Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 www.sprigerlik.com/cotet/1738-494 DOI 10.1007/s106-01-083-7 A DXDR large deflectio aalysis of uiformly loaded square, circular ad elliptical orotropic plates usig o-uiform rectagular fiite-differeces M. Kadkhodaya 1,*, A. Erfai Moghadam 1, G.J. Turvey ad J. Alamatia 3 1 Departmet of Mechaical Egieerig, The Uiversity of Ferdowsi, 91775-1111, Mashhad, Ira Egieerig Departmet, Lacaster Uiversity, Bailrigg, Lacaster LA1 4YR, U.K. 3 Departmet of Civil Egieerig, Azad Uiversity, Mashhad, Ira (Mauscript Received October 17, 011; Revised Jauary 7, 01; Accepted April 30, 01) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract A fiite-differece aalysis of e large deflectio respose of uiformly loaded square, circular ad elliptical clamped ad simplysupported orotropic plates is preseted. Several types of o-uiform (graded) mesh are ivestigated ad a mesh suited to e curved boudary of e orotropic circular ad elliptical plate is idetified. The DXDR meod a variat of e DR (dyamic relaatio) meod is used to solve e fiite-differece forms of e goverig orotropic plate equatios. The DXDR meod ad irregular rectiliear mesh are combied alog wi e Cartesia coordiates to treat all types of boudaries ad to aalyze e large deformatio of o-isotropic circular/elliptical plates. The results obtaied from plate aalyses demostrate e potetial of e o-uiform meshes employed ad it is show at ey are i good agreemet wi oer results for square, circular ad elliptical isotropic ad orotropic clamped ad simply-supported plates i bo fied ad movable cases subjected to trasverse pressure loadig. Keywords: Orotropic circular ad elliptical plates; No-uiform rectagular mesh; Large deflectio; Dyamic relaatio ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Itroductio The icreasig use of composite materials i i-walled structures, which may be regarded as assemblies of plate ad shell elemets, provides bo e catalyst ad justificatio for e cotiual developmet ad refiemet of umerical procedures for e solutio of orotropic ad oer plate fleure problems. The solutios of particular iterest here are for e large deflectio respose of clamped ad simply-supported circular ad elliptical plates wi rectagular Cartesia orotropy. Such plates commoly arise i practice as ispectio covers ad oer types of ed closures. Bo e small ad large deflectio resposes of orotropic plates have bee ivestigated by a umber of researchers durig e course of e past half-cetury. For eample, i e 1970 s Chia [1, ] studied e oliear fleural behavior of rectagular orotropic plates usig a approimate aalytical solutio. Dalaei ad Kerr [3] ivestigated e small deflectio respose of clamped rectagular orotropic plates subjected to uiform trasverse pressure usig e eteded Katorovich meod. I later studies, Mbakogu ad Pavlovica [4] ad Tabalov et al. [5] also ivestigated e small deflectio * Correspodig auor. Tel.: +98 9153111869, Fa.: +98 511 8763304 E-mail address: kadkhoda@um.ac.ir Recommeded by Associate Editor Heug Soo Kim KSME & Spriger 01 respose of clamped rectagular orotropic plates. Mistou et al. [6] ivestigated e behavior of clamped lamiated rectagular orotropic plates subjected to uiform trasverse pressure bo eperimetally ad umerically usig a Ritz approimate aalysis. Bhaskar ad Kaushik [7] used a double Fourier series approach to aalyze usymmetric cross-ply lamiated clamped rectagular plates. Recetly, Salehi ad Sobhai [8] used Midli plate eory i cojuctio wi e DR meod to ivestigate bo e small ad large deflectio respose symmetrically lamiated fiber-reiforced sector plates. I e majority of e aforemetioed papers e plate was eier of rectagular or circular geometry. Therefore, for umerical mesh-based solutios, a uiform square or rectagular mesh would be adequate for rectagular plates wi rectagular Cartesia orotropy. Similarly, for circular plates wi polar orotropy, a orogoal polar mesh would suffice but would ot readily accommodate rectagular Cartesia orotropy. Here, however, e focus of iterest is o e use of a sigle o-uiform rectagular mesh at may be used to aalyze square, circular ad elliptical plates wi rectagular Cartesia orotropy. It is e mai objective of e curret study to show if e DXDR meod ad irregular rectiliear mesh are combied alog wi e Cartesia coordiates oe ca treat all types of
33 M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 Fig. 1. Square plate wi its pricipal aes of orotropy parallel to e ad y aes: pla view; cross-sectio. boudaries which is also etedable to e large deformatio of o-isotropic plates. The mai applicatio of e curret meod ca be e aalysis of orotropic materials which are made iitially i rectagular dies ad e fibers are iserted i rectiliear dictios ad e are cut ito circular ad elliptical forms. The large deflectio respose of clamped ad simplysupported square, circular ad elliptical orotropic plates subjected to uiform trasverse pressure is ivestigated usig several types of o-uiform rectagular fiite-differece mesh. Cosequetly, e same goverig equatios may be used for square, circular ad elliptical plates. Moreover, e mesh could be used to aalyze orotropic plates wi more geeral boudary shapes. A variat of e DR meod, amely e DXDR meod [9] is employed to solve e fiite-differece forms of e large deflectio orotropic plate equatios. The results are preseted for bo deflectios ad stress couples i order to demostrate e accuracy of e preset o-uiform mesh large deflectio solutios by compariso wi couterpart solutios obtaied wi uiform meshes ad by oer approimate aalysis techiques.. Mesh geeratio Cosider a square orotropic plate wi sides of leg a ad ickess h, as show i Fig. 1. For coveiece, e origi of e co-ordiate system, y, z is located at e ceter of e mid-plae of e plate. I order to carry out a fiitedifferece aalysis of a square or rectagular plate it is geerally sufficiet to use a uiform square or rectagular mesh etedig over e plate domai. However, for plates wi curved boudaries it is ot usually possible to use a uiform square or regular rectagular mesh because e odes of e mesh do ot coicide wi curved boudary of e plate. Differet types of o-uiform rectagular meshes may be geerated i eier or bo of e co-ordiate directios ad y. The mesh may be uiform i oe directio ad ouiform i e oer directio. I is case e mesh is deoted as a Type 1 mesh. Fig. shows a eample of a Type 1 mesh, i which e mesh is uiform i e y directio ad o-uiform i e directio. Here, e mesh size i e directio reduces as e odal distace from e ceter of e plate icreases (it is assumed at symmetry is eploited ad oly e positive quadrat of e plate is modeled). Clearly, e mesh may be arraged to be o-uiform wi respect to bo co-ordiate directios. Oe eample of ese o-uiform meshes, desigated Type, is illustrated i Fig.. Such o-uiform bi-directioal meshes may be geerated i a variety of ways. Here, simple power laws are employed. The co-ordiates of e odes of e mesh i e ad y directios are defied as follows: i = a k Fig.. Eamples of o-uiform meshes: uiform i e y directio y = 1 ad o-uiform i e directio 1 [Type 1]; o- = y = 0.5 [Type ]. uiform i bo ad y directios ( ) (1a)
M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 333 j y= a k y. (1b) I Eq. (1) a is e half side leg of e square plate or e radius of e circular plate, i ad j are e ode umbers i e ad y directios, respectively, where 0 i, j k, k+ 1 is e total umber of odes i e ad y directios, respectively, ad ad y are bo positive costats. If oe of e costats or y is equal to oe ad e oer is ot equal to oe, e a Type 1 mesh is defied, i.e. = 1 ad y 1 or 1 ad y = 1. O e oer had, e mesh will be o-uiform i bo ad y directios whe = 1 ad y 1. Moreover, whe = y= 1, a uiform mesh wi respect to bo e ad y aes is defied. For plates wi a curved boudary, ad y caot be selected idepedetly. I oer words, ese quatities must be determied so at e mesh odes coicide wi curved boudary of e plate. For eample, e boudary odes of a circular plate must satisfy followig equatio: + y = a () where a is e radius of plate. Substitutig Eqs. (1a) ad (1b) ito Eq. () leads to e followig result: y i j + = 1. k k For a Type 1 mesh (uiform i oe directio ad ouiform i e oer directio), e ukow power or y i e o-uiform directio may be determied by solvig Eq. (3). I oer words, (or y ) is determied so at e mesh lies itersect e circle boudary. It is clear at (or y ) will have a specific value for each mesh lie. However, a Type mesh is defied by = y= 0.5, where e mesh has e same degree of o-uiformity i bo e ad y directios. Oer values of ad y, betwee 0 ad 1 or greater a 1, are ulikely to be of iterest because ey oly icrease e rate of directioal o-uiformity of e mesh. I such cases, ad y would have differet values for each lie of e mesh for e oer values, i.e. whe y 0.5. Nevereless, for e sake of completeess, ey are ivestigated to a limited etet ad discussed later i e paper. Similar remarks apply to oer plates wi curviliear boudaries, e.g. elliptical plates. I order to ivestigate e accuracy of ese meshes, e circular ad elliptical forms of Eq. (3) are used first to geerate meshes for square ad rectagular plates, respectively. This mesh geeratio meod may be eplaied by a simple eample. Cosider a square plate of side leg a. To defie a Type 1 mesh over a square plate, a circle of radius a wi its ceter at e ceter of e square plate is defied, as show i Fig.. The, a Type 1 mesh is geerated i oe directio ( or y ) over e (3) circle by, for eample, settig = 1 ad determiig y from Eq. (3). Here, e total umber of mesh odes i e ad y directios is assumed to be 13 (k = 1). The mesh lies over circle are e eteded to e sides of e square to produce a Type 1 mesh over e square plate. Oer mesh types (for eample, Type mesh which is show i Fig. ), may be geerated i a similar maer. For rectagular plates, a iscribed ellipse should be used. To demostrate e mesh distributio i e curret study, e used mesh for e circular ad elliptical plates are show i Figs. 3 ad 3, respectively. 3. Goverig orotropic plate equatios The Cartesia forms of e equilibrium equatios for a orotropic plate may be writte i e followig form [10]: N N y + = 0 y N y Ny + = 0 y M M y M y w N + + + + y y Fig. 3. The geeral form of mesh distributio of: circular plate; elliptical plate. w w Ny + N y + q= 0. y y (4) (5) (6)
334 M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 It should be appreciated at Eqs. (4) ad (5) are e iplae equilibrium equatios ad Eq. (6) is e out-of-plae equilibrium equatio. I ese equatios, N, N y ad N y are e stress resultats, M, M y ad M y are e stress couples, w is e deflectio ad q is e trasverse pressure. The stress resultats ad stress couples may be epressed as: (, y, y ) h( σ, σ y, τ y ) h N N N dz = (7a) ( y y ) h ( σ σ y τ y ) h M, M, M z,, dz. = (7b) I Eq. (7) σ, σ y ad τ y are e direct ad shear stress compoets. The stresses uder e itegrals o e right-had sides of Eq. (7) are give i Ref. [11] as: σ Q11 Q 1 σ y = Q1 Q τ 0 0 y 0 ε 0 ε y. Q66 γ y I Eq. (8), ε, ε y ad γ y are e direct ad shear strais ad e terms Qij( i, j = 1,, 6) are e reduced stiffess which may be epressed as: Q11= E11 ( 1 υ1υ1) υ υ 1 E 1 E Q 11 1= = Q= Q ( 1 υ1υ1) ( 1 υ1υ1) E ( 1 υ1υ1) 66 1. (8) (9a) (9b) (9c) = G (9d) I Eq. (9), E11 ad E are e elastic moduli wi respect to e major ad mior aes of orotropy, respectively, ad G 1 is e shear modulus wi respect to e same aes. For orotropic plates, which are cosidered here, e pricipal aes of orotropy coicide wi e ad y aes. The terms υ 1 ad υ 1 deote e major ad mior Poisso s ratios. The strais of poits at a distace z above e mid-plae of e plate are give as: 0 0 k, k y ad 0 k y are e curva- of e plate mid-plae ad tures ad twist of e plate mid-plae. The relatioships betwee e mid-plae strais ad displacemets are give as: 0 u 1 w ε = + 0 v 1 w ε = + y y 0 u v w w γ y = + +. y y (11a) (11b) (11c) I Eq. (11) u ad v are e i-plae displacemets i e ad y directios, respectively. Moreover, it should also be appreciated at geometric oliearity, due to stretchig of e plate mid-plae, is reflected by e presece of e quadratic terms i Eq. (11). Similar relatioships eist betwee e mid-plae curvatures ad deflectio of e plate as follows: 0 w k = (1a) 0 w k y y = (1b) 0 w k y. y = (1c) The boudary coditios i e plate aalyses for e square plate are: Fied clamped edge Alog =± a : w u= 0, v= 0, w= 0, = 0. Alog y =± a : w u= 0, v= 0, w= 0, = 0. y Movable clamped edge Alog =± a : (13a) (13b) 0 0 ε ε k 0 0 εy = ε y + z k y. γ 0 0 y γ y k y (10) w v= 0, w= 0, N = 0, = 0. Alog y =± a : (13c) I Eq. (10) 0 0 y ε, ε ad 0 γ y are e direct ad shear strai w u= 0, w= 0, N y = 0, = 0. y (13d)
M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 335 (c) Fied simply-supported edge Alog =± a : u= 0, v= 0, w= 0, M = 0. (13e) ( σ ) = 0 & ε = 0 membrae u u u = 0 cosθ + siθ = 0. y t (13k) Alog y =± a : u= 0, v= 0, w= 0, M y = 0. (13f) (d) Movable simply-supported edge Alog =± a : v= 0, w= 0, M = 0, N = 0. (13g) Alog y =± a : u= 0, w= 0, M = 0, N = 0. (13h) y y Because bo e plate geometry ad e loadig are symmetric about e ad y aes oly oe quarter of e plate has to be aalyzed. It is clear at w, N, N y, M ad M y are symmetric about e ad y aes, M y is atisymmetric about e ad y aes, u is symmetric about e ais ad atisymmetric about e y ais ad υ is atisymmetric about e ais ad symmetric about e y ais. Hece, e foregoig symmetry/atisymmetry coditios have to be eforced alog e ad y aes. I e case of e circular ad elliptical plates, e same symmetry/atisymmetry coditios have to be eforced alog e ad y aes. However, alog e circumferetial boudary e coditios are somewhat more complicated. For eample i e fied clamped boudary coditio, at all odal poits aroud e circumferece of e plate u, v ad w are set to zero. I additio, e slope ormal to e circumferece, i.e. i e directio (see Fig. ), has also to be set to zero. This boudary coditio is defied as: w w w y w = + = cosθ+ w si θ. y y (13i) I Eq. (13i) e agle θ is defied as show i Fig.. It is easy to show at e coditio for simply-supported cases (fied ad movable) would be: w w M = 0 = 0 cos θ + w w siθ cosθ + si θ = 0. y y (13j) Moreover, e coditio for movable cases (simplysupported ad clamped) is: 4. DXDR solutio of e goverig large deflectio orotropic plate equatios The DXDR meod is a modified versio of e origial DR meod. I is modified meod e startig vector ad dampig factor are selected more deliberately compared to e origial DR [1]. It is a iterative time-steppig solutio techique for modelig e respose of damped dyamic systems. However, it has maily bee used to obtai e static respose by artificially dampig out e oscillatios of equivalet quasidyamic systems. The startig poit of ay DXDR aalysis to determie e static structural respose is e formulatio of e quasi-dyamic system of goverig equatios. Thus, e secod-order quasi-dyamic plate equatios describig e large deflectio respose of e square, circular ad elliptical orotropic plates, i.e. Eqs. (4)-(6), may be epressed as follows for e time icremet [1]: m D&& c D& f p (14) ii i + ii i + i = i i i where D & ad 3 1 vectors of velocity ad acceleratio, fi ad p i are vectors of e iteral forces (correspodig pricipally to e LHS s of Eqs. (4)-(6)) ad e applied pressure q at e i ode ad m ii ad c ii are ( 3 3) diagoal mass ad dampig matrices for e ree displacemet compoets u, v ad w at ode i durig e iteratio, respectively. Now, because oly e static large deflectio respose of e orotropic plate is of iterest, Eq. (14) may be trasformed ito time-steppig iitial value format for e plate velocities [1]. The equatio e becomes: D && are ( ) ( τ ςi ) τ D ( + τ ςi ) ( + τ ςi ) 1 1 + r D i i = i +. mii & & (15) I Eq. (15), τ is e fictitious time step, ς are e odal dampig factors ad ri = ( pi fi ) is e force vector at ode i. 1 The odal displacemets D i + are determied from e velocities calculated i Eq. (15) by meas of e followig simple itegratio rule: 1 + + 1 + 1 Di Di τ Di. = + & (16) Furermore, it is coveiet to relate e dampig ad mass matrices as follows [1]: i
336 M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 ii i ii c = ς m. (17) I e DXDR meod, e mass matri, odal dampig factors ad e time icremet should be defied i such a way as to guaratee e stability ad covergece of e iterative procedure. The most commo meod of achievig is objective is to determie m ii by meas of Gerschgöri eorem. Accordig to is eorem, e followig iequality must be satisfied i order to guaratee e stability of e iteratios [13]: Table 1. Elastic costats for e ree types of orotropic materials used i e large deflectio plate aalyses. Elastic costats Glass-epoy Graphite-epoy Boro-epoy E11/E G1/E υ 1 3 0.5 0.5 40 0.6 0.5 10 0.33 0. ii ( τ ) k S, m ij T. 4 (18) j= 1 I Eq. (18) k is e umber of degrees of freedom of e structure ad S ij, T is e ij elemet of e taget stiffess matri durig e iteratio ad is give by: f S i ij, T. D j = (19) Alteratively, by applyig Rayleigh's Priciple [14] at each ode, e istataeous critical dampig factor for ode i durig e iteratio [1] may be epressed as: ς 1 T ( Di ) fi Di i =. T ( Di ) mii D i (0) Oer approaches have also bee suggested [13]. The fictitious time icremet of e iteratio is usually assumed to be costat ad equal to uity. However, some researchers have proposed at is parameter should also be based o e Rayleigh quotiet [15]. Techiques have also bee proposed to determie e iitial displacemet vector i order to reduce e computatioal time [1, 16, 17]. 5. Large deflectio results for square, circular ad elliptical plates usig o-uiform meshes Differet results are preseted i is sectio. The total umber of odes i e ad y directios, is equal to 13 (k = 1). The first set of results has bee computed for a square clamped isotropic plate subjected to uiform trasverse pressure loadig. The results are iteded to demostrate at coverged ad accurate deflectios ca be obtaied wi ouiform fiite-differece meshes. I Fig. 4 e plate ceter deflectios wi Type 1 ad Type o-uiform meshes are compared wi deflectios obtaied from Chia s [18] approimate aalytical solutio. It is clear at e ceter deflectios obtaied for bo meshes are i reasoably good agreemet wi e deflectios predicted by Chia s approimate aalytical solutio. Fig. 4. Compariso of load-ceter deflectio resposes of clamped square isotropic plates. A secod set of compariso aalyses has bee udertake for clamped square orotropic plates subjected to uiform trasverse pressure i order to demostrate at e DXDR aalysis produces coverged ad accurate results for orotropic materials. For ese comparisos it has bee ecessary to defie e elastic moduli of e orotropic plate materials to be used i e aalyses. Three materials were selected wi properties represetative of uidirectioal glass, carbo ad boro fiber reiforced polymers. The elastic costats, o-dimesioalized wi respect to e mior elastic modulus, are preseted i Table 1. The pressure-ceter deflectio resposes of glass-epoy, carbo-epoy ad boro-epoy plates are preseted i Figs. 5, 5 ad 5(c), respectively. Agai, Chia s [18] approimate aalytical solutio is used as e bechmark solutio for e compariso. It is evidet at e ceter deflectios predicted wi e Type 1 mesh (o-uiform i e directio) are i closer agreemet wi Chia s [18] values a e ceter deflectios predicted wi e Type mesh (ouiform i bo directios). It is presumed at e additioal o-uiformity of e mesh is resposible for e uderestimatio of e ceter deflectio respose by e Type mesh. It has to be poited out here at i orotropic plates e depedecy of results to e type of mesh is more severe a at for e isotropic plates. However, usig e Type 1 mesh is always more reliable i all cases. I order to eamie furer e differeces betwee e results obtaied wi bo uiform ad o-uiform meshes, e stress couple distributio alog e ais is computed for a square glass-epoy clamped plate whe e major orotropic ais is ormal to e ais. It is clear at ere is very good agreemet betwee e stress couples for bo uiform ad Type 1 o-uiform meshes. The Type o-uiform mesh appears to predict sigificatly smaller (i e egative sese) values for
M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 337 Fig. 6. Compariso of stress couple profiles alog e ais of a square glass-epoy plate subjected to uiform trasverse pressure (computed usig uiform ad Type 1 ad meshes) pricipal ais of orotropy parallel to e y ais. (c) Fig. 5. Compariso of load-ceter deflectio resposes of square orotropic clamped plates subjected to uiform trasverse pressure (computed usig Type 1 ad meshes): glass-epoy plate; carbo-epoy plate; (c) boro-epoy plate. Fig. 7. Deflectio profiles alog e ad y aes of square orotropic clamped plates subjected to uiform trasverse pressure computed usig uiform ad Type 1 meshes: major ais of orotropy i e directio; major ais of orotropy i e y directio. e stress couple whe / a 0.7, as show i Fig. 6. It is perhaps wor emphasizig at i e Type 1 mesh aalysis, show i Fig. 6, e major orotropic ais was parallel to y ais (e uiform mesh directio). This choice of directio for e major orotropic ais may be justified by comparig e results obtaied for e deflectio profiles alog e ad y aes. The clamped square orotropic plate was re-aalyzed twice wi bo e uiform ad ouiform Type 1 meshes, i.e. oce wi e major ais of orotropy parallel to e y ais ad oce wi major ais parallel to e ais. It is clear from e deflectio profiles alog e ad y aes, show i Fig. 7, at e results obtaied wi e major ais of orotropy parallel to e y ais are more accurate. It is well kow at for orotropic plates subjected to uiform trasverse pressure over e cetral regio of e plate e deflectio profile is flatter trasverse to e pricipal orotropic ais a it is parallel to is ais [18]. Moreover, for e Type 1 mesh, wi e pricipal ais of orotropy i e y directio, e mesh i e directio, i.e. trasverse to e major orotropic ais, is coarser earer to e ceter ad fier earer e edge of e plate. Hece, because e mesh is coarser i e flat regio of e plate, e error is smaller a if e coarse mesh is used i e directio of e major orotropic ais (e y directio). Oer o-uiform meshes may be geerated by usig differet values of e epoet y i Eq. (1b). Square glassepoy clamped plates subjected to uiform trasverse pressure
338 M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 Table. The values correspodig to each mesh lie for y = 1., 1.3 ad 1.4 (k = 1). i j y = 1. y = 1.3 y = 1.4 0 1 1 1 1 1 11 0.335796 0.31378 0.308149 10 0.89474 0.71840 0.5583 3 9 0.50980 0.3154 0.13571 4 8 0.1607 0.194986 0.176410 5 7 0.183100 0.161456 0.14700 6 6 0.15156 0.13003 0.111811 7 5 0.1103 0.100496 0.08360 8 4 0.091613 0.07995 0.05850 9 3 0.063537 0.047938 0.03608 10 0.037457 0.061 0.0189 11 1 0.014788 0.00899 0.005469 1 0 0 0 0 Fig. 9. A compariso betwee e covergece rate of two differet meshes: Type 1 mesh; mesh wi y = 1.4. y Fig. 8. Deflectio profiles alog e y ais of a square orotropic clamped plate subjected to uiform trasverse pressure computed usig uiform ad a rage of o-uiform meshes: pricipal ais of orotropy i e directio; pricipal ais of orotropy i e y directio. have bee aalyzed usig uiform ad o-uiform meshes defied by y = 1., 1.3 ad 1.4. Here, 1 ad it is calculated from Eq. (3) for each mesh lie (k = 1). Table shows e values for y = 1., 1.3 ad 1.4. The deflectio profiles alog e y ais obtaied wi e uiform ad o-uiform meshes are show i Fig. 8. It is clear at as e epoet icreases e o-uiformity of e mesh icreases ad e calculated deflectios are over-estimated compared to e values obtaied from e uiform mesh aalyses. For eample, whe y = 1.4 e mid-spa deflectio is about twice of at predicted wi e uiform mesh. Geerally e mesh used to aalyze e orotropic curved boudaries should have adequate accuracy as well as satisfactory covergece rate. As aforemetioed, e Type 1 mesh could provide sufficiet accuracy i almost all cases. Furermore, is type of mesh could also give higher covergece rate. The Figs. 9 ad 9 show e variatio of residual eergy wi iteratio umber for Type 1 mesh ad for e mesh wi y = 1.4, respectively. The o-uiform mesh may ot just be used to aalyze square isotropic ad orotropic plates. It may also be used to aalyze isotropic ad orotropic circular ad elliptical plates. It is assumed at e aes of orotropy are parallel to e ad y aes. I order to demostrate e fleibility ad accuracy of e Type 1 mesh e large deflectio respose of a clamped isotropic circular plate subjected to uiform trasverse pressure is aalyzed. The ceter deflectio-pressure respose is show i Fig. 10 ad compared wi e results give i Ref. [18] ad also wi e results obtaied by usig a DXDR polar co-ordiate fiite-differece aalysis of e problem. I order to demostrate e ability of e Type 1 mesh for bo e small ad large deflectio resposes of clamped orotropic circular plates, two furer results comparisos are preseted for a glass-epoy plate. A small deflectio aalysis of e load-deflectio respose for e case of uiform tras-
M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 339 Fig. 10. Load-ceter-deflectio respose of a clamped isotropic circular plate subjected to uiform trasverse pressure, compariso of Type 1 mesh ( y = 1, 1 ) ad oer approimate resposes. Fig. 13. Large deflectio of e load-ceter deflectio respose of a movable clamped isotropic circular plate subjected to uiform trasverse pressure, compariso of Type 1 mesh ( y = 1, 1 ) ad approimate resposes. Fig. 11. Small deflectio aalysis of e load-ceter deflectio respose of a clamped orotropic circular plate subjected to uiform trasverse pressure, compariso of Type 1 mesh ( y = 1, 1 ) ad eact resposes. Fig. 1. Deflectio profiles alog e ad y aes of a clamped orotropic circular plate subjected to uiform trasverse pressure, profiles computed wi a Type 1 mesh ( y = 1, 1 ) wi e major ais of orotropy parallel to e y ais. verse pressure loadig is show i Fig. 11. As epected, e load-deflectio respose is liear ad e Type 1 mesh results are i good agreemet wi e eact results give i Ref. [19]. Large deflectio aalysis results for e Type 1 mesh are show i Fig. 1. The deflectio profiles i e ad y directios are preseted for e pricipal aes of orotropy parallel to e y directio wi e o-uiformity of e mesh i e directio, Type 1 mesh. The Type oe mesh is also capable to ivestigate e circular ad elliptical plates wi movable boudary coditio i bo fied ad simply-supported cases. Fig. 13 shows e large deflectio aalysis of a movable clamped isotropic circular plate subjected to uiform trasverse pressure. As it is see, ere is a good agreemet betwee e results obtaied usig e Type 1 mesh ( y = 1, 1 ) ad e Baerjee results [0]. Moreover, e employed mesh ca also be used to aalyze e isotropic elliptical plates. The results obtaied by e aid of curret study ad ose of Baerjee [1] are compared to each oer i Figs. 14-15 for movable clamped ad simplysupported edges, respectively. Figs. 14-15 ad 14(c)- 15(c) show e profiles alog e ad y aes ad e stress couple profiles alog e ais for e elliptical plate for e metioed boudary coditios. Furermore, e meod ca be eteded to aalyze e large orotropic circular ad elliptical plates wi clamped ad simply-supported movable edges. Figs. 16 ad 17 show e results obtaied from curret study for glass-epoy orotropic plates. Geerally, by e aid of proposed mesh ad meod eplaied above it is ot oly possible to aalyze plates wi differet boudaries but e adequate covergece ad accuracy are also achievable. For istace, e Table 3 shows e accuracy of e results which is attaiable i is meod compared to oer published data. A quick glace at Figs. 4-17 displays e capability of e proposed meod ad at it could properly treat e circular ad elliptical boudaries while Cartesia aes are used. The employed proposed irregular mesh may be applied to ay oer curved boudaries wi more complicated shapes which are especially useful to aalyze e orotropic plates whe e reiforced fibers are placed i rectiliear directios. I fact, it quite happes at e reiforced sheets are made iitially i large scale ad e e plates are cut out wi a arbitrary shape ad boudary form.
340 M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 (c) Fig. 14. Large deflectio aalysis of a movable clamped isotropic elliptical plate subjected to uiform trasverse pressure (a/b = ): loaddeflectio curve ad compariso of Type 1 mesh ( y= 1, 1 ) ad approimate resposes; profiles alog e ad y aes; (c) stress couple profiles alog e ais. (c) Fig. 15. Large deflectio aalysis of a movable simply-supported isotropic elliptical plate subjected to uiform trasverse pressure (a/b = ): load-deflectio curve ad compariso of Type 1 mesh ( y = 1, 1) ad approimate resposes; profiles alog e ad y aes, (c) stress couple profiles alog e ais.
M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 341 Table 3. A geeral compariso betwee e curret results wi ose of oers published data. Large deflectio aalysis Curret result Referece result Average differece (%) Square orotropic clamped plates (Glass-epoy) Fig. (5) Chia [18].74 Clamped isotropic circular plate Fig. (10) Chia [18].6 Movable clamped isotropic circular plate Fig. (13) Baerjee [0].5 Movable clamped isotropic elliptical plate Fig. (14) Baerjee [1] 1.96 Movable simply-supported isotropic elliptical plate Fig. (15) Baerjee [1] 0.68 best results whe e o-uiformity is trasverse to e major ais of orotropy. Fig. 16. Large deflectio aalysis of e load-ceter deflectio respose of a movable simply-supported ad clamped glass-epoy orotropic circular plate subjected to uiform trasverse pressure usig Type 1 mesh ( y = 1, 1 ). Fig. 17. Large deflectio aalysis of e load-ceter deflectio respose of a movable simply-supported ad clamped glass-epoy orotropic elliptical plate subjected to uiform trasverse pressure usig compariso of Type 1 mesh ( y = 1, 1 ). 6. Cocludig remarks Clamped ad simply-supported square, circular ad elliptical plates wi Cartesia orotropy subjected to uiform trasverse pressure have bee aalyzed usig e DXDR techique i cojuctio wi bo uiform ad o-uiform Cartesia fiite-differece meshes. Bo fied ad movable boudary coditios have bee ivestigated. It has bee show at e use of o-uiform fiite-differece meshes permits square, circular ad elliptical plate geometries to be aalyzed usig e same Cartesia mesh. Moreover, bo e small ad large deflectio resposes of e plates may be predicted wi good accuracy provided e o-uiformity of e mesh is ot ecessive. It appears at, for orotropic plates, e Type 1 mesh (o-uiform i oe directio oly) gives Ackowledgmet The research reported i is paper was carried out durig part of e first auor s sabbatical leave spet i e Egieerig Departmet at Lacaster Uiversity. The first auor wishes to record his appreciatio to e Iraia Miistry of Higher Educatio ad Ferdowsi Uiversity for providig fiacial support durig his sabbatical leave. Auors wish to record eir appreciatio to e Egieerig Departmet for supportig is research collaboratio. Refereces [1] C. Y. Chia, Large deflectio of rectagular orotropic plates, ASCE Proceedigs, Joural of e Egieerig Mechaics Divisio, 98 (5) (197) 185-198. [] C. Y. Chia ad M. K. Prabhakara, Noliear aalysis of orotropic plates, proceedigs of e istitutio of mechaical egieers, Joural of Mechaical Egieerig Sciece, 17 (3) (1975) 133-138. [3] M. Dalaei ad A. D. Kerr, Aalysis of clamped rectagular orotropic plates subjected to a uiform lateral load, Iteratioal Joural of Mechaical Scieces, 37 (5) (1995) 57-535. [4] F. C. Mbakogu ad M. N. Pavlovic, Bedig of clamped orotropic rectagular plates: a variatioal symbolic solutio, Computers ad Structures, 77 () (000) 117-18. [5] P. Tabakov, V. Verijeko ad B. Verijeko, Refied eory for e aalysis of lamiated orotropic structures, Composite Structures, 6 (3-4) (003) 435-441. [6] S. Mistou, M. Karama ad M. Sabarots, Eperimetal ad umerical simulatios of e static behavior of orotropic plates, Measuremet, 30 (3) (001) 197-10. [7] K. Bhaskar ad B. Kaushik, Aalysis of clamped usymmetric cross-ply rectagular plates by superpositio of simple eact double fourier series solutios, Composite Structures, 68 (3) (005) 303-307. [8] M. Salehi ad A. R. Sobhai, Elastic liear ad o-liear aalysis of fibers-reiforced symmetrically lamiated sector midli plate, Composite Structures, 65 (1) (004) 65-79. [9] M. Kadkhodaya, L. C. Zhag ad R. Sowerby, Aalysis of wriklig ad bucklig of elastic plates by DXDR meod,
34 M. Kadkhodaya et al. / Joural of Mechaical Sciece ad Techology 6 (10) (01) 331~34 Computers ad Structures, 65 (4) (1997) 561-574. [10] M. Kadkhodaya, Usig iterlacig ad irregular meshes i plate aalysis, Iraia Joural of Sciece ad Techology, 6 (B3) (00) 419-430. [11] J. M. Whitey, Structural aalysis of lamiated aisotropic plates, Techomic Publishig Compay, New York (1987). [1] L. C. Zhag, M. Kadkhodaya ad Y. W. Mai, Developmet of e madr meod, Computers ad Structures, 5 (1) (1994) 1-8. [13] P. Uderwood, Dyamic Relaatio, i T. Belytshko ad T.J.R. Hughes, eds., Computatioal Meod for Trasiet Aalysis, Chapter 5, Elsevier, Amsterdam (1983) 45-56. [14] W. T. Thomso, Vibratio eory ad applicatios, Pretice-Hall, 5 Editio, New York (1997). [15] S. Qiag, A adaptive dyamic relaatio meod for oliear problems, Computers ad Structures, 30 (4) (1988) 855-859. [16] L. C. Zhag ad T. X. Yu, Modified adaptive dyamic relaatio meod ad its applicatio to elastic-plastic bedig ad wriklig of circular plates, Computers ad Structures, 33 () (1989) 609-614. [17] S. E. Ha ad K. S. Lee, A study of e stabilizig process of ustable structures by dyamic relaatio meod, Com- puters ad Structures, 81 (17) (003) 1677-1688. [18] C. Y. Chia, Noliear Aalysis of Plates, McGraw-Hill, New York (1980). [19] S. G. Lekhitskii, Aisotropic Plates, Gordo ad Breach Sciece, New York (1968). [0] B. Baerjee, Large deflectios of circular plates of variable ickess, Iteratioal Joural of Solids ad Structures, 19 () (1983) 179-18. [1] G. C. Siharay ad B. Baerjee, A modified approach to large deflectio aalysis of i elastic plates, Iteratioal Joural of Mechaical Scieces, 8 (3) (1986) 173-177. Mehra Kadkhodaya received his B.Sc. ad M.Sc. degrees from Tehra Uiversity, Ira i 1987 ad Ph.D degree from e Uiversity of Sydey, Australia i sheet metal formig area i 1996. Curretly as a full professor he is workig i Ferdowsi Uiversity of Mashhad, Ira. He has published about 36 joural papers ad more a 50 coferece papers.