Browse > Article
http://dx.doi.org/10.12989/sem.2021.80.4.455

Free vibration analysis of thin rectangular plates with two adjacent edges rotationally-restrained and the others free using finite Fourier integral transform method  

Zhang, Jinghui (Key Laboratory of Green Construction and Intelligent Maintenance for Civil Engineering of Hebei Province, Yanshan University)
Lu, Jiale (Faculty of Infrastructure Engineering, Dalian University of Technology)
Ullah, Salamat (Department of Civil Engineering, Abasyn University)
Gao, Yuanyuan (Key Laboratory of Green Construction and Intelligent Maintenance for Civil Engineering of Hebei Province, Yanshan University)
Zhao, Dahai (Key Laboratory of Green Construction and Intelligent Maintenance for Civil Engineering of Hebei Province, Yanshan University)
Jamal, Arshad (Department of Civil and Envirnmental Engineering, King Fahd University of Petroleum and Minerals)
Civalek, Omer (Research Center for Interneural Computing, China Medical University)
Publication Information
Structural Engineering and Mechanics / v.80, no.4, 2021 , pp. 455-462 More about this Journal
Abstract
For the first time, the finite Fourier integral transform approach is extended to analytically solve the free vibration problem of rectangular thin plates with two adjacent edges rotationally-restrained and others free. Based on the fundamental transform theory, the governing partial differential equations (PDEs) of the plate is converted to ordinary linear algebraic simultaneous equations without assuming trial function for deflection, which reduces the mathematical complexity caused by both the free corner and rotationally-restrained edges. By coupling with mathematical manipulation, the analytical solutions are elegantly achieved in a straightforward procedure. In addition, the vibration characteristics of plates under classical boundary conditions are also studied by choosing different rotating fixed coefficients. Finally, more than 400 comprehensive analytical solutions were well validated by finite element method (FEM) results, which can be served as reference data for further studies. The advantages of the present method are that it does not need to preselect the deformation function, and it has general applicability to various boundary conditions. The presented approach is promising to be further extended to solve the static and dynamic problems of moderately thick plates and thick plates.
Keywords
finite Fourier integral transform method; free edges; free vibration; rotationally-restrained edges; thin rectangular plate;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Civalek, O. (2017), "Free vibration of carbon nanotubes reinforced (CNTR) and functionally graded shells and plates based on FSDT via discrete singular convolution method", Compos. Part B-Eng., 111, 45-59. https://doi.org/10.1016/j.compositesb.2016.11.030.   DOI
2 Senjanovic, I., Tomic, M., Vladimir, N. and Hadzic, N. (2015), "An approximate analytical procedure for natural vibration analysis of free rectangular plates", Thin Wall. Struct., 95, 101-114. https://doi.org/10.1016/j.tws.2015.06.015.   DOI
3 Di Sciuva, M. and Sorrenti, M. (2019), "Bending, free vibration and buckling of functionally graded carbon nanotube-reinforced sandwich plates, using the extended Refined Zigzag Theory", Compos. Struct., 227, 111324. https://doi.org/10.1016/j.compstruct.2019.111324.   DOI
4 Civalek, O. (2019), "Vibration of functionally graded carbon nanotube reinforced quadrilateral plates using geometric transformation discrete singular convolution method", Int. J. Numer. Meth. Eng., 121(5), 990-1019. https://doi.org/10.1002/nme.6254.   DOI
5 Civalek, O., Korkmaz, A. and Demir, C. (2010), "Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two-opposite edges", Adv. Eng. Softw., 41(4), 557-560. https://doi.org/10.1016/j.advengsoft.2009.11.002.   DOI
6 Datta, N. and Verma, Y. (2018), "Analytical scrutiny and prominence of beam-wise rigid-body modes in vibration of plates with translational edge restraints", Int. J. Mech. Sci., 135, 124-132. https://doi.org/10.1016/j.ijmecsci.2017.11.019   DOI
7 Dozio, L. and Carrera, E. (2011), "A variable kinematic Ritz formulation for vibration study of quadrilateral plates with arbitrary thickness", J. Sound Vib., 330(18), 4611-4632. https://doi.org/10.1016/j.jsv.2011.04.022.   DOI
8 Duan, G. and Wang, X. (2014), "Vibration analysis of stepped rectangular plates by the discrete singular convolution algorithm", Int. J. Mech. Sci., 82, 100-109. https://doi.org/10.1016/j.ijmecsci.2014.03.004.   DOI
9 Gupta, A., Jain, N.K., Salhotra, R. and Joshi, P.V. (2015), "Effect of microstructure on vibration characteristics of partially cracked rectangular plates based on a modified couple stress theory", Int. J. Mech. Sci., 100, 269-282. https://doi.org/10.1016/j.ijmecsci.2015.07.004.   DOI
10 Huang, C.S., Lee, M.C. and Chang, M.J. (2018), "Vibration and buckling analysis of internally cracked square plates by the MLS-Ritz approach", Int. J. Struct. Stab. Dyn., 18(9), 1850105. https://doi.org/10.1142/S0219455418501055.   DOI
11 Li, R., Wang, B. and Li, P. (2014), "Hamiltonian system-based benchmark bending solutions of rectangular thin plates with a corner point-supported", Int. J. Mech. Sci., 85, 212-218. https://doi.org/10.1016/j.ijmecsci.2014.05.004.   DOI
12 Ilkhani, M.R., Bahrami, A. and Hosseini-Hashemi, S.H. (2016), "Free vibrations of thin rectangular nano-plates using wave propagation approach", Appl. Math. Model., 40(2), 1287-1299. https://doi.org/10.1016/j.apm.2015.06.032.   DOI
13 Joshi, P.V., Jain, N.K. and Ramtekkar, G.D. (2015), "Analytical modelling for vibration analysis of partially cracked orthotropic rectangular plates", Eur. J. Mech.-A/Solid., 50, 100-111. https://doi.org/10.1016/j.euromechsol.2014.11.007.   DOI
14 ABAQUS (2013), Analysis User'S Guide V6.13, Dassault Systemes, Pawtucket, RI.
15 Abolghasemi, S., Eipakchi, H.R. and Shariati, M. (2016), "An analytical procedure to study vibration of rectangular plates under non-uniform in-plane loads based on first-order shear deformation theory", Arch. Appl. Mech., 86(5), 853-867. https://doi.org/10.1007/s00419-015-1066-8.   DOI
16 Zhang, J., Ullah, S. and Zhong, Y. (2020), "Accurate free vibration solutions of orthotropic rectangular thin plates by straightforward finite integral transform method", Arch. Appl. Mech., 90(2), 353-368. https://doi.org/10.1007/s00419-019-01613-1   DOI
17 Li, R., Zheng, X., Wang, P., Wang, B., Wu, H., Cao, Y. and Zhu, Z. (2019b), "New analytic free vibration solutions of orthotropic rectangular plates by a novel symplectic approach", Acta Mechanica, 230(9), 3087-3101. https://doi.org/10.1007/s00707-019-02448-1.   DOI
18 Li, R., Tian, B. and Zhong, Y. (2013), "Analytical bending solutions of free orthotropic rectangular thin plates under arbitrary loading", Meccanica, 48(10), 2497-2510. https://doi.org/10.1007/s11012-013-9764-1.   DOI
19 Kumar, S., Ranjan, V. and Jana, P. (2018), "Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method", Comput. Struct., 197, 39-53. https://doi.org/10.1016/j.compstruct.2018.04.085.   DOI
20 Lai, S.K. and Xiang, Y. (2009), "DSC analysis for buckling and vibration of rectangular plates with elastically restrained edges and linearly varying in-plane loading", Int. J. Struct. Stab. Dyn., 09(03), 511-531. https://doi.org/10.1142/S0219455409003119.   DOI
21 Li, R., Wang, P., Yang, Z., Yang, J. and Tong, L. (2018a), "On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space", Appl. Math. Model., 53, 310-318. https://doi.org/10.1016/j.apm.2017.09.011.   DOI
22 Li, Y., Zhou, M. and Li, M. (2020), "Analysis of the free vibration of thin rectangular plates with cut-outs using the discrete singular convolution method", Thin Wall. Struct., 147, 106529. https://doi.org/10.1016/j.tws.2019.106529   DOI
23 Civalek, O. and Baltacioglu, A.K. (2019), "Free vibration analysis of laminated and FGM composite annular sector plates", Compos. Part B-Eng., 157, 182-194. https://doi.org/10.1016/j.compositesb.2018.08.101.   DOI
24 Canales, F.G. and Mantari, J.L. (2018), "An assessment of fluid compressibility influence on the natural frequencies of a submerged plate via unified formulation", Ocean Eng., 147, 414-430. https://doi.org/10.1016/j.oceaneng.2017.08.026.   DOI
25 Chen, Y., Jin, G. and Liu, Z. (2014), "Flexural and in-plane vibration analysis of elastically restrained thin rectangular plate with cutout using Chebyshev-Lagrangian method", Int. J. Mech. Sci., 89, 264-278. https://doi.org/10.1016/j.ijmecsci.2014.09.006.   DOI
26 Civalek, O. and Akgoz, B. (2013), "Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix", Comput. Mater. Sci., 77, 295-303. https://doi.org/10.1016/j.commatsci.2013.04.055.   DOI
27 Eftekhari, S.A. and Jafari, A.A. (2013), "Accurate variational approach for free vibration of variable thickness thin and thick plates with edges elastically restrained against translation and rotation", Int. J. Mech. Sci., 68, 35-46. https://doi.org/10.1016/j.ijmecsci.2012.12.012.   DOI
28 Huang, C.S. and Chan, C.W. (2013), "Vibration analyses of cracked plates by the ritz method with moving least-squares interpolation functions", Int. J. Struct. Stab. Dyn., 14(2), 1350060. https://doi.org/10.1142/S0219455413500600.   DOI
29 Thai, C.H., Ferreira, A.J.M., Lee, J. and Nguyen-Xuan, H. (2018), "An efficient size-dependent computational approach for functionally graded isotropic and sandwich microplates based on modified couple stress theory and moving Kriging-based meshfree method", Int. J. Mech. Sci., 142, 322-338. https://doi.org/10.1016/j.ijmecsci.2018.04.040.   DOI
30 Lim, C.W., Lu, C.F., Xiang, Y. and Yao, W. (2009), "On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates", Int. J. Mech. Sci., 47(1), 131-140. https://doi.org/10.1016/j.ijengsci.2008.08.003.   DOI
31 Thai, C.H., Nguyen, T.N., Rabczuk, T. and Nguyen-Xuan, H. (2016), "An improved moving Kriging meshfree method for plate analysis using a refined plate theory", Comput. Struct., 176, 34-49. https://doi.org/10.1016/j.compstruc.2016.07.009.   DOI
32 Thai, C.H. and Nguyen-Xuan, H. (2019), "A moving kriging interpolation meshfree method based on naturally stabilized nodal integration scheme for plate analysis", Int. J. Comput. Meth., 16(4), 1850100. https://doi.org/10.1142/S0219876218501001.   DOI
33 Thang, P.T., Nguyen-Thoi, T., Lee, D., Kang, J. and Lee, J. (2018), "Elastic buckling and free vibration analyses of porous-cellular plates with uniform and non-uniform porosity distributions", Aerosp. Sci. Technol., 79, 278-287. https://doi.org/10.1016/j.ast.2018.06.010.   DOI
34 Malekzadeh, P. and Shojaee, M. (2018), "A unified formulation for free vibration of functionally graded plates", Sci. Eng. Compos. Mater., 25(1), 109-122. https://doi.org/10.1515/secm2016-0031.   DOI
35 Liu, B. and Xing, Y. (2011a), "Exact solutions for free vibrations of orthotropic rectangular Mindlin plates", Compos. Struct., 93(7), 1664-1672. https://doi.org/10.1016/j.compstruct.2011.01.014.   DOI
36 Liu, B. and Xing, Y. (2011b), "Exact solutions for free in-plane vibrations of rectangular plates", Acta Mechanica Solida Sinica, 24(6), 556-567. https://doi.org/10.1016/S0894-9166(11)60055-4.   DOI
37 Malekzadeh, P. and Karami, G. (2008), "Large amplitude flexural vibration analysis of tapered plates with edges elastically restrained against rotation using DQM", Eng. Struct., 30(10), 2850-2858. https://doi.org/10.1016/j.engstruct.2008.03.016.   DOI
38 Papkov, S.O. (2016), "A new method for analytical solution of inplane free vibration of rectangular orthotropic plates based on the analysis of infinite systems", J. Sound Vib., 369, 228-245. https://doi.org/10.1016/j.jsv.2016.01.025.   DOI
39 Papkov, S.O. and Banerjee, J.R. (2015), "A new method for free vibration and buckling analysis of rectangular orthotropic plates", J. Sound Vib., 339, 342-358. https://doi.org/10.1016/j.jsv.2014.11.007.   DOI
40 Ullah, S., Zhang, J. and Zhong, Y. (2019), "Accurate buckling analysis of rectangular thin plates by double finite sine integral transform method", Struct. Eng. Mech., 72(4), 491-502. http://doi.org/10.12989/SEM.2019.72.4.491.   DOI
41 Wang, Z., Xing, Y., Sun, Q. and Yang, Y. (2019), "Highly accurate closed-form solutions for free vibration and eigenbuckling of rectangular nanoplates", Compos. Struct., 210, 822-830. https://doi.org/10.1016/j.compstruct.2018.11.094.   DOI
42 Watts, G., Pradyumna, S. and Singha, M.K. (2018),"Free vibration analysis of non-rectangular plates in contact with bounded fluid using element free Galerkin method", Ocean Eng., 160, 438-448. https://doi.org/10.1016/j.oceaneng.2018.04.056.   DOI
43 Ugurlu, B. (2016), "Boundary element method based vibration analysis of elastic bottom plates of fluid storage tanks resting on Pasternak foundation", Eng. Anal. Bound. Elem., 62, 163-176. https://doi.org/10.1016/j.enganabound.2015.10.006.   DOI
44 Xing, Y., Sun, Q., Liu, B. and Wang, Z. (2018), "The overall assessment of closed-form solution methods for free vibrations of rectangular thin plates", Int. J. Mech. Sci., 140, 455-470. https://doi.org/10.1016/j.ijmecsci.2018.03.013.   DOI
45 Xu, T. and Xing, Y. (2016), "Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation", Acta Mechanica Sinica, 2(6), 1088-1103. https://doi.org/10.1007/s10409-016-0600-4.   DOI
46 Zeng, H.C., Huang, C.S., Leissa, A.W. and Chang M.J. (2016), "Vibrations and stability of a loaded side-cracked rectangular plate via the MLS-Ritz method", Thin Wall. Struct., 106, 459-470. https://doi.org/10.1016/j.tws.2016.05.013.   DOI
47 Zhang, S. and Xu, L. (2017), "Bending of rectangular orthotropic thin plates with rotationally restrained edges: A finite integral transform solution", Appl. Math. Model., 46, 48-62. http://doi.org/10.1016/j.apm.2017.01.053.   DOI
48 Zheng, X., Sun, Y., Huang, M., An, D., Li, P., Wang, B. and Li, R. (2019), "Symplectic superposition method-based new analytic bending solutions of cylindrical shell panels", Int. J. Mech. Sci., 152, 432-442. https://doi.org/10.1016/j.ijmecsci.2019.01.012.   DOI
49 Li, R., Zheng, X., Wang, H., Xiong, S., Yan, K. and Li, P. (2018b), "New analytic buckling solutions of rectangular thin plates with all edges free", Int. J. Mech. Sci., 144, 67-73. https://doi.org/10.1016/j.ijmecsci.2018.05.041.   DOI
50 Zhang, Y., Du, J., Yang, T. and Liu, Z. (2014), "A series solution for the in-plane vibration analysis of orthotropic rectangular plates with elastically restrained edges", Int. J. Mech. Sci., 79, 15-24. https://doi.org/10.1016/j.ijmecsci.2013.11.018   DOI
51 Zhou, Y. and Wang, Z. (2014), "Application of the differential quadrature method to free vibration of viscoelastic thin plate with linear thickness variation", Meccanica, 49(12), 2817-2828. https://doi.org/10.1007/s11012-014-0043-6.   DOI
52 Xing, Y., Wang, Z. and Xu, T. (2018), "Closed-form analytical solutions for free vibration of rectangular functionally graded thin plates in thermal environment", Int. J. Appl. Mech., 10(3), 1850025. https://doi.org/10.1142/S1758825118500254.   DOI
53 Liu, T., Hu, G., Wang, A. and Wang, Q. (2019), "A unified formulation for free in-plane vibrations of arbitrarily-shaped straight-sided quadrilateral and triangular thin plates", Appl. Acoust., 155, 407-422. https://doi.org/10.1016/j.apacoust.2019.06.014.   DOI
54 Wang, X. and Yuan, Z. (2017), "Discrete singular convolution and Taylor series expansion method for free vibration analysis of beams and rectangular plates with free boundaries", Int. J. Mech. Sci., 122, 184-191. https://doi.org/10.1016/j.ijmecsci.2017.01.023.   DOI
55 Malekzadeh, P. and Beni, A.A. (2015), "Nonlinear free vibration of in-plane functionally graded rectangular plates", Mech. Adv. Mater. Struct., 22(8), 633-640. https://doi.org/10.1080/15376494.2013.828818.   DOI
56 Hashemi, S.H., Karimi, M. and Taher H.R.D. (2010), "Vibration analysis of rectangular Mindlin plates on elastic foundations and vertically in contact with stationary fluid by the Ritz method", Ocean Eng., 37(2-3), 174-185. https://doi.org/10.1016/j.oceaneng.2009.12.001.   DOI
57 Hwu, C., Hsu, H.W. and Lin, Y.H. (2017), "Free vibration of composite sandwich plates and cylindrical shells",Compos. Struct., 171, 528-537. https://doi.org/10.1016/j.compstruct.2017.03.042.   DOI
58 Kumar, S. and Jana, P. (2019), "Application of dynamic stiffness method for accurate free vibration analysis of sigmoid and exponential functionally graded rectangular plates", Int. J. Mech. Sci., 163, 105105. https://doi.org/10.1016/j.ijmecsci.2019.105105.   DOI
59 Li, R., Wang, H., Zheng, X., Xiong, S., Hu, Z., Yan, X., Xiao, Z., Xu, H. and Li, P. (2019a), "New analytic buckling solutions of rectangular thin plates with two free adjacent edges by the symplectic superposition method", Eur. J. Mech. A-Solid., 76, 247-262. https://doi.org/10.1016/j.euromechsol.2019.04.014.   DOI
60 Li, R., Zheng, X., Yang, Y., Huang, M. and Huang, X. (2019c), "Hamiltonian system-based new analytic free vibration solutions of cylindrical shell panels", Appl. Math. Model., 76, 900-917. https://doi.org/10.1016/j.apm.2019.07.020.   DOI
61 Najarzadeh, L., Movahedian, B. and Azhari, M. (2018), "Free vibration and buckling analysis of thin plates subjected to high gradients stresses using the combination of finite strip and boundary element methods", Thin Wall. Struct., 123, 36-47. https://doi.org/10.1016/j.tws.2017.11.015.   DOI
62 Razavi, S. and Shooshtari, A. (2015), "Nonlinear free vibration of magneto-electro-elastic rectangular plates", Comput. Struct., 119, 377-384. https://doi.org/10.1016/j.compstruct.2014.08.034.   DOI