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Differential cubature method for buckling analysis of arbitrary quadrilateral thick plates

  • Wu, Lanhe (Department of Mechanics and Engineering Science, Shijiazhuang Railway Institute) ;
  • Feng, Wenjie (Department of Mechanics and Engineering Science, Shijiazhuang Railway Institute)
  • Received : 2002.05.21
  • Accepted : 2003.06.27
  • Published : 2003.09.25

Abstract

In this paper, a novel numerical solution technique, the differential cubature method is employed to study the buckling problems of thick plates with arbitrary quadrilateral planforms and non-uniform boundary constraints based on the first order shear deformation theory. By using this method, the governing differential equations at each discrete point are transformed into sets of linear homogeneous algebraic equations. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations of the plate. Detailed formulation and implementation of this method are presented. The buckling parameters are calculated through solving a standard eigenvalue problem by subspace iterative method. Convergence and comparison studies are carried out to verify the reliability and accuracy of the numerical solutions. The applicability, efficiency, and simplicity of the present method are demonstrated through solving several sample plate buckling problems with various mixed boundary constraints. It is shown that the differential cubature method yields comparable numerical solutions with 2.77-times less degrees of freedom than the differential quadrature element method and 2-times less degrees of freedom than the energy method. Due to the lack of published solutions for buckling of thick rectangular plates with mixed edge conditions, the present solutions may serve as benchmark values for further studies in the future.

Keywords

References

  1. Alexander Chajes, (1974), Principles of Structural Stability Theory, Prentice Hall, Inc.
  2. Bellman, R.E. and Casti, J. (1971), "Differential quadrature and long term integration", Journal of Mathematical Analysis and Applications, 34, 235-238. https://doi.org/10.1016/0022-247X(71)90110-7
  3. Bellman, R.E. and Casti, J. (1972), "Differential quadrature: A technique for the rapid solution of nolinear partialdifferential equations", Journal of Computational Physics, 10, 40-52. https://doi.org/10.1016/0021-9991(72)90089-7
  4. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: A review",Applied Mechanics Review, 48, 1-28.
  5. Brunelle, E.J. (1971), "Buckling of transversely isotropic Mindlin plates", AIAA J., 9, 1018-1022. https://doi.org/10.2514/3.6326
  6. Brunelle, E.J. and Robertson, S.R. (1974), "Initially stressed Mindlin plates", AIAA J., 12, 1036-1045. https://doi.org/10.2514/3.49407
  7. Civan, F. (1994), "Solving multivariable mathematical models by the quadrature and cubature methods",Numerical Method for Partial Differential Equations, 10, 545-567. https://doi.org/10.1002/num.1690100503
  8. Dawe, D.J. and Roufaeil, O.L. (1982), "Buckling of rectangular Mindlin plates", Comput. Struct., 15, 461-471. https://doi.org/10.1016/0045-7949(82)90081-5
  9. Edwardes, R.J. and Kabaila, A.P. (1978), "Buckling of simply supported skew plates", Int. J. Numer. MethodsEngrg., 12, 779-785. https://doi.org/10.1002/nme.1620120504
  10. Hamada, M., Inoue, Y. and Hashimoto, H. (1967), "Buckling of simply supported but partially clampedrectangular plates uniformly compressed in one direction", Bull. JSME, 10, 35-40. https://doi.org/10.1299/jsme1958.10.35
  11. Hegedus, T. (1988), "Finite strip buckling analysis of skew plates under combined loading", in: Evayi M,Stability of Steel Structures, Akademi Kiado, Budapest, 633-641.
  12. Hinton, E. (1978), "Buckling of initially stressed Mindlin plates using a finite strip method", Comput. Struct., 8,99-105. https://doi.org/10.1016/0045-7949(78)90164-5
  13. Keer, L.M. and Stahl, B. (1972), "Eigenvalue problems of rectangular plates with mixed edge conditions", Trans.ASME J. Appl. Mech., 39, 513-520. https://doi.org/10.1115/1.3422709
  14. Kennedy, J.B. and Prabhakara, M.K. (1979), "Combined-load local buckling of orthotropic skew plates", J.Engrg. Mech. Div., ASCE, 105, 71-79.
  15. Kitipornchai, S., Xiang, L., Wang, C.M. and Liew, K.M. (1993), "Buckling of thick skew plates", Int. J. Numer.Methods Engrg, 36, 1299-1310. https://doi.org/10.1002/nme.1620360804
  16. Liew, K.M., Han, J.B. and Xiao, Z.M. (1996), "Differential quadrature method for Mindlin plates on Winklerfoundations", Int. J. Mech. Sci., 38, 405-421. https://doi.org/10.1016/0020-7403(95)00062-3
  17. Liew, K.M. and Liu, F.L. (1997), "Differential cubature method: A solution technique for Kirchhoff plates ofarbitrary shape", Comput. Meth. Appl. Mech. Eng., 13, 73-81.
  18. Liu, F.L. and Liew, K.M. (1998), "Differential cubature method for static solutions of arbitrary shaped thickplates", Int. J. Solids Struct., 35(28-29), 3655-3674. https://doi.org/10.1016/S0020-7683(97)00215-1
  19. Liu, F.L. (2001), "Differential quadrature element method for buckling analysis of rectangular Mindlin plateshaving discontinuities", Int. J. Solids Struct., 38, 2305-2321. https://doi.org/10.1016/S0020-7683(00)00120-7
  20. Luo, J.W. (1982), "A hybrid/mixed model finite element analysis for buckling of moderately thick plates",Comput. Struct., 15, 359-364. https://doi.org/10.1016/0045-7949(82)90070-0
  21. Malik, M. and Bert, C.W. (1998), "Three dimensional elasticity solutions for free vibrations of rectangular platesby the differential quadrature method", Int. J. Solids Struct., 35, 299-381. https://doi.org/10.1016/S0020-7683(97)00073-5
  22. Mizusawa, T., Kajita, T. and Naruko, M. (1980), "Analysis of skew plate problems with various constraints", J.Sound Vib., 73(4), 575-584. https://doi.org/10.1016/0022-460X(80)90669-0
  23. Navin, J., Knight, N.F. and Ambur, D.R. (1995), "Buckling of arbitrary quadrilateral anisotropic plates", AIAA J.,33, 938-944. https://doi.org/10.2514/3.12512
  24. Rao, V.G., Venkataramana, J. and Raju, K.K. (1975), "Stability of moderately thick rectangular plates using ahigh precision triangular finite element", Comput. Struct., 5, 257-259. https://doi.org/10.1016/0045-7949(75)90028-0
  25. Saadatpour, M.M., Azhari, M. and Bradford, M.A. (1998), "Buckling of arbitrary quadrilateral plates withintermediate supports using the Galerkin method", Comput. Methods Appl. Mech. Engrg., 164, 297-306. https://doi.org/10.1016/S0045-7825(98)00030-9
  26. Sakiyama, T. and Matsuda, H. (1987), "Elastic buckling of rectangular Mindlin plates with mixed boundaryconditions", Comput. Struct., 25, 801-808. https://doi.org/10.1016/0045-7949(87)90172-6
  27. Srinivas, S. and Rao, A.K. (1969), "Buckling of rectangular thick plates", AIAA J., 7, 1645-1746. https://doi.org/10.2514/3.5463
  28. Wang, C.M., Liew, K.M. and Alwis, W.A. (1992), "Buckling of skew plates and corner condition for simplysupported edges", J. Engrg. Mech., ASCE, 118, 651-662. https://doi.org/10.1061/(ASCE)0733-9399(1992)118:4(651)
  29. Wittrick, W.H. (1956), "On the buckling of oblique plates in shear", Aircraft Engrg., 28, 25-27. https://doi.org/10.1108/eb032652
  30. Xiang, Y. (1993), "Numerical developments in solving the buckling and vibration of Mindlin plates", Ph.Dthesis, The University of Queensland, Australia.

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