New analytic buckling solutions of moderately thick clamped rectangular plates by a straightforward finite integral transform method

  • Salamat Ullah
  • Haiyang Wang
  • Xinran Zheng
  • Jinghui Zhang
  • Yang Zhong
  • Rui LiEmail author


A first endeavor is made in this paper to explore new analytic buckling solutions of moderately thick rectangular plates by a straightforward double finite integral transform method, with focus on typical non-Lévy-type fully clamped plates that are not easy to solve in a rigorous way by the other analytic methods. Solving the governing higher-order partial differential equations with prescribed boundary conditions is elegantly reduced to processing four sets of simultaneous linear equations, the existence of nonzero solutions of which determines the buckling loads and associated mode shapes. Both numerical and graphical results confirm the validity and accuracy of the developed method and solutions by favorable comparison with the literature and finite element analysis. The succinct but effective technique presented in this study can provide an easy-to-implement theoretical tool to seek more analytic solutions of complex boundary value problems.


Analytic solution Thick plate Buckling Finite integral transform method 



The authors gratefully acknowledge the support from the Young Elite Scientists Sponsorship Program by CAST (No. 2015QNRC001), Opening Fund of State Key Laboratory of Nonlinear Mechanics, Chinese Academy of Sciences, and Fundamental Research Funds for the Central Universities of China (No. DUT18GF101).


  1. 1.
    Kirchoff, G.: Uber das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. (Crelle’s J.) 40, 51–88 (1850)CrossRefGoogle Scholar
  2. 2.
    Reissner, E.: The Effect of Transverse Shear deformation on the bending of elastic plates. J. Appl. Mech. Trans. ASME 12(2), A69–A77 (1945)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. Trans. ASME 18(1), 31–38 (1951)zbMATHGoogle Scholar
  4. 4.
    Srinivas, S., Rao, A.: Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. Int. J. Solids Struct. 6(11), 1463–1481 (1970)CrossRefzbMATHGoogle Scholar
  5. 5.
    Wang, C., Xiang, Y., Kitipornchai, S., Liew, K.: Buckling solutions for Mindlin plates of various shapes. Eng. Struct. 16(2), 119–127 (1994)CrossRefGoogle Scholar
  6. 6.
    Liew, K.: Solving the vibration of thick symmetric laminates by Reissner/Mindlin plate theory and thep-Ritz method. J. Sound Vib. 198(3), 343–360 (1996)CrossRefGoogle Scholar
  7. 7.
    Wang, C.M., Lim, G.T., Reddy, J.N., Lee, K.H.: Relationships between bending solutions of Reissner and Mindlin plate theories. Eng. Struct. 23, 838–849 (2001)CrossRefGoogle Scholar
  8. 8.
    Ghannadpour, S., Ovesy, H., Zia-Dehkordi, E.: Buckling and post-buckling behaviour of moderately thick plates using an exact finite strip. Comput. Struct. 147, 172–180 (2015)CrossRefGoogle Scholar
  9. 9.
    Jafari, N., Azhari, M.: Buckling of moderately thick arbitrarily shaped plates with intermediate point supports using a simple hp-cloud method. App. Math. Comput. 313, 196–208 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Civalek, Ö.: Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method. Int. J. Mech. Sci. 49(6), 752–765 (2007)CrossRefGoogle Scholar
  11. 11.
    Bui, T., Nguyen, M., Zhang, C.: Buckling analysis of Reissner-Mindlin plates subjected to in-plane edge loads using a shear-locking-free and meshfree method. Eng. Anal. Bound. Elements 35(9), 1038–1053 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bodaghi, M., Saidi, A.R.: Thermoelastic buckling behavior of thick functionally graded rectangular plates. Arch. Appl. Mech. 81(11), 1555–1572 (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Nazarimofrad, E., Zahrai, S.M., Kholerdi, S.E.S.: Effect of rotationally restrained and Pasternak foundation on buckling of an orthotropic rectangular Mindlin plate. Mech. Adv. Mater. Struct. 25(7), 592–599 (2018)CrossRefGoogle Scholar
  14. 14.
    Yiotis, A.J., Katsikadelis, J.T.: Buckling analysis of thick plates on biparametric elastic foundation: a MAEM solution. Arch. Appl. Mech. 88(1–2), 83–95 (2018)CrossRefGoogle Scholar
  15. 15.
    Li, R., Ni, X., Cheng, G.: Symplectic superposition method for benchmark flexure solutions for rectangular thick plates. J. Eng. Mech. 141(2), 04014119 (2015)CrossRefGoogle Scholar
  16. 16.
    Li, R., Wang, P., Tian, Y., Wang, B., Li, G.: A unified analytic solution approach to static bending and free vibration problems of rectangular thin plates. Sci. Rep. 5, 17054 (2015)CrossRefGoogle Scholar
  17. 17.
    Li, R., Zheng, X., Wang, H., Xiong, S., Yan, K., Li, P.: New analytic buckling solutions of rectangular thin plates with all edges free. Int. J. Mech. Sci. 144, 67–73 (2018)CrossRefGoogle Scholar
  18. 18.
    Yao, W., Zhong, W., Lim, C.W.: Symplectic Elasticity. World Scientific, Singapore (2009)CrossRefzbMATHGoogle Scholar
  19. 19.
    Lim, C.W., Lu, C.F., Xiang, Y., Yao, W.: On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. Int. J. Eng. Sci. 47(1), 131–140 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lim, C.W.: Symplectic elasticity approach for free vibration of rectangular plates. Adv. Vib. Eng. 9(2), 159–163 (2010)Google Scholar
  21. 21.
    Lim, C.W., Xu, X.S.: Symplectic elasticity: theory and applications. Appl. Mech. Rev. 63(5), 050802 (2010)CrossRefGoogle Scholar
  22. 22.
    Li, R., Zhong, Y., Tian, B., Liu, Y.: On the finite integral transform method for exact bending solutions of fully clamped orthotropic rectangular thin plates. Appl. Math. Lett. 22(12), 1821–1827 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tian, B., Li, R., Zhong, Y.: Integral transform solutions to the bending problems of moderately thick rectangular plates with all edges free resting on elastic foundations. Appl. Math. Model. 39(1), 128–136 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhang, S., Xu, L., Li, R.: New exact series solutions for transverse vibration of rotationally-restrained orthotropic plates. Appl. Math. Model. 65, 348–360 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nwoji, C.U., Onah, H.N., Mama, B.O., Ike, C.C., Ikwueze, E.U.: Elastic buckling analysis of simply supported thin plates using the double finite Fourier sine integral transform method. Explor. J. Eng. Technol. 1(1), 37–47 (2017)Google Scholar
  26. 26.
    Mama, B.O., Nwoji, C.U., Ike, C.C., Onah, H.N.: Analysis of simply supported rectangular Kirchhoff plates by the finite Fourier sine transform method. Int. J. Adv. Eng. Res. Sci. 4(3), 285–291 (2017)CrossRefGoogle Scholar
  27. 27.
    Xing, Y., Xiang, W.: Closed-form solutions for eigenbuckling of rectangular Mindlin plate. Int. J. Struct. Stab. Dyn. 16(8), 1550079 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    ABAQUS: Analysis User’s Guide V6.13. Dassault Systèmes, Pawtucket (2013)Google Scholar
  29. 29.
    Teo, T., Liew, K.: A differential quadrature procedure for three-dimensional buckling analysis of rectangular plates. Int. J. Solids Struct. 36(8), 1149–1168 (1999)CrossRefzbMATHGoogle Scholar
  30. 30.
    Xiang, Y.: Numerical developments in solving the buckling and vibration of Mindlin plates. Ph.D. Thesis, The University of Queensland, Australia (1993)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Salamat Ullah
    • 1
  • Haiyang Wang
    • 2
  • Xinran Zheng
    • 2
  • Jinghui Zhang
    • 1
  • Yang Zhong
    • 1
  • Rui Li
    • 2
    • 3
    Email author
  1. 1.Faculty of Infrastructure EngineeringDalian University of TechnologyDalianChina
  2. 2.State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, and International Research Center for Computational MechanicsDalian University of TechnologyDalianChina
  3. 3.State Key Laboratory of Nonlinear Mechanics, Institute of MechanicsChinese Academy of SciencesBeijingChina

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