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1 The Ratchet-Shakedown Diagram for a Thin Pressurised Pipe Subject to Additional Axial Load and Cclic Secondar Global Bending R.A.W.Bradford and D.J.Tipping Universit of Bristol, Mechanical Engineering, Queen's Building, Universit Walk, Bristol BS8 TR, UK, RickatMerlinHaven@hotmail.com, tel ; EDF Energ, Barnett Wa, Barnwood, Gloucester, GL4 3RS, David.Tipping@edf-energ.com Abstract The ratchet and shakedown boundaries are derived analticall for a thin clinder composed of elastic-perfectl plastic Tresca material subject to constant internal pressure with capped ends, plus an additional constant axial load, F, and a ccling secondar global bending load. The analtic solution in good agreement with solutions found using the linear matching method. When F is tensile, ratcheting can occur for sufficientl large cclic bending loads in which the pipe gets longer and thinner but its diameter remains the same. When F is compressive, ratcheting can occur in which the pipe diameter increases and the pipe gets shorter, but its wall thickness remains the same. When subject to internal pressure and cclic bending alone (F = ), no ratcheting is possible, even for arbitraril large bending loads, despite the presence of the axial pressure load. The reason is that the case with a primar axial membrane stress exactl equal to half the primar hoop membrane stress is equipoised between tensile and compressive axial ratcheting, and hence does not ratchet at all. This remarkable result appears to have escaped previous attention.. Introduction A structure subject to two or more tpes of loading, at least one of which is primar and at least one of which is ccling, ma potentiall accumulate deformation which increases ccle on ccle. This is ratcheting. Rather less severe loading ma result in parts of the structure undergoing plastic ccling, involving a hsteresis loop in stressstrain space, but without accumulating ratchet strains. Still less severe loading ma result in purel elastic ccling, perhaps after some initial plasticit on the first few ccles. This is shakedown. It is desirable that engineering structures be in the shakedown regime since ratcheting is a severe condition leading potentiall to failure. The intermediate case of stable plastic ccling ma be structurall acceptable but will involve the engineer in non-trivial assessments to demonstrate acceptablit, probabl involving the possibilit of cracks being initiated b the repeated plastic straining (b fatigue, and possibl b creep). Deciding which of the three tpes of behaviour results from a given loading sequence on a given structure is, therefore, of considerable importance. Unfortunatel ratcheting/shakedown problems are difficult to solve analticall in the general case. However, analtical solutions for sufficientl simple geometries and loadings do exist. One of the earliest, and undoubtedl the most influential, of these is the Bree problem, Ref.[]. Bree's analtic solution addresses uniaxial loading of a rectangular cross section in an elastic-perfectl plastic material, the loading consisting of a constant primar membrane stress and a secondar bending load which ccles between zero and some maximum. When normalised b the ield stress, the primar membrane stress is denoted whilst the normalised secondar elastic outer fibre bending stress range is denoted. The ratchet boundar is defined as the curve on an, plot above which ratcheting occurs. Similarl, the shakedown boundar is defined as the curve on the, plot below which shakedown to elastic ccling occurs. The two curves ma or ma not be separated b a region of stable plastic ccling. In obvious notation, the three tpes of

2 region are denoted R, S and P. Variants on the Bree problem which have been solved analticall include, (i) the case when the primar membrane load also ccles, either strictl in-phase or strictl in anti-phase with the secondar bending load, Refs.[-4], (ii) the Bree problem with different ield stresses at the two ends of the load ccle, Ref.[5], and, (iii) the Bree problem with biaxial stressing of a flat plate, an extra primar membrane load being introduced perpendicular to the Bree loadings, Ref.[6]. The analses in Refs.[-6] used the same approach as Bree's original analsis, Ref.[]. However, alternative, "non-ccling", methods for analtical ratchet boundar determination are also emerging, e.g., Refs.[7,8]. The difficult of obtaining analtic solutions for more complicated geometries or loadings has prompted the development of numerical techniques to address ratcheting and shakedown. For example, direct cclic analsis methods, e.g., Ref.[9], can calculate the stabilised stead-state response of structures with far less computational effort than full step-b-step analsis. A technique which is now being used widel is the Linear Matching Method (LMM), e.g., Ref.[]. LMM is distinguished from other simplified methods in ensuring that both equilibrium and compatibilit are satisfied at each stage. This paper presents a Bree-tpe analsis of the ratchet and shakedown boundaries for the case of a thin clinder composed of elastic-perfectl plastic material with internal pressure and capped ends, plus an additional axial load ( F ), together with a global bending load. The pressure and additional axial loads are constant primar loads. The global bending is secondar in nature and ccles. The global bending load is envisaged as arising from a uniform diametral temperature gradient with bending of the pipe being restrained. The temperature gradient ccles between zero and some maximum value. After developing the analtic solution for the ratchet and shakedown boundaries, the solution is verified b use of the LMM technique. (Alternativel this ma be seen as a validation of the LMM technique). Section formulates the equations which specif the problem. Section 3 defines normalised, dimensionless quantities which will be used throughout the rest of the paper. Section 4 describes the method of solution. Sections 5 and 6 present the solution for the case of tensile ratcheting in the axial direction. Section 7 completes the solution, considering shakedown and stable plastic ccling as well as compressive ratcheting in the axial direction. Section 8 describes the numerical analses carried out using the LMM method, and finall the ke results are summarised in the Conclusions, Section 9. Remarkabl it will be shown that ratcheting cannot occur if the additional axial load is zero, F =, despite the primar pressure load acting in the axial direction as well as the hoop direction. This behaviour is probabl specific to a straight pipe since ratcheting of pipe bends due to constant pressure and cclic global bending has been analsed in Refs.[,].. Formulation of the Problem The notation for stresses and strains in this section will include a tilde, e.g., σ ~, to distinguish them from the normalised, dimensionless quantities which will be used hereafter. The problem considers a thin clinder so that through-wall stress variations ma be neglected. The clinder is under internal pressure, P, and an axial load, F. Note that capped ends ensure that the pressure load also contributes to the total axial load. Both

3 these primar loads are constant (i.e., not ccling). The clinder wall is therefore subject to a constant hoop stress, which is uniform around the circumference, of, Pr σ ~ H = () t where r, t are the clinder radius and thickness respectivel. The integral of the axial stress around the clinder circumference equilibrates the applied axial load plus the axial pressure load, F = F + r P = σ ~ TOT rtdθ where σ ~ is the axial stress at the angular position θ around the circumference, and the integral is carried out over the whole circumference. Equ.() holds at all times since the pressure and the additional axial load are constant. The axial stress is not uniform around the circumference as a consequence of the ccling secondar bending load. This bending load is envisaged as arising due to a uniform diametral temperature gradient, i.e., a temperature which varies linearl with the Cartesian coordinate ~ x perpendicular to the clinder axis. Bending of the clinder is taken to be restrained so that the temperature gradient generates a secondar bending stress. The origin of ~ x is taken to be the clinder axis. The elasticall calculated bending stress is denoted σ ~ b, and its tensile side is taken to be ~ x >. Hence ~σ ~ the elastic bending stress distribution across the pipe diameter would be b x / r where r ~ x r. This secondar bending load ccles between zero and its maximum value and back again repeatedl. The material is taken to be elastic-perfectl plastic with ield strength σ. This is a common simplifing assumption in such ratcheting analses, without which the problem would not be analticall tractable. The Tresca ield criterion is assumed, again for analtic simplicit. Throughout it will be assumed that the compressive radial stress on the inner surface is negligible compared with the other stresses, i.e., the thin shell limit. Stressing is therefore biaxial. The hoop stress is necessaril less than ield, σ ~ < σ otherwise the clinder collapses. It is worth spelling out wh this remains true for this situation of biaxial stressing. Possible cases are, If the axial stress, σ ~, is positive but less than σ ~ H then the Tresca ield criterion is just σ ~ = σ ; H If the axial stress, σ ~, is positive and greater than σ ~ H then the Tresca ield criterion is just σ=σ ~ and hence ielding occurs with σ ~ H < σ ; If the axial stress, σ ~, is negative then the Tresca ield criterion is σ ~ H σ ~ = σ and hence ielding again occurs with σ ~ < σ. In all cases, therefore, avoidance of collapse requires σ ~ H < σ as a necessar but not sufficient condition. Consequentl, if σ ~ H < σ is assumed, in regions where the axial stress, σ ~, is positive, the ield criterion can be taken to be simpl σ=σ ~. In regions H H ()

4 where the axial stress, σ ~, is negative, the ield criterion becomes σ ~ = σ + σ~ H. Defining the positive quantit σ = σ σ~, the ield criteria are, H For σ ~ > : For σ ~ < : σ=σ ~ (3a) σ ~ = σ (3b) The dimensionless parameter α is defined as, α σ H = = (4) Thus, α quantifies the influence of the hoop stress on the ratcheting behaviour. σ Consistent with common practice for similar ratcheting analses the results will be expressed in terms of the following dimensionless load parameters, FTOT σ ~ b = = (5) rtσ σ Because bending of the clinder is restrained b assumption, the axial strain, ε ~, is uniform around the circumference. Despite no bending deformation being possible, nevertheless it is possible for the clinder to be subject to net axial ratcheting, i.e., if ε ~ increases ccle on ccle. This total axial strain is composed of four parts: the elastic strain due to the axial stress, the elastic Poisson strain due to the hoop stress, the plastic strain, and the thermal strain. Hence, if the thermal load is acting, ~ ~ ~ ~ b x Thernal Load On: ~ σ νσ H ~ σ ε = + ε p (6a) E E E r where E and ν are the elastic moduli and ε ~ p is the axial component of plastic strain. Note that the thermal strain is negative where the thermal stress is tensile, i.e., for ~ x >. There will also be either radial or hoop components of plastic strain, but these pla no part initiall in the analsis. The significance of the radial and hoop plastic strains to the ratcheting behaviour is discussed in 7. When the thermal load is removed, (6a) becomes, σ~ νσ~ H Thermal Load Off: ε ~ = + ε ~ p (6b) E E Equations (6a,b) together with the equilibrium condition, (), and the ield criteria, (3a,b), suffice to solve the problem. 3. Dimensionless Form It is convenient to work with dimensionless quantities, normalising all stresses b σ and all strains b ε = σ / E, and the position coordinate b the radius, r. For example, σ = σ ~ / σ, σ H= σ ~ H / σ, ε = Eε ~ / σ, ε p= Eε ~ p / σ, x = ~ x / r. Using also the normalised loads, (5), the equilibrium equation, (), and the expressions (6a,b) for the axial strain become, Thermal Load On: σ~ σ ε = σ νσ + ε x (7a) H p

5 Thermal Load Off: ε = σ νσ H + ε (7b) p Equilibrium: Whilst the ield criteria are, For σ > : σ = For σ < : = σ dθ (8) (9a) σ = α (9b) From here on all quantities are understood to be in this normalised form. 4. Solution Method The method follows the now traditional approach of Refs.[-6]. It is simplest to implement b translating the algebraic relations, (7-9), into a geometric description of the piece-wise linear stress and plastic strain distributions. The ke to this is the requirement that ε be independent of x. The rules that permit construction of the distributions are, ) If the thermal load is acting, then, at all points x, Either, the slope of the σ versus x graph is zero and the slope of the versus x graph is, Or, the slope of the σ versus x graph is and the slope of the x graph is zero. ) If the thermal load is not acting, then, at all points x, ε p ε p versus Either, the slope of the σ versus x graph is zero and the slope of the versus x graph is also zero, Or, the slope of the σ versus x graph is ε p and the slope of the ε p versus x graph is, 3) If the stress is in the elastic range, α < σ <, then the plastic strain, ε p, is unchanged from its value on the last half-ccle. 5. Solution for Figure Figure illustrates the stress distributions in the case that ielding occurs on both diametricall opposite points x = and x =. Because the primar loads are constant, the stress distributions with and without the thermal load present are mirror images. The qualitative form of the plastic strain distributions follows from the rules of 4 and have been illustrated for the first three half-ccles (noting that half ccle has the thermal load acting, half ccle has no thermal load, half ccle 3 has the thermal load reinstated, etc). The rules of 4 are sufficient to demonstrate that the assumed stress distributions lead to ratcheting, the ratchet strain being illustrated in Figure. Algebraicall the ratchet strain is, ε rat = b () For ratcheting to occur it is therefore required that b <, which, referring to Figure, simpl means that all parts of the cross-section ield in tension under one or other of the loading conditions. The ratchet boundar corresponding to Figure is thus specified b, b = ()

6 To solve for the unknown dimensions a and b note that the slope of the elastic part of the stress distribution on the first half-ccle must be so that the stress in this region is given explicitl b, For At < : = + ( x b) a x< b σ () x= a this implies, α = + ( a b) (3) The angular positions corresponding to x= a and x= b are θ a and θ b respectivel, where, for dimensions normalised b r, we have simpl θ a= sin a and θ b= sin b. The equilibrium equation (8) becomes, Carring out the integrals gives, θ θ a θ b / = ( α ) dθ + [ + ( sinθ b) ] dθ +. dθ (4) / θ a θb θ θ θ b a ( b) + ( cosθ cosθ ) b a = α + + a b (5) In general the dimensions a and b could be found b numerical solution of the simultaneous equations (3) and (5) for an given loads and consistent with the ratcheting condition of Figure. Equ.() would then provide the ratchet strain per ccle. Our present purpose is onl to determine the ratchet boundar, given b (). On the ratchet boundar ( b = ) (3) and (5) become, R ratchet boundar: + α a = a and ( ) ( + α ) + ( cosθ ) θ α (6) = a Substituting the first of equs.(6) into the second provides an equation for in terms of which is the ratchet boundar on the (, ) diagram. The resulting ratchet boundaries are plotted as the pink curves in Figures 3-6 for α values of.7,.5,.3 and. respectivel. (These correspond to hoop stresses of 3%, 5%, 7% and 9% of ield respectivel). For Figure to appl, a must lie in the range - to. The condition a > implies, due to (6), On R ratchet boundar, a > + α < and > + α (7) Hence the ratchet boundaries corresponding to Figure and Equs.(6) appl onl in the parameter range indicated b (7), i.e., the pink part of the curves in Figures 3-6. Following Bree, the region above this ratchet boundar is denoted R. The left-hand extreme of the pink curves is given b a and (6) then gives, On R ratchet boundar, a ( α ) and (8) Hence the ratchet boundar has a vertical asmptote at = ( α ). This is the case that there is no additional axial load, F =, when the axial load is that due to pressure onl, thus,

7 σ H P ( α ) = = σ P where σ a is the axial membrane stress due to pressure (normalised b ield). Figures 3-6 show this vertical asmptote as the dashed pink line. This leads to the remarkable conclusion that ratcheting is not possible if no axial load other than that due to pressure is applied. The phsical reason for this is discussed in 7. Figures 3-6 also show the ratchet boundar for the case α = (i.e., zero pressure) for comparison (green dashed curves). Note that this case is similar to the original Bree problem, Ref.[], except for the clindrical geometr. For small the ratchet boundar for the case α = tends to = / and hence gives onl as. We expect to find another ratcheting region, corresponding to R on the Bree diagram, for which Figure illustrates the candidate stress and plastic strain distributions. The boundar between the two occurs when a = on Figure (or equivalentl, σ = α on Figure ). From (3) and (5) this gives, = + α + b b b and + ( ) + b a (9) θ θ = α cosθ () The R/R boundar curve is determined parametricall over b b (). This boundar between the R and R regions is shown on Figures 3-6 as the continuous blue curve. (The corresponding R/R boundar for the case α = is the dashed blue curve). 6. Solution for Figure It is clear from Figure that the ratchet strain is again given b () though b will be different. The slope of the elastic part of the stress distribution for the first half-ccle in Figure gives, σ = + b This provides one relationship between the unknown quantities b and σ. A second relationship is provided b the equilibrium equation, (8), which in this case gives, () θ b / = [ + ( sinθ b) ] dθ +. dθ (a) / θb θ θ = b cosθ (b) b b Hence, + ( ) + b Eqn.(b) determines b in terms of and, and the ratchet strain is then given b () and σ b (). Figures and become the same when σ = α and this condition when substituted into () and (b) reproduces the R/R boundar given b () - as it should for consistenc. It is clear from Figure that ratcheting occurs if and onl if b <. Hence the ratchet boundar in the R region is given b substituting b = into () and (b) giving, R Ratchet Boundar: + = (3)

8 This compares to the original Bree ratchet boundar in region R which is + 4 =. Unlike in region R, in region R the ratchet boundar is independent of α, i.e., it is independent of the hoop stress. Figure is applicable onl if σ > α and substituting this condition into = σ and (3) gives the complement of (7), i.e., On R ratchet boundar: + α > and < + α (4) which again confirms consistenc with the R region ratchet boundar. The ratchet boundaries for α values of.,.,.3... to are compared on Figure 7. Note that Figures 3-7 show onl the regions where ratcheting produces tensile plastic straining in the axial direction, for > ( α )/. For values of less than ( α )/ the additional axial load is compressive, F <. Ratcheting in this region is discussed in The Complete 'Universal' Ratchet-Shakedown Diagram So far we have derived onl the ratcheting solution but not the shakedown solution, nor what regions of the, diagram give rise to stable plastic ccling. Moreover, the ratcheting solution has been found onl for the case of tensile ratcheting in the axial direction. We ma complete the solution ver quickl b appeal to the similarit of the present problem with that of Ref.[6], the two problems differing onl in the clindrical geometr of the present problem in contrast to the flat plate considered in Ref.[6]. It ma be observed that our eqns.(7-9) are essentiall the same as the corresponding equations of Ref.[6], except that the linear integrations over dz in the latter are replaced b circular (trignometric) integrals over d θ in the present case. A rather elegant solution to the complete problem in Ref.[6] was found b allowing to take negative values. This corresponds in the present problem to considering additional axial loads which are compressive and exceed the axial pressure load, so that the net axial stress is also compressive. Ref.[6] identified nine qualitativel different stress and plastic strain distributions corresponding to nine different regions of the ratchet-shakedown diagram. This showed that for < ( α )/ there are ratcheting regions where the axial ratchet strain is compressive. Moreover the overall ratchet-shakedown diagram has a mirror plane of smmetr at = ( α )/. Since we alread have the solution for > ( α )/ this allows us to deduce immediatel the complete solution. In addition, Ref.[6] showed that the ratchet-shakedown diagrams for different α become a single, 'universal', diagram applicable for all α if the, axes are suitabl redefined. The redefinition used in Ref.[6] to accomplish this was, + α = and + α = (5) + α This same redefinition also removes the α dependence from the ratchet boundar curves for the present problem, as ma readil be proved b substitution of (5) into (3) and (6) which respectivel become,

9 R ratchet boundar: + = (6) R ratchet boundar: = + + sin independent of α, as claimed. Similarl, the boundar between the R and R regions, eqn.() becomes, + R/R boundar: = ( ) sin + again independent of α. The common point of intersection of the R ratchet boundar, (6), the R ratchet boundar, (7), and the boundar between the R and R regions, (8), which defines point B, is, Point B: (7) (8) = = (9) + α which is equivalent to = and = + α. The boundaries of the regions exhibiting compressive axial ratcheting follow immediatel from mirror smmetr in the plane = ( α )/ which is just the plane =. 5. Hence, replacing the LHS of (6) and (7) with gives these ratchet boundaries, and the same replacement on the LHS of (8) provides the division into ratcheting regions R3 and R4. Their common point of intersection (point A) is, Point A: = = (3) The final boundar is that between the stable plastic ccling region and the shakedown region, which is given simpl b = between points A and B. The resulting complete, 'universal' ratchet-shakedown diagram, plotted on, axes, is shown in Figures 8a,b. Following Ref.[6] we identif regions R and R as having tensile ratchet strains in the axial direction and equal and opposite compressive ratchet strains through the wall-thickness, but no ratchet straining in the hoop direction. So, in these regions the pipe gets longer and thinner but its diameter remains the same. In contrast, in regions R3 and R4, the axial ratchet strain is compressive and there is an equal and opposite tensile ratchet strain in the hoop direction, but no ratchet strains in the thickness direction. In these regions the pipe diameter increases and the pipe gets shorter, but its wall thickness remains the same. Now we see the phsical reason wh a pressurised clinder with no additional axial load ( F = ) cannot ratchet even for an arbitraril large cclic secondar global bending load. It is because this case represents the perfect balance between tensile and compressive ratcheting in the axial direction - and hence cannot ratchet at all. 8. Comparison with LMM The problem has also been analsed using LMM. For reasons of computational convenience, however, the Mises ield criterion was used in the LMM, as contrasted with the use of the Tresca condition in the analtic solution. Consequentl care needs

10 to be taken in making the comparison, and it is not to be expected that the two solutions will agree precisel. The LMM can be easil applied using the Abaqus CAE plug-in developed b Ure, Ref.[3]. This takes an Abaqus linear-elastic analsis model from Abaqus CAE and automaticall modifies it to allow a LMM assessment to be performed. This is done via a simple user interface that permits the user to specif the materials properties, the loading sequence, the tpe of LMM analsis required and the loads to be factored as part of the LMM process. There are three tpes of LMM assessment available in this software: shakedown, stead cclic state and ratchet limit. In this application both the shakedown and ratchet solutions were requested. The LMM assessment is performed in three stages. Firstl a linear-elastic analsis is performed for each of the loads separatel. This stage provides the elastic stresses, σ e ij (x), that are used to define the required loading for the LMM iterations. Secondl the loading ccle is defined as a sequence of scaled combinations of the elastic stresses and the LMM iterativel finds the stead state response of the structure. This stage calculates an initial constant residual stress field, ρ, and a varing residual stress field, ρ, for the specified level of loading. Finall, starting from the solution r ij from stage, the LMM finds the scaling factor that can be applied to an additional constant loading that keeps the structure within the ratchet boundar at the required level of cclic loading. This updates the constant residual stress ρ and calculates the load factors λ UB and λ LB on the constant additional load that will keep the structure within global shakedown (cclic plasticit with no ratcheting). These factors provide upper and lower bounds to the solution, the strateg being to ensure that these bounds are ver close, and hence bracket the true result tightl. The meshes used for the evaluation of the ratchet boundar for the thin clinder, subject to constant internal pressure, constant axial load and cclic secondar global bending load, were modelled as a half clinder. Four meshes were used to perform sensitivit studies. Three of the meshes had r/t=9.5, the fourth had r/t=39.5. The LMM results presented here are mainl calculated with a x3x mesh of C3DR elements with r/t=9.5. Four loads were applied: the internal pressure on the clindrical surface, the axial pressure end-load, the additional axial load, F, and the cclic bending load. The secondar global bending load was applied as a linear temperature variation across the diameter of the clinder and constraining the axiall loaded end of the mesh to remain parallel to the smmetr plane defined at the other end of the mesh. Mesh sensitivit checks were performed against the Bree problem. The clinder problem was then addressed firstl with zero pressure. The resulting ratchetshakedown diagram is shown in comparison with the analtic solution in Figure 9. For this uniaxial case there is no distinction between the Tresca and Mises ield conditions and the LMM results agree ver well indeed with the analtic solution. For non-zero pressure loading, in order to compare the Tresca analtical results with the Mises LMM results, the value of the LMM results is effectivel re-scaled so that there is agreement at =. The LMM ratchet-shakedown diagrams plotted on this basis are shown in Figures -3 for α =.9,.5,.4 and.. Bearing in mind that the LMM and analtic solutions for differing ield criteria cannot be made exactl comparable b an renormalisation, the agreement between the two is good. c ij c ij

11 Note in particular that Figures and 3 show the downward sloping ratchet boundar between the R3 and S3 regions when the net compressive axial load is sufficientl great (tending towards ). Finall, Figure 4 shows an LMM result for extremel large elastic secondar bending stresses up to = 6, confirming that there is indeed a vertical asmptote at F =. This confirms numericall that ratcheting cannot occur if the onl primar load is pressure, however large is the secondar global bending load. 9. Conclusions The ratcheting and shakedown boundaries have been derived analticall for a thin clinder composed of elastic-perfectl plastic Tresca material subject to constant internal pressure with capped ends, plus an additional constant axial load, F, and a ccling secondar global bending load. The analticall derived ratchet-shakedown boundaries are shown for various pressure loads in Figures 3-7, for which attention is confined to tensile axial ratcheting. However, the complete solution, covering also net compressive axial loads and the possibilit of compressive axial ratcheting, is displaed b Figure 8 which is plotted on axes such that the ratchet-shakedown diagram is applicable to an level of pressure (the 'universal' diagram). The analtic solution has been found to be in good agreement with solutions for the same problems addressed using the linear matching method (LMM) and the Mises criterion, Figures 9-4. When ratcheting occurs for an additional axial load, F, which is tensile, the pipe gets longer and thinner but its diameter remains the same. When ratcheting occurs with compressive F, the pipe diameter increases and the pipe gets shorter, but its wall thickness remains the same. When subject to internal pressure and cclic bending alone (F = ), no ratcheting is possible, even for arbitraril large bending loads, despite the presence of the axial pressure load. The phsical reason for this remarkable result is that the case with a primar axial membrane stress exactl equal to half the primar hoop membrane stress is equipoised between tensile and compressive axial ratcheting, and hence does not ratchet at all.. References [] J.Bree, Elastic-Plastic Behaviour of Thin Tubes Subject to Internal Pressure and Intermittent High-Heat Fluxes with Application to Fast Nuclear Reactor Fuel Elements, Journal of Strain Analsis (967) [] H.W.Ng and D.N.Moreton, Ratchetting rates for a Bree clinder subjected to inphase and out-of-phase loading, Journal of Strain Analsis for Engineering Design (986) -6. [3] H.W.Ng and D.N.Moreton, Alternating plasticit at the surfaces of a Bree clinder subjected to in-phase and out-of-phase loading, Journal of Strain Analsis for Engineering Design (987) 7-3. [4] R.A.W.Bradford, "The Bree problem with primar load ccling in-phase with the secondar load", International Journal of Pressure Vessels and Piping 99- () [5] R.A.W.Bradford, J.Ure and H.F.Chen, "The Bree Problem with Different ield Stresses On-Load and Off-Load and Application to Creep Ratcheting", International Journal of Pressure Vessels and Piping 3C (4) 3-39.

12 [6] R.A.W.Bradford, "Solution of the Ratchet-Shakedown Bree Problem with an Extra Orthogonal Primar Load", International Journal of Pressure Vessels and Piping (in press, available on-line March 5). [7] Reinhardt, W., "A Non-Cclic Method for Plastic Shakedown Analsis". ASME J. Press. Vessel Technol. 3(3) (8) paper 39. [8] Adibi-Asl, R., Reinhardt, W., "Non-cclic shakedown/ratcheting boundar determination part : analtical approach". International Journal of Pressure Vessels and Piping 88 () 3-3. [9] Jiang W, Leckie FA. A direct method for the shakedown analsis of structures under sustained and cclic loads. Journal of Applied Mechanics 59 (99) 5-6. [] Chen H, Ponter ARS. Linear matching method on the evaluation of plastic and creep behaviours for bodies subjected to cclic thermal and mechanical loading. International Journal of Numerical Methods in Engineering 68 (6) 3-3. [] Haofeng Chen, James Ure, Tianbai Li, Weihang Chen, Donald Mackenzie, "Shakedown and limit analsis of 9 pipe bends under internal pressure, cclic in-plane bending and cclic thermal loading", International Journal of Pressure Vessels and Piping 88 () 3-. [] Han F. Abdalla, "Shakedown boundar determination of a 9 back-to-back pipe bend subjected to stead internal pressures and cclic in-plane bending moments", International Journal of Pressure Vessels and Piping 6 (4) -9. [3] J.M.Ure, An Advanced Lower and Upper Bound Shakedown Analsis Method to Enhance the R5 High Temperature Assessment Procedure, Universit of Strathclde, Thesis submitted for Doctor of Engineering (EngD) in Nuclear Engineering, (3).

13 Table FE Geometr and Loading (assuming a ield stress of MPa). r/t Inner Radius r i (mm) Thickness t (mm) Outer Radius r o (mm) Additional Axial Membrane Stress F axial (MPa) Pressure Loads Internal P (MPa) Axial P axial (MPa) Outer Surface Global Bending Stress σ b (MPa) (Induced b T(x)) ± ± Figure A Distribution of Axial Stress and Plastic Strain which Produces Ratcheting (corresponding to region R on the Bree diagram). These distributions define the ratchet boundar for α < < + α and > + α. ratchet strain x = a b -b -a x α Ke First half-ccle stress; Second half-ccle stress First half-ccle plastic strain Second half-ccle plastic strain Third half-ccle plastic strain (Subsequent half-ccle strains not shown but obtained from the above b displacing upwards b the same ratchet strain per complete ccle)

14 Figure A Distribution of Axial Stress and Plastic Strain which Produces Ratcheting (corresponding to region R on the Bree diagram). These distributions + α define the ratchet boundar for > and < + α. ratchet strain x = - b -b σ α Ke First half-ccle stress; Second half-ccle stress First half-ccle plastic strain Second half-ccle plastic strain Third half-ccle plastic strain (Subsequent half-ccle strains not shown but obtained from the above b displacing upwards b the same ratchet strain per complete ccle)

15 Figure 3: x-tensile Ratchet Diagram, α =.7 (asmptote at =.5) Fig. (alpha=.7) Fig. (alpha=.7) R/R boundar (alpha=.7) Fig. (alpha=) Fig. (alpha=) R/R boundar (alpha=) asmptote (alpha=.7) 6 R R

16 Figure 4: x-tensile Ratchet Diagram, α =.5 (asmptote at =.5) Fig. (alpha=.5) Fig. (alpha=.5) R/R boundar (alpha=.5) Fig. (alpha=) Fig. (alpha=) R/R boundar (alpha=) asmptote (alpha=.5) 6 5 R 4 3 R

17 Figure 5: x-tensile Ratchet Diagram, α =.3 (asmptote at =.35) Fig. (alpha=.3) Fig. (alpha=.3) R/R boundar (alpha=.3) 9 Fig. (alpha=) Fig. (alpha=) 8 R/R boundar (alpha=) asmptote (alpha=.3) R 4 3 R

18 Figure 6: x-tensile Ratchet Diagram, α =. (asmptote at =.45) Fig. (alpha=.) Fig. (alpha=.) R/R boundar (alpha=.) Fig. (alpha=) Fig. (alpha=) R/R boundar (alpha=) asmptote (alpha=.) R R Figure 7: x-tensile Ratchet Diagrams for α =.,.,.3... (α = has no pressure load) alpha = (no pressure) alpha =.9 alpha =.8 alpha =.7 alpha =.6 alpha =.5 alpha =.4 alpha =.3 alpha =. alpha =

19 Figure 8a: The complete, universal ratchet/shakedown diagram (on re-scaled, axes, applicable for an α) In R3 and R4 the hoop ratchet is tensile and there is a balancing reduction in pipe length In R and R the axial ratchet is tensile and there is a balancing reduction in wall thickness 8 '=/(+alpha) 6 4 R4 R R3 P R S3 S S E '=(+alpha)/(+alpha) Figure 8b: The complete, universal ratchet/shakedown diagram (on re-scaled, axes, applicable for an α) 3 In R3 and R4 the hoop ratchet is tensile and there is a balancing reduction in pipe length In R and R the axial ratchet is tensile and there is a balancing reduction in wall thickness R4 R '=/(+alpha) P R3 A B R S3 S C S E '=(+alpha)/(+alpha)

20 Figure 9: LMM Ratchet and Shakedown Boundaries Compared with the Analtic Solution, on, axes and confined to > for Zero Pressure (α = ). LMM (LB): Ratchet Boundar 9. LMM (UB): Ratchet Boundar LMM (UB): Shakedown Boundar Thin Clinder (alpha=.) 8. Rectangular Section (Bree)

21 Figure : LMM Ratchet and Shakedown Boundaries Compared with the Analtic Solution, on, axes and confined to > (α =.9) alpha= LMM Shakedown (UB) LMM Ratchet (UB) Analtic Ratchet R/R Zero Additonal Axial Load Figure : LMM Ratchet and Shakedown Boundaries Compared with the Analtic Solution, on, axes and confined to > (α =.5) alpha= LMM Shakedown (UB) LMM Ratchet (UB) Analtic Ratchet R/R Zero Additonal Axial Load

22 Figure : LMM Ratchet and Shakedown Boundaries Compared with the Analtic Solution, on, axes and confined to > (α =.4) alpha= LMM Shakedown (UB) LMM Ratchet (UB) Analtic Ratchet R/R Zero Additonal Axial Load Figure 3: LMM Ratchet and Shakedown Boundaries Compared with the Analtic Solution, on, axes and confined to > (α =.) alpha= LMM Shakedown (UB) LMM Ratchet (UB) Analtic Ratchet R/R Zero Additonal Axial Load

23 Figure 4: LMM solution for α =.5 showing the asmptote chased to an extremel large value of 6 (thus demonstrating that there is no ratcheting at zero additional axial load) alpha= LMM (UB) LMM (LB) Analtic R/R Tresca Von Mise