RPT nite-element formulation for linear dynamic analysis of orthotropic plates

Transcription

1 Scintia Iranica B (01) 5(), 13{3 Sharif Univrsity of Tchnology Scintia Iranica Transactions B: Mchanical Enginring RPT nit-lmnt formulation for linar dynamic analysis of orthotropic plats J. Rouzgar and M. Saydain Dpartmnt of Mchanical and Arospac Enginring, Shiraz Univrsity of Tchnology, Shiraz, P.O. Box , Iran. Rcivd 1 Novmr 016; rcivd in rvisd form 10 January 017; accptd 6 March 017 KEYWORDS Astract. This papr prsnts a nit-lmnt formulation for dynamic analysis of 1. Introduction shar corrction factor to r n th ovrstimatd shar strain nrgy of th plat. Anothr drawack to this thory is th constant transvrs shar strss across th plat thicknss which is inconsistnt with strssfr conditions on th plat surfacs. Highr-ordr Shar Dformation Thoris (HSDTs) wr dvlopd using mor unknown varials in thir formulations to ovrcom shortcomings of CPT and FSDT [4-10]. Rcntly, svral simpl and cint HSDTs hav n proposd and mployd in study of various structurs such as functionally gradd, laminatd, and sandwich plats and shlls [11-13]. In 00, Shimpi [13] prsntd a simpl HSDT calld two-varial R nd Plat Thory (RPT), which involvs only two unknown functions. H sparatd th transvrs d ction into th shar and nding parts which rsultd in two uncoupld govrning quations for nding analysis of isotropic plats. Th two-varial r nd plat thory, which can usd for thin and thick plats, prdicts paraolic variation of transvrs shar strsss across th plat thicknss and dos not nd shar corrction factor. In this thory, zro traction conditions ar Orthotropic; Finit-lmnt mthod; Dynamic analysis; Two-varial r nd plat thory; Rctangular plat lmnt. orthotropic plats using two-varial R nd Plat Thory (RPT). Hamilton's principl is mployd to otain th govrning quations, and th smi-discrt approach is utilizd for solving th quations. Aftr constructing spatial wak form quations, a 4-nod rctangular plat lmnt with six Dgrs Of Frdom (DOFs) pr nod is introducd for discrtization of th domain. An unconditionally stal implicit Nwmark schm is usd for tmporal discrtization. A MATLAB cod with th capaility of modlling oth static and dynamic plat prolms with various oundary conditions is gnratd. Svral numrical prolms ar solvd, and th otaind displacmnts and strsss ar compard with th xisting rsults in th litratur. Th rsults dmonstrat th accuracy, simplicity, and cincy of th prsnt mthod in dynamic analysis of plat prolms. 01 Sharif Univrsity of Tchnology. All rights rsrvd. Plats ar structural lmnts commonly usd in many industrial applications,.g. arospac, marin, nuclar industris, tc. Thy ar classi d in th litratur as thin, modratly thick, or thick plat. Th classi cation of a rctangular plat into on of th thr aformntiond catgoris is asd on rlation ha : h ing th plat thicknss and a ing th shortst sid lngth. In th framwork of Classical Plat Thory (CPT) [1], which is th simplst plat thory, th rsults ar unralistic in th cas of thick plats caus it dos not tak into account shar dformation cts. First-ordr Shar Dformation plat Thory (FSDT) prsntd y Mindlin [] and Rissnr [3] considrs th shar cts in its formulation, ut it nds a *. Corrsponding author. Tl./Fax: addrss: rouzgar@sutch.ac.ir (J. Rouzgar) doi: /sci

2 14 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 satisd on th plat surfacs. This thory was rapidly dvlopd and xtndd to various applications. In 006, Shimpi and Patl [14] mployd this this mthod for analysis of orthotropic plats. In 00, Kim t al. [15] xtndd th application of this thory to analysis of laminatd composit plat. Thy invstigatd th nding of antisymmtric cross-ply and angl-ply laminats using Navir solution. In 01, Thai and Kim [16] usd th two-varial rnd plat thory, Lvy-typ solution procdur, and stat spac concpt to otain a closd-form solution for orthotropic rctangular plats with two opposit dgs simply supportd and th othr two dgs having aritrary oundary conditions. Rouzgar and Adoli prsntd nit-lmnt formulations for fr viration [17] and uckling [1] analysis of isotropic and orthotropic plats using this thory. By adding two in-plan displacmnts to this thory, a four-varial rnd plat thory was introducd. Rouzgar and Aad mployd this thory for nding [1] and fr viration [0] analysis of cross-ply laminatd plats intgratd with pizolctric layrs. Rouzgar and Gholami [1] studid th non-linar nding analysis of rctangular plats y four-varial rnd plat thory and Dynamic rlaxation mthod. Th practical and typically complicatd prolms could solvd in an approximat mannr mploying various numrical approachs, such as nit-layr mthod, collocation mthod, nit-lmnt mthod, nit strip mthod, and mshlss mthods. Among diffrnt numrical approachs, Finit-Elmnt Mthod (FEM) is an cint and attractiv tchniqu stalishd in th 140's. Rcntly, som invstigations hav n prformd on isogomtric nit-lmnt approach utilizing shar dformation plat thoris. Nguyn-Xuan t al. [] mployd isogomtric nitlmnt mthod with a rnd plat thory for static, fr viration and uckling analysis of FG plats. Thai t al. [3] prsntd a gnralizd shar dformation thory for static, dynamic and uckling analysis of isotropic and sandwich FG plats. An invrs trigonomtric shar dformation thory along with an isogomtric approach was mployd in this study. Nguyn and Nguyn-Xuan [4] prsntd an isogomtric nit-lmnt approach to thr-dimnsional static and dynamic analyss of FG plats. Nguyn t al. [5] introducd a novl unid framwork on HSDTs for th modling and analysis of laminatd composit plats. Thy found that th proposd formulation with a polynomial form can thortically covr all xisting HSDTs modls and is thus sucint to dscri th nonlinar and paraolic variations of transvrs shar strss. Nguyn t al. [6] prsntd an isogomtric nit-lmnt formulation asd on Bzir xtraction of th non-uniform rational B-splins in comination with a gnralizd unconstraind highrordr shar dformation thory for laminatd composit plats. Prvious rsarchs on two-varial rnd plat thory ar mostly focusd on analytical solutions to som plat prolms with spcic gomtry, loading and oundary conditions. A nw nit-lmnt formulation of shar dformation plat was introducd y Patl and Shimpi [7]. Katori and Okada [] dvlopd a nit-lmnt formulation asd on th rnd plat thory using triangular and quadrilatral lmnts for discrtization of th domain. Rcntly, Rouzgar and Adoli [] prsntd a nit-lmnt formulation for static analysis of isotropic and orthotropic plats using a two-varial rnd plat thory. Although dynamic analysis of plat structurs is th sujct of a larg numr of rsarch works, thr ar still som hypothss ithr not introducd or not sucintly tstd. In this study, a nw nit-lmnt formulation asd on th two-varial rnd plat thory is dvlopd for dynamic analysis of isotropic and orthotropic plats. Th smi-discrt mthod is usd to simulat th dynamic havior of prolms, and an implicit Nwmark schm is mployd for tmporal discrtization. Th prformanc of th cod in simulation of isotropic and orthotropic plats undr static and dynamic loadings is provd y solving svral nchmark prolms.. Thory of prolm.1. Two-varial rnd plat thory Considr a homognous, orthotropic, thick lastic plat of thicknss, h, which is sujctd to a transvrs dynamic load, q(x; y; t), pr unit ara. According to Figur 1, axs x, y, and z ar th Cartsian coordinat systms locatd at th cornr of th plat. Basd on two-varial rnd plat thory, th displacmnt ld is assumd as follows [13]: u(x; y; z; t) v(x; y; z; ; s ; w(x; y; t) w (x; y; t) w s (x; y; t); (1) Figur 1. Gomtry and coordinat systm of th prolm.

3 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 15 whr u and v ar th displacmnts in x and y dirctions, w and w s ar nding and shar componnts of transvrs displacmnt w, and: f(z) z 4 5z z : () 3 h Rgarding th assumption of small displacmnts, th innitsimal strain componnts ar otaind as follows: " x z x f(z) s x; " y z y f(z) s y; xy z xy f(z) s xy; yz g(z) s yz; xz g(z) s xz; " z 0; (3a) whr: x y xy s ; g(z) ; s w w ; s w s ; s w ; df(z) dz s s ; 5 4 z 5 : (3) h Nglcting normal strss, z, th othr strss componnts for orthotropic plat ar otaind y th following constitutiv rlations: >< >: x y xy yz zx 6 4 whr: Q 11 > >; Q 11 Q Q 1 Q Q Q Q 55 E ; 3 >< 7 5>: " x " y xy yz zx > >; : (4) Q 1 1E E ; Q E ; Q 44 G 3 ; Q 55 G 31 ; Q 66 G 1 ; (5) whr E 1, E, G 1, G 3, and G 31 ar lastic moduli, and 1 and 1 ar Poisson's ratios... Govrning quations for dynamic prolms Th govrning quations and oundary conditions ar otaind using Hamilton's principl: 0 t t 1 (T )dt: (6) Total potntial nrgy of th plat,, is th sum of strain nrgy and work don y xtrnal loads: zh y xa z h y0 x0 a yz yz zx zx dxdydz 1 x " x y " y xy xy 0 0 qwdxdy; (7) whr q is th normal xtrnal prssur applid onto th top surfac of plat. Total kintic nrgy of th plat (T ) can writtn as follows: zh y T z h @w dxdydz: By sustituting strss, strain, and displacmnt lds into Eqs. (6)-() and taking into account th indpndnt variations of w and w s, th govrning dirntial quations ar otaind as follows: D 4 4 (D 1 D w 4 h (r w w w )q; () 4 w s D 11 4 (D 1 D 66 w 4 w s w s A w s 44 h (r w s w q; (10)

4 16 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 whr D 11, D, D 1, D 66, A 44, and A 55 ar matrial stinsss of th plat: D 11 Q 11h 3 1 ; D Q h 3 1 ; D 1 Q 1h 3 1 ; D 66 Q 66h 3 1 ; A 44 5Q 44h 6 v ; A 55 5Q 55h : (11) 6.3. Finit-lmnt formulation for dynamic prolms Appling th Hamilton's principl for a plat lmnt with volum, v, and mid-plan surfac,, th following quation will @t 1 v ( xx " xx yy " yy xy xy xz xz yz yz )dv I V nw qwdxdy ds 0; (1) whr V n and M nn ar th ctiv shar forc and nding momnt, rspctivly. Sustituting th strss and strain and displacmnts lds into Eq. (1), w will hav: (T U) h @t w w D w D w 4D 1 w s D w w w s D w w s 4D w A w dxdy s I q(w w s )dxdy V n (w w s s w M (w w s ds: (13) Th wak form of govrning quations is otaind y quating variation of sum of kintic and total potntial nrgis to zro: I 0 (w ) T :: w I (D 1 w ) T :: (D 1 w ) I 0 (w s ) T :: w s I 4 (D 1w s ) T :: (D 1 w s ) dxdy :: I0 ( w w :: s )(w w s ) dxdy (D w ) T D(D w ) 1 4 (D w s ) T D(D w s ) (D 1 w s ) T A(D 1 w s ) dxdy h i (w w s ) T q dxdy I (@(w w s T M nn [(w w s )] T V n ds 0; in which I 0, I, A, D, D 1, and D ar: I 0 h; I h3 1 ; (14)

5 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 17 A D 1 A44 0 ; D 0 A 4 D 11 D 1 0 D 1 D D ; i T : (15) In ach lmnt, th nding and shar componnts of transvrs displacmnt ar computd as follows: w (x; y) w s (x; y) nx j1 nx j1 j' j (x; y) N T ; s j' j (x; y) N T s ; (16) whr, s, ' j, and N ar nding DOFs, shar DOFs, intrpolating functions, and shap functions, rspctivly. Th two-varial RPT nit-lmnt quations ar otaind y sustituting Eq. (16) into Eq. (14): M 11 M 1 M 1 whr: M F F >< >: :: :: s > >; K K s ; (17) K 11 (B T DB )dxdy; K 1 4 B T DB B T 1 AB 1 dxdy; M 11 I 0 NN T I B T 1 B 1 dxdy; M I 0 NN T I 4 B 1 T B 1 dxdy; M 1 F Nqdxdy I 0 NN T dxdy; I whr B 1 and M nsnq n ; (1) B 1 D 1 N; B D N: (1).4. Linar rctangular lmnt A non-conforming four-nod rctangular lmnt with th following six dgrs of frdom pr nod is dnd for discrtizing th plat domain: DOFs : @w w s s o : (0) If a and a s dnot lmntal nding and shar DOFs and a i and as i dnot nding and shar nodal DOFs, w can writ that: a a s >< >: >< >: a i a j a k a l a i s a j s a k s a l s > >; whr : a i > >; whr: as i < : < : /@x w s s /@x ; ; ; : (1) Th shap functions for discrtizing nding and shar componnts of transvrs dction ar dnd as follows [30]: N T [' 1 ' ' 3 ::: ' 1]; ' i g i1 (i 1; 4; 7; 10); whr: g i1 1 (1 0)(1 0 )( 0 0 ); ' i g i (i ; 5; ; 11); whr: g i 1 1( 0 1)(1 0 )(1 0 ) ; ' i g i3 (i 3; 6; ; 1); whr: g i3 1 1( 0 1)(1 0 )(1 0 ) ; (x x c) ; a (y y c) / ; 0 i ; i : () As illustratd in Figur, a and ar half-width and half-lngth of lmnt, and x c and y c ar th coordinats of th mid-point of th lmnt.

6 1 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 Figur. Rctangular plat lmnt..5. Nwmark formulation for tmporal discrtization By ignoring th damping cts, th following hyprolic dirntial quation can writtn for lastodynamic havior of many nginring structurs: M d u dt Ku F; (3) whr M and K ar th mass and stinss matrics, rspctivly, and F is th xtrnal forc vctor. Th dynamic rspons of structur is solvd using th wllknown Nwmark-ta mthod. Th constant avrag acclration mthod, which is unconditionally stal for linar prolms, is adoptd in this rsarch [31]. 3. Rsults and discussions A FE cod asd on th prsntd formulation is gnratd using MATLAB softwar. Som nchmark prolms ar solvd y th cod, and th otaind rsults ar compard with th xisting rsults in th litratur. Exampl 1. Considr a simply supportd squar plat sujctd to a uniformly distriutd load of p 0 10 lf/ft which is suddnly applid to plat and rmains constant during tim. Th matrial and gomtric paramtrs ar considrd as: Young's modulus E psi, Poisson's ratio 0:3, Wight dnsity 157:5 l/ft 3, Thicknss h 0:5 in, sid lngth, Width a ft. Th static rspons of plat can otaind analytically using Navir solution. In this mthod, two trigonomtric sris ar usd for indpndnt varials w and w s as in Eqs. (4) and (5): w (x; y) w s (x; y) m1 n1 m1 n1 w mn sin w s mn sin mx ny sin ; (4) a mx ny sin : (5) a Ths xprssions satisfy oundary conditions, automatically. Th distriutd load is stimatd as doul Fourir xpansion sris according to Eq. (6): q(x; y) m1 n1 q mn sin( mx a Cocint q mn is otaind y: q mn a 0 0 q(x; y) sin( mx a ) sin(ny ): (6) ) sin(ny )dxdy: (7) By sustituting displacmnt and load functions into govrning Eqs. () and (10), th following rlations ar otaind: w mn w s mn q mn D 4 (m a n ) ; () q mn D 4 4 (m a n ) 5Gh 6 (m a n ) ; () whr D and G ar dnd as follows: D Eh 3 1(1 ) ; G E (1 ) : (30) Static displacmnt of cntral point of plat can otaind analytically y sustituting Eqs. () and () into Eqs. (4) and (5) and taking into account suf- cint numr of trms of sris to fulll convrgd rsult: w(4; 4) w (4; 4) w s (4; 4) 1:76 : :7606 in: (31) Th static analysis of prolm is also prformd y Static FF-RPT cod. Th ct of numr of lmnts on cntral dction of th plat is invstigatd in Tal 1. By incrasing th numr of lmnts, th otaind rsults convrg to analytical valu. According to this tal, considring 10 lmnts in ach plat sid is sucint to achiv convrgd rsult. Th dynamic rspons of th plat undr prscrid loading is prformd using Dynamic FE-RPT cod. Th computational tim is considrd 0.15 s which is dividd into 1500 tim stps of 0.1 ms; th otaind transvrs dction of cntral point of th

7 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 1 Tal 1. Convrgnc study for Exampl 1. Msh dsign w (in) Figur 5. Tim history of dction of cntral point of rctangular plat sujctd to suddnly applid patch load. Young's modulus E 1 l/in, Poisson's ratio 0:3, Mass dnsity 1 l.sc /in 4, Thicknss h 0:in. Figur 3. Comparison of cntral dction of squar plat sujctd to uniform loading. plat is shown in Figur 3. It is sn that th rsults of th prsntd formulation ar in good agrmnt with thos of nit strip mthod rportd in [3], and th maximum dirnc twn ths two mthods is lss than 10%. Furthrmor, as xpctd, for th applid loading, th dynamic rspons of plat oscillats twn 0 and twic th static valu. Exampl. Considr a simply supportd rctangular plat undr uniformly distriutd patch load of p 0 0:01 psi, as shown in Figur 4. Th load is suddnly applid to th patch ara and rmains constant during applid tim. Th gomtry and matrial paramtrs ar considrd as: Figur 5 compars th tim history of transvrs dction of cntral point of th plat otaind y th proposd Dynamic FE-RPT formulation with th rsult of nit strip mthod, and good agrmnt is osrvd twn th rsults of two mthods. Th analysis is prformd y considring tim stp of 1 ms, and ach plat sid is discrtizd into 10 lmnts. Exampl 3. A simply supportd rctangular plat of (60 in 40 in 1 in) with matrial proprtis of E psi, 0:3, and 0:00073 l.sc /in 4 is considrd. Th plat is sujctd to a uniform prssur having triangular tim variations as shown in Figur 6(a). Th dynamic rspons of th plat is otaind using th dynamic FE-RPT cod considring a tim stp of 0.01 ms. Du to th symmtry of gomtry and loading, half plat is discrtizd y a msh of 1 : Figur 6() compars th otaind dynamic rspons of plat with th rsults of oundary lmnt mthod rportd in [33], showing good agrmnt twn th rsults. Figur 4. Rctangular plat sujctd to suddnly applid patch load [3]. Figur 6. Cntral dction of rctangular plat sujctd to a uniform prssur having triangular tim variation.

8 0 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 Tal. Matrial proprtis of orthotropic plat. E1 (GPa) EE1 G1E1 G13E1 G3E1 1 1 (kg/m 3 ) Figur 7. Tim variation of air last prssur acting on a squar plat [34]. Figur. Normalizd cntral dction of orthotropic plat considring dirnt thicknss ratios, whr a 1. Figur 10. Normalizd cntral strss, x, of orthotropic plat considring dirnt thicknss ratios, whr a 1. Figur. Tim history of cntral dction of squar plat sujctd to air last. Exampl 4. A squar plat with lngth of 50 mm, thicknss of 3.4 mm, and matrial proprtis of E 06:4 GPa, 0:3, and 700 kg/m 3 is considrd. All dgs ar clampd, and th plat is sujctd to air last prssur shown in Figur 7. Th otaind tim history of cntral dction of plat is compard with xprimntal [34] and nit strip mthod [3] rsults in Figur, whr good agrmnt is osrvd twn th rsults. A msh of 1 1 is usd for discrtization of half plat, and tim stp is st to 0.01 ms Paramtric study In this sction, an orthotropic plat undr uniformly distriutd loading with various oundary conditions, including fully simply support (SSSS), fully clamp (CCCC), and simply-clamp (SSCC), is considrd, and cts of dirnt paramtrs, such as aspct ratios, orthotropy ratio (E 1 E ), and thicknss ratio, ar studid on normalizd transvrs dctions, in-plan normal strsss, and transvrs shar strsss. Th analyss ar prformd using a msh of for th half plat and a tim stp of 0.01 ms. Matrial proprtis ar chosn according to Tal, and th following normalizd paramtrs ar introducd as follows: w wq 11 /hq 0 ; x x /q 0 ; y y /q 0 ; xz xz /q 0 : (3) Figurs and 10 show th normalizd transvrs dction and normal strss of plat for dirnt thicknss ratios. As sn, th dynamic rsponss oscillat twn zro and twic th static valus, and th incrmnt of th thicknss ratio incrass th stinss of plat, which rsults in th incras of frquncy and dcras of amplitud of rsults. Th ct of oundary condition on normalizd transvrs dction and shar strss is invstigatd, as shown in Figurs 11 and 1. As xpctd, y changing oundary conditions from fully simply supportd to fully clampd and, consquntly, incrasing th constraints on plat, th amplitud and frquncy of rsults dcras and incras, rspctivly.

9 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 1 Figur 11. Normalizd cntral dction of orthotropic plat considring dirnt oundary conditions, whr ha 0:1 and a 1. Figur 14. Normalizd cntral normal strss, x, of orthotropic plat considring dirnt aspct ratios, whr ha 0:1. Figur 1. Normalizd cntral out-of-plan shar strss, xz, of orthotropic plat considring dirnt oundary conditions, whr ha 0:1 and a 1. Figur 15. Normalizd cntral out-of-plan shar strss, xz, of orthotropic plat considring dirnt aspct ratios, whr ha 0:1. Figur 13. Normalizd cntral dction of orthotropic plat considring dirnt aspct ratios, whr ha 0:1. In Figurs 13-15, cts of plat aspct ratio on normalizd dctions and strsss ar studid. Paramtr a is constant in all analyss; y incrasing a ratio and, consquntly, dcrasing th plat dimnsion (), frquncy and amplitud of rsults oviously incras and dcras, rspctivly. Th ct of orthotropy ratio (E 1 E ) is invstigatd on dynamic rspons of plat in Figurs 16 and 17. Paramtr E 1 is kpt constant in all anal- Figur 16. Normalizd cntral dction of orthotropic plat considring dirnt orthotropy ratios, whr ha 0:1 and a 1. yss. Thrfor, y incrasing E 1 E ratio, E and, consquntly, th ovrall stinss of structur dcras; morovr, frquncy and amplitud of x and xz dcras and incras, rspctivly. 4. Conclusion In this study a nit-lmnt formulation for dynamic analysis of isotropic and orthotropic plats is dvlopd

10 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{3 Figur 17. Normalizd cntral out-of-plan shar strss, xz, of orthotropic plat considring dirnt orthotropy ratios, whr ha 0:1 and a 1. using th two-varial rnd plat thory. This thory, which can usd for oth thin and thick plats, prdicts paraolic variation of transvrs shar strsss across th plat thicknss. In comparison with FSDT with thr unknown paramtrs and th HSDTs with mor than thr paramtrs, this thory has only two paramtrs; thrfor, th simplicity and accuracy ar th main faturs of th prsnt formulation. Th main novlty of this work is constructing a dynamic nit-lmnt modl asd on th two-varial rnd plat thory for analysis of plat prolms. Aftr constructing wak-form quations using th Hamilton's principl, a nw 4-nod rctangular plat lmnt with six dgrs of frdom at ach nod is introducd for discrtization of th domains. Th smidiscrt approach is adoptd to dal with th dynamic analysis, and an unconditionally stal implicit Nwmark schm is usd for tmporal discrtization. Th cincy and accuracy of th prsntd formulation ar provd y solving som nchmark isotropic and orthotropic plat prolms. Th otaind rsults ar in good agrmnt with analytical solutions of common plat thoris. Morovr, th cts of aspct ratio, thicknss-to-sid ratio, matrial proprtis and typ of oundary conditions on dctions and strsss ar invstigatd. Th rsults show that th two-varial rnd plat thory is simpl and cint in comparison with prvious highr-ordr shar dformation plat thoris; morovr, th prsnt nit-lmnt formulation can a vry usful tool for simulation of oth thin and thick plat prolms undr static and dynamic loadings. Th prsnt study can xtndd to othr plat structurs, such as functionally gradd, laminatd composit, and smart plats. Rfrncs 1. Kirchho, G. \Equilirium and motion of an lastic disc [ AUr das glichgwicht und di wgung inr lastischn schi", J. RinAngw. Math., 15(40), pp. 51- (150).. Rissnr, E. \Th ct of transvrs shar dformation on th nding of lastic plats", J. Appl. Mch., 1(), pp (145). 3. Mindlin, R.D. \Inunc of rotary inrtia and shar on xural motions of isotropic lastic plats", J. Appl. Mch., 1(1), pp (151). 4. Whitny, J.M. and Sun, C.T. \A highr ordr thory for xtnsional motion of laminatd composits", Sound Vi., 30(1), pp. 5-7 (173). 5. Hanna, N.F. and Lissa, A.W. \A highr ordr shar dformation thory for th viration of thick plats", Sound Vi., 170(4), pp (14). 6. Rddy, J.N. \A simpl highr-ordr thory for laminatd composit plats", ASME. J. Appl. Mch., 51(4), pp (14). 7. Rddy, J.N. and Phan, N.D. \Staility and viration of isotropic, orthotropic and laminatd plats according to a highr-ordr shar dformation thory", Sound Vi., (), pp (15).. Bhimaraddi, A. and Stvns, L.K. \A highr ordr thory for fr viration of orthotropic, homognous, and laminatd rctangular plats", J. Appl. Mch., 51(1), pp (14).. Kant, T. \Numrical analysis of thick plats", Comput. Mthods Appl. Mch. Eng., 31(1), pp. 1-1 (1). 10. Lo, K.H., Christnsn, R.M., and Wu, E.M. \A highordr thory of plat dformation, Part 1: Homognous plats", J. Appl. Mch., 44(4), pp (177). 11. Ghugal, Y.M. and Sayyad, A.S. \A static xur of thick isotropic plats using trigonomtric shar dformation thory", J. Solid. Mch., (1), pp. 7-0 (010). 1. El Mich, N., Tounsi, A., ian, N., Mcha, I., and Adda Bdia, E.A. \A nw hyprolic shar dformation thory for uckling and viration of functionally gradd sandwich plat", Int. J. Mch. Sci., 53(4), pp (011). 13. Shimpi, R.P. \Rnd plat thory and its variants", AIAA J., 40(1), pp (00). 14. Shimpi, R.P. and Patl H.G., \A two varial rnd plat thory for orthotropic plat analysis", Int. J. Solids Struct., 43(-3), pp (006). 15. Kim, S.E., Thai, H.T., and L, J. \A two varial rnd plat thory for laminatd composit plats", Compos. Struct., (), pp (00). 16. Thai, H.T. and Kim, S.E. \Analytical solution of a two varial rnd plat thory for nding analysis of orthotropic Lvy-typ plats", Int. J. Mch. Sci., 54(1), pp (01). 17. Rouzgar, J. and Adoli Sharifroor, R. \Finit lmnt formulations for fr viration analysis of isotropic and orthotropic plats using two-varial rnd plat thory", Sci. Iran., 3(4), pp (016).

11 J. Rouzgar and M. Saydain/Scintia Iranica, Transactions B: Mchanical Enginring 5 (01) 13{ Rouzgar, J. and Adoli Sharifroor, R. \Finit lmnt formulations for uckling analysis of isotropic and orthotropic plats using two-varial rnd plat thory", Iran. J. Sci. Tchnol. -Trans. Mch. Eng., 41(3), pp (017). 1. Rouzgar, J. and Aad, F. \Analysis of cross-ply laminats with pizolctric r-rinforcd composit actuators using four-varial rnd plat thory", J. Thor. Appl. Mch., 53(), pp (015). 0. Rouzgar, J. and Aad, F. \Fr viration analysis of FG plat with pizolctric layrs using four-varial rnd plat thory", Thin Wall. Struct., (1), pp (015). 1. Rouzgar, J. and Gholami, M. \Non linar nding analysis of thick rctangular plats y four-varial R- nd plat thory and Dynamic Rlaxation mthod", Modars Mch. Eng., 15(), pp (015) (In Prsian).. Nguyn-Xuan, H., Tran, L.V., Thai, C.H., Kulasgaram, S., and Bordas, S.P.A. \Isogomtric analysis of functionally gradd plats using a rnd plat thory", Compos. Part B-Eng., 64, pp. -34 (014). 3. Thai, C.H., Kulasgaram, S., Tran, L.V., and Nguyn- Xuan, H. \Gnralizd shar dformation thory for functionally gradd isotropic and sandwich plats asd on isogomtric approach", Comput. Struct., 141, pp (014). 4. Nguyn, K.D. and Nguyn-Xuan, H. \An isogomtric nit lmnt approach for thr-dimnsional static and dynamic analysis of FGM plats structurs", Compos. Struct., 13, pp (015). 5. Nguyn, T.N., Thai, C.H., and Nguyn-Xuan, H. \On th gnral framwork of high ordr shar dformation thoris for laminatd composit plat structurs: A novl unid approach", Int. J. Mch. Sci., 110, pp (016). 6. Nguyn, L.B., Thai, C.H., and Nguyn-Xuan, H. \A gnralizd unconstraind thory and isogomtric nit lmnt analysis asd on Bzir xtraction for laminatd composit plats", Eng. Comput., 3(3), pp (016). 7. Patl, H. and Shimpi, R.P. \A Nw Finit Elmnt for a Shar Dformal Plat", 47th AIAA/ASME/ASCE/AHS/ASC Structurs, Structural Dynamics, and Matrials Confrnc., Nwport, Rhod Island, USA (006).. Katori, H. and Okada, T. \Plat lmnt asd on a highr-ordr shar dformation thory", Transactions of th Japan Socity of Mchanical Enginrs Sris A, 7(716), pp (006).. Rouzgar, J. and Adoli Sharifroor, R. \A nit lmnt formulation for nding analysis of isotropic and orthotropic plats asd on two-varial rnd plat thory", Sci. Iran., (1), pp (015). 30. Mlosh, R.J. \Structural analysis of solids", J. Struct. Div. ASCE, (4), pp (163). 31. Rouzgar, S.J. and Mirzai, M. \Modling dynamic fractur in Kirchho plats and shlls using th xtndd nit lmnt mthod", Sci. Iran., 0(1), pp (013). 3. Shikh, A.H. and Mukhopadhyay, M. \Linar and nonlinar transint viration analysis of stind plat structurs", Finit Elm. Anal. Ds., 3(14), pp (00). 33. Providakis, C.P. and Bskos, D.E. \Fr and forcd viration of plats y oundary lmnts", Comput. Mthods Appl. Mch. Eng., 74(3), pp (1). 34. Houlston, R. and Slatr, J.E. \A summary of xprimntal rsults on squar plat and stind plats sujctd to air last loading", Shock Vir. Bull., (57), pp (16). Biographis Jafar Rouzgar is currntly an Assistant Profssor at th Dpartmnt of Mchanical and Arospac Enginring of Shiraz Univrsity of Tchnology, Iran. H rcivd his BSc dgr in Mchanical Enginring from Shiraz Univrsity, Iran in 00. H also rcivd his MSc and PhD dgrs in Mchanical Enginring from Tariat Modars Univrsity, Iran in 004 and 010, rspctivly. His rsarch intrsts includ FEM and XFEM, thoris of plats and shlls, and fractur mchanics. Mohammad Saydain rcivd his BSc in Mchanical Enginring from Vali--Asr Univrsity, Rafsanjan, Iran in 013. H also rcivd his MSc dgr in Mchanical Enginring from Shiraz Univrsity of Tchnology, Iran in 015. His rsarch intrsts includ FEM, thoris of plats and shlls and composit matrials.