QUASI-STATIC TRANSIENT THERMAL STRESSES IN A DIRICHLET S THIN HOLLOW CYLINDER WITH INTERNAL MOVING HEAT SOURCE

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1 Iteatoal Joual of Pyss ad Mateatal Sees ISSN: (Ole) A Oe Aess, Ole Iteatoal Joual Aalable at tt:// 014 Vol. 4 (1) Jauay-Ma, /Solae ad Duge Resea Atle QUASI-STATIC TRANSIENT THERMAL STRESSES IN A DIRICHLET S THIN HOLLOW CYLINDER WITH INTERNAL MOVING HEAT SOURCE *D. T. Solae 1 ad M. H. Duge 1 Sudaa Na ad Uasaa Keta College Aola, Maaasta State, Ida Aad Neta College, Waoa, Maaasta State, Ida *Auto fo Coesodee ABSTRACT Ts ae oes wt taset o-oogeeous teoelast oble wt Dlet s bouday odto t ollow ylde of soto ateal of e adus a, oute adus b ad egt, ouyg te ego R: a b,, z 0 ag tal teeatue f (,, z) laed a abet teeatue zeo. Te ylde s subjeted to te atty of og eat soue alog ula tajetoy of adus 0, wee a 0 b, aoud te ete of te ylde wt ostat agula eloty. Te eat oduto equato otag eat geeato te s soled by alyg tegal tasfo teque ad Gee s teoe s adoted dedug te soluto of eat oduto equato. Te soluto s obtaed a sees fo of Bessel futo ad tgooet futo ad deed teal stesses. Keywods: Dlet s T Hollow Cylde, Mog Heat Soue, Teal Stesses, Gee s Teoe INTRODUCTION Dug te seod alf of 0 t etuy, o-soteal obles of te teoy of elastty beae easgly otat. Ts s due to te wde alato dese felds. Te g elotes of ode aaft ge se to aeodya eatg, w odues tese teal stesses tat edue te stegt of aaft stutue. I ts eset ae we detee teeatue, teal stesses, a Dlet s t ollow ylde, deteed by R:, a b z,0 wt teal og eat soue. Heat oduto equato wt eat geeato te s soled by alyg tegal tasfo teque ad Gee s teoe. Soluto s obtaed sees fo of Bessel futo ad tgooet futo. Te det obles s ey otat ew of ts eleae to aous dustal eas subjeted to eatg su as a saft of late, tubes ad te ole of ollg ll fo base of fuae bole of teal owe lat, gas owe lat ad easueet of aeodya eatg. Foulato of te Heat oduto oble Cosde a t ollow ylde of soto ateal of e adus a, oute adus b ad tess ouyg te ego R: a b,, z 0 ag tal teeatue f (,, z) laed a abet teeatue zeo. Te ylde s subjeted to te atty of og eat soue w ages ts lae alog ula tajetoy of adus 0, wee a 0 b, aoud te ete of te ylde wt ostat agula eloty. Te atty of og eat soue ad tal teeatue of te ylde ay ause te geeato of eat due to ulea teato tat ay be a futo of osto ad te te fo g(,, z, t) w/s 3. Te teeatue dstbuto of te t ollow ylde s desbed by te dffeetal equato of eat oduto wt eat geeato te as [5] age o.8 s ge by 1 1 T T g t Coygt 014 Cete fo Ifo Bo Teology (CIBTe) 188

2 Iteatoal Joual of Pyss ad Mateatal Sees ISSN: (Ole) A Oe Aess, Ole Iteatoal Joual Aalable at tt:// 014 Vol. 4 (1) Jauay-Ma, /Solae ad Duge Resea Atle Wee T T(,, z, t) s teeatue dstbuto, s teal odutty of te ateal of te ylde, s teal dffusty, s desty, C s sef eat of te ateal ad g s te C oluet eegy geeato te te ylde. Wee s Lalaa oeato yldal oodates ad 1 1 z Now osde a stataeously og eat soue g s loated at a ot ( 0,, ) ad eleasg ts eegy sotaeously at te. Su oluet eat soue yldal oodates s ge by 1 g(,, z, t) gs ( 0 ) ( ) ( z ) ( t ) Hee aboe equato edues to 1 1 T T gs ( 0 ) ( ) ( z ) ( t ) t (3.1) t (3.) Wt tal ad oogeeous bouday odtos, T 0 at a ad b (3.3) T 0 at z (3.4) T 0 at z (3.5) T f (,, z) at t 0, (3.6) Foulato of teoelast Poble Let us todue a teal stess futo elated to ooet of stess te yldal oodates syste as [3] 1 1 (4.1) (4.) 1 (4.3) Te bouday odto fo a tato fee body ae 0, 0 at a o b (4.4) Wee (4.5) Wee s oleetay soluto ad satsfes te equato satsfes te equato 0 (4.6) 4 Wee s teeatue age E (4.7) 4 s atula soluto ad T T T, s tal teeatue Coygt 014 Cete fo Ifo Bo Teology (CIBTe) 189

3 Iteatoal Joual of Pyss ad Mateatal Sees ISSN: (Ole) A Oe Aess, Ole Iteatoal Joual Aalable at tt:// 014 Vol. 4 (1) Jauay-Ma, /Solae ad Duge Resea Atle 1 1 se ylde s t, z ooet s eglgble Soluto We defe tegal tasfo of T(,, z, t) by T(,,, ) (,,, ) ( )os ( ')s ( t T z t R z ) d (5.1) Ad ts ese by R T(,,, t) R ( )os ( ')s ( z ) T(,, z, t) (5.) N( ),, 0 Wee R ( ) J ( ) Y ( b) J ( b) Y ( ) (5.3) b J ( a) a J ( a) J ( b) N( ) R ( ) d (5.4) s oot of te tasedetal equato J( a) Y ( b) J( b) Y ( a) 0 (5.5) 0,1,,3,... (5.6) 0,1,,3,... (5.7) By tag tegal tasfo of equato (3.1) ad usg followg Gee s teoe N T T d T d T ds (5.8) R R 1 s W yeld as dt ( ) T gs R ( 0 )s ( ) ( t ) dt Ts s lea dffeetal equato of fst ode wose soluto by alyg tal odto (3.6) s R ( )os ( )s ( z ) ( ) T f (,, ) gs R ( 0 )s ( ) e.,, 0 N( ) e ( (5.9) R ( )os ( )s ( z ) ( ) f (,, ) gs R ( 0 )s ( ) e.,, 0 N( ) ( e 1 (5.10) Soluto of Teoelast Poble: Let sutable fo of satsfyg equato (4.6) be 0 A B os( ) C D s ( ) (6.1) Let sutable fo of satsfyg equato (4.7) be E R ( )os ( )s ( z ) ( ) f (,, ) g R ( )s ( ) e s 0,, 0 N( ) Coygt 014 Cete fo Ifo Bo Teology (CIBTe) 190

4 Iteatoal Joual of Pyss ad Mateatal Sees ISSN: (Ole) A Oe Aess, Ole Iteatoal Joual Aalable at tt:// 014 Vol. 4 (1) Jauay-Ma, /Solae ad Duge Resea Atle ( e 1 (6.) Fo (4.5) (6.1) ad (6.) we obta A B os( ) C D s ( ) 0 E R ( )os ( )s ( z ) ( ) f (,, ) g R ( )s ( ) e N s 0,, 0 ( ) ( e 1 (6.3) Fo (4.1) ad (6.3) we obta A( ) B( ) os ( ) C( ) D( ) s( ) 0 E 1 1 N( ),, 0 R 1( ) ( ) R ( ) os ( )s ( z ). ( ) ( f (,, ) gs R ( 0 )s ( ) e e 1 (6.4) Fo (4.) ad (6.3) we obta A( 3 ) B( 3 ) os ( ) C( 3 ) D( 3 ) s( ) 0 E 1 R 1( ) R ( ) os ( )s ( z ).,, 0 N( ) ( ) ( f (,, ) gs R ( 0 )s ( ) e e 1 (6.5) Fo (4.3) ad (6.3) we obta A( ) B( ) s ( ) C( ) D( ) os( ) 0 E 1 R 1( ) (1 ) R ( ) s ( )s ( z ).,, 0 N( ) ( ) ( f (,, ) gs R ( 0 )s ( ) e e 1 (6.6) Alyg odto (4.4o (6.4) ad (6.6) we obta E b R 1( b)s ( z ) ( ) ( A f (,, ) gs R ( 0 )s ( ) e e 1 (6.5) bn( ) E b R 1( b)s ( z ) ( ) ( B f (,, ) gs R ( 0 )s ( ) e e 1 bn( ) (6.6) CD 0 (6.7) Substtutg te alue of ostats aboe equatos we obta E R 1( b) R 1( ) ( ) R ( ) b b.,, 0 b Coygt 014 Cete fo Ifo Bo Teology (CIBTe) 191

5 Iteatoal Joual of Pyss ad Mateatal Sees ISSN: (Ole) A Oe Aess, Ole Iteatoal Joual Aalable at tt:// 014 Vol. 4 (1) Jauay-Ma, /Solae ad Duge Resea Atle os ( )s ( ) z ( ) ( f (,, ) gs R ( 0 )s ( ) e e 1 N( ) E 3 3 R 1( b) b b R 1( ) R ( ).,, 0 b os ( )s ( z ) ( ) ( f (,, ) gs R ( 0 )s ( ) e e 1 N( ) (6.9) E R 1( b) R 1( ) (1 ) R ( ) (1 ) b (1 ) b.,, 0 b s ( )s ( z ) ( ) ( f (,, ) gs R ( 0 )s ( ) e e 1 N( ) (6.10) Coluso I ts ae we deteed te teeatue dstbuto tee desos ad teal stesses a Dlet s t ollow ylde wt og eat soue wt aalytal aoa o te sufae s establsed. By gg atula alues to te aaete oe a obta te desed esults by uttg alues of te aaetes te equatos (5.9), (6.8), (6.9), (6.10). Fo tese equato we obsee tat tally ( t 0 ) all stesses ases. REFERENCES Solae DT ad Duge MH (014). Quas-Stat taset Teal Stesses a Dlet s t Sold ylde wt teal og eat soue (IOSR-JM) 10() Solae DT ad Duge MH. Quas-Stat taset Teal Stesses a Neua s t Sold ylde wt teal og eat soue. AJER 3(3) Solae DT ad Duge MH (014). Quas-Stat taset Teal Stesses a Rob s t Sold ylde wt teal og eat soue AJCEM 3() Ozs M Neat (No date). `Heat oduto, Seod Edto, A Wley-Itesee Publato Jo Wley ad Sos,. New-Yo. Noda N, Hetas RB, Tagawa Y (00). Teal Stesses, Seod edto. (6.8) Coygt 014 Cete fo Ifo Bo Teology (CIBTe) 19