Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 52, 3, pp. 731-744, Warsaw 2014 NUMERICAL VERIFICATION OF ANALYTICAL SOLUTION FOR AUTOFRETTAGED HIGH-PRESSURE VESSELS Andrzej Trojnacki, Maciej Krasiński Cracow University of Technology, Department of Mechanical Engineering, Kraków, Poland e-mail: atroj@mech.pk.edu.pl; mkr@mech.pk.edu.pl Thick-walled cylinders are widely used in various engineering applications. In an optimal design of pressurized thick-walled cylinders, an increase in the allowable internal pressure can be achieved by an autofrettage process. In the paper, analysis is carried out to develop a procedure in which the autofrettage pressure is determined analytically. The obtained equivalent stress distribution is compared with those of the conventional solid wall and of several multi-layer vessels. The results of the analytical approach are verified by FEM modelling. Tensile tests have been carried out to determine the real mechanical properties of thematerial of thevessel andto createamaterialmodel.Thepresentedexample illustrates the advantages of the autofrettage technique. Keywords: autofrettage, optimal solution, finite element verification 1. Introduction Modern power engineering systems and advanced chemical technologies require the use of large thick-walled pressure vessels sustaining the pressure of hundreds MPa. Conventional methods of one thick shell manufacturing (forging or rolling of thick sheets) resulted in technological difficulties and became too expensive. These problems forced engineers to search for new po- ssibilities of increasing load bearing capacity of the vessel. Now, the autofrettage technique is commonly applied to improve the resultant stress distribution and to increase the capacity of the vessel. A number of contributions to the autofrettage technique have appeared recently. Solutions have been obtained either in an analytical form or with numerical implementations. A proce- dure for elastic-plastic analysis of a thick-walled cylinder under internal pressure was proposed by Zhao et al. (2003). It involves two parametric functions and piecewise linearization of the stress-strain curve. The method provides a general elastic-plastic solution which accounts for the effect of deformed geometry due to high operating pressure. The optimumautofrettage pres- sure was determined by Ayob and Elbasheer (2007) analytically. A validation by a numerical simulation shows that the analytical approach and numerical results correlate well. Majzoobi et al. (2003) used both numerical and experimental techniques for the investigations of the autofrettage process and its influence on the pressure capacity. A finite element analysis was performed by Alegre et al. (2006) to obtain the residual stresses after the autofrettage for the vessel made of the material which shows strong Bauschinger’s effect. The simulation procedure may be applied for other autofrettage designs that need the Bauschinger effect of the material to be considered. An autofrettage model considering the material strain-hardening relation- ship and the Bauschinger effect, based on the actual tensile-compressive stress-strain curve of material, plane-strain, and modified yield criterion was proposed by Huang and Cui (2006). The predicted residual stress distributions of autofrettaged tubes from the considered model were compared with the numerical results and the experimental data. The influence of Bau- schinger effect and yield criterion on the residual stress was discussed based on the introduced 732 A. Trojnacki,M. Krasiński model. An analytical study of spherical autofrettage-treated pressure vessels, considering the Bauschinger effect was presented by Adibi-Asl and Livieri (2007), where a general analytical solution for stress and strain distributions was proposed for both loading and unloading phases. The optimization procedure, numerical simulation and experiments were employed by Majzo- obi and Ghomi (2006) to determine the minimumweight of a compound cylinder for a specific pressure. Many researchers have focused onmethods to extend vessels lifetimes. Fatigue analysis was performedbyKoh (2000) to predict the fatigue life of an autofrettaged pressurevessel containing radial holes subjected to cyclic internal pressure. Finite element analysis was used to calculate the residual and operating stress distributions and to determine numerical stress concentration factors at the hole. Analysis of the combined effect of autofrettage and shrink-fit in amulti-layer vessel was carried out by Kumar et al. (2011). Thicknesses of layers, autofrettage percentage and radial interference for the shrink-fit were assumed as design variables, whereas hoop stress throughout the thickness was the objective function. Calculation of fatigue life for several ca- ses was studied. The optimum design of a similar 3-layered vessel for maximum fatigue life expectancy under the combined effects of autofrettage and shrink-fitwas performed by Jahed et al. (2006). The numerical optimization procedure was employed to obtain the optimum size of each layer and to optimize the initial stress distribution. The results showed that with a proper combination of operations, a significant life enhancement could be achieved. The Tresca-Guest yield condition (the T-G condition) and the Huber-Mises-Hencky yield condition (the H-M-H condition) are used in the present paper to develop a procedure in which the autofrettage pressure is determined analytically. The obtained reduced equivalent stress distribution is compared with distributions for a solid virgin cylinder and for the multi-layer vessel withmodified initial stress distribution. A finite element method (FEM) using ANSYS RO simulation is carried out on the cylinder to develop a procedure in which the autofrettage process is determined numerically. The real properties of the vessel material are introduced into numerical simulation with the Bauschinger effect implemented. The numerical example illustrates the advantages of the autofrettage technique. 2. Analytical solution of the autofrettaged vessel Theoretical distributions of radial and circumferential stresses in cylindrical vessels within the elastic range must satisfy Lame’s equations (Timoshenko and Goodier, 1951). For this reason, the corresponding distribution of equivalent stress σeq in the solid wall is precisely determined. In a solid nonpressurized cylinder subjected to the inner operating pressure popr, themaximum equivalent stress σ′eqmax appears at the inner radius while the outer parts of the wall are less loaded. Inmany industrial applications it is important todecrease themaximumequivalent stress in the wall or to reduce σeq at the outermost surfaces, which may be additionally subjected to action of aggressive fluids (Fig. 1). Autofrettage is often used to introduce advantageous residual stresses into thick-walled pres- sure vessels and to enhance their pressure bearing capacity. In this technique, the vessel is subjected to an internal pressure large enough to cause yielding within the wall near the inner surface. Large scale yielding occurs in the autofrettaged cylinder wall. Upon the release of this pressure, a compressive residual circumferential stress is developed to a certain radial depth at the bore. These residual stresses serve to reduce the stresses obtained as a result of subsequent application of the operating pressure, thus increasing the load bearing capacity. The degree of autofrettage in the thick-walled cylinder of inner and outer radii ri and ro, respectively, subjected to the inner autofrettage pressure pa, is defined as a junction (limit) radius rj of wall thickness occupied by the plastic zone. The radial σelr and circumferential σ el ϕ Numerical verification of analytical solution for autofrettaged high-pressure vessels 733 Fig. 1. Condition of minimum equivalent stress σeq at the junction radius rj under the operating pressure popr. Distribution of stress σeq: (a) – in a solid virgin wall, (b) – in the optimum autofrettaged vessel stresses within the elastic region rj ¬ r¬ ro are given in terms of the radius r by well-known Lame’s formulation σelr =C1− C2 r2 σelϕ =C1+ C2 r2 (2.1) In the plastic region ri ¬ r ¬ rj, theoretical distributions of stress depend on the applied assumptions and the yield theory defining the equivalent stress σeq. For the elastic-perfectly plasticmaterialmodel andwhen thematerial is incompressible in plastic deformation (ν =0.5), the more complex H-M-H [Φf] yield condition may by linearized for plain strain (εz = 0) to that similar to the T-G [τmax] one |σplr −σplϕ |=CSy (2.2) where Sy stands for the yield stress. The coefficient C depends on the applied yield theory, and for the T-G yield condition C = 1, and for the H-M-H yield condition C = 2/ √ 3. Finally, in both cases, the stresses in the plastic region are expressed by the same equations σplr =CSy ln ( r ro ) +C3 σ pl ϕ =CSy [ ln ( r ro ) +1 ] +C3 (2.3) Three constants of integrationmaybedetermined from the appropriate boundary conditions and thecondition of stress continuity across the elastic-plastic boundaryat the junction radius rj r= ri σplr =−pa r= rj σplr =σ el r σ pl ϕ =σ el ϕ r= ro σelr =0 (2.4) where the additional fourth condition relates the autofrettage pressure pa to the limit radius rj pa =CSy [ ln (rj ri ) + r2o −r2j 2r2o ] (2.5) Equation (2.5) is valid for both yield conditions after substituting for C the appropriate value. If the autofrettage pressure is removed after a part of the cylinder thickness has become plastic, a residual stress is setup in thewall.Assumingthatduringunloading thematerial follows Hooke’s law, the residual stress can be easily obtained. Themaximum equivalent stress σeqmax in the autofrettaged vessel subjected to the operating pressure popr appears at the junction 734 A. Trojnacki,M. Krasiński radius rj (Fig. 1) and does not depend on the used yield criterion. If the cylinder is loaded again with the operating pressure, by superposing the residual stresses σresr and σ res ϕ due to autofrettage procedure and stresses σr and σϕ produced by the operating pressure, the final equivalent stress distribution in the wall versus radius r becomes σeq(r)= 1 C ∣ ∣ ∣(σr+σ res r )− (σϕ+σresϕ ) ∣ ∣ ∣ (2.6) and at the junction radius rj takes the form σeq(rj)= 1 C 2r2ir 2 o (r2o −r2i )r2j [popr−pa]+Sy (2.7) Equivalent stress (2.7) reaches the minimum with respect to r as dσeq(rj)/drj = 0 which leads to the optimum value of the limit radius rjopt = riexp (popr CSy ) (2.8) It appears that optimization of the equivalent stress along the vessel wall causes that the difference of equivalent stresses at the outermost radii σeq(ri)= 1 C 2r2o r2o −r2i [popr−pa]+Sy σeq(ro)= 2r2j r2o −r2j ( 1 C pa−Sy ln rj ri ) + 1 C 2r2i r2o −r2i [popr−pa] (2.9) is minimum, and for rj = rjopt their difference is zero σeq ∣ ∣ ∣ r=rjopt (ri)=σeq ∣ ∣ ∣ r=rjopt (ro)= k=min (2.10) where k stands for theminimum value of the equivalent stress. Junction radius (2.8) is optimal when for any other value of it an increase in the equivalent stress occurs and the condition of equal andminimumequivalent stresses at theoutermost radii under appliedoperatingpressure is disturbed.Theautofrettagepressurewhich isnecessary toobtain theoptimumlimit radius, (2.8), may be determined from Eq. (2.5). In the autofrettaged vessel, the maximum strength σeqmax under the operating pressure popr appears at the junction radius. Application the optimum autofrettage pressure and partial yielding of the wall to the optimum junction radius causes minimum equivalent stress under the operating pressure and equal and minimum values of the equivalent stress at the outermost radii. The maximum operating pressure is equal to the autofrettage pressure pmax = paopt. Under this pressure, full yielding of the vessel wall in the range r < rjopt appears again and the decay of the equivalent stress in the range rjopt ¬ r¬ ro. Themaximumautofrettage pressure paextr corresponds to the internal pressure required for the wall thickness of cylinder to yield completely rj extr = ro. Such a cylinder may be subjected to the maximum operating pressure pextr = paextr. 3. Finite element modelling of the autofrettaged vessel The autofrettage process has been simulated by the finite element method (FEM) making use of elastic-plastic analysis. Special crude alloy steel 16Mo3 (1.5415) according to PN-EN 10028- 2:2010 has bsen applied for the vessel wall. Thismaterial is assigned to high-temperature appli- cations and is often used in the high-pressure technology. Actual mechanical properties of this Numerical verification of analytical solution for autofrettaged high-pressure vessels 735 material have been determined in tensile tests. A set of cylindrical specimens has been cut from a segment of a tube in the circumferential direction. The specific values of material data have been calculated as arithmetical averages of 7 tests: ultimate strength Sut = 518.43MPa, yield limit Sy =317.29MPa,maximum strain εut =0.1750 and plastic strain εpl =0.0127. They are defined in Fig. 2 together with the elastic strain which for Young’s modulus E =2.1 ·105MPa adopted in the elastic range becomes εel =0.0015. Fig. 2. Parabolic and segmental approximation of the real stress-strain curve (in the stretched scale) The shape of experimental stress-strain curves σ = f(ε) suggests parabolic approximation beyond the yield limit. The parabola containing the point of coordinates εpl, Sy and reaching themaximum value at the point εut, Sut (Fig. 2) was applied to describe the tensile behaviour ofmaterial. For thenumerical calcuations the parabolawas replaced byfive segments of different slopes but of equal length in the orthogonal projection at the ε axis. Such an approximation enables direct introduction of the nonlinearmaterial properties in the softwaremoduleANSYS RO which was used in the paper. Moreover, it was assumed that the relationship between the equivalent stress (stress inten- sity) σeq and equivalent strain (strain intensity) εeq under complex stress states σeq = f(εeq) was the same as the stress-strain relationship under uniaxial tensile loading. The stress intensity was derived from the Huber-Mises-Hencky yield criterion and the strain intesity was defined (Życzkowski, 1981) as εeq = 2√ 3 √ (εr −εϕ)2+(εϕ−εz)2+(εz −εr)2 (3.1) where εr, εϕ and εz are principal strains at a certain point of the cross-section. Thefinite element calculationswere carriedouton theassumptionofplain strainwhich seems justified as the cylindrical parts of the vessels are usually of considerable length. Thematerial of the vessel was described by multi-linear kinematic strain hardening with the Baushinger effect included. FEM modelling of the vessel wall geometry is simplified because of the axial symmetry. The numerical model of the wall was built of layers (rectangular slices) of the assumed radii divided into higher order finite elements PLANE183 adapted to the axial symmetry. The size of quadratic 8-node finite elements of 0.4mm creates the mesh of sufficient density as an increase of the density in five times produces the difference in the equivalent stress less than 0.4%. A typical finite element mesh with boundary conditions applied is presented in Fig. 3. 736 A. Trojnacki,M. Krasiński Fig. 3. Computational model of the vessel, mesh of finite elements and illustration of introduced boundary conditions (not to scale) 4. Numerical example The detailed analytical and numerical calculations were carried out for a cylinder of the outer diameter 2ro = 800mm subjected to internal pressure. The thickness of the wall was to =200mm. The vessel was made of material 16Mo3 with the experimentally confirmed yield point Sy =317.29MPa.Thevessel was designed for the operating pressure popr underwhich the equivalent stress reaches the yield limit Sy at a certain point of thewall. The pressure popr deri- ved from theT-G [τmax] yield criterion is pTopr =118.98MPa and that based on theH-M-H [Φf] yield criterion is pHopr =137.39MPa. On the applied assumptions, the T-G andH-M-H yield conditions may be expressed by the same equation (2.2) and, therefore, both criteria lead to similar results of the autofrettage. For this reason, the distributions of equivalent stress at some characteristic radii versus the junction radius rj for the vessel subjected to the operating pressure pTopr or p H opr, respectively, are the same (Fig. 4). The only difference is in relation (2.5) describing the autofrettage pressure pa versus the autofrettage radius rj. Fig. 4. Equivalent stress σeq under pressure pTopr or p H opr at the radii: rj, ri and ro, respectively, and the autofrettage pressure pa versus junction radius rj The optimum autofrettage radius is the same rjopt = 290.99mm for both criteria, but the autofrettage pressure is different. For the T-G yield condition, pTaopt =193.66MPa and for the H-M-H yield condition, pHaopt = 223.63MPa. Under the pressure p T opr or p H opr, the equivalent stress at the outer radii is the same and reaches minimum σeq(ri) = σeq(ro) = 118.14MPa. The maximum equivalent stress in the considered vessel occurs at the junction radius σeq(rjopt) = 223.22MPa. The distributions of equivalent stress at the outermost radii σeq|r=ri and σeq|r=ro are plotted in Fig. 4 versus the junction radius rj (solid fine lines) as well as the Numerical verification of analytical solution for autofrettaged high-pressure vessels 737 autofrettage pressure versus the corresponding junction radius (dashed lines). Any other mo- dification of the residual stress resulting in the junction radius different from rjopt causes an increase in the maximum equivalent stress σeq(rjopt) which can be seen in Fig. 4 (bold solid line). Analytical results for the autofrettaged vessel are gathered inTable 1. The calculations were carried out for the vessel autofrettaged to the optimum limit radius rj opt and to themaximum limit radius rj extr =400mm. In both cases, the vessel was subjected first to the pressure popr (pTopr or p H opr) and then to the maximum pressure (paopt for rjopt or paextr for rj extr). The relative decrease of the maximum equivalent stress ∆σeq with respect to the solid wall was determined in the first case and the relative increase of the load bearing capacity ∆pmax was determined in the second case. The total yielding of thewall calculated analytically based on the Tresca-Guest yield criterion occurs under the autofrettage pressure pTaextr = 219.93MPa and this is the maximum operating pressure which may be applied to the vessel. The calculations carried out for the Huber-Mises-Hencky yield condition give pHaextr =253.95MPa. Table 1.Decrease of the equivalent stress and increase of the load capacity for several types of vessels Type of the wall Number of layers or junction radius rj [mm] Calcula- tions Under operating Under maximum pressure pressure Maximum Decrease Maximum Increase of strength of strength capacity capacity σeq [MPa] ∆σeq [%] pmax [MPa] ∆p [%] Layered optim. 2 layers analytical 239.63 24.48 171.02 24.48 under popr 25 layers analytical 176.65 44.33 207.71 51.18 Autofrettaged rjopt =290.99 analytical 223.22 29.65 223.63 62.77 to rjopt rjopt =291.60 FEM 223.16 29.67 222.51 62.24 Layered optim. 2 layers analytical 271.59 14.40 181.91 32.40 under pmax 25 layers analytical 251.54 20.72 246.77 79.61 Autofrettaged to rj extr rj extr =400 analytical 249.99 21.21 253.95 84.84 rj extr =4001) FEM 248.56 21.66 251.38 83.29 rj extr =4002) FEM 235.51 25.77 266.79 94.52 rj extr =4003) FEM 251.46 20.75 311.38 127.04 1) autofrettage pressure pFEMaextr1 =251.38MPa, strain intensity εeq(ri)= 0.0067, εeq(ro)= 0.0015 2) autofrettage pressure pFEMaextr3 =266.79MPa, strain intensity εeq(ri)= 0.0501, εeq(ro)= 0.0127 3) autofrettage pressure pFEMalim =311.38MPa, strain intensity εeq(ri)= 0.1750, εeq(ro)= 0.0488 The distributions of equivalent stress σeq versus the radius r for the vessels autofrettaged to the radius rjopt (dashed lines) and to the radius rj extr (solid lines) under the pressure popr (fine lines) andunder the pressure paopt or paextr (bold lines) are presented inFig. 5. Thedotted line corresponds to the solid non-pressurized wall subjected to the maximum in this case pressure popr =137.39MPa. The analytical solution of the autofrettaged vessel was compared with the analytical results obtained for the optimum shrink-fit multi-layer vessels (Krasiński et al., 2013). The results are gathered inTable 1.TheH-M-Hyield criterionwas applied to solve thewall composed of 2 layers and of 25 layers of the same thickness. The layered vessels were designed on the assumption of equal equivalent stress at the inner surfaces of layers under the pressure popr =137.39MPa.The distributions of the equivalent stress versus the radius r areplotted inFig. 6 (solid fine lines) and 738 A. Trojnacki,M. Krasiński Fig. 5. Equivalent stress σeq of the wall autofrettaged to the radii: rjopt or rj extr, respectively, versus radius r. Fine lines – under the pressure popr, bold lines – under the maximum pressure paopt or paextr Fig. 6. Distribution of σeq versus radius r for the vessel autofrettaged to rj opt and for the layered vessel with σeq equalized for pressure popr =137.39MPa may be compared with the distribution for the vessel autofrettaged to the junction radius rjopt (dashed fine lines). Bold lines correspond to the equivalent stresses which appear in these vessels under themaximumpressure pmax for the shrink-fit vessels or paopt for the autofrettaged vessel. In this case, themaximumpressure pmax for themulti-layer vessels is the loading for which the yield limit Sy is first reached at the inner surface of radius ri. Variations in the equivalent stress across the wall determined for the multi-layer vessels under the pressure pmax which leads in this case to the condition of maximum strength σeq equal to the yield limit Sy at the inner surfaces of all layers are presented in Fig. 7. They are compared with the distributions obtained for the vessel autofrettaged to the radius rj extr. The residual radial stresses σresr calculated analytically for the autofrettaged vessel are il- lustrated graphically in Fig. 8 together with the interlayer initial compressive stresses qiopt which must be introduced into the layered vessel wall in order to equalize the equivalent stress at the inner surfaces of layers under the operating pressure. The autofrettage process was so- lved using the Huber-Mises-Hencky yield theory. The residual stress which occurs in the vessel wall autofrettaged to the optimum junction radius rjopt can be compared with the interlayer stresses qiopt which causes the same and equal equivalent stress under the operating pressure popr = 137.39MPa. The residual radial stress which appears in the vessel wall autofrettaged Numerical verification of analytical solution for autofrettaged high-pressure vessels 739 Fig. 7. Distribution of σeq versus radius r for the vessel autofrettaged to rjextr and for the layered vessel with σeq equalized to the maximum value Sy Fig. 8. Dashed lines – distributions of radial residual stress σresr for the vessels autofrettaged to rjopt and to rj extr, respectively. Discrete distributions refer to the initial interlayer compressive stresses in layered walls: 2-layer: • – qiextr, ◦ – qiopt, and 25-layer: N – qiextr,△ – qiopt to the maximum radius rj extr = 400mm may be compared with the interlayer stresses qiextr, which ensures the maximum equivalent stress σeq equal to Sy under the maximum operating pressure pmax. The autofrettage process was modelled by applying pressure to the inner surface of the vessel, removing it and then calculating the residual stress field, followed by reloading with the operating pressure. The finite element procedure was carried out first on the vessel auto- frettaged with the pressure pFEMaopt = 222.51MPa which, under the assumed operating pressu- re pFEMopr = 137.15MPa, ensures the equality of the equivalent stress at the outermost radii σeq(ri) = σeq(ro) = 121.16MPa and at the limit radius produces σeq(rj opt) = 223.16MPa. Themaximum pressure which may be applied to this vessel is the autofrettage pressure pFEMaopt . The results presented in Table 1 are closed to those predicted by the analytical approach. The optimum radius rjopt of the elastic-plastic junction in the autofrettaged cylinder derived for the elastic-perfectly plastic material does not differ significantly compared with that obtained using an elastic-plastic with a strain hardeningmaterialmodel. The reason is that themaximum strain intensity which appears at the inner radius εeq(ri) = 0.0035 is less than εpl = 0.0127. Accordingly, only the first two rectilinear fragments of the stress-strain relationship were used in the numerical procedure, likewise in analytical calculations. A certain small discrepancy is caused by numerical errors. 740 A. Trojnacki,M. Krasiński Moreover, the FEM and analytical results coincide on the assumption that the junction radius rj extr =400mm is achievedwhen the equivalent stress reaches the first the yield limit Sy there. Such a situation occurs under the autofrettage pressure pFEMaextr1 =251.38MPa. The strain hardening of thematerial was not engaged in thenumerical procedure.While the strain intensity at the outer radius εeq(ro)= 0.0015 is associated with the yield limit Sy, the strain intensity at the inner radius εeq(ri) = 0.0067 is less than εpl =0.0127 and corresponds to the Sy too. The strain hardening has no influence on the numerical solution even for the autofrettage pressure pFEMaextr2 =252.30MPaunderwhich the strain intensity at the inner radius εeq(ri)= εpl =0.0127 because at the outer radius there is still εeq(ro)= 0.0030<εpl. The influence of the parabolic part of the stress-strain curve on the finite element solution is revealed only for theautofrettagepressuregreater than pa >pFEMaextr2.Adistinct effect is observed on the assumption that the strain intensity at the outer radius reaches εeq(ro) = 0.0127 = εpl, which occurs for the autofrettage pressure pFEMaextr3 = 266.79MPa. The strain intensity at the inner radius becomes εeq(ri)= 0.0501>εpl. Examination of the equivalent stress distributions depicted in Fig. 9 leads to the conclusion that in thewhole cross-section, the equivalent stress is beyond the yield limit Sy, however, at the same time pFEMaextr3

p FEM aextr3 > p FEM aextr1, the corresponding distributions of residual radial stress tend to decrease. Moreover, the relative decrease of the Numerical verification of analytical solution for autofrettaged high-pressure vessels 741 maximum values of σresr (15.36% and 32.27%) is greater than the relative increase of the au- tofrettage pressure pa (6.13% and 23.87%). Such a relation may occur as the strain hardening was taken into account in the FEM procedure. Fig. 10. Distributions of radial residual stress σresr versus radius r for vessels autofrettaged to the radii: rjopt and rj extr, respectively The finite element calculations were carried out for the 16Mo3 material for which a stress- strain approximationwas created using the data derived fromthe tensile tests. In particular, this steel seems to be a goodmaterial to fabricate thick-walled cylinderswhich are to be subjected to theautofrettage process,mainly onaccount of ahighultimate strain value εut andaconsiderable difference between εel and εpl. This material is plastic enough and able to withstand large deformations, which is the key requirement in the autofrettage process. Owing to the latter property, the favourable effects of strain hardening on the vessel performance manifest at high autofrettage pressures exceeding pFEMaextr2 =252.30MPa. The horizontal junction segment (plastic plateau) of the stress-strain curve approximation (Fig. 2) is called the perfectly plastic flow and corresponds to the yield pointSy. The influence of the length of the plastic plateau on some selected strength parameters is shown in Fig. 11. The total length of the plateau was subdivided into five segments of equal length ∆ε=0.0022, and FEM calculations were carried out for the appropriate five approximations of the stress-strain curve. The parabolic part of the approximation had a maximum (as previously) at the point with the coordinates εut, Sut but in each case the parabola was passing through a different point with coordinates depending on the plastic plateau division. The results are summarized in Fig. 11 where the bold solid line represents the autofrettage pressure pFEMaextr3 which increases with the increase in the plastic plateau length. The fine solid line corresponds to the operating pressure popr underwhich the equivalent stress at the outermost radiiwas set equal to the values indicated in Fig. 11 by the dashed line. It should be noted that all considered parameters of the autofrettaged vessel vary linearly versus the length of the plastic plateau. It appears that an increase in length of the plastic plateau, from zero to its maximum value, gives rise to a slight increase (by 4.77%) in the autofrettage pressure and to a significant reduction (by 11.40%) of the operating pressure popr equalizing the equivalent stress at the outermost radii. The autofrettage has obvious advantages when applied to thickwalls characterized by a high thickness coefficient definedas β= ro/ri. Several strengthparameters of thevessel autofrettaged with the optimumpressure versus the coefficient β are presented in Fig. 12. The examination of these relationships leads to the conclusion that a decrease in the coefficient β in 40% gives rise to reduction of the maximum pressure paextr by 73.70%, while the maximum equivalent stress at the junction radius increases by 22.48%. 742 A. Trojnacki,M. Krasiński Fig. 11. Autofrettage pressure pFEMaextr3and the pressure popr equalizing the equivalent stress at the outermost radii (along dashed line) versus the length of the plastic plateau Fig. 12. Some chracteristic parameters of the autofrettaged vessel versus the thickness coefficient β 5. Final remarks The presented investigations confirm the advantages of the autofrettage technique applied to thick-walled, high-pressure vessels. The advantages were demonstrated using an example of a cylindrical vessel with the outer diameter of 800mm regarded to be the maximum value with respect to costs for the solid vessel. For vessels with greater outer diameters, a better solution is a layered wall composed of thin layers either shrink-fitted, bent along the screw or spiral line or fabricated using the Smith technology. The maximum value of the thickness coefficient β = 2.00 was assumed which is admissible in the strength analysis of the pressure vessels under the regulations of the Polish Office of Technical Inspection (OTI). The vessel is made of ductile 16Mo3 steel which is appropriate for autofrettage processes because of its mechanical behaviour, in particular of the high value of the ultimate strain εut. The analytical well-known approach based on Lame’s solution reveals that the autofrettage optimum pressure results in a 30% decrease of the equivalent stress under operating pressure. However, the strength capacity of this vessel increases by 63% with respect to the solid virgin wall. Even greater spectacular strength effect may be achieved for the cylinder autofrettaged across its wall giving rise to the load bearing capacity by 85%. Residual stresses can be also generated by introducing the interlayer interference fit into the multi-layer cylinder.Theadvantages of autofrettage applied to the solidwallwere comparedwith Numerical verification of analytical solution for autofrettaged high-pressure vessels 743 the results obtained for the optimum designed layered vessels. The wall composed of 2 layers exhibits a 24% decrease of the equivalent stress under the operating pressure, and this result is similar to that obtained for the solidwall subjected tooptimumautofrettage, but theappropriate increase of the load-bearing capacity by 25% is even less. The strength possibilities of the wall made of a large number thin layers may be utilised in a larger degree. The results of analytical calculations compiled in Table 1 suggest that the advantages of layered wall composed of 25 layers are comparable to those of the autofrettaged wall. The results of FEM simulation and their comparison with the analytical approach is of particular importance. Since the vessel wall ismadeof a typicalmaterial used in the autofrettage technology, the strain hardening effect occuring beyond the yield limit cannot be utilised. As the plastic plateau is of considerable length, the results of FEM calculations for autofrettaged vessel under the optimum andmaximum pressures coincide (within the admissible error limits) with the analytical results. Major differences are revealed for a cylinder autofrettaged with a pressure which produces the equivalent stress equal to the ultimate stress at the inner surface. Such a vessel exhibits themaximumpossible load carrying capacity for the assumed dimensions and the material which exceeds by 127% the pressure that can be withstood by a solid wall of the same dimensions andmaterial. In conclusion, it should be mentioned that the well-designed autofrettage technique has obvious advantages over other technologies of thick-walled vessels, in particular over multi- layer technologies. The design is material-saving and cost-effective. In the case of shrink-fit cylinders, care must be taken to ensure precise fits between the layers, which presents serious difficulties when handling elements of considerable length and diameters. Moreover, the shrink- fit cylinders have to be heated (or cooled down), which requires large furnace installations. The Smith method utilises the thermal shrinkage of the longitudinal welds whose magnitude can be only approximately determined. In the light of the current expertise in the field of forging machines (presses and rollers), only high-efficiency pumps are required in fabrication of a typical thick-walled autofrettaged vessel. References 1. 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