Module 9 Thin and thick cylinders

Transcription

1 Mdule 9 Thn and thck cylndes Vesn 2 ME, IIT Khaagu

2 Lessn 3 Desgn ncles f thck cylndes Vesn 2 ME, IIT Khaagu

3 Instuctnal Objectves: At the end f ths lessn, the students shuld have the knwledge f: Falue thees aled t thck walled essue vessels. Vaatn f wall thckness wth ntenal essue based n dffeent falue thees. Falue cten f estessed thck cylndes. Cmasn f wall thckness vaatn wth ntenal essue f sld wall, sngle jacket and lamnated thck walled cylndes. Falue cten f thck walled cylndes wth autfettage Alcatn f thees f falue f thck walled essue vessels. Havng dscussed the stesses n thck walled cylndes t s mtant t cnsde the falue cten. The fve falue thees wll be cnsdeed n ths egad and the vaatn f wall thckness t ntenal adus at t/ adus at / wth / y f dffeent falue thees wuld be dscussed. A numbe f cases such as =0, =0 bth nn-ze and ae ssble but hee nly the cylndes wth clsed ends and subjected t an ntenal essue nly wll be cnsdeed, f an examle Maxmum Pncal Stess they Accdng t ths they falue ccus when maxmum ncal stess exceeds the stess at the tensle yeld nt. The falue envele accdng t ths falue mde s shwn n fgue and the falue ctea ae gven by 1 = 2 = ± y. If =0 the maxmum values f ccumfeental and adal stesses ae gven by = θ(max) = + = (1) (max) = Vesn 2 ME, IIT Khaagu

4 Hee bth θ and ae the ncal stesses and θ s lage. Thus the cndtn f falue s based n θ and we have + =y whee y s the yeld stess. Ths gves (2) t = 1 1+ y y y - y + y 1 - y F- Falue envele accdng t Maxmum Pncal Stess They Maxmum Shea Stess they Accdng t ths they falue ccus when maxmum shea stess exceeds the maxmum shea stess at the tensle yeld nt. The falue envele accdng t ths cten s shwn n fgue and the maxmum shea stess s gven by τ max = whee the ncal stesses 1 and 2 ae gven by Vesn 2 ME, IIT Khaagu

5 = = 1 θ 2 = = + Hee 1 s tensle and 2 s cmessve n natue. τ max may theefe be gven by τ = 2 max (3) and snce the falue cten s τ max = y / 2 we may wte t 1 = 1 2 y 1 (4) 2 1 yc - =1 yt - yc + yt 2 2 = yt 1 = yt + yt 1 1 = yc 2 = yc - yc 1 2 yt - =1 yc F- Falue envele accdng t Maxmum Shea Stess they Maxmum Pncal Stan they Accdng t ths they falue ccus when the maxmum ncal stan exceeds the stan at the tensle yeld nt. Vesn 2 ME, IIT Khaagu

6 1 ε1 1 ν 2 3 ε E = { ( + )} = y and ths gves ν ( ) + = y whee ε y and y ae the yeld stan and stess esectvely. Fllwng ths the falue envele s as shwn n fgue Hee the thee ncle stesses can be gven as fllws accdng t the standad 3D slutns: = = 1 θ (5) +, 2 = = and The falue cten may nw be wtten as 2 + ν + ν = y and ths gves 2 = = 3 z ( ) ( ) t ν y = ν y (6) 2 + y + y - y 1 - y F- Falue envele accdng t Maxmum Pncal Stan they Vesn 2 ME, IIT Khaagu

7 Maxmum Dsttn Enegy They Accdng t ths they f the maxmum dsttn enegy exceeds the dsttn enegy at the tensle yeld nt falue ccus. The falue envele s shwn n fgue and the dsttn enegy E d s gven by {( ) 2 ( ) 2 2 ( 1 ) } 1+ ν Ed = E Snce at the unaxal tensle yeld nt 2 = 3 = 0 and 1 = y E d at the tensle yeld nt = 1+ ν 3E We cnsde 1 = θ, 2 = and 3 = z and theefe = y = (7) 2 = z The falue cten theefe educes t 1 = 2 3 y t 1 = 1 3 y 1 whch gves (8) Vesn 2 ME, IIT Khaagu

8 2 y - y y 1 - y F- Falue envele accdng t Maxmum Dsttn Enegy They Plts f / y and t/ f dffeent falue ctea ae shwn n fgue Maxmum ncal stess they Dsttn enegy they Maxmum stan they Maxmum shea stess they t F- Cmasn f vaatn f aganst t f dffeent falue cten. The ctea develed and the lts aly t thck walled cylndes wth ntenal essue nly but smla ctea f cylndes wth extenal Vesn 2 ME, IIT Khaagu

9 essue nly n case whee bth ntenal and extenal essues exst may be develed. Hweve, n the bass f these esults we nte that the ate f ncease n / y s small at lage values f t/ f all the falue mdes cnsdeed. Ths means that at hghe values f small ncease n essue eques lage ncease n wall thckness. But snce the stesses nea the ute adus ae small, mateal at the ute adus f vey thck wall cylndes ae neffectvely used. It s theefe necessay t select mateals s that / y s easnably small. When ths s nt ssble estessed cylndes may be used. All the abve thees f falue ae based n the edctn f the begnnng f nelastc defmatn and these ae stctly alcable f ductle mateals unde statc ladng. Maxmum ncal stess they s wdely used f bttle mateals whch nmally fal by bttle factue. In sme alcatns f thck cylndes such as, gun baels n nelastc defmatn can be emtted f e functnng and thee desgn based n maxmum shea stess they maxmum dsttn enegy they ae accetable. F sme essue vessels a satsfacty functn s mantaned untl nelastc defmatn that stats fm the nne adus and seads cmletely thugh the wall f the cylnde. Unde such ccumstances nne f the falue thees wuld wk satsfactly and the cedue dscussed n sectn lessn 9.2 s t be used Falue ctea f e-stessed thck cylndes Falue ctea based n the thee methds f e-stessng wuld nw be dscussed. The adal and ccumfeental stesses develed dung shnkng a hllw cylnde ve the man cylnde ae shwn n fgue Vesn 2 ME, IIT Khaagu

10 s s Jacket Cylnde θ s F- Dstbutn f adal and ccumfeental stesses n a cmste thck walled cylnde subjected t an ntenal essue. Fllwng the analyss n sectn 9.2 the maxmum ntal (esdual) ccumfeental stess at the nne adus f the cylnde due t the cntact essue s s = 2 s θ s = s 2 and the maxmum ntal (esdual) ccumfeental stess at the nne adus f the jacket due t cntact essue s s + s θ s = s s = Suesng the ccumfeental stesses due t (cnsdeng the cmste cylnde as ne) the ttal ccumfeental stesses at the nne adus f the cylnde and nne adus f the jacket ae esectvely Vesn 2 ME, IIT Khaagu

11 = s θ s = s 2 s s + s + θ = 2 s s s = + 2 These maxmum stesses shuld nt exceed the yeld stess and theefe we may wte 2 + 2s + s s = y (9) s s s 2 s s 2 = y (10) It was shwn n sectn-9.2 that the cntact essue s s gven by Eδ = s s s s s s (11) Fm (9), (10) and (11) t s ssble t elmnate s and exess t/ n tems f / y and ths s shwn gahcally n fgue Lamnated Sngle jacket Sld wall t Vesn 2 ME, IIT Khaagu

12 F- Plt f / y vs t/ f lamnated multlayeed, sngle jacket and sld wall cylndes. Ths shws that even wth a sngle jacket thee s a cnsdeable eductn n wall thckness and thus t cntbutes t an ecnmc desgn. As dscussed eale autfettage causes yeldng t stat at the nne be and wth the ncease n essue t seads utwads. If nw the essue s eleased the ute elastc laye exets adal cmessve essue n the nne tn and ths n tun causes adal cmessve stess nea the nne tn and tensle stess at the ute tn. F a gven flud essue dung autfettage a gven amunt f nelastc defmatn s duced and theefe n sevce the same flud essue may be used wthut causng any addtnal elastc defmatn. The self hng effect eaches ts maxmum value when yeldng just begns t sead t the ute wall. Unde ths cndtn the cylnde s sad t have eached a fully lastc cndtn and the cesndng ntenal flud essue s knwn as fully lastc essue, say, f. Ths essue may be fund by usng the educed equlbum equatn (3) n sectn whch s educed hee f cnvenence d θ = + d (12) Anthe equatn may be btaned by cnsdeng that when the maxmum shea stess at a nt n the cylnde wall eaches shea yeld value τ y t emans cnstant even afte futhe yeldng. Ths s gven by 1 2 ( ) θ =τ y (13) Hweve exements shw that fully lastc essue s eached befe nelastc defmatn has sead t evey nt n the wall. In fact Lude s lnes aea fst. Lude s lnes ae sal bands acss the cylnde wall such that the mateal between the bands etans elastcty. If the cylnde Vesn 2 ME, IIT Khaagu

13 s ket unde fully lastc essue f seveal hus unfm yeldng acss the cylnde wall wuld ccu. Ths gves d d = 2τ ylg+ c = 2 τ y / and n ntegatn we have Alyng the bunday cndtn at = = 0 we have and = 2τylg (14) θ = τ + 2 y 1 lg Als alyng the bunday cndtn at = = - f we have 2 lg f = τy (15) Snce the basc equatns ae ndeendent f whethe the cylndes ae en clsed ends, the exessns f and θ aly t bth the cndtns. The stess dstbutns ae shwn n fgue θ τ τ Tensle τ τ Cmessve F- Stess dstbutn n a thck walled cylnde wth autfettage If we ughly assume that 2τ y = y we have Vesn 2 ME, IIT Khaagu

14 f = lg y (16) The esults f maxmum ncal stess they and maxmum shea stess they alng wth the fully lastc esults ae eltted n fgue whee we may cmae the elatve mets f dffeent falue ctea. It can be seen that cylndes wth autfettage may endue lage ntenal essue at elatvely lw wall thckness. 2.0 Maxmum autfettage Maxmum ncal stess they 0.8 Maxmum shea stess they F- Plts f / y vs f maxmum shea stess they, maxmum ncal stess they and maxmum autfettage. Vesn 2 ME, IIT Khaagu

15 Fnally t must be emembeed that f tue essue vessel desgn t s essental t cnsult Ble Cdes f me cmlete nfmatn and gudelnes. Pessue vessels can be extemely dangeus even at elatvely lw essue and theefe the methdlgy stated hee s a ugh gude and shuld nt be cnsdeed t be a cmlete desgn methdlgy Pblems wth Answes Q.1: Detemne the necessay thckness f the shell lates f 2.5m damete ble wth the ntenal essue f 1MPa. The mateal s mld steel wth a tensle stength f 500MPa. Assumng an effcency f the lngtudnal welded jnt t be 75% and a fact f safety f 5 fnd the stess n the efated steel late. A.1: Cnsdeng that the ble desgn s based n thn cylnde ncles the shell thckness s gven by t = whee s the ble adus and η s the jnt effcency. η ty Ths gves t = 6 10 x x10 6 x = m = 16.6 mm,say 20mm. The stess n the efated late s theefe gven by =.e. 62.5MPa t Q.2: A hydaulc cylnde wth an ntenal damete 250mm s subjected t an ntenal essue f 10 MPa. Detemne the wall thckness based n (a) Maxmum ncal stess they, b) Maxmum shea stess they and c) Vesn 2 ME, IIT Khaagu

16 A.2: Maxmum dsttn enegy they f falue. Cmae the esults wth wall thckness calculated based n thn cylnde assumtn. Assume the yeld stess f the cylnde mateal t be 60 MPa. Cnsdeng that the hydaulc cylndes ae nmally desgned n the thck cylnde assumtn we have fm sectn f Maxmum Pncal stess They we have t = Hee 1+ 1 y y y 1 =10 / and = 125 mm. Ths gves t = 22.9mm, say 23 mm Fm sectn f Maxmum Shea Stess they we have t = Wth y y and = 125 mm, t = 28.2 mm, say 29 mm. Fm sectn f maxmum dsttn enegy they we have 1 t = y wth y and = 125mm t = 23.3 mm, say 24 mm. Vesn 2 ME, IIT Khaagu

17 Cnsdeng a thn cylnde t = y and ths gves t = mm, say 21 mm. The thn cylnde aach yelds the lwest wall thckness and ths s bably nt safe. The lagest wall thckness f 29mm edcted usng the maxmum shea stess they s theefe adted. Q.3: A cylnde wth extenal damete 300mm and ntenal damete 200mm s subjected t an ntenal essue f 25 MPa. Cmae the elatve mets f a sngle thck walled cylnde and a cmste cylnde wth the nne cylnde whse ntenal and extenal dametes ae 200mm and 250 mm esectvely. A tube f 250 mm ntenal damete and 300mm extenal damete s shunk n the man cylnde. The safe tensle yeld stess f the mateal s 110 MPa and the stess set u at the junctn due t shnkage shuld nt exceed 10 MPa. A.3: We fst cnsde the stesses set u n a sngle cylnde and then n a cmste cylnde. Sngle cylnde The bunday cndtns ae at = 150mm = 0 and at = 100mm = - 20MPa Usng equatn (10) n sectn C2 C = and C C = Ths gves C 1 = 16 and C 2 = The h stess at = 100mm and = 150 mm ae 52 MPa and 32 MPa esectvely. Vesn 2 ME, IIT Khaagu

18 Stess n the cmste cylnde The stesses n the cylnde due t shnkage nly can be fund usng the fllwng bunday cndtns at = 150mm = 0 and at = 125mm = -10MPa Fllwng the abve cedue the h stess at = 150 mm and = 125mm ae 45.7MPa and 55.75MPa esectvely. The stess n the nne cylnde due t shnkage nly can be fund usng the fllwng bunday cndtns at = 100mm = 0 and at = 125mm = -10MPa Ths gves the h stess at = 100mm and = 125mm t be MPa and MPa esectvely. Cnsdeng the ntenal essue nly n the cmlete cylnde the bunday cndtns ae at = 150mm = 0 and at = 100mm = -25 MPa Ths gves ( θ ) =150mm = 40MPa ( θ ) =125mm = 49 MPa ( θ ) =100mm = 65MPa. Resultant stess due t bth shnkage and ntenal essue Oute cylnde ( θ ) =150mm = = 85.7 MPa ( θ ) =125mm = = MPa Inne cylnde ( θ ) =125mm = = 3.3 MPa ( θ ) =100mm = = 9.25 MPa The stesses n bth the sngle cylnde and the cmste ae wthn the safe tensle stength f the mateal. Hweve n the sngle cylnde the stess gadent s lage acss the wall thckness wheeas n the cmste cylnde the stess vaatn s gentle. These esults ae llustated n fgue Vesn 2 ME, IIT Khaagu

19 104 MPa 85.7 MPa MPa 3.3 MPa MPa 9.25 MPa 200 mm 250 mm 300 mm F- Stess gadents (ccumfeental) n the nne and ute cylndes as well as the gadent acss the wall f a sngle cylnde Summay f ths Lessn The lessn ntally dscusses the alcatn f dffeent falue thees n thck walled essue vessels. Falue cten n tems f the at f wall thckness t the ntenal adus and the at f ntenal essue t yeld stess have been deved f dffeent falue cten. Falue cten f estessed cmste cylndes and cylndes wth autfettage have als been deved. Fnally cmasns f dffeent falue cten have been dscussed Refeences f Mdule-9 1) Desgn f machne elements by M.F.Stts, Pentce hall f Inda, ) Machne desgn-an ntegated aach by Rbet L. Ntn, Peasn Educatn Ltd, Vesn 2 ME, IIT Khaagu

20 3) A textbk f machne desgn by P.C.Shama and D.K.Agawal, S.K.Kataa and sns, ) Mechancal engneeng desgn by Jseh E. Shgley, McGaw Hll, ) Fundamentals f machne cmnent desgn, 3 d edtn, by Rbet C. Juvnall and Kut M. Mashek, Jhn Wley & Sns, ) Advanced stength and aled stess analyss, 2 nd Edtn, by Rchad G. Budynas, McGaw Hll Publshes, ) Mechancs f Mateals by E.J. Hean, Pegamn Pess, Vesn 2 ME, IIT Khaagu