FREE VIBRATION ANALYSIS OF COMPOSITE LAMINATED PLATES USING HOST 12

E This paper presents an application of a Higher Order Shear Deformation Theory (HOST 12) to problem of free vibration of simply supported symmetric and antisymmetric angle-ply composite laminated plates. The theoretical model HOST12 presented incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate – thus modeling the warping of transverse crosssections more accurately and eliminating the need for shear correction coefficients. Solutions are obtained in closed-form using Navier’s technique by solving the eigenvalue equation. Plates with varying number of layers, degrees of anisotropy and slenderness ratios are considered for analysis. The results compared with those from exact analysis and various theories from references. ةصلاخلا ةيرظنل قيبطتلا مدقي ثحبلا اذه ) Higher Order Shear Deformation Theory (HOST 12) ( رحلا زازتهلأا ةلأسمل لل حئافص لا ة يقبط ة بآرملا و ة لثامتملا ر يغلا ة لثامتم ) ةد ماعتم ر يغلا فا يللأا تاذ .( ة مدّقملا ة يرظنلا 12 HOST رسّ فت ي تلا تا قبطلا تاهيوش ت جمد ت ضرعتسم يعيبط داهجإ ،ضرعتسملا صّقلا هيوشت تاريثأت / كمسلل ةبسنلاب يوتسملا تاحازلأ يطّخلالا عيزوتلا و داهجإ ش ت لكّش ت اذكه هيو ّ صقلا حيحصت تلاماعمل ةجاحلا ليزتو رثآأ ةقّدب ةضرعتسملا ةيضرعلا عطاقملا . ة يقبطلا حئافص لل طوبضملا لحلا تنمضت تلاداعملل لولحلا ةبآرملا ) Navier Solution ( . رابتعلأاب ذخأُ ريثأت يونت ضرعلا ى لا كمس لا ةبس نو ي بورتنلأا ةجردو تا قبطلا ددعآ حئافصلا تافصاوم ع . تنلا عونتم تايرظنو ةطوبضم لولح عم تنروق جئا ة ةددعتم رداصم نم .

which ignores the effect of transverse shear deformation becomes inadequate for the analysis of multilayer composites.The First Order Shear Deformation Theories (FSDTs) based on (Reissner E., 1945) and (Mindlin RD., 1951) assume linear in-plane stresses and displacements respectively through the laminate thickness.Since FSDTs account for layerwise constant states of transverse shear stress, shear correction coefficients are needed to rectify the unrealistic variation of the shear strain/stress through the thickness.In order to overcome the limitations of FSDTs, higher order shear deformation theories (HSDTs) that involve higher order terms in the Taylor's expansions of the displacement in the thickness coordinate were developed.(Hildebrand et al., 1949) were the first to introduce this approach to derive improved theories of plates and shells.Using the higher order theory of (Reddy, 1984) free vibration analysis of isotropic, orthotropic and laminated plates was carried out by (Reddy and Phan, 1985).A generalized Levy-type solution in conjunction with the closed form solution was developed for the bending, buckling and vibration of antisymmetric angle-ply laminated plates by A. (Khdeir A., 1989).The exact solutions were obtained for the classical Kirchhoff theory and the numerical results were compared with their counterparts using the first order transverse shear deformation theory.The comparisons showed that the results obtained within the classical laminated theory could be significantly inaccurate.A selective review of the various analytical and numerical methods used for the stress analysis of laminated composite and sandwich plates was presented by (Kant and Swaminathan, 2001).Using the higher order refined theories already reported in the literature by (Kant, 1982), (Pandya andKant, 1988) and(Kant andManjunatha, 1988), analytical formulations, solutions and comparison of numerical results for the buckling, free vibration and stress analyses of cross-ply composite and sandwich plates were presented by (Kant and Swaminathan, 2002).
Recently the theoretical formulations and solutions for the static analysis of antisymmetric angle-ply laminated composite and sandwich plates using various higher order refined computational models were presented by (Swaminathan and Ragounadin, 2004), (Swaminathan et al., 2006) and (Swaminathan and Patil, 2008).

THEORETICAL FORMULATION Higher Order Shear Deformation Theory (HSDT 12)
For the first time, derived the equation of motion based on higher-order shear deformation theory (HOST 12) in the present study.
The assumptions of a higher order plate theory can also be used within equivalent single layer formulation from (Swaminathan and Patil, 2008): (1) 1.The plate may be moderately thick.
2. The in-plane displacement u (x, y, z, t) and v (x, y, z, t) are cubic functions of z.The strain components will be derived, based on the displacement, as: (2) where: (3) Substituting eq. ( 2) in the stress-strain relation of the lamina, the constitutive relations for any layer in the (x, y) can be expressed in the form: where: And the transformation matrix [T] is given by the transformation equations: The entire collection of forces and moments resultants for N-layered laminated are defined as:

Laminate Constitutive Equations
where the overall laminate stiffnesses A ij , B i j , D i j , E i j , F i j , G i j a n d H i j a r e : (n =1, 2, 3, 4, 5, 6, 7)

Laminated Plates
The equilibrium differential equations in  For the first time, the equations of motion to the HOST12 eq. ( 1) can be expressed in terms of displacements ( )

Free Vibration Solution by HOST12
The following form of solution satisfies the differential the equations of motion and the boundary condition eq. ( 13), when the applied load q (x, y, t) on the right hand side of the equations of motion is set to zero.
The elements of the matrix [K] (Stiffness Matrix) [M] (Mass Matrix) are given in (Salam Ahmed A., 2008).

RESULT AND DISCUSSION
In the following, it is assumed that the material is fiber-reinforced and remains in the     • Kant T., Mallikarjuna., 1989, "A higher order theory for free vibration of unsymmetrically laminated composite and sandwich plates -finite element evaluation", Comput Struct,32,1125-32.• Mallikarjuna, Kant T., 1989, "Free vibration of symmetrically laminated plates using a higher order theory with finite element technique", Int J. Numer Methods Eng,28,1875-89.

INTRODUCTION
Laminated composite plates and shells are finding extensive usage in the aeronautical and aerospace industries as well as in other fields of modern technology.It has been observed that the strength and deformation characteristics of such structural elements depend upon the fiber orientation, stacking sequence and the fiber content in addition to the strength and rigidities of the fiber and matrix material.Though symmetric and antisymmetric laminates are simple to analyze and design, some specific application of composite laminates requires the use of symmetric and antisymmetric laminates to fulfill certain design requirements.Symmetric and antisymmetric angle-ply laminates are the special form of symmetric and antisymmetric laminates and the associated theory offers some simplification in the analysis.The Classical Laminate Plate Theory (Reissner E.

3.
The transverse displacement w (x, y, z, t)of any point (x, y) cubic functions of z. 4. The transverse shear stress XZ σ , YZ σ are parabolic in z. 5.The in-plane stresses X σ , Y σ and XY τ are cubic functions of z. 6.The normal to the mid-surface before deformation are straight, but not necessarily remaining normal to the mid-surface after deformation.7. The transverse normal strain Z σ is not zero.The parameters u o , v o are the in-plane displacements and w o is the transverse displacement of a point (x, y) on the middle plane.The functions θ x , θ y are rotations of the normal to the middle plane about y and x axes respectivelythe parabolic variation of transverse shear stresses through the thickness of the plate (Swaminathan and Patil, 2008).
etc, are written in terms of the ply stiffness ( ) k ij Q and the ply coordinates z k and z k-1 , the following is obtained:

Fig. 1
Fig. 1 Geometry and the co-ordinate system of a rectangular plate of thickness h Equation of Motion in Terms of Displacements HOST12 elastic range.The boundary conditions are SSSS, and the analytical procedure (HOST 12) is used in this work.The material properties are :-E 2 =6.92 x10 9 N/m 2 , E 1 = 40E 2 , G 12 =G 13 =0.5E 2 , G 23 =0.6E 2 , v 12 =0.25 Dimensions of plate: a=1 m , b=1 m , h=0.02 m Fig. 2 Effect of degree of orthotropy of individual layers on the fundamental frequency of simply supported symmetric square laminates using (HOST 12): a/h=5,

Table 1
Effect of degree of orthotropy of

Table 2
Analytical effect of degree of orthotropy and (a/h) ratio of individual layers on the fundamental