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1 Citation: Thai, Huu-Tai, Nguyen, Trung-Kien, Vo, Thuc and Ngo, Tuan (017) A ne simple shear deformation plate theory. Composite Structures, 171. pp ISSN Pulished y: Elsevier URL: < This version as donloaded from Northumria Research Link: Northumria University has developed Northumria Research Link (NRL) to enale users to access the University s research output. Copyright and moral rights for items on NRL are retained y the individual author(s) and/or other copyright oners. Single copies of full items can e reproduced, displayed or performed, and given to third parties in any format or medium for personal research or study, educational, or not-for-profit purposes ithout prior permission or charge, provided the authors, title and full iliographic details are given, as ell as a hyperlink and/or URL to the original metadata page. The content must not e changed in any ay. Full items must not e sold commercially in any format or medium ithout formal permission of the copyright holder. The full policy is availale online: This document may differ from the final, pulished version of the research and has een made availale online in accordance ith pulisher policies. To read and/or cite from the pulished version of the research, please visit the pulisher s esite (a suscription may e required.)

2 A ne simple shear deformation plate theory Huu-Tai Thai a,, Trung-Kien Nguyen, Thuc P. Vo c, Tuan Ngo d a School of Engineering and Mathematical Sciences, La Troe University, Bundoora, VIC 3086, Australia Faculty of Civil Engineering, Ho Chi Minh City University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Vietnam c Faculty of Engineering and Environment, Northumria University, Necastle upon Tyne, NE1 8ST, UK d Department of Infrastructure Engineering, The University of Melourne, Parkville, VIC 3010, Australia Astract This paper proposes a ne simple shear deformation theory for isotropic plates. The present theory involves one unknon and one governing equation as in the classical plate theory, ut it is capale of accurately capturing shear deformation effects. The displacement field of the present theory as ased on a to variale refined plate theory in hich the transverse displacement is partitioned into the ending and shear parts. Based on the equilirium equations of three-dimensional (3D) elasticity theory, the relationship eteen the ending and shear parts as estalished. Therefore, the numer of unknons of the present theory as reduced from to to one. Closed-form solutions ere presented for oth Navier- and Levy-type plates. Numerical results indicate that the otained predictions are comparale ith those generated y ABAQUS and availale results predicted y 3D elasticity theory, first-order and third-order shear deformation theories. Keyords: Shear deformation theory; isotropic plate; analytical solution Corresponding author. Tel.: address: tai.thai@latroe.edu.au (H.T. Thai). 1

3 1. Introduction Plates and shells are common structural elements in civil engineering structures such as uildings, ridges, tunnels, retaining alls and other infrastructure. In general, the ehaviour of plate and shell structures can e predicted using either D plate theories or 3D elasticity theory. The classical plate theory (CPT) is the simplest plate theory developed y Love [1] ased on the assumptions proposed y Kirchhoff []. Hoever, this theory is only applicale for thin plates in hich the shear deformation effects are negligily small. For thick plates, the CPT underestimates deflections and overestimates uckling loads as ell as natural frequencies ecause of neglecting these effects. A large numer of shear deformation theories have een proposed to take into account the shear deformation effects. One of the earliest shear deformation theories as proposed y Reissner [3] and Mindlin [4]. It should e noted that Mindlin s theory as ased on an assumption of a linear displacement variation across the plate thickness. It as therefore referred to as the first-order shear deformation theory (FSDT). This assumption leads to constant transverse shear strains and transverse shear stresses across the thickness. A shear correction factor is therefore needed to account for the discrepancy eteen the constant shear stresses and the paraolic distriution of shear stresses in the 3D elasticity theory. On the other hand, Reissner's theory as ased on the assumptions of a linear variation of ending stresses and a paraolic distriution of transverse shear stresses across the thickness. These assumptions lead to a displacement field hich is not necessarily linear across the thickness, and the shear correction factor is not required as in the case of Mindlin's theory. Higher-order shear deformation theories (HSDTs) ere proposed to eliminate the use of the shear correction factor in the FSDT, and to otain a etter prediction of the responses of very thick plates. The

4 HSDT is developed ased on a higher-order displacement variation across the plate thickness using either polynomial functions (e.g. the third-order shear deformation theory (TSDT) of Reddy [5]) or non-polynomial functions (e.g. the sinusoidal theory of Touratier [6], hyperolic theory of Soldatos [7], exponential theory of Karama et al. [8] and among others). Several typical shear deformation theories developed from 010 for composite structures can e found in Refs. [9-4]. A comprehensive revie of plate theories as reported y Ghugal and Shimpi [5] for isotropic and laminated plates and Thai and Kim [6] for functionally graded plates. Although the existing HSDTs provide a etter prediction compared ith the CPT, they are much more complicated and computationally expensive than the CPT ecause of introducing additional dependent unknons into the theory. Therefore, this paper aims to propose a simple HSDT hich contains the same numer of unknons and governing equations of motion as in the case of the CPT. The present theory as ased on the refined plate theory (RPT) of Shimpi [7] and 3D elasticity theory. Analytical solutions of the present theory ere also presented. The otained predictions ere then compared ith the availale results predicted y the FSDT, TSDT and 3D elasticity theory as ell as those generated y ABAQUS for validation.. Kinematics The displacement field of the present theory as derived ased on the displacement field of the RPT [7] and the equilirium equations of 3D elasticity theory. According to Shimpi [7], the displacement field of the RPT is given as follos: 5 z 1s u x, y, z, t z z x 3h 4 x (1a) 3

5 5 z 1s vx, y, z, t z z y 3h 4 y (1),,,,,,, x y z t x y t x y t (1c) s here ( u, v, ) are the total displacement along the coordinates ( x, y, z ); and s are the ending and shear components of transverse displacement, respectively; and h is the plate thickness. The equilirium equations of 3D elasticity theory in the asence of ody forces are ritten as: x y z x xy xz u (a) xy y yz v x y z () x y z xz yz z (c) here the dot-superscript convention indicates differentiation ith respect to time t ; i are the stress components of the stress tensor; and is the density. Sustituting Eq. (1) into Eq. (), the equilirium equations are reritten as: x xy xz 5 z 1 s z z x y z x 3h 4 x (3a) z z x y z y 3h 4 y xy y yz 5 z 1 s x y z xz yz z s (3) (3c) The equilirium equations in Eq. (3) can e reritten in terms of stress resultants for a plate under a transversely distriuted load q as shon in Fig. 1 y multiplying the first to equations y z and then integrating all three equations ith respect to z, and 4

6 applying several oundary conditions, i.e. the transverse shear stresses xz and yz equal to zero at z h/ and the normal stress through the thickness 0 at z h/ and z q at z h/. The resulting equations are: z M M x xy Qx I x y x (4a) M M Q I x y y xy y y (4) Q x x Q y 0 y q I s (4c) here the moments M, shear forces Q and mass inertias I are defined as x h/ M z dz (5a) h/ x y h/ M z dz (5) h/ y xy h/ M z dz (5c) h/ xy Q x h/ dz (6a) h/ xz Q y h/ dz (6) h/ yz I 0 h/ dz (7a) h/ I h/ z dz (7) h/ According to Shimpi [7], the moments and shear forces can e expressed in terms of 5

7 the dependent unknons, as s Mx D x y (8a) My D y x (8) M xy D1 xy (8c) s Qx As x s Qy As y (9a) (9) here D Eh 3 1(1 ) and A s 5Eh 1 1 (10) here E and are the Young s modulus and Poisson s ratio of isotropic materials, respectively. Sustituting Eqs. (8) and (9) into Eq. (4a) or Eq. (4), the relationship eteen the ending component and shear component s is otained as: I D (11) s As As here. The displacement field of the present theory can e otained x y y sustituting Eq. (11) into Eq. (1): 5 z 1I D u x, y, z, t z z x 3h 4 As x As x 5 z 1I D vx, y, z, t z z y 3h 4 As y As y (1a) (1) 6

8 D I x, y, z, t A A (1c) s s It can e seen from Eq. (1) that the displacement field of the present theory involves only one unknon. If the underlined terms in Eq. (1) are neglected, the present theory ecomes the CPT. 3. Governing equations of motion Sustituting Eqs. (4a) and (4) into Eq. (4c), the equations of motion of the present theory is otained from the equilirium equations of 3D elasticity theory as: M Mxy M x y q I I x xy y 0 s (13) Eq. (13) can e expressed in terms of the displacement y sustituting Eqs. (8) and (11) into Eq. (13): D q I I I D I I As As (14) here x 4 x y y 4 4. For a static analysis, Eq. (14) as simplified as: 4 D q (15) Eq. (15) is similar to the governing equation of the CPT. The only difference is that the governing equation of the present theory involves the ending component of the transverse displacement instead of the total transverse displacement as in the case of the CPT. The oundary conditions of the present theory are in a similar form as the CPT as [7]: For free edges: 7

9 M xy M x 0 and Vx Qx 0 y M xy M y 0 and Vy Qy 0 x on the edges x 0, a (16a) on the edges y 0, (16) For simply supported edges: 0 and M 0 on the edges x 0, a (17a) x 0 and M 0 on the edges y 0, (17) y For clamped edges, there are to possile types of clamped oundary conditions as mentioned in [7]: 0 and 0 x 0 and 0 y on the edges x 0, a (18a) on the edges y 0, (18) or 0 and 0 and u z z 0 v z z on the edges x 0, a (19a) on the edges y 0, (19) 4. Analytical solutions 4.1. Navier-type plates Consider a Navier-type plate sujected to a transverse load q as shon in Fig. 1. The expansion of hich satisfies the simply supported oundary conditions of the Navier-type plate as selected as follos: it x, y, t W e sinxsin y (0) m1 n1 mn 8

10 here m / a; n / ; is the natural frequency; and W mn are coefficients. The transversely distriuted load q can e also expressed as: q x, y Q sinxsin y (1) m1 n1 mn here a q0 for sinusoidally distriuted load 4, Qmn q x y sin xsin ydxdy 16q0 a () 00 for uniformly distriuted load mn Sustituting Eqs. (0) and (1) into Eq. (14), the folloing equation is otained I D I I D I I W Q mn mn As As (3) For a static analysis ( 0 ), the analytical solution of the ending component is otained as: Qmn sinxsin y (4) m1 n1 D The closed-form solutions for the deflections, stresses, ending moments M and shear forces Q are susequently otained ased on as follos: D Qmn 1 sin xsin y (5) m1 n1 As D Ez D z Q x 1 A 3h 4 D 5 1 mn 1 sin sin x y (6a) m1 n1 s Ez D z Q y 1 A 3h 4 D 5 1 mn 1 sin sin x y (6) m1 n1 s 9

11 Ez D 5z 1 Qmn xy 1 cos xcos y (6c) 1 m1 n1 As 3h 4 D 61 z Qmn xz cos xsin y (6d) h4 h m1 n1 61 z Qmn yz sin xcos y (6e) h4 h m1 n1 Q M x x y mn sin sin (7a) m1 n1 Q M y x y mn sin sin (7) m1 n1 Qmn M xy 1 cos xcos y (7c) m1 n1 Q Q x y x mn cos sin (8a) m1 n1 Q Q x y y mn sin cos (8) m1 n1 It should e noted that the CPT does not account for the underlined terms in Eqs. (5) and (6). For a free viration analysis ( q 0 ), the analytical solution for the natural frequency is otained from in Eq. (3) as: I0D I0D I0ID I0 ( I )( ) I0 ( I )( ) 4 ( ) A s A s As (9) I I / A 0 s If the time derivative term in Eq. (11) is neglected, the natural frequency of a simply supported plate is simplified as: 10

12 I 0 D ID 0 I A s (30) If the underlined shear deformation term in Eq. (30) is neglected, the natural frequency of the present theory yields the frequency otained from the CPT. Fig. illustrates the comparison eteen the nondimensional fundamental frequencies a h / D predicted y CPT (using Eq. (30) ithout the underline term), FSDT given y Thai et al. [8] and the present theory ith the exact and simplified formulations. It can e oserved that the difference in the frequencies predicted y the exact and simplified formulations in Eqs. (9) and (30) is negligile, and the predictions of the present theory are close to those predicted y the FSDT [8]. 4.. Levy-type plates Consider a rectangular plate ith simply supported oundary conditions at the edges x 0, a and aritrary oundary conditions at the remaining edges y 0,. The folloing expansion of as chosen to satisfy the oundary conditions of the Levytype plate as follos: it x, y, t W y e sinx (31) m1 The applied load q can e expressed as: here m q x, y Q y sin x (3) m1 q for sinusoidally distriuted load Qm y q x y xdx q a m m a 0, sin 4 (33) 0 0 for uniformly distriuted load Sustituting Eqs. (31) and (3) into the governing equations of motion Eq. (14), the 11

13 folloing equation is otained: W CW C W Q D (34) '''' m 1 m m m / here the prime notations denote the derivatives ith respect to y and the coefficients C i are define y: I0 I I0 I0I 4 4 I I0 C1, C D D As DAs D As (35) For a static analysis 0, the governing equations of the present theory are similar to those of the CPT as in Eq. (15). Therefore, the analytical solution of Eq. (34) is otained as: Qm Wm y Am Bm ycoshy Cm Dm ysinhy (36) 4 D here the constants ( Am, Bm, Cm, D m ) can e otained using oundary conditions at the edges y 0,. Sustituting the stress resultants (M and Q) and deflection into Eqs. (16)-(19), the oundary conditions at the edges y 0, can e reritten in terms of W m as Free (F): Simply supported (S): D Wm Wm 0 (37a) D I Wm DWm 0 (37) D I D 1 Wm Wm 0 As As As (38a) D Wm Wm 0 (38) Clamped (C): 1

14 D I D 1 Wm Wm 0 As As As (39a) D I D 1 Wm Wm 0 As As As (39) or D I D 1 Wm Wm 0 As As As (40a) D I D 1 Wm Wm 0 4As 4As 4As (40) For a free viration analysis q 0, the state-space approach can e used to solve Eq. (34) for the natural frequency. More details on the application of this approach can e found in Refs. [9-3]. 5. Numerical examples A numer of examples ere presented in this section to illustrate the accuracy and efficiency of the present theory and its analytical solutions. The ending response and natural frequency predicted y the present one unknon shear deformation theory ere compared ith availale results predicted y 3D elasticity theory and ell-knon plate theories such as the CPT, FSDT and TSDT. It should e noted that the CPT has only a single unknon as in the case of the present theory, hilst the FSDT and TSDT involve three unknons. In addition, the numerical solutions generated in this study using the shell element S4R of the commercial finite element softare ABAQUS [33] ere also used for validation. S4R is a roust quadrilateral 4-node element ith reduced integration hich is applicale for oth thin and thick plate/shell structures. Based on a convergence study, the mesh sizes of and ere respectively selected for 13

15 the numer of element along the edges of square plates and rectangular plates ith aspect ratios of Bending analysis The first ending example aims to validate the present theory for Navier-type plates. Three values for the length-to-thickness ratio a/h of 5, 10 (corresponding to thick and moderately thick plates) and 100 (corresponding to thin plates) ere taken into account. The otained predictions ere compared ith availale results reported y Reddy [34] using the CPT, FSDT, and TSDT in Tale 1 in hich the normalized quantities ere defined as follos: h i qa a h i,,, ( i x, y) h 0,0, xy xy yz xz h qa h a yz,0,0 qa h xz 0,,0 qa Eh qa a 3, 4 (41) The otained predictions ere calculated using up to 19 terms in the series as performed y Reddy [34]. It should e noted that the values of the transverse shear stresses given y Reddy [34] ere ased on integrating the equilirium equations of 3D elasticity theory ith respect to the thickness coordinate as xz x z x xy dz (4a) h/ y yz z xy y dz (4) h/ x y It can e oserved from Tale 1 that the deflections and in-plane stresses otained y 14

16 the present theory agree ell ith those predicted y the FSDT and TSDT. Since the shear deformation effects ere included in the present theory, FSDT and TSDT, their predictions are similar each other and agree ell ith the finite element solutions of ABAQUS. On the contrary, the CPT underestimates deflections of thick plates ecause of ignoring shear deformation effects. For example, for thick plates ith a/h = 5, the CPT underestimates the deflections y 17.10% and 11.38% for square and rectangular plates, respectively. It is also seen from Tale 1 that the transverse shear stresses otained from the present theory are identical ith those predicted y the CPT and FSDT. As stated y Reddy [34], oth CPT and FSDT give more accurate predictions of the transverse shear stresses than the TSDT hen the stress equilirium equations of 3D elasticity theory are used. It is also noted that the present theory has only one unknon as in the CPT, hilst oth FSDT and TSDT involves three unknons. Although the CPT can give the same accuracy in predicting the transverse shear stress as in the case of the present theory, it as ased on an indirect and lengthy process through the use of equilirium equations in Eq. (4). In contrast to the CPT, the present theory predicts the transverse shear stresses in a direct manner through the use of constitutive relations in Eqs. (6d) and (6e). It should e noted that the transverse shear stresses predicted y the CPT using constitutive relations are alays equal to zero. A comparison of the variation of ith respect to a/h as also plotted in Fig. 3 for square plates. Again, an excellent agreement eteen the results generated y the present theory, TSDT and ABAQUS as oserved. To further illustrate the accuracy of the present theory in predicting the transverse shear stresses, Fig. 4 compared the distriution of the transverse shear stress predicted y the present theory ith that given y Auricchio and Sacco [35] ased on the exact 3D 15

17 elasticity solutions derived y Pagano [36]. For the plate strip under a sinusoidal load, the transverse shear stress given in Eq. (6d) as simplified as xz 6qa 1 z h 4 h. It can e seen from Fig. 4 that the otained prediction agrees ell ith the exact 3D solutions. This is expected since the present theory as ased on equilirium considerations. The next ending example aims to verify the present theory for Levy-type plates. Tale compared the nondimensional deflection at the centre of Levy-type plates under uniformly distriuted loads. The aspect ratio is taken as.0, hilst the nondimensional 100D deflection is defined as ˆ ith 4 qa 3 Eh D 1 1. Four values of a/h of 5, 10 (corresponding to thick and moderately thick plates), 5 and 1000 (corresponding to thin plates) ere considered. For the plates involving the clamped oundary conditions (CC, SC and FC plates), to types of the clamped oundary conditions descried in Eqs (39) and (40) ere used. The otained solutions ere compared ith the availale solutions of the FSDT reported y Zenkour [37] and the FE solutions of ABAQUS. The plate deformations generated y ABAQUS ere also illustrated in Fig. 5 for the case of moderately thick plates ith a/h = 10. In general, the deflections predicted y the present theory are in good agreement ith those predicted y the FSDT and ABAQUS, except for the case of the thick CC plate ith a/h = 5 here a slightly difference of 6.43% eteen the results as found if the clamped oundary conditions in Eq. (39) ere used. Hoever, this difference ecomes negligile hen the clamped oundary conditions in Eq. (40) ere used. This indicated that for the plate involving the clamped oundary conditions, the oundary conditions descried in Eq. (40) should e used to give a etter prediction. 16

18 5.. Free viration analysis The first verification for free viration analysis as carried out for SS square plates. This example aims to verify the present theory for a ide range of Navier-type plates ith a thickness-to-length ratio h/a covering from (very thin plates) to 0.4 (very thick plates). Tale 3 shoed a comparison of the first eight nondimensional frequencies otained in this study ith availale results predicted y the CPT [38], FSDT [39], TSDT [40-41] and 3D elasticity theory [4-43]. The otained results ere also compared ith the FE results computed independently in this study using ABAQUS. The first eight mode shapes generated y ABAQUS ere also plotted in Fig. 6 for very thick plates ith h/a = 0.4. It is noted that the 3D results given y Lie et al. [4] ere ased on the Ritz method, hilst Malik and Bert [43] used the differential quadrature (DQ) method to solve for the frequencies. The TSDT results reported y Shufrin and Eisenerger [40] ere ased on the extended Kantorovich numerical method, hilst Hosseini-Hashemi et al. [41] employed the Levy method to derive exact TSDT solutions. Both CPT and FSDT results given y Leissa [38] and Hosseini-Hashemi and Arsanjani [39], respectively, ere ased on an analytical approach. Good agreement eteen the results as found in Tale 3 for all models of very thin to very thick plates. The next verification for free viration analysis aims to verify the present theory for a ide range of square and rectangular plates ith various oundary conditions. Tale 4 contains the nondimensional frequencies of plates for various values of the thickness-to-length ratio h/a. It is noted that the frequencies of the CC, SC and FC plates ere otained ased on the clamped oundary conditions descried in Eq. (40) due to its accuracy. The results otained in this study ere compared ith availale results reported y Malik and Bert [43], Hosseini-Hashemi and Arsanjani [39] and 17

19 Hosseini-Hashemi et al. [41] ased on 3D elasticity theory, FSDT and TSDT, respectively. Good agreement eteen the results is oserved in Tale 4 confirming the accuracy of the present theory. 6. Conclusions A simple and accurate shear deformation theory has een proposed for thick isotropic plates. The governing equations of motion of the present theory ere derived ased on 3D elasticity theory and RPT. The accuracy of the present theory in predicting the ending ehaviour and natural frequencies as verified for a ide range of Navier- and Lery-type plates. Numerical results indicated that the present theory is not only much more accurate than the CPT ut also comparale ith the FSDT and TSDT hen compared ith 3D elasticity theory and ABAQUS. Although the present theory has only one unknon and one governing equations of motion as in the CPT, its predictions are comparale ith those generated y the FSDT and TSDT hich have three unknons and three governing equations. Acknoledgements This ork in this paper as supported y the School of Engineering and Mathematical Sciences at La Troe University. This financial support is gratefully acknoledged. References [1] Love AEH. The small free virations and deformation of a thin elastic shell. Philosophical Transactions of the Royal Society of London. A 1888;179: [] Kirchhoff GR. Uer das gleichgeicht und die eegung einer elastischen scheie. 18

20 1850;40: [3] Reissner E. The effect of transverse shear deformation on the ending of elastic plates. Journal of Applied Mechanics 1945;1:69-7. [4] Mindlin RD. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 1951;18: [5] Reddy JN. A simple higher-order theory for laminated composite plates. Journal of Applied Mechanics 1984;51: [6] Touratier M. An efficient standard plate theory. International Journal of Engineering Science 1991;9: [7] Soldatos KP. A transverse shear deformation theory for homogeneous monoclinic plates. Acta Mechanica 199;94: [8] Karama M, Afaq KS, Mistou S. Mechanical ehaviour of laminated composite eam y the ne multi-layered laminated composite structures model ith transverse shear stress continuity. International Journal of Solids and Structures 003;40: [9] Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Polit O. Analysis of laminated shells y a sinusoidal shear deformation theory and radial asis functions collocation, accounting for through-the-thickness deformations. Composites Part B: Engineering 011;4: [10] Mantari JL, Guedes Soares C. Generalized hyrid quasi-3d shear deformation theory for the static analysis of advanced composite plates. Composite Structures 01;94: [11] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN, et al. A quasi-3d hyperolic shear deformation theory for the static and free viration 19

21 analysis of functionally graded plates. Composite Structures 01;94: [1] Neves AMA, Ferreira AJM, Carrera E, Roque CMC, Cinefra M, Jorge RMN, et al. A quasi-3d sinusoidal shear deformation theory for the static and free viration analysis of functionally graded plates. Composites Part B: Engineering 01;43: [13] Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN, et al. Static, free viration and uckling analysis of isotropic and sandich functionally graded plates using a quasi-3d higher-order shear deformation theory and a meshless technique. Composites Part B: Engineering 013;44: [14] Thai HT, Choi DH. A simple first-order shear deformation theory for laminated composite plates. Composite Structures 013;106: [15] Thai HT, Choi DH. A simple first-order shear deformation theory for the ending and free viration analysis of functionally graded plates. Composite Structures 013;101: [16] Thai HT, Kim SE. A simple quasi-3d sinusoidal shear deformation theory for functionally graded plates. Composite Structures 013;99: [17] Thai HT, Kim SE. A simple higher-order shear deformation theory for ending and free viration analysis of functionally graded plates. Composite Structures 013;96: [18] Mantari JL, Guedes Soares C. A trigonometric plate theory ith 5-unknons and stretching effect for advanced composite plates. Composite Structures 014;107: [19] Mantari JL, Guedes Soares C. Four-unknon quasi-3d shear deformation theory for advanced composite plates. Composite Structures 014;109:

22 [0] Mantari JL, Granados EV. A refined FSDT for the static analysis of functionally graded sandich plates. Thin-Walled Structures 015;90: [1] Vo TP, Thai HT, Nguyen TK, Inam F, Lee J. A quasi-3d theory for viration and uckling of functionally graded sandich eams. Composite Structures 015;119:1-1. [] Mantari JL, Ramos IA, Carrera E, Petrolo M. Static analysis of functionally graded plates using ne non-polynomial displacement fields via Carrera Unified Formulation. Composites Part B: Engineering 016;89: [3] Sarangan S, Singh BN. Higher-order closed-form solution for the analysis of laminated composite and sandich plates ased on ne shear deformation theories. Composite Structures 016;138: [4] Farzam-Rad SA, Hassani B, Karamodin A. Isogeometric analysis of functionally graded plates using a ne quasi-3d shear deformation theory ased on physical neutral surface. Composites Part B: Engineering 017;108: [5] Ghugal YM, Shimpi RP. A revie of refined shear deformation theories of isotropic and anisotropic laminated plates. Journal of Reinforced Plastics and Composites 00;1: [6] Thai HT, Kim SE. A revie of theories for the modeling and analysis of functionally graded plates and shells. Composite Structures 015;18: [7] Shimpi RP. Refined plate theory and its variants. AIAA Journal 00;40: [8] Thai HT, Park M, Choi DH. A simple refined theory for ending, uckling, and viration of thick plates resting on elastic foundation. International Journal of Mechanical Sciences 013;73:40-5. [9] Thai HT, Kim SE. Levy-type solution for uckling analysis of orthotropic plates 1

23 ased on to variale refined plate theory. Composite Structures 011;93: [30] Thai HT, Kim SE. Levy-type solution for free viration analysis of orthotropic plates ased on to variale refined plate theory. Applied Mathematical Modelling 01;36: [31] Thai HT, Kim SE. Analytical solution of a to variale refined plate theory for ending analysis of orthotropic Levy-type plates. International Journal of Mechanical Sciences 01;54: [3] Thai HT, Choi DH. Analytical solutions of refined plate theory for ending, uckling and viration analyses of thick plates. Applied Mathematical Modelling 013;37: [33] ABAQUS. Standard user's manual, version 6.15: Dassault Systemes Corp., Providence, RI (USA); 016. [34] Reddy JN. A refined nonlinear theory of plates ith transverse shear deformation. International Journal of Solids and Structures 1984;0: [35] Auricchio F, Sacco E. Refined first-order shear deformation theory models for composite laminates. Journal of Applied Mechanics 003;70: [36] Pagano N. Exact solutions for composite laminates in cylindrical ending. Mechanics of Composite Materials: Springer; p [37] Zenkour AM. Exact mixed-classical solutions for the ending analysis of shear deformale rectangular plates. Applied Mathematical Modelling 003;7: [38] Leissa AW. The free viration of rectangular plates. Journal of Sound and Viration 1973;31: [39] Hosseini-Hashemi S, Arsanjani M. Exact characteristic equations for some of

24 classical oundary conditions of virating moderately thick rectangular plates. International Journal of Solids and Structures 005;4: [40] Shufrin I, Eisenerger M. Staility and viration of shear deformale plates first order and higher order analyses. International Journal of Solids and Structures 005;4: [41] Hosseini-Hashemi S, Fadaee M, Rokni Damavandi Taher H. Exact solutions for free flexural viration of Levy-type rectangular thick plates via third-order shear deformation plate theory. Applied Mathematical Modelling 011;35: [4] Lie KM, Hung KC, Lim MK. A continuum three-dimensional viration analysis of thick rectangular plates. International Journal of Solids and Structures 1993;30: [43] Malik M, Bert CW. Three-dimensional elasticity solutions for free virations of rectangular plates y the differential quadrature method. International Journal of Solids and Structures 1998;35:

25 Tale Captions Tale 1. Deflections and stresses of SS plates under uniformly distriuted loads Tale. Deflections ŵ of rectangular plates ith various oundary conditions under uniformly distriuted loads ( a) Tale 3. The first eight frequencies of square SS plates Tale 4. Fundamental frequencies of square and rectangular Levy-type plates 4

26 Tale 1. Deflections and stresses of SS plates under uniformly distriuted loads /a a/h Methods x y xy yz xz 1 5 ABAQUS CPT [34] FSDT [34] TSDT [34] Present ABAQUS CPT [34] FSDT [34] TSDT [34] Present ABAQUS CPT [34] FSDT [34] TSDT [34] Present ABAQUS CPT [34] FSDT [34] TSDT [34] Present ABAQUS CPT [34] FSDT [34] TSDT [34] Present ABAQUS CPT [34] FSDT [34] TSDT [34] Present

27 Tale. Deflections ŵ of rectangular plates ith various oundary conditions under uniformly distriuted loads ( a) a/h Methods Boundary conditions CC SC SS FC FS FF 5 ABAQUS FSDT [37] Present (1.0186)* (1.0799) (1.193) 10 ABAQUS FSDT [37] Present (0.890) (0.9664) (1.1014) 5 ABAQUS FSDT [37] Present (0.8519) (0.9334) (1.0671) 1000 ABAQUS FSDT [37] Present (0.8445) (0.970) (1.0605) * Deflection values otained using Eq. (40) for the clamped oundary conditions. 6

28 Tale 3. The first eight frequencies of square SS plates h/a Methods Modes ABAQUS D-Ritz [4] TSDT [40] Present ABAQUS D-Ritz [4] TSDT [40] Present ABAQUS D-Ritz [4] D-DQ [43] FSDT [39] TSDT [40] Present ABAQUS D-Ritz [4] D-DQ [43] FSDT [39] TSDT [40] Present ABAQUS TSDT [41] Present ABAQUS CPT [38] FSDT [39] TSDT [41] Present

29 Tale 4. Fundamental frequencies of square and rectangular Levy-type plates /a h/a Methods Boundary conditions CC SC SS FC FS FF ABAQUS TSDT [41] Present ABAQUS TSDT [41] Present ABAQUS D-DQ [43] FSDT [39] TSDT [41] Present ABAQUS D-DQ [43] FSDT [39] TSDT [41] Present ABAQUS FSDT [39] TSDT [41] Present ABAQUS TSDT [41] Present ABAQUS TSDT [41] Present ABAQUS FSDT [39] TSDT [41] Present ABAQUS FSDT [39] TSDT [41] Present ABAQUS FSDT [39] TSDT [41] Present

30 Figure Captions Fig. 1. Geometry and coordinates of a rectangular plate Fig.. Comparison of fundamental frequencies of SS square plates Fig. 3. Comparison of deflections of SS square plates Fig. 4. Transverse shear stresses for SS plate strips under sinusoidal loads Fig. 5. Deflections of Levy-type plates ith a/h = 0.1 and = a Fig. 6. Mode shapes of SS square thick plates ith a/h = 0.4 9

31 0 x N = N 0 N x = N 0 0 y, v x z, q a/ 0 y N = N 0 a/ h/ h/ O x, u a Fig. 1. Geometry and coordinates of a rectangular plate CPT, Eq. (30) ithout underlined term FSDT [8] Present, Eq. (9) Present, Eq. (30) a/h Fig.. Comparison of fundamental frequencies of SS square plates 30

32 CPT TSDT ABAQUS Present a /h Fig. 3. Comparison of deflections of SS square plates z/h D solution [35] Present xz / q q L >> a a Fig. 4. Transverse shear stresses for SS plate strips under sinusoidal loads 31

33 CC SC SS FC FS FF Fig. 5. Deflections of Levy-type plates ith a/h = 0.1 and = a Fig. 6. Mode shapes of SS square thick plates ith a/h = 0.4 3