov 2008 to 129.5.224.57. Redistribution subject to ASMEmay be treated independently. To match the material used in theANSYS model, linear strain hardening was selected. It is recog-nized that, following initial plastic deformation, many materials ofinterest exhibit significant nonlinearity during unloading. How-ever, the purpose of this work is to validate other aspects of themodel and the nonlinear unloading complexity is deliberatelyavoided.

Although Huangs formulation is restricted to an incompress-ible material under plane strain, it does nonetheless allow forrapid analysis of an autofrettaged tube. Other methods such asAvitzur 6,7 and Bland 8 are less direct. Huangs method istherefore well suited for comparison to, or validation of, anothermethod.

Table 1 contains the three parameters relating to the loadingphase. They are selected with knowledge of the materials elastic-plastic loading curve and hence are the same for both wall ratios.The elastic stresses part 0-1 in Fig. 2 are given by Eq. 3, theplastic stresses part 1-2 in Fig. 2 by Eq. 4

= E1 3

= A1 + A2B1 4Table 2 contains the three parameters relating to the unloading

phase, which depend on the prior plasticity in the tube. Accord-ingly, two sets of unloading parameters must be defined, one foreach wall ratio. These account for the fact that for a given over-strain, the bore plastic strain, on which strain hardening and theBauschinger effect depend, increases with wall ratio. Huangsmethod assumes a uniform response to plastic strain in the initialyield zone, as does the ANSYS model. The elastic stresses part 2-3in Fig. 2 are given by Eq. 5, the plastic stresses part 3 in Fig.3 by Eq. 6

UL = UL E2 5

UL = A3 + A4 ULB2 6

2.3 Hencky Program. The Hencky program allows accuratesolution of hydraulic autofrettage for a wide range of end condi-tions and materials. The program was derived from a basic for-mulation by Jahed and Dubey 9 and further developed by Parkeret al. 4. It allows radial variation of unloading properties whichare crucial when determining residual stresses based on priorplastic strain. This feature is beyond the capability of the ANSYSFE model 10 employed here. Notably, the Hencky program cansimulate the often significant nonlinearity exhibited by variouscandidate gun steels e.g., A723, HY180, PH 13-8Mo during un-loading following initial plastic deformation.

2.4 ANSYS FE Model. Two forms of ANSYS models were cre-ated, one to simulate the plane stress condition and the other theplane strain conditions true plane strain z=0, plus open and

A1=990 MPa A1=990 MPaA2=20.9 GPa A2=20.9 GPa

B1=1 B1=1

Table 2 Huang unloading parameters

K=2.0 K=2.5

A3=1623 MPa A3=1999 MPaA4=20.9 GPa A4=20.9 GPa

B2=1 B2=1

Transactions of the ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Downlclosed ended. Both forms utilized an eight node, two-dimensional2D, element, namely, PLANE82 which allows accurate results tobe obtained in the 2D analyses conducted.

Autofrettage pressures were determined by using the corre-sponding value from the Hencky program as an initial figure; theprecise autofrettage pressures for the relevant ANSYS model werethen obtained by interpolating the autofrettage stresses and iterat-ing to give equal overstrain.

2.5 Plane Stress Model. The plane stress model Fig. 3 rep-resents a small angle 0.5 deg of an r , section of the tube. Theplane stress element property was employed to achieve the re-quired end condition. A range of angles and mesh sizing factorswere varied in an error analysis, to obtain a low-complexity modelstill generating accurate results.

The results of the error analysis were used to develop a mesh inwhich the radial dimension of the elements was varied throughoutthe tube wall; smaller elements were used at the bore to properlymodel plastic deformation and associated stresses.

2.6 Plane Strain Models. The plane strain models representan r ,z section of the tube in which the longitudinal z- axis isplaced on the global ANSYS Y-axis, allowing the elements axi-symmetric property to be used to reduce model size. Again, arange of models was tested, altering model geometry and meshsizing factors, until a small but accurate model was obtained.

The three different plane strain conditions were implementedby applying a variety of end constraints to the r ,z section. Thetrue plane strain condition is achieved by specifying zero move-ment in the Y direction of the end surfaces AB and CD, as Fig. 4

Fig. 3 Plane stress model diagram

Fig. 4 True plane strain model diagram

Journal of Pressure Vessel Technologyoaded 11 Nov 2008 to 129.5.224.57. Redistribution subject to ASMEshows.Figure 5 shows the open-ended condition, which is achieved by

constraining end surface CD to zero movement in the Y directionand coupling all nodes on the opposite surface AB to ensureequal Y deflection, such that the surface deflection is determinedby the equilibrium condition zero net axial force.

The closed-ended condition, shown in Fig. 6, is designated byconstraining as in the open-ended condition but applying an addi-tional axial load, Fz, to surface AB the node-coupled surfaceequal to

Fz = pAF ra2 7

2.7 Modeling Process. Autofrettage is modeled by applyingthe chosen pressure to the loading surface representing the boreof the tube in the first load step and solving. The residual stresses

Fig. 5 Open-ended model diagram

Fig. 6 Closed-ended model diagram

MAY 2006, Vol. 128 / 219 license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

Downlare then determined by removing the pressure in the second loadstep and solving again. However, to model the Bauschinger ef-fect, between the loading and unloading load steps the tube mate-rial must be altered from the default kinematic hardening model;the Bauschinger effect is represented by changing the yield stressof the region of the tube that underwent plastic deformationtheprimary yield zone rarrp. This is achieved by defining twotemperature profiles for the material, one matching the initial ma-terial state and the other the deformed material state whose yieldstress has been preselected using the peak equivalent stress expe-rienced by the tube bore. Note that the material properties areotherwise unchanged between the two temperature profiles i.e., achange in temperature is not being simulated; the titular changeof temperature is merely a convenient method of tailoring thematerial properties to simulate the Bauschinger effect.

The mesh used for the three r ,z section models is shown in Fig.7. The plane strain condition allows considerable reduction in thenumber of elements, as the section length may be kept short. Theend constraints are selected to simulate the midsection of the tube,by maintaining plane strain conditions, making a long section un-necessary. As in the r , mesh, an error analysis was conducted toidentify an accurate but not overly fine mesh, and element sizeswere decreased at the bore.

3 ComparisonsThe FE model was tested in two sets of comparisons, to 70%

overstrain: first, against both Huangs model and the Hencky pro-gram in the incompressible, true plane strain condition; second,against the Hencky program in a variety of end conditions trueplane strain, plane stress, open and closed ended for a more stan-dard material, =0.3. For each, two wall ratios were used, K=2.0 and K=2.5. As stated above, the focus of this paper is ongeometric accuracy rather than material fidelity; accordingly, theYoungs and Tangent moduli were kept constant in loading andunloading i.e., E1=E2=E, H1=H2=H.

The input parameters are summarized as follows refer to Fig. 2and Tables 1 and 2 to compare:

Y =1100 MPaE=209 GPaH=E /10=20.9 GPa=0.7, K=2.5=0.45, K=2.0

The values of were chosen to give similar amounts of reyield-ing for both wall ratios.

3.1 Normalization of Results. The radial position is normal-ized using the following expression to relate it to the tube wallthickness:

rNorm =r ra

rb ra8

The stresses are normalized with respect to the yield stress Y.

3.2 First Comparison: Huang-Hencky-ANSYS, =0.5. Theresults from the first stage of the comparison are shown below.Figure 8 plots the autofrettage stresses from the ANSYS model

Fig. 7 r ,z Section mesh

220 / Vol. 128, MAY 2006oaded 11 Nov 2008 to 129.5.224.57. Redistribution subject to ASMEagainst those predicted by Huangs method and the Hencky Pro-gram for K=2.0; Fig. 9 plots the residual hoop stresses from theANSYS model against those predicted by Huangs method and theHencky Program for both K=2.0 and 2.5. In both figures, theresults from the Hencky Program and Huangs method are shownas lines generally too close to be distinguishable and the resultsfrom the ANSYS model are shown as symbols overlaying the cor-responding plots from the other two methods.

All three principal stresses show good agreement with theHencky Program and Huangs model. The plots for K=2.5showed similar agreement and are omitted in the interest of brev-ity. Experience has shown that, for validation and subsequentchecking, it is important to summarize autofrettage stresses sepa-

Fig. 8 Comparison of autofrettage stresses, =0.5, K=2.0, =0.45

Fig. 9 Comparison of residual hoop stresses, =0.5, for K=2.0, =0.45, and K=2.5, =0.7


Downlrately from residual stresses.When assessing the increase in fatigue lifetime of an autofret-

taged tube, the key property is bore residual hoop stress. Figure 9shows a comparison of the ANSYS calculated values shown bysymbols against those from the Hencky Program and Huangsmethod. A very close agreement can be seen throughout the tubewall, including at the bore. This indicates the ANSYS model canaccurately reproduce results from Huangs model, when using abilinear material.

3.3 Second Comparison: Hencky-ANSYS, =0.3. The resultsfrom the second comparison, of the ANSYS model against theHencky Program for the four specified end conditions, are shownin Figs. 1013 for K=2.0 and 2.5. The graphs plot the residualhoop stresses throughout the tube walls; they show the ANSYSresults as symbols overlaid on the continuous lines generated us-ing the Hencky Program results. The autofrettage pressures re-

Fig. 10 Residual hoop stresses for the open-ended tube

Fig. 11 Residual hoop stresses for the closed-ended tube

Journal of Pressure Vessel Technologyoaded 11 Nov 2008 to 129.5.224.57. Redistribution subject to ASMEquired to achieve 70% overstrain are listed in Tables 3 and 4.

4 DiscussionThe key property of all the residual hoop stress plots is the

presence of secondary yielding near the bore, which ultimatelyrestricts the degree of prestressing possible in tubes. The reyield-ing depends on the autofrettage pressure due to the compressiveeffects of its removal and the strength of the material alteredfrom initial properties by deformation during loading.

For the incompressible conditions, Figs. 8 and 9 show excellentagreement between the ANSYS model, the Hencky program andHuangs model. This indicates that the ANSYS model can accu-rately predict stresses in such a case. The comparisons betweenthe ANSYS model and Hencky program for the wider range of endconditions and a more realistic Poissons ratio, are given in Figs.1013. Again, a close match is exhibited. A slight variation may

Fig. 12 Residual hoop stresses for the plane strain tube

Fig. 13 Residual hoop stresses for the plane stress tube

MAY 2006, Vol. 128 / 221 license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

here is an approximation. To obtain more realistic values of re-sidual stress, a more accurate material model is required that prop-erly follows the nonlinear unloading and also encompasses thevarying degree of plastic strain experienced by the materialthroughout the tube wall. These enhancements will be the focus offuture work.

NomenclatureA14 material model parametersB1,2 material model exponentsE1,2 loading and unloading Youngs moduliH1,2 loading and unloading reverse tangent moduli

Table 3 Autofrettage pressures, K=2.0

PressureMPa

End state ANSYS Hencky

Open ended 877.4 879.1Closed ended 913.2 914.7Plane strain 906.5 908.4Plane stress 877.3 878.8

Table 4 Autofrettage pressures, K=2.5

Pressure

Downlbe seen in the reyield zones rarrs away from the bore, moreclearly visible for K=2.0 and =0.3; the ANSYS results show asmall decrease in the magnitude of residual hoop stresses and aslight, commensurate, increase of reyield radius rs.

The autofrettage pressures in Tables 3 and 4 show two sets ofvery similar values. The pressures for the open-ended conditionare most similar to those for the plane stress. This is not surprisingsince the former represents a tube in which the axial stresses sumto zero net force and the latter one in which all axial stresses arezero. The autofrettage pressures for the closed-ended condition aremost similar to those for the plane strain condition. This is like-wise unsurprising since they reduce to the same problem for thecase of an incompressible material. Finally, the fact that autofret-tage pressure for plane strain and open ends exceeds that for planestress reflects the observations in 11. This could also be inferred,using von Mises criterion, from the presence of an axial stresswhich, in the near-bore region, generally varies between 0.3 and0.5 times hoop stress, depending on Poissons ratio.

From Tables 3 and 4, it can be seen that the pressure requiredfor a given depth of autofrettage is affected by the chosen endcondition. This is controlled by the von Mises yield criterion thatconsiders all three principal stresses. The axial stress is the inter-mediate principal stress and is influenced by the chosen end con-dition; it, therefore, influences the degree of yielding. The varia-tion between the various models in the near-bore reyield zone ismodest. The effectiveness of autofrettage lies in its ability to cre-ate compressive residual bore stresses to enhance service life byinhibiting the growth of surface cracks and increasing the safeworking pressure of the tube.

5 SummaryGood agreement was observed between the results generated by

the ANSYS FE model, the Hencky numerical program and Huangsanalytical model. This demonstrates that given a correctly calcu-lated degree of plastic strain at the bore, an accurate value ofresidual stress may be predicted by the ANSYS model for a numberof end conditions. However, the bilinear stress-strain profile used

MPaEnd state ANSYS Hencky

Open ended 1220 1223Closed ended 1254 1257Plane strain 1246 1249Plane stress 1224 1227

222 / Vol. 128, MAY 2006oaded 11 Nov 2008 to 129.5.224.57. Redistribution subject to ASMEK wall ratio, rb /raNorm superscript indicating a normalized valuepAF autofrettage pressure at ra

ra ,rb inner and outer tube radiirp ,rs primary and secondary yield radii

UL unloading superscript Bauschinger effect factor, Y

/Y+

Poissons ratioE elastic stress range between peak plastic strain

and onset of reverse yieldingY

+, forward and reverse yield stresses, in simpletension

Y+, forward and reverse yield strains, in simple

tension

AcknowledgmentThe first author would like to acknowledge Rosamund Gibson

and Cleveland Gibson, John Reynolds, and Darina Fierov fortheir help in the preparation of the paper.

References1 Bauschinger, J., 1881, ber die Vernderung der Elasticittsgrenze und des

Elasticittsmodulus Verschiedener Metalle, Zivilingenieur, 27, pp. 289348.2 Milligan, R. V., Koo, W. H., and Davidson, T. E., 1966, The Bauschinger

Effect in a High-Strength Steel, ASME J. Basic Eng., 88, pp. 480488.3 Huang, X. P., and Cui, W., 2005, Effect of Bauschinger Effect and Yield

Criterion on Residual Stress Distribution of Autofrettaged Tube, Gun Tubes2005 Conference, Oxford, April.

4 Parker, A. P., Troiano, E., Underwood, J. H., and Mossey, C., 2003, Charac-terization of Steels Using a Revised Kinematic Hardening Model NLKHIncorporating Bauschinger Effect, ASME J. Pressure Vessel Technol., 125,pp. 277281.

5 Liu, C. K. undated, Stress and Strain Distributions in a Thick-Walled Cyl-inder of Strain-Hardening Material, Elastic-Plastically Strained by InternalPressure, NASA TN D-2941.

6 Avitzur, B., 1988, Determination of Residual Stress Distributions in Autof-rettaged Tubing: A Discussion, Bent Laboratories Technical Report No.ARCCB-MR-88034.

7 Avitzur, B., 1989, AutofrettageStress Distribution Under Load and RetainedStresses After Depressurization, Bent Laboratories Technical Report No.ARCCB-TR-89019.

8 Bland, D. R., 1956, Elastoplastic Thick-Walled Tubes of Work-HardeningMaterial Subject to Internal and External Pressures and to Temperature Gradi-ents, J. Mech. Phys. Solids, 4, pp. 209229.

9 Jahed, H., and Dubey, R. N., 1997, An Axisymmetric Method of Elastic-Plastic Analysis Capable of Predicting Residual Stress Field, ASME J. Pres-sure Vessel Technol., 119, pp. 264273.

10 ANSYS 9.0, ANSYS, Inc., Canonsburg, PA, http://www.ansys.com11 Parker, A. P., 2001, Autofrettage of Open-End Tubes Pressures, Stresses,

Strains and Code Comparisons, ASME J. Pressure Vessel Technol., 123, pp.271281.


same

Documents