EVALUATION OF SPRINGBACK PREDICTION CAPABILITY USING UNIFORM PURE BENDING. A Thesis by. Kunal Indravadan Patel

Transcription

1 EVALUATION OF SPRINGBACK PREDICTION CAPABILITY USING UNIFORM PURE BENDING A Thess by Kunal Indavadan Patel Bachelo of Engneeng, Sada Patel Unvesty, Inda, 00 Submtted to the College of Engneeng and the faculty of Gaduate School of Wchta State Unvesty n patal fulfllment of the equements fo the degee of Maste of Scence May 006

2 EVALUATION OF SPRINGBACK PREDICTION CAPABILITY USING UNIFORM PURE BENDING I have examned the fnal copy of ths thess fo fom and content and ecommend that t be accepted n patal fulfllment of the equements fo the degee of Maste of Scence, wth a majo n Industal and Manufactung Engneeng. Vswanathan Madhavan, Commttee Cha We have ead ths thess and ecommend ts acceptance: Don Malzahn, Commttee Membe Hamd Lankaan, Commttee Membe

3 DEDICATION To HIM and all I love

4 ACKNOWLEDGEMENTS I would lke to expess my snceest thanks and appecaton to my advso D. Vs Madhavan. Ths would have been an mpossble task wthout hs untng gudance and patent help thoughout my thess wok. I am tuly thankful to hm fo the tme that he devoted to my wok. He s among the few to dectly o ndectly nfluence my thoughts and chaacte. I would also lke to expess heaty thanks to D. Hamd Lankaan and D. Don Malzahn fo the tme and effot n evewng ths wok and suggestng constuctve nputs. I would lke to thank Ms. Kste Bxby fo takng the tme to evew ths manuscpt. My fends Panath Aedla and Rajen Pandya helped me thoughout my stay at Wchta State Unvesty, and I wll be eve thankful to them. I would also lke to acknowledge the help of all my fends who have dectly o ndectly helped me thoughout my gaduate caee. v

5 ABSTRACT The am of ths study s to develop unfom pue bendng as an objectve test fo detemnng the accuacy of spngback pedcton by employng dffeent FEA technques. A complete theoetcal soluton fo the bendng moment and change n sheet thckness s avalable only fo unfom pue bendng of pefectly plastc sheets. Howeve, plastc hngng develops natually n smulatons of bendng pefectly plastc sheets. We have developed a method to pevent plastc hngng and acheve unfom pue bendng of sheets by applyng constant equatons to nodes along the cente fbe. The eo n the bendng moment fo fomng (E ), the eo due to ncomplete unloadng dung spngback (E ), and eo n the change of cuvatue coespondng to the change n bendng moment dung unloadng (E 3 ) ae consdeed ndependently to get nsghts nto the easons fo dscepances between fnte element analyss and theoetcal esults fo spngback. Unfom pue bendng s also used to study the bendng moment and spngback expeenced wth wok-hadenng mateals. Compasons have been made wth analytcal solutons contanng mno appoxmatons n tems of the behavo of the mateal nea the cente fbe, whch s subject to evese loadng. The fact that two dffeent theoetcal models fo the mateal undegong evese defomaton yeld esults that dffe by less than % leads to a hgh degee of confdence n the theoetcal models. We have used unfom pue bendng to study the nheent spngback pedcton capablty of dffeent types of element analyss, convegence paametes, and dscetzaton level n two dffeent fnte element analyss packages, namely MARC and ABAQUS. v

6 Fo smulatons n ABAQUS usng two dmensonal elements and a pefectly plastc mateal model, the bendng moment gven by FEA s less than that pedcted by the theoetcal model by about -3%, ndcatng lesse spngback than that pedcted by theoy. Howeve fo thee dmensonal elements, the bendng moment s hghe by about 0% fo a elatve cuvatue () of 0.. Fo a coase dscetzaton (about 4 elements aound a 90 bend), ths eo nceases to about 37%. Fo a wok-hadenng mateal model, two dmensonal elements pedct % less bendng moment than the theoy, ndcatng an unde-pedcton n spngback. Shell elements wth educed ntegaton gve an unde-pedcton of spngback and show a negatve eo value between % and 0% fo the smulatons wth dffeent ntegaton ponts, whle shell wth full ntegaton show a postve total eo of abut 3%, ndcatng a hghe spngback than pedcted by theoy. Changng the convegence toleance value by 00 fom the default value shows a % change n calculated esults. Fo MARC, the two dmensonal elements unde-pedcts spngback by 0%, whle the thee dmensonal elements have shown ove-pedcton up to 30% n moment calculatons. Based on the fndngs, unfom pue bendng s ecommended as a benchmak test fo dentfyng the ntnsc accuacy wth whch spngback can be pedcted by FEA smulatons usng dffeent smulaton paametes. Unfom pue bendng can be used to develop effectve gudelnes fo elable fnte element smulatons of spngback. v

7 TABLE OF CONTENTS Chapte Page. INTRODUCTION..... Need fo Objectve Test fo Spngback Pedcton Capablty..... Sheet Metal Bendng Pocesses n Industy Objectves of ths Study Oganzaton of ths Thess...4. LITERATURE REVIEW Stesses and Stans n Pue Bendng Theoetcal Models fo Pue Bendng Hll s Theoy Cafood s Theoy Dadas and Majless s Model Fnte Element Analyss of Sheet Bendng Pesent Study TECHNICAL APPROACH Smulaton of Unfom Pue Bendng Analyss n MARC and ABAQUS/Standad Mateal Popetes Mult Pont Constants fo Unfom Bendng of Pefectly Plastc Sheets Results Post Pocessng Eo Analyss RESULTS AND DISCUSSION Vefcaton of Unfom Pue Bendng Compason of FEA Results fo Pefectly Plastc Mateal wth the Pedctons of Hll s model Compason of FEA Results wth Theoetcal Models fo Wok-Hadenng Mateals CONCLUSIONS AND FUTURE WORK Conclusons Futue Wok REFERENCES...5 v

8 7. APPENDICES...55 A. Cafood s Pocedue to Calculate,, and...55 B. FEA Models...59 C. Rks Ramm Pocedue...6 D. MPC Suboutne...63 E. Tmoshenko-Goode Theoy...65 v

9 LIST OF FIGURES Fgue Page. Dffeent bendng pocesses commonly used n ndusty ( Analyss of pue bendng (Hll, 950) Dffeent zones n a sheet bend Movement of mateal n layes of the sheet bend (Wolte, 950) Schematc epesentaton of stess-stan hstoy fo dffeent fbes subjected to evese loadng Compason of theoetcal models fo the bendng of pefectly plastc and wok hadenng mateals wth expemental esults obtaned by Cafood (970). (condton: alumnum sheet thckness = 4 mm) FEA model fo D elements Bounday condtons fo 3D smulatons Stess-stan cuve of alumnum epesented usng the Voce equaton Geometc elatonshps between the locaton of the fee node and the othe nodes along the centelne of the sheet, used n devng the constant equatons on the centelne nodes of sheet Plot of the nodal postons gven by the constant equatons (calculaton n MAPLE), at an ntemedate stage n the bend showng that the equatons would ndeed constan the nodes to le on a ccle (a) Paametes to calculate moment; (b) Schematc descbng esults post-pocessng to obtan the bendng moment, membane foce and shea foce at each secton by summaton of elevant moments and stesses ove the coss-secton The cuvatue of the centelne nodes of the sheet at dffeent stages of bendng (condton coespondng to smulaton J n Table 4) (a) The adus of cuvatue of centelne nodes wth ncease n ; (b) adus of the centelne nodes at = x

10 5. (a) Maxmum pncpal stess (D elements); = (condton coespondng to smulaton J n Table 4); (b)maxmum Pncpal stess (3D shell elements); =0.36. (condton coespondng to smulaton R n Table 4) Ccumfeental and adal stesses (aveaged ove the fst 70 sectons) nduced along the thckness of the sheets (Smulaton J n Table 4) Compason of the vaaton of moment wth elatve cuvatue the theoetcal model of Hll (950) Compason of the vaaton of moment wth elatve cuvatue gven by FEA wth the theoetcal model of Hll (950) Compason of the vaaton of moment wth elatve cuvatue gven by ABAQUS usng fve dffeent element types, wth the theoetcal models of Cafood (970) and Dadas and Majless (98) Compason of the vaaton n sheet thckness wth elatve cuvatue gven by ABAQUS usng fve dffeent element type, wth the theoetcal models of Cafood (970) Compason of total eo (E) as a functon of elatve cuvatue gven by ABAQUS usng fou dffeent element types Compason of the vaaton of moment wth elatve cuvatue gven by MARC usng thee dffeent element types, wth the theoetcal models of Cafood (970) and Dadas and Majless (98) Effect of suboutne on the change n sheet thckness gven by MARC x

11 LIST OF TABLES Table Page. Coeffcents of Ludwk mateal model that epesent the behavo of alumnum Vaous combnatons of paametes unde whch smulatons have been caed out n ABAQUS usng a elastc pefectly plastc mateal model Results fo smulatons caed out n ABAQUS usng an elastc pefectly plastc mateal model Vaous combnatons of paametes unde whch smulatons have been caed out n ABAQUS usng a wok hadenng mateal model Results fo smulatons caed out n ABAQUS usng D contnuum elements and a wok-hadenng mateal model Results fo smulatons caed out n ABAQUS usng 3D shell elements and a wok-hadenng mateal Vaous combnatons of paametes unde whch smulatons have been caed out n MARC usng a wok-hadenng mateal Results fo wok-hadenng mateal model smulatons D elements (MARC) Results fo wok-hadenng mateal model smulaton 3D elements (MARC) x

12 CHAPTER INTRODUCTION.. Need fo An Objectve Test fo Spngback Pedcton Capablty Sheet metal fomng s wdely used n most ndustes fo fomng sheets nto appopate shapes by plastcally defomng the sheet mateal beyond ts yeld stength to acheve pemanent defomaton. The desgn and manufactung of the tools that accomplsh ths eques sgnfcant nvestment. Fnte Element Analyss (FEA) s cuently the most commonly used numecal method fo analyzng metal fomng (Paulsen et al., 998). FEA based vtual manufactung allows fo a sgnfcant cost educton n the development of the manufactung pocess by substtutng tal and eo pocedues wth moe effectve numecal smulaton technques (Focelesse et al., 996). Numeous studes have valdated the accuacy of FEA esults wth sheet metal fomng data. Focellese et al. (996), studed V-punch and U-de bendng by usng explct and mplct FEM codes. Zhou et al., (996) studed convegence and CPU tme equements fo pedctng spngback and wnklng. Howeve, accuate calculaton of spngback, especally fo cases nvolvng lage cuvatues, s outnely stll not feasble. Accodng to Xu et al. (004), analyss of the spngback phase s complcated snce t nvolves mateal and geometc non-lneaty. It s theefoe desable to have a standad test that can help to povde gudance n developng accuate fnte element smulatons fo pedctng spngback... Sheet Metal Bendng Pocesses n Industy Hgh stength alumnum alloys ae constantly beng developed and adopted by the acaft and automoble ndustes fo nceasng numbes of pats n ode to satsfy hghe stength to weght ato equements. Many of these pats ae made by pocesses that nvolve

13 Fgue. Dffeent bendng pocesses commonly used n ndusty ( the bendng of sheets. Sheet metal and ppe-fomng pocesses that nvolve a sgnfcant amount of bendng nclude v-bendng, ppe/tube bendng, and stetch bendng. Fgue llustates the knematcs of some of these pocesses. Developng the tools to cay out these pocesses usually eques estmatng the spngback of the pat and compensatng fo the tool geomety, whch ensues that the fomed sheet afte spngback matches the equed shape of the pat. It s vey mpotant to pedct spngback quanttatvely, as well as qualtatvely, n ode to educe tal and eo, ewok and scap. Although bendng s techncally a smple pocess, t causes a complex hstoy of nduced stesses and movement of mateal n the blank. The pocesses llustated n Fgue typcally mpose bendng, stetchng, and shea stesses on the blank. Fcton, between the punch and the blank and between the blank and the de, plays a vtal ole n the defomaton of the blank.

14 Afte bendng s completed, the tool s acton ceases. At ths tme, the bent sheet s not held by any constanng foce and s fee to eleve the ntenal elastc stans. Ths s seen as spngback n the sheet. Pedctng spngback s complcated due to the complex stess and stan hstoy the pat undegoes dung the bendng pocess. Fo most components, spngback s compensated n the toolng that s used fo bendng. These tools ae desgned to bend the blank moe than equed so that the blank attans the desed shape afte spngback. The accuacy wth whch spngback s compensated fo detemnes the scap ate of a pocess. Tadtonally, empcal fomulae and ules of thumb based on expemental tal-and-eo have been used to desgn the pocessng oute, pofle geomety, and toolng. Ths s an expensve appoach to the poblem, waantng the use of numecal methods such as FEA to gan nsghts nto the sheet fomng and spngback pocess (Paulsen et al., 995). An accuate analyss of the elastc spngback n bendng s extemely mpotant when detemnng the ove-bendng angle equed to compensate fo the spngback effect (Focellese et al., 996). Fnte element analyss s nceasngly beng used to compensate spngback n tools used fo stampngs, extusons and stetched pats. The fomng phase s nceasngly smulated usng the explct tme ntegaton method (Make, 998) because t smplfes complex non-lnea analyses by usng contacts. Spngback s almost always analyzed usng an mplct tme ntegaton scheme, whch eques an teatve solve to convege wth a soluton wthn a specfed convegence toleance..3. Objectves of ths Study The am of ths study s to develop an objectve test that detemnes the accuacy of spngback pedcton when usng dffeent FEA technques, element fomulatons, convegence 3

15 toleances, and dscetzaton levels that can also povde nsghts nto causes fo cetan eos. Snce only bendng stesses ae mpotant n spngback, unfom pue bendng of sheet metal s a good canddate pocess to seve as an objectve test fo spngback. Pue bendng efes to the applcaton of a pue moment at each secton, wth mnmum membane o shea stesses supemposed. Unfom bendng efes to sheet bendng pocesses whee the cuvatue of the sheet emans constant fo all the stages of fomng so that end-effects asng fom appled loads do not nfluence the esults. The advantage of unfom pue bendng s that FEA solutons can be compaed to avalable theoetcal solutons fo the bendng moment and thckness as a functon of bend adus, especally fo a pefectly plastc mateal whee an exact soluton s avalable. Even fo wok-hadenng mateals, analytcal solutons ae avalable, wth only mno appoxmatons n tems of the behavo of mateal nea the cente fbe whch s subject to evese loadng. Choosng unfom pue bendng fo testng spngback pedcton capablty, the objectves of ths study ae as follows: () developng a technque fo achevng unfom pue bendng n FEA smulatons, () analyzng the bendng and spngback stages ndependently and developng eo estmates that povde nsghts nto the accuacy of the smulaton of each of the stages, and () compang the pefomance of dffeent types of FEA, wth dffeent combnatons of nput paametes, to pedct spngback..4 Oganzaton of ths Thess Chapte hghlghts elevant studes n pue bendng. Hll s theoy (950) fo pefectly plastc mateal s dscussed, as ae wok hadenng mateal models poposed by Cafood (970) and Dadas and Majless (98). Recent developments n the pedcton of spngback usng FEA ae also dscussed. 4

16 Chapte 3 dscusses the development of a methodology to smulate unfom pue bendng of sheets by enfocng appopate nodal constants though use suboutnes. By studyng the eo n bendng moment pedcton fo the fomng stage, the eo due to ncomplete unloadng dung the unloadng stage, and the eo n pedctng the change of cuvatue coespondng to the change n the bendng moment, nsghts egadng the souces of eo and the elatve magntude s obtaned. In chapte 4, esults obtaned fom two FEA packages (ABAQUS and MARC), ae compaed wth theoetcal esults by Hll (950), Cafood (970), and Dadas and Majless (98). Gudelnes to developng elable fnte element smulatons fo pedctng spngback ae povded n chapte 5, based on obseved eo values fo smulatons wth dffeent element fomulatons, dscetzaton level, loadng type, and convegence toleance values. 5

17 CHAPTER LITERATURE REVIEW Spngback occus when the tool foces ae eleased fom a fomed sheet. Too much spngback and poo dmensonal accuacy causes pats to be scapped. Accuate pedcton and effectve compensaton of spngback ae dffcult ssues n sheet metal fomng (Xu et al., 004). Most plate sectons defom unde the acton of a pue bendng moment (Delannay et al., 003). Johnson and Yu (98) studed spngback n beams, subjectng them to unaxal and baxal elastc-plastc pue bendng. Leu (995) used unfom pue bendng to study the effect of ansotopy and stan hadenng exponent on spngback... Stesses and Stans n Pue Bendng In a pue bendng pocess, the sheet s bent solely by the applcaton of a moment wthout any othe extenal loads on the sheet. Pue bendng s a theoetcal concept and can only be appoxmated n pactce. Fgue shows a sketch that ncludes the appled moments and the nduced tangental and adal stesses n the sheet. Vaous paametes of nteest n sheet bendng ae also ndcated. M M Fgue. Analyss of pue bendng (Hll, 950). 6

18 Fo hgh cuvatues, the compessve adal stesses ( ) nduced n the bend, due to the natue of stess equlbum equatons fo cuved beams, cannot be gnoed. The ccumfeental stesses ( ) ae tensle along the top laye and compessve along the bottom laye. Snce yeldng of the sheet mateal depends on both and, the compessve causes the aveage n the layes subjected to compesson to be hghe n magntude than the aveage n the layes subjected to tenson. In ode fo the esultant membane stess to be zeo, the neutal axs s close to the nne adus than to the oute adus ( (b-c) > (c-a) ). Whle t s suffcent to dvde the sheet nto two zones, one n tenson and the othe n compesson, fo the analyss of bendng of a pefectly plastc mateal, ths s not the case fo the bendng of a wok-hadenng mateal. Followng Wolte (950) as descbed by Cafood (970), fo wok-hadenng mateals the bend needs to be dvded nto thee zones though the thckness (shown schematcally n Fgue 3) based on the stans nduced n the sheet.. Zone conssts of layes of mateal wth a adus lage than that of the ognal cental laye. The mateal n ths egon s subjected to monotoncally nceasng ccumfeental tenson (elongaton) dung bendng.. Zone conssts of layes of mateal wth a adus smalle than that of the neutal laye. The mateal n ths egon s subjected to monotoncally nceasng compesson along the ccumfeence.. Zone 3 conssts of layes of mateal that le between the neutal laye and the ognal cental laye. The mateal n ths egon s subjected ntally to ccumfeental compesson and late evese-loaded n ccumfeental tenson due to the mgaton of the neutal laye towads the nne adus. The evesal of stesses causes the mateal to show educed yeld stess whch s a manfestaton of the Bauschnge effect. 7

19 Fgue 3. Dffeent zones n a sheet bent to hgh cuvatue. Fgue 4 shows the defomaton of the dffeent layes as the sheet bends. The ognally flat sheet (shown on the ght) s dvded nto ten layes of equal thckness, wth the boundaes maked fom 0 to 0. Accodng to elementay bendng theoy, nteface 5 s the cente fbe of the sheet and the neutal laye fo bendng. Layes above ths nteface ae n tenson, whle layes below ths ae n compesson. The bent confguaton of the sheet shown n the left sde of Fgue 4 shows the thnnng of the top layes and the thckenng of the bottom laye by the tme the sheet s bent to a elatve cuvatue (=t/ m ) of appoxmately.0. The esultng dffeence between the adus of the cente fbe of the sheet ( c ) and the mean adus of cuvatue o ( m ( ) ) can be easly peceved. As dscussed above, due to the hghe level of ccumfeental stesses n the bottom laye than the tensle stess n the top laye, the neutal laye les below the mean laye as shown n Fgue 4. Consdeng the fact that the layes above n ae now n tenson, and the layes below c wee ognally n compesson, t can be deduced that Zone 3 extends fom c to n. 8

20 o c Zone Zone Zone 3 m n Fgue 4. Movement of mateal n layes of the bent sheet (Wolte, 950)... Theoetcal Models fo Pue Bendng Ludwk (904) poposed a theoy of pue bendng. He assumed the neutal laye to concde wth the cental laye of the sheet at dffeent stages of bendng. Ths theoy gnoes the adal stess dstbuton though the sheet thckness. The study povdes a bass fo futhe development of bendng theoy usng pefectly plastc and wok-hadenng mateal models.... Hll s Theoy In 950, Hll poposed a complete soluton fo pue bendng of sheets fo a pefectly plastc mateal model. The theoy consdes the adal stesses and the movement of layes n the sheet. It poves that the sheet thckness emans unchanged and the bendng moment s constant thoughout the bendng opeaton. Hll s soluton fo a pefectly plastc mateal poposes a 9

21 constant moment of 4 3 Yt 0 pe unt wdth of the bend fo all cuvatues of the bend. It s pedcted (wthout poof), that wok-hadenng causes thnnng of the sheet.... Cafood s Theoy Cafood (970) fst developed an analyss of the bendng of wok-hadenng mateals, coectly satsfyng the equlbum equatons, the bounday condtons and takng nto account the change n thckness of the sheet. He used the stess-stan elatons suggested by Voce (948) (Equaton 3 on page 9) to descbe the stan hadenng behavo of the mateal. Cafood s theoy (970) assumes that the yeld stength of the mateal n the evese bent egon (Zone 3) s a constant value equal to the yeld stength of the vgn mateal. Cafood developed equatons fo the stans and stesses n the thee zones n tems of the adal locaton, also takng nto consdeaton the yeld cteon (ncludng adal stesses) and the bounday condtons. The elatve sheet thckness ( = t/t 0 ), the elatve adus of the neutal axs (= n / o ) and the bendng moment ( d M. ) ae obtaned as functons of the elatve cuvatue (=t/ m ). The equaton deved by Cafood that elates the nomalzed bendng moment (M/t ) to the elatve cuvatue () s as follows: 4 ln ln ln t M () 0

22 Whee,, and ae the paametes n the Voce mateal model. An explct numecal calculaton s employed to tack and as a functon of to calculate M. Ths s explaned befly n Appendx A. Cafood s wok povdes esults fo the dependence of M,, and on, fo dffeent mateal popetes epesentng fou dffeent mateals. Cafood (970) pefomed expements that appoxmated unfom pue bendng and obtaned a close coelaton between theoetcal pedctons and expemental esults...3. Dadas and Majless s Model Dadas and Majless (98) expanded Cafood s wok by ntoducng two dffeent assumptons fo the behavo of the fbes n Zone 3 subjected to evesed loadng. The fst model consdes a lnea stess-stan behavo fo fbes n Zone 3 that ae subject to evesed bendng. The second model consdes a knematc hadenng model usng Ludwk s equaton n ode to smplfy the stess equlbum equatons. Fgue 5 (Dadas and Majless, 98) shows the stess-stan hstoes of dffeent ponts, A, B, D, E and F between c and n and clealy shows the dffeent assumptons fo the mateal stength n zone 3. The subscpts 0, and ndcate thee bends of nceasng cuvatue at whch the stan and stesses ae compaed. Fo nstance, the stess at pont D at tme accodng to model would be D, wheeas the stess accodng to model would be D. Dadas and Majless (98) deved the followng equaton fo the bendng moment pe unt wdth: k0 M ( n ku 0 0 m0 n k n ) ( o 0 m m ( n ) m! m 0 n m n n m0 ) m ( ) m ( n ) m! m n m ()

23 They adopted a pocedue smla to that adopted by Cafood fo fomulatng the dffeental equatons to deve the evoluton of, and M wth. They show plots of the adal stess and tangental stess fo bendng wth dffeent mateal popetes. The adal stess s found to be compessve and contnuous thoughout the thckness. The tangental stess esembles the stess-stan cuve of the mateal. Dadas and Majless also found that the bendng moment ntally nceases wth the defomaton and then deceases as the bend pogesses. Cafood s model Model Model. D ` Fgue 5. Schematc epesentaton of stess-stan hstoy fo dffeent fbes subjected to evese loadng.

24 Nomalzed Moment (m/t 0 ) Relatve Cuvatue () Cafood's Expement Cafood's Theoy Dadas- Majless Theoy Hll's Theoy Fgue 6. Compason of theoetcal models fo the bendng of pefectly plastc and wok hadenng mateals wth expemental esults obtaned by Cafood (970) (condton: alumnum, sheet thckness=4mm). Fgue 6 compaes the bendng moment pedcted by Hll s, Cafood s and Dadas and Majless s model fo one condton. It also shows the expemental esults obtaned by Cafood fo the same condton..3. Fnte Element Analyss of Sheet Bendng Sheet metal bendng can be smulated by fnte element analyss usng the mplct and explct ntegaton schemes. Implct ntegaton eques a foce, o dsplacement based convegence cteon, to be satsfed. In the fomng stage, ths leads to hgh analyss tme. Theefoe fomng smulatons fo fnte element analyses ae typcally done usng the explct tme ntegaton method. Explct tme ntegaton s nethe accuate no effcent fo the evaluaton of elastc spngback because, wth the contact sufaces emoved, the defomed sheet stats to oscllate aound the new equlbum poston untl the knetc enegy s totally dsspated (Mca et al., 997). Spngback s usually analyzed usng the mplct tme ntegaton scheme, wheeby accuate esults can be obtaned. 3

25 Bendng defomaton n numecal modelng has been appoached n thee ways: the membane appoach, the shell appoach, and the 3-D (sold) element appoach (Zhou et al., 996). The selecton of the numbe of ntegaton ponts though the thckness fo shell elements affects the accuacy of esults. Xu et al. (004) suggests a mnmum of fve ntegaton ponts and ecommends seven fo elable spngback analyss. The ncease n ntegaton ponts comes wth an nceased CPU tme equement. Mattasson et al. (995) poposed that the magntude of spngback s nvesely popotonal to the element sze used n the FEA model. Stess elaxaton s accuate fo smalle element szes such as 0.5 mllmetes. Accodng to Cao et al. (999), the most sgnfcant factos nclude mateal consttutve law (ntal yeld cteon, stan hadenng, and hadenng law), element type, contact model, fcton law, and mesh densty. Typcally, the study of the accuacy of spngback uses test pats fomed unde ndustal condtons. The fcton and contacts, as well as the pat geomety, geneally esult n membane and shea stesses supemposed on the bendng stesses, whch makes the stess dstbuton complex. Snce the theoetcal models fo these pocesses ae mathematcally nvolved, the tadtonal appoach has been to compae FEA esults wth expemental valdaton. Compason of the total spngback can mask dffeent eos that counteact one anothe. Ths makes t dffcult to sepaate tue accuacy fom fotutous concdence. Snce an exact soluton exsts only fo pue bendng of pefectly plastc mateals, by compang FEA esults wth theoetcal esults fo the pue bendng pocess, the pefomance of dffeent element fomulatons, analyss paametes, and so on, can be objectvely studed. 4

26 .4. Pesent Study Pue bendng s selected as a canddate pocess that can be used as an objectve test fo compang the pefomance of dffeent FEA paametes n sheet fomng and spngback. Such tests can be used as good benchmak to dentfy the appopateness of the esults obtaned by a patcula combnaton of paametes n the analyss. Spngback can be measued wthout eo f an accuate bendng moment s obtaned fom these analyses. Howeve, when a sheet made of a pefectly plastc mateal s bent, t s pone to developng a plastc hnge n the sheet. It eques the use of suboutnes to enfoce mult-pont constants on the sheet nodes n ode to acheve unfom pue bendng. Ths has been accomplshed and sheets have been bent to a elatve cuvatue of appoxmately 0.5. FEA esults ae compaed wth theoetcal models and avalable expemental data. Compang the bendng moment gven by FEA wth that pedcted by the appopate theoetcal soluton eveals the accuacy wth whch the type of FEA and the nput paametes used can epesent states of pue bendng. The effect of the numbe of ntegaton ponts, element fomulatons, convegence testng, and dscetzaton s studed n ths wok. 5

27 CHAPTER 3 TECHNICAL APPROACH Fnte element analyss of unfom pue bendng s used to compae dffeent types of analyses and FEA packages fo the spngback pedcton capablty. Unfom pue bendng s ealzed by applyng a couple at one end of the sheet wth symmetc bounday condtons on the othe. The cuent study coves values of elatve cuvatue ( = t/ m ) angng up to 0.8, whch s n the ange of ndustal bendng opeatons. Fo these values of, FEA esults of smulatons wth an elastc pefectly plastc mateal model ae compaed wth Hll s analytcal soluton fo pefectly plastc mateals and FEA esults fo smulatons wth wok-hadenng mateal models, wth the models of Cafood (970) and Dadas and Majless (98). 3.. Smulaton of Unfom Pue Bendng The basc equements fo ealzng unfom pue bendng ae as follows:. Bend the sheet usng a pue bendng moment.. Do not use any shea foces, nomal foces, o contacts wth toolng to bend the sheet, snce they lead to membane and/o shea stesses n the sheet. Vaous models wee developed to ealze dffeent levels of pue bendng. A detaled descpton of the development hstoy s compled n Appendx B. In bendng the oute layes of the sheet ae subjected to tensle stan and the nne layes to compessve stan. To smulate a pue bendng moment, two equal and opposte foces ae appled on two adjacent nodes at one end (the ght end, see Fgue 7) of the sheet. As the sheet bends, the decton of acton of these foces follows the otaton of the end face (ght sde) of the beam. Symmetc bounday condtons ae appled to the othe (left) end of the sheet. The left-most nodes of the sheet ae constaned fom movng n the X-decton. The cental node n 6

28 Symmetc bounday condtons Rgd Elements couple Fgue 7. FEA model fo elements. ths set s also constaned fom movng n the Y-decton. Fgue 7 shows a schematc of the model. The foces compsng the couple esult n sgnfcant local effects on the elements nea the nodes to whch they ae appled. Ths makes the convegence of non-lnea teatons used n mplct solves dffcult. Rgd elements ae used to pevent localzed defomaton and dstbute the foce ove a lage aea of the sheet. Rgd beam elements ae ntduced fom the nodes on whch the couple foces ae appled to othe neghbong nodes, as shown n Fgue 7, among the elements. The sheet thckness used n each model s 4 mm. In ths study of pue bendng, the length of the beam s not a sgnfcant facto because unfom pue bendng s ndependent of the length of sheet. Two dmensonal smulatons ae caed out usng a vaety of contnuum elements, ncludng 4 node and 8 node plane stan elements. Shell elements ae used fo 3D smulatons. The z-dsplacement and the X and Y otatons of all the nodes n the model ae constaned, as shown n Fgue 8 n ode to enfoce plane stan condtons. Note that one ow of elements along the X-decton would have been suffcent, though thee ae used. 7

29 Constaned -6 DOF Y Constaned 3-5 DOF Z X Fgue 8. Bounday condtons fo 3D smulatons. The Rks-Ramm loadng pocedue s used n the analyss. Although the moment equed to bend the sheet ntally nceases wth cuvatue, as the elatve cuvatue appoaches unty the moment equed to contnue bendng deceases. The magntude of appled couple should decease at ths tme to avod convegence eos. The Rks-Ramm pocedue offes the equed automatc loadng steppng algothm. It s typcally used fo snap-though poblems whee smla educton of load wth dsplacement s encounteed. A bef descpton of the Rks- Ramm pocedue, adapted fom the MARC (005) and ABAQUS theoy manuals, s povded n Appendx C. Equvalent models ae developed n MARC and ABAQUS fo compaatve analyss. 3.. Analyss n MARC and ABAQUS/Standad Both MARC and ABAQUS/Standad cay out the bendng smulaton by dect (mplct) tme ntegaton. The D plane stan analyses n MARC ae caed out usng fully ntegated plane stan Quad4 elements and Quad8 elements (element types and 7 n MARC, espectvely). Quad4 elements employ lnea ntepolaton functons, whle Quad8 elements employ quadatc ntepolaton functons to calculate the dsplacements based on the nodal values. MARC element 75, a fou node, thck shell element wth global dsplacements and 8

30 otatons as degees of feedom (MARC manual, Vol. B, 005) s used n the 3D model wth shell elements. Fo the D plane stan models n ABAQUS, CPE4 and CPE8 element fomulatons (equvalent to Quad4 and Quad8 elements n MARC) ae used. S4 elements ae used fo smulatons wth shell elements. They nclude assumed stan fomulaton to avod shea lockng. S4R, CPE8R and CPE4R elements ae used n ABAQUS to study the effect of educed ntegaton Mateal Popetes FEA esults ae valdated wth the exact theoetcal model gven by Hll (950) fo a pefectly plastc mateal popety. A mateal wth a constant yeld stess of 58. MPa s used n the pefectly plastc smulatons. Because the theoy assumes a gd, pefectly plastc mateal, an analyss was caed out wth the Young s modulus scaled up by a facto of 0 to detemne the effect of educed elastc stans. FEA esults obtaned usng the wok-hadenng mateals ae compaed wth the models gven by Cafood (970) and Dadas and Majless (98). The same effectve stess vs. effectve stan cuve s used n computng the bendng moment accodng to both these models. The Voce (948) equaton fo mateal popetes s A B Ce D (3) whee the constants fo alumnum ae A = 3.6, B = 3., C = 73.5, and D = 3.75, as gven by Cafood (970). Dadas and Majless (98) use Ludwk s gd wok hadenng mateal law whch can be stated as n y (4) 9

31 TABLE COEFFICIENTS OF LUDWIK MATERIAL MODEL THAT REPRESENT THE BEHAVIOR OF ALUMINUM AS SHOWN IN FIGURE 9. Mateal Young s Modulus (MPa) Intal Yeld Stess (MPa) Stength Coeffcent (MPa) Posson s Rato Stan Hadenng Exponent Alumnum Cafood's mateal model DM 0 0 Stess (MPa) Stan 0.3 Fgue 9. Stess-stan cuve of alumnum epesented usng the Voce equaton. Equvalent paametes fo both these models to epesent the same stan hadenng cuve ae obtaned by cuve fttng. In the sheet bendng opeatons consdeed, the maxmum stans eached ae about 0.3. Hence the cuves ae ft to ths value of stan, whch can be noted n Fgue 9. The coeffcents of Ludwk model coespondng to the coeffcents of the Voce model shown below equaton 3 ae lsted n Table Mult Pont Constants fo Unfom Bendng of Pefectly Plastc Sheets Pefectly plastc mateals ae pone to developng plastc hnges caused by plastc nstablty. Plastc hngng n bendng s compaable to the neckng n tenson tests. 0

32 In ths study, mult-pont constants ae employed to constan the nodes along the cental fbe of the sheet to le on a ccle. As noted eale, the fst node (on the left sde) along the cente fbe s subject to symmetc bounday condtons. The second node s fee to move as dctated by the loadng. All the nodes on the cental fbe of the sheet to the ght of the second node follow the defomaton of the second node. Ths s acheved by usng use suboutnes UFORMS n MARC and MPC n ABAQUS. The followng s a succnct explanaton of the govenng equatons fo the suboutne. Fgue 0 shows a dagammatc epesentaton fo the deved equatons. The equatons specfed n the use suboutne need to constan the movement of the nodes n the cental ow of the beam to appopately scale and follow the movement of the second node (Node ) of the same ow. Gven the dsplacement of Node, the adus, R, can be calculated usng the fomula L cos sn R (5) The length L of the segment between Node and Node, s gven by L (( x x) ( y y ) ) (6) and the angle s calculated as tan y x y x (7) x X u (8) y Y v (9)

33 whee x and y denote the cuent coodnates of the node ndcated by the subscpt, X and Y denote the ognal coodnates, and u and v denote the dsplacements n the X and Y dectons espectvely. Node s constaned n X and Y dectons, and n the FEA model used, s also located at the ogn of the coodnate system. Ths smplfes the fomulae of L and as shown below: L tan x y y x (0) () Unfom bendng eques that the nodes on the cental ow be equally spaced, and that at all subsequent nodes afte Node be placed at an angula ncement of about the cente of cuvatue, as shown n Fgue. The X and Y coodnates of any node (along the cental ow) can be obtaned usng x R sn( ) () y R(cos( ) ) (3) A plot of the coodnate locatons obtaned fom these equatons (at fxed ) fo all the nodes n the cental ow (ntally at Y = 0) s shown n Fgue. It valdates the equatons to be able to povde the appopate constants when used n the suboutne to ealze unfom bendng. Note that Node s fee to move, and ts movement foces the othe centelne nodes to move appopately, esultng n a unfom bend along the length of sheet. The eacton due to the appled MPCs s not expected to be lage because t just takes a gentle gudance to have the sheet contnue to bend unfomly. Futhemoe, the constant equatons do not constan the adus of cuvatue of bend and ae not expected to cause any sgnfcant eacton foces on the sheet.

34 These equatons have been ncopoated n the MPC suboutne (fo ABAQUS) and UFORMS suboutne (fo MARC). L 3 Fee 4 Ted L R 3 R Y X Fgue 0. Geometc elatonshps between the locaton of the fee node and the othe nodes along the centelne of the sheet, used n devng the constant equatons on the centelne nodes of the sheet. 3

35 X-Coodnate (mm) Y-Coodnate (mm) X-Y coodnates -5-0 Fgue. Plot of the nodal postons gven by the constant equatons (calculaton n MAPLE), at an ntemedate stage n the bend showng that the equatons would ndeed constan the nodes to le on a ccle Results Post Pocessng Post-pocessng of the esults fo each of the smulatons s pefomed usng Mcosoft Excel. The unfomty of bend s vefed by fttng a ccle though the coodnates of the fst 70 nodes along the cental ow. The cente of the ccle s the cente of cuvatue of the sheet, fom whch the adal and angula locatons of each of the nodes s defned and used n subsequent computaton. The stess components n the coodnate dectons ae tansfomed to the ccumfeental stesses and adal stesses accodng to cos sn cos sn xx xy yy (4) xx sn xy sn cos yy cos (5) The total bendng moment pe unt wdth nto the plane s obtaned by ntegatng (summng) the acton of the tangental stesses though the thckness of the sheet as follows: 4

36 B d d M.. ) (.. 0 (6) Fgue (a) epesents the paametes nvolved n ths calculaton. Fgue (b) epesents the pocedue schematcally. The esultng summaton shown above can be aved at ethe by pefomng the ntegaton o by consdeng ndvdually the contbutons of the mean stess ove a segment and the gadent of stess ove t, as shown n Fgue (a) d d d d (7) node node gadant M + ( mean M 5

37 F s F m M d F s F F s m d d F m Fgue (a) Paametes to calculate moment; (b) Schematc descbng esults postpocessng to obtan the bendng moment, membane foce and shea foce at each secton by summaton of elevant moment and stesses ove the coss-secton. The stesses ae also summed up ove each of the sectons to obtan the total membane and shea foces. F F m f o o. d n ( n d ). ( If the membane foce s non-neglgble, the tue bendng moment s obtaned by subtactng the contbuton of the membane foce fom the moment calculated n equaton 6 as follows: ) (7) (8) M FEA M B F m n (9) The values of M FEA, F m and F s ae aveaged ove coss-sectons 0 though 70 to obtan the aveage M FEA, F m and F s whch ae used fo futhe pocessng. Moments fo the shell 6

38 elements ae output by the softwae. An aveage of these moment values ove nodes of the sheet s calculated and used as the value of bendng moment. The same pocessng s epeated fo the esults obtaned at the end of the spngback smulaton. The cuvatue afte spngback s obtaned by fttng a ccle to the centelne nodes. The esdual moment, membane foce, and shea foce ae also obtaned as descbed above. Subsequently, afte post-pocessng, the eo n bendng moment calculaton s obtaned. Spngback depends dectly on the calculated bendng moment. A systematc method of eo analyss n spngback pedcton s dscussed n the followng secton Eo Analyss The change n angle due to spngback s dectly elated to the change n cuvatue by the elaton = * BA, whee BA s the bend allowance, the length of the unstetched fbe, whch, by defnton, emans constant. The total eo n spngback pedcton by FEA can be pattoned nto the followng sepaate eos:. Eo n the bendng moment equed fo fomng at the fomng stage.. Eo dung the unloadng stage, f the sheet does not unload completely. 3. Eo n change n cuvatue of the stuctue n esponse to the unloadng of the bendng moment, as compaed to the change n cuvatue expected theoetcally. These thee eo components ae defned as M FEA M theoy E 00% M (0) theoy M E 00% () M theoy FEA theoy E3 00% theoy () 7

39 Mspngback whee theoy = E'I M s the esdual bendng moment afte spngback, M theoy s the bendng moment expected based on theoy, M FEA s the bendng moment computed fom FEA esults as descbed n the pevous subsecton and M spngback s gven by: M spngback wk ' t n n c n (3) It can be seen fom the above defntons that the total eo, E=E +E +E 3 s the ato of the eo n spngback angle pedcton to the actual spngback angle. A negatve E ndcates less spngback than theoetcal esults, whle a postve value of E ndcates a hghe spngback as compaed to theoy. The theoy obtaned usng the above mentoned equaton s tue fo staght beams and s not accuate fo cuved beams, especally of hghe cuvatues. A moe accuate theoy poposed by Tmoshenko and Goode (970) s used to calculate the change n cuvatue of the bent sheet ( theoy ). Ths s explaned n detal n Appendx E. 8

40 CHAPTER 4 RESULTS AND DISCUSSION 4.. Vefcaton of Unfom Pue Bendng The sheet s found to bend unfomly unde the effect of the appled couple and constant equatons. Fgue 3 shows the poston of the centelne nodes of the sheet wth ncease n bend cuvatue (.e. the geomety of the cental laye of the bent sheet at dffeent tme ncements), as obtaned fom fnte element analyss. The change n thckness of the sheet wth bendng s also shown n the legend. 0 Cuent X Co-odnate (mm) X = 0.5, t = X = 0.46, t = Cuent Y Co-odnate (mm) X = 0.37, t = X = 0.484, t = X = 0.597, t = Fgue 3. The cuvatue of the centelne nodes of the sheet at dffeent stages of bendng. (condton coespondng to smulaton J n Table 4). 9

41 Radus Relatve Cuvatue () X=0.5, t=3.993 X=0.46, t=3.987 X=0.37, t=3.987 X=0.484, t=3.956 X=0.597, t= E E+00 Radus (mm) 6.76E E E E E Ognal X-coodnate (mm) Fgue 4(a). The adus of cuvatue of centelne nodes wth ncease n, (b) adus of the centelne nodes at = Fgue 4(a) shows the vaatons n calculated ad along the centelne nodes of the sheet. Fgue 4(b) shows the vaatons n a magnfed manne fo the tghtest bend n fgue (a) to show that the devaton fom a tue ccle s neglgble. The standad devaton fo ths set of calculated ad at dffeent sectons s mm. Fgues 5(a) and 5(b) show typcal dstbutons of maxmum pncpal stess along the sheet fo smulatons wth D (4-noded) and 3D (4-noded) elements and hghlght the fact that the stesses ae also unfom aound the bend. 30

42 Fgue 5(a). Maxmum pncpal stess (D elements); = (Condton coespondng to smulaton J n Table 4); (b)maxmum pncpal stess (3D shell elements); =0.36. (condton coespondng to smulaton R n Table 4). 3

43 Fgue 6 shows the dstbuton of ccumfeental and adal stess though the thckness of the bent sheet. The ccumfeental stess s found to follow the patten of stess-stan model of the mateal as poposed by Dadas and Majless (98). The adal stess s compessve thoughout the thckness of the sheet, but show some devaton fom the dstbuton expected based on theoy. 4.. Compason of FEA Results fo Pefectly Plastc Mateal wth the Pedctons of Hll s Model Usng the appoach mentoned n the pevous chapte, unfom pue bendng has been caed out unde the condtons mentoned n Table (page 37) fo elastc pefectly plastc mateal. The changes made to the model between one analyss and the next (efeed to by the next ndex and lsted n the ow below) ae hghlghted n these tables. Fo nstance, n Table, t can be seen that the effect of thee dffeent convegence toleances ae nvestgated n analyses A and B, fo CPE4 and CPE4R elements subject to couple foces whle F and G coesponds to analyses wth S4 and S4R elements subjected to moment loads Ognal Y coodnates of cente nodes (mm) Stess (MPa) Ccumfeental Stess Radal Stess Fgue 6. Ccumfeental and adal stesses (aveaged ove the fst 70 sectons) nduced along the thckness of the sheet (smulaton J n Table 4). 3

44 Fgue 7 shows a compason of the bendng moment gven by ABAQUS wth that gven by Hll s model fo pefectly plastc mateals. Wth all plane stan element smulatons, the bendng moment s coect to wthn 3% of the theoetcal value up to a elatve cuvatue of 0.3. The smulaton wth E nceased ten tmes (smulaton C n Table ), to study the effect of gd pefectly plastc mateal, dd not show much dffeence n the esults at lowe values ( < 0.4)), although the bendng moment nceased steeply at hghe values of (> 0.4). The change n thckness of the sheet fo a pefectly plastc mateal s shown n Fgue 8. The thckness change s about 0.5%. Smulatons F and G use fully ntegated and educed ntegated shell elements espectvely and show an eo close to -0%. The eo nceases to about 37% fo these elements when a coase dscetzaton wth only 4 and 9 elements along a 90 degee bend s used (smulaton H and I). Hll's mateal model Moment (N.mm) CPE4 elements CPE4 elements; gd plastc (E=730 GPa) CPE8 elements CPE8R elements Relatve Cuvatue () CPE4R elements Fgue 7. Compason of the vaaton of moment wth elatve cuvatue gven by FEA wth the theoetcal model of Hll (950). 33

45 Change n thckness (mm) Relatve Cuvatue () CPE4 elements CPE4 elements; gd plastc (E=730 GPa) CPE8 elements CPE8R elements CPE4R elements Fgue 8. Compason of the vaaton n sheet thckness wth elatve cuvatue gven by ABAQUS usng fve dffeent element types, wth the constant sheet thckness expected based on the theoetcal models of Hll (950). Table 3 (page 38) shows esults of these analyses wth ABAQUS. The CPE4 and CPE4R elements do not show much dffeence n bendng moment at close to 0.3. The CPE4 elements show a negatve eo of about 3% n bendng moment, ndcatng a lowe bendng moment than that detemned by theoy whch wll lead to educed spngback. The 8-noded plane stan elements show an eo close to 0% at vey low cuvatues. The 3D shell element shows a negatve eo of about 0%. As the dscetzaton of the sheet s made coase, the eo n bendng moment ses to about 37%. The lowe bendng moment pedcted by FEA unde most condtons also ndcates unde pedcton of spngback Compason of FEA Results wth Theoetcal Models fo Wok-Hadenng Mateals Fgue 9 shows the elatonshp between nomalzed moments and elatve cuvatue fo wok-hadenng mateal obtaned fom ABAQUS. CPE8 and S4R elements show values close to Cafood s model at lowe, whle the devaton nceases wth nceasng. S4R elements ae able to attan hgh cuvatues wth neang.0; howeve, at hghe cuvatues, S4R elements 34

46 Fgue Compason of the vaaton of moment wth elatve cuvatue gven by ABAQUS CPE8 elements; conv. usng fve dffeent element types, wth the theoetcal models of Cafood (970) tol=5.0e-3 and Dadas and Majless (98). ) Nomalzed Moment (M/t Relatve Cuvatue () Cafood Model CPE4 elements; conv tol=5.0e-3 S4R elements; 5 ntegaton ponts; conv tol=5.0e-3 DM Model CPE8R elements; conv tol=5.0e-3 CPE4R elements; conv tol=5.0e-3 S4 elements; 7 ntegaton ponts Fgue 9. Compason of the vaaton of moment wth elatve cuvatue gven by ABAQUS usng fve dffeent element types, wth the theoetcal models of Cafood (970) and Dadas and Majless (98). show nceased bendng moment values compaed to the theoetcal models. CPE4 elements show a good accuacy n calculatng bendng moment. The total eo values obtaned ae about - 5% ndcatng a smalle spngback pedcton fo smulatons. In all, close coelaton s obtaned usng 4 noded plane stan elements. The educed ntegaton and full ntegaton elements dsplay smla esults othe then a mno devaton of about % at cuvatues close to 0.5. Fgue 0 shows the vaaton of sheet thckness wth nceasng cuvatue. It shows a good coelaton wth Cafood s theoetcal model. The decease n thckness s steady as the cuvatue nceases. As the moment calculaton s dectly popotonal to the squae of thckness, the decease n thckness educes the total calculated moment. The S4R elements show constant thckness of 4mm. 4 noded and 8 noded elements show smla tends, although the CPE4 and CPE4R elements ae close to the theoetcal esults. 35

47 Change n sheet thckness (mm) Relatve Cuvatue () CPE4 elements; conv tol=5.0e-3 CPE4R elements; conv tol=5.0e-3 CPE8R elements; conv tol=5.0e-3 CPE8 elements; conv tol=5.0e-3 Cafood's theoy S4R elements, conv tol=5.0e-3 Fgue 0. Compason of the vaaton n sheet thckness wth elatve cuvatue gven by ABAQUS usng fve dffeent element type, wth the theoetcal models of Cafood (970) Total Eo E (%) Relatve Cuvatue () CPE4 elements CPE4R elements S4 elements S4R elements Fgue. Compason of Total eo (E) wth elatve cuvatue pedcted by ABAQUS usng fou dffeent element types. Fgue shows the total eo as the sheet bends to hghe cuvatue. D plane stan elements show an eo tend gong fom postve eo towads negatve eo ndcatng a hghe pobablty of unde-estmatng spngback at hghe cuvatues, whle the evese s obtaned fo 3D shell elements whee the eo nceases wth the ncease n cuvatue, lkely attbutable to the fact that the thckness of the shell elements eman constant. 36

48 Fgue s a plot of the nomalzed bendng moment wth fo the wok-hadenng mateal usng MARC softwae. Smla to ABAQUS, the D 4-noded elements gve close esults at lowe, whle the devaton fom theoy nceases wth ncease n cuvatue. 3D fully ntegated shells show a close appoxmaton to theoy at lowe cuvatues but the eo nceases at hghe cuvatues. The smulaton wth use suboutne shows a peak n calculated bendng moment at close to. and deceases shaply beyond ths. The tend s n lne wth theoetcal esults whee the moment deceases afte nceases beyond. The plane stan elements show an eo of about 5% as compaed to the theoetcal models. Hence, t s seen that element type n MARC and CPE4 elements n ABAQUS show a close coelaton wth the theoetcal model. Element type 75 vey closely follows the theoetcal model untl a elatve cuvatue of 0.6. Beyond that, smla to S4R elements n ABAQUS, the bendng moment pedcted by them s geate than the theoetcal models. Cafood model 40 DM Model Nomalzed Moment (M/t0 ) El type ; conv tol=.0e-5 El type 75; 7 ntegaton pts; conv tol=.0e-5 El type ; UFORMS=yes; conv tol=.0e Relatve Cuvatue () Fgue. Compason of the vaaton of moment wth elatve cuvatue gven by MARC usng thee dffeent element types, wth the theoetcal models of Cafood (970) and Dadas and Majless (98). 37

49 Relatve Cuvatue () Change n thckness (mm) Element type ; conv tol=.0e-5 Element type ; UFORMS Cafood's theoy Fgue 3. Effect of the use suboutne on the change n sheet thckness gven by MARC. Fgue 3 shows the effect of usng suboutne on the change n sheet thckness. It can be noted that the thckness vaaton between the two smulatons s not sgnfcant. Howeve, consstently deceasng tend devates about 5% fom the theoetcal esults at a of 0.6. The deceasng sheet thckness tend contnues at hghe cuvatue fo smulaton wth suboutnes. Table 4 (page 43) lsts the smulatons caed out n ABAQUS, wth dffeent element fomulatons, numbe of ntegaton ponts, dscetzaton levels, loadng types, and convegence toleance values fo wok-hadenng mateals. Table 7 (page 46) s a smla lst fo smulatons caed out n MARC. Dffeent analyss nputs ae chosen such that the compason of the esults fo bendng moment, completeness of unloadng, and coelaton between changes n cuvatue and unloadng wll povde a clea ndcaton of the effect of changes n these paametes on the eo n spngback pedcton. The esults fo analyses wth ABAQUS D 4-noded and 8-noded elements ae shown n Table 5 (page 44) whle Table 6 (page 45) shows the esults fo 3D shell elements. Smlaly, Table 8 (page 47) shows MARC esults fo D elements whle Table 9 (page 48) shows esults fo 3D fully ntegated shell elements. 38

50 The esults fom Table 5 ndcate an unde-pedcton of spngback angle usng the D elements. E s about % fo CPE4 elements (smulatons J, K and L), ndcatng a good coelaton wth theoy. E s about -6% fo CPE8 elements (smulatons M, N and O) ndcatng smalle spngback calculaton. The smulaton wth CPE4 and hghe dscetzaton (Smulaton Q) show an E of less than %. E fo the 4-noded D elements (smulatons J, K and L) s less than -%, whle t s about -9% fo equvalent cases wth CPE8 elements (smulatons M, N and O). E 3 fo mostly all the smulatons wth plane stan elements esult n less than -0% eos (smulatons J though Q except Smulaton K). The secton stess befoe spngback fo 4-noded elements s about 3% of the yeld stess of the mateal (smulatons J, K and L), whle t s moe than 0% fo CPE8 elements (smulatons M, N and O). Total eo s negatve fo all the examned cases. The total eo n actual calculaton of spngback s about -% fo CPE4 elements (smulatons J, K and L), whle t s about -0% fo CPE8 elements (smulatons M, N and O), ndcatng an unde-pedcton n bendng moment calculaton. An ncease n convegence toleance by 00 dd not vay the esult by moe than % (smulatons J, K, L and M, N, O). Table 6 shows a compason of esults fo 3D shell elements wth full and educed ntegaton fo dffeent numbes of ntegaton ponts. E s about 4% fo most of the cases wth S4 and S4R elements (smulatons R though AG except smulaton Z). E fo all the smulatons wth 3D shell elements s less than 0.%, ndcatng a complete unloadng of the sheet. E 3 fo S4R elements anges between % and 0% (smulaton R though AA) n the negatve decton fo most of the cases, whle t s less than -% fo most cases usng S4 elements (smulatons AB though AG). Oveall, the shell elements wth educed ntegaton gve an unde-pedcton of spngback and show a negatve eo value fo the smulatons wth dffeent ntegaton ponts 39

51 (smulaton R, S, V, W), whle shells wth full ntegaton show a postve total eo of about 3% (smulatons AA though AG except smulaton AC), ndcatng a hghe spngback than pedcted by theoy. Smla to plane stan elements dscussed pevously, S4R elements do not show a dffeence n esults by changng the convegence toleance by a facto of 00. The moment loadng n smulaton T show a negatve total eo -% ndcatng unde-pedcton of moments. In Table 8, the D plane stan 4-noded and 8-noded elements n MARC, E s about -3%, whle E s about -%. E 3 shows a negatve eo of about 7% (smulaton AH, AI and AJ). The membane stesses afte unloadng fo element ae about 5% of the ntal yeld stess, whle they ae about 4% fo element 75. The change n oveall eo between smulatons AH, AI, and AJ s not moe than %, ndcatng a smalle nfluence of change n convegence toleance fo these analyses. Table 9 shows an eo estmate esults fo usng 3D shell elements n MARC and vayng the numbe of ntegaton ponts though the thckness of the element. E s less than 7% fo most of the cases except smulaton AP (wth ntegaton ponts and tghte convegence toleance), whee the eo s close to 6%. Wth a smalle E, a faly good unloadng, about % s seen n all cases. Smulaton AG (wth 5 ntegaton ponts) and AP show an nceased eo n E 3. Fo othe cases, the eo s found to be less than 4%. The least oveall eo s pedcted by element 75 wth 7 ntegaton ponts. The D plane stan elements closely appoxmate the bendng pocess. An eo value of about -6% s found fo the fst stage. At lowe cuvatues, the FEA closely appoxmates theoy. As cuvatue nceases, the bendng moment calculaton devates fom the theoy. Wthn shell elements, the educed ntegaton elements do bette as they do not cause shea o membane 40

52 lockng. Ths enables the shell elements to acheve defomaton to hgh values of. Wthn plane stan elements, the analyss wth CPE8R fomulaton (smulaton L) s seen to gve esults wthn 0% of accuate moment calculaton. Inceased dscetzaton deceases the total eo n calculaton. Typcally, shell elements do not nclude the calculaton of adal stesses. As the adal stesses become sgnfcant wth the ncease n cuvatue, the effect s seen magnfed n the devaton of moment fom theoy at hghe cuvatues. Oveall, fo MARC, 4-noded plane stan elements ae seen to gve about -0% eo n moment calculaton, whle 8-noded plane stan elements show an eo of about -8%. Equvalent element fomulaton n ABAQUS shows an eo of about -0% and -0% espectvely ndcatng lowe spngback than pedcted by theoy. The decease n dscetzaton level fo the shell elements dd not affect the eo fo the case of wok-hadenng mateal but, as epoted, the eo fo pefectly plastc mateal nceased to about 37%. Changng the convegence values by 00 tmes fom the default made about % change n the bendng moment pedcton. The educed ntegaton element fomulaton s typcally pefeed fo fomng at hgh cuvatues. Use of fully ntegated elements causes shea lockng of elements whch typcally poduces ncoect esults. Reduced ntegaton elements do not show ths poblem and ae pefeed ove fully ntegated elements fo such poblems. In ths study, the educed ntegaton elements dd not pove to be sgnfcantly helpful than the fully ntegated elements. Ths can be due to fne meshng of the sheet. Oveall the esults show 4-node plane stan elements to show smalle devaton fom theoetcal esults than the shell elements. They pove to be bette as they nclude the fomulaton fo ncludng adal stesses as the shell elements typcally do not nclude adal stess n calculatons. 4

53 TABLE VARIOUS COMBINATIONS OF PARAMETERS UNDER WHICH SIMULATIONS HAVE BEEN CARRIED OUT IN ABAQUS USING A ELASTIC PERFECTLY PLASTIC MATERIAL MODEL INDEX No. of elements Thckness (mm) Fnal Radus (mm) Relatve Cuvatue (X) Element type Young's Modulus No. of Convegence ntegaton pts LOADS Fomng Spngback Toleance COUPLE A 0 * CPE4 73 GPa NA FORCE FORCE FORCE 5.00E-03 COUPLE B 0 * CPE4R 73 GPa NA FORCE FORCE FORCE 5.00E-03 COUPLE C 0 * CPE4 730 Gpa NA FORCE FORCE FORCE 5.00E-03 COUPLE D 0 * CPE8R 73 GPa NA FORCE FORCE FORCE 5.00E-03 COUPLE E 0 * CPE8 73 GPa NA FORCE FORCE FORCE 5.00E-03 F * S4 73 GPa 7 MOMENT FORCE FORCE 5.00E-03 G * S4R 73 GPa 7 MOMENT FORCE FORCE 5.00E-03 H * S4R 73 Gpa 7 MOMENT FORCE FORCE 5.00E-03 I * S4R 73 GPa 7 MOMENT FORCE FORCE 5.00E-03 4

54 TABLE 3 RESULTS FOR SIMULATIONS CARRIED OUT IN ABAQUS USING AN ELASTIC PERFECTLY PLASTIC MATERIAL At the end of fomng Index c Fnal adus (mm) Relatve Cuvatue (X) Bendng moment (N-mm) gven by HILL's model FEM Fm/t Membane Stess (Mpa) Eo E (%)* A B C D E F G H I * Eo calculated wth espect to Hll's model 43

55 TABLE 4 INDEX VARIOUS COMBINATIONS OF PARAMETERS UNDER WHICH SIMULATIONS HAVE BEEN CARRIED OUT IN ABAQUS USING A WORK-HARDENING MATERIAL Convegence No. of elements Thckness(mm) Fnal Radus (mm) Relatve Cuvatue () Element type No. of ntegaton pts LOADS Fomng Spngback Toleance J 0 * CPE4 NA COUPLE FORCE FORCE FORCE 5.00E-03 K 0 * CPE4 NA COUPLE FORCE FORCE FORCE.00E-0 L 0 * CPE4 NA COUPLE FORCE FORCE FORCE 5.00E-05 M 0 * CPE8 NA COUPLE FORCE FORCE FORCE 5.00E-03 N 0 * CPE8 NA COUPLE FORCE FORCE FORCE.00E-0 O 0 * CPE8 NA COUPLE FORCE FORCE FORCE 5.00E-05 P 0 * CPE4R NA COUPLE FORCE FORCE FORCE 5.00E-03 Q 0 * CPE4 NA COUPLE FORCE FORCE FORCE 5.00E-03 R 3 * S4R 5 COUPLE FORCE FORCE FORCE 5.00E-03 S 3 * S4R 7 COUPLE FORCE FORCE FORCE 5.00E-03 T * S4R 7 MOMENT FORCE FORCE 5.00E-03 U * S4R 7 MOMENT FORCE FORCE 5.00E-03 V 3 * S4R 9 COUPLE FORCE FORCE FORCE 5.00E-03 W 3 * S4R COUPLE FORCE FORCE FORCE 5.00E-03 X 3 * S4R COUPLE FORCE FORCE FORCE.00E-0 Y 3 * S4R COUPLE FORCE FORCE FORCE 5.00E-05 Z 3 * S4R 5 COUPLE FORCE FORCE FORCE 5.00E-03 AA 3 * S4R 5 MOMENT FORCE FORCE 5.00E-03 AB 3 * S4 5 COUPLE FORCE FORCE FORCE 5.00E-03 AC 3 * S4 7 COUPLE FORCE FORCE FORCE 5.00E-03 AD 3 * S4 9 COUPLE FORCE FORCE FORCE 5.00E-03 AE 3 * S4 COUPLE FORCE FORCE FORCE 5.00E-03 AF 3 * S4 COUPLE FORCE FORCE FORCE.00E-0 AG 3 * S4 COUPLE FORCE FORCE FORCE 5.00E-05 44

56 TABLE 5 RESULTS FOR SIMULATIONS CARRIED OUT IN ABAQUS USING D CONTINUUM ELEMENTS AND A WORK-HARDENING MATERIAL MODEL At the end of fomng At the end of spngback Index c Fnal adus (mm) Relatve Cuvatue () Bendng moment (N-mm) gven by DM s model Cafood s model FEM Eo E (%)* Fm/t Membane Stess (Mpa) c Fnal adus (mm) MB FEM BM (Nmm) Eo E (%) Change n cuvatue () Fm/t Membane Stess (Mpa) FEM Theoetcal Eo E3 (%) Total Eo (%) J E-04.06E K E-04.E L E-04.06E M E-04.06E N E-04.07E O E-04.06E P E-04.05E Q E-04.0E * Eo calculated wth espect to Cafood's model 45

57 TABLE 6 RESULTS FOR SIMULATIONS CARRIED OUT IN ABAQUS USING 3D SHELL ELEMENTS AND A WORK-HARDENING MATERIAL Index c Fnal adus (mm) Relatve Cuvatue () At the end of fomng At the end of spngback Bendng moment (N-mm) gven by DM s model Cafood s model FEM Eo E (%)* c Fnal adus (mm) MB FEM BM (Nmm) Change n cuvatue () Eo E (%) FEM Theoetcal R E-03.94E-04.07E-03.09E S E E-07.06E-03.0E T E E E-04.09E U E E-07.00E-03.05E V E-03.99E-03.08E-03.0E W E E-04.07E-03.0E X E E-04.07E-03.0E Y E E-04.07E-03.0E Z E E-04.08E-03.4E AA E-03.45E-03.05E-03.07E AB E E-07.09E-03.0E AC E E-07.07E-03.09E AD E E-07.08E-03.09E AE E E-07.07E-03.09E AF E E-07.08E-03.0E AG E E-07.08E-03.09E * Eo calculated wth espect to Cafood's model Eo E3 (%) Total Eo (%) 46

58 TABLE 7 VARIOUS COMBINATIONS OF PARAMETERS UNDER WHICH SIMULATIONS HAVE BEEN CARRIED OUT IN MARC USING A WORK-HARDENING MATERIAL Index No. of elements Thckness Fnal adus Relatve Cuvatue () Element type Numbe of Integaton Ponts Assumed stan Convegence testng Fomng Spngback Toleance AH 0 * ELEMENT NA YES FORCE DISPLACEMENT.00E-05 AI 0 * ELEMENT NA YES FORCE DISPLACEMENT.00E-06 AJ 0 * ELEMENT NA YES FORCE DISPLACEMENT 5.00E-04 AK 0 * ELEMENT 7 NA YES FORCE DISPLACEMENT 5.00E-04 AL 4 * ELEMENT 75 5 YES FORCE DISPLACEMENT.00E-05 AM 4 * ELEMENT 75 7 YES FORCE DISPLACEMENT.00E-05 AN 4 * ELEMENT 75 9 YES FORCE DISPLACEMENT.00E-05 AO 4 * ELEMENT 75 YES FORCE DISPLACEMENT.00E-05 AP 4 * ELEMENT 75 YES FORCE DISPLACEMENT.00E-06 AQ 4 * ELEMENT 75 YES FORCE DISPLACEMENT 5.00E-04 AR 4 * ELEMENT 75 5 YES FORCE DISPLACEMENT.00E-05 47

59 TABLE 8 RESULTS FOR WORK-HARDENING MATERIAL MODEL SIMULATIONS D ELEMENTS (MARC) Index c Fnal adus (mm) Relatve Cuvatue (X) At the end of fomng At the end of spngback Bendng moment (N-mm) gven by DM s model Cafood s model FEM Eo E (%)* Fm/t Membane Stess (Mpa) c Fnal adus (mm) MB FEM BM (Nmm) Eo E (%) Change n cuvatue () Fm/t Membane Stess (Mpa) FEM Theoetcal Eo E3 (%) Total Eo (%) AH E-04.03E AI E-04.0E AJ E-04.03E AK E E * Eo calculated wth espect to Cafood's model 48

60 TABLE 9 RESULTS FOR WORK-HARDENING MATERIAL MODEL SIMULATION 3D ELEMENTS (MARC) Index c Fnal adus (mm) Relatve Cuvatue (X) At the end of fomng At the end of spngback Bendng moment (N-mm) gven by DM s model Cafood s model FEM Eo E (%)* c Fnal adus (mm) MB FEM BM (Nmm) Change n cuvatue () Eo E (%) FEM Theoetcal Eo E3 (%) Total Eo (%) AL E-04.8E AM E-03.7E AN E-03.7E AO E-03.7E AP E-04.7E AQ E-03.7E AR E-03.7E * Eo calculated wth espect to Cafood's model 49

61 CHAPTER 5 CONCLUSIONS AND FUTURE WORK 5.. Conclusons The use of fnte element smulatons fo desgnng sheet metal fomng pocesses has ecently become well establshed. Howeve, accuate pedcton of spngback, especally fo cases nvolvng lage cuvatues, s stll not outnely feasble. A complete theoetcal soluton s avalable only fo unfom pue bendng of pefectly plastc sheets. In ths wok, a methodology s developed to acheve unfom pue bendng by the applcaton of mult pont constants. The total eo (E) n spngback pedcton s pattoned nto thee components: eo n bendng moment pedcton fo the fomng stage (E ), eo due to ncomplete unloadng dung the unloadng stage (E ), and eo n pedctng the change of cuvatue coespondng to the change n bendng moment (E 3 ). Fo pefectly plastc smulatons, D elements show good coelaton fo up to a of 0.3, whle the 3D elements show eo E of about 0% at a value of = 0.. Reducng the numbe of elements along the bend causes the eo to ncease to 37%. Usng ABAQUS fo a wok-hadenng mateal model, the D elements show eo E of about -%, whle E s found to be less than %. The 8-noded elements show a E about -6%, whle E eaches -0%. Oveall, D elements show a negatve total eo ndcatng less spngback than pedcted by theoy. The 3D shell elements wth educed ntegaton show a total eo of about -% n most cases, whle the fully ntegated elements show an eo of about % ndcatng a nomnal ove-pedcton of spngback. The 4-noded plane stan elements n MARC showed a total eo of -%, ndcatng lesse spngback than that poposed by the theoy. Fo fully ntegated shells, the total eo 50

62 obtaned s of the ode of 3% fo 7 ntegaton ponts, nceasng up to 8% fo ntegaton ponts. Smulaton wth ntegaton ponts show an nceased eo of about 8%. The 3D elements n MARC ndcate an eo E of less than 6% wth dffeent ntegaton ponts. Whle the E eo was about %. The ncease n numbe of ntegaton ponts fo the smulaton dd not affect E and E much, whle E 3 nceased wth the ncease n numbe of ntegaton ponts. Loweng the dscetzaton nceases the eo n bendng moment calculaton as s noted fom the pefectly plastc smulaton. The use of shell elements wth fne dscetzaton yelds good coelaton fo < 0.4. Shell element fomulaton handles the shea lockng phenomenon effcently at fne dscetzaton. Tghtenng the convegence values moe than 00 tmes fom the default values cause a small change of about % n the esults. D plane stan elements show a close coelaton wth theoetcal esults and can be used fo cases nvolvng bendng at hgh cuvatues as they nclude the effect of adal stess n the fomulatons. It s ecommended that unfom pue bendng can be used as a benchmak fo dentfyng the nheent accuacy of esults that can be expected when usng any patcula combnaton of paametes n a sheet bendng analyss. 5.. Futue Wok The followng ecommendatons ae made fo futue wok:. Repeat the compaatve studes wth softwae such as LS-Dyna, ADINA, ABAQUS/Explct, etc... v. Thck shell elements, that nclude the effect of adal stesses, can be studed. Use FEA to study stetch-bendng and shea-bendng. Extend the appled concept to doubly cuved geometes. v. Estmate the effect of the elastc coe on the theoetcal bendng moment calculaton. 5

63 REFERENCES 5

64 REFERENCES ABAQUS Documentaton 6.4, ABAQUS Inc. Boogaad, A. H., Mendes, T., Huetnk, J., 003, Effcent Implct Fnte Element Analyss of Sheet Fomng Pocesses, Intenatonal Jounal fo Numecal Methods n Engneeng, 56, pp Cao, J., Lu, Z., Lu, W. K., 999, Pedcton of Spngback n Staght Flangng Opeaton, Symposum on Advances n Sheet Metal Fomng, ASME Intenatonal Mechancal Engneeng Congess and Exposton, MED-Vol. 0, pp Chu, C. C., 99, The Effect of Restanng Foce on Spngback, Intenatonal Jounal of Solds Stuctues, 7, No. 8, pp Cafood, R., 970, Plastc Sheet Bendng, Ph. D. dssetaton, Chalmes Teknska Högskola, Götebog. Dadas, P.,and Majless, S. A., 98, Plastc Bendng of Wok Hadenng Mateals, Tansactons of ASME, Vol. 04, pp Delannay, L., Loge, R.E., Chastel, Y., Van Houtte, P., 003, Pedcton of esdual stesses and spngback afte bendng of a textued alumnum plate, Jounal De Physque. IV : JP, 05, pp Focellese, A., Fatn, L., Gabell, F., and Mca, F., 996, Compute Aded Engneeng of the Sheet Bendng Pocess, Jounal of Mateals Pocessng Technology, 60, pp Hll, R., 950, The Mathematcal Theoy of Plastcty, Oxfod Unvesty Pess, London. Leu, D., 997, A Smplfed Appoach fo Evaluaton of Bendablty and Spngback n Plastc Bendng of Ansotopc Sheet Metals, Jounal of Mateals Pocessng Technology, 66, pp 9 7. Ludwk, P., 904, Technologsche Stude ube Blechbegung, Velag des Deutschen Polytechnschen Veens n Bohmen, Paha. Make, B. N., 998, Use s Gude to Statc Spngback Smulaton usng LS-DYNA, Lvemoe Softwae Technology Copoaton. Mattasson, K., Stange, A., Thldekvst, P., and Samuelsson, A., 995, Smulaton of Spngback n Sheet Metal Fomng, Shen and Dawson, Rottedam. 53

65 Mca, F., Focellese, A., Fatn, L., Gabell, F., and Albet, N., 997, Spngback Evaluaton n Fully 3-D Sheet Metal Fomng Pocesses, Annals of CIRP, Vol. 46, pp MSC MARC Manual Volume A: Theoy and Use Infomaton, MSC Softwae Copoaton. MSC MARC :Manual Volume B: Element Lbay, MSC Softwae Copoaton Paulsen, F., and Welo, T., 996, Applcaton of Numecal Smulaton n the Bendng of Alumnum-alloy Pofles, Jounal of Mateals Pocessng Technology, 58, pp Tmoshenko, S. P. and Goode, J. N., 970, Theoy of Elastcty, McGaw Hll Book Company, New Yok Voce, E., 948, The elatonshp between stess and stan fo homogeneous defomatons. Jounal of Instumentaton and metallugy, 74, Wollte K., 950, Bldsames Begen von Blechen um geade Kanten. Dss. TH Hannove Xu, W. L., and Ma, C. H., L, C. H., Feng, W. J., 004, Senstve Factos n Spngback Smulaton fo Sheet Metal Fomng, Jounal of Mateals Pocessng Technology, 5, pp. 7. Zhou, D., and 996, Bendng and Spngback Analyss usng Membane Elements, Engneeng Systems Desgn and Analyss, Vol. 3, pp

66 APPENDICES 55

67 APPENDIX A CRAFOORD S PROCEDURE TO CALCULATE,, AND Cafood (970) povdes an teatve pocedue to dffeental equatons fo,, and M n tems of. These paametes as adapted fom Cafood (970) ae as follows: Relatve sheet thckness, t 0 t Relatve cuvatue, m t Dmensonless quantty, u n Usng Cafood s theoy, the change n sheet thckness () elatve to the change n cuvatue () s gven by d d The adus of a neutal axs can be calculated usng 0.5 ln 0.5 ln 0.5 ln 0.5 ln ) (ln ) ln ( (A.) Whee,,, and ae mateal constants. Bendng moment s calculated accodng to the equaton 56

68 M ln t ln ln The equatons ae solved n MAPLE. The code s as follows: ##Ths s based on the algothm by Cafood estat; wth(lnalg): ##conveson fom A,B,C,D to alpha... fo bascally plane stan elatons between sgma(ph)-sgma()=/sqt(3)*sgma(eq) Convg_tol := ; delta_kappa := 0.000; Amat:=3.6: Bmat:=3.: Cmat:=73.5: Dmat:=3.75: alpha:=*amat/sqt(3): beta:=4*bmat/3: gama:=*cmat/sqt(3): delta:=*dmat/sqt(3): t0 := 4; ##equaton 4.36 fom Cafood's thess also equaton 6 n DM pape deqn:=dff(eta(kappa),kappa)= -0.5*(((- 0.5*kappa^)/(eta^*ho^))-)*eta/kappa; ##ths s eqn 4.49 fom Cafood's thess - elatng ho to kappa and eta (ho to kappatmp and etatmp eqn:=(gama- *alpha)*ln(ho)+(beta*ln(ho)^/)+(gama/delta*(ho^delta))= gama/delta*(-(((+0.5*kappatmp)/etatmp)^(-delta))+(((- 0.5*kappatmp)/etatmp)^delta))-alpha*(ln((- 0.5*kappatmp)/etatmp)+ln((+0.5*kappatmp)/etatmp))- beta/*((ln((-0.5*kappatmp)/etatmp)^)- ln((+0.5*kappatmp)/etatmp)^); d_eta := -0.5*(((-0.5*kappa^)/(eta^*ho^))-)*eta/kappa * delta_kappa; kappatmp:=delta_kappa; hotmp:=.0;kappa[0] := 0; kappa[] := delta_kappa; eta[0] := ; eta[] := ;etatmp := ; fd := fopen(`c:/adus/ntemed.txt`,append, TEXT); # Assumng that kappa maxmum of nteest s.0 - to decde numbe of ncements n kappa. DO ths j loop only f tou want a cuve n tems of kappa. Othewse can dectly jump to any kappa of nteest by settng kappatmp := kappa fo j fom to /delta_kappa do fo fom to 500 do 57

69 eta[j] := eta[j-] + subs({kappa = kappa[j],eta = eta[j], ho = hotmp}, d_eta); #Nume_k45:=dsolve({subs(ho=hotemp,deqn),eta(e- 00)=},eta(kappa),type=numec); #eta[]:=op(,op(,nume_k45(algsubs(temp=kappatmp,temp)))); etatmp := eta[j]; ho[j]:=fsolve(eqn,ho=0.5..); f abs(hotmp-ho[j])< Convg_tol then hotmp :=ho[j]; beak; f; hotmp :=ho[j]; od; kappa[j]:=kappatmp; fpntf(fd,`%4.0f %4.0f %4.0f %d %d \n`,kappa[j],eta[j],ho[j],j, ): c[j] := /*eta[j]/kappa[j]*t0*sqt(4+kappa[j]^); #pnt(kappa[j],eta[j],ho[j],c[j], j, ): kappatmp:=kappatmp+delta_kappa; kappa[j+]:=kappatmp;eta[j+] := eta[j]; od: fclose(fd); ##Afte the eta, kappa and ho values ae obtaned, they ae used to calculate the moment and othe equed paametes fd:=fopen(`c:/adus/ntemed.txt`,read,text); alldata:=eaddata(fd,3): fclose(fd); convet(alldata,aay): fd:=fopen(`c:/adus/fnal.txt`,append,text); fo k fom 4845 to owdm(alldata) do ## settng the values of kappa, eta and ho fom the ead aay kappav:=alldata[k,]; etav:=alldata[k,]; hov:=alldata[k,3]; y := etav * t0/kappav * (+kappav/); c := /*etav/kappav*t0*sqt(4+kappav^); := etav * t0/kappav * (-kappav/); t := y - ; c := sqt((y^ + ^)/); m := (y+)/; o := (y^-^)/(*t0); n:=sqt(y*); BM[k]:=evalf((/kappav+0.5)^*(alpha/4- beta/8+beta/4*ln(/etav+0.5*kappav/etav)+((gama/(*delta- 4))*((+0.5*kappav)/etav)^(-delta)))+(/kappav- 0.5)^*((alpha/4+beta/8-(beta/4*ln(/etav-0.5*kappav/etav))- (gama/(*delta+4))*((- 0.5*kappav)/etav)^(delta)))+(etav/kappav)^*(beta/8-58

70 gama/4+gama/(4-*delta))+(hov*etav/kappav)^*(-alpha/- beta/8+gama/4+beta/4*ln(hov)+((gama/(4+*delta))*hov^delta))); #ths s M/t0^ actbm[k]:=etav^*bm[k]; fpntf(fd,`%4.0f %4.0f %4.0f %5.0f %5.0f %5.0f %5.0f %5.0f %5.0f %5.0f %6.0f %6.0f\n`,kappav,etav,hov,y,m,o,n,,c,c,BM[k],actBM[k] ); od; fclose(fd); 59

71 APPENDIX B FEA MODELS In the ntal phase, dsplacement bounday condtons wee gven to the sheet. The govenng equatons fo sheet bendng ae x t f 0 x0 sn tme 4 f tme x 0 (B.) y t f 0 x0 cos tme 4 f tme y 0 (B.) Compessve stesses wee geneated because of the enfoced constant. Late, the same set of dsplacement bounday condtons wee appled to a dummy beam, as shown n Fgue B. The dummy beam was made a lttle longe than the sheet so that the contact exsts even at hgh cuvatues. Contact was gven at the end of the dummy beam and the last 5 nodes on the cente fbe of the sheet. Fo spngback, the contact was eleased. Even these smulatons wee found to nduce a sgnfcant amount of compessve stesses. Cafood s theoy suggested usng a long dummy beam of nfnte length compaed to the sheet to be bent. A small foce s appled on the end of the long dummy od. Ths gves a neglgble shea stess to the sheet whle the sheet s bent mostly due to the effect of the moment caused by the foce appled to the end of the sheet. The Smulaton has lmtatons n tems of the convegence ctea. Sheet bends only to a of about 0.05 befoe temnaton. 60

72 Subsequently, model employed n ths study was developed. Bendng s acheved usng a couple foce at the ght end nodes of the sheet. The egon of sheet whee the loads ae appled defoms seveely. Rgd elements ae used to dstbute the foce equally. These elements mantaned a gd stuctue whle the foces wee tansmtted though the steet. Fgue B3 shows the schematcs of the model. Fgue B. Model wth a dummy beam (length of sheet=00 mm; thckness of sheet = 4 mm; length of dummy beam = 0 mm). Fgue B. Model wth a long am (dummy beam) used fo bendng (Sheet length= 50 mm, length of dummy beam = 500 mm. Fgue B3. Model used fo cuent study (Length=50 mm, thckness of sheet=4 mm). 6

73 APPENDIX C RIKS RAMM PROCEDURE Modfed Rks-Ramm analyss s used to povde an automatc steppng load fo the analyss. A geneal descpton of Rks-Ramm pocedue adapted fom MARC manuals (005) and ABAQUS 6.4. documentaton (005) s as follows: a. Rks analyss s a dsplacement-based steppng pocedue. b. Loadng n an ncement s based on the calculated length of the dsplacement vecto fo that ncement. c. The ac-length (length of the dsplacement vecto) s gven by the fomula: C = U T * U (C.) d. Fo the analyss, we gve an ntal value of load (as a factonal value of total load appled fo that loadcase). Ths s the statng load fo the analyss (.e., loadng fo the fst cycle of the fst ncement of the analyss). e. The dsplacements ae calculated (fo the cuent cycle) usng the followng equaton: U= K - () (P) (C.) f. Usng ths foce, convegence s calculated and the end of that patcula ncement s attaned. g. The ac-length s ncemented (Ths s decded by the numbe of ecycles equed to complete the pevous ncement). The analyss uns ae based on the ntal loadng and the load equed at the end of the loadcase. Sometmes the ac-lengths can also be negatve leadng to a negatve, foce applcaton. A typcal loadng pofle fo Rks analyss s shown n Fgue C. It can be seen 6

74 Fgue C. Load appled by FEA softwae wth change n dsplacement (MARC manuals 005). that the load nceases and deceases wth the ac length. It shows negatve ac-lengths wth deceasng loads to mantan the equlbum of the stuctue. Also, the loadng ncements ae constaned by the maxmum loadng allowed n any patcula ncement o the maxmum allowed ac-length n the ncement. The loadcase ends when the maxmum tme (fo the loadcase) o the maxmum numbe of ncements specfed (fo the loadcase) s eached. Typcally Rks method s used to solve snap-though poblems, whee the equed foce abuptly deceases afte the metal snaps. It s also useful fo solvng ll-condtoned poblems such as lmt load poblems o almost unstable poblems that exhbt softenng (ABAQUS Analyss Uses Manual, 005). 63