Reseach Jounal of Applied Sciences, Engineeing and Technology 7(15): 3114-315, 014 DOI:10.1906/jase.7.650 ISSN: 040-7459; e-issn: 040-7467 014 Maxwell Scienific Publicaion Cop. Submied: Ocobe 09, 013 Acceped: Ocobe 4, 013 Published: Apil 19, 014 Reseach Aicle Sess Analysis of Nonhomogeneous Roaing Disc wih Abiaily Vaiable Thickness Using Finie Elemen Mehod Abdu Rosyid, Mahi Es-Saheb and Faycal Ben Yahia Mechanical Engineeing Depamen, College of Engineeing, King Saud Univesiy, P.O. Box 800, Riyadh 1141, Saudi Aabia Absac: Roaing discs wih vaiable hickness and nonhomogeneous maeial popeies ae fequenly used in indusial applicaions. The nonhomogeniy of maeial popeies is ofen caused by empeaue change houghou he disc. The govening diffeenial equaion pesening his poblem conains many vaiable coefficiens so ha no possible analyical closed fom soluion fo his poblem. Many numeical appoaches have been poposed o obain he soluion. Howeve, in his sudy he Finie Elemen Mehod (FEM), which pesens a poweful ool fo solving such a poblem, is used. Thus, a ubine disc modeled by using ax symmeic finie elemens was analyzed. Bu, in ode o avoid inaccuacy of he sess calculaion quie fine meshing is implemened. The analysis showed ha maximum displacemen occus a he bounday of he disc, eihe a he oue o inne bounday, depending on he loadings. The maximum adial sess occus a an aea in he middle of he disc which has he smalles hickness. In his sudy, oaional blade load was shown o give he lages conibuion o he oal displacemen and sess. Also, he adial displacemen and sess in a disc wih vaiable hickness ae found o be affeced by he conou of he hickness vaiaion. In geneal, he esuls obained show excellen ageemen wih he published woks. Keywods: Finie elemen mehod, nonhomogeneous maeial popeies, oaing disc, vaiable hickness INTRODUCTION Roaing discs have many pacical engineeing applicaions such as in seam and gas ubine discs, ubo geneaos, inenal combusion engines, casing ship popelles, uboje engines, ecipocaing and cenifugal compessos jus o menion a few. Bake disk can be an example of solid oaing disk whee only body foce is involved. In ealiy, he hickness of he disc is fequenly no consan, such as ubine discs, due o economic consideaions and in ode o impove mechanical pefomance. Tubine discs ae usually hick nea hei hub and ape down o a smalle hickness owad he peiphey, since a consan hickness ponounces sess concenaion nea he cene, even fo solid disc. Fuhemoe, i has been shown ha he sesses in vaiable hickness oaing annula and solid discs ae much lowe han hose in consan hickness discs a he same angula velociy. In many applicaions, he disc is woking unde high empeaue which pesens hemal loading. Beside he fac ha, he empeaue houghou he disc is usually no consan, i.e., hee is empeaue gadien pesen houghou he disc. This empeaue gadienusually esuled in changes in maeial popeies houghou he disc and heoaing disc becomes nonhomogeneous in he adial diecion. Geneally, hee ae wo appoaches fo he soluion of oaing discs, namely, heoeical and numeical mehods. Fo homogeneous oaing disc wih consan hickness, closed-fom analyical soluion is available. Howeve, fo nonhomogeneous oaing disc wih vaiable hickness, analyical soluion is no possible o obain. Hence, many numeical aemps has been pesened o solve such a poblem. Timoshenko and Goodie (1970) was he fis o obain a closed fom soluion fo homogeneous consan-hickness oaing discs wihou any empeaue gadien. Sodola (197) obained an analyical soluion of he poblem of homogeneous oaing discs wih hypebolic pofile hickness. Den Haog (1987) epoed he closed fom fomula fo homogeneous oaing discs wih consan and hypebolic hickness unde seveal mechanical loadings. Boesi and Richad (003) included hemal sess in he closed fom fomula. Game (1985) achieved a good adapive numeical soluion fo he consan hickness solid discs wih a linea hadening maeial. Recenly, Shama e al. (011) conduced analysis of sesses and sains in a oaing homogeneous hemoplasic cicula disk using FEM. Coesponding Auho: Mahi Es-Saheb, Mechanical Engineeing Depamen, College of Engineeing, King Saud Univesiy, P.O. Box 800, Riyadh 1141, Saudi Aabia This wok is licensed unde a Ceaive Commons Aibuion 4.0 Inenaional License (URL: hp://ceaivecommons.og/licenses/by/4.0/). 3114
Guven (1997, 1998) obained an analyical soluion of he poblem of oaing vaiable hickness discs wih igid inclusion fo he elasic-plasic (Guven, 1997) and fully plasic sae (Guven, 1997). Leopold (1984) calculaed elasic sess disibuions in oaing discs wihvaiable hickness by using a semi-gaphical mehod. Lae, semi-analyical analysis of Funcionally Gaded Maeials (FGM) oaing discs wih vaiable hickness was poposed by Baya e al. (008). Hojjai and Jaffai (007) poposed Vaiaion Ieaion Mehod (VIM) o obain he elasic analysis of non-unifom hickness and densiy oaing discs subjeced o only cenifugal loadings. Howeve, You e al. (000) poposed numeical analysis using Runge-Kua mehod compaed o FEM fo elasic-plasic oaing discs wih abiay vaiable hickness and densiy. Likewise, Hojjai and Hassani (008) poposed Vaiablemaeial Popeies (VMP) mehod and numeical analysis using Runge Kua s mehod compaed o FEMfo oaing discs of non-unifom hickness and densiy. Recenly, hey also poposed semi-exac soluion fo hemomechanical analysis of FGM elasic-sain hadening oaing discs (Hassani e al., 01). Fuhemoe, o solve nonhomogeneous oaing disc wih vaiable hickness, Jahed e al. (005) poposed disceizaion of he disc ino a finie numbe of ings, whee each ing has consan hickness and maeial popeies, bu diffeen ings may have diffeen hickness and maeial popeies. In his sudy, heefoe, he poblem of nonhomogeneous oaing disc wih abiaily vaiable hicknessis addessed and ohe elaed issues ae discussed and pesened. I is demonsaed ha he analyical soluion fo such a poblem is no possible o be obained. Consequenly, o solve his poblem he FE echnique is used. The complee deails of he FE fomulaion and analysis of he poblemalong wih numeical examples o veify he soluion ae pesened in he nex secions. Fig. 1: Infiniesimal elemen of he disc Fig. : Displacemens in infiniesimal elemen of he disc 1 dh dσ σ σ γ σ h d d g + + + ω = 0 () Because he poblem is axisymmeic, sess componens in angenial diecion vanish: dθ dθ σ cos ddh σ cos ddh = 0 (3) By simplificaion and aking sin (dθ/) = dθ/ due o small angle Ө, we obained: METHODOLOGY Analyical soluion fo elasic nonhomogeneous disc wih vaiable hickness unde mechanical and hemal loading: The poblem can be consideed as plane sess wih vaiable hickness. Hence, σ z = 0, as his is a saic poblem, he soluion has o saisfy equilibium, compaibiliy and consiuve law of he maeial popeies. Equilibium: Assuming ha he weigh of he disc is negleced, equilibium of infiniesimal elemen of he disc, as shown in Fig. 1, in he adial diecion is: h + h + + + h =0 (1) 3115 Compaibiliy: Displacamens occus in he disc is shown in Fig.. The compaibiliy ends up wih saindisplacemen elaion as follows: du u + d u d du ε = = (4) d d ( ) + u dθ dθ u ε = = (5) dθ Popeies of maeial, which ae expessed by consiuive law of maeial: Because he poblem is elasic, Hooke s law is used o ge he sess-sain elaion. Due o only mechanical loading and σ z = 0: 1 ε = ( σ νσ) (6) E
1 ε = ( σ νσ ) (7) E Because hee is also hemal loading; hen, and ε oal = ε elasic + ε (8) hemal ε hem al = αt (9) Hence, he oal sains become: 1 ε σ νσ αt E = ( ) + (10) 1 ε = ( σ νσ ) + αt (11) E Re-aanging Eq. (10) and (11), we ge: σ = Eε + νσ EαT (1) Now subiuing Eq. (9) and (31) ino (), which is he equilibium equaion in adial diecion, we obained a second ode Odinay Diffeenial Equaion (ODE) of he fom: d u d whee, f f du + f + f u = f d 1 3 1 d he = + ln d 1 ν 1 1 ν d he 1 dν = + ln + d 1 ν d d 1 ν d he (0) f 3= (( 1+ ν) αt ) γω + ( 1+ ν) αt ln 1 d Eg d ν The coefficiens of he ODE conain vaiable paamees:, h = h( ), E = E ( ), ν = ν ( ), α = α( ), γ = γ ( ), T = T ( ) and σ = Eε + νσ EαT (13) Subiuing Eq. (13) ino (1) o ge σ : E EαT σ = ( ε + νε ) 1 ν 1 ν Now subiuing Eq. (1) in o (13) o ge σ : E EαT σ = ( ε + νε ) 1 ν 1 ν Subiuing Eq. (4) and (5) ino (14) we ge: E du u EαT σ = + ν 1 ν d 1 ν σ E du u ( 1 ) T 1 ν d ν ν α = + + (14) (15) (16) (17) Similaly, subsiuing Eq. (4) and (5) ino (15) we ge: E u du EαT σ = + ν 1 ν d 1 ν σ E u du ( 1 ) T 1 ν ν d ν α = + + (18) (19) 3116 Because he coefficiens of he ODE conain many vaiable paamees, hee ae no exac soluions fo his ODE. Howeve, as saed ealie, hee have been numeous numeical appoaches o solve such a poblem. Bu in his sudy, he FE appoach is uilized. The deails of he FE analysis and soluion ae pesened below. Fem soluion fo elasic nonhomogeneous disc wih vaiable hickness unde mechanical and hemal loading: FEM is one of he mos successful and dominan numeical mehod in he las cenuy. I is exensively used in modeling and simulaion of engineeing and science due o is vesailiy fo complex geomeies of solids and sucues and is flexibiliy fo many non-linea poblems. Mos discs wih vaiable hickness in he applicaions ae ax symmeic. Fo such a case, ax symmeic elemen is he mos economical bu adequae o use in he finie elemen analysis. Fo any ohe cases in which he disc is no ax symmeic and heefoe no adequae o be modeled by axisymmeic elemen, cyclic elemen and 3D solid elemen can be used. The axisymmeic symmeic elemen has anslaional degees of feedom pe node. Using cylindical coodinae sysem whee z is he axis, is he adius and Ө is he cicumfeence angle, he sesses in he axisymmeic poblem is shown in Fig. 3. Inenal and exenal pessue woking a he inne and oue suface of he disc ae suface foces. The elemen suface foce veco is given by:
Fig. 3: Sesses in he axisymmeic poblem Fig. 4: A half coss secional view of he disc wih dimensions in mm p f = N ds p { } [ ] T s s S z Cenifugal foces due o oaional speed of he disc ae body foces. The elemen body foce veco is given by: T Rb f = N. d. dz Z A b { b} π [ ] The elemen hemal foces woking on he disc ae given by: { } T ft π ε Dε T da A = The oal elemen foces {f} ae sum of he elemen suface foces, he elemen body foces and he elemen hemal foces. The global foce vecos {F} ae obained by assembling all he elemen foces. The elemen siffness maix is given by: [ ] = π [ ] [ ][ ] T k B D B. d. dz A Assembling all of he elemen siffness maices, he global siffness maix [K] is obained. The global displacemen vecos ae given by {F} = {K}{d}. The displacemen funcion vecos ae given by: { ψ } = [ N]{ d} The global sain vecos ae given by: { ε } = [ B]{ d} Finally, he global sess veco ae given by: { σ } = [ D][ B]{ d} (1) () (3) (4) (5) (6) (7) Numeical example: In his example, an axisymmeic gas ubine disc was analyzed. Noncommecial ANSYS Wokbench was used o conduc he finie elemen 3117 analysis. The hickness of he disc vaies along he adial diecion. The empeaue vaies along he adial diecion as well. Due o he empeaue vaiaion, Young modulus, Poisson aio and hemal coefficien change as funcion of he empeaue. Because he empeaue changes as funcion of adius, hen Young modulus, Poisson aio and he hemal coefficien can also be expessed as funcion of adius. Because analyical soluions of such a poblem is no available as aleady explained, finie elemen analysis was caied ou o obain he soluions. Geomey and maeial popeies: The disc is axisymmeic. The geomey of he disc is shown in Fig. 4. Fo convenience, he change of empeaue, Young Modulus, Poisson aio and hemal coefficien is assumed linea. The densiy of 80 kg/m 3 is assumed o be consan. The seady sae empeaue a he oue bounday is equal o he empeaue of high empeaue gas. Based on he daa given by Baack and Domas (1976), Liu e al. (00), Claudio e al. (00), Jahed e al. (005), Mauhi e al. (01) and Elhefny and Guozhu (01), his empeaue has a ange of 550-900 C. In his sudy, a value of 800 C is used. Thoughou he disc, he empeaue gadually deceases as he adius ges close o he inne bounday. The empeaue houghou he disc neve eaches he same value as a cooling sysem is applied inside he disc. Based on daa published by Baack and Domas (1976) and Jahed e al. (005) and ecenly by Mauhi e al. (01), he empeaue a he inne bounday has a ange of 450-500 C. In his sudy, a value of 500 C is used. Following he afoemenioned bounday empeaues along wih assumpion ha he empeaue changehoughou he disc is linea, he following elaion is used o expess he empeaue change: T () = 3 + 34; in mm (8) The empeaue disibuion is shown in Fig. 5. Due o he empeaue vaiaion, Young Modulus, poisson aio and hemal coefficien vay houghou he disc as follows:
Fig. 5: Tempeaue disibuion houghou he disc Fig. 6: Young modulus vaiaion houghou he disc Fig. 8: Themal coefficien vaiaion houghou he disc Fig. 7: Poisson s aio vaiaion houghou he disc E ( T ) = - 0.05 T + 00; E in GPa (9) ν ( T ) = (1.3333x10-4 ) T + 0.3 (30) α (T) = (6.6667x10-9 ) T + (10.6667x10-6 ) (31) As he empeaue is a funcion of adius, Eq. (9) o (31) can be expessed as funcions of adius, such ha E, v and α ae given by: 3118 E () = - 0.15 + 188.5; in mm, E in GPa (3) ν () = (4.0816 x 10-4 ) + 0.645; in mm (33) α () = (.0408 x 10-8 ) + (1.14 x 10-6 ); in mm (34) The change of Young modulus, Poisson aio and hemal coefficien is shown in Fig. 6 o 8. Elemen ype, loads and bounday condiions: Axisymmeic model is used due o he axisymmeic geomey of he disc. Fuhemoe, because he model is symmeic in longiudinal diecion, hen half of he secion aea was used. Shink fi wih he oo shaf causes a adial foce in ouwad diecion on he inne suface of he disc. This foce esuls in 300 MPa compessive pessues on he inne suface of he disc. The oaion of he disc causes a cenifugal body foce in ouwad diecion. The oaion velociy is 15,000 pm = 1570 ad/s. The
Fig. 9: FEM model wih he loads and bounday condiions (a) absolue value of he adial sess occus a he inne suface of he disc. As he adius inceases, he displacemen and sess decease unil i eaches he minimum value on he oue suface of he disc. An abup incease of he sess disibuion occus a he smalles hickness of disc as sess concenaion occus hee. The ounded-off value of he minimum sess is zeo, which saisfies zeo pessue bounday condiion on he oue suface of he disc. (b) Fig. 10: (a) Coase and (b) efined meshings of he model blades aached on he oue suface of he oaing disc causes a cenifugal foce in he ouwad diecion on he oue suface of he disc. This cenifugal foce esuls in 500 MPa ensile pessues on he oue suface of he disc. Fuhemoe, high empeaue of he ho blades is ansfeed by conducion hough he disc, causing linea empeaue vaiaion hough he disc adius. This esuls in linealy vaiable hemal load. As bounday condiion, ficionless suppo (i.e., olles) is pu on he disc side which funcions as symmey plane. Thus, no displacemen pependicula o his plane is allowed, ye displacemen along he plane is allowed. The model is shown in Fig. 9. Meshing: Coase and efined meshing qualiies wee applied o he model as shown in Fig. 10. Fo boh, nodes ae only a he elemen cones. RESULTS AND DISCUSSION Case 1: The disc is subjeced o shink fi load only: By aking only he shink fi load ino accoun, he adial displacemen and sess ae shown in Fig. 11. The displacemen and sess have ouwad diecion due o he shink fi load coming fom he shaf. The maximum displacemen as well as he maximum 3119 Case : The disc is subjeced o oaional body load only: The cenifugal body foce causes adial displacemen and sess in ouwad diecion, as shown in Fig. 1. I is shown ha he displacemen cuve follows he hickness conou of he disc. Inceasing hickness ends o esul in inceasing displacemen. The maximum displacemen occus a he inne suface of he disc wheeas he minimum occus a he oue suface. The maximum adial sess occus a he middle of he disc. Case 3: The disc is subjeced o oaional blade load only: Beside he oaional body load, he oaing aached blades cause addiional cenifugal load woking on he oue suface of he disc. This load causes displacemen and sess in ouwad diecion, as shown in Fig. 13. The maximum displacemen occus a he oue suface of he disc, wheeas he minimum occus nea he hub of he disc. Lage adial sess occus as he adius inceases. Wih he given geomey of he disc, he maximum sess occus a he smalles hickness of he disc as sess concenaion occus hee. Case 4: The disc is subjeced o hemal load only: The hemal load causes adial displacemen and sess as shown in Fig. 14. The displacemen has negaive values a mos of he adius values. The negaive values epesen displacemen in inwad diecion. Nea he oue suface, he displacemen has posiive values, which means ha he displacemen has ouwad
(a) Fig. 11: (a) Radial displacemen (mm) and (b) sess (MPa) due o shink fi only (b) (a) 310
Fig. 1: (a) Radial displacemen (mm) and (b) sess (MPa) due o oaional body load only (b) (a) Fig. 13: (a) Radial displacemen (mm) and (b) sess (MPa) due o oaional body load only (b) 311
(a) Fig. 14: (a) Radial displacemen (mm) and (b) sess (MPa) due o hemal load only (b) Fig. 15: Radial displacemen due o combined loads (mm) 31
(a) Fig. 16: Radial sess disibuion, (a) wih coase meshing, (b) wih efined meshing diecion. Thus, due o he hemal load, he disc expands o inwad and ouwad diecion. The adial sess has ouwad diecion. The ounded-off sess a he inne and oue suface of he disc is zeo. The maximum sess occus a he middle of he disc. Case 5: The disc is subjeced o combined loads: By combining all he loads, he adial displacamen of he disc wih efined meshingis shown in Fig. 15. Lage adial displacemen occus a he inne and oue suface of he disc. The displacemens a he inne and oue suface ae 1.31 and 1.33 mm, especively. The (b) 313 minimum displacemen of 1.136 mm occus a he middle of he disc. The adial sess a he inne suface is negaive, which means ha i is compessive wih ouwad diecion. As he adius incease, his compessive sess deceases unil i eaches zeo a some adius value, hen i sas being posiive, which means ha i is ensile wih ouwad diecion. Nea he smalles hickness of he disc, he maximum sess occus. The sessa he inne and oue suface ae 99. MPa (compessive, ouwad diecion) and 500 MPa (ensile, ouwad diecion) especively.
Table 1: Maximum adial displacemen and sess Maximum displacemen ---------------------------------------------------------- Maximum adial sess ----------------------------------------------------------------------- Load Value (mm) Posiion Value (MPa) Posiion Shink fi load 0.51 Inne suface 99.80 Inne suface Roaional body load 0.8 Inne suface 19.0 Middle, nea he smalles hickness Roaional blade load 0.71 Oue suface 104.90 Middle, nea he smalles hickness Themal load 0.11 Middle of he disc 190.68 Middle, nea he smalles hickness Combined loads 1.33 Oue suface 145.90 Middle, nea he smalles hickness Figue 16 shows he adial sess disibuion by using boh he coase and efined meshing. I is shown ha diffeen plo go nea he smalles hickness of he disc whee sess concenaion occus. The maximum values of he adial displacemen and sess due o each load as well as combined loads ae shown in Table 1. I is shown ha he oaional blade load conibues he mos o he oal displacemen and sess. To educe is value, educing he weigh of he blades and/o he oo speed can be poposed. The maximum displacemen occus a he oue suface, ye i is only slighly lage han ha a he inne suface. Reducing he oaional blade load can be poposed o educe he displacemen a he oue suface, as excessive displacemen a ha pa may cause ubbing agains he saionay pas. Assuming ha he shink fi is fixed, educing he oo speed will educe he displacemen a he inne suface. The oo speed should no exceed a ceain speed which causes looseness beween he shaf and he disc hub. The maximum oal sess occus a he middle, nea he smalles hickness of he disc. Thickening his pa can be poposed o educe he maximum oal sess as i was shown ha sess concenaion occus hee. CONCLUSION Analyical soluion of oaing disc wih vaiable hickness and nonhomogeneous maeial popeies can no be obained because hee ae many vaiable paamees in he coefficiens of is govening diffeenial equaion. Fo his eason, numeous numeical appoaches have been poposed o obain he appoximae soluions. One of he appoaches having been widely used ecenly is FEM. Mos oaing discs used in applicaions ae ax symmeic. Theefoe, ax symmeic elemen is he mos economical bu adequae o use fo he FE analysis. A ubine disc was analyzed as an example in his sudy. Seveal loads wee applied. I was shown ha maximum displacemen occus a he bounday of he disc, eihe a he oue bounday o he inne bounday, depending on he loadings. The maximum adial sess occus a he aea in he middle of he disc which has he smalles hickness. Fuhemoe, he analysis showed ha each load gives diffeen conibuion o he oal adial displacemen and sess of he disc. The oaional blade load was shown o give he lages conibuion. Howeve, his does no apply o 314 all ubine discs as i depends on he values of diffeen loads fo any specific ubine disc. The adial displacemen and sess in a disc wih vaiable hickness ae affeced by he conou of he hickness vaiaion. The middle aea of he disc is poposed o have smalle hickness, bu should no be oo hin as sess concenaion will occu hee. In he sess analysis, a quie fine meshing paiculaly in aea wih sess concenaion is equied o avoid inaccuacy of he sess calculaion due o concenaed sess which canno be spead well o suounding aea. NOMENCLATURE : Displacemen : Radial sain : Tangenial sain : Radial sess : Tangenial sess : Specific weigh : Gaviaional acceleaion : Densiy : Young modulus : Poisson aio : Themal coefficien h : Thickness of he disc : Radius of he disc : Inne adius of he disc : Oue adius of he disc : Shape funcion : Elemen suface foce veco : Elemen body foce veco : Elemen hemal foce veco : Suface foce veco elemen in adial diecion : Suface foce veco elemen in axial diecion : Body foce veco elemen in adial diecion : Body foce veco elemen in axial diecion : B maix : Sess-sain (consiuive) maix : Elemen sifness maix : Nodal displacemen veco : Displacemen funcion veco : Sain veco : Sess veco
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