Composite Structures

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1 Composite Structures () 5 Contents lists availale at ScienceDirect Composite Structures journal homepage: A two variale refined plate theor for laminated composite plates Seung-Eock Kim a, Huu-Tai Thai a, Jaehong Lee, * a Department of Civil and Environmental Engineering, Sejong Universit, Kunja Dong Kwangjin Ku, Seoul -, Repulic of Korea Department of Architectural Engineering, Sejong Universit, Kunja Dong, Kwangjin Ku, Seoul -, Repulic of Korea article info astract Article histor: Availale online Jul Kewords: Laminated composite plates Shear deformation theor Refined plate theor Higher-order theor Navier solution A two variale refined plate theor of laminated composite plates is developed in this paper. The theor accounts for paraolic distriution of the transverse shear strains, and satisfies the ero traction oundar conditions on the surfaces of the plate without using shear correction factor. Equations of motion are derived from the Hamilton s principle. The closed-form solutions of antismmetric cross-pl and anglepl laminates are otained using Navier solution. Numerical results of present theor are compared with three-dimensional elasticit solutions and results of the first-order and the other higher-order theories reported in the literature. It can e concluded that the proposed theor is accurate and simple in solving the static ending and uckling ehaviors of laminated composite plates. Crown Copright Ó Pulished Elsevier Ltd. All rights reserved.. Introduction * Corresponding author. Tel.: + ; fa: +. address: jhlee@sejong.ac.kr (J. Lee). Fier reinforced composite are widel used in the aerospace, automotive, marine and other structural applications. In the past three decades, researches on laminated composite plates have received great attention, and a variet of laminated theories has een introduced. The classical laminate plate theor (CLPT), which neglects the transverse shear effects, provides reasonale results for thin plates. However, the CLPT underpredicts deflections and overpredicts frequencies as well as uckling loads with moderatel thick plates. Man shear deformation theories account for transverse shear effects have een developed to overcome the deficiencies of the CLPT. The first-order shear deformation theories (FSDTs) ased on Reissner [] and Mindlin [] account for the transverse shear effects the wa of linear variation of in-plane displacements through the thickness. Since FSDT violates equilirium conditions at the top and ottom faces of the plate, shear correction factors are required to rectif the unrealistic variation of the shear strain/stress through the thickness. In order to overcome the limitations of FSDT, higher-order shear deformation theories (HSDTs), which involve higher-order terms in Talor s epansions of the displacements in the thickness coordinate, were developed Lirescu [], Levinson [], Bhimaraddi and Stevens [5], Redd [], Ren [], Kant and Panda [], and Mohan et al. []. A good review of these theories for the analsis of laminated composite plates is availale in Refs. [ ]. A two variale refined plate theor (RPT) using onl two unknown functions was developed Shimpi [5] for isotropic plates, and was etended Shimpi and Patel [,] for orthotropic plates. The most interesting feature of this theor is that it does not require shear correction factor, and has strong similarities with the classical plate theor in some aspects such as governing equation, oundar conditions and moment epressions. The purpose of this paper is to develop the RPT for laminated composite plates. The present theor satisfies equilirium conditions at the top and ottom faces of the plate without using shear correction factors. Governing equations are derived from the Hamilton s principle. Navier solution is used to otain the closed-form solutions for simpl supported antismmetric cross-pl and angle-pl laminates. To illustrate the accurac of the present theor, the otained results are compared with three-dimensional elasticit solutions and results of the first-order and the other higher-order theories.. Refined plate theor for laminated composite plates.. Basic assumptions Consider a rectangular plate of total thickness h composed of n orthotropic laers with the coordinate sstem as shown in Fig.. Assumptions of the RPT are as follows: (i) The displacements are small in comparison with the plate thickness and, therefore, strains involved are infinitesimal. (ii) The transverse displacement W includes three components of etension w a, ending w, and shear w s. These components are functions of coordinates,, and time t onl. Wð; ; ; tþ ¼w a ð; ; tþþw ð; ; tþþw s ð; ; tþ ðþ (iii) The transverse normal stress r is negligile in comparison with in-plane stresses r and r. -/$ - see front matter Crown Copright Ó Pulished Elsevier Ltd. All rights reserved. doi:./j.compstruct...

2 S.-E. Kim et al. / Composite Structures () 5 c ¼ c a þ gcs c ¼ c a þ gcs e ¼ ð5þ where h h n k k+ k n n+ e ¼ ou ; e ¼ ov ; c ¼ ou þ ov ; c a ¼ ow a ; w j ¼ o ; w s js ¼ o w j ¼ o ; w s js ¼ o j ¼ o w ; cs c a ¼ ow a ; cs f ¼ þ 5 ; 5 g ¼ h 5 h js ¼ o w s ðþ.. Constitutive equations Fig.. Coordinate sstem and laer numering used for a tpical laminated plate. (iv) The displacements U in -direction and V in -direction consist of etension, ending, and shear components. U ¼ u þ u þ u s ; V ¼ v þ v þ v s ðþ The ending components u and v are assumed to e similar to the displacements given the classical plate theor. Therefore, the epression for u and v can e given as u ¼ ow ; v ¼ ow ðaþ The shear components u s and v s give rise, in conjunction with w s, to the paraolic variations of shear strains c, c and hence to shear stresses r, r through the thickness of the plate in such a wa that shear stresses r, r are ero at the top and ottom faces of the plate. Consequentl, the epression for u s and v s can e given as u s ¼ 5 ows h ; v s ¼ 5 ows h ðþ.. Kinematics Based on the assumptions made in the preceding section, the displacement field can e otained using Eqs. () () as Uð; ; ; tþ ¼uð; ; tþ ow þ 5 ows h Vð; ; ; tþ ¼vð; ; tþ ow þ 5 ows ðþ h Wð; ; ; tþ ¼w a ð; ; tþþw ð; ; tþþw s ð; ; tþ The etension component w a of transverse displacement is negligile small in most cases. It can e neglected for a simpler version of the present theor named RPT. The strains associated with the displacements in Eq. () are e ¼ e þ j þ fjs e ¼ e þ j þ fjs c ¼ c þ j þ fjs Under the assumption that each laer possesses a plane of elastic smmetr parallel to the plane, the constitutive equations for a laer can e written as r r r r r Q Q Q Q ¼ Q Q 5 Q 55 e e c c c where Q ij are the plane stress-reduced stiffnesses, and are known in terms of the engineering constants in the material aes of the laer: E Q ¼ ; Q m m ¼ m E ; Q m m ¼ ; m m Q ¼ G ; Q ¼ G ; Q 55 ¼ G Since the laminate is made of several orthotropic laers with their material aes oriented aritraril with respect to the laminate coordinates, the constitutive equations of each laer must e transformed to the laminate coordinates (,, ). The stress-strain relations in the laminate coordinates of the kth laer are given as r r r r r ðkþ Q Q Q Q Q Q ¼ Q Q Q Q Q 5 5 Q 5 Q 55 ðkþ E e e c c c where Q ij are the transformed material constants given as Q ¼ Q cos h þ ðq þ Q Þsin hcos h þ Q sin h Q ¼ðQ þ Q Q Þsin hcos h þ Q ðsin h þ cos hþ Q ¼ Q sin h þ ðq þ Q Þsin hcos h þ Q cos h Q ¼ðQ Q Q Þsinhcos h þðq Q þ Q Þsin hcosh Q ¼ðQ Q Q Þsin hcosh þðq Q þ Q Þsinhcos h Q ¼ðQ þ Q Q Q Þsin hcos h þ Q ðsin h þ cos hþ Q ¼ Q cos h þ Q 55 sin h Q 5 ¼ðQ 55 Q Þcoshsinh Q 55 ¼ Q 55 cos h þ Q sin h ðkþ ðþ ðþ ðþ ðþ

3 S.-E. Kim et al. / Composite Structures () 5 in which h is the angle etween the gloal -ais and the local -ais of each lamina... Equation of motions The strain energ of the plate can e written as U ¼ Z r ij e ij dv V ¼ Z ðr e þ r e þ r c þ r c þ r c ÞdV V ðþ ða ij ; B ij ; D ij ; E ij ; F ij ; H ij Þ¼ B s ij ¼ B ij þ 5 h E ij D s ij ¼ D ij þ 5 h F ij Z h= h= ði; j ¼ ; ; Þ Q ij ð; ; ; ; ; Þd ði; j ¼ ; ; Þ H s ij ¼ D ij 5 h F ij þ 5 h H ij ða ij ; D ij ; F ij Þ¼ Z h= h= Q ij ð; ; Þd ði; j ¼ ; ; Þ ði; j ¼ ; 5Þ ði; j ¼ ; ; Þ Sustituting Eqs. (5) and () into Eq. () and integrating through the thickness of the plate, the strain energ of the plate can e rewritten as U ¼ Z fn e þ N e þ N c þ M j þ M j þ M j A þ M s j s þ Ms j s þ Ms j s þ Q a c a þ Q a c a þ Q s c s þ Q s c s gdd where the stress resultants N, M, and Q are defined Z h= Z ðn ; N ; N Þ¼ ðr ; r ; r Þd ¼ XN kþ ðr ; r ; r Þd h= k¼ k ðm ; M ; M Þ¼ Z h= h= ðm s ; Ms ; Ms Þ¼ Z h= h= ðq a ; Q a ; Q s ; Q s Þ¼ Z h= ðr ; r ; r Þd ¼ XN ðr ; r ; r Þf d ¼ XN h= Z ¼ XN kþ k¼ Z kþ k¼ k Z kþ k¼ k ðr ; r ; gr ; gr Þd k ðr ; r ; gr ; gr Þd ðr ; r ; r Þd ðr ; r ; r Þf d ðþ ðþ Sustituting Eq. () into Eq. () and integrating through the thickness of the plate, the stress resultants are given as A a ij ¼ 5 A ij 5 h D ij ði; j ¼ ; 5Þ A s ij ¼ 5 A ij 5 h D ij þ 5 h F ij ði; j ¼ ; 5Þ The work done applied forces can e written as Z V ¼ qðw a þ w þ w s Þdd þ Z N o ðw a þ w þ w s Þ A A þ N o ðw a þ w þ w s Þ þ N o ðw a þ w þ w s Þ dd ð5þ ðþ where q and N ; N ; N are transverse and in-plane distriuted force, respectivel. The kinetic energ of the plate can e written as T ¼ Z q u ii dv ¼ Z h i I u þ v þð w a þ w þ w s Þ dd V A þ Z ( o w I þ o w þ I o w s þ o w ) s dd A ðþ where q is mass of densit of the plate and I i (i =,) are the inertias defined ði ; I Þ¼ Z h= h= ð; Þqd ðþ Hamilton s principle [] is used herein to derive the equations of N N N M M M M s M s M s A A A A A A 5 A A A B B B ¼ B B B 5 B B B B s B s B s B s B s B s 5 B s B s B s B B B B B B 5 B B B D D D D D D 5 D D D D s D s D s D s D s D s 5 D s D s D s B s B s B s B s B s B s 5 B s B s B s D s D s D s D s D s D s 5 D s D s D s H s H s H s H s B s H s 5 5 H s H s H s e e c j j j j s j s j s ðaþ Q a Q a Q s Q s A A 5 A a A a 5 A 5 A 55 A a 5 A a 55 ¼ A a A a 5 A s A s 5 5 A a 5 A a 55 A s 5 A s 55 c a c a c s c s where A ij, B ij, etc., are the plate stiffness, defined ðþ motion appropriate to the displacement field and the constitutive equation. The principle can e stated in analtical form as ¼ Z t dðu þ V TÞdt ðþ where d indicates a variation with respect to and. Sustituting Eqs. (), () and () into Eq. () and integrating the equation parts, collecting the coefficients of du, dv,

4 S.-E. Kim et al. / Composite Structures () 5 dw, dw s, and dw a, the equations of motion for the laminate plate are otained as follows: du : on þ on ¼ I u on dv : þ on ¼ I v dw : o M þ o M þ o M þ q þ NðwÞ ¼ I ð w a þ w þ w s Þ I r w o M s dw s : þ o M s þ o M s þ oq s þ oq s þ q þ NðwÞ ¼ I ð w a þ w þ w s Þ I r w s oq a dw a : þ oq a þ q þ NðwÞ ¼I ð w a þ w þ w s Þ where N(w) is defined NðwÞ ¼N o ðw a þ w þ w s Þ þ N o ðw a þ w þ w s Þ þ N o ðw a þ w þ w s Þ ðþ ðþ Eq. () can e epressed in terms of displacements (u,v,w,w s,w a ) sustituting for the stress resultants from Eq. (). For homogeneous laminates, the equations of motion () take the form o u A þ A o u þ A o u þ A o v þða þ A Þ o v þ A o v o w B þ B o w þðb þ B Þ o w þ B o w B s o w s þ o w s Bs þðbs þ Bs Þ o w s þ o w s Bs ¼ I u ðaþ o u A þða þ A Þ o u þ A o u þ A o v þ A o v þ A o v o w B þðb þ B Þ o w þ B o w þ B o w B s o w s þðbs þ Bs Þ o w s þ o w s Bs þ o w s Bs ¼ I v o u B þ B o u þðb þ B Þ o u þ B o u o v þ B þðb þ B Þ o v þ B o v þ B o v o w D þ D o w þ ðd þ D Þ o w o w þ D þ D o w D s o w s þ o w s Ds þ ðds þ Ds Þ o w s þ D s o w s þ o w s Ds þ q þ NðwÞ ¼I ð w a þ w þ w s Þ I r w B s o u þ o u Bs þðbs þ Bs Þ o u þ o u Bs ðþ ðcþ þ B s o v þðbs þ Bs Þ o v þ o v Bs þ o v Bs D s o w þ o w Ds þ ðds þ Ds Þ o w þd s o w þ o w Ds H s o w s þ o w s Hs þ ðhs þ Hs Þ o w s þ H s o w s þ o w s Hs þ A a o w a 55 þ o w a Aa þ o w a Aa 5 þ A s o w s þ o w s As 5 þ q þ NðwÞ þ As 55 o w s ¼ I ð w a þ w þ w s Þ I r w ðdþ o w a A 55 þ A o w a þ A o w a 5 þ A a o w s 55 þ o w s Aa þ o w s Aa 5 þ q þ NðwÞ ¼ I ð w a þ w þ w s Þ ðeþ. Analtical solutions for antismmetric cross-pl laminates The Navier solutions can e developed for rectangular laminates with two sets of simpl supported oundar conditions (Fig. ). For antismmetric cross-pl laminates, the following plate stiffnesses are identicall ero: A ¼ A ¼ D ¼ D ¼ D s ¼ Ds ¼ F ¼ F ¼ H ¼ H ¼ H s ¼ Hs ¼ B ¼ B ¼ B ¼ B ¼ B s ¼ Bs ¼ Bs ¼ Bs ¼ E ¼ E ¼ E ¼ E ¼ A 5 ¼ A a 5 ¼ As 5 ¼ D 5 ¼ F 5 ¼ ; B ¼ B ; B s ¼ Bs ; E ¼ E ðþ The following SS- oundar conditions for antismmetric crosspl laminates can e written as vð; Þ ¼w a ð; Þ ¼w ð; Þ ¼w s ð; Þ ¼ ow a ð; Þ ¼ow ð; Þ ð; Þ ¼ vða; Þ ¼w a ða; Þ ¼w ða; Þ ¼w s ða; Þ ¼ ow a ða; Þ ¼ow ða; Þ ða; Þ ¼ N ð; Þ ¼M ð; Þ ¼Ms ð; Þ ¼N ða; Þ ¼M ða; Þ ¼Ms ða; Þ ¼ uð; Þ ¼w a ð; Þ ¼w ð; Þ ¼w s ð; Þ ¼ ow a ð; Þ ¼ow ð; Þ ð; Þ ¼ uð; Þ ¼w a ð; Þ ¼w ð; Þ ¼w s ð; Þ ¼ ow a ð; Þ ¼ow ð; Þ ð; Þ ¼ N ð; Þ ¼M ð; Þ ¼Ms ð; Þ ¼N ð; Þ ¼M ð; Þ ¼Ms ð; Þ ¼ ðþ

5 S.-E. Kim et al. / Composite Structures () 5 a at = and = a wa w ws v= wa = w = ws = = = = s N = M = M = SS- at = and = wa w ws u = wa = w = ws = = = = s N = M = M = a SS- at = and = a wa w ws u = wa = w = ws = = = = s N = M = M = at = and = wa w ws v= wa = w = ws = = = = s N = M M = Fig.. Two tpes of simpl supported oundar conditions. The oundar conditions in Eq. () are satisfied the following epansions u ¼ X v ¼ X w ¼ X w s ¼ X w a ¼ X U mn cos a sin V mn sin a cos W mn sin a sin W smn sin a sin W amn sin a sin ð5þ where a = mp/a, = np/, and (U mn,v mn,w mn,w smn,w amn ) are coefficients. The applied transverse load q is also epanded in the doule- Fourier series as q ¼ X Q mn sin a sin ðþ Sustituting Eqs. (), (5) and () into Eq. (), the Navier solution of antismmetric cross-pl laminates can e determined from equations s s s s U mn s s s s V mn s s s þ k s þ k k W mn s s s þ k s þ k s 5 þ k 5 W smn k s 5 þ k s 55 þ k W amn m U mn m V mn þ m m m 5 W mn ¼ Q mn ðþ m m m 5 5 W smn Q mn m 5 m 5 m 55 W amn Q mn where s ¼ A a þ A ; s ¼ aða þ A Þ s ¼ B a ; s ¼ B s a s ¼ A a þ A ; s ¼ B ; s ¼ B s s ¼ D a þ ðd þ D Þa þ D s ¼ D s a þ ðd s þ Ds Þa þ D s s ¼ H s a þ ðh s þ Hs Þa þ H s þ A s 55a þ A s s 5 ¼ A a 55a þ A a ; s 55 ¼ A 55 a þ A m ¼ m ¼ m ¼ m 5 ¼ m 5 ¼ m 55 ¼ I m ¼ I þ I ða þ Þ; m ¼ I þ I ða þ Þ k ¼ N a þ N. Analtical solutions for antismmetric angle-pl laminates ðþ For antismmetric angle-pl laminates, the following plate stiffnesses are identicall ero: A ¼ A ¼ D ¼ D ¼ D s ¼ Ds ¼ F ¼ F ¼ H ¼ H ¼ H s ¼ Hs ¼ B ¼ B ¼ B ¼ B ¼ B s ¼ Bs ðþ ¼ B s ¼ Bs ¼ E ¼ E ¼ E ¼ E ¼ A 5 ¼ A a 5 ¼ As 5 ¼ D 5 ¼ F 5 ¼ The following SS- oundar conditions (Fig. ) for antismmetric angle-pl laminates can e written as uð; Þ ¼w a ð; Þ ¼w ð; Þ ¼w s ð; Þ ¼ ow a ð; Þ ¼ow ð; Þ ð; Þ ¼ uða; Þ ¼w a ða; Þ ¼w ða; Þ ¼w s ða; Þ ¼ ow a ða; Þ ¼ow ða; Þ ða; Þ ¼

6 S.-E. Kim et al. / Composite Structures () 5 N ð; Þ ¼M ð; Þ ¼Ms ð; Þ ¼N ða; Þ ¼M ða; Þ ¼Ms ða; Þ ¼ vð; Þ ¼w a ð; Þ ¼w ð; Þ ¼w s ð; Þ ¼ ow a ð; Þ ¼ow ð; Þ ð; Þ ¼ vð; Þ ¼w a ð; Þ ¼w ð; Þ ¼w s ð; Þ ¼ ow a ð; Þ ¼ow ð; Þ ð; Þ ¼ N ð; Þ ¼M ð; Þ ¼Ms ð; Þ ¼N ð; Þ ¼M ð; Þ ¼Ms ð; Þ ¼ ðþ The oundar conditions in Eq. () are satisfied the following epansions u ¼ X v ¼ X w ¼ X w s ¼ X w a ¼ X U mn sin a cos V mn cos a sin W mn sin a sin W smn sin a sin W amn sin a sin ðþ Sustituting Eqs. (), () and () into Eq. (), the equations of the form in Eq. () are otained with the following coefficients s ¼ A a þ A ; s ¼ aða þ A Þ; s ¼ ðb a þ B Þ s ¼ ðb s a þ B s Þ; s ¼ A a þ A ; s ¼ ðb a þ B a Þ s ¼ ðb s a þ B s a Þ; s ¼ D a þ ðd þ D Þa þ D s ¼ D s a þ ðd s þ Ds Þa þ D s s ¼ H s a þ ðh s þ Hs Þa þ H s þ A s 55a þ A s s 5 ¼ A a 55a þ A a ; s 55 ¼ A 55 a þ A m ¼ m ¼ m ¼ m 5 ¼ m 5 ¼ m 55 ¼ I m ¼ I þ I ða þ Þ; m ¼ I þ I ða þ Þ k ¼ N a þ N ðþ the CLPT, FSDT, HSDT, and eact solution of three-dimensional elasticit. In order to investigate the efficienc of the present theor, a simpler version of proposed theor (RPT) is also developed omitting the etension component w a of transverse displacement. The description of various displacement models is given in Tale. In all eamples, a shear correction factor of 5/ is used for FSDT. The following lamina properties are used: Material (Pagano []) E ¼ 5E ; G ¼ G ¼ :5E ; G ¼ :E ; m ¼ :5 Material (Ren []) E ¼ E ; G ¼ G ¼ :5E ; G ¼ :E ; m ¼ :5 Material (Noor []) ðaþ ðþ E ¼ E ; G ¼ G ¼ :E ; G ¼ :5E ; m ¼ :5 ðcþ For convenience, the following nondimensionaliations are used in presenting the numerical results in graphical and taular forms:! E h w ¼ wða=; =Þ qa! ðþ a N ¼ N cr E h 5.. Bending The static ending solution can e otained from Eq. () setting the time derivative terms and in-plane forces to ero: s s s s U mn s s s s V mn s s s s W mn ¼ Q mn ð5aþ s s s s s 5 5 W smn Q mn s 5 s 55 W amn Q mn For the case of RPT (Tale ), the static ending solution of can e simplified as s s s s U mn s s s s V mn ¼ ð5þ s s s s 5 W mn Q mn s s s s W smn Q mn 5. Numerical results and discussion In this section, various numerical eamples are descried and discussed for verifing the accurac of the RPT in predicting the static ending and uckling ehaviors of simpl supported antismmetric cross-pl and angle-pl laminates. For the verification purpose, the results otained RPT are compared with those of Eample. A simpl supported two-laer antismmetric crosspl (/) square laminate under sinusoidal transverse load is considered. Material set is used. The numerical results and corresponding errors with respect to three-dimensional elasticit solution of nondimensionalied deflection are given in Tale. From the results, it can e seen that the results otained using HSDT and RPT (a simpler version of present theor) are identical. Tale Displacement models Model Theor Unknown function CLPT Clasical laminate plate theor FSDT First-order shear deformation theor (Whitne and Pagano []) 5 HSDT Higher-order shear deformation theor (Redd []) 5 Ren Higher-order shear deformation theor (Ren []) RPT Refined plate theor without w a (Present) RPT Refined plate theor with w a (Present) 5

7 S.-E. Kim et al. / Composite Structures () 5 Tale Nondimensionalied deflections of simpl supported two-laer (/) square laminates under sinusoidal transverse load a/h Source w Error (%) Eact []. HSDT.5.5 FSDT 5.. RPT.5.5 RPT.. 5 Eact []. HSDT..5 FSDT.. RPT..5 RPT.5.5 Eact []. HSDT.. FSDT.. RPT.. RPT.. Eact []. HSDT.. FSDT.. RPT.. RPT.. Eact []. HSDT.5.5 FSDT.5. RPT.5.5 RPT.5.5 CLPT...5 w.5 CLPT FSDT HSDT RPT RPT Tale Nondimensionalied deflections of simpl supported two-laer (5/ 5) square and rectangular laminates under sinusoidal transverse load a/h Source w Square plate (a = ) Rectangular plate ( = a) Ren []..5 HSDT..5 FSDT.5. RPT.. RPT..5 Ren []..5 HSDT.55. FSDT.5. RPT.55.5 RPT.5.5 Ren [].5. HSDT.. FSDT.. RPT.. RPT.. CLPT w.5 CLPT FSDT HSDT RPT RPT E /E.5 5 a/h Fig.. The effect of modulus ratio on nondimensionalied deflection of simpl supported two-laer (5/ 5) square laminates under sinusoidal transverse load (a/h = ). Fig.. The effect of side-to-thickness ratio on nondimensionalied deflection of simpl supported two-laer (/) square laminates under sinusoidal transverse load. Tale Nondimensionalied uniaial uckling load of simpl supported antismmetric crosspl (//...) square laminates (a/h = ) Numer of laers Source N Error (%) In general, the present theor (RPT) gives more accurate results in predicting deflections when compared to HSDT. Compared to the elasticit solution (Pagano []), RPT underpredicts deflection.5% while HSDT underpredicts deflection.5% for a/h ratio equal to 5. Fig. shows the variation of nondimensionalied deflection with respect to different side-to-thickness ratios using all models. Eample. A simpl supported two-laer antismmetric anglepl (5/ 5) laminate under sinusoidal transverse load is considered. Material set is used. The numerical results of nondimensionalied deflection for the square and rectangular plates are shown in Tale. In the case of thick plates, there is a considerale difference eists etween the results otained using the various models and the values reported Ren []. For a/h ratio equal to, the deflections predicted FSDT, HSDT, and RPT are Eact []. HSDT.5. FSDT.. RPT.5. RPT.5. CLPT.5. Eact []. HSDT.5. FSDT.5. RPT.5. RPT.5. CLPT.5. Eact []. HSDT 5.5. FSDT RPT 5.5. RPT 5.5. CLPT 5..

8 S.-E. Kim et al. / Composite Structures () 5.%,.%, and.5% lower for a square plate and.%,.%, and.55% lower for a rectangular plate as compared to the values otained Ren []. The results computed using all the five models are in good agreement with those reported Ren [] for thin plates (a/h = ). The nondimensionalied deflections of two-laer (5/ 5) square laminates under sinusoidal transverse load are presented in Fig. for various ratio of modulus E /E (G = G =.5E, G =.E, m =.5, a/h = ). 5.. Buckling For uckling analsis, the applied loads are assumed to e inplane forces N ¼ N ; N ¼ cn ; c ¼ N ; N N ¼ ðþ The uckling solution can e otained from Eq. () setting the time derivative terms and transverse forces to ero: s s s s s s s s s s s N ða þ c Þ s N ða þ c Þ N ða þ c Þ s s s N ða þ c Þ s N ða þ c Þ s 5 N ða þ c Þ5 N ða þ c Þ s 5 N ða þ c Þ s 55 N ða þ c Þ U mn V mn W mn ¼ ðþ W smn W amn Following the condensation of variales procedure to eliminate the in-plane displacements U mn and V mn, the following sstem is otained: s N ða þ c Þ s N ða þ c Þ N ða þ c Þ 5 s N ða þ c Þ s N ða þ c Þ s 5 N ða þ c Þ N ða þ c Þ s 5 N ða þ c Þ s 55 N ða þ c Þ W mn W smn ¼ where W amn s ¼ s s s ; s ¼ s s s s ¼ s s s ; s ¼ s s s ¼ s s s ; ¼ s s s s ; ¼ s s s s ¼ s s s s ; ¼ s s s s ðþ ðþ For nontrivial solution, the determinant of the coefficient matri in Eq. () must e ero. This gives the following epression for uckling load: N ¼ a þc ðs s s s Þs 55 s s 5 s ðs þs 55 s 5 Þþðs þs Þðs 5 s 55 Þþs s 55 s 5 s s ðaþ For the case of RPT (Tale ), the uckling load can e simplified as s N ¼ s s s a þ c s þ s s s ðþ Eample. A simpl supported antismmetric cross-pl (/) n (n =,, 5) square laminate sujected to uniaial compressive load on sides =,a is considered. Material set is used. Tale shows a comparison etween the results otained using the various models and the three-dimensional elasticit solutions given Noor []. The results clearl indicate that the present theor (RPT) gives more accurate results in predicting the uckling loads when compared to HSDT, and results otained using HSDT and RPT (a simpler version of present theor) are identical. Compared to the three-dimensional elasticit solution, the uckling loads predicted RPT, HSDT, and FSDT are.%,.%, and.%, respectivel, for four-laer antismmetric cross-pl (///) square laminates. The effect of side-to-thickness ratio on uckling load of simpl supported four-laer (///) square laminates is also presented in Fig. 5. Eample. A simpl supported two-laer antismmetric anglepl (h/ h) square laminate sujected to uniaial compressive load on sides =,a is considered. Material set is used. The numerical values of nondimensionalied uckling load are given in Tale 5. The results are compared with the values reported Ren []. For all values of side-to-thickness ratio and fier orientation, the uckling loads predicted the RPT and HSDT are almost identical. For a/h ratio equal to and the fier orientation equal to, the uckling load values predicted FSDT, HSDT, and RPT are N CLPT FSDT HSDT RPT RPT 5 a/h Fig. 5. The effect of side-to-thickness ratio on nondimensionalied uniaial uckling load of simpl supported four-laer (///) square laminates. Tale 5 Nondimensionalied uniaial uckling load of simpl supported two-laer (h/ h) square laminates a/h Source N h = h =5 Ren [].5. HSDT.. FSDT.55.5 RPT.5. RPT.5. Ren [] HSDT..5 FSDT..55 RPT.5.5 RPT.. Ren [].. HSDT.5. FSDT..5 RPT.5. RPT.5. CLPT.5.

9 S.-E. Kim et al. / Composite Structures () 5 5 N CLPT FSDT HSDT RPT RPT ehaviors of antismmetric cross-pl and angle-pl laminated composite plates. Acknowledgements The support of the research reported here Ministr of Commerce, Industr and Energ through Grant -- and Korea Ministr of Construction and Transportation through Grant -CA-A5 are gratefull acknowledged. E /E Fig.. The effect of modulus ratio on nondimensionalied uniaial uckling load of simpl supported two-laer (5/ 5) square laminates (a/h = )..%,.%, and.% lower as compared to the values otained Ren []. The results computed using all the five models are in a good agreement with those reported Ren [] for thin plates (a/h = ). The effect of modulus ratio on nondimensionalied uniaial uckling load of simpl supported two-laer (5/ 5) square laminate is presented in Fig. (G = G =.E, G =.5E, m =.5, a/h = ).. Concluding remarks A two variale refined theor is developed for laminated composite plates. The theor gives paraolic distriution of the transverse shear strains, and satisfies the ero traction oundar conditions on the surfaces of the plate without using shear correction factors. The accurac and efficienc of the present theor has een demonstrated for static ending and uckling ehaviors of antismmetric cross-pl and angle-pl laminates. The conclusions of this theor are as follows: () The deflection and uckling load otained using RPT (a simpler version of present theor with four unknowns) and HSDT (five unknowns) are almost identical. () Compared to the three-dimensional elasticit solution, the present theor (RPT) gives more accurate results of deflection and uckling load than the HSDT. () Since the etension component w a of transverse displacement is negligile small in most cases, RPT is sufficient for predicting the ending and uckling ehaviors. In conclusion, it can e said that the proposed theor RPT is accurate and simple in solving the static ending and uckling References [] Reissner E. The effect of transverse shear deformation on the ending of elastic plates. J Appl Mech, Trans ASME 5;():. [] Mindlin RD. Influence of rotar inertia and shear on fleural motions of isotropic, elastic plates. J Appl Mech, Trans ASME 5;():. [] Lirescu L. On the theor of anisotropic elastic shells and plates. Int J Solids Struct ;():5. [] Levinson M. An accurate, simple theor of the statics and dnamics of elastic plates. Mech Res Commun ;(): 5. [5] Bhimaraddi A, Stevens LK. A higher order theor for free viration of orthotropic, homogeneous and laminated rectangular plates. J Appl Mech, Trans ASME ;5():5. [] Redd JN. A simple higher-order theor for laminated composite plates. J Appl Mech, Trans ASME ;5:5 5. [] Ren JG. A new theor of laminated plate. Compos Sci Technol ;:5. [] Kant T, Panda BN. A simple finite element formulation of a higher-order theor for unsmmetricall laminated composite plates. Compos Struct ;():5. [] Mohan PR, Naganaraana BP, Prathap G. Consistent and variationall correct finite elements for higher-order laminated plate theor. Compos Struct ;():5 5. [] Noor AK, Burton WS. Assessment of shear deformation theories for multilaered composite plates. Appl Mech Rev ;():. [] Redd JN. A review of refined theories of laminated composite plates. Shock Vi Dig ;():. [] Redd JN. An evaluation of equivalent-single-laer and laerwise theories of composite laminates. Compos Struct ;5( ): 5. [] Mallikarjuna M, Kant T. A critical review and some results of recentl developed refined theories of fier-reinforced laminated composites and sandwiches. Compos Struct ;():. [] Dahsin L, Xiau L. An overall view of laminate theories ased on displacement hpothesis. J Compos Mater ;:5. [5] Shimpi RP. Refined plate theor and its variants. AIAA J ;():. [] Shimpi RP, Patel HG. A two variale refined plate theor for orthotropic plate analsis. Int J Solids Struct ;():. [] Shimpi RP, Patel HG. Free virations of plate using two variale refined plate theor. J Sound Vi ;( 5):. [] Redd JN. Energ and variational methods in applied mechanics. New York: John Wile and Sons;. [] Pagano NJ. Eact solution for rectangular idirectional composites and sandwich plates. J Compos Mater ;():. [] Ren JG. Bending, viration and uckling of laminated plates. In: Cheremisinoff NP, editor. Handook of ceramics and composites, vol.. New York: Marcel Dekker;. p. 5. [] Noor AK. Stailit of multilaered composite plate. Fire Sci Technol 5;:. [] Whitne JM, Pagano NJ. Shear deformation in heterogeneous anisotropic plates. J Appl Mech, Trans ASME ;():.