FREE VIBRATION ANALYSIS OF LAMINATED COMPOSITES BY A NINE NODE ISO- PARAMETRIC PLATE BENDING ELEMENT

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1 FREE VIBRATION ANALYSIS OF LAMINATED COMPOSITES BY A NINE NODE ISO- PARAMETRIC PLATE BENDING ELEMENT Kanak Kalita * Ramachandran M. Pramod Raichurkar Sneha D. Mokal and Salil Haldar Department of Aerospace Engineering & Applied Mechanics Indian Institute of Engineering Science and Technology Shibpur Howrah India 703. MPSTME SVKM S NMIMS University Dhule Maharashtra India *Author to whom correspondence should be addressed: kanakkalita0@gmail.com; Tel: Received 8 February 06; accepted 8 September 06 ABSTRACT Composite laminates are being widely used in engineering industry primarily due to their high strength-to-weight ratio. Considerable research has been carried out to understand the static and dynamic behaviour of laminated composite plates. There is much demand for developing efficient finite element codes which can predict the dynamic responses of laminated structures at affordable computational cost. In this paper a nine node isoparametric plate bending element has been used for free vibration analysis of laminated composite plate. The first-order shear deformation theory (FSDT) has been incorporated in the element formulation. Composite plates with different side-to-thickness ratio (a/h) ply orientations and number of layers have been analysed. Based on comparison with literature data we propose that the present formulation is capable of yielding highly accurate results. Laminated composites with central cut-outs are also studied. Novel data is reported for skew laminated composites. It is found that the natural frequency increases with the increase in skew angle (α) and decreases with increase in aspect ratio (b/a) and thickness (h). Keywords: Natural frequency laminated composites FEM rotary inertia FSDT.. INTRODUCTION In the last two decades there has been unprecedented growth in the use of laminated composites. In U.S alone the composite industry has grown by 6.3% in 04 to reach $8. billion in value and 5.5 billion pounds in terms of annual shipment []. This is due to the flexibility to tailor-make composite as per specific requirements. Being lightweight and having high strength to weight ratio composites have been the ideal choice to replace conventional materials where weight is a prime concern. More than 50% of the structure of 787 Dreamliner by Boeing and the A350XWB by Airbus is made up of composites. Composites have come a long way since its modest beginning in 50 s when only -3% of the Boeing 707 s structure was made of composite []. Composites have already become a part of everyday life in form of car bumpers made of glass fibre composite tennis and badminton rackets made of carbon fibre composite and glass fibre boats and dive boards [3]. To use the laminated composite plates efficiently it is necessary to develop appropriate theories to accurately predict the structural and dynamic behaviour. A number of theories proposed for evaluating the characteristics of composite laminates have been reviewed and discussed in detail by Noor and Burton [4] [5] Reddy [6] Mallikarjuna and Kant [7] Varadan and Bhaskar [8] and very recently by Khandan et al []. These literatures provide a good understanding of the different theories and discuss the advantages and disadvantages of the different theories. In general all these theories can be classified as equivalent single layer theory (ESLT) and discrete layer theory (DLT). The main difference between the two is that ESLT assume the whole composite plate to be made up one equivalent layer whereas in DLT each layer is considered in the analysis. Significant amount of studies has been conducted on the dynamic behaviour of composite plates. Thai and Kim [0] used two variable refined plate theories to calculate the natural frequencies of laminated composite plates. Thai and Choi [] conducted vibration analyses of thick rectangular plates with various boundary conditions using two variable refined plate theories. Xiang et al. [] presented a meshless method based on thin plate spline radial basis functions and higher-order shear deformation theory (HSDT) to analyse the free vibration of clamped laminated composite plates. Adim et al. [3] recently demonstrated the utility of a simplified HSDT. Asadi et al. [4] demonstrated the use of FSDT for composite shells. In their work first order polynomials for in-plane displacements in the z- direction were utilized allowing for the inclusion of shear deformation and rotary inertia effects. In a similar work they [5] used general differential quadrature to solve free vibration problems of isotropic cross-ply angle-ply and general lay-up cylindrical shells. Higher-order shear deformation theory has been used by Asadi and co-workers to calculate natural frequencies of composite plates [6] and shells [7]. Shi et al. [8] used the Galerkin method (taking the transverse shear effects) to perform dynamic analysis of clamped laminated plate. Zhen et al. [] proposed an accurate higher-order theory for free vibration analysis of laminated composite and sandwich plates based on the Hamilton s principle and Navier s technique. Aagaah et al. [0] presented natural frequencies of square laminated composite plates for different supports at edges using a third order shear deformation theory of plates (TSDT). Advanced Composites Letters Vol. 5 Iss

2 Recently an energy-oriented modified Fourier method has been used to perform dynamics analysis of isotropic and composite plates and shells with general boundary conditions. Dynamic analysis of laminated cylindrical shells [] laminated plates [] composite cylindrical shells with general elastic boundary conditions [3] functionally graded cylindrical shells [4] composite laminated structure elements of revolution [5] has been carried out by this method. In the present work a nine node isoparametric element is used to calculate the non-dimensional natural frequencies of composite plates. A generalized finite element computer code is written using first order shear deformation theory because it is simple to implement and gives better results than the CPT since the generalized displacement field not only requires any derivative but also includes transverse shear strains. Further the computational cost of using the FSDT is cheaper than that HSDT. To make the study more comprehensive two mass lumping schemes are used in this work to compute the natural frequencies by considering the effect of rotary inertia and without considering rotary inertia. The existing literature lacks such a thorough comparison on natural frequencies calculated with- and without- considering rotary inertia for composite plates.. FINITE ELEMENT FORMULATION In the current formulation the finite element method was used for free vibration of the plate. The mid plane of the plate of the element is regarded as the reference plane. Since composites are weak in shear the shear deformation effect has been accounted for here. This was done by using the theory of Mindlin plate where it is assumed that the normal to the central plane of the plate before bending remains straight but not necessarily normal to the deformed middle surface after bending. A nine-node isoparametric plate bending element (Fig. ) is used in the current finite element formulation. One of the main advantages of the element is that any form of plate can be well managed with a simple mapping technique that can be defined as: xx = NN rr xx rr andyy = NN rr yy rr () Where (xy) are the coordinates of any point within the element (x r y r ) is the coordinate of rth nodal point and N r is the corresponding interpolation function of the element. In this element Lagrangian interpolation function has been used for N r.. The Lagrange interpolation formula is obtained by multiplying Lagrange interpolation shape function in x-direction and Lagrange interpolation shape function in the y-direction then we get the shape function N r for the particular rectangular or square element as: NN rr = (xx xx )(xx xx ) (xx xx nn ) (xx rr xx )(xx rr xx ) (xx rr xx nn ) (yy yy )(yy yy ) (yy yy nn ) (yy rr yy )(yy rr yy ) (yy rr yy nn ) The shape functions for - node isoparametric element are: NN = ξξξξ (ξξ )(ηη ) NN 4 = ξξξξ (ξξ + )(ηη ) NN 4 3 = ξξξξ (ξξ + )(ηη + ) 4 NN 4 = ξξξξ (ξξ )(ηη + ) NN 4 5 = ηη (ξξ )(ηη ) NN 6 = ξξ (ξξ + )(ηη ) NN 7 = ηη (ξξ )(ηη + ) NN 8 = ξξ (ξξ )(ηη ) NN = (ξξ )(ηη ) The elegance of the formulation lies in the treatment of the inclusion of the shear deformation effect by taking the bending rotations as independent variables in the field which are as follows: { φφ θθ xx xx xx φφ } = yy θθ { yy } Where ϕ x and ϕ y are the average shear rotation over the entire plate thickness and θ x and θ y are the total rotations in bending. Other independent field variables are u v and w where w is the transverse displacement while u and v are the corresponding in-plane displacements. The interpolation functions used for the representation of element geometry equations () are used to express the displacement field at a point within the element in terms of nodal variables as: uu = NN rr uu rr ; vv = NN rr vv rr ; ww = NN rr ww rr ; θθ xx = NN rr θθ xxrr ; () θθ yy = NN rr θθ yyrr Fig.: Nine node isoparametric plate bending element Advanced Composites Letters Vol. 5 Iss.5 06 For a laminate the generalized stress strain relationship with respect to its reference plane may be expressed as: 0

3 [σσ] = [DD]{εε} (3) The generalized stress vector {σ} in the above equation is: {σσ} TT = [NN xx NN yy NN xxxx MM xx MM yy MM xxxx QQ xx QQ yy ] Where N x N y N xy are in-plane force resultants; M x M y bending moments in x and y direction; Mxy is the twisting moment resultant. Q x Q y are the transverse shear force resultants. In the first-order shear deformation theory a shear correction factor (k c ) is required to adjust the transverse shear stiffness for studying the static or dynamic problems of plates. The accuracy of solutions of the FSDT is strongly dependent on predicting better estimates for the shear correction factor. In this case the shear correction factor is assumed to be 5/6. The generalized strain in terms of displacement is written as: and where: A ij B ij D ij are the extensional extensional-bending and bending stiffness coefficients which are defined in terms of the lamina stiffness coefficients. Here n denotes the number of the laminas. (Q ij ) k are the material coefficients. For any orthotropic material they are known in terms of the engineering constants of the k th layer and given as: Where E is the longitudinal modulus and E is the transverse modulus μ is the major Poisson s ratios G G 3 G 3 are the shear moduli. μ is determined by using Advanced Composites Letters Vol. 5 Iss.5 06 (4) {εε} TT = {( ) (dddd dddd ) ( + ) ( θθ xx ) ( θθ yy ) ( θθ xx θθ yy ) ( θθ xx) ( θθ yy)} [DD] = AA AA AA 6 BB BB BB AA AA AA 6 BB BB BB AA 6 AA 6 AA 66 BB 6 BB 6 BB BB BB BB 6 DD DD DD BB BB BB 6 DD DD DD BB 6 BB 6 BB 66 DD 6 DD 6 DD kk cc. AA 55 kk cc. AA 54 [ kk cc. AA 45 kk cc. AA 44 ] nn AA iiii = (QQ iiii ) kk (ZZ kk+ ZZ kk ) ; BB iiii = (QQ iiii) kk (ZZ kk+ ZZ kk ) ; kk= nn DD iiii = 3 (QQ 3 iiii) kk (ZZ kk+ ZZ 3 kk ) kk= QQ kk EE = ; QQ kk EE μμ μμ = QQ kk 44 = GG 3 ; QQ kk 55 = GG 3 ; QQ kk 66 = GG nn kk= μμ μμ ; QQ kk = QQ kk = μμ EE μμ μμ the relation μ E =μ E. With the help of equation () and equation (5) the strain vector may be written as: {εε} = Where [B] is the strain matrix containing interpolation functions and their derivatives and {δ} is the nodal displacement vector having order 45 Once the matrices [B] and [D] are obtained the stiffness matrix of the plate element [K] can be easily derived by the virtual work method and it may be expressed as: In the above equation the Jacobian matrix J is derived from equation () by taking the derivatives of the co-ordinates equation (6). The integration has been carried out numerically following Gauss quadrature technique. In the similar manner the consistent mass matrix of an element can be derived and it may be expressed as: Where [ or{εε} = NN rr NN rr ] or {εε} = [BB]{δδ} + + [BB] rr {δδ rr } ee [KK] ee = [BB] TT [DD][BB] JJ dddd dddd [MM] = Where [N 0 ] = null matrix of the order 0 uu rr vv rr ww rr θθ xxrr { θθ yyrr } ρρh [[NN uu ]TT [NN uu ] + [NN vv ] TT [NN vv ] + [NN ww ] TT [NN ww ] + h [NN θθ xx ] TT [NN θθxx ] + h [NN θθ yy ] TT + + [NN θθyy ]] JJ dddddddd [NN uu ] = [[NN rr ][NN 0 ][NN 0 ][NN 0 ][NN 0 ]] [NN vv ] = [[NN 0 ][NN rr ][NN 0 ][NN 0 ][NN 0 ]] [NN ww ] = [[NN 0 ][NN 0 ][NN rr ][NN 0 ][NN 0 ]] [NN θθxx ] = [[NN 0 ][NN 0 ][NN 0 ][NN rr ][NN 0 ]] [NN θθyy ] = [[NN 0 ][NN 0 ][NN 0 ][NN 0 ][NN rr ]] (6) (7) 0

4 In equation (8) the first two terms of the mass matrix are associated with in-plane movements of mass and the third term indicates transverse movement of mass (which is usually found to contribute the major inertia) whereas the last two terms are associated with rotary inertia and their contribution becomes significant only in a plate having higher thickness. Two types of mass lumping schemes are recommended. In the first lumping scheme LSWORI (lumping scheme without rotary inertia) the effect of in-plane and transverse movements of mass is considered. In the second mass lumping scheme LSWRI (lumping scheme with rotary inertia) the effect of rotary inertia as well as transverse and in-plane movements of mass are considered. The element stiffness matrix and mass matrix having an order of forty five are evaluated for all the elements and they are assembled together to form the overall stiffness matrix [K 0 ] and mass matrix [M 0 ]. Once [K 0 ] and [M 0 ] are obtained the equations of motion of the plate may be expressed as: [KK 0 ] = ωω [MM 0 ] After incorporating the boundary conditions in the above equation it is solved by the simultaneous iterative technique to get frequency ω. Fig. shows a typical 5 layer composite plate used in the study and a skew plate with all geometric dimensions. 3. CONVERGENCE AND VALIDATION STUDY Example : Simply supported anti-symmetric crossply (0 0 /0 0 ) n square laminates A simply supported anti-symmetric cross-ply (0 0 /0 0 ) n square laminates with different side-to-thickness ratios (a/ h= to 0) has been considered. The fundamental frequencies obtained by the present formulation are presented in Table along with the published results by Thai [0] and Reddy [6]. Reddy [6] used third-order shear deformation theory while refined plate theory was used by Thai [0]. The relative material properties of each layer are E / E = 40 G = G 3 = 0.6E G 3 = 0.5E ν = 0.5. A mesh convergence test was carried out. Excellent mesh convergence is seen at 4x4 which represents 4 divisions in X and Y directions each. Hence throughout the study a 4 X 4 mesh is maintained unless otherwise specified. The percentage error between the present solutions and RPT [0] and TSDT [6] and reported here as % error () and % error () respectively. % error is calculated as: % error = (ωω RRRRRR ωω pppppppppppppp ) 00 ωω RRRRRR In all cases excellent agreement with the published results is seen. In general it is seen that the computer code developed with the current formulation is very accurate for small and moderately thick plates. Also the current results are validated against first-order shear deformation theory results of Whitney and Pagano [7]. In most cases the current formulation has predicted the exact frequency reported by FSDT [7]. TSDT assumes that the displacements vary as cubic functions; and the in-plane strains are cubic and the shear strains are quadratic. This allows the line elements normal to the mid-surface not only to rotate but also to deform and not necessarily remain straight whereas FSDT used in the current formulation and Whitney and Pagano [7] assumes the displacements to vary linearly across the thickness. Though FSDT is not as accurate as TSDT for very thick plates it is simple to implement and gives much better results than the CPT and hence is reliable for thin and moderately thick plates. Also the computational cost using the FSDT is cheaper than that using the TSDT. Fig.: (a) A typical composite plate with dimensions (b) Skew plate Advanced Composites Letters Vol. 5 Iss.5 06 Example : Laminated square plate with different boundary conditions A three layer (0/0/0) laminated square plate has been analysed with different mesh divisions using the mass lumping schemes mentioned earlier. The study has been made for the plates having different thickness ratios (h/a = and 0.0) and different boundary conditions as SSSS SCSS and CCSS. The relative material properties of a layer are E = 40 G = G 3 = 0.6E G 3 = 0.5E ν = ν 3 =ν 3 =0.5. The fundamental frequencies obtained in all the cases are presented in Table along with the fundamental frequencies of Liew et al. [8] and exact funda-

5 mental frequencies [6] obtained from their closed-form solution. The percent variation between natural frequencies calculated with- and without- rotary inertia at mesh 4x4 is also shown. Liew et al. [8] had solved the problem using the moving least squares differential quadrature. It can be seen that the present results for SSSS plate are almost as good as the exact solutions by Reddy [6]. For very thin plate the present formulation give better results than Liew et al. [8] in case of CCSS composite plate. Also it is worth note that the previous results have been calculated only by considering rotary inertia. To highlight the effect of rotary inertia natural frequencies calculated without considering rotary inertia are also presented here. As expected frequencies calculated without considering rotary inertia are higher than those calculated using rotary inertia. Table : Non-dimensional fundamental frequencies λλ = ωωaa hh ρρ EE of cross-ply anti-symmetric square laminates with E =40 Lamination (0/0) (0/0) (0/0) 3 (0/0) 5 Source a/h LSWRI(6x6) LSWRI(0x0) LSWRI(4x4) LSWRI(6x6) FSDT [7] RPT [0] TSDT [6] % error () % error () LSWRI (6x6) LSWRI (0x0) LSWRI(4x4) LSWRI (6x6) FSDT [7] RPT [0] TSDT [6] % error () % error () LSWRI(6x6) LSWRI(0x0) LSWRI(4x4) LSWRI(6x6) FSDT [7] RPT [0] TSDT [6] % error () % error () LSWRI(6x6) LSWRI(0x0) LSWRI(4x4) LSWRI(6x6) FSDT [7] RPT [0] TSDT [6] % error () % error () Table : Non-dimensional fundamental frequency λλ = ωωaa hh ρρ EE oof a three-layer (0/0/0) laminated square plate with various boundary conditions and thickness ratios h/a Source Boundary Conditions SSSS SCSS CCSS LSWORI (6X6) LSWORI (0X0) LSWORI (4X4) LSWRI (6X6) LSWRI (0X0) LSWRI (4x4) % variation..6.8 Liew et al. [8] Exact [6] LSWORI (6X6) LSWORI (0X0) LSWORI (4X4) LSWRI (6X6) LSWRI (0X0) LSWRI (4x4) % variation Liew et al. [8] Exact [6] LSWORI (6X6) LSWORI (0X0) LSWORI (4X4) LSWRI (6X6) LSWRI (0X0) LSWRI (4x4) % variation Liew et al. [8] Exact [6] LSWORI (6X6) LSWORI (0X0) LSWORI (4X4) LSWRI (6X6) LSWRI (0X0) LSWRI (4x4) % variation Liew et al. [8] Exact [6] Advanced Composites Letters Vol. 5 Iss.5 06

6 Example 3: Cross-ply square laminates with different orthotropy A simply supported cross-ply (0/0) n square laminates with different modulus ratios (E ) has been considered where the number of layers have been taken as two to five. The relative material properties of each layer are G = G 3 = 0.6E G 3 = 0.5E v = 0.5. The fundamental frequencies obtained in all the cases are presented in Fig. 3 with the fundamental frequencies presented by Thai [0]; TSDT [6] and exact solution [] obtained from their closedform solution. It is seen that the present results are within the acceptable limit of the published results. In comparison with the results of Thai [0] and those obtained by TSDT [6] the present results provide satisfactory accuracy. It is observed that as E ratio increases the fundamental frequencies increases monotonically irrespective of the number of layers involved in laminated square plates. Table 4 and Table 5 presents the non-dimensional frequencies for laminated SSSS and CSCS rectangular plates. The relative material properties of a layer are E = 5 G = G 3 = 0.5E G 3 = 0.E v = 0.5. Table 3: First six non-dimensional frequencies λλ = ωωaa hh ρρ EE for the rectangular CCCC laminated plates (θ 0 /-θ 0 /θ 0 /-θ 0 ). Lamination b/a a/h (5/-5/5/-5) (30/-30/30/-30) (45/-45/45/-45) Source Modes (Non dimensional form) LSWRI FSDT [8] HSDT [] LSWRI FSDT [8] HSDT [] LSWRI LSWRI LSWRI LSWRI LSWRI FSDT [8] HSDT [] LSWRI FSDT [8] HSDT [] LSWRI LSWRI LSWRI LSWRI LSWRI FSDT [8] HSDT [] LSWRI FSDT [8] HSDT [] LSWRI LSWRI LSWRI LSWRI Table 4: First six non-dimensional frequencies λλ = ωωaa hh ρρ EE for rectangular SSSS laminated plates (θ 0 /-θ 0 /θ 0 /-θ 0 ). Fig. 3: Non-dimensional fundamental frequency λλ = ωωaa hh ρρ EE o of simply supported cross ply square laminates with different E/E ratios (a/h=5) with n= 3 and RESULTS AND DISCUSSION 4. Anti-symmetric laminates with different boundary condition Table 3 lists the first six non-dimensional frequencies of clamped supported rectangular composite plate (θ 0 /-θ 0 / θ 0 /-θ 0 ) with various side-to-thickness ratio a/h. Wherever possible the results have been compared with the existing literature and good agreement with the results of Shi et al. [8] and Xiang et al. [] is seen. The present results are in more agreement with those of Shi et al. [8] as compared to Xiang et al. [] who has used higher order deformation theory (HSDT). Shi et al. [8] incorporated the transverse shear effects by considered first order shear deformation theory of Mindlin. Advanced Composites Letters Vol. 5 Iss.5 06 Lamination (5/-5/5/-5) (30/-30/30/-30) (45/-45/45/-45) Modes (Non dimensional form) b/a a/h Source LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI

7 Table 5: First six non-dimensional frequencies λλ = ωωaa hh ρρ EE for rectangular CSCS laminated plates (θ 0 /-θ 0 /θ 0 /-θ 0 ). Lamination (5/-5/5/-5) (30/-30/30/-30) (45/-45/45/-45) Modes (Non dimensional form) b/a a/h Source LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI Anti-symmetric skew laminates A clamped supported anti-symmetric angle-ply plate (θ 0 /-θ 0 /θ 0 /-θ 0 ) is considered. The relative material properties of a layer are E = 5 G = G 3 = 0.5E G 3 = 0.E v = 0.5. The effect of skew angle α is studied and the six non-dimensional frequencies are presented in Table 6. The natural frequencies decreases with increase in aspect ratio for all skew angles and laminate stacking sequences. Also the natural frequencies increase with an increase in the skew angle. These findings are in excellent agreement with Srinivasa et al. [30]. 4.3 Anti-symmetric laminates with cutouts A simply supported square anti-symmetric angle-ply plate (45 0 /-45 0 ) is considered. The relative material properties of each layer are E = 40 G = G 3 = 0.6E G 3 = 0.5E ν = ν 3 =0.5 ρ=500kg/m 3. Table 7 lists the non-dimensional first four frequencies (calculated with- and withoutrotary inertia) of a simply supported square plate having square and rectangular cut-outs of different sizes with various side-to-thickness ratio a/h. The present results are in good agreement with the results of Sivakumar et al. [3]. For the sake of brevity the mode shapes are not included here. 4.4 Anti-symmetric laminates with different number of layers Table 8 lists the non-dimensional frequencies of a simply supported square anti-symmetric angle-ply plate (45/-45) n with various aspect ratio b/a. The relative material properties of a layer are E = 40 G = G 3 = 0.5E G 3 = 0.6E v = 0.5. It is observed that the frequency increases with increase in number of layers. Table 6: First six non-dimensional frequencies λλ = ωωaa hh ρρ EE for rectangular CCCC laminated plates (θ 0 /-θ 0 /θ 0 /-θ 0 ). a/h b/a α Modes (Non dimensional form) Source (5/-5/5/-5) 5 LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI (30/-30/30/-30) 5 LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI (45/-45/45/-45) 5 LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI Advanced Composites Letters Vol. 5 Iss

8 Table 7: First four non-dimensional frequencies λλ = ωωaa hh ρρ EE for rectangular SSSS laminated plates with central cutout (θ 0 /-θ 0 /θ 0 /-θ 0 ). Cut out a/h Source Square 0.a x 0.a Rectangular 0.4a x 0.a Modes 3 4 LSWRI LSWORI FSDT [3] LSWRI LSWORI FSDT [3] LSWRI LSWORI FSDT [3] LSWRI LSWORI FSDT [3] Table 8: First four non-dimensional frequency for rectangular SSSS laminated plates with different layers. b/a Number of layers Source LSWRI LSWORI FSDT [3] LSWRI LSWORI FSDT [3] Modes LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI LSWRI CONCLUSIONS Based on the above results it is seen that the present formulation is very simple yet accurate and does not involve the use of complicated mathematics. The following conclusions are drawn from this study: For thick plates rotary inertia is very significant; for thin plates and shells rotary inertia has no effect. The rotary inertia has the effect of decreasing the frequencies. For all other parameters unchanged as the aspect ratio increases the fundamental frequency decreases. The frequency increases with increase in skew angle and in general decreases with increase in thickness due to the fact that the stiffness-to-mass ratio decreases with the increase in thickness. The orthotropy ratio (E ) have significant influ- Advanced Composites Letters Vol. 5 Iss.5 06 ence on fundamental frequency. Stronger the orthotropy higher is the fundamental frequency. References:. P. K. Mallick Fiber-reinforced composites: materials manufacturing and design CRC press R. Stewart Carbon fibre composites poised for dramatic growth Reinforced Plastics 53/4 pp S. K. Kassegne and K.-S. Chun Buckling characteristic of multi-laminated composite elliptical cylindrical shells International Journal of Advanced Structural Engineering (IJASE) 7/ pp A. K. Noor and W. S. Burton Computational models for high-temperature multilayered composite plates and shells Applied Mechanics Reviews 45/0 pp A. K. Noor and W. S. Burton Assessment of shear deformation theories for multilayered composite plates Applied Mechanics Reviews 4/ pp J. N. Reddy On refined theories of composite laminates Meccanica 5/4 pp T. Kant and others A critical review and some results of recently developed refined theories of fiber-reinforced laminated composites and sandwiches Composite Structures 3/4 pp T. K. Varadan and K. Bhaskar Review of different laminate theories for the analysis of composites JOURNAL- AERONAUTICAL SOCIETY OF INDIA 4/ R. Khandan S. Noroozi P. Sewell and J. Vinney The development of laminated composite plate theories: a review Journal of Materials Science 47/6 pp H.-T. Thai and S.-E. Kim Free vibration of laminated composite plates using two variable refined plate theory International Journal of Mechanical Sciences 5/4 pp H.-T. Thai and D.-H. Choi Analytical solutions of refined plate theory for bending buckling and vibration analyses of thick plates Applied Mathematical Modelling 37/8 pp S. Xiang H. Shi K.-m. Wang Y.-t. Ai and Y.-d. Sha Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates European Journal of Mechanics-A/Solids /5 pp B. Adim T. H. Daouadji and A. Rabahi A simple higher order shear deformation theory for mechanical behavior of laminated composite plates International Journal of Advanced Structural Engineering (IJASE) 8/ pp E. Asadi W. Wang and M. S. Qatu Static and vibration analyses of thick deep laminated cylindrical shells using 3D and various shear deformation theories Composite Structures 4/ pp E. Asadi and M. S. Qatu Free vibration of thick laminated cylindrical shells with different boundary conditions using general differential quadrature Journal of Vibration and Control p E. Asadi and S. J. Fariborz Free vibration of composite plates with mixed boundary conditions based on higher-order shear deformation theory Archive of Applied Mechanics 8/6 pp M. Yaghoubshahi E. Asadi and S. J. Fariborz A higher-order shell model applied to shells with mixed boundary conditions Proceedings of the Institution of Mechanical 5

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