Elastoplastic buckling analysis of thick rectangular plates by using the differential quadrature method

The paper investigates the elastoplastic buckling behavior of thick rectangular plates by using the Differential Quadrature (DQ) method. Mindlin plate theory is adopted to take the transverse shear effect into considerations. Both incremental theory and deformation theory are employed. Due to the material non-linearity, iteration processes are involved for obtaining solutions. Detailed methodology and procedures are derived. The elastoplastic buckling behavior of thick rectangular plates with ten combinations of boundary conditions and under various loadings is studied. To verify the DQ solution procedures, DQ results are compared with existing analytical solutions for plates with two boundaries simply supported and the others simply supported, clamped, or free. Then the DQ method is used to obtain solutions of rectangular thick plates with other combinations of boundary conditions. Since no analytical solutions for such cases are available, the buckling loads obtained by the DQ method could serve as a reference. The phenomenon reported in the literature, namely, the deformation theory generally gives consistently lower buckling loads than the incremental theory and large discrepancy in predictions between the two theories exists with increasing of plate thickness. E/@s"0, and c in the Ramberg-Osgood relations, is also observed for the cases studied herein. Apart from the phenomenon reported earlier by Durban that deformation theory predicted a progressively lower in-plane shear modulus as the level of plasticity increased, thus predicted lower buckling loads, another reason is given herein to explain the large discrepancy in predictions for thicker plates with the deformation theory and incremental theory.

[1]  E. J. Brunelle,et al.  Buckling of transversely isotropic Mindlin plates , 1971 .

[2]  A. Zinober.  Matrices: Methods and Applications , 1992 .

[3]  John W. Hutchinson,et al.  Bifurcations at a spherical hole in an infinite elastoplastic medium , 1980 .

[4]  Chang-New Chen,et al.  Differential quadrature element analysis using extended differential quadrature , 2000 .

[5]  Huu-Tai Thai,et al.  BUCKLING ANALYSIS OF PLATES USING THE TWO VARIABLE REFINED PLATE THEORY , 2009 .

[6]  A. B. Rami Shani,et al.  Matrices: Methods and Applications , 1992 .

[7]  Xinwei Wang,et al.  Elastoplastic buckling analyses of rectangular plates under biaxial loadings by the differential qudrature method , 2009 .

[8]  Philippe Le Grognec,et al.  Some new analytical results for plastic buckling and initial post-buckling of plates and cylinders under uniform compression , 2009 .

[9]  Xinwei Wang,et al.  New approaches in application of differential quadrature method to fourth‐order differential equations , 2004 .

[10]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[11]  Yang Xiang,et al.  Exact buckling solutions for composite laminates: proper free edge conditions under in-plane loadings , 1996 .

[12]  David Durban,et al.  Elastoplastic buckling of axially compressed circular cylindrical shells , 1992 .

[13]  C. Shu,et al.  Free Vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method , 2000 .

[14]  Zvi Zuckerman,et al.  Elastoplastic buckling of rectangular plates in biaxial compression/tension , 1999 .

[15]  Kevin T. Chu,et al.  A direct matrix method for computing analytical Jacobians of discretized nonlinear integro-differential equations , 2007, J. Comput. Phys..

[16]  James Rhodes,et al.  Buckling of thin plates and members - and early work on rectangular tubes , 2002 .

[17]  Xinwei Wang,et al.  Buckling analysis of stiffened circular cylindrical panels using differential quadrature element method , 2008 .

[18]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[19]  W. Chen,et al.  The application of special matrix product to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates , 1999, ArXiv.

[20]  C. Wang,et al.  Elastic/plastic buckling of thick plates , 2001 .

[21]  D. Durban,et al.  Bifurcation of elastoplastic pressure-sensitive spheres , 2008, Comput. Math. Appl..

[22]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[23]  C. Bert,et al.  Two new approximate methods for analyzing free vibration of structural components , 1988 .

[24]  Chien Ming Wang,et al.  Allowance for prebuckling deformations in buckling load relationship between Mindlin and Kirchhoff simply supported plates of general polygonal shape , 1995 .

[25]  John P. Boyd,et al.  A numerical comparison of seven grids for polynomial interpolation on the interval , 1999 .

[26]  Xinwei Wang,et al.  Differential quadrature for buckling analysis of laminated plates , 1995 .

[27]  Xinwei Wang,et al.  Differential quadrature buckling analyses of rectangular plates subjected to non-uniform distributed in-plane loadings , 2006 .

[28]  W. Chen,et al.  A Lyapunov Formulation for Efficient Solution of the Poisson and Convection-Diffusion Equations by the Differential Quadrature Method , 2002, ArXiv.

[29]  Tun Myint Aung,et al.  Plastic buckling analysis of thick plates using p-Ritz method , 2007 .

[30]  Xinwei Wang,et al.  Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method , 2003 .

[31]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[32]  David Durban,et al.  Elastoplastic Buckling of Annular Plates in Pure Shear , 1989 .

[33]  J. Peterson,et al.  PLASTIC BUCKLING OF PLATES AND SHELLS UNDER BIAXIAL LOADING , 1968 .