On Accurate Stress Determination in Laminated Finite Length Cylinders Subjected to Thermo Elastic Load

Transcription

1 Intenatonal Jounal of Mechancs and Solds ISSN Volume 6, Numbe 1 (2011), pp Reseach Inda Publcatons On Accuate Stess Detemnaton n Lamnated Fnte Length Cylndes Subjected to Themo Elastc Load *Payal Desa and Taun Kant Depatment of Cvl Engneeng, Indan Insttute of Technology Bombay, Powa, Mumba , Inda *Coespondng Autho E-mal: payaldesa79@gmal.com Abstact In ths pape, we analye the bounday value poblems (BVPs) of a fnte length lamnated cylndes unde themoelastc loads usng a sem analytcal cum numecal appoach. Exact elastcty equatons ae used n the analyss wthout any assumptons to detemne the accuate stesses. Examples coveed ae daphagm suppoted sotopc, othotopc and lamnated composte cylnde unde symmetc themal load whch s consdeed as a two dmensonal (2D) plane stan poblem of themoelastcty n (, ) decton. The bounday condtons ae satsfed exactly by takng an analytcal expesson n axal () decton n tems of Foue sees expanson. Fundamental (basc) dependent vaables ae chosen n the adal coodnate of the cylnde. Fst ode smultaneous odnay dffeental equatons ae obtaned as mathematcal model whch ae ntegated though an effectve numecal ntegaton technque by fst tansfomng the BVP nto a set of ntal value poblems (IVPs). The numecal esults obtaned ae also fst valdated fo the accuacy wth 1D soluton of an nfntely long cylnde. Keywods: A. Lamnate; B. Themal popetes; C. Computatonal mechancs; C. Defomaton; C. Stuctual compostes. Intoducton Themal stesses ae of geat pactcal mpotance, especally n lage composte cylndes such as steam-tubne otos, heavy shafts and lage tubne dscs. In all these cases, heatng o coolng must be gadual n ode to educe the tempeatue gadent n the adal decton. Moeove, detemnaton of the stesses and

2 8 Payal Desa and Taun Kant defomaton n thn and thck othotopc lamnated cylndcal shells subject to themal loadng s a majo actvty n the desgn of equpments such as pessue vessels, nuclea eactos, ppng, boles and heat exchanges. Composte cylndes ae also wdely used n vaous aeospace engneeng applcatons such as aeospace vehcles, heat shelds fo e-enty vehcles, etc. and need accuate analyss of defomatons and stesses nduced by themal loadng. The classc themal stess poblem of nfntely long cylndes made of elastc sotopc mateals has been studed by many long back. Kent [1] pesented solutons fo the stesses n sold and hollow sphees and long cylndes n whch the tempeatue s a functon of both adal coodnate and tme and obtaned soluton by substtutng n the aleady avalable fomulae. Potsky [2] gave complete 2D analytcal soluton elated to steady-state tempeatue stesses n a cylndcal tube of mecuy boles whee the tubes ae exposed to the flame and eceve adaton on one sde only. Jaege [3] gave analytcal esults fo themoelastc poblems of nfntely long sold and hollow cylndes at constant tempeatue and fo the case of peodc suface tempeatue dstbuton unde the condton of plane stess usng sees of Bessel functons. Yang and Lee [4] obtaned a sees soluton fo thck- walled cylndes subjected to a tempeatue dstbuton whch vaed both adally and axally. The soluton s based on 3D lnea theoy of themoelastcty wth appopate appoxmatons by neglectng small tems. Kalam and Tauchet [5] analysed stesses n a hollow othotopc elastc cylnde due to a steady-state plane tempeatue dstbuton T(,) usng the Ay stess functon. Iyenga and Chandashekhaa [6] gave a thee-dmensonal goous soluton fo detemnng themal stesses n a fnte length sold cylnde due to a steady state axsymmetc tempeatue feld ove one of ts end sufaces. In ths pape, govenng dffeental equatons fom exact theoy of 3D themoelastcty, whch goven the behavou of a fnte length ccula othotopc/lamnated cylnde n a state of plane stan n (, ) unde tempeatue loadng whch s a functon of both adal and axal coodnates, ae taken. By assumng a global analytcal soluton n the longtudnal decton whch satsfes the two end bounday condtons exactly, the 2D genealed plane stan poblem s educed to a 1D poblem n the adal decton. The equatons ae efomulated to enable applcaton of an effcent and accuate numecal ntegaton technque fo the soluton of the BVP of a cylnde. In addton, one dmensonal elastcty equatons of an nfntely long axsymmetc cylnde ae utled to efomulate the 1D mathematcal model sutable fo numecal ntegaton. These equatons ae summaed n the Appendx I and II. Ths has been done wth a vew to check and compaes the esults of the pesent fomulaton of fnte length cylnde unde unfom ntenal/extenal themal load, when the length of the cylnde tends to nfnty. Fomulaton Basc govenng equatons of an symmetc cylnde [7] n cylndcal coodnates ae (Fg.1). Equlbum equatons

3 On Accuate Stess Detemnaton 9 τ + + = 0 τ τ + + = 0 Stan dsplacement elatons u u w w u ε = ε = ε = γ = + Stess-stans-tempeatue elatons fo cylndcally othotopc mateal ε = ν ν +αt E E E, τ ε = ν ν + +α T, γ = E E E ε = ν + ν +α T E E E G (1a) (1b) (1c) Stesses n tems of stans can be wtten as follows C C C ε α T C21 C22 C = 23 ε αt C31 C32 C 33 ε αt (1d) τ = Gγ whee, ν ν ν ν = E, ν = E, ν = E E E E E(1 υ υ) E( υ +υυ ) E( υ +υ υ) C11 =, C12 =, C13 = Δ Δ Δ E(1 υυ ) E( υ +υυ ) E (1 υυ ) C22 =, C32 = C33 = Δ Δ Δ Δ=(1 ν ν ν ν ν ν 2ν ν ν ) C = C, C = C, C = C (1e) Stesses n tems of dsplacement components can be cast as follows: u u w = C11 α T + C12 α T + C13 αt u u w = C21 α T + C22 α T + C23 αt (1f)

4 10 Payal Desa and Taun Kant u u w = C31 α T + C32 α T + C33 αt w u τ = Gγ = G + and bounday condtons n the longtudnal and adal dectons ae, at = 0, l, u = = 0; at =, =τ =0; at = o, = τ =0 (2) n whch l s the length, s the nne adus and o s the oute adus of a hollow cylnde. Radal decton s chosen to be a pefeed ndependent coodnate. Fou fundamental dependent vaables, v., dsplacements, u and w and coespondng stesses, and τ that occu natually on a tangent plane = constant, ae chosen n the adal decton. Ccumfeental stess and axal stess ae teated hee as auxlay vaables [8] snce these ae found to be dependent on the chosen fundamental vaables. A set of fou fst ode patal dffeental equatons n ndependent coodnate whch nvolve only fundamental vaables s obtaned though algebac manpulaton of Eqs. (1a) and (1f). These ae, u C u C w = +α + α + α C C C T T T w 1 u = τ G τ C 21 C21C 12 αt u C21C 13 αt 1 w = c22 + c 2 23 C11 C11 C11 τ τ C31 ( αt) C12C 31 u C13C 31 w = C31 + C32 αt + C33 αt C11 C11 C11 (3a) (3b) (3c) (3d) and the auxlay vaables, u u w = C21 α T + C22 α T + C23 αt u u w = C31 α T + C32 α T + C33 αt (4a) (4b) A longtudnally snusodal/unfom and though thckness logathmc vaaton of tempeatue s assumed as follows [14],

5 On Accuate Stess Detemnaton 11 Type-I π T(,) = T msn whee T l m o T0 log = o log (5a) Type-II o N Tlog 1 π 4 0 T(,) = Tm sn whee Tm = (5b) = 1,3,5,... l π o log A longtudnally snusodal/unfom and though thckness lnea type vaaton of tempeatue s assumed as follows [15], Type-III π T(,) = Tmsn whee Tm = T 1 + T 2 l h (6a) Type-IV 1 π 4 T(,) = T sn whee T T T N m m = = 1,3,5,... l π h (6b) whee T 0 s ntal efeence tempeatue, T 1 and T 2 ae aveage and dffeence n se n tempeatue of top and bottom sufaces of cylnde. Vaatons of the fou fundamental dependent vaables whch completely satsfy the bounday condtons of smple (daphagm) suppots at = 0, l can then be assumed as, N π u (, ) = U ()sn = 1,3,5,... l N π w (, ) = W ()cos l = 1,3,5,... N = 1,3,5,... N π (, ) = ()sn l π τ (, ) = τ ()cos l = 1,3,5,... Substtuton of Eq. (7) n Eqs. (3a-d) and Eqs. (4a-b) and smplfcaton esultng fom othogonalty condtons of tgonometc functons leads to the followng fou smultaneous odnay dffeental equatons nvolvng only fundamental vaables. These ae, ' () C12 U() C13 π U () = +α Tm + αtm + α Tm + W() (8a) C C C l (7)

6 12 Payal Desa and Taun Kant ' 1 π W () = τ() U() G l ' π C 21 () C21. C 12 α U() () = τ () C22 Tm 2 l C11 C11 C 21C 13 α πw ( ) + C23 Tm + C11 l ' τ() πc31 π C 12C 31 π πu ( ) τ () = () C31 α Tm + C32 α Tm l C11 l C11 l l 2 CC π π + C33 α Tm+ W( ) C 11 l l (8b) (8c) (8d) and the auxlay vaables, N C C C U () CC π π = + α + α + = 1,3,5,... C11 C11 C11 l l [ () C22 Tm C23 Tm W() ] sn (9a) N C C C U() C C π π = + α m + α m+ = 1,3,5... C11 C11 C11 l l [ () C32 T C33 T W() ] sn (9b) Soluton The above system of fst ode smultaneous odnay dffeental equatons (8a-d) togethe wth the appopate bounday condtons at the nne and oute edges of the cylnde (Eq (2)), foms a two-pont BVP. Howeve, a BVP n ODEs cannot be numecally ntegated as only a half of the dependent vaables (two) ae known at the ntal edge and numecal ntegaton of an ODE s ntnscally an IVP. It becomes necessay to tansfom the poblem nto a set of IVPs. The ntal values of the emanng two fundamental vaables must be selected so that the complete soluton satsfes the two specfed condtons at the temnal boundaes dung the pocess of ntegaton [8]. Ths technque has been successfully appled and explaned n detal to solutons of plate s poblems [9, 10, 11, 12, and 13] n the past. Howeve, poblems elated to cylndcal coodnates ae uncoveed n that lteatue. Fouth ode Runge-Kutta algothm wth modfcatons suggested by Gll [16] s used fo the numecal ntegaton of the IVPs. A Fotan code s wtten fo analyng the poblems. Results and dscusson Non dmensonaled paametes ae defned as follows fo themal loadng, v.,

7 On Accuate Stess Detemnaton 13 1 = R = 0 R 2 ( + ) 1 1 (u,w) = (u,w) (,,, τ ) = (,,, τ ) α TR 0 αte 0 (11) A hollow cylnde s analysed by takng two h/r atos of 1/5 and 1/50 whch cove geometcally thck and thn cases. T 0 and T 2 ae assumed as 1 C whee as T 1 s assumed as 0 C. Mateal popetes fo cylndcally othotopc mateal ae taken as follows. (Kalam and Tauchet [5]) E = KN/m, E = KN/m, E = KN/m ν = 0.25, ν = 0.25, ν = (12) α = / C, α = / C, α = / C Radal and hoop quanttes ae maxmum at = l/2 wheeas axal quanttes ae maxmum at = 0, l. Radal stesses and adal dsplacements, pesented n Tables 1-2, fo tempeatue vaatons descbed by Eqs. (5) and (6) fo sotopc and othotopc cylndes ae compaed wth the plane stan elastcty soluton fo nfntely long cylnde, the soluton gven by Tmoshenko and Goode [14]. Eq. (13) shows analytcal soluton fo adal stess, hoop stess and adal dsplacement fom exact theoy of elastcty fo nfntely long cylnde unde plane stan state gven n Tmoshenko and Goode [14]. These ae used to valdate and check the pesent esults thoughout wheeve applcable. 2 2 o E 1 = Td + Td υ α ( ) o α 2 2 o E υ 1 = Td + Td T α ( ) o α α u = Td+ C 1 + C2 υ 1 α υ whee, C o 2 o (1 + υ)(1 2 υ) 1 1+ υ 1 = Td, C = υ 1 o υ o α υε αtd In these numecal computatons, the followng popetes fo sotopc mateal ae used. E E E = = = 2 10 KN/m ν = ν = ν = 0.3 α = α = α = / C Thee sets of numecal esults ae pesented n tables 1-3 fo clea compason,. e., (1) esults fom the pesent 2D fnte length cylnde fomulaton, (2) Computatons on the analytcal fomulae avalable fo nfntely long cylnde unde plane stan condton gven n Tmoshenko and Goode [14] and (3) esults fom the (13)

8 14 Payal Desa and Taun Kant pesent 1D nfntely long cylnde fo whch fomulaton s gven n Appendx whch foms BVP of plane stan stuaton. When the cylnde s subjected to a snusodal pessue load, the esults wthn the lmted cental length one only ae compaed wth the plane stan one dmensonal solutons, whle n the case of an unfomly dstbuted load ove the ente length of the cylnde, such compasons ae vald fo most of the length of the cylnde except nea the end suppots. A few cases out of the extensve numecal expements caed out n the pesent study. Though thckness vaatons of basc dependent vaables fo snusodal tempeatue vaaton of Type-I ae shown n Fgs. 2 fo h/r = 1/5 fo fnte length othotopc cylndes of l/r atos = 4 and 40. Fgs. 3-4 shows though thckness vaaton of quanttes fo unfomly dstbuted themal load of Type-II fo h/r = 1/5 and n Fg. 5 fo h/r=1/50. Fg. 6 shows though thckness vaaton of quanttes fo unfomly dstbuted themal load of Type-IV fo h/r = 1/5. Though thckness vaaton of basc dependent vaables fo snusodal tempeatue vaaton of Type-I ae shown n Fg. 7 fo h/r = 1/5 fo fnte length (0 0 /90 0 ) layeed othotopc cylndes of l/r atos = 4 and 40. Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-I and h/r=1/5 ae shown n Fg. 8. Dstbuton of adal dsplacement u and adal stess though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-I and h/r=1/50 ae shown n Fg. 9. Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-I and h/r=1/50 ae shown n Fg. 10. Dstbuton of adal dsplacement u and adal stess though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/5 ae shown n Fg. 11. Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/5 ae shown n Fg. 12. Dstbuton of adal dsplacement u and adal stess though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/50 ae shown n Fg.13. Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/50 ae shown n Fg. 14. Dstbutons of adal and shea stesses ae paabolc wheeas adal

9 On Accuate Stess Detemnaton 15 dsplacement, hoop stess and axal stess ae lnea though thckness. Axal dsplacement s constant though the thckness fo all cases. It s seen n Tables 1-3 that fo sotopc and othotopc cylndes wth l/r = 40, epesentng an nfntely long cylnde, the esults ae close to the plane stan analytcal elastcty soluton gven by Tmoshenko and Goode [14]. Fo unfomly dstbuted themal load, a convegence study s caed out by takng dffeent numbe of hamonc tems n Foue sees. A detaled convegence study s shown n Tables 4-5 fo l/r atos = 4, 40 and fo h/r = 1/5 fo tempeatue vaaton Type-II. Fom obsevaton of the numecal esults obtaned n the pesent analyss, t was found that l/r atos affect the ate of convegence. Fast convegence s obtaned fo l/r = 4 compaed to 0. It can be seen fom Table 3 that fo an sotopc cylnde of h/r=1/5, adal dsplacement conveges at N = 23 fo l/r = 4, and N = 40 fo l/r = 40; axal dsplacement at N = 27 fo l/r = 4 and N = 65 fo l/r = 40. It can futhe be seen fom Table 6 that fo an othotopc cylnde of h/r = 1/5, adal dsplacement conveges at N = 19, axal dsplacement at N = 27 fo l/r = 4. Poo convegence s seen fo shea and adal stesses n both the cases. Results pesented hee ae fo hamoncs N = 103 tems both fo l/r = 4 and l/r = 40. Table 1: Non-dmensonal adal stess (=l/2) and adal dsplacement u (=l/2) though thckness fo daphagm suppoted elastc sotopc cylnde unde tempeatue vaaton Type-I fo υ=0.3 and h/r=1/5. Pesent-Fnte length cylnde Pesent nfntely long cylnde and [14] Pesent-Fnte length cylnde (=l/2) u (=l/2) l/r l/r Pesent nfntely long cylnde and [14]

10 16 Payal Desa and Taun Kant Table 2: Compason of non-dmensonal adal dsplacement u (=l/2) and adal stess (=l/2) though thckness fo daphagm suppoted elastc sotopc cylnde unde themal loadng of Type-I and II and fo h/r=1/5 wth elastcty plane stan soluton fo nfntely long cylnde and fnte cylnde fom the pesent wok. Pesent - Fnte length cylnde u (=l/2) Snusodal tempeatue () Unfomly dstbuted tempeatue () Pesent nfntely long cylnde and [14] Pesent- Fnte length cylnde Snusodal tempeatue () (=l/2) Unfomly dstbuted tempeatue () Pesent nfntely long cylnde and [14] Table 3: Compason of non-dmensonal adal dsplacement u (=l/2) and adal stess (=l/2) though thckness fo daphagm suppoted elastc othotopc cylnde unde themal loadng of Type-I and II and fo h/r=1/5 wth elastcty plane stan soluton fo nfntely long cylnde and fnte cylnde fom the pesent wok. Pesent u (=l/2) Snusodal tempeatue () Unfomly dstbuted tempeatue () Pesent nfntely long cylnde Pesent (=l/2) Pesent nfntely Snusodal Unfomly long tempeatue () dstbuted cylnde tempeatue ()

11 On Accuate Stess Detemnaton 17 Table 4: Values of non-dmensonal basc dependent vaables at = R fo daphagm suppoted fnte length sotopc cylnde of ato h/r = 1/5 unde unfomly dstbuted loadng fo tempeatue vaaton Type-II-a convegence study fo l/r atos = 4, 40. l/r = 4 l/r = 40 N u w τ u w τ Table 5: Values of non-dmensonal basc dependent vaables at = R fo daphagm suppoted fnte length othotopc cylnde of ato h/r = 1/5 unde unfomly dstbuted loadng fo tempeatue vaaton Type-II-a convegence study fo l/r atos = 4, 40. w 0 w N u τ u τ

12 18 Payal Desa and Taun Kant , u, w o l Fgue 1a: Coodnate system and geomety of cylnde. u = 0, = 0 u = 0, = 0 Fgue 1b: Fnte cylnde unde snusodal extenal themal loadng.

13 On Accuate Stess Detemnaton 19 u = 0, = 0 u = 0, = 0 Fgue 1c: Fnte cylnde unde unfomly dstbuted extenal themal loadng. Radal dsplacement at =l/ Radal stess at =l/ Radal dstance /R Radal dstance /R Fgue 2: Dstbuton of adal dsplacement u and adal stess though thckness n othotopc cylnde fo tempeatue vaaton Type-I and h/r=1/ adal dsplacement at =l/ Radal stess at =l/ Radal dstance /R Radal dstance /R Fgue 3: Dstbuton of adal dsplacement u and adal stess though thckness n othotopc cylnde fo tempeatue vaaton Type-II and h/r=1/5.

14 20 Payal Desa and Taun Kant Axal dsplacement at =0,l Shea stess at =0,l Radal dstance /R Radal dstance /R Fgue 4: Dstbuton of axal dsplacement w and shea stess τ though thckness n othotopc cylnde fo tempeatue vaaton Type-II and h/r=1/5. Radal dsplacement at =l/ Radal stess at =l/ Radal dstance /R Radal dstance /R Fgue 5: Dstbuton of adal dsplacement u and adal stess though thckness n othotopc cylnde fo tempeatue vaaton Type-II and h/r=1/50. 1 Radal dsplacement at =l/ Radal stess at =l/ Radal dstance /R -8 Radal dstance /R Fgue 6: Dstbuton of adal dsplacement u and adal stess though thckness n othotopc cylnde fo tempeatue vaaton Type-IV and h/r=1/5.

15 On Accuate Stess Detemnaton Radal dsplacement at =l/ Radal stess at =l/ Radal dstance /R Radal dstance /R Fgue 7: Dstbuton of adal dsplacement u and adal stess though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-I and h/r=1/ Axal dsplacement at =0,l Shea stess at =0,l Radal dstance /R Radal dstance /R Fgue 8: Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-I and h/r=1/ Radal dsplacement at =l/ Radal dstance /R Radal stess at =l/ Radal dstance /R Fgue 9: Dstbuton of adal dsplacement u and adal stess though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-I and h/r=1/50.

16 22 Payal Desa and Taun Kant Axal dsplacement at =0,l Shea stess at =0,l Radal dstance /R Radal dstance /R Fgue 10: Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-I and h/r=1/ Radal stess at =l/ Radal dstance /R Radal stess at =l/ Radal dstance /R Fgue 11: Dstbuton of adal dsplacement u and adal stess though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/ Axal dsplacement at =0,l Radal dstance /R Shea stess at =0,l Radal dstance /R Fgue 12: Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/5.

17 On Accuate Stess Detemnaton 23 Radal dsplacement at =l/ Radal stess at =l/ Radal dstance /R Radal dstance /R Fgue 13: Dstbuton of adal dsplacement u and adal stess though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/ Axal dsplacement at =0,l Radal dstance /R Shea stess at =0,l Radal dstance /R Fgue 14: Dstbuton of axal dsplacement w and shea stess τ though thckness n two laye othotopc composte cylnde (0/90) fo tempeatue vaaton Type-III and h/r=1/50. Conclusons Numecal analyss of elastc sotopc, othotopc and lamnated fbe enfoced composte cylndes unde snusodal and unfomly dstbuted themal loadngs of two types ae pesented. Mathematcal model s based on the exact theoy of elastcty wthout any knematc and knetc assumptons. Basc equatons ae cast n a fom sutable fo numecal ntegaton n the adal decton. Numecal ntegaton technque adopted hee s found to be vey effectve and accuate; as (1) t ncopoates mxed vaables both stesses and dsplacements n the analyss (2) contnuty condtons though thckness fo layeed mateals ae dectly satsfed whle pefomng the ntegaton.

18 24 Payal Desa and Taun Kant Acknowledgements Patal suppot of a USIF collaboatve poject gant IND104 s gatefully acknowledged. Nomenclatue,, Cylndcal coodnates u,v,w Dsplacement components,, Nomal stess components paallel to,, and axs τ ε, ε, ε, γ E α T ν, 0 l T 0,T 1,T 2 p u,w Sheang stess n cylndcal coodnates Unt elongatons (nomal and shea stan) components n cylndcal coodnates Young s modulus of elastcty Coeffcent of themal expanson pe degee centgade Tempeatue se at any pont n a cylnde Posson s ato Inne and oute adus of the cylnde Length of the cylnde Intal efeence tempeatue Unfom extenal pessue Nondmensonaled dsplacement components,, Nondmensonaled nomal stess components paallel to,, and, axs Nondmensonaled sheang stess n cylndcal coodnates τ R Nondmensonaled adus ( 0 + ) Mean adus 2 Refeences [1] Kent, C. H., 1932, Themal Stesses n Sphees and Cylndes, ASME J. of Appl. Mech., 54, [2] Potsky, H., 1937, Themal Stesses n Cylndcal Ppes, Phl.Mag. S., 7(24), 160, [3] Jaege, J. C. 1945, On Themal Stesses n Ccula Cylndes, Phl. Mag., 36: [4] Yang, K.W., Lee CW., 1971 Themal Stesses n Thck-Walled Ccula Cylndes unde Axsymmetc Tempeatue Dstbuton, ASME J. of Engg Industy., 93(B),

19 On Accuate Stess Detemnaton 25 [5] Kalam, M.A., Tauchet TR. 1978, Stesses n othotopc elastc cylnde due to a plane tempeatue dstbuton T(,), J. of Them. Stess; 1, [6] Sundaa Raja Iyenge, K.T., Chandashekhaa K. 1966, Themal Stesses n a fnte sold cylnde due to an axsymmetc tempeatue feld at the end suface, Nucl. Engg. Desgn; 3, [7] Boley, B.A., Wene J.H., 1960, Theoy of Themal Stesses, New Yok, Wley. [8] Kant, T., Ramesh, C.K., 1981, Numecal Integaton of Lnea Bounday Value Poblems n Sold Mechancs by Segmentaton Method, Int J Num Meth. n Engg, 17, [9] Kant, T., Setlu, A.V. 1973, Compute analyss of clamped-clamped and clamped suppoted cylndcal shells, J. Aeo. Soc. of Inda; 25, [10] Ramesh, CK, Kant T, Jadhav VB. 1974, Elastc analyss of cylndcal pessue vessels wth vaous end closues, Int. J. of Pess. Vess. and Pp., 2, [11] Kant, T. 1981, Numecal analyss of elastc plates wth two opposte smply suppoted ends by segmentaton method, Comp. and Stuct., 14, [12] Kant, T. 1982, Numecal analyss of thck plates, Comp. Meth. Appl. Mech. and Engg., 31, [13] Kant, T, Hnton E. 1983, Mndln plate analyss by segmentaton method, ASCE J. Engg. Mech., 109, [14] Tmoshenko, S., Goode J.N., 1951, Theoy of Elastcty, New Yok, McGaw-Hll. [15] Khae, R.K., Kant, T., Gag, A.K. 2003, Closed-fom Themo-mechancal Solutons of Hghe-ode Theoes of Coss-ply Lamnated Shallow Shells, Comp. Stuct., 59, [16] Gll, S., 1951, A pocess fo the step-by-step ntegaton of dffeental equatons n an automatc dgtal computng machne, Poc. Camb. Phl. Soc., 47(1), Appendx I 1d and 2d Fomulaton fo sotopc cylnde unde themal loadng + = 0 u u ε = ε = (A1) E α ( 1 ) TE, E α = ( 1 ) (1 )(1 2 ) υ ε + υε = + TE (1 + )(1 2 ) υ ε υε υ υ υ υ υ 1 2υ

20 26 Payal Desa and Taun Kant du υ u αte = + d λ(1 υ) 1 υ λ(1 υ)(1 2 υ) d υ u λ(1 2 υ) αte υ = d 1 υ (1 υ) (1 2 υ) 1 υ 2D fomulaton fo sotopc cylnde unde themal loadng s, u 1 ν u ν w αet = +, (1 ν ) 1 ν (1 ν) λ(1 ν)(1 2 ν) w 1 u = τ G τ u (1 2 ν ) υ λ wυ(1 2 υ) αet = + λ (1 υ) 1 υ (1 υ) (1 υ) (A2) (A3) 2 τ 1 λ( 1 2υ) w ν νλ( 1 2ν) u (2υ-1) = τ E 2 α ( 1 υ) 1 ν 1 ν (1- υ)(1-2 υ) E E whee λ= G= (1+ ν)(1-2 ν) and 2(1+ ν) { T } Appendx II 1d Fomulaton fo othotopc cylnde unde themal loadng d 1 u u + ( ) = 0, ε = ε = d = C ( ε α T) + C ( ε α T) = C ( ε α T) + C ( ε α T) du u du u = C11 C11αT + C12 C12αT = C21 C21αT + C22 C22αT d d du C12 u C12 = + αt + αt d C11 C11 C11, d C 21 u C21C 12 α T C C = + C C22 (A5) d C11 C11 C11 (A4) Whee, ν E υ E E ν = E, C =, C =, C =, C = C E (1 υυ ) (1 υυ ) (1 υυ )