Magneto-Elastic Analysis of an Annular FGM Plate Based on Classical Plate Theory Using GDQ Method

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1 736 Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method Abstact Using GDQ method, the adial and cicumfeential stesses in an annula FGM plate with a unifom thickness unde a tansvese axisymmetic load is investigated. It is assumed that a unifom adial magnetic field acts on the top suface of the plate. The modulus of elasticity E and the magnetic pemeability coefficient of the plate along its thickness ae assumed to vay accoding to the volume distibution function. The Poisson s atio is consideed to be constant. Based on the classical plate theoy (CPT), equilibium equations ae deduced and the displacement fields ae detemined. The adial and cicumfeential stesses as well as tansvese and adial displacements ae obtained accodingly. The effect of volume faction function powe m on the maximum deflection in the absence and pesence of the magnetic field is also investigated. Moeove, the effect of t/a and b/a atios on displacements, stesses, induction magnetic field intensity and the esulting Loent foce ae also investigated. Accoding to the esults, fo diffeent points along the adial diection, the application of adial magnetic field to the top suface of the plate completely changes the state of stess in both tangential and adial diections, esulting in tensile and compessive stesses in these two diections. The esults also indicate that in pesence of magnetic field, the plate displacement and stess components ae loweed consideably. M. Shishesa a A. Zakipou b A. Jafaadeh c a Depatment of Mechanical Engineeing, Shahid Chaman Univesity of Ahva, Ahva, Ian addess: mshishehsa@scu.ac.i b Mechanical Engineeing Depatment, Islamic Aad Univesity, Behbahan Banch, Behbahan, Ian addess: ali.kipou0@gmail.com c Mechanical Engineeing Depatment, Amikabi Univesity of Technology, Tehan, Ian addess: jafaadeh-ayat@aut.ac.i Received In evised fom Accepted Available online Keywods Functionally Gaded Mateial, Magneto Elastic, GDQ method 1 INTRODUCTION Functionally gaded mateials (FGM) ae heteogeneous composites which ae usually a combination of ceamic and metal, o a combination of two o moe diffeent metals. Popeties of FG mateials change unifomly and continuously acoss the thickness, fom point to point,

2 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 737 accoding to a function. These mateials can be used when good themal and mechanical esistances ae equied simultaneously. Fo example, in a FGM plate made of ceamic and metal, the ceamic secto is a good insulato against heat and themal shocks while the metallic secto acts as a good esistance against mechanical loads. The unique popeties of these mateials ae due to thei composite natue and thei physical popeties which gadually change acoss the thickness of the specimen. This in effect educes the hysteesis stesses, as well as the stess concentation, with a significant impovement in mechanical popeties of the oveall mateial, especially at the ceamic and metallic junction. In ecent yeas, consideable eseach has been oiented towad FGM plates to bette undestand thei behavio unde vaious loading conditions. Bayat et al. (014) investigated the magneto-themo-mechanical esponse of a FGM annula otating disc with vaiable thickness and obseved that unlike the positive adial stesses developed in a mechanically loaded FGM disk, in a FGMM (functionally gaded magneto-elastic mateial) disk, the adial stesses due to magneto - themal load can be both tensile and compessive. Behavan Rad and Shaiyat (015) analyed a poous cicula FG plate with vaiable thickness subjected to nonaxisymmetic and non-unifom shea along with a nomal taction and a magnetic actuation. The plate was suppoted on a non-unifom Ke elastic foundation. They evaluated the effect of mateial, loading, bounday and elastic foundation on the esulting displacement, stess, Loent foce, electomagnetic stess and magnetic petubation quantities. Chi and Chung (006) studied the mechanical behavio of FGM plates unde tansvese load using a numeical method. Ma and Wang (003) studied nonlinea buckling and bending behavio of FGM cicula plates unde mechanical and themal loads based on the classical nonlinea von Kaman plate theoy. They discussed the effects of mateial powe distibution function and bounday conditions on the tempeatue distibution, nonlinea bending, citical buckling tempeatue and themal post-buckling behavio of the plate in details. Najafiadeh and Heydai (004) analyed themal buckling of FGM cicula plates unde vaious themal loads based on highe ode shea defomation theoy (HSDT) and compaed thei esults with those obtained using fist ode shea defomation theoy (FSDT) and classical plate theoy (CPT). They showed that HSDT theoy pedicts the behavio of FGM cicula plates with highe pecision compaed to FSD and CP theoies. Paveen and Reddy (1998) studied the themo elastostatic and themo-elastodynamic esponse of FGM plates subjected to vaying pessue and tempeatue loading. They showed that the combination popotion of mateials in FGM plates plays an impotant ole on detemining thei esponse. Reddy et al. (1999) analyed axisymmetic bending and stetching of FGM annula plates using the fist-ode shea defomation Mindlin plate theoy and concluded that this theoy gives good esults on FGM annula plates wheneve the Kichhoff solution is not applicable. Reddy (000) pesented a theoetical solution and finite element model of FGM ectangula plates based on thid ode shea defomation theoy (TSDT) and studied the effect of mateial distibution on deflections and stesses. Saidi et al. (009), investigated the bending and buckling behavio of FGM cicula plates, based on the thid-ode shea defomation theoy and compaed thei esults with those of fist ode shea defomation theoy. Applying finite Hankel integal tansfoms, Wang and Dai (004) obtained analytical expessions Latin Ameican Jounal of Solids and Stuctues 13 (016)

3 738 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method fo magneto dynamic stess and petubation esponse of an axial magnetic field vecto in an othotopic cylinde unde themal and mechanical shock loads. They deduced the esponse histoies of dynamic stesses and the petubation of the field vecto. Yuda and Jing (009) obtained the electodynamic equations and electomagnetic foce expessions of a cuentconducting thin plate in an electomagnetic field, based on Maxwell equations. They investigated the nonlinea sub-hamonic esonance of the thin plate with two simply suppots on opposite sides unde a mechanical live load. In thei analysis, an inconstant tansvese magnetic field load was applied as well. Ghobanpou Aani et al. (010) pesented a semi-analytical solution fo magnetothemo-elastic poblem in a functionally gaded (FG) hollow otating disks with vaiable thickness unde unifom magnetic and themal fields. They showed that imposing a magnetic field on the disk significantly deceases tensile cicumfeential stesses. Using GDQ method, Lal and Saini (015) analyed the effect of two-dimensional non-homogeneity on tansvese vibation of othotopic ectangula plates of bidiectional thickness vaiation on the basis of Kichhoff s plate theoy. They investigated the effect of non-homogeneity paametes, density paametes, thickness paamete and the aspect atio on natual fequencies, fo the fist thee modes of vibation. In anothe pape which was published in the same yea, they also investigated the effect of unifom tensile in-plane foces on the adially symmetic vibatoy chaacteistics of functionally gaded cicula plates with linealy vaying thickness along adial diection. The plate was esting on a Winkle foundation. In this eseach, the effect of a adial magnetic field on a FGM annula plate is analyed. The plate is subjected to a tansvese mechanical load as well as a tansvese load f deduced fom the adial magnetic field applied to the top suface of the plate. Due to complexity of the poblem, the deduced diffeential equations ae solved using DQM method. This is a numeical scheme which can be accuately applied to poblems with o without a close fom solution. Applying the classical plate theoy, the effect of volume faction function powe m and the petubation of magnetic field vecto ae studied on the induced displacement and stess components. Moeove, the effect of t/a and b/a atios on the displacement and stess fields ae studied. BASIC FORMULATIONS Conside a adial magnetic field vecto H as shown in Figue 1. The esulting Loent foce (f) and the petubation of electic field vecto e act along and diections espectively. Now assume an annula cicula plate with a unifom tansvese load Po acting on its top suface (see Fig. ) is exposed to this magnetic field. As a esult, the total tansvese load acting on the plate, along diection, would be q (q=po+ f). This will induces a displacement field vecto U in the plate. Latin Ameican Jounal of Solids and Stuctues 13 (016)

4 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 739 Figue 1: Radial magnetic field vecto. Figue : Geomety, loading and coodinate system of the annula plate. Assuming the magnetic pemeability () of the plate is equal to the magnetic pemeability of its suounding, neglecting the displacement electic cuents, the Maxwell s electodynamics equations fo the plate may be witten as (Wang and Dai, 004); j h, h ( U H ) cul( U H ), div h 0 (1) Latin Ameican Jounal of Solids and Stuctues 13 (016)

5 740 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method U e ( )( H), h e ( ) () t t o whee j is the suface density vecto of the electic cuent, e is the petubation of the electic field vecto, h is the petubation of the magnetic field vecto and to is the time. On using cylindical coodinates (,, ), application of the magnetic field vecto H ( H,0,0) to Eqs. (1) and (), esults in; U ( u,0, w), h ( 0,0, h ) (3) o w w e ()(0,0, H ( )) T (4) w w j 0, H ( ), 0 (5) f ( )( j H ) ( ) H w ( w ) (6) h w H ( w ) (7) To deduce the equilibium equations, it is assumed that the plate popeties ae distibuted along the thickness accoding to equations (8) - (10). 1 g ( ) ) t m ( (8) E ( ) g ( ) E1 (1 g ( )) E (9) ( ) g ( ) 1 (1 g ( )) (10) Hee, m is the volume faction function powe, t is the plate thickness which is assumed to be unifom and E 1, E, 1 and ae the elastic moduli and magnetic pemeability coefficients of phases 1 and of the FG mateial, espectively (see Table 1, Ghobanpou Aani et al. 010). Moeove, the Poisson s atio of the plate is consideed to be constant. FGM plate popeties Phase Mateial E (GPa) µ (H / m) ν 1 ( Ceamic ) Ziconia E ( Metal ) Aluminum E Table 1: Mechanical popeties of the individual constituents used fo the annula FGM plate (Ghobanpou Aani et al. 010). Latin Ameican Jounal of Solids and Stuctues 13 (016)

6 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 741 In axisymmetic poblems, deflection w, stesses and stains ae independent of the cicumfeential diection and hence, thei deivatives with espect to ae equal to eo. Consequently, in the pesent study, due to axisymmetic loading, deflection w becomes only a function of and one may wite; 0 dw( ) u (, ) u ( ) (11a) d u (, ) 0 (11b) w(, ) w( ) (11c) Additionally: 0 u du d w( ) (1a) d d 0 u 1 u u dw( ) (1b) d 0 (1c) 0 whee u is the displacement of the middle suface of the FGM plate. Using Hooke's law, the adial and cicumfeential stesses ae; E 1 [ ] (13a) E [ ] (13b) 1 Expessing the total potential enegy as U V f, then the stain enegy U and the potential enegy of the extenal foces Vf ae equal to; U a t b t [ ] d d (14a) V f a ( q w) d Application of minimum total potential enegy ( 0 (15). b (14b) ) to the plate, esults in equation a b d d d d d d N N u M M Q Q q w d 0 (15) Latin Ameican Jounal of Solids and Stuctues 13 (016)

7 74 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method In Eq. (15), and, M i i, Q and q (q=po+f) ae the shea foce esultant and tansvese load,, ae the esultant foces and moments, and is the small tansvese nomal otation about axis, espectively. Using equation (15), the equilibium equations may be witten as follow: whee Simplifying Eqs. (16), one may wite; d u : N N 0 (16a) d d : M M Q 0 (16b) d d w : Q q 0 (16c) d dn N N 0 (17a) d d M dm 1 dm q ( ) 0 (17b) d d d N i N N M M t / d t / t / t / d (18a) (18b) On using Eq. (1), Eq. (13) may be witten as; 0 0 E( ) du u d w( ) dw( ) [ ( )] (19a) 1 d d d 0 0 E( ) u du 1 dw( ) d w( ) [ ( )] (19b) 1 d d d Theefoe; 0 du d w() N ˆ ˆ ˆ ˆ A11 A1 B11 B d 1 d N ˆ ˆ 0 A ˆ ˆ 1 A11 u B1 B11 1 dw( ) d 0 du d w() M ˆ ˆ ˆ ˆ B11 B1 C11 C d 1 d M ˆ ˆ 0 B ˆ ˆ 1 B11 u C1 C11 1 dw( ) d (0a) (0b) Latin Ameican Jounal of Solids and Stuctues 13 (016)

8 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 743 whee; t / ˆ ˆ ˆ 1 ( A, B, C ) E( )(1,, ) d (1a) t / t / ˆ ˆ ˆ ( A, B, C ) E( )(1,, ) d (1b) t/ Based on Eqs. (17a) and (0a), one may wite; d u 0 0 ˆ 3 () 1 du () () 1 () 1 () () B d w d w dw u 0 ˆ 3 d d A d 11 d d () On using Eqs. (17b) and (0b) and substituting P o f fo (), one can obtain: q 4 ˆ 3 ˆ ˆ ˆ d w() C 11 d w() C11 d w() C11 ( ) H dw() C11 ( ) 4 H 3 3 d d d d 3 0 ˆ 0 ˆ 0 ( ) H ( ) ˆ ˆ d u B 11 d u( ) B 11 du( ) B 11 0 w () B 11 u ( 3 3 d d d ) p 0 (3) To solve the above equilibium equations, fou types of bounday conditions ae consideed as follow; 1. Clamped-Clamped (C-C) suppots; Hee, the oute and inne edges of the plate ae clamped. In this case, the bounday conditions ae; dw ( b) dw ( a) w ( b) 0, 0, w ( a) 0, 0 d d. Simply suppoted-clamped (SS-C); Hee, the oute edge of the plate is clamped while the inne edge is simply suppoted. In this case, the bounday conditions ae; dw( a) w( b ) 0, M ( b ) 0, w( a ) 0, 0 d 3. Fee-Clamped (F-C); Hee, the oute edge of the plate is clamped while the inne edge is fee. In this case, the bounday conditions ae; dw( a) N ( b) 0, M ( b) 0, w( a) 0, 0 d 4. Simply suppoted-simply suppoted (SS-SS): Base on this type of bounday condition, the oute and inne edges of the plate ae simply suppoted. In this case, the bounday conditions ae; w( b ) 0, M ( b ) 0, w( a ) 0, M ( a) 0 To solve the equilibium equations () and (3), genealied diffeential quadatue (GDQ) is used, as will be descibed next. Latin Ameican Jounal of Solids and Stuctues 13 (016)

9 744 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 3 GENERALIZED DIFFERENTIAL QUADRATURE (GDQ) GDQ is a numeical method that appoximates the deivative of a function with espect to a vaiable as the sum of weighted linea function values at all domain points. This may be witten as; d k w( ) d i k n j1 k ij C w( ) j i 1,,..., n (4) whee n is the numbe of gid points in diection. In this wok, fo inteval b a, Chebyshev polynomials ae used to identify the gid points. These point ae defined by Eq. (5). 1 ( i 1) i b 1cos ab n 1 (5) Additionally, the Lagange intepolating polynomials ae used fo the test functions as: whee; theefoe; j1 g M ( ) ) (1 (6) ( ) M ( ) i ( ) i i n n (1) M ( ) ( j ), M ( i ) ( i j ) (7) j1, ji k 1 C k k 1 ij Cij k Cii Cij fo i j: k,3,..., n 1 and i, j 1,,..., n i (8) and; n k k Cii Cij fo i j: i, j 1,,..., n (9) j1, ji In Eqs. (8) and (9), the tems Cij and Cii ae defined as; M ( i ) Cij fo i j, i, j 1,,..., n (1) (30) ( ) M ( ) i j (1) n j Cii Cij fo i j, i, j 1,,..., n j1, ji Based on these notations, application of GDQ method to Eqs. () and (3), esults in Eqs. (3) and (33), which may be witten as; (31) Latin Ameican Jounal of Solids and Stuctues 13 (016)

10 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 745 C ˆ C C 0 fo i 1,,..., n n 0 n ij 0 ui B11 3 ij ij Cij u j C ij w ˆ j j 1 i i A 11 j1 i i (3) n 3 ˆ C 4 ij 1 ( q) H ( ) 1 q H C 11 Cij C ij C ˆ 3 ij wj ˆ j1 i i C 11 i i C 11 ( ) H C C 1 fo i 1,,..., n, q 1,,..., nn n q ˆ 3 ij ij 0 w i B11 Cij u 3 j P0 i j1 i i i (33) Fo simplicity, we define each tem in Eq. (3) as a sepaate paamete defined in Eqs. (34)- (38). 0 u 1 w1 0 (34) w n 0 K K un C ij if i j : K11 Cij fo i, j 1,,..., n nn i C ij 1 if i j : K11 Cij fo i, j 1,,..., n nn i i (35) (36) Bˆ C 11 3 ij C ij K1 Cij fo i, j 1,,..., n nn ˆ A 11 i i Similaly, the tems in Eq. (33) ae defined in Eqs. (39)- (44). K K 1 0 un (37) F1 0 (38) 0 u 1 w1 0 (39) w n ˆ Cij C ij if i j : K B11 Cij fo i, j 1,,..., n nn i i (40) Latin Ameican Jounal of Solids and Stuctues 13 (016)

11 746 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 3 1 ˆ Cij C 1 ij if i j : K B11 Cij fo i, j 1,,..., n nn 3 i i i 3 C 4 ij 1 ( q ) H Cij C ij ˆ i i C 11 : ˆ if i j K C nn 11 1 ( q) H 3 C ij ˆ i C i 11 fo i, j 1,,..., n, q 1,,..., nn (41) (4) 3 C 4 ij 1 ( q ) H Cij C ij ˆ i i C 11 : ˆ if i j K C nn 11 1 ( q) H ( q) H C 3 ij ˆ ˆ i C i 11 i C 11 fo i, j 1,,..., n, q 1,,..., nn F P 1 0 (43) (44) Accoding to Eqs. (3) to (44), equilibium Eqs. (3) and (33) may be witten in a matix fom as; K11 K1 0 P 0 u K K w 1 nn 0 n1 n 1 Hee, n and nn ae the total numbe of gid points in and diections espectvely. In Eqs. (45), the deteminant of matix [K] is equal to eo. Application of the peviously descibed bounday conditions to Eq. (45) esults in values of wi (i = 1, n) and ui (i = 1, n). Using these values, one can calculate the adial and tangential stesses in each case.. (45) 4 NUMERICAL RESULTS AND DISCUSSION To investigate the effect of plate geometic paametes on the induced displacements and stesses, it is assumed that the annula FGM plate is expeiencing a unifom mechanical load 4 N 6 A P0 10 ( ) and a magnetic field of H 310. The bottom and top sufaces of the m m plate ae made fom pue metal and pue ceamic, espectively. The mateial popeties used to extact the esults ae given in Table 1. Also, the dimensionless paametes used in the upcoming figues ae defined as; Latin Ameican Jounal of Solids and Stuctues 13 (016)

12 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 747 t, Po a t, P a o f h f, h Po H Figues 3 shows the vaiation in midplane deflection in tems of a, based on fou diffeent types of bounday conditions given in section. In all cases, the tansvese load q is the sum of mechanical and magnetic loads. Accoding to this figue, fo all values of m, the plate deflection is maximum at 0.66 fo case (a), while fo case (b), it occus at 0.5 a a. Fo cases (c), and (d), this value occus at 0.60 a, and fee inne edge of the plate, espectively. (a) (C-C) suppots (b) (SS-C) suppots (c) (SS-SS) suppots (d) (F-C) suppots Figue 3: The effect of paamete m on tansvese deflection of the plate fo thee diffeent suppot conditions. Latin Ameican Jounal of Solids and Stuctues 13 (016)

13 748 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method As expected, the (C-C) condition expeiences the least deflection among all cases. In all cases, a pue ceamic plate with m = 0, expeiences the least deflection compaed to othe values of m. This is due to an incease in flexual igidity of the plate which is caused by a decease in m. To investigate the effect of paamete m on adial deflection, a typical (C-C) plate with simila loads and dimensions was selected. Accoding to the esults shown in Fig. 4, any incease in powe atio m has a simila effect on midplane adial displacement. Accoding to these esults, fo a homogeneous plate with m = 0, the adial displacement of the midplane is equal to eo. This is due to symmety in mechanical popeties of the plate which makes B 11 equal to eo. In addition, it is ealied that eo adial deflection occus at a point whee tansvese deflection w is maximum ( 0.66 a ). Figue 4: The effect of paamete m on the midplane adial displacements, (C-C) suppots. Futhe examination of the esults evealed that as long as a b atio emained equal to 5, eo adial displacement occued at 0.66, egadless of the value fo m (see Fig. 5). a Figues 6-9 show the vaiation in adial and tangential stesses in midplane due to a combination of tansvese load Po and a adial magnetic field H. The plate dimensions fo diffeent values of m ae given in each figue. Accoding to these figues, fo all types of bounday conditions discussed befoe, highe values of m ( 0 m ) esult in highe absolute magnitude of both stess components. Fo m = 0 (a homogeneous plate with B 11 0 ), both midplane stess components ae equal to eo. Latin Ameican Jounal of Solids and Stuctues 13 (016)

14 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 749 Figue 5: The effect of vaiation in plate dimensions on adial displacement, (C-C) suppots. (a) Radial stess distibution (b) Tangential stess distibution Figue 6: The effect of paamete m on the mid-plane adial stesses based on (C-C) suppots. Additionally, the natue of vaiation in both stess components appeas to be highly dependent on the type of bounday conditions imposed on the plate. The esults indicate that (SS-C) bounday conditions poduce the least adial stess, while a plate with (F-C) bounday conditions expeiences the highest tangential stess in the midplane. Latin Ameican Jounal of Solids and Stuctues 13 (016)

15 750 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method (a) Radial stess Distibution (b) Tangential stess distibution Figue 7: The effect of paamete m on the mid-plane adial stesses based on (SS-C) suppots. (a) Radial stess distibution (b) Tangential stess distibution Figue 8: The effect of paamete m on the mid-plane adial stess based on (SS-SS) suppots. The effect of magnetic field vaiation on displacement and stess components ae shown in Figs , fo a typical plate with clamped suppots. These esults ae based on the value of m = 1. Although not shown, simila behavios wee obseved fo othe values of m. Accoding to Fig. 10, exposue of the plate to a adial magnetic field has a substantial effect on its deflection. Any incease in magnetic field H causes a decease in plate deflection in pesence of a tansvese load Po. The location of maximum displacement seems to baely depend on the magnitude of H. Latin Ameican Jounal of Solids and Stuctues 13 (016)

16 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 751 (a) Radial stess distibution (b) Tangential stess distibution Figue 9: The effect of paamete m on the midplane adial stesses based on (F-C) suppots. Figue 10: The effect of adial magnetic field vaiation H on plate tansvese deflection w, (C-C) suppots. u Simila effect was obseved fo adial deflection, in pesence and absence of the magnetic field H (see Fig. 11). In eithe case, the magnitude of the tansvese mechanical load is a Po. These esults ae extacted fo m = 1. Accoding to this figue, the pesence of the adial magnetic field shifts the location of eo adial displacement u fom 0.57 to 66 a a 0., while substantially deceasing its extema. Figues 1 and 13 show the additional effect of magnetic field H on adial and tangential stess components induced in the plate, based on m = 1. The intoduction of magnetic field Latin Ameican Jounal of Solids and Stuctues 13 (016)

17 75 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method vecto substantially affects the location and values of both stesses at the inne and oute edges of the plate, as well as othe locations. Additionally, any change fom positive to negative stesses in both diections seems to be highly affected by the pesence of magnetic field. Figue 11: The effect of adial magnetic field vaiation H on plate adial deflection, (C-C) suppots. Figue 1: The effect of adial magnetic field vaiation H on plate adial stess, (C-C) suppots. Latin Ameican Jounal of Solids and Stuctues 13 (016)

18 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 753 Figue 13: The effect of adial magnetic field vaiation H on plate tangential stess, (C-C) suppots. Figues show the vaiation in Loent foce and the induction magnetic field intensity along the plate adius, fo the fou types of bounday conditions discussed befoe. The esults ae given fo the plate midplane. Unlike the Loent foce, distibution of h along the plate adius appeas to be a function of m. Highe values of m esult in highe values of h. (a) Loent foce distibution (b) Induction magnetic field intensity distibution Figue 14: The effect of vaiation in powe m on (a) Loent foce f and (b) induction magnetic field intensity fo (C-C) suppots. Latin Ameican Jounal of Solids and Stuctues 13 (016)

19 754 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method (a) Loent foce distibution (b) Induction magnetic field intensity distibution Figue 15: The effect of vaiation in powe m on (a) Loent foce f and (b) induction magnetic field intensity fo (SS-C) suppots. (a) Loent foce distibution (b) Induction magnetic field intensity distibution Figue 16: The effect of vaiation in powe m on (a) Loent foce f and (b) induction magnetic field intensity fo (SS-SS) suppots. Accoding to these figues, the (SS-SS) bounday conditions appea to exhibit less complex distibution fo both Loent foce and the induction magnetic field intensity along the plate adius. Latin Ameican Jounal of Solids and Stuctues 13 (016)

20 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 755 (a) Loent foce distibution (b) Induction magnetic field intensity distibution Figue 17: The effect of vaiation in powe m on (a) Loent foce f and (b) induction magnetic field intensity fo (F-C) suppots. Figues 18-1 show the effect of vaiation in plate geomety on the esulting stess and displacement components fo a plate with a typical (C-C) bounday conditions. Hee, the thee geometic paametes a, b and t ae changed in the same popotion. Othe paametes ae kept the same as befoe. Accoding to Figs. 18 and 19, location of maximum w and eo adial displacement baely depends on the plate dimensions. Figue 18: The effect of vaiations in plate dimensions on tansvese deflection, (C-C) suppots. Latin Ameican Jounal of Solids and Stuctues 13 (016)

21 756 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method Additionally, educing the plate dimensions will lowe the magnitude of displacement components, and hence elevant stesses ae educed, as shown in Figs. 0 and 1. This is due to plate stiffening as a esult of lowe dimensions and lowe plate s inne adius. Figue 19: The effect of vaiation in plate dimensions on adial displacement, (C-C) suppots. Figue 0: The effect of vaiation in plate dimensions on adial stess distibution, (C-C) suppots. Latin Ameican Jounal of Solids and Stuctues 13 (016)

22 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 757 Figue 1: The effect of vaiation in plate dimensions on tangential stess distibution, (C-C) suppots. The effect plate dimensions on Loent foce and the induction magnetic field intensity ae shown in Figs. and 3, fo a typical plate with (C-C) suppots. Accoding to Fig., in contast with h, the Loent foce distibution along the plate adius does not seem to depend much on the plate dimensions. Figue 3 indicates that any change in the plate dimensions baely affects the location of eo value fo h. Figue : The effect of plate dimensions on Loent foce distibution, (C-C) suppots. Latin Ameican Jounal of Solids and Stuctues 13 (016)

23 758 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method Figue 3: The effect of plate dimensions on the induction magnetic field intensity distibution, (C-C) suppots. Figue 4 shows the tansvese displacement of diffeent points acoss the plate thickness at 0.66 a (location of maximum tansvese deflection). The esults ae plotted fo diffeent values of m in Fig. 4(a) and diffeent values of H in Fig. 4(b). The applied loads and bounday conditions, as well as the plate dimensions ae as shown. Accoding to Fig. 4(a), fo an isotopic plate with m=0, all points acoss the plate thickness have the same deflection. Fo highe values of m, the tansvese deflection becomes moe gadual acoss the plate thickness. Accoding to Fig. 4(b), any vaiation in H does not seem to highly affect the tansvese deflection acoss the plate thickness. Figues 5 and 6 show the vaiation in Loent foce and the induction of magnetic field intensity acoss the plate thickness. Accoding to these Figues, fo m=0, f and h ae constant along the plate thickness. Fo othe values of m, the distibution of Loent foce and the induced magnetic field intensity acoss the plate thickness appea to highly depend on m. The esults on these figues ae based on (location of maximum tansvese deflection). Accoding to figue 5, the magnitude of h a is the highest on the top suface of the plate while Fig. 6 indicates that the peak value of the Loent foce appeas to be at the bottom. Latin Ameican Jounal of Solids and Stuctues 13 (016)

24 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 759 (a) Vaiation in m (b) vaiation in H Figue 4: The effect of paamete m and H on tansvese deflection of the plate along diection, (C-C) suppots. Figue 5: The effect of powe m on distibution of the induction magnetic field intensity acoss the plate thickness, (C-C) suppots. Latin Ameican Jounal of Solids and Stuctues 13 (016)

25 760 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method Figue 6: The effect of powe m on Loent foce distibution along the plate thickness, (C-C) suppots. Vaiations in adial and tangential stess components along the plate thickness ae shown in Figs. 7 and 8, espectively. The esults ae fo a typical plate with clamped suppots. It is assumed that m = 1 and (location of maximum tansvese deflection). Fo small a values of H both stess components seem to change almost linealy acoss the plate thickness. The maximum values of both stess components appea to be at top suface of the plate. Figue 7: The effect of adial magnetic field on adial stess component accos the plate thickness, (C-C) suppots. Latin Ameican Jounal of Solids and Stuctues 13 (016)

26 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method 761 Figue 8: The effect of adial magnetic field on tangential stess component acoss the plate thickness, (C-C) suppots. It is woth to mention that based on the values given in Figs and 7-8, fo all values of H, eo adial and tangential stesses occu both in adial and tansvese diections. The locations of eo stess components acoss the plate thickness appea to be on a suface which is slightly above the =0 plane. 5 CONCLUSIONS In this wok, the adial and tansvese displacements, as well as the adial and cicumfeential stess components in an annula FGM plate subjected to a combination of tansvese load Po and Loent foce f, wee calculated based on fou types of bounday conditions. The elastic modulus of the plate as well as the magnetic pemeability coefficient acoss the plate thickness wee assumed to vay accoding to the volume distibution function, while the Poisson s atio was taken to be constant. Classical plate theoy was applied to analye the poblem. The deduced equilibium equations wee solved using genealied diffeential quadatue method. Accoding to the esults, the effect of additional load f induced by the adial magnetic field H seems to be substantial on plate adial and tansvese deflections. Additionally, in pesence of a tansvese load Po, application of a adial magnetic field to the plate top suface, completely changes the state of tangential and adial stesses along its adius, esulting in positive and negative stesses in these two diections. The effect of additional tansvese load f induced by the magnetic field vecto H on both stess components acoss the plate thickness seems to be moe on bottom suface (pue metal). Moeove, the magnitude of the Loent foce and the induced magnetic field intensity h acoss the plate thickness appea to highly depend on the volume faction function powe m. Intoduction of a Latin Ameican Jounal of Solids and Stuctues 13 (016)

27 76 M. Shishesa et al. / Magneto-Elastic Analysis of an Annula FGM Plate Based on Classical Plate Theoy Using GDQ Method magnetic field H to a plate which is aleady loaded by a tansvese load Po educes the plate displacement and stess components, and hence, esulting in a highe facto of safety in the plate. Refeences Bayat, M., Rahimi, M., Saleem, M., Mohaab, A.H., Wudtke, I., Talebi, H., (014). One-dimensional analysis fo magneto-themo-mechanical esponse in a functionally gaded annula vaiable-thickness otating disk. Applied Mathematical Modelling 38: Behavan Rad, A., Shaiyat, M., (015). Thee-dimensional magneto-elastic analysis of asymmetic vaiable thickness poous FGM cicula plates with non-unifom tactions and Ke elastic foundations. Composite Stuctues, 15: Chi, S.H., Chung, Y.L., (006). Mechanical behavio of functionally gaded mateial plates unde tansvese load Pat I: Analysis. Intenational Jounal of Solids and Stuctues, 43: Ghobanpou Aani, A., Loghman, A., Shajai, A. R., Ami, S. (010). Semi-analytical solution of magnetothemo-elastic stesses fo functionally gaded vaiable thickness otating disks. Jounal of Mechanical Science and Technology 4 (10): Lal, R. and Saini, R. (015). Buckling and Vibation of Functionally Gaded Non-unifom Cicula Plates Resting on Winkle Foundation. Latin Ameican Jounal of Solids and Stuctues 1(1): Lal, R. and Saini, R. (015). On the use of GDQ fo vibation chaacteistic of non-homogeneous othotopic ectangula plates of bilinealy vaying thickness. Acta Mechanica, 6: Ma, L.S., Wang, T.J., (003). Nonlinea bending and post-buckling of a functionally gaded cicula plate unde mechanical and themal loadings. Intenational Jounal of Solids and Stuctues 40: Najafiadeh, M.M., Heydai, H.R., (004). Themal buckling of functionally gaded cicula plates based on highe ode shea defomation plate theoy. Euopean Jounal of Mechanics A/Solids 3: Paveen, G.N., Reddy, J.N., (1998). Non-linea tansient themoelastic analysis of functionally gaded Ceamic-Metal plates. Intenational Jounal of Solids and Stuctues 35: Reddy, J.N., (000). Analysis of functionally gaded plates. Int. J. Nume. Methods Eng 47: Reddy, J.N., Wang, C.M., Kitiponchai, S., (1999). Axisymmetic bending of functionally gaded cicula and annula plates. Euopean Jounal of Mechanics A/Solids 18: Saidi, A.R., Rasouli, A., Sahaee, S., (009). Axisymmetic bending and buckling analysis of thick functionally gaded cicula plates using unconstained thid-ode shea defomation plate theoy. Composite Stuctues 89: Wang, X., Dai, H.L. (004). Magneto themodynamic stess and petubation of magnetic field vecto in an othotopic themoelastic cylinde. Intenational Jounal of Engineeing Science 4: Yuda, Hu. Jing, Li., (009). The magneto-elastic sub hamonic esonance of cuent-conducting thin plate in magnetic field. Jounal of Sound and Vibation 319: Latin Ameican Jounal of Solids and Stuctues 13 (016)