1 VIBRATION ANALYSIS OF SYMMETRIC LAMINATED COMPOSITE PLATE WITH FULLY CLAMPED BOUNDARY CONDITION Vimal Kumar Tiwari 1, Manindra Kumar Singh 2 Kirti Chandraul 3 1 Research scholar Department of Civil Engineering Javaharal nehru College of Technology and Science Rewa, M.P. 2,3 Ass. Professor Department of Civil Engineering Jawaharlal nehru College of Technology and Science Rewa, M.P. Abstract This paper presents a vibration analysis of laminated composite plates. The model has been developing using an appropriate eight node isoperimetric element (SHELL281) from the ANSYS element library. In this paper Numerical results have been computed for the effect of material properties, thickness ratio of plate, different aspect ratio, and different number of layer of laminated composite plate. The natural frequencies and mode shapes are compared for fully simply supported and fully clamped condition. Comparisons are made with the result for composite laminated plate. Keywords: vibration analysis, Boundary condition, composite plate. 1. Introduction composite plates are mostly used in many engineering applications such as mechanical, aerospace, marine, automobile, sports, biomedical, heavy machinery, agricultural equipment and health instrument due to their high strength to weight ratio, low specific density, long fatigue life, high stiffness to weight ratio, low weight, high modulus, good electrical and thermal conductivity and other superior material properties. [1-3] Sharma studied, Free vibration analysis of laminated composite plates with elastically restrained edges using FEM and Free vibration analysis of moderately thick anti symmetric crossply laminated rectangular plates with elastic edge constraints. [4] Aydogdu studied Vibration analysis of cross ply laminated square plates with general boundary condition. [5] Karami studied DQM free vibration analysis of moderately thick symmetric laminated plates with elastically restrained edges, Composite Structures. [6] Ashour AS. Studied Vibration of angle-ply symmetric laminated composite plates with edges elastically restrained[7] Reddy studied. Mechanics of Laminated Anisotropic Plates: 2. Laminated composite plate element SHELL 281 is a eight-node linear shell element with six degrees of freedom at each node. Those are translation in x, y, z direction and rotation about x, y, z axis. It is well-suited for linear, large rotation, and large strain nonlinear applications. The element formulation is based on logarithmic strain and true stress measures. Fig. 1 shows the idea regarding the SHELL281 element. The details of the element can be seen in reference [8].
2 Figure 1 SHELL 281 3. Numerical Result and Discussion To show the accuracy of our results we compare the results with the previous reference papers as in literature. It is clear from convergence study that there is up to 2 to 4% variation in result. Reference 1 2 3 4 5 6 7 8 M=N [1] 1.86 3.33 3.48 3.63 4.88 5.40 5.51 6.44 [4] 1.87 3.37 3.69 4.96 5.48 5.60 6.54 6.91 Table 1, Comparisons of non-dimensional frequencies for a angle-ply laminate for boundary condition i.e. SSSS h/b 0.001 35.07 51.38 82.78 92.87 118.80 125.12 162.43 174.46 181.95 208.27 0.01 32.51 40.74 59.47 84.35 89.42 89.72 102.29 126.84 132.76 160.89 0.05 23.79 32.04 48.27 48.88 53.95 65.50 70.99 78.80 82.41 84.20 0.2 9.42 14.45 15.90 19.44 21.37 23.35 25.06 25.98 29.14 30.44 Table 2, Comparison of non-dimensional frequencies with respect to the results given by [9]the thickness ratio (h/b = 0.001, 0.01, 0.05, 0.2) of plate for cross-ply laminate (0 0 /90 0 /0 0 ) for CCCC external condition (h/b=0.01)
3 a/b 1 32.51 40.73 59.46 84.34 89.42 89.72 102.29 126.84 132.76 160.89 1.5 16.84 28.98 39.63 47.77 51.34 66.53 75.44 82.16 83.76 97.14 2 12.12 23.99 26.20 34.92 44.23 49.51 53.48 56.73 72.68 73.95 2.5 10.40 17.14 25.28 29.83 30.05 41.24 48.32 48.86 68.04 59.66 Table 3, Comparison of non-dimensional frequencies with respect to the results given by [9]the aspect ratio (a/b= 1, 1.5, 2, 2.5) of plate for cross-ply laminate (0 0 /90 0 /0 0 ) for CCCC external condition (h/b=0.01) a/b 1 29.26 57.97 57.97 77.50 102.03 102.20 115.16 115.16 144.18 157.28 2 21.23 25.44 35.25 50.84 53.97 56.16 61.83 71.89 72.62 89.55 3 20.61 21.67 24.54 30.08 38.78 51.03 53.65 54.27 55.83 59.05 4 20.48 20.89 21.98 24.28 28.29 34.49 43.17 53.57 53.85 54.52 Table 4, Variation of non-dimensional frequency ( ω = ωb 2 ρ ) with aspect ratio (a/b = 1, 2, 3, 4) h/b = 0.02 for four layers symmetric for fully clamped boundary condition
4 E1/E2 10 20.20 40.70 41.13 55.88 73.47 74.46 84.47 85.11 108.08 116.63 20 26.49 52.53 53.18 70.98 93.19 94.63 105.70 106.64 133.47 144.81 30 31.21 61.0 61.79 81.81 106.40 108.13 120.26 121.38 150.78 162.48 40 35.05 67.60 68.49 90.23 116.16 118.08 131.16 132.40 163.63 174.92 Table 5, Variation of non-dimensional frequency ( ω = ωb 2 ρ ) with material properties (e1/e2 =10 20 30 40) h/b = 0.02 for four layers symmetric (α/-α/-α/α) for fully clamped boundary condition h/b 0.02 29.26 57.97 57.97 77.50 102.03 102.20 115.16 115.16 144.18 157.28 0.4 25.84 47.18 47.18 61.94 76.36 76.52 86.66 86.66 106.02 109.84 0.6 22.11 37.93 37.93 49.12 58.65 58.77 66.66 66.66 80.60 81.66 0.8 18.86 31.15 31.15 40.01 47.08 47.15 53.49 53.49 64.23 64.38 Table 6, Variation of non-dimensional frequency ( ω = ωb 2 ρ ) with thickness ratio (h/b = 0.02 0.1 0.2 0.5)for four layers symmetric for fully clamped boundary condition N 4 29.26 57.97 57.97 77.50 102.03 102.20 115.16 115.16 144.18 157.28 6 30.50 60.57 60.57 80.94 106.93 107.09 120.55 120.55 150.99 165.28 8 31.31 62.16 62.16 83.05 109.74 109.9 123.68 123.68 154.90 169.92 10 31.55 62.66 62.66 83.72 110.68 110.84 124.72 124.72 156.21 171.15 Table 7, Table Variation of non-dimensional frequency ( ω = ωb2 ρ ) with no of layer (n = 4, 6, 8, 10) h/b = 0.02 for four layers symmetric for fully clamped boundary condition
5 4. Conclusion Effect of thickness ratio In this study shows that if we increases thickness ratio the non dimensional frequency decreases in fully clamped conditions. Effect of material properties In this study shows that if we increases material properties the non dimensional frequency increases in fully clamped conditions. Effect of aspect ratio The result shows that non dimensional Frequency decreases as aspect ratio increases in fully clamped conditions. Effect of number of layer In this study result shows that if we increases number of layer the non dimensional frequency increases in fully clamped conditions. 4. References [1] Sharma A. K., Mittal N. D., Free vibration analysis of laminated composite plates with elastically restrained edges using FEM, Central European journal of engineering (2013) 306-315. [2] Sharma A.K., Mittal N.D., Review on stress and vibration analysis of composite plates, Journal of Applied Sciences, 10(23), 2010, 3156-3166. [3] Sharma A.K., Mittal N.D., Sharma A., Free vibration analysis of moderately thick anti symmetric crossply laminated rectangular plates with elastic edge constraints, International Journal of Mechanical Sciences, 53, 2011, 688 695. [4] Aydogdu M., Timarci T., Vibration analysis of cross ply laminated square plates with general boundary conditions, Comp Sci Tech, 63, 2003, 1061 70. [5] Karami G., Malekzadeh P., Mohebpour S.R., DQM free vibration analysis of moderately thick symmetric laminated plates with elastically restrained edges, Composite Structures, 74, 2006, 115-125. [6] Ashour AS. Vibration of angle-ply symmetric laminated composite plates with edges elastically restrained. Compos Struct (in press). [7] Reddy JN. Mechanics of Laminated Anisotropic Plates: Theory and Analysis. Boca Raton (FL): CRC Press; 1997. [8] ANSYS Inc., ANSYS 14.0 reference manual, 2009. [9] Pushpendra k. kushwaha1 and jyoti vimal. Study of Vibration Analysis of Laminated Composite Plates Using FEM rippublication ISSN 2250-3234 Volume 4, Number 6 (2014), pp. 675-680