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30120140505004

  1. 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 45 ANALYSIS FOR FREE VIBRATION OF LAMINATED COMPOSITE & SANDWICH PLATES WITH THE HELP OF EULER-LAGRANGE EQUATION BASED ON FIRST ORDER SHEAR DEFORMATION THEORY, FORMULATIONS AND SOLUTION TO THE NATURAL FREQUENCY & COMPARE TO OBTAINED RESULTS Prabhat Kumar Sinha, *Saurabha Kumar, Vijay Kumar Yadav Mechanical Engineering Department, Shepherd School of Engineering and Technology, Sam Higginbottom Institute of Agriculture, Technology and Sciences, (Formerly Allahabad Agriculture Institute) Allahabad 211007 (INDIA) ABSTRACT This paper present the analytical solution for free vibration of Laminated composite and sandwich plates with the help of Euler-Lagrange Equation based on first order shear deformation theory to analytical formulations and solution to the natural frequency, analysis of simply supported composite and sandwich plates on compared to obtained results, nondimensionalized fundamental frequencies. Theoretical model presented herein incorporates laminate deformations which accounts for the effect of transverse shear deformation, transverse normal strain/stress and none-linear variation of In-plane displacement with respect to the thickness coordinate-thus modeling warping of transverse cross-section more accurately and eliminating the need for shear correction coefficients. Inthis present article the result is compared from obtained result of previous literature. INTRODUCTION Laminated composite plates are being increasingly used in aeronautical and aerospace industry as well as in other field of technology. To use them efficiently a good understanding of their structural and dynamical behavior and also accurate knowledge of the deformation characteristics, stress distribution, natural frequencies and buckling load under the various load conditions are needed. The classical Laminates Plate Theory [1], which is an extension of Classical Plate Theory [2,3] neglects the effect of out-of-plane strains. The greater differences in elastic properties between fiber filaments and matrix material lead to a high ratio of in-plane young’s modulus to transverse shear modulus for the most of the composite laminate developed to date. Because of this reason the INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2014): 7.5377 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
  2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 46 transverse shear deformations are much pronounced for laminated plate than for isotropic plate. Thus the Classical Laminates Plate Theory (CLPT) which ignores the effects of transverse shear deformation becomes inadequate for the analysis of multilayer composites. In general the CLPT often underpredicts deflections and overpredict the natural frequencies and buckling loads. The first- order theories (FSTDs) based on Reissnr [4] and Mindlin [7] assume linear in-plane stresses and displacement respectively through the laminate thickness. Since the FSTD accounts for layerwise constant states of transverse shear, stress, shear correction coefficient are needed to rectify the unrealistic variation of the shear strain/stress through the thickness and which ultimately define the shear strain/stress through the thickness and which ultimately define the shear strain energy. Many studies have been carried out using FSTD for the free vibration analysis of composite plates [9-14]. In order to overcome the limitation of the Lamination of FSTD the analytical solution for free vibration of Laminated composite and sandwich plates with the help of Euler-Lagrange Equation based on first order shear deformation theory to Analytical formulations and solution to the natural frequency analysis of simply supported composite and sandwich plates on compared to obtained result, nondimensionalized fundamental frequencies develop that involve comparative studies of T.Kant, Kant-Manjunatha and Pandaya Kant and Those of Reddy, Senthilnathan and Whiteny-Pagno theories[5].The theoretical model presented herein incorporates laminate deformations which accounts for the effect of transverse shear deformation transverse normal strain/stress and nonlinear variation of in plane displacement with respect to thickness coordinate. MATHEMATICAL FORMULATION A laminated composite plate of length Lx, width Ly and thickness h is considered with the coordinate system placed at the mid-plane of the laminate, as shown in Fig. 1. The fiber direction is indicated by angle h, which represent the positive rotation angle of the principlematerial axes of the layer from the x-axes in the xy-plane. (u1, u2, u3) are defined as the displacements of a point (x, y, z) in the laminate. The displacement field based on the first order shear deformation theory is written as [27]: uଵሺx, y, z, tሻ ൌ uሺx, y, tሻ ൅ z‫׎‬ଵሺx, y, tሻ uଶሺx, y, z, tሻ ൌ uሺx, y, tሻ ൅ z‫׎‬ଶሺx, y, tሻ(1) uଷሺx, y, z, tሻ ൌ wሺx, y, tሻ Fig. 1: Geometry and coordinate system of a rectangular laminated composite plate where (u, v, w) denote the displacements of a point on the mid-plane z = 0. Also, ‫׎‬ଵ and ‫׎‬ଶ are the rotations of a transverse normal about the y and x-axis, respectively, and t is time. Using the dynamic version of principle of the virtual displacements, the following Euler–Lagrange equations of motion of the FSDT will be derived using the calculus of variations [24]. z Lx Ly y z x x h/2 h/2 Z2 Z1 zk Zk+1 ZN ZN+1
  3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 47 ∂Nଵ ∂x ൅ ∂N଺ ∂y ൌ I଴ ∂ଶ u ∂tଶ ൅ Iଵ ∂ଶ ‫׎‬ଵ ∂tଶ ∂N଺ ∂x ൅ ∂Nଶ ∂y ൌ I଴ ∂ଶ v ∂tଶ ൅ Iଵ ∂ଶ ‫׎‬ଶ ∂tଶ ∂Qଵ ∂X ൅ ∂Qଶ ∂y ൅ N ൅ q ൌ I଴ ∂ଶ w ∂tଶ ∂Mଵ ∂x ൅ ∂M଺ ∂y െ Qଵ ൌ Iଶ ∂ଶ ‫׎‬ଵ ∂tଶ ൅ Iଵ ∂ଶ u ∂tଶ ப୑ల பଡ଼ ൅ ப୑మ ப୷ െ Qଶ ൌ Iଶ பమ‫׎‬మ ப୲మ ൅ Iଵ பమ୴ ப୲మ (2) Nሺwሻ ൌ ப ப୶ ቀNଵ ப୵ ப୶ ൅ N଺ ப୵ ப୷ ቁ ൅ ப ப୷ ሺN଺ ப୵ ப୶ ൅ Nଶ ப୵ ப୷ ሻ(3) where q is the resultant distributed force at the surface of laminate, (Ni, Mi) are the force and moment resultants respectively, Qi are the transverse shear forces and Ii are the mass inertia. ሺNଵ, Nଶ, N଺ሻ ൌ න ሺσଵ, σଶ, σ଺ሻdz ୦/ଶ ି୦/ଶ ሺMଵ, Mଶ, M଺ሻ ൌ න ሺσଵ, σଶ, σଷ ୦/ଶ ି୦/ଶ ሻzdz ሺQଵ, Qଶ, ሻ ൌ ‫׬‬ ሺσହ, σସሻdz ୦/ଶ ି୦/ଶ ሺI଴,, Iଵ, Iଶሻ ൌ ‫׬‬ ሺ1, z, zଶ୦/ଶ ି୦/ଶ ሻρ଴dz(4) with: ߪଵ, ൌ ߪ௫௫, ߪଶ ൌ ߪ௬௬,Substituting for the force and moment resultants in terms of displacements into Eq. (2), one can express the equations of motion in terms of displacements as follows [23]: ப ப୶ ቂAଵଵ ப୳ ப୶ ൅ Aଵଶ ப୴ ப୷ ൅ Aଵ଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Bଵଵ ப‫׎‬భ ப୶ ൅ Bଵଶ ப‫׎‬మ ப୷ ൅ Bଵ଺ ቀ ப‫׎‬భ ப୷ ൅ ப‫׎‬మ ப୶ ቁቃ ൅ ப ப୷ ቂAଵ଺ ப୳ ப୶ ൅ Aଶ଺ ப୴ ப୷ ൅ A଺଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Bଵ଺ ப‫׎‬భ ப୶ ൅ Bଶ଺ ப‫׎‬మ ப୷ ൅ B଺଺ ቀ ப‫׎‬భ ப୷ ൅ ப‫׎‬మ ப୶ ቁቃ ൌ I଴ பమ୳ ப୲మ ൅ Iଵ பమ‫׎‬భ ப୲మ (3a) ப ப୶ ቂAଵ଺ ப୳ ப୷ ൅ Aଶ଺ ப୴ ப୷ ൅ A଺଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Bଵ଺ ப‫׎‬భ ப୶ ൅ Bଶ଺ ப‫׎‬మ ப୷ ൅ B଺଺ ቀ ப‫׎‬భ ப୶ ൅ ப‫׎‬మ ப୶ ቁቃ ൅ ப ப୷ ቂAଵଶ ப୳ ப୶ ൅ Aଶଶ ப୴ ப୷ ൅ Aଶ଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Bଵଶ ப‫׎‬భ ப୶ ൅ Bଶଶ ப‫׎‬మ ப୷ ൅ Bଶ଺ ቀ ப‫׎‬భ ப୷ ൅ ப‫׎‬మ ப୶ ቁቃ ൌ I଴ பమ୴ ப୲మ ൅ Iଵ பమ‫׎‬మ ப୲మ (3b) ப பଡ଼ ሾAହହ ቀ ப୵ ப୶ ൅ ‫׎‬ଵቁ ൅ Aସହ ቀ ப୵ ப୷ ൅ ‫׎‬ଶቁሿ ൅ ப ப୷ ቂAସହ ቀ ப୵ ப୶ ൅ ‫׎‬ଵቁ ൅ Aସସ ቀ ப୵ ப୷ ൅ ‫׎‬ଶቁቃ ൅ q ൅ Nሺwሻ ൌ I଴ பమ୵ ப୲మ (3c) ப ப୶ ሾBଵଵ ப୳ ப୶ ൅ Bଵଶ ப୴ ப୷ ൅ Bଵ଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Dଵଵ ப‫׎‬భ ப୶ ൅ Dଵଶ ப‫׎‬మ ப୷ ൅ Dଵ଺ ቀ ப‫׎‬భ ப୷ ൅ ப‫׎‬మ ப୶ ቁ ൅ ப ப୷ ሾBଵ଺ ப୳ ப୶ ൅ Bଶ଺ ப୴ ப୷ ൅ B଺଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Dଵ଺ ப‫׎‬భ ப୶ ൅ Dଶ଺ ப‫׎‬మ ப୷ ൅ D଺଺ ቀ ப‫׎‬భ ப୷ ൅ ப‫׎‬మ ப୶ ቁ_ ቂAହହ ቀ ப୵ ப୶ ൅ ‫׎‬ଵቁ ൅ Aସହ ቀ ப୵ ப୷ ൅ ‫׎‬ଶቁቃ ൌ Iଶ பమ‫׎‬మ ப୲మ ൅ Iଵ பమ୴ ப୲మ (3d) ப ப୶ ቂBଵ଺ ப୳ ப୶ ൅ Bଶ଺ ப୴ ப୷ ൅ B଺଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Dଵ଺ ப‫׎‬భ ப୶ ൅ Dଶ଺ ப‫׎‬మ ப୷ ൅ D଺଺ ቀ ப‫׎‬భ ப୷ ൅ ப‫׎‬మ ப୶ ቁቃ ൅ ப ப୷ ቂBଵଶ ப୳ ப୶ ൅ Bଶଶ ப୴ ப୷ ൅ Bଶ଺ ቀ ப୳ ப୷ ൅ ப୴ ப୶ ቁ ൅ Dଵଶ ப‫׎‬భ ப୶ ൅ Dଶଶ ப‫׎‬మ ப୷ ൅ Dଶ଺ ቀ ப‫׎‬భ ப୷ ൅ ப‫׎‬మ ப୶ ቁቃ െ ሾAସହ ቀ ப୵ ப୶ ൅ ‫׎‬ଵቁ ൅ Aସସ ቀ ப୵ ப୷ ൅ ‫׎‬ଶቁ ൌ Iଶ பమ‫׎‬మ ப୲మ ൅ Iଵ பమ୴ ப୲మ (3e) where the extensional stiffnesses Aij, the bending stiffnesses Dij and the bending-extensional coupling stiffnesses Bij are given by [27]
  4. 4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 48 A୧୨ ൌ ෍ Qഥ ୧୨ ሺ୩ሻ ୒ ୩ୀଵ ሺZ୩ାଵ െ Z୩ሻ ሺi, j ൌ 1,2,6ሻ A୧୨ ൌ K୧୨ ∑ Qഥ ୧୨ ሺ୩ሻ୒ ୩ୀଵ ሺZ୩ାଵ െ Z୩ሻ ሺi, j ൌ 4,5ሻ(4) B୧୨ ൌ 1 2 ෍ Qഥ ୧୨ ሺ୩ሻ ୒ ୩ୀଵ ൫Z୩ାଵ ଶ െ Z୩ ଶ ൯ , D୧୨ ൌ 1 3 ෍ Qഥ ୧୨ ሺ୩ሻ ୒ ୩ୀଵ ൫Z୩ାଵ ଷ െ Z୩ ଷ ൯ where ܳത ௜௝ ሺ௞ሻ are the transformed material constants [27]. Constants Kij appearing in Eq. (4) are the shear correction factors which are taken as 5/6 to account for the parabolic variation of transverse shear stresses. FINITE ELEMENT FORMULATION In developing the finite element model of the FSDT, it is assumed that the variables (u, v, w, /1, /2) each can be represented as a product of a function of time t and a function of coordinates (x, y) as [23]: ‫ݑ‬ሺx, y, tሻ ൌ ෍ u୨ሺtሻ ୬ ୨ୀଵ ψ୨ ୣሺx, yሻ, vሺx, y, tሻ ൌ ෍ v୨ሺtሻ ୬ ୨ୀଵ ψ୨ ୣሺx, yሻ , wሺx, y, tሻ ൌ ∑ w୨ሺtሻ୬ ୨ୀଵ ψ୨ ୣሺx, yሻ, ‫׎‬ଵሺx, y, tሻ ൌ ∑ S୨ ଵ ሺtሻ୬ ୨ୀଵ ψ୨ ୣሺx, yሻ ,(5) ‫׎‬ଶሺx, y, tሻ ൌ ෍ S୨ ଶ ሺtሻ ୬ ୨ୀଵ ψ୨ ୣሺx, yሻ . Where ሺu୨,v୨, w୨, S୨ ଵ , S୨ ଶ ሻ denote the nodal value of ሺu, v, w, ‫׎‬ଵ, ‫׎‬ଶሻ, respectively, associated with element ௘ .Moreover,߰௝ ௘ are the linear Lagrange interpolation function in this paper (see A). Substituting Eq.(5) into the weak form of Eq. (3), Finite element model of the first order theory can be obtained as[23}: ሾKୣሿሼ∆ୣ ሽ ൅ ሾMୣሿ൛∆ሷୣ ൟ ൌ ሼFୣ ሽ(6) where the explicit expression for calculation of the stiffness, mass and force matrices in above equation is presented in A. The plate has been modeled by 4-node rectangular elementswith five degrees of freedom assigned to each node. Hence, the element stiffness matrix has an order of 20 × 20. To overcoming the problem of shear locking, reduced integration of transverse shear stiffnesses utilizing the one point Gauss–Legenderquadriture formula [34] is used in this paper. 4. NUMERICAL RESULT AND DISCUSSION The various models compared in the current study are given in tables. A shear correction factor 5/6 is used in computing the result using T. Kant theory. The nondimensionalized neutral frequencies ߱ഥof general rectangular composite and sandwich plates with simple supports are considered for comparison. The nonedimensionlized natural frequencies computed using various model for two, four, six and 10 layer antisymmetric cross-ply laminate with layers of equal thickness are given Tables. Is T.kant theories is less compared to the other theories. T.kant gives the better result in comparison to other theories. Pandya- Kant theories are in good agreement gives better accurate result in comparison to other theories. The variation of natural frequencies with respect to
  5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 49 side to thickness ratio a/h is presented in table no 1. The result show that for thick plates the result of Kant-Manjunatha and Pandya-Kant theories are in good agreement and considerable difference exist between the result obtained using these theories and models of Reddy, Senthilnathan et al. and Whitney-Pagano. The variation of fundamental frequency with respect to the various parameters like the side- to-thickness ratio (a/h), thickness of the core to thickness of flange (tc/tf) and aspect ratio (a/b) of five layer sandwich plate with antisymmetric cross-ply face sheets using all models are given in tabular form in Table1 to 3. The following material properties are used for the face sheets and the cores. Face sheets (Graphite-Epoxy T300/934) E1=19 ×106 psi(131 GPa),E2=1.5 × 106 psi (10.34 GPa), E2=E3 G12=1 ×106 psi (6.895 GPa), G13=0.90 ×106 psi (6.205 Gpa), G23=1 × 106 psi (6.895 GPa), ʋ12=0.22, ʋ13=0.22, ʋ23=0.49ൌ 0.057 ݈ܾ/݄݅݊ܿଷ ሺ1627 ‫݃ܭ‬ ݉ଷ⁄ ሻ . Core properties (Isotropic) E1=E2=E3=2G=1000 psi (6.89 × 10-3 GPa),G12=G13=G23=500 psi (3.45 × 10-3 GPa), ʋ12= ʋ13= ʋ23=0,ߩ ൌ 0.3403 × 10-2 lb/inch3 (97 Kg/m3 ) . The result clearly show that for all the parameters considered, the frequency values predicted by models of T.Kant[5], and Kant-Manjunatha are in good agreement and those of Pandya-Kant, Reddy and Senthilnathan et al.and Whitney-Pango theories are higher than that predicted Kant- Manjunatha and T.Kant models. For the same sandwich Plate, the variation of fifth mode natural frequencies with respect to the various parameter are shown in Fig. 2 to 4. The result clearly indicate that that even at higher modes of vibration, the natural frequency values obtained using the theory of T.Kant and Kant-Manjunatha theory is good agreement but this present work is near abut all others values. Table 1 Nonedimensionalized fundamental frequencies ߱ഥ ൌ ܾ߱ଶ ݄⁄ ඥሺߩ ‫ܧ‬ଶ⁄ ሻ௙ of an antisymmetric (0/90/Core/0/90) sandwich plate with a/b=1 tc /tf =10 a/h Present Model-1 Present Model-2 Present Model-3 [22]a [23]a [8]a 2 4 10 20 30 40 50 60 70 80 90 100 2.1639 3.8857 7.5350 11.0483 13.0841 14.0546 14.7077 15.1195 15.3934 15.5826 15.7170 15.8191 1.1941 2.1036 4.8594 8.5955 11.0981 12.6821 13.6899 14.3497 14.7977 15.1119 15.3380 15.5093 1.1734 2.0913 4.8519 8.5838 11.0788 12.6555 13.6577 14.3133 14.7583 15.0702 15.2946 15.4647 1.6252 3.1013 7.0473 11.2664 13.6640 14.4390 15.0323 15.3868 15.6134 15.7660 15.8724 15.9522 1.6252 3.1013 7.0473 11.2664 13.6640 14.4390 15.0323 15.3868 15.6134 15.7660 15.8724 15.9522 5.2017 9.0312 13.8694 15.5295 15.9155 16.0577 16.1264 16.1612 16.1845 16.1991 16.2077 16.2175 a Results using these theories are computed independently and are reported newly as benchmark results for sandwich plates.
  6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 50 Table 2 Nonedimensionalized fundamental frequencies ߱ഥ ൌ ܾ߱ଶ ݄⁄ ඥሺߩ ‫ܧ‬ଶ⁄ ሻ௙ of an antisymmetric (0/90/core/0/90) sandwich plate with a/b=1 and a/h=10 tc/tf Present Model-1 Present Model-2 Present Model-3 [22]a [23]a [8]a 4 10 20 30 40 50 100 10.6729 7.5350 5.5812 4.9267 4.6377 4.4783 4.1183 8.9948 4.8594 3.1435 2.8481 2.8266 2.8625 3.0290 8.9690 4.8519 3.1407 2.8466 2.8255 2.8614 3.0276 10.7409 7.0473 4.3734 3.4815 3.1664 3.0561 3.0500 10.7409 7.0473 4.3734 3.4815 3.1664 3.0561 3.0500 13.9190 13.8694 12.8946 11.9760 11.2036 10.5557 8.4349 a Result using these theories are computed independently and are to be same as the result reported in earlier references. Table 3 Nondimensionalized fundamental frequency ߱ഥ ൌ ܾ߱ଶ ݄⁄ ඥሺߩ ‫ܧ‬ଶ⁄ ሻ௙ of an antisymmeric (0/90/core/0/90) sandwich plate with withtc/tf =10 and a/h=10 a/b Present Model-1 Present Model- 2 Present Model-3 [22]a [23]a [8]a 0.5 1.0 1.5 2.0 2.5 3.0 5.0 22.5290 7.5350 4.6908 4.4742 2.8635 2.4212 1.5244 15.0316 4.8594 2.8188 2.4560 1.5719 1.3040 0.8187 15.0128 4.8519 2.8130 2.4469 1.5660 1.2976 0.8180 21.4500 7.0473 4.1587 3.6444 2.3324 1.9242 1.1541 21.6668 7.0473 4.1725 3.6582 2.3413 1.9216 1.1550 39.4840 13.8694 9.4910 10.1655 6.5059 5.6588 3.6841 a Results using these theories are computed independently and are found to be same as result reported in the earlier references. Fig. 2: Nonedimensionalized fifth mode natural frequency (߱ഥ) versus side-to-thickness ratio (a/h) of simply supported five-layer sandwich plate with antisymmetric cross-ply face sheets. 0 2 4 6 8 10 12 14 16 18 2 4 10 20 30 40 50 60 70 80 90 100 ߱߱߱߱̅̅̅̅=ܾܾܾܾ^2⁄݄݄݄݄√(ߩߩߩߩ⁄‫ܧ‬‫ܧ‬‫ܧ‬‫ܧ‬2)f a/h Present Model-1 Present Model-2 Present Model-3 [22]a [23]a [8]a
  7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 51 Fig. 3: Nonedimensionalized fifth mode natural frequency (߱ഥ) versus thickness of core to thickness of face sheet ratio(tc/tf ) of simply supported five-layer sandwich plate with antisymmetric cross-ply- face sheets Fig. 4: Nonedimensionalized fifth mode natural frequency (߱ഥ) versus aspect ratio (a/b) of simply supported five-layer sandwich plate with antisymmetric cross-ply face sheets in Figure 2-4. The result clearly indicate that even at higher mode of vibration, the natural frequencies values obtained using the theories of T.Kant, Kant-Manjunatha and Pandya-Kant are in good agreement and is very much lesser compared to other higher order and first order theories considered in the present investigation. 0 2 4 6 8 10 12 14 16 4 10 20 30 40 50 100 ߱߱߱߱̅̅̅̅=ܾܾܾܾ^2⁄݄݄݄݄√(ߩߩߩߩ⁄‫ܧ‬‫ܧ‬‫ܧ‬‫ܧ‬2)f tc /tf Present Model-1 Present Model-2 Present Model-3 [22]a [23]a [8]a 0 5 10 15 20 25 30 35 40 45 0.5 1 1.5 2 2.5 3 5 ߱߱߱߱̅̅̅̅=ܾܾܾܾ^2⁄݄݄݄݄√(ߩߩߩߩ⁄‫ܧ‬‫ܧ‬‫ܧ‬‫ܧ‬2)f a/b Present Model-1 Present Model- 2 Present Model-3 [22]a [23]a [8]a
  8. 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 52 CONCLUSION In this article, analytical solution for free vibration of Laminated composite and sandwich plates with the help of Euler-Lagrange Equation based on first order shear deformation theory to analytical formulations and solution to the natural frequency, analysis of simply supported composite and sandwich plates on compared to obtained results, nondimensionalized fundamental frequencies has been investigated. Theoretical model presented herein incorporates laminate deformations which accounts for the effect of transverse shear deformation, transverse normal strain/stress and none- linear variation of In-plane displacement with respect to the thickness coordinate-thus modeling warping of transverse cross-section more accurately and eliminating the need for shear correction coefficients.Finite element method based on the first order shear deformationtheory has been used to perform the equations of motion of the plate. The presented method can be employed to perform the parametric studies about various dynamic and structural properties of the vehicle–bridge systems, which are useful for the practical design problems. REFERENCES [1] S.R. Mohebpour a, P. Malekzadeh a, A.A. Ahmadzadeh b, Dynamic analysis of laminated composite plates subjected to a moving oscillator by FEM 93(2011)1574-1583. [2] Ayre RS, Jacobsen LS. Transverse vibration of a two-span beam under action of a moving alternating force. J Appl Mech, Trans ASME 2010; 17:283–90. [3] Wang RT. Vibration of multi-span Timoshenko beams to a moving force. J Sound Vib 1997; 207:731–42. [4] Reissner E. The effect of Transverse shear deformation on the bending of elastic pates. ASME J Appl Mech 1945;12(2)69-77. [5] T.Kant K. Swaminathan, Analytical Solution of free vibration of laminated composite and sandwich plates based on higher-order refined theory 53 (2001) 73-85. [6] Stanisik MM, Hardin JC. On the response of beams to an arbitrary number of concentrated moving masses. J Franklin Inst 1975; 287:115–23. [7] Mindlin RD. Influence of rotatory inertia shear on flexural motion of isotropic, elastic platesASME j Appl Mech 2012; 18:31-8. [8] Whitney J.M pango NJ shear deformation in the heterogeneous anisotropic plates. ASME J; Appl Mech 1970;37(4):1031-6. [9] Yang PC, Norris CH, Stavsky Y. Elastic wave propagation in heterogeneous anisotropic plates. ASME J Appl Mech 1970; 37(4):1031-6. [10] Ambartsumyan SA. Theory of anisotropic Plates. Westport Connecticut: Technomic Publishing company; 1979. [11] Sun CT, Whitney JM. Theories for the dynamic response of laminated plates. AIAA J 1973;1:178. [12] Bert CW, Chen TLC, Effect of shear deformation on vibration of antisymmetricangle ply laminated rectangular plates. Int J Solids Struct 1978; 14:465-73. [13] Reddy JN. Free vibration of antisymmetric angle ply laminated plates including transverse shear deformation by the finite element method. J Sound Vibration 1979; 4:565-76. [14] Noor AK, Burton WS. Stress and free vibration analysis of multilayer composite plates. Compos Struct 1989; 11; 183-204. [15] Hildebrand FB, Reissner E, Thomas GB. Note on the foundations of the theory of small displacements of orthotropic shells. NACA TN-1833, 2001.
  9. 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online), Volume 5, Issue 5, May (2014), pp. 45-53 © IAEME 53 [16] Ghafoori E, Kargarnovin MH, Ghahremani AR. Dynamic responses of a rectangular plate under motion of an oscillator using a semi-analytical method. J Vib Control, doi:10.1177/1077546309358957. [17] Wu JJ. Free vibration characteristics of a rectangular plate carrying multiple three-degree-of freedom spring–mass systems using equivalent mass method. Int J Solids Struct 2006;43:72746. [18] Zhu XQ, Law SS. Dynamic behavior of orthotropic rectangular plates under moving loads. J Eng Mech 2003;129:79. [19] Lee SY, Yhim SS. Dynamic analysis of composite plates subjected to multi moving loads based on a third order theory. Int J Solids Struct 2004;41: 4457–72. [20] Kant T, Varaiya JH, Arora CP. Finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration schemes. Comput Struct 1990;36:401–20. [21] Kadivar MH, Mohebpour SR. Finite element dynamic analysis of unsymmetric composite laminated beams with shear effect and rotary inertia under the action of moving loads. Finite Elem Anal Design 1998; 29:259–73. [22] ReddynJN A Simple higher Order Theory for laminated composite plates .ASME J Appl. Mech 1984; 51:745-52 [23] Senthilnathan NR, Lim KH, Lee KH, Chow ST .Buckling of shear deformable plates. AIAA J1987;25(9);1268-71 [24] Reddy JN. Energy and vibration method in applied Mechanics. New York: John Wiley and Sones.; 2006. [25] Yang YB, Lin BH. Vehicle–bridge interaction analysis by dynamic condensation method. J Struct Eng, ASCE. [26] Inbanatham MJ, Wieland M. Bridge vibrations due to vehicle moving over rough surface. J Struct Eng, ASCE 1987;113:1994–2008. [27] Reddy JN. Mechanics of laminated composite plates: theory and analysis. Boca Raton, FL: CRC Press; 1997. [28] Wu JS, Dai CW. Dynamic response of multispannon uniform beam due to moving loads. J Struct Eng, ASCE 2010;113:458–74. [29] Chang D, Lee H. Impact factors for simple-span highway girder bridges. J StructEng, ASCE 1994;120:704–15. [30] Yang YB, Wu YS. A versatile element for analyzing vehicle–bridge interaction response. Eng Struct 2001;23:452–69. [31] Mohebpour SR, Fiouz AR, Ahmadzadeh AA. Dynamic investigation of laminated composite beams with shear and rotary inertia effect subjected to the moving oscillators using FEM. Compos Struct 2011;93:1118–26. [32] Ghafoori E, Asghari M. Dynamic analysis of laminated composite plates traversed by a moving mass based on a first-order theory. Compos Struct 2010;92:1865–76. [33] Guyan RJ. Reduction of stiffness and mass matrices. AIAA J 2009;3: 380. [34] Reddy JN. An introduction to the finite element method. New York: McGraw- Hill, Inc.; 2010. [35] Bathe KJ. Finite element procedure in engineering analysis. Englewood Cliffs, NJ: Prentice-Hall; 2012. [36] Ansari Fatima-uz-Zehra and S.B. Shinde, “Flexural Analysis of Thick Beams using Single Variable Shear Deformation Theory”, International Journal of Civil Engineering & Technology (IJCIET), Volume 3, Issue 2, 2012, pp. 292 - 304, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316. [37] Prabhat Kumar Sinha and Rohit, “Analysis of Complex Composite Beam by using Timoshenko Beam Theory & Finite Element Method”, International Journal of Design and Manufacturing Technology (IJDMT), Volume 4, Issue 1, 2013, pp. 43 - 50, ISSN Print: 0976 – 6995, ISSN Online: 0976 – 7002.

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