Free vibration and buckling analysis of third-order shear deformation plate theory using exact wave propagation approach

[1]           S. P. Timoshenko, S. Woinowsky-Krieger, 1959, Theory of plates and shells, McGraw-hill,

[2]           R. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. appl. Mech., Vol. 18, pp. 31, 1951.

[3]           J. N. Reddy, A simple higher-order theory for laminated composite plates, Journal of applied mechanics, Vol. 51, No. 4, pp. 745-752, 1984.

[4]           M. Zakeri, R. Attarnejad, Numerical free vibration analysis of higher-order shear deformable beams resting on two-parameter elastic foundation, Journal of Computational Applied Mechanics, Vol. 46, No. 2, pp. 117-131, 2015.

[5]           K. Bell, A refined triangular plate bending finite element, International journal for numerical methods in engineering, Vol. 1, No. 1, pp. 101-122, 1969.

[6]           K. Liew, F.-L. Liu, Differential cubature method: a solution technique for Kirchhoff plates of arbitrary shape, Computer Methods in Applied Mechanics and Engineering, Vol. 145, No. 1-2, pp. 1-10, 1997.

[7]           G. Wei, Y. Zhao, Y. Xiang, The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution, International Journal of Mechanical Sciences, Vol. 43, No. 8, pp. 1731-1746, 2001.

[8]           C. Lü, Z. Zhang, W. Chen, Free vibration of generally supported rectangular Kirchhoff plates: State‐space‐based differential quadrature method, International journal for numerical methods in engineering, Vol. 70, No. 12, pp. 1430-1450, 2007.

[9]           S. Papargyri-Beskou, D. Beskos, Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates, Archive of Applied Mechanics, Vol. 78, No. 8, pp. 625-635, 2008.

[10]         L. Dozio, On the use of the trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates, Thin-Walled Structures, Vol. 49, No. 1, pp. 129-144, 2011.

[11]         S. Shojaee, E. Izadpanah, N. Valizadeh, J. Kiendl, Free vibration analysis of thin plates by using a NURBS-based isogeometric approach, Finite Elements in Analysis and Design, Vol. 61, pp. 23-34, 2012.

[12]         S. C. Brenner, L.-y. Sung, H. Zhang, Y. Zhang, A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates, Journal of Computational and Applied Mathematics, Vol. 254, pp. 31-42, 2013.

[13]         F. Millar, D. Mora, A finite element method for the buckling problem of simply supported Kirchhoff plates, Journal of Computational and Applied Mathematics, Vol. 286, pp. 68-78, 2015.

[14]         A. Çetkin, S. Orak, The free vibration analysis of point supported rectangular plates using quadrature element method, Journal of Theoretical and Applied Mechanics, Vol. 55, No. 3, pp. 1041-1053, 2017.

[15]         J. Reddy, A. Khdeir, Buckling and vibration of laminated composite plates using various plate theories, AIAA journal, Vol. 27, No. 12, pp. 1808-1817, 1989.

[16]         H.-S. Shen, J. Yang, L. Zhang, Free and forced vibration of Reissner–Mindlin plates with free edges resting on elastic foundations, Journal of sound and vibration, Vol. 244, No. 2, pp. 299-320, 2001.

[17]         L. Qian, R. Batra, L. Chen, Free and forced vibrations of thick rectangular plates using higher-order shear and normal deformable plate theory and meshless Petrov-Galerkin (MLPG) method, Computer Modeling in Engineering and Sciences, Vol. 4, No. 5, pp. 519-534, 2003.

[18]         S. H. Hashemi, M. Arsanjani, Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates, International Journal of Solids and Structures, Vol. 42, No. 3, pp. 819-853, 2005.

[19]         G. Shi, A new simple third-order shear deformation theory of plates, International Journal of Solids and Structures, Vol. 44, No. 13, pp. 4399-4417, 2007.

[20]         S. Hosseini-Hashemi, K. Khorshidi, M. Amabili, Exact solution for linear buckling of rectangular Mindlin plates, Journal of sound and vibration, Vol. 315, No. 1, pp. 318-342, 2008.

[21]         S. Hosseini-Hashemi, M. Fadaee, H. R. D. Taher, Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory, Applied Mathematical Modelling, Vol. 35, No. 2, pp. 708-727, 2011.

[22]         S. Eftekhari, A. Jafari, Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions, Applied Mathematical Modelling, Vol. 37, No. 12, pp. 7398-7426, 2013.

[23]         D. Shi, Q. Wang, X. Shi, F. Pang, Free vibration analysis of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions, Shock and Vibration, Vol. 2014, 2014.

[24]         K. K. Pradhan, S. Chakraverty, Transverse vibration of isotropic thick rectangular plates based on new inverse trigonometric shear deformation theories, International Journal of Mechanical Sciences, Vol. 94, pp. 211-231, 2015.

[25]         I. Senjanović, N. Vladimir, D. S. Cho, A new finite element formulation for vibration analysis of thick plates, International Journal of Naval Architecture and Ocean Engineering, Vol. 7, No. 2, pp. 324-345, 2015.

[26]         W. Xiang, Y. Xing, A new first-order shear deformation theory for free vibrations of rectangular plate, International Journal of Applied Mechanics, Vol. 7, No. 01, pp. 1550008, 2015.

[27]         S. M. Mousavi, J. Paavola, J. Reddy, Variational approach to dynamic analysis of third-order shear deformable plates within gradient elasticity, Meccanica, Vol. 50, No. 6, pp. 1537-1550, 2015.

[28]         Q. Wang, D. Shi, X. Shi, A modified solution for the free vibration analysis of moderately thick orthotropic rectangular plates with general boundary conditions, internal line supports and resting on elastic foundation, Meccanica, Vol. 51, No. 8, pp. 1985-2017, 2016.

[29]         Y. Zhou, J. Zhu, Vibration and bending analysis of multiferroic rectangular plates using third-order shear deformation theory, Composite Structures, Vol. 153, pp. 712-723, 2016.

[30]         P. N. Babagi, B. N. Neya, M. Dehestani, Three dimensional solution of thick rectangular simply supported plates under a moving load, Meccanica, pp. 1-18.

[31]         R. Javidi, M. Moghimi Zand, K. Dastani, Dynamics of Nonlinear rectangular plates subjected to an orbiting mass based on shear deformation plate theory, Journal of Computational Applied Mechanics, 2017.

[32]         H. Makvandi, S. Moradi, D. Poorveis, K. H. Shirazi, A new approach for nonlinear vibration analysis of thin and moderately thick rectangular plates under inplane compressive load.

[33]         A. Daneshmehr, A. Rajabpoor, A. Hadi, Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories, International Journal of Engineering Science, Vol. 95, pp. 23-35, 2015.

[34]         M. Hosseini, H. H. Gorgani, M. Shishesaz, A. Hadi, Size-Dependent Stress Analysis of Single-Wall Carbon Nanotube Based on Strain Gradient Theory, International Journal of Applied Mechanics, Vol. 9, No. 06, pp. 1750087, 2017.

[35]         M. Hosseini, M. Shishesaz, K. N. Tahan, A. Hadi, Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials, International Journal of Engineering Science, Vol. 109, pp. 29-53, 2016.

[36]         M. Z. Nejad, A. Rastgoo, A. Hadi, Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique, Journal of Solid Mechanics, Vol. 6, No. 4, pp. 366-377, 2014.

[37]         M. Z. Nejad, A. Hadi, Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 106, pp. 1-9, 2016.

[38]         M. Z. Nejad, A. Hadi, Non-local analysis of free vibration of bi-directional functionally graded Euler–Bernoulli nano-beams, International Journal of Engineering Science, Vol. 105, pp. 1-11, 2016.

[39]         M. Z. Nejad, A. Hadi, A. Farajpour, Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials, Structural Engineering and Mechanics, Vol. 63, No. 2, pp. 161-169, 2017.

[40]         M. Z. Nejad, A. Hadi, A. Rastgoo, Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory, International Journal of Engineering Science, Vol. 103, pp. 1-10, 2016.

[41]         M. Z. Nejad, A. Rastgoo, A. Hadi, Exact elasto-plastic analysis of rotating disks made of functionally graded materials, International Journal of Engineering Science, Vol. 85, pp. 47-57, 2014.

[42]         M. Shishesaz, M. Hosseini, K. N. Tahan, A. Hadi, Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory, Acta Mechanica, Vol. 228, No. 12, pp. 4141-4168, 2017.

[43]         A. Hadi, M. Z. Nejad, A. Rastgoo, M. Hosseini, Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory, Steel and Composite Structures, Vol. 26, No. 6, pp. 663-672, 2018.

[44]         M. Zamani Nejad, M. Jabbari, A. Hadi, A review of functionally graded thick cylindrical and conical shells, Journal of Computational Applied Mechanics, Vol. 48, No. 2, pp. 357-370, 2017.

[45]         A. Hadi, M. Z. Nejad, M. Hosseini, Vibrations of three-dimensionally graded nanobeams, International Journal of Engineering Science, Vol. 128, pp. 12-23, 2018.

[46]         X. Zhang, Frequency analysis of submerged cylindrical shells with the wave propagation approach, International Journal of Mechanical Sciences, Vol. 44, No. 7, pp. 1259-1273, 2002.

[47]         B. Kang, C. Riedel, C. Tan, Free vibration analysis of planar curved beams by wave propagation, Journal of sound and vibration, Vol. 260, No. 1, pp. 19-44, 2003.

[48]         T. Natsuki, M. Endo, H. Tsuda, Vibration analysis of embedded carbon nanotubes using wave propagation approach, Journal of Applied Physics, Vol. 99, No. 3, pp. 034311, 2006.

[49]         S.-K. Lee, B. Mace, M. Brennan, Wave propagation, reflection and transmission in curved beams, Journal of sound and vibration, Vol. 306, No. 3, pp. 636-656, 2007.

[50]         M. N. Bahrami, M. K. Arani, N. R. Saleh, Modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform beams, Scientia Iranica, Vol. 18, No. 5, pp. 1088-1094, 2011.

[51]         A. Bahrami, A. Teimourian, Free vibration analysis of composite, circular annular membranes using wave propagation approach, Applied Mathematical Modelling, Vol. 39, No. 16, pp. 4781-4796, 2015.

[52]         A. Bahrami, A. Teimourian, Nonlocal scale effects on buckling, vibration and wave reflection in nanobeams via wave propagation approach, Composite Structures, Vol. 134, pp. 1061-1075, 2015.

[53]         A. Bahrami, A. Teimourian, Small scale effect on vibration and wave power reflection in circular annular nanoplates, Composites Part B: Engineering, Vol. 109, pp. 214-226, 2017.

[54]         A. Bahrami, Free vibration, wave power transmission and reflection in multi-cracked nanorods, COMPOSITES PART B-ENGINEERING, Vol. 127, pp. 53-62, 2017.

[55]         A. Bahrami, A. Teimourian, Study on the effect of small scale on the wave reflection in carbon nanotubes using nonlocal Timoshenko beam theory and wave propagation approach, Composites Part B: Engineering, Vol. 91, pp. 492-504, 2016.

[56]         M. Ilkhani, A. Bahrami, S. Hosseini-Hashemi, Free vibrations of thin rectangular nano-plates using wave propagation approach, Applied Mathematical Modelling, Vol. 40, No. 2, pp. 1287-1299, 2016.

[57]         A. Bahrami, A wave-based computational method for free vibration, wave power transmission and reflection in multi-cracked nanobeams, Composites Part B: Engineering, Vol. 120, pp. 168-181, 2017.