A Diect Methd f the Evaluatin f Lwe and Uppe Bund Ratchet Limits J. Ue a, H. Chen a, T. Li a, W. Chen a, D. Tipping b, D. Macenzie a a Dept f Mechanical Engineeing, Univesity f Stathclyde, Glasgw, Sctland, G1 1XJ b Cental Engineeing Suppt, Existing Nuclea, ED Enegy, Banwd, Gluste, GL4 3RS Abstact The calculatin f the atchet limit is ften vital f the assessment f the design and integity f cmpnents which ae subject t cyclic lading. This w descibes the additin f a lwe bund calculatin t the existing Linea Matching Methd uppe bund atchet analysis methd. This lwe bund calculatin is based n Melan's theem, and maes use f the esidual and elastic stess fields calculated by the uppe bund technique t calculate the lwe bund atchet limit multiplie. By ding this, the methd cmbines the stable cnvegence f the uppe bund methd but etains the cnsevatism ffeed by the lwe bund. These advantages ae als cmplemented by the ability f the Linea Matching Methd t cnside eal 3D gemeties subject t cmplex lad histies including the effect f tempeatue dependent yield stess. The cnvegence ppeties f this lwe bund atchet limit ae investigated thugh a benchma pblem f a plate with a cental hle subject t cyclic themal and mechanical lads. T demnstate the effectiveness f the methd, the atchet limit f a thic walled pipe-pipe intesectin, als subject cyclic themal and mechanical lads, is cnsideed. Validatin f these esults is pvided by full elastic-plastic EA in ABAQUS. Keywds: Ratchet limit, shaedwn limit, lwe and uppe bund, cyclic lading 1. Intductin Duing peatin unde cyclic lading cnditins, a stuctue will shw ne f thee behavius: shaedwn, evesed plasticity ( plastic shaedwn) and atcheting. Ratcheting, the accumulatin f stain with each lad cycle, is banned by all pessue vessel design cdes and theefe must be designed against [1]. Knwledge f the atchet bunday is als desiable whee the stuctue includes acute stess aises, such as cacs, which vilate the shaedwn cnditin. With these facts in mind, it is highly desiable t be able t calculate the atchet limit f a stuctue. The calculatin f the atchet limit has been studied by many eseaches. The cmplex natue f the atcheting mechanism means that clsed fm analytical slutins ae esticted t the simplest situatins.
Incemental inite Element Analysis (EA) can nly pedict if shaedwn, evesed plasticity atcheting ccu, and s a tial and e pcess is equied t detemine the shaedwn and atchet bundaies. As a esult f this, many diect methds have been develped which can calculate the shaedwn limit f a stuctue [2-5]. The Linea Matching Methd (LMM) has been extended beynd mst the diect methds t include calculatin f the atchet limit [6,7]. The cuent methd has been implemented successfully in ABAQUS thugh use subutines, and is capable f calculating the shaedwn and atchet limits f stuctues with cmplex lad histies and tempeatue dependent yield stess. Hweve, the existing atchet analysis methd nly pvides uppe bund limits. This w descibes the additin f a lwe bund analysis which is based n the stess slutins geneated by the uppe bund calculatin. Implementing the lwe bund in this way means that the stable cnvegence ffeed by the uppe bund is etained, but is still cmplemented by the cnsevatism f a lwe bund slutin. 2. The Linea Matching Methd Cnside a bdy f vlume V and suface aea S. A cyclic tempeatue histy θ(x i acts within the vlume and vaying mechanical lads (x i act n pat f the suface S T. The emainde f the suface is cnstained t have ze displacement ate. These lads act ve a time cycle f t t, and can be decmpsed int thei cnstant and cyclic cmpnents: (xi,t) λ(xi ) θ(x i,t) (x i,t) (1) Whee λ is a lad paamete, (x i ) is a cnstant lad distibutin and θ(x i,t) and (x i,t) ae the cyclic histy f themal and mechanical ve time t. The linea elastic stess histy assciated with these lads is: ˆ ( x ˆ ˆ ( x whee ˆ ˆ ˆ (2) Whee ˆ epesents the vaying stesses due t (x i,t) and θ(x i,t). The lad paamete λ allws a ange f lading histies t be cnsideed. this cyclic pblem definitin, the stesses and stain ates will asymptte t a steady cyclic state whee ( t) ( t t), ( t) ( t t) (3) This stess state can be decmpsed int thee cmpnents as shwn belw in equatin 4: the elastic slutin ˆ, the tansient slutin accumulated up t the beginning f the cycle,, and a esidual slutin which epesents the changes duing the cycle. ( x ˆ ( x ( x ) ( x (4) a stable cyclic slutin thee is n accumulatin f stess stain fm ne cycle t the next, and theefe: ( x,) ( x (5) Based n this stable cyclic fmulatin, the evaluatin f the atchet limit becmes pssible if the applied lading can be decmpsed int cnstant and vaying cmpnents. Because the stuctue is subjected t stable cyclic lad cnditins, the changing esidual stess is caused diectly by this cyclic lad. The cnstant esidual stess field caused by the cnstant lading can then be evaluated afte the changing esidual stess field has been cmputed. The calculatin f the atchet limit cnsists f tw minimizatin pcesses. The fist is an incemental minimizatin f the evaluatin f the cyclic histy f esidual stess and plastic stain ange. The secnd is a glbal minimizatin f the atchet limit due t an exta cnstant lad. In essence, nce the stable
espnse f the stuctue t the cyclic lading is detemined, the LMM calculates the maximum additinal cnstant lading which will nt cause the cmpnent t atchet. 2.1. Uppe Bund mulatin The uppe bund fmulatin f the LMM has been descibed in detail in the ws [6,7]. A bief utline is given hee as an intductin t the lwe bund calculatin. Once the esidual stess histy assciated with the cyclic cmpnent f the applied lading has been detemined, the pblem educes t a taditinal shaedwn analysis whee the linea elastic slutin is augmented by this vaying esidual stess field. This shaedwn analysis is based n Kite's uppe bund theem [8]. This theem states that if 1) any inematically admissible stain ate can be fund such that the stain field is cmpatible with the applied displacements and 2) that the plastic dissipatin within the bdy is less than equal t the applied w, then shaedwn des nt ccu. The uppe bund pcedue f the LMM uses linea elastic slutins f the applied lads, the lad histy being cnstucted within the use subutine. Using supepsitin, the ttal linea elastic stess at each lad instance in the cycle is calculated. At the pints whee this stess is geate than the yield stess, the yung s mdulus is educed. Subsequent iteatins then use the mdified shea mdulus fm the pevius iteatin, which allws the stesses t edistibute in the stuctue. In paallel with the educing shea mdulus, the applied cnstant lad is als scaled using the multiplie, λ UB, which gives the stess field f equatin 6. At the end f each iteatin, the apppiate enegy ttals ae calculated t detemine the uppe bund atchet limit multiplie λ UB which is t be used in the next iteatin. The cmbinatin f the mdified mdulus and the lad multiplie pduces an unambiguus uppe bund atchet limit f the applied lading. 2.2. Lwe Bund Calculatin UB ( x ˆ ˆ ( x ( x Melan's theem states that f a given lad set the stuctue will exhibit shaedwn if a cnstant esidual stess field can be fund such that f any cmbinatin f cyclic elastic and esidual stesses the yield cnditin is nt vilated [9]. The evaluatin f the atchet limit descibed hee is essentially a shaedwn assessment augmented by the changing esidual stess field. This means that Melan's shaedwn theem can be extended t the assessment f the atchet limit if the stesses used in its calculatin include the changing esidual stess, whee the stess is cmpaed t the tempeatue dependent yield stess t give the lwe bund atchet multiplie. In tems f the LMM, the stess slutins calculated in stage 2 f the uppe bund methd cmpaed with the tempeatue dependent yield cnditin: LB UB ˆ y ˆ ( x T ( x Numeically, this means that the yield stess f each integatin pint is divided by the effective stess at that pint t btain the lcal λ LB. The minimum λ LB calculated fm the entie mdel is used as the lwe bund atchet multiplie. It is wth nting that the yield stess used in this cmpaisn is the lcal yield stess at that integatin pint, and theefe tempeatue dependent yield stess can be used in the calculatin. This gives the lwe bund stess field: (6) (7)
θ a) b) c) igue 1 - Hled late a) Gemety b) Mesh and c) Cyclic Themal Lad t 3. Numeical Examples ˆ LB UB ˆ ˆ ( x ( x (8) 3.1. late with Cental Hle igue 1 shws a schematic f the plate gemety and the finite element mesh used. A quate mdel is used with apppiate symmety bunday cnditins. The ati between the diamete f the hle D and the length f the plate L is.2. The ati between the thicness T and the length L is.5. The element type selected f this mdel was ABAQUS type C3D2R, a 2-nde quadatic bic element with educed integatin. The tp and ight sufaces f the plate ae cnstained t emain in-plane when displaced. The plate is subject t a cnstant uniaxial tensin, σ p, alng ne side f the plate and a cyclic tempeatue diffeence between the be f the hle and the edge f the plate, θ. The cnveged uppe and lwe bund atchet limits ae shwn in the inteactin diagam in igue 2a. The themal stess is nmalised against the efeence themal stess at θ = θ = 1 C. claity the evese plasticity limit (als calculated by the LMM) is shwn in the figue and thus shws the lad dmains whee the plate will exhibit diffeent cyclic behaviu. The atchet limit cuve fllws the classic bee-lie fm, and the lwe and uppe bunds cnvege vey clsely. The maximum diffeence between the lwe and uppe bund in this example is less than 2%. igue 2b shws the cnvegence f the lwe and uppe bunds at the pint A. The uppe bund cnveges quicly t the final slutin, wing t the fact that the stess cncentatin at the hle is 3.5 3 2.5 2 1.5 1.5 A Uppe Bund Lwe Bund R Limit 1.6 1.4 1.2.8.6.4.2.2.4.6.8 1 5 1 15 2 a) b) igue 2 - Results f Hled late a) Inteactin Diagam and b) Cnvegence f int A 1 Lad Multiplie Uppe Bund Lwe Bund Iteatin Numbe
12CM9-1 G235GH aveaged ut ve the whle vlume. The lwe bund cnveges me slwly, because this high stess is the limiting fact f the lwe bund calculatin. 3.2. ipe Intesectin igue 3 - ipe Intesectin gemety and Mesh igue 3 shws a diagam f the pipe intesectin and the E mesh used. This gemety is taen fm the EERC Design-by-Analysis [1] and, due t symmetic lading, half symmety is used in the mdel. ABAQUS element DC3D2 was used f the themal analysis and ABAQUS element C3D2R was used f the stuctual analysis. The pipe intesectin is made fm tw mateials. The main pipe is G235GH and the small pipe is 12CM9-1. The small pipe is mdelled as being "set-n", and s the weld egin has the same mateial ppeties as the main pipe. Bth tempeatue dependent and independent analyses ae pesented with tempeatue dependent yield stesses being taen fm BSEN 128-29 [11]. In bth cases a tempeatue independent Yung's mdulus and themal expansin cefficient is used. The pipe intesectin is subject t a cnstant intenal pessue, (with the clsed end cnditin) and cyclic tempeatue diffeence between the inne and ute sufaces, θ. The tempeatue vaies linealy fm ambient tempeatue at the ute suface t θ at the inne suface. This themal lad is cycled in the same way as that f the themal lading applied t the hled plate. One end f the main pipe is cnstained axially and the fee ends f bth pipes ae cnstained t expand in-plane, which eplicates the expansin f lng pipes. The atchet inteactin diagams f tempeatue dependent and independent yield stess ae given in 6 5 4 3 D E Uppe Bund Lwe Bund R Limit UB Temp Dependent LB Temp Dependent Step By Step.12.1.8.6 lastic Stain (%) C E 2 1 B C.5 1 1.5 2 2.5 3 5 1 15 2 25 a) b) igue 4 - ipe Intesectin Results a) Inteactin Diagam b) lastic Stain fm ull Elastic-lastic E at ints B, C, D and E.4.2 B D Cycle Numbe
igue 4a. The cyclic themal lad is nmalised against the initial applied θ f 1 C and the cnstant intenal pessue is nmalised against the initial applied pessue f 1Ma. The tempeatue independent evese plasticity limit calculated by the LMM shaedwn methd is included f claity. Once again the lwe and uppe bund limits have cnveged vey clse t ne anthe. igue 4a shws that, with the mateial ppeties used, tempeatue dependency des nt have a significant effect n the atchet inteactin diagam. The tempeatue independent atchet cuve has been validated by fu full elastic plastic finite element analyses. The plts f plastic stain (EMAG) against numbe f cycles f these pints ae shwn in igue 4b. int B, just inside the bunday, shws shaedwn behaviu whilst the pint C, just utside the bunday, shws clea atcheting behaviu. A simila situatin is bseved with pints D and E which shw evesed plasticity and atcheting behaviu espectively. 4. Cnclusin The linea Matching Methd has been pven t give accuate bunds t shaedwn and atcheting f cmpnents subject t cyclic lading. With the additin f the lwe bund t the cuent atchet analysis methd, it is nw pssible t calculate an accuate yet cnsevative atchet bund. This is a pweful tl t enginees invlved in the assessment and design f stuctual cmpnents. Acnwledgements The auths wuld lie t than the Engineeing and hysical Sciences Reseach Cuncil (ESRC) f the United Kingdm, ED Enegy and The Univesity f Stathclyde f thei suppt duing this w. Refeences [1] Bitish Enegy Geneatin Ltd. R5, An Assessment cedue f the High Tempeatue Respnse f Stuctues, Issue 3, 23. [2] Macenzie D, Byle JT, Hamiltn R. The elastic cmpensatin methd f limit and shaedwn analysis: a eview. Junal f Stain Analysis 2;35(3):171-188 [3] Seshadi R. Inelastic evaluatin f mechanical and stuctual cmpnents using the genealized lcal stess stain methd f analysis. Nuclea Engineeing and Design 1995;153:287-33 [4] Muscat M, Macenzie D, Hamiltn R. Evaluating shaedwn unde pptinal lading by nn-linea static analysis. Cmputes and Stuctues 23;81:1727-1737 [5] Chen H. Lwe and uppe bund shaedwn analysis f stuctues with tempeatue dependent yield stess. Junal f essue Vessel Technlgy 21;132:1122 1-8 [6] Chen H, nte ARS. A diect methd n the evaluatin f atchet limit. Junal f essue Vessel Technlgy 21;132:4122 1-8 [7] Chen H, nte ARS. A methd f the evaluatin f a atchet limit and the amplitude f plastic stain f bdies subjected t cyclic lading. Eu J Mech A/Slids 21;2:555-571 [8] Kite WT. Geneal theems f elastic plastic slids. gess in slid mechanics, Sneddn JN and Hill R, eds., Nth Hlland, Amstedam 196;1:167-221 [9] Melan E. Theie statisch unbestimmte systeme aus ideal-plastichem baustff. Sitzungsbe. d. Aad. d. Wiss. 1936;Wien 2A(145):195 218 [1] Eupean essue Equipment Reseach Cuncil. Design by analysis manual 1999. [11] Bitish Standads Institute. BS EN 128-2:29, lat pducts made f steel f pessue pupses.