The Complementary Functions Method (CFM) Solution to the Elastic Analysis of Polar Orthotropic Rotating Discs

Transcription

1 J. Appl. Comput. Mech., 4(3) (08) 6-30 DOI: 0.055/JACM ISSN: jacm.scu.ac.i The Complementay Functions Method (CFM) Solution to the lastic Analysis of Pola Othotopic Rotating Discs Vebil Yıldıım Depatment of Mechanical ngineeing, Çukuova Univesity, Adana, 0330, Tukey, Received August 30 07; Revised Octobe 3 07; Accepted fo publication Novembe Coesponding autho: Vebil Yıldıım, vebil@cu.edu.t Copyight 08 Shahid Chaman Univesity of Ahvaz. All ights eseved. Abstact. This study pimaily deals with intoducing an efficient numeical technique called the Complementay Functions Method (CFM) fo the solutions of the initial value poblem fo the linea elastic analysis of anisotopic otating unifom discs. To bing the pefomance of the method to light, fist, closed fom fomulas ae deived fo such discs. The govening equation of the poblem at stake is solved analytically with the help of the ule-cauchy technique unde thee types of bounday conditions namely fee-fee, fixed-fee, and fixed-guided constaints. Secondly, the CFM is applied to the same poblem. It was found that both numeical and analytical esults coincide with each othe up to a desied numeical accuacy. Thid, afte veifying the esults with the liteatue, a paametic study with CFM on the elastic behavio of discs made up of five diffeent mateials which physically exist is pefomed. And finally, by using hypothetically chosen anisotopy degees fom 0.3 though 5, the effects of the anisotopy on the elastic esponse of such stuctues ae investigated analytically. Useful gaphs ae povided fo eades. Keywods: Initial value poblem (IVP); xact elasticity solution; Pola othotopic; Rotating disc.. Intoduction Rotating discs ae essential stuctual components and continue to be used in a vaiety of engineeing applications. Fo a vey long time, eseaches have paid close attention to the stess and defomation analyses of stationay and otating discs with unifom and vaiable thickness in mechanical engineeing designs. These studies oiginated with the application of isotopic and homogeneous mateials as a disc's pincipal mateial. In pocess of time, enginees became familia with anisotopic mateials [-3] which enable enginees to incease the citical and bust speeds of such discs [4-9]. Tang [4] obtained closed-fom stess solutions fo otating anisotopic discs with constant thickness fo thee cases: a solid disc, a disc mounted on a cicula igid shaft, and a disc with a cicula hole at the cente. Muthy and Sheboune [5] pesented analytical solutions fo otating stess-fee anisotopic annula discs with vaiable thickness and a disc mounted on a cicula igid shaft (fixed-fee). Reddy and Sinath [6] obtained closed-fom solutions fo stesses and displacement in an anisotopic otating cicula disc of vaiable thickness and vaiable density. They found that the stesses and displacement wee lowe when the mass density of mateial of the disc inceased adially. They also pesented gaphically the degee of anisotopy and density vaiation on stesses and displacement. Chang [7] obtained closed-fom stesses in a otating disc o cylinde made of othotopic mateial. He employed a semi-invese technique fo the mateial consideed to be linea elastic and macoscopically homogeneous. Chang [8], in his anothe study, analytically obtained the stesses in otating discs o cylindes made of geneal othotopic mateial. The adial and cicumfeential stesses in a otating cicula disc wee functions of adial coodinates only, and shea stess was equal to zeo in Chang s [8] study. By using a laminated plate theoy on layeed plates with extension-bending coupling, Bet [9] obtained appoximate solutions with stess-fee boundaies. Guushanka [0] Published Online: June 3 08

2 CFM Solution to the lastic Analysis of Pola Othotopic Rotating Discs 7 studied themal stesses in a otating nonhomogeneous anisotopic disc of vaying thickness and density. Chistensen and Wu [] used anisotopic mateials in the optimal design of flywheels. By defining the displacement field by means of both an algebaic and tigonometic seies, Belingadi et al [] appoximately calculated the stess distibution in otating discs made of othotopic mateial. Fo unifom discs with and without a cental hole, Belingadi et al. [] concluded that the pevious analytical solution fo a disc with a cental hole could give ise to unacceptable esults. Genta and Gola [3] developed a paticula closed-fom solution fo the constant thickness disc with a cental hole made of an othotopic mateial. Lekhnitskii [4] poposed a closed-fom solution fo the stess analysis in othotopic disc and cylindes unde pessue. lishakoff [5] studied buckling of pola othotopic cicula plates esting on an elastic foundation in the context of Rayleigh's method. Oilin et al. [6] calculated the bust speed of aeo engine disks. mploying the classical laminated plate theoy, Tütüncü [7] detemined the stesses and defomations esulting fom centifugal foces in otating especially othotopic cicula plates. Tütüncü [7] also studied the effect of anisotopy with a stiffness atio, which was defined as the atio of cicumfeential stiffness to adial stiffness. The plate was assumed to be igidly fixed to a concentic od allowing no defomation in its cental egion. In Tütüncü s [7] study, the oute bounday was fee of any constaints, o the plate was placed in a stiff casing which pevented adial defomation. Anoldi et al. [8] addessed two majo aspects of failue analysis, that is, the static and cyclic limit (bust) speeds fo composite disks with constant thickness subjected to both intenal pessue loading (as one would see in a hydo bust test) and pue otation (as in the case of a fee spinning disk). They gaphically displayed thei esults (inteaction diagams) in a designe fiendly fomat. Zhou and Ogawa [9] obtained elastic solutions of a otating solid disc made of a cubic anisotopic mateial using diect displacement method. Displacement, stain, and stess distibutions within the disc wee expessed as simple functions of pola coodinates in the study. Çallıoğlu [0] conducted a stess analysis on a otating hollow disc made of ectilinealy glass-fibe/epoxy pepeg unde themal loading. In anothe study, Çallıoğlu [] caied out a themal stess analysis of cuvilinealy othotopic otating discs. Çallıoğlu et al. [, 3] analytically investigated the elasticplastic stess analysis of a cuvilinealy othotopic otating annula disc fo stain hadening mateial behavio. Saye et al. [4] handled themoelastic stess analysis in a themoplastic cuvilinealy fibe einfoced composite disk. Tahani et al. [5] analyzed defomation and stess fields in cicumfeentially fibe-einfoced composite otating disks based on a semi-analytical method. Koo [6, 7] studied vibation analysis and citical speeds of pola othotopic annula discs and composite laminated otating discs. Alexandova and Vila Real [8] offeed a mathematical model fo displacement and stess fields in elasticpefectly plastic anisotopic annula disks. Nie et al. [9] investigated the mateial tailoing poblem fo a otating disc composed of a adially inhomogeneous mateial to attain a constant hoop stess, a constant adial stess, o a constant in-plane shea stess thoughout the disc. By the development of technology, and because of thei impeccable heat-esistance featues, othotopic mateials wee supplanted by functionally gaded mateials (FGM). A functionally gaded mateial may also be modelled as a non-homogeneous othotopic mateial [30-38]. Zeng et al. [38] consideed vaiable thickness discs, while othes concentated on unifom discs [30-37]. In this wok, the elastostatic behavio of pola othotopic otating annula discs unde thee diffeent bounday conditions is studied both analytically and numeically. The complementay functions method (CFM) is intoduced fo the numeical solution technique. The pesent analytical and numeical esults ae veified with the exact and numeical ones in the available liteatue. A compehensive paametic study is conducted to conside the effects of pola othotopic mateial types which physically exist, bounday conditions, and anisotopy degee on the adial vaiation of the adial displacement, adial stess, hoop stess, and von-mises equivalent stess. Moeove, basic chaacteistics and each step of the CFM ae moe clealy explained in detail though the poblem coveed in the pesent study. In these espects, this study also offes a distinguished analysis in the elated ealm.. Govening quation By epesenting the deivatives of the elated quantity with espect to the adial coodinate by the pime symbol, unde the axisymmetic loading and plain-stess conditions, the stain- displacement elations in the infinitesimal theoy of elasticity, u, ae witten as: ij whee ij u indicates the adial displacement, and u, u () and stand fo the adial and tangential stain components, espectively. The stain-displacement elations in q. () ae used fo both isotopic and pola othotopic mateials in the linea elastic analyses. By using the plain stess assumption fo pola othotopic mateials, the stess-stain elations, that is, Hooke s law, take the following fom: u '( ) ( ) Q Q ( ) u ( ) ( ) Q Q ( ) whee Q tems ae the tansfomed on-axis in-plane stiffness tems ( 0) o the elements of the tansfomed in-plane ij stiffness matix of a lamina when a ply is otated by an angle about the adial coodinate in case coss-ply symmetic laminates with espect to the adial coodinate o balanced symmetic laminates (see Appendix). The elements of the Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30 ()

3 8 Vebil Yıldıım, Vol. 4, No. 3, 08 tansfomed off-axis in-plane stiffness tems 0 ae to be evisited in Section 5. In q. () Q and Q ae efeed to as the adial stiffness and the cicumfeential stiffness, espectively. In q. (), and also imply the adial and hoop stesses; and come to mean Young s modulus in the and θ diections, espectively. Poisson s atios, v and v, ae not independent but ae elated to each othe as follows: v v (3) quilibium equation, in case a unifom disc o a cicula annulus otates at a constant angula velocity,, is (Fig. ) whee denotes the mateial density. By substituting stesses in q. () and the fist deivative of the adial stess with espect to the adial coodinate into the equilibium equation given by q. (4) and eplacing stain components with thei countepat adial displacement expessions given in q. (), the following second ode Navie diffeential equation with constant coefficients is deived in tems of the adial displacement: u u Q 0 u (5) whee the atio of denotes the anisotopy/pola degee of the disc mateial and indicates the atio of cicumfeential stiffness to the adial stiffness. In the pesent study, an additional stiffness atio,, is defined to make the paametic study easie and plausible as follows: Q v v Q Q v Q v (4) (6). Analytical Solution By using ule-cauchy solution technique, the govening equation with constant coefficients in q. (5) is solved as follows: whee Q u C C C C, Q 9 In the absence of both the inne and oute pessues ( pa pb 0), the bounday conditions ae to be used in the detemination of integation constants in q. (7) to conside just centifugal foce effects. Let a, and b be the inne and oute adii of the disc, espectively (Fig. ). As easily guessed, taction-fee bounday conditions fo otating discs (Fig. c) ae studied moe extensively in the available liteatue, but the fixed-guided constaints have attacted less attention. Afte detemining the integation constants C and C accoding to the bounday conditions (Fig. ), the adial displacement is obtained in the fist place. Substitution of the fist deivative and the adial displacement in q. (7) into Hooke s law in q. () gives both the exact adial and hoop stesses. (7) (8) (a) (b) (c) Fig.. a) A otating disc with fixed-guided sufaces, b) A otating disc mounted on a otating igid shaft, c) A otating cicula annulus Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

4 CFM Solution to the lastic Analysis of Pola Othotopic Rotating Discs 9 u Q u Q C C Q C C Q Q 3 u Q u Q C C Q C C Q Q 3 Integation constants in q. (7) ae found fo fee-fee suppoted discs a b fo fixed-fee suppoted discs u a b and finally fo fixed-guided u a u b C C C 6 6 a b b a 3 3 C a b 3a b 6 6 a b 0, 0 as (Fig. b), a b b b a a b a b a 0, 0 suppoted discs as follows (Fig. a): C b a a 6 6 a b b 0, 0 as (Fig. c), 6 6 b a C a b It may be noted that solution (7) becomes unbounded when 9 0 fo odinay othotopic mateials. (9) (0) () (). Numeical Solution with CFM The complementay functions method (CFM) is one of the methods pofitably and tuthfully used fo the solution of initial value poblems (IVP) [39-40]. Its cental advantage ove many methods is that it allows stable accuate numeical solution of odinay diffeential equation sets even with vaiable coefficients up to any desied pecision afte conveting the two-point bounday value poblem (BVP) such as q. (5) into IVP such as the following. ( ) u '( ) u ( ) 0 '( ) ( ) ( ) ( ) ( ) ( ) Unde axisymmetic loading and plane-stess conditions, qs. (5) and (3) goven the elastostatic behavio of unifom discs made of pola othotopic mateials, which otate at a constant angula velocity,, about the axis passing though its cente. As one may easily guess, fo isotopic and homogeneous mateials:. The authos have also employed this method diectly o used it as an assistive method in some static, buckling, and dynamic poblems [4-46]. In the pesent study, CFM is to be fist diectly applied to the elastic behavio of otating pola othotopic unifom discs. In the IVP, it is necessay to tansfom a second ode diffeential equation into two fist-ode diffeential equation sets. The pincipal vaiables in the IVP may be chosen in diffeent ways. Fo example, in geneal, the IVP tansfomation with vaiables z u and z u is fequently adopted instead of using vaiables z u and z, which is the most convenient fom of the poblem available, as in the pesent study. To explain CFM, the fequently chosen vaiables, z u and z u ae used as the pincipal vaiables of the IVP. This labelling also means that z u z and z u. To tansfom BVP into IVP, the govening equation, q. (5), is ewitten in tems of only z, z, and thei fist deivatives. Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30 (3)

5 0 Vebil Yıldıım, Vol. 4, No. 3, 08 Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30 z z z (4) Q Then, the following, which descibes the IVP fom of q. (5), may be easily obtained: 0 0 z '( x ) z ( x ) (5) z '( x ) z ( x ) Q whee the last tem at the ight side is called the nonhomogeneous tem. Unde the chosen physical bounday conditions, in CFM, the geneal solution of q. (5) ove the inteval [a, b] is given by [39-40]: u x u x b u x b u x (6) whee 0 0 u x is the paticula solution of q.(5), while u x and u x ae the homogeneous solutions of the same equation, which should be linealy independent. As expected, b and b ae constants which should be detemined fom the given physical bounday conditions of the chosen poblem. In all the fou solution stages in CFM, all tems at the ight side u 0 x, u x, u x, b and b ae to be detemined. The pescibed bounday conditions consisting of eithe initial zeo conditions o initial Konecke delta conditions ae fist, u x with the help of eithe the paticula o the homogeneous solution of q. (5) in the used to find u x u x and 0 fist thee stages. In the final stage, the given physical bounday conditions such as u a 0 and 0 u b ae consideed to detemine unknown constants b and b by using the solution given in q. (6). Then, the numeical solution pocedue becomes complete. In the fist stage let z u 0, z u 0, solve q. (5) with non-homogeneous tem unde the fist pescibed bounday conditions that is unde zeo initial conditions, z ( a) 0, z ( a) 0 and get the paticula solution, u 0 x, and its deivative u 0 x, at the same time. In the second stage, let z u, z u. Solution of q. (5) without non-homogeneous tem unde the second pescibed bounday conditions z ( a), z ( a) 0 gives the fist homogeneous solution ux and its deivative u x, simultaneously. In the thid stage, afte letting z u, z u, again solve q. (5) without non-homogeneous tem unde pescibed bounday conditions z ( a) 0, z ( a) to each the last homogeneous solution u x and its deivative u x, at once. Afte imposing u 0 x, ux and u x into the geneal solution, q. (6), the physical bounday conditions of the diffeential equations such as q. (7) should now be applied to detemine the emaining unknown coefficients b and b in a staight fowad manne. Fo example, fo fixed-guided discs (Fig. ) the following may be witten. u a u 0 a bu a bu a 0 (7) u b u b b u b b u b This gives: 0 0 u ( a) u ( a) b u ( a) o u( b) u( b) b u o ( b) This is a linea equation set of ode two; theefoe, the solution may be witten eadily in an explicit fom as follows: u o ( a) u( a) u o ( b) u( b) u o ( a) u( b) uo ( b) u( a) b u( a) u( a) u( a) u( b) u( b) u( a) u ( b) u ( b) u( a) u o ( a) u( b) u o ( b) u o ( b) u( a) uo ( a) u( b) b u( a) u( a) u( a) u( b) u( b) u( a) u ( b) u ( b) The solution pocess comes to an end by substitution of and in the geneal solution of the adial displacement, q. (6). As undestood fom the above, z u ( ) and z u ( ) ae concuently obtained at the end of the fou stages of CFM. This may be helpful fo much moe advanced studies. Afte detemining the adial displacement and its deivatives as explained above, the adial stess and the tangential stess (hoop stess) ae achieved fom Hooke s law in q. (). As a final emak, it may be noted that, in the fist thee stages, which equie the numeical solution of the fist degee odinay (8) (9)

6 CFM Solution to the lastic Analysis of Pola Othotopic Rotating Discs diffeential equations which may be eithe nonhomogeneous o homogeneous fom, a pocedue like RK4 o RK5 fom Runge- Kutta family may be used. 3. Veification of the Results As a compaative example, a disc with a=cm and b=5cm is chosen [35]. The disc is made of an injection molded Nylon 6 composite containing 40wt% shot glass fibe (MAT-5). The mateial popeties ae given in Table. The dimensionless elastic stess and displacements ae defined as:,, u u (0) 3 b b b b b b Compaisons of dimensionless adial displacements, adial stesses, and hoop stesses with the liteatue fo MAT-5 ae pesented in Table fo taction-fee discs. Table depicts a pefect synchonization of the esults. Table also veifies that RK4 is basic but a sufficiently poweful technique in the numeical solution pocedue to be used in the solution of these types of poblems. Table. Anisotopic mateials and thei popeties Label (GPa) (GPa) ( kg ) Polycabonate [6] MAT GFRP (-Glass/poxy) [6] MAT CFRP (T300/N508) [6] MAT A Glass Fibe/poxy Pepeg [-3] MAT An injection molded Nylon 6 composite containing 40wt% shot glass fibe [-3, 35] MAT Table. Compaison of dimensionless adial displacement, adial stess, and hoop stess with the liteatue fo MAT-5 unde stess-fee bounday condition a/b 4. xamples with CFM xact (Pesent) xact [35] Numeical [35] Numeical (Pesent) Numeical [36] _ _ _ In this section, some poblems elated to unifom otating discs made of physical anisotopic mateials whose popeties ae given in Table ae to be consideed. The anisotopy degees of the physical mateials ae pesented in Table 3. The same inne and oute adii, a=cm and b=5cm, ae to be used fo the disc geomety as in the pevious section. The equivalent stess based on von-mises failue citeia unde plane-stess assumption is defined as: () eq Numeical values of the dimensionless tangential stesses at the inne suface fo diffeent types of physical mateials and Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

7 Vebil Yıldıım, Vol. 4, No. 3, 08 bounday conditions ae exhibited in Table 4. This table shows that the tangential stesses at the inne suface ae highe than the oute suface unde fee-fee conditions. The minimum cicumfeential stesses ae obtained at the inne suface fo the disk having fixed-guided sufaces. Table 3. and atios of physical mateials pesented in Table = = / MAT- 0.3 MAT MAT MAT MAT Table 4. Dimensionless tangential stesses at the inne suface fo diffeent mateials and bounday conditions _ () Fee-Fee Fixed-Fee Fixed-Guided MAT MAT MAT MAT MAT Stess-Fee Boundaies Vaiations of the dimensionless adial displacement, adial, hoop, and von-mises equivalent stesses along the adial diection with fee-fee bounday condition and physical anisotopic mateials ae illustated in Fig.. The following ae obseved in Fig. : * The mateials MAT-, MAT-4, and MAT-5, which have, show all symmetic wave-like adial displacement vaiations along the adial diection. This was also obseved in Peng and Li s study [35]. When <, this wave-like vaiation becomes too small to be noticed as in MAT- and MAT-3. Fig.. Vaiation of the elastic quantities with mateial types fo fee-fee bounday conditions * MAT-3 offes maximum adial displacements. * Maximum adial displacements ae mostly at the inne suface fo all mateials except fo MAT-4. * Radial stesses at both sufaces vanish. This situation is consistent with the suface fee bounday conditions. In the Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

8 CFM Solution to the lastic Analysis of Pola Othotopic Rotating Discs 3 intemediate sufaces, adial stesses ae all in tension. Maximum adial stess occus aound the middle suface slightly close to the inne suface fo all types of mateials. Although they ae vey close to each othe, the adial stesses ae highe to some extent fo othotopic mateials with than the othes. * Maximum adial stess is aound one sixth of the maximum tangential stess. * The tangential stesses and equivalent stesses which ae all in tension show familia adial vaiation. They incease monolithically by inceasing the adial coodinate. The maximum hoop stess and equivalent stess ae at the inne suface. * The cuves of equivalent stesses ae almost in the same chaacte of the cuves of hoop stesses, since cicumfeential stesses ae dominant fo fee-fee boundaies. * Maximum hoop and equivalent stesses ae obseved in mateials MAT- and MAT-3, whose othotopic degees ae less than unit, <. 4. Fixed-Fee Boundaies Vaiations of the dimensionless adial displacement, adial, hoop, and von-mises equivalent stesses along the adial diection with fixed-fee bounday condition and the mateials in Table ae illustated in Fig. 3. The following conclusions may be dawn fom Fig. 3: * Radial displacements diminish at the inne suface, while the adial stesses vanish at the oute suface in accodance with the applied bounday conditions. * The adial displacements pogessively incease fom the inne suface towads the oute suface. * Both the adial stesses and equivalent stesses decease by inceasing adial coodinate. * Fo this bounday condition, the influence of the adial stesses ove the hoop stesses seems to be inceased. Fig. 3. Vaiation of the elastic quantities with mateial types fo fixed-fee bounday conditions * The anisotopic mateials having geate anisotopy degee exhibit much highe adial displacements. In othe wods, MAT-5 has the highest adial displacement, followed by MAT-4, MAT-, MAT-, and MAT-3, espectively. * Vaiations of the hoop stesses along the adial diection may be diffeent with espect to the type of mateials. Fo example, the maximum hoop stess is aound the middle suface fo both MAT-5 and MAT-4. Among them, MAT-5 has the maximum hoop stess. The maximum hoop stess in MAT- is obseved at the egion vey close to the inne suface. Fo MAT- and MAT-3, the diffeences in the magnitudes of the hoop stesses at both sufaces ae vey small. * MAT-5 shows almost symmetic behavio in the vaiation of the hoop stesses. * The maximum adial stesses occu at the inne suface. MAT-3 having the smallest and atios in all mateials in Table, offes the highest maximum adial stess, followed by MAT-, MAT-4, and MAT-, espectively. MAT-5 has the smallest maximum adial stess. * It may be obseved that when the anisotopy degee inceases, the numeical value of the maximum adial stess Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

9 4 deceases. Vebil Yıldıım, Vol. 4, No. 3, 08 * The vaiations of both the adial stess and equivalent stess cuves show simila chaacteistics with the same ode of matei al types. 4.3 Fixed-Fixed Boundaies Vaiations of the dimensionless adial displacement, adial, hoop, and von-mises equivalent stesses along the adial coodinate with fixed-fixed bounday condition and the mateials in Table ae illustated in Fig. 4. The outcomes that can be extacted fom Fig. 4 may be as follows: * The adial stesses vanish at both the inne and oute sufaces as compaed with the bounday conditions. * The maximum adial displacements occu at the vicinity of the middle suface. MAT-5 followed by MAT-4, MAT-, MAT-, and MAT-3, espectively, gives the highest adial displacement. * MAT-3 pesents almost no adial displacement. * The absolute values of the maximum adial stess ae almost twice of the maximum hoop stesses. * MAT- followed by MAT-3 has the highest adial stesses. * Radial stesses may have almost symmetic vaiation with espect to the middle suface. They may be in both tension and compession. Tensions ae located at the inne suface of the disc. * The tension-compession behavio is also obseved in the vaiation of hoop stesses. Howeve, this is not full symmetic. The maximum hoop stesses ae eithe at the inne suface o at the vicinity of the inne suface of the disc. Compessions occu at the oute suface. * The highest hoop stesses ae obseved in MAT-5 followed by MAT-, MAT-4, MAT-, and MAT-3, espectively. * MAT-3 has almost stable vaiation fo the hoop stesses. * The vaiation of the equivalent stesses takes almost a symmetic fom due to the existence of compessions at the oute suface fo both the adial and hoop stesses. * MAT-3 followed by MAT-, MAT-, MAT-4, and MAT-5, espectively, gives the smallest equivalent stesses at the middle suface of the disc. MAT-3 followed by MAT-, MAT-, MAT-, MAT-4, and MAT-5, espectively, also shows the highest equivalent stesses. Fig. 4. Vaiation of the elastic quantities with mateial types fo fixed-fixed bounday conditions 5. Investigation of Anisotopy ffects To study the effect of the fibe oientation angle, econside Q ij which is an element of the tansfomed in-plane stiffness matix of a lamina when a ply is otated by an angle about the adial axis. By efeing to diections along the fibe Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

10 CFM Solution to the lastic Analysis of Pola Othotopic Rotating Discs 5 and tansvese to the fibe by subscipts x and y, espectively, and denoting the in-plane shea by subscipt s, the tansfomed off-axis in-plane stiffness tems ae given in Appendix ( 0). As expected, diffeent tansfomed stiffness tems and eventually diffeent esultant stesses though the thickness of a laminate may be developed accoding to the chosen anisotopic mateial types and/o fibe oientations of each lamina fo the same mateial. Stess discontinuities at ply intefaces may also occu at some fibe oientations in each ply. As stated above, is efeed to as the anisotopy degee of the inhomogeneous mateial and indicates the atio of cicumfeential stiffness to the adial stiffness. As a numeical example, conside an anisotopic mateial (CFRP (T300/N508) of = 8.0 ; = 0.3 ; = 7.7 ; = 0.8, and labelled by MAT-3) in Table. Table 5. Coesponding fibe angles fo a cetain anisotopy degee (0 90 ) ± ( ) ± ( ) Coesponding fibe angles fo a cetain anisotopy degee fo the chosen mateial ae given in Table 5 fo (0 90 ). The angles may be oiginated fom eithe the coss-ply symmetical laminates (namely all the plies ae oiented at 0 and 90 with espect to the adial coodinate axis) o the balanced symmetical laminates (plies oiented at ±, that is, fo each ply oiented + thee must exists a ply oiented at ). As stated above, the same anisotopy degees fo specially othotopic laminates may also be obtained by placing stiffenes in the vey adial and cicumfeential diections. Vaiation of the atios of the tansfomed stiffness tems with espect to the fibe oientation angle (0 90 ) is pesented in Figue 5. Fig. 5. Vaiation of the atios of the tansfomed stiffness tems with espect to the fibe oientation angle As shown in Fig. 5 and Table 6, = 9 coesponds to the fibe oientation angle of =±67.974, in othe wods, (± ) laminate configuations. As stated above, fo odinay othotopic mateials, solution (7) becomes unbounded at = 9. It is also clealy seen in Fig. 5 that the anisotopy degee cuve has an inflection point when = 9. The oveweight of cicumfeential stiffness on the adial stiffness inceases at a snail's pace fom = 0 though =. Howeve, when > the supemacy of the cicumfeential stiffness on the adial stiffness inceases by leaps and bounds. In the pesent study, fo some values of the anisotopy atios between 0.3 and 5 (0.3 5), vaiations of the adial displacements, the adial stess, and the hoop stess with the anisotopy degees and bounday conditions ae gaphically illustated in Figs. 6, 7 and 8. As stated above, the anisotopy degees ae chosen hypothetically in this section. Howeve, the values of the othe atio,, ae chosen as = 0.8 in accodance with the pola othotopic chaacte. Conside Fig. 6. * Fo fixed-guided ends, the maximum adial displacement is placed almost at the mid-way suface when <. The maximum adial displacement begins to shift towads the oute suface by inceasing anisotopy degees geate than. Radial displacement inceases by inceasing anisotopy degee fom 0.3 to 5. * Fo fixed-fee ends, the adial displacements steadily buildup by inceasing anisotopy degees towads the oute suface of the disc. * Fo fee-fee ends, while the maximum adial displacement comes about at the inne suface of the disk fo 0.3 < < 0.6, it occus at the oute suface at highe degees of > 0.6 in accodance with the wave-like vaiation with the adial coodinate. Fig. 7 shows that: * Fo fixed-guided ends, the adial stess may be eithe in tension o compession in natue. Fo both and 5, the adial stess deceases by inceasing anisotopy degees. The diffeences in the decease ae moe notable fo the anisotopy degee geate than unit. While the tension stesses ae placed at the inne suface, the compession stesses ae located at the oute suface. * Fo fixed-fee ends, adial stesses decease by inceasing anisotopy degee. The maximum adial stess occus at the inne suface. It may be concluded fom the figue that, fo anisotopy degees geate than 5, the adial stess at the inne suface stats to be diminished, wheeas the maximum adial stess begins to emege at the vicinity of the inne suface. This means that when cicumfeential stiffness is moe than five times highe than adial stiffness, the hoop fibes cay moe load Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

11 6 Vebil Yıldıım, Vol. 4, No. 3, 08 than the adial fibes; eventually, the adial stess at the inne suface is weakened. Fig. 6. Vaiation of the adial displacements with the anisotopy degees and bounday conditions (a=cm, b=0 cm) (0.3 5) * Fo fee-fee ends, inceasing anisotopy degees makes maximum adial stesses which ae close to the inne suface much smalle. The following conclusions may be achieved fom Fig. 8: * Fo fixed-guided sufaces, as the anisotopy degee inceases, the maximum hoop stesses incease. While tensions ae at the inne suface, compessions ae at the oute suface. Maximum hoop stesses ae in the vicinity of the inne suface. As the anisotopy degee inceases, the location of the maximum hoop stess shifts to the middle suface. * In both fixed-fee and fee-fee bounday conditions, hoop stesses ae all in tension. * As the anisotopy degee inceases, the magnitudes of hoop stesses incease unde all bounday conditions. * Maximum hoop stesses ae obseved at the inne suface fo fee-fee, at eithe inne o oute suface fo fixed-fee, and at eithe the vicinity of the inne o mid-way sufaces fo fixed-fixed discs. Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

12 CFM Solution to the lastic Analysis of Pola Othotopic Rotating Discs 7 Fig. 7. Vaiation of the adial stesses with the anisotopy degees and bounday conditions (a=cm, b=0 cm) (0.3 5) Fig. 8. Vaiation of the hoop stesses with the anisotopy degees and bounday conditions (a=cm, b=0 cm) (0.3 5) Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

13 8 6. Conclusion Vebil Yıldıım, Vol. 4, No. 3, 08 The fact that otating discs made of a pola othotopic mateial have not been studied compehensively was the motivation of the pesent study. In this wok, compact analytical fomulas ae offeed fo detemination of the elastic esponse of otating pola othotopic unifom discs unde thee diffeent bounday conditions. The same poblem is also oiginally solved numeically by an efficient initial value poblem solution technique called the complementay functions method (CFM). The pefect accodance is obseved between the analytical and numeical solutions. Afte validation of the esults, paametic studies by both analytical fomulas and numeical CFM esults ae conducted to study the elastic behavio of such discs made of eithe physically anisotopic o hypothetically anisotopic mateials. Quite useful esults about the geneal elastic behavio of such discs ae obtained, which ae exhibited by gaphs fo the eades. The pesent study also eveals that CFM is a poweful and an accuate numeical method to be applied to advanced studies of such stuctues. Appendix The tansfomed off-axis in-plane stiffness tems ae ( 0) whee Q c Q s Q c s Q 4c s Q 4 4 xx yy xy ss Q c s Q c s Q ( c s ) Q 4c s Q 4 4 xx yy xy ss Q s Q c Q c s Q 4c s Q 4 4 xx yy xy ss The tansfomed on-axis in-plane stiffness tems ae ( = 0) Q Q xx (A) y c cos, s sin, v yx v xy (A) yv x v v v v xy x xy y yx xy yx x v y xy xy xv yx y v xyv yx v xy x Q yy y v v v xy Q y xy yx x ss s y (A3) Refeences [] Bidgoli, A.M.M., Daneshmeh, A.R., Kolahchi, R., Analytical Bending Solution of Fully Clamped Othotopic Rectangula Plates Resting on lastic Foundations by The Finite Integal Tansfom Method, Jounal of Applied and Computational Mechanics, (), 05, [] Akano, T.T., Fakinlede, O.A., Olayiwola, P.S., Defomation Chaacteistics of Composite Stuctues, Jounal of Applied and Computational Mechanics, (3), 06, [3] Katsikadelis, J.T., Tsiatas G.C., Saint-Venant Tosion of Non-Homogeneous Anisotopic Bas, Jounal of Applied and Computational Mechanics, (), 06, [4] Tang, S., lastic Stesses in Rotating Anisotopic Discs, Intenational Jounal of Mechanical Sciences,, 969, [5] Muthy, D., Sheboune, A., lastic Stesses in Anisotopic Discs of Vaiable Thickness, Intenational Jounal of Mechanical Sciences,, 970, [6] Reddy, T.Y., Sinath, H., lastic Stesses in a Rotating Anisotopic Annula Disc of Vaiable Thickness and Vaiable Density, Intenational Jounal of Mechanical Sciences, 6(), 974, [7] Chang, C.I., A Closed-Fom Solution fo an Othotopic Rotating Disc, Jounal of Applied Mechanics, 4(4), 974, 3. [8] Chang, C.I., The Anisotopic Rotating Discs, Intenational Jounal of Mechanical Sciences, 7(6), 975, [9] Bet, C.W., Centifugal Stesses in Abitaily Laminated, Rectangula-Anisotopic Cicula Discs, Jounal of Stain Analysis fo ngineeing Design, 0, 975, [0] Guushanka, G.V., Themal Stesses in a Rotating Nonhomogeneous, Anisotopic Disc of Vaying Thickness and Density, Jounal of Stain Analysis fo ngineeing Design, 0, 975, [] Chistensen, R.M., Wu,.M., Optimal Design of Anisotopic (Fibe-Reinfoced) Flywheels, Jounal of Composite Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

14 CFM Solution to the lastic Analysis of Pola Othotopic Rotating Discs 9 Mateials,, 977, [] Belingadi, G., Genta, G., Gola, M., A Study of the Stess Distibution in Rotating, Othotopic Discs, Composites, 0(), 979, [3] Genta, G., Gola, M., The Stess Distibution in Othotopic Rotating Discs, Jounal of Applied Mechanics, 48, 98, [4] Lekhnitskii, S.G., Theoy of lasticity of an Anisotopic Body, Mi Publishes, Moscow, 98. [5] lishakoff, I., Buckling of Pola Othotopic Cicula Plates on lastic Foundation by Computeized Symbolic Algeba, Compute Methods in Applied Mechanics and ngineeing, 68(), 988, [6] Oilin, H., Oixuan, Z., Ping, W., Guicang, H., A New Method fo Calculating Bust Speed of Aeoengine Disks, ASM-9- GT-, 99, -4. [7] Tutuncu, N., ffect of Anisotopy on Stesses in Rotating Discs, Intenational Jounal of Mechanical Sciences, 37, 000, [8] Anoldi, S.M., Saleeb, A.F., Al-Zoubi, R., Defomation and Life Analysis of Composite Flywheel Disk and Multi-Disk Systems, NASAffM , 00, -56. [9] Zhou, F., Ogawa, A, lastic Solutions fo a Solid Rotating Disc with Cubic Anisotopy, Jounal of Applied Mechanics, 69, 00, [0] Callioglu, H., Stess Analysis of an Othotopic Rotating Disc unde Themal Loading, Jounal of Reinfoced Plastics and Composites, 3(7), 004, [] Callioglu, H., Themal Stess Analysis of Cuvilinealy Othotopic Rotating Discs, Jounal of Themoplastic Composite Mateials, 0, 007, [] Callioglu, H., Topcu, M., Altan, G., Stess Analysis of Cuvilinealy Othotopic Rotating Discs unde Mechanical and Themal Loading, Jounal of Reinfoced Plastics and Composites, 4, 005, [3] Callioglu, H., Topcu, M., Taakçıla, A.R., lastic-plastic Stess Analysis of an Othotopic Rotating Disc, Intenational Jounal of Mechanical Sciences, 48, 006, [4] Saye, M., Topcu, M., Bektas, N.B., Taakcila, A.R., Themoelastic Stess Analysis in a Themoplastic Composite Disc, Science and ngineeing of Composite Mateials, (4), 005, [5] Tahani, M., Nosie, A., Zebajad, S.M., Defomation and Stess Analysis of Cicumfeentially Fibe-Reinfoced Composite Disks, Intenational Jounal of Solids and Stuctues, 4(9 0), 005, [6] Koo, K.N., Vibation Analysis and Citical Speeds of Pola Othotopic Annula Discs in Rotation, Composite Stuctues, 76, 006, [7] Koo, K.N., Mechanical Vibation and Citical Speeds of Rotating Composite Laminate Discs, Micosystem Technologies, 4, 008, [8] Alexandova, N., Vila Real, P.M.M., Defomation and Stess Analysis of an Anisotopic Rotating Annula Disk, Intenational Jounal fo Computational Methods in ngineeing Science and Mechanics, 9(), 008, [9] Nie, G.J., Zhong, Z., Bata, R.C., Mateial Tailoing fo Othotopic lastic Rotating Discs, Composites Science and Technology, 7, 0, [30] Duodola, J., Attia, O., Defomation and Stesses in Functionally Gaded Rotating Discs, Composites Science and Technology, 60, 000, [3] Chen, J., Ding, H., Chen, W., Thee-Dimensional Analytical Solution fo a Rotating Disc of Functionally Gaded Mateials with Tansvese Isotopy, Achive of Applied Mechanics, 77, 007, 4-5. [3] Wang, X., Sudak, L.J., Thee-Dimensional Analysis of Multi-Layeed Functionally Gaded Anisotopic Cylindical Panel unde Themomechanical Loading, Mechanics of Mateials, 40(4), 008, [33] Kansal, G., Pavez, M., Themal Stess Analysis of Othotopic Gaded Rotating Discs, Intenational Jounal of Moden ngineeing Reseach, (5), 0, [34] Lubada, V.A., On Pessuized Cuvilinealy Othotopic Cicula Disc, Cylinde and Sphee Made of Radially Nonunifom Mateial, Jounal of lasticity, 09, 0, [35] Peng, X.L., Li, X.F., lastic Analysis of Rotating Functionally Gaded Pola Othotopic Discs, Intenational Jounal of Mechanical Sciences, 60, 0, [36] Boga, C., Analytical and Numeical Axisymmetic lastic Stess Analyses of Stationay/Rotating Discs Made of Isotopic/Othotopic Functionally Gaded Mateials by The Tansfe Matix Method, Ph.D. Thesis, Depatment of Mechanical ngineeing, Çukuova Univesity: No 698, 06. [37] Kaca, I., Yıldıım, V., ffect of the Anisotopy Ratios on the xact lastic Behavio of Functionally Powe-Gaded Pola Othotopic Rotating Unifom Discs Unde Vaious Bounday Conditions, Digital Poceeding of ICOC Cappadocia, Nevsehi, Tukey, , May 8-0, 07. [38] Zheng, Y., Bahaloo, H., Mousanezhad, D., Vazii, A., Nayeb-Hashemi, H., Displacement and Stess Fields in a Functionally Gaded Fibe-Reinfoced Rotating Disk with Nonunifom Thickness and Vaiable Angula Velocity, Jounal of ngineeing Mateials and Technology, 39, 07, [39] Aktas, Z., Numeical Solutions of Two-Point Bounday Value Poblems, Ankaa, Tukey, MTU, Depatment of Compute ngineeing, 97. [40] Robets, S., Shipman, J., Fundamental Matix and Two-Point Bounday-Value Poblems, Jounal of Optimization Theoy and Applications, 8(), 979, [4] Haktanı, V., Kıal,., Diect Application of Complementay Functions Method to Axisymmetical Shells and Cylindical Vaults (Baels), Jounal of Ispata ngineeing Faculty of Akdeniz Univesity, 6, 99, Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30

15 30 Vebil Yıldıım, Vol. 4, No. 3, 08 [4] Haktanı, V., A New Method fo the lement Stiffness Matix of Abitay Plana Bas, Composite Stuctues, 5(4), 994, [43] Haktanı, V., The Complementay Functions Method fo the lement Stiffness Matix of Abitay Spatial Bas of Helicoidal Axes, Intenational Jounal fo Numeical Methods in ngineeing, 38(6), 995, [44] Yıldıım, V., Fee Vibation Analysis of Non-Cylindical Coil Spings by Combined Use of the Tansfe Matix and the Complementay Functions Methods, Communications in Numeical Methods in ngineeing, 3(6), 997, [45] Kaca, İ., Yıldıım, V., Fee Vibation/Buckling Analyses of Non-Cylindical Initially Compessed Helical Composite Spings, Mechanics Based Design of Stuctues and Machines, 44 (4), 06, [46] Yıldıım, V., Kaca, I., Intoducing a Compute Package Pogam fo lastic Analysis of Functionally Gaded Rotating Thick-Walled Annula Stuctues, Digital Poceeding of ICOC CAPPADOCIA, Nevşehi, Tukey, , May 8-0, 07. Jounal of Applied and Computational Mechanics, Vol. 4, No. 3, (08), 6-30