Discrete Singular Convolution and Differential Quadrature Method for Buckling Analysis of Laminated Composite Plates

Abstract

Two different numerical methods for buckling analysis of laminated composite plates are presented. Main

formulations are based on the first-order shear deformation theory (FSDT) have been given. The method of

discrete singular convolution (DSC) and differential quadrature (DQ) are employed for numerical solution. The

results obtained by DSC and DQ methods were compared.

Keywords

• Differential quadrature,discrete singular convolution,buckling,laminated composite

References

1. [1] Reddy, J.N., Mechanics of laminated composite plates and shells: theory and analysis, 2nd ed. New York: CRC Press, 2003.
2. [2] Ventsel, E., Krauthammer, T., Thin Plates and Shells: Theory: Analysis, and Applications, 1st ed. CRC Press, 2001.
3. [3] Qatu, M., Vibration of Laminated Shells and Plates. Academic Press, U.K., 2004.
4. [4] Bathe, K.J., Finite element procedures in engineering analysis. Englewood Cliffs. NJ: Prentice-Hall, 1982.
5. [5] Civalek, Ö., Finite Element analysis of plates and shells. Elazığ: Fırat University 1988 (in Turkish).
6. [6] Civalek, Ö., Geometrically non-linear static and dynamic analysis of plates and shells resting on elastic foundation by the method of polynomial differential quadrature (PDQ) [Ph. D. thesis]. Elazig: Firat University; 2004 (in Turkish).
7. [7] Bert, C.W. and Malik, M., Differential quadrature method in computational mechanics: a review, Applied Mechanics Review, 49(1), 1-28, 1996.
8. [8] Civalek, Ö., Linear and nonlinear dynamic response of multi-degree-of freedom-systems by the method of harmonic differential quadrature (HDQ) [Ph. D. thesis]. İzmir: Dokuz Eylül University; 2003 (in Turkish).

9. [9] Wei G.W., A new algorithm for solving some mechanical problems, Computer Methods in Applied Mechanics and Engineering, 190,2017-2030, 2001.
10. [10] Wei, G.W., Vibration analysis by discrete singular convolution, Journal of Sound and Vibration, 244, 535-553, 2001.
11. [11] Wei, G.W., Discrete singular convolution for beam analysis, Engineering Structures, 23, 1045-1053, 2001.
12. [12] Wei, G.W., Zhou Y.C., Xiang, Y., Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: Theory and algorithm. International Journal for Numerical Methods in Engineering, 55,913-946, 2002.
13. [13] Wei, G.W., Zhou Y.C., Xiang, Y., The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution, International Journal of Mechanical Sciences, 43,1731-1746, 2001.
14. [14] Wei, G.W., Zhou Y.C., Xiang, Y., A novel approach for the analysis of high-frequency vibrations, Journal of Sound and Vibration, 257(2), 207-246, 2002.
15. [15] Zhao, Y.B., Wei, G.W. and Xiang, Y., Discrete singular convolution for the prediction of high frequency vibration of plates, International Journal of Solids and Structures, 39, 65-88, 2002.
16. [16] Zhao, Y.B., and Wei, G.W., DSC analysis of rectangular plates with non-uniform boundary conditions, Journal of Sound and Vibration, 255(2), 203-228, 2002.
17. [17] Civalek, Ö., Gürses, M., Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique, International Journal of Pressure Vessels and Piping, 86, 677-683, 2009.
18. [18] Civalek, Ö., Vibration Analysis of Laminated Composite Conical Shells by the Method of Discrete Singular Convolution Based on the Shear Deformation Theory, Composite Part-B: Engineering, 45, 1001-1009, 2013.
19. [19] Civalek, Ö., Free vibration analysis of single isotropic and laminated composite conical shells using the discrete singular convolution algorithm, Steel and Composite Structures, 6(4),353-366, 2006.
20. [20] Civalek, Ö., The determination of frequencies of laminated conical shells via the discrete singular convolution method, Journal of Mechanics of Materials and Structures, 1(1), 163-182, 2006.
21. [21] Civalek, Ö., A four-node discrete singular convolution for geometric transformation and its application to numerical solution of vibration problem of arbitrary straight-sided quadrilateral plates, Applied Mathematical Modelling, 33(1), 300-314, 2009.
22. [22] Civalek, Ö., Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method, International Journal of Mechanical Sciences, 49, 752–765, 2007.
23. [23] Demir, Ç., Mercan, K., Civalek, Ö., Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel, Composites Part B: Engineering, 94, 1-10, 2016.
24. [24] Civalek, Ö., Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method, Applied Mathematical Modelling, 33(10), 3825-3835, 2009.
25. [25] Civalek, Ö., Free vibration analysis of symmetrically laminated composite plates with first-order shear deformation theory (FSDT) by discrete singular convolution method, Finite Elements in Analysis and Design, 44(12-13)725-731, 2008.
26. [26] Civalek, Ö., Vibration analysis of conical panels using the method of discrete singular convolution, Communications in Numerical Methods in Engineering, 24, 169-181, 2008.
27. [27] Civalek, Ö., Korkmaz, A.K., Demir, Ç., Discrete Singular Convolution Approach for Buckling Analysis of Rectangular Kirchhoff Plates Subjected to Compressive Loads on Two Opposite Edges, Advance in Engineering Software, 41, 557-560, 2010.
28. [28] Xin, L., Hu, Z., Free vibration of simply supported and multilayered magnetoelectro-elastic plates, Composite Structures, 121, 344-350, 2015.
29. [29] Civalek, Ö., Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory. Journal of Composite Materials, 42(26), 2853-2867, 2008.
30. [30] Seçkin, A., Sarıgül, A.S., Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: algorithm and verification. Journal of Sound and Vibration, 315,197-211, 2008.
31. [31] Seçkin, A., Modal and response bound predictions of uncertain rectangular composite plates based on an extreme value model. Journal of Sound and Vibration, 332, 1306-1323, 2013.
32. [32] Xin, L., Hu, Z., Free vibration of layered magneto-electro-elastic beams by SSDSC approach. Composite Structures, 125, 96-103, 2015.
33. [33] Wang, X., Xu, S., Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution. Journal of Sound and Vibration, 329, 1780-1792, 2010.
34. [34] Baltacıoglu, A.K., Civalek, Ö., Akgöz, B., Demir, F., Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution. International Journal of Pressure Vessels and Piping, 88, 290-300, 2011.
35. [35] Civalek, Ö., Akgöz, B., Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix. Computational Materials Science, 77, 295-303, 2013.
36. [36] Gürses, M., Civalek, Ö., Korkmaz, A., Ersoy, H., Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first-order shear deformation theory. International Journal for Numerical Methods in Engineering, 79, 290-313, 2009.
37. [37] Baltacıoglu, A.K., Akgöz, B., Civalek, Ö., Nonlinear static response of laminated composite plates by discrete singular convolution method. Composite Structures, 93, 153-161, 2010.
38. [38] Gürses, M., Akgöz, B., Civalek, Ö., Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation. Applied Mathematics and Computation, 219, 3226–3240, 2012.
39. [39] Mercan, K., Civalek, Ö., DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix, Composite Structures, 143, 300-309, 2016.
40. [40] Wang, X., Wang, Y., Xu, S., DSC analysis of a simply supported anisotropic rectangular plate. Composite Structures, 94, 2576-2584, 2012.
41. [41] Civalek, Ö., Mercan, K., Demir, Ç., Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique. Curved and Layered Structures, 3, 82-90, 2016.
42. [42] Striz, A.G., Wang, X., Bert, C.W., Harmonic differential quadrature method and applications to analysis of structural components. Acta Mechanica, 111, 85-94, 1995.
43. [43] Shu, C. and Xue, H., Explicit computations of weighting coefficients in the harmonic differential quadrature, Journal of Sound and Vibration, 204(3), 549-555, 1997.
44. [44] Han, J.B., Liew, K.M., An eight-node curvilinear differential quadrature formulation for Reissner/Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 141, 265-280, 1997.
45. [45] Civalek, Ö., Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns, Engineering Structures, 26(2), 171-186, 2004.
46. [46] Civalek, Ö., and Ülker, M., Harmonic differential quadrature (HDQ) for axisymmetric bending analysis of thin isotropic circular plates, International Journal of Structural Engineering and Mecanics, 17(1), 1-14, 2004.
47. [47] Liew, K.M., Han, J-B., Xiao, Z.M., and Du, H., Differential quadrature method for Mindlin plates on Winkler foundations, International Journal of Mechanical Sciences 38(4), 405-421, 1996.
48. [48] Liu, F.L., Liew, K.M., Free vibration analysis of Mindlin sector plates: numerical solutions by differential quadrature method. Computer Methods in Applied Mechanics and Engineering, 177, 77-92, 1999.
49. [49] Shu, C., Chen, W., Du, H., Free vibration analysis of curvilinear quadrilateral plates by the differential quadrature method. Journal of Computational Physics,163, 452-466, 2000.
50. [50] Civalek, Ö., A four-node discrete singular convolution for geometric transformation and its application to numerical solution of vibration problem of arbitrary straight-sided quadrilateral plates. Applied Mathematical Modeling, 33, 300-314, 2009.
51. [51] Civalek, Ö., Use of Eight-Node Curvilinear Domains in Discrete Singular Convolution Method for Free Vibration Analysis of Annular Sector Plates with Simply Supported Radial Edges. Journal of Vibration and Control, 16, 303-320, 2010.
52. [52] Khdeir, A.A., Librescu, L., Analysis of symmetric cross-ply elastic plates using a higher-order theory, Part II: buckling and free vibration, Composite Structures, 9, 259-277, 1988.
53. [53] Noor, A.K., Stability of multilayered composite plates, Fibre Sciences Technology, 8(2):81-89, 1975.
54. [54] Akgöz, B., Civalek, Ö., A new trigonometric beam model for buckling of strain gradient microbeams. International Journal of Mechanical Sciences, 81, 88-94, 2014.
55. [55] Baltacıoğlu, A., Civalek, Ö., Akgöz, B., Demir, F., Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution. International Journal of Pressure Vessels and Piping, 88(8-9), 290-300, 2011.
56. [56] Civalek, Ö., Finite Element analysis of plates and shells. Elazığ: Fırat University, 1998.
57. [57] Civalek, Ö., Geometrically non-linear static and dynamic analysis of plates and shells resting on elastic foundation by the method of polynomial differential quadrature (PDQ). 2004, Ph. D. Thesis, Firat University, Elazig, 2004 (in Turkish).
58. [58] Civalek, Ö., Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory. Journal of Composite Materials, 42(26), 2853-2867, 2008.
59. [59] Civalek, Ö. and Akgöz, B., Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix. Computational Materials Science, 77, 295-303, 2013.
60. [60] Civalek, Ö., Demir, Ç., and Akgöz, B., Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory. International Journal of Engineering and Applied Sciences, 2(1), 47-56, 2009.
61. [61] Civalek, Ö., Korkmaz, A., and Demir, Ç., Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges. Advances in Engineering Software, 41(4), 557-560, 2010.
62. [62] Demir, Ç., Civalek, Ö. Nonlocal Finite Element Formulation for Vibration. International Journal of Engineering and Applied Sciences, 8, 109–117, 2016.
63. [63] Civalek, Ö., Demir, Ç. A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation, 289, 335–352, 2016.
64. [64] Akgöz, B., Civalek, Ö. A microstructure-dependent sinusoidal plate model based on the strain gradient elasticity theory. Acta Mechanica, 226, 2277–2294, 2015.
65. [65] Akgöz, B., Civalek, Ö. Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory. Acta Astronautica, 119, 1–12, 2016.
66. [66] Akgöz, B., Civalek, Ö. A novel microstructure-dependent shear deformable beam model. International Journal of Mechanical Sciences, 99, 10–20, 2015.
67. [67] Akgöz, B., Civalek, Ö. Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity. Structural Engineering and Mechanics, 48, 195–205, 2013.
68. [68] Akgöz, B., Civalek, Ö. Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Composite Structures, 134, 294–301, 2015.