Shakedown and Limit Analysis of 90 o Pipe Bends Under Internal Pressure, Cyclic In-plane Bending and Cyclic Thermal Loading

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1 Shakedown and Limit Analyi of 9 o ipe Bend Under Internal reure, Cyclic In-plane Bending and Cyclic Thermal Loading Haofeng Chen*, Jame Ure, Tianbai Li, Weihang Chen, Donald Mackenzie Department of Mechanical Engineering, Univerity of Strathclyde, Glagow, G1 1XJ Abtract The Linear Matching Method i ued to create the hakedown limit and limit load interaction curve of 9 degree pipe bend for a range of bend factor. Two load cae are conidered i) internal preure and in-plane bending (which include opening, cloing and revered bending) and ii) internal preure and a cyclic through wall temperature difference giving rie to thermal tree. The effect of the ratio of bend radiu to pipe mean radiu (R/r) and mean radiu to wall thickne (r/t) on the limit load and hakedown behaviour are preented. Keyword: ipe bend, hakedown, limit load, revere platicity, ratchetting 1. Introduction 9 o pipe bend are common in piping ytem, and it i vital to enure that thee component are deigned againt platic collape. Additionally, deign againt incremental platic collape (ratchetting) or low cycle fatigue failure (reulting from alternating platicity) mut alo be performed. The conervatim reulting from limiting the tructure to purely elatic behaviour i often not acceptable and component are allowed to hakedown, which allow platic deformation that doe not reult in either alternating platicity or ratchetting. The hakedown of tructure ha been tudied by many reearcher. The complexity of the phenomenon mean that analytical olution exit for only the implet of geometrie and loading. Incremental Finite Element Analyi (FEA) can be ued to predict if elatic hakedown, alternating platicity or ratchetting occur but doe not evaluate limit or boundarie between thee different repone. Creation of the Bree-like [1] hakedown boundary diagram require many FEA calculation to be performed, which become very computationally expenive with complex geometrie. Conequently, many direct method have been created baed upon the Koiter [2] kinematic and Melan [3] tatic theorem to calculate the hakedown limit. Typical of thee are the GLOSS R-node method [4], the Elatic Compenation Method [5], mathematical programming method [6] and the Linear Matching Method (LMM) [7,8]. The LMM ha been hown to give accurate reult for many geometrie with complex load hitorie and temperature dependent *Correponding Author 1

2 material propertie. LMM ABAQUS ubroutine have been conolidated by the Britih Energy Generation Ltd R5 reearch program and are ued in the aement of plant component [9]. The LMM i ued in thi work to analye the hakedown behaviour of 9 degree pipe bend. Firt a ummary of current literature regarding the hakedown of thi geometry i given, followed by an explanation of the LMM procedure, the Finite Element model ued and finally the reult, dicuion and concluion. 2. Summary of Limit and Shakedown Analyi of ipe Bend Figure 1 how a typical 9 o pipe bend. Such bend are commonly decribed in term of two ratio: r/t and R/r, where r i the mean pipe radiu, R i the bend radiu and t i the wall thickne. When thee ratio are combined, they give the bend characteritic of the pipe, h: h R r r t Rt 2 r (1) 2.1 Limit Load Solution Coniderable attention ha been given to limit analyi of pipe bend under in-plane bending and internal preure in recent year. Calladine [1] derived a limit moment equation for pipe bend with no attached traight ection uing thin hell theory: M 1.19π tς h 3 L 2 y 2 where h.5 (2) where σ y i the yield tre of the material. Goodall [11] derived a limit preure expreion for bend with no attached traight, alo uing thin hell theory and the Treca yield criterion: L ς y r 1 t R r r 1 2R (3) Other than thee equation, no further theoretical olution could be found for the platic limit of thi geometry. Intead, reearcher have ued extenive FEA to generate data for variou ratio of R/r and r/t. Thee data point have then been plotted and equation derived from curve-fit of the data. A recent review of the literature by Lei [12] ha highlighted ome important concluion. Firtly, the attached traight ection provide a ignificant level of reinforcement to the bend by 2

3 allowing the platic zone to pread. Thi increae the limit load above thoe calculated by Calladine and Goodall. The literature preent many equation for the limit load of a pipe bend under in-plane bending, internal preure and combination loading baed on both mall and large deformation FEA. Whilt thee equation provide a good fit to the data they are derived from, it i found that in many cae the equation do not uccefully predict the reult from other author, epecially when combined loading i concerned. Additionally, the range of pipe bend characteritic which thee equation decribe are too narrow to be applied to the range analyed in thi work (a general maximum found wa h =.7 compared to h = 1 in thi paper). A a reult of thi, a well a the fact that the geometrie with r/t = 5 cannot be conidered "thin" a an approximation, the reult preented here are normalied againt the limit preure and moment for a thick walled traight pipe with the ame mean radiu and wall thickne: L 2 ro ς yln 3 ri (4) M L 4 ς 3 y 3 o r r 3 i (5) where r o and r i are the outer and inner radii of the pipe repectively. All limit analye in the literature concern mechanical loading only a thermal loading and thermal tree have no effect on the limit load of a component. 2.2 Shakedown Analyi Depite the extenive volume of work regarding limit analyi and limit load olution for mechanical loading, very little work ha been publihed regarding the hakedown behaviour of thi geometry. H.F. Abdalla et al [13] preented the hakedown behaviour of a ingle bend with R = 48mm, r = 133.5mm and t = 3mm (giving R/r = 3.6 and r/t = 44.5). Further calculation are carried out by C.S Oh et al [14] for the hakedown of bend covering a large range of h for internal preure and in-plane cloing and opening bending (revered bending and thermal loading i not conidered). In [14], the ame lower bound numerical technique i ued a in [13], which i thought by the author of thi work to give olution which are too conervative when compared to the exact hakedown limit olution. Unlike limit analyi, cyclic thermal loading may play an important role in the hakedown limit, however, no publihed work could be found regarding the hakedown of pipe bend under thermal and mechanical loading. 3

4 The objective of thi paper i to ue the Linear Matching Method to analye the hakedown and limit load of 9 o pipe bend ubject to both mechanical and thermal loading in a ytematic manner. Two cae are conidered. The firt i when the pipe i ubjected to an internal preure and in-plane cloing, opening and revered bending. The mechanim of revere platicity and ratchetting are explored and a comparion between the hakedown behaviour of opening, cloing and revered bending i preented. The econd cae i where the pipe i ubjected to an internal preure and a cyclic temperature gradient through the wall thickne. The effect of temperature dependent yield tre i conidered and the hakedown behaviour of pipe bend i compared to that of a traight pipe. Although large deformation calculation introduce geometric trengthening and weakening effect, which for pipe bend take the form of the ovaliation of the ection due to the applied moment, mall deformation analyi i hown to give conitent and conervative reult. Therefore mall deformation aumption are ued throughout. 3. Overview of LMM The Linear Matching Method ha been decribed in detail in other work [7,8]. A baic overview i preented here for convenience. The bai of the Linear Matching Method i that limit tate olution can be developed by ytematically reducing the modulu in region of high tre of a linear elatic olution. Doing o reduce the maximum tre in thee region and allow the tre to re-ditribute in the body, which in turn increae the load at which all tree lie within yield. The material i aumed to be linear elatic-perfectly platic and to atify both the von-mie yield criterion and the platic incompreibility condition. An iteration of thi procedure begin with calculation of the new hear modulu µ baed upon the tre from the previou linear olution and the train rate i ε,given by 2ς y μ i 3ε (6) where σ y i the yield tre of the material. Conider a body of volume V, with urface area, S. The body i ubjected to mechanical load λ(x,t), and a thermal load λθ(x,t) which are applied over a part of the urface S T. The remainder of the urface area i contrained to have zero diplacement rate u i. Thee load are applied over the time cycle t t. The linear elatic tre olution λˆ ς, i calculated from thee load a 4

5 λςˆ θ x,t λςˆ x,t λςˆ x,t i i i (7) where θ λˆ ς and λˆ ς are the elatic olution correponding to λθ(x,t) and λ(x,t) repectively. λ i introduced a a load parameter o that a range of loading hitorie can be conidered. For cyclic loading, the tre hitory over the time cycle t t i given by ς λςˆ r x, t ρ x ρ x, t i i i (8) where ρ x i i the elf-equilibrating contant reidual tre field and correpond to the reidual r tate of tre at the beginning and end of each cycle. ρ x, t i denote the changing component of reidual tre through the cycle, which for hakedown condition mut equal zero, meaning that the cyclic tre hitory at hakedown i given by ς λςˆ x,t ρ x i i (9) Baed on Koiter theorem [2], the upper bound hakedown limit, λ UB i given by: λ UB y V Δt V Δt ς ε ε ςˆ ε dtdv dtdv (1) where ε i a kinematically admiible train rate and ε 2 ε 3 ε i the effective train rate. Conecutive iteration of the procedure i hown to converge to the leat upper bound, λ UB λ S where λ S i the exact hakedown multiplier. The lower bound i found by Melan theorem [3], which tate that the tructure will hakedown if the uperpoition of the applied tree and reidual tree do not violate the temperature dependent yield condition at any point: f λ ςˆ x, t ρ x LB i i (11) where λ LB i the lower bound hakedown multiplier. Thi procedure can be implemented in the commercial finite element oftware package ABAQUS [15] through the ue of the uer ubroutine UMAT. In each iteration, the linear problem i olved for tre, train and diplacement. The UMAT ubroutine compute the varying hear modulu, the Jacobian matrix, the reidual tre and the updated tre for a given train increment. The upper bound hakedown limit i found by integrating equation (1) over the entire volume and thi i ued 5

6 in the next iteration. The lower bound hakedown multiplier i found by checking the condition for all integration point. Shakedown analyi deal with cyclic load condition, which may have any number of load extreme over the time cycle. Hence, the limit load can be determined uing the procedure decribed above a a pecial cae of hakedown, where the number of load extreme in the cyclic load domain reduce to one; i.e. a monotonic load condition. 4. FEA Model A three dimenional olid model wa contructed in ABAQUS. Due to the ymmetry of the geometry, a one quarter model wa ued and ymmetry boundary condition were applied (ee Figure 2). ABAQUS type C3D2R quadratic element with reduced integration were ued for the tructural analyi and ABAQUS type DC3D2 element were ued for the heat tranfer analyi. To meh the bend, at leat three element were ued through the thickne, ten around the radiu of the bend and twenty element around the circumference of the pipe. A refinement tudy wa conducted to validate the accuracy of the meh ued. Such a meh wa choen to give ufficient denity around the area of interet and to maintain reaonable element apect ratio. The attached traight ection wa mehed with twenty element along it length, which were biaed to be maller in the region of the bend. The internal preure,, wa applied at the inner urface auming the cloed end condition, with an equivalent axial tenion applied at the free end of the traight ection to replicate the axial tre. The moment wa applied a a linear preure ditribution at the free end of the traight ection, which wa applied uing the uer ubroutine DLOAD in ABAQUS (Figure 3). When conducting a limit analyi, the preure and moment load were applied monotonically. When hakedown analyi wa performed, the preure wa held at a contant value and the initial bending moment wa cycled i) from zero to a maximum of M for opening bending ii) from zero to a minimum of -M for cloing bending or iii) from -.5M to +.5M for revered bending, hown in Figure 4. When the cae of internal preure and cyclic thermal loading wa conidered, a teady tate heat tranfer analyi wa conducted with T o = 1 o C applied at the inner urface and o C at the outer urface to contruct the mot evere thermal hitory. The temperature ditribution calculated by thi heat tranfer analyi wa then applied to the model, which give rie to initial maximum tree in a thermal cycle. An additional contraint wa added - the free end of the pipe wa contrained via 6

7 equation to expand in-plane along it length, imulating the thermal expanion of a long pipe. The temperature gradient through the wall thickne wa cycled from zero to a maximum and back to zero over the time tep. Analyi of 9 geometrie i preented with R/r = 2, 3 and 5, each of which with r/t = 5, 1 and 2, which give.1 h 1. In each cae the attached traight ection were a length uch that L/R = 8, which wa found to be more than adequate to alway provide the maximum reinforcement poible to the bend. In each cae a Young' Modulu of 2Ga and a oion ratio of.3 wa ued. The teel i aumed to have a thermal conductivity of 43 W/m 2 K and a thermal expanion coefficient of 1x1-5 o C -1. The yield tre i aumed to be temperature dependent, with data for thi taken from D55 Britih Standard for unfired welded preure veel deign [16], a hown in Table 1. For each analyi, two linear elatic olution were performed: one for the internal preure loading and the econd for either the bending moment or thermal loading. Thee two tre olution are then ued to contruct the initial tre tate in equation (7) to begin the iterative proce to find the upper and lower bound hakedown limit multiplier. 5. Internal reure and Bending Moment Figure 5 how the two elatic olution for R/r = 2 and r/t = 5 ued to contruct the initial tre condition for the hakedown and limit analyi. The internal preure caue high tree at the inner urface of the intrado of the bend. The bending moment caue high tree at the flank. Figure 6 preent the hakedown limit and limit load interaction curve for R/r = 2 and r/t = 5 under internal preure and opening bending. It can be een that the hakedown boundary follow a claic Bree-like hape whereby the revere platicity boundary i a contant value of cyclic bending. The limit load urface calculated by the LMM i hown and thi ha been verified at 5 point uing ABAQUS elatic-platic incremental analyi (uing the Rik' method). The difference between the LMM and ABAQUS limit olution i le than 1%, providing confidence that the LMM produce accurate reult. The contour of platic train reulting from the cyclic loading at point A and B in Figure 6 are hown in Figure 7, which correpond to the revere platicity limit and the ratchet limit repectively. The platic train correponding to the revere platicity limit i retricted to a very mall area at the flank of the pipe, with the train accumulating at the inner urface. The platic train correponding to the ratchet limit occur globally at the intrado of the bend, with the train initiating at the outer urface. The location of ratchetting train matche thoe found by Chen, Gao and Chen in [17], giving further confidence in the accuracy of the method. The revere platicity 7

8 mechanim occur mainly due to the location of the peak tre caued by the bending moment. The ratchetting mechanim occur at the location of peak preure tre, at the intrado of the bend. Figure 8 how a comparion between the reult obtained by C.S. Oh et al [14] and the reult obtained here for R/r=2, r/t=1 ubject to internal preure and cyclic opening bending. The data i normalied againt equation (2) and the Goodall limit preure uing the von-mie criterion (equation (3) multiplied by 2/ 3). The method employed by C. S. Oh et al i a lower bound method and both the lower and upper bound calculated by the LMM are hown in the Figure. Comparion of the two hakedown curve reveal that they produce comparable revere platicity boundarie but that the LMM predict a ignificantly larger ratchetting boundary. In order to verify the accuracy of the LMM, the limit load for preure loading wa calculated uing ABAQUS Rik' analyi and two point were choen (labelled C and D in Figure 8) for full cyclic analyi in ABAQUS. The limit preure given by the Rik' analyi correlate well with the equivalent value calculated by the LMM, with a difference of le than 1% between the two olution. The platic train hitorie at the for the cyclic loading at point C and D are hown in Figure 9. The train in Figure 9 were taken from the poition of maximum platic train, which wa at the inner urface at the intrado of the bend. The platic train hitory of point C how that hakedown i achieved after approximately 1 cycle. oint D how claic ratchetting behaviour, with the platic train increaing every cycle. Thee three point all agree well with the hakedown boundary calculated by the LMM, giving confidence in the accuracy of the reult produced by thi method. Further benefit of the LMM can be found with the computing time taken to generate the hakedown curve. The time taken for the LMM to generate the point on the ratchetting boundary wa le than 1% of that taken for the two analye for point A and B to complete. The following ection preent reult which demontrate the effect of h, r/t and R/r on the hakedown limit and limit load urface for both opening and cloing bending. The applied preure and moment are normalied againt the limit preure and moment for an equivalent thick walled traight pipe repectively (equation (4) and (5)) to allow eae of comparion between different geometrie. It i clear from Figure 6 that the LMM produce lower and upper bound hakedown limit that converge very cloe to each other. Thi i repreentative of all reult obtained and for the remainder of thi paper only the upper bound limit will be hown for clarity in the Figure. 5.1 Effect of Bend Characteritic h 8

9 Figure 1 how the hakedown boundarie for all 9 geometrie conidered ubject to contant internal preure and cyclic opening bending. A clear trend i een in which the normalied moment (correponding to the revere platicity limit) increae with h. lotting the normalied cyclic moment correponding to the revere platicity limit againt h how a quadratic relationhip, given by equation (12): R Limit h h.732 (12) where R Limit i the cyclic moment correponding to the revere platicity limit normalied againt the limit moment of a thick walled traight pipe (equation (5)). Other than thi trend, no further correlation can be found with h: the limit preure, the preure at which ratchetting begin and the lope of the ratchetting boundarie do not how a correlation with h. 5.2 Effect of the ratio r/t Figure 11 how the hakedown limit and limit load urface for fixed R/r value of 2, 3 and 5, and varying r/t value for an opening bending moment. The general trend for each geometry conidered i imilar. A r/t decreae (i.e. the pipe become thicker), the limit load urface expand, indicating larger load to failure. Large increae in the limit moment are een with decreaing r/t but comparatively little effect i oberved in the normalied limit preure. The graph alo reveal that the alternating platicity boundary occur at higher value of normalied moment with decreaing r/t. One intereting obervation i the margin between the limit load and the hakedown limit urface. In general, there i a ignificant margin between both line at low preure, which reduce to a minimum at around the point where alternating platicity change to ratchetting. At preure larger than thi the margin increae once more before converging to the limit load for preure loading. A r/t become maller, the margin between the two line become maller to the extent that there i almot no margin between the hakedown and limit load for a large normalied preure range for R/r = 3, r/t = 5 and R/r = 5 and r/t = 5. Thi phenomenon i particularly important for the deigner. The hakedown behaviour for cyclic loading of the bend may, for ome loading condition, be very cloe to the limit load for monotonic loading. In uch cae, the conervatim in the deign i low when compared to loading condition which offer a larger (and afer) margin between the hakedown and limit curve. The ame tudy wa conducted for R/r = 3 under cloing bending, hown in Figure 12. The trend oberved for opening bending apply equally to cloing bending. 9

10 5.3 Effect of the ratio R/r Figure 13 preent the ame reult a in Figure 11 for pipe bend ubject to internal preure and opening bending but in each graph r/t i fixed and the effect of changing R/r are oberved. Each geometry exhibit imilar trend. The limit load curve expand with increaing R/r, and how an increae in both the limit moment and limit preure toward a normalied value of 1. Thi correlate with the expected behaviour that a R/r the behaviour of the bend will tend toward that of a traight pipe. In term of the hakedown limit, increaing R/r increae the normalied moment with which alternating platicity occur. It i alo oberved that the margin between the limit load and hakedown limit curve reduce with increaing R/r in all cae. The ame tudy wa undertaken with the cloing bending cae, uing a fixed value of r/t = 1. Figure 14 how that the ame effect and trend are oberved a per the opening bending cae. 5.4 Comparion of Opening, Cloing and Revered Bending Figure 15 how a comparion between cloing, opening and revered bending for r/t = 1 and R/r = 2, 3 and 5. The bending moment range, M, i plotted rather than the peak value of moment, M, ued in previou figure. In each cae the limit load urface for cloing bending i larger than that for opening bending but give the ame limit load for bending and preure loading alone. Limit analyi of revered bending i not poible due to the conflicting direction of opening and cloing bending. The hakedown limit curve how that the normalied moment at which revere platicity occur doe not change for any mode of bending, i.e. equation (12) applie for all three mode of bending. Cloing bending ha an increaed ratchetting boundary over opening bending. The difference between the opening and cloing bending ratchetting boundarie reduce with increaing R/r. The ratchetting boundary of revered bending ret between that of opening and cloing bending apart from the cae of R/r=5, where revere bending lie outide cloing bending at the tranition region between revere platicity and ratchetting. Examination of the contour plot reveal that the location of the alternating platicity mechanim doe not change for opening, cloing or revered bending, and thu it i the range of the bending tre which determine the alternating platicity boundary rather than if it i tenile or compreive. The cae of the revered bending hakedown limit being greater than cloing bending for R/r=5 (Figure 15c) can be explained ince both the cyclic opening bending (Figure 4a) and cloing bending (Figure 4b) are equivalent to the revere bending (Figure 4c) uperimpoed on a contant mean bending moment. Thi mean moment caue a tenile tre at the intrado for opening bending and 1

11 a compreive tre for cloing bending. In the majority of the analye performed in thi paper, the addition of the tree from the internal preure and mean moment at the intrado i alway tenile along the axi of the pipe, which reult in the ratchetting mechanim beginning at the outer urface of the intrado. In the cae of R/r=5 for cloing bending at high value of bending moment, the compreive tre from the mean moment i large enough to caue the um of thee tree in the region to be compreive. Thi change to the tre caue the ratchetting mechanim to originate from the inner urface of the intrado. It i thi change in ratchetting mechanim which reduce the hakedown envelope of cloing bending below that of revered bending for preure value of.45< <.66. L 6. Internal reure and Cyclic Thermal Loading The applied temperature gradient through the wall (T o =1 o C at the inner urface, o C at the outer urface) combined with the contraint that the free end expand in-plane create the thermal tre which i ued a the initial value for the cyclic tre in equation (7). The initial tre value for the contant internal preure loading i the ame a that in Figure 5a. The reult preented here how the effect of R/r and r/t on the hakedown limit interaction curve. The internal preure are normalied againt equation (4) with a yield tre of 2Ga and the cyclic thermal loading, T, i normalied againt the applied initial inner urface temperature T o = 1 o C. 6.1 Temperature Dependent Yield Stre Figure 16 give the hakedown boundarie for R/r=3 and r/t=1, howing the difference between temperature dependent and temperature independent yield tre (temperature dependent yield value are given in Table 1). It i clear that where temperature dependency i trong, or where large temperature are involved, it i important to conider thi in the analyi. A ignificant reduction in the entire hakedown boundary i oberved and the remainder of the reult preented here therefore take temperature dependent yield tre into conideration. 6.2 Effect of R/r Figure 17 give a comparion of the hakedown boundarie with changing R/r and fixed r/t=1. Alo included i the hakedown boundary for a traight pipe under the ame loading. It can be een that the hape of the hakedown envelope for the pipe bend i very imilar to that of the traight pipe. A R/r of the bend increae, the hakedown envelope tend toward that of the traight pipe. Thi i becaue a the radiu of the bend become larger with repect to the pipe radiu, the effect of the 11

12 bend decreae, reulting in a lower tre concentration. When R/r (i.e. a traight pipe) the concentration i zero. Overall, for the range of bend conidered here, the reduction are relatively mall when compared to a traight pipe. The mot evere bend conidered here, R/r = 2, ha a revere platicity limit which i more than 9% that of the traight pipe and a ratchetting boundary which i at leat 8% of the traight pipe. 6.3 Effect of r/t Figure 18 how the hakedown boundarie for fixed R/r = 3 and varying value of r/t = 5, 1 and 2, and alo that of a traight pipe with r/t=1. Thi how that only very marginal change occur to the hakedown envelope over the thickne range conidered. Thi i primarily becaue the magnitude of thermal tre created i relatively independent of thickne, with only thick pipe howing an increae in thermal tre (which caue the very lightly reduced revere platicity boundary for r/t=5 in Figure 18). Overall, the pipe thickne ha little effect on the normalied hakedown limit interaction diagram (clearly the abolute value will be affected). 7. Concluion Shakedown and limit urface for pipe bend covering a large range of bend characteritic are preented. Non-dimenionaliing the reult againt limit moment and preure for an equivalent traight pipe allow trend to be identified with changing R/r, r/t and cloing and opening bending. Baed on the reult preented, the following obervation and deign recommendation are propoed: The Linear Matching Method ha been verified by incremental full cyclic FEA and ABAQUS Rik' analye, howing that it give very accurate hakedown limit and limit load envelope. When ubject to internal preure and cyclic in-plane bending, the normalied moment correponding to the revere platicity limit i related to h by equation (12) for all three mode of bending. The hakedown envelope how no further trend with h; the ratio R/r and r/t mut be conidered to acertain the behaviour. Decreaing r/t increae the normalied cyclic bending moment correponding to the revere platicity limit. Decreaing r/t alo decreae the margin between the hakedown envelope (for cyclic loading) and the limit load urface (for monotonic loading). Deigner mut therefore take care to enure that a ufficient margin i preent to prevent an unexpected increae in the cyclic load cauing platic collape. 12

13 Increaing R/r increae the normalied cyclic bending moment correponding to the revere platicity limit. Additionally, increaing R/r decreae the margin between the hakedown and limit urface, and o care hould be taken to enure ufficient margin i preent. Opening and cloing bending how no difference in the trend diplayed, and all thoe tated above can be equally applied to either opening or cloing bending. Direct comparion of opening and cloing bending how that cloing bending give a larger limit load urface. The normalied bending moment which caue revere platicity i not changed for opening, cloing or revered bending. Cloing bending how an increaed ratchetting limit over opening bending, with thee curve converging with increaing R/r. At high R/r value, cloing bending may have a maller hakedown envelope than revered bending due to a change in the cloing bending ratchetting mechanim at high value of bending moment. Temperature dependent yield tre i an important deign conideration a ignificant reduction in the hakedown limit envelope can reult. Smaller R/r ratio reduce the hakedown limit envelope when the pipe i ubject to contant internal preure and a cyclic thermal tre, but the lowet R/r conidered in thi paper till had more than 8% of the hakedown trength of a traight pipe. r/t i hown to have very little impact on the hakedown limit envelope for normalied value of internal preure and cyclic thermal tre (clearly the abolute value of thee load will change with changing r/t). Acknowledgment The author gratefully acknowledge the upport of the Engineering and hyical Science reearch Council of the United Kingdom, The Univerity of Strathclyde and Britih Energy Generation Ltd during the coure of thi work. The author would alo like to thank Mr David Tipping of Britih Energy Generation Ltd for hi advice and upport. Reference [1] Bree J. Elatic-latic Behaviour of thin tube Subjected to Internal reure and Intermittent High-Heat Fluxe with Application to Fat Nuclear Reactor Fuel Element. Journal of train Analyi 1967;2(3): [2] Koiter WT. General theorem for elatic platic olid. rogre in olid mechanic, Sneddon JN and Hill R, ed., North Holland, Amterdam, 196;1:

14 [3] Melan E. Theorie tatich unbetimmter yteme au ideal-platichem bautoff. Sitzungber. d. Akad. d. Wi., 1936;Wien 2A(145): [4] Sehadri R. Inelatic evaluation of mechanical and tructural component uing the generalized local tre train method of analyi. Nuclear Engineering and Deign, 1995;153(2-3): [5] Mackenzie D, Boyle JT, Hamilton R. The elatic compenation method for limit and hakedown analyi: a review. Tran IMechE, Journal of Strain Analyi for Engineering Deign, 2;35(3): [6] Liu YH, Carvelli V, Maier G. Integrity aement of defective preurized pipeline by direct implified method. International Journal of reure Veel and iping, 1997;74:49 57 [7] Chen HF, onter ARS. Shakedown and limit analye for 3-D tructure uing the Linear Matching Method. International Journal of reure Veel and iping, 21;78: [8] Chen HF. Lower and upper bound hakedown analyi of tructure with temperature-dependent yield tre. ASME Journal of reure Veel Technology 21;132(1): [9] Ainworth RA (editor). R5: an aement procedure for the high temperature repone of tructure. Britih Energy Generation Ltd., 23;(3) [1] Calladine CR. Limit Analyi of Curved Tube. Journal of Mechanical Engineering Science 1974;16(2):85-87 [11] Goodall IW. Lower Bound Limit Analyi of Curved Tube Loaded by Combined Internal reure and In-lane Bending Moment. Central Electricity Generation Board (1978) Report RD/B/N436. [12] Lei Y. Review of Limit Load Solution for Defect Free ipe Bend. Britih Energy Generation Ltd (29) E/RE/BBGB/34/GEN/8. [13] Abdalla F, Megahed MM, Younan MYA. Shakedown Limit of a 9 Degree ipe Bend Uing Small and Large Diplacement Formulation. Journal of reure Veel Technology 27;129: [14] Oh CS, Kim YJ, ark CY. Shakedown Limit Load for Elbow under Internal reure and Cyclic Inplane Bending. International Journal of reure Veel and iping 28;85: [15] ABAQUS. Uer' Manual Verion [16] Britih Standard Intitute. Specification for "Unfired Fuion Welded reure Veel". D 55:26. 14

15 [17] Chen X, Gao B, Chen G. Ratcheting Study of reuried Elbow Subjected to Revered In-lane Bending. Journal of reure Veel Technology 26;128:

16 Table 1 - Temperature Dependent Yield Stre Temperature ( o C) Yield Stre (Ma)

17 Figure Caption Figure 1 - ipe Bend with Attached Straight Section Figure 2 - Typical Mehed FEA Model Figure 3 - Application of ure Bending Moment Uing the DLOAD Subroutine Figure 4 - Application of Cyclic Bending Moment During Shakedown Analyi for a) Opening Bending, b) Cloing Bending and c) Revered Bending Figure 5 - Elatic olution for a) Internal reure and b) Opening Bending Figure 6 - Shakedown Limit and Limit Load curve for R/r = 2, r/t = 5, Internal reure and Opening Bending Figure 7 - Contour lot of latic Strain to Show Failure Mechanim of a) Alternating laticity at oint A and b) Ratcheting at oint B Figure 8 - Comparion of LMM with Data from C.S. Oh et al for R/r=2, r/t=1 Subject to Internal reure and Opening Bending Figure 9 - latic Strain Hitorie for oint C and B by Full Cyclic FEA Figure 1 - Shakedown Limit Interaction Curve for h=.1 to h=1, Internal reure and Opening Bending Figure 11 - Effect of changing r/t with a) R/r=2, b) R/r=3 and c) R/r=5, opening bending Figure 12 - Effect of changing r/t with fixed R/r=3, cloing bending Figure 13 - Effect of Changing R/r with a) r/t=5, b) r/t=1 and c) r/t=2, opening bending Figure 14 - Effect of changing R/r with fixed r/t=1, cloing bending Figure 15 - Comparion of Cloing and Opening Bending for fixed r/t=1 and a) R/r=2, b) R/r=3 and c) R/r=5 Figure 16 - Effect of Temperature Dependent Yield Stre, R/r=3 and r/t=1 Figure 17 - Effect of changing R/r with R/r=2, 3 and 5 and Straight ipe with fixed r/t=1 Figure 18 - Effect of Changing r/t with r/t=5, 1 and 2 with fixed R/r=3 and traight pipe with r/t=1 17

18 Figure 1 - ipe Bend with Attached Straight Section Symmetry lane Figure 2 - Typical Mehed FEA Model 18

19 Figure 3 - Application of ure Bending Moment Uing the DLOAD Subroutine M Moment Moment.5M Moment time t 1 t 2 time t 1 t 2 t 1 t 2 time -.5M -M a) b) c) Figure 4 - Application of Cyclic Bending Moment During Shakedown Analyi for a) Opening Bending, b) Cloing Bending and c) Revered Bending 19

20 a) b) Figure 5 - Elatic olution for a) Internal reure and b) Opening Bending M M L Upper Bound Shakedown Limit Lower Bound Shakedown Limit Limit Load Limit Load by ABAQUS Rik' Analyi.5.4 oint A oint B L Figure 6 - Shakedown Limit and Limit Load curve for R/r = 2, r/t = 5, Internal reure and Opening Bending 2

21 a) oint A b) oint B Figure 7 - Contour lot of latic Strain to Show Failure Mechanim of a) Alternating laticity at oint A and b) Ratcheting at oint B M M L LMM Upper Bound LMM Lower Bound Data from C.S. Oh et al Limit Load by ABAQUS Rik' Analyi.4.2 oint C oint D L Figure 8 - Comparion of LMM with Data from C.S. Oh et al for R/r=2, r/t=1 Subject to Internal reure and Opening Bending 21

22 latic Strain (%) oint C oint D Number of Cycle Figure 9 - latic Strain Hitorie for oint C and B by Full Cyclic FEA M M L h value L Figure 1 - Shakedown Limit Interaction Curve for h=.1 to h=1, Internal reure and Opening Bending 22

23 M M L L r/t=5 hakedown limit r/t=5 Limit Load r/t=1 Shakedown Limit r/t=1 Limit Load r/t=2 Shakedown Limit r/t=2 Limit Load M M L a) r/t=5 Shakedown Limit r/t=5 Limit Load r/t=1 Shakedown Limit r/t=1 Limit Load r/t=2 Shakedown Limit r/t=2 Limit Load L M M L b) r/t=5 Shakedown Limit r/t=5 Limit Load r/t=1 Shakedown Boundary r/t=1 Limit Load r/t=2 Shakedown Boundary r/t=2 Limit Load L c) L Figure 11 - Effect of changing r/t with a) R/r=2, b) R/r=3 and c) R/r=5, opening bending 23

24 M M L r/t=5 Shakedown Limit r/t=5 Limit Load r/t=1 Shakedown Limit r/t=1 Limit Load r/t=2 Shakedown Limit r/t=2 Limit Load L Figure 12 - Effect of changing r/t with fixed R/r=3, cloing bending 24

25 M M L R/r=5 Shakedown Limit R/r=5 Limit Load R/r=3 Shakedown Limit R/r=3 Limit Load R/r=2 Shakedown Limit R/r=2 Limit Load M M L a) L R/r=5 Shakedown Limit R/r=5 Limit Load R/r=3 Shakedown Limit R/r=3 Limit Load R/r=2 Shakedown Limit R/r=2 Limit Load b) L M M L R/r=5 Shakedown Limit R/r=5 Limit Load R/r=3 Shakedown Limit R/r=3 Limit Load R/r=2 Shakedown Limit R/r=2 Limit Load c) L Figure 13 - Effect of Changing R/r with a) r/t=5, b) r/t=1 and c) r/t=2, opening bending 25

26 M M L R/r=5 Shakedown Limit R/r=5 Limit Load R/r=3 Shakedown Limit R/r=3 Limit Load R/r=2 Shakedown Limit R/r=2 Limit Load L Figure 14 - Effect of changing R/r with fixed r/t=1, cloing bending 26

27 .6 M M L Cloing Bending Shakedown Limit Cloing Bending Limit Load Revered Bending Shakedown Limit Opening Bending Shakedown Limit Opening Bending Limit Load M M L a) L Cloing Bending Shakedown Limit Cloing Bending Limit Load Revered Bending Shakedown Limit Opening Bending Shakedown Limit Opening Bending Limit Load M M L b) L Cloing Bending hakedown Limit Cloing Bending Limit Load Revered Bending Shakedown Limit Opening Bending Shakedown Limit Opening Bending Limit Load c) L Figure 15 - Comparion of Cloing and Opening Bending for fixed r/t=1 and a) R/r=2, b) R/r=3 and c) R/r=5 27

28 3.5 3 T T o Temperature Independent Yield Temperature Dependent Yield L Figure 16 - Effect of Temperature Dependent Yield Stre, R/r=3 and r/t= T T o Straight ipe, r/t=1 R/r=5 2 R/r=3 1.5 R/r= L Figure 17 - Effect of changing R/r with R/r=2, 3 and 5 and Straight ipe with fixed r/t=1 28

29 T T o Straight ipe, r/t=1 r/t=5 r/t=1 r/t= L Figure 18 - Effect of Changing r/t with r/t=5, 1 and 2 with fixed R/r=3 and traight pipe with r/t=1 29