Abstract
The bending problem of a thin rectangular plate with in-plane variable stiffness is studied. The basic equation is formulated for the two-opposite-edge simply supported rectangular plate under the distributed loads. The formulation is based on the assumption that the flexural rigidity of the plate varies in the plane following a power form, and Poisson’s ratio is constant. A fourth-order partial differential equation with variable coefficients is derived by assuming a Levy-type form for the transverse displacement. The governing equation can be transformed into a Whittaker equation, and an analytical solution is obtained for a thin rectangular plate subjected to the distributed loads. The validity of the present solution is shown by comparing the present results with those of the classical solution. The influence of in-plane variable stiffness on the deflection and bending moment is studied by numerical examples. The analytical solution presented here is useful in the design of rectangular plates with in-plane variable stiffness.
Similar content being viewed by others
References
Tatting, B. F. and Gürdal, Z. Analysis and Design of Variable Stiffness Composite Cylinders, Ph. D. dissertation, Virginia Polytechnic Institute and State University, 1–200 (1998)
Cheng, Z. Q. and Batra, R. C. Three-dimensional thermoelastic deformations of a functionally graded elliptic plate. Composites: Part B, 31, 97–106 (2000)
Chakrabortya, A., Gopalakrishnana, S., and Reddy, J. N. A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences, 45, 519–539 (2003)
Dai, K. Y., Liu, G. R., Lim, K. M., Han, X., and Du, S. Y. A meshfree radial point interpolation method for analysis of functionally graded material (FGM) plates. Computational Mechanics, 34, 213–223 (2004)
Ferreira, A. J. M., Batra, R. C., Roque, C. M. C., Qian, L. F., and Martins, P. A. L. S. Static analysis of functionally graded plates using third-order shear deformation theory and a meshless method. Composite Structures, 69, 449–457 (2005)
Yu, T. and Zhong, Z. Vibration of a simply supported functionally graded piezoelectric rectangular plate. Smart Materials and Structures, 15, 1404–1412 (2006)
Shang, E. T. and Zhong, Z. Closed-form solutions of three-dimensional functionally graded plates. Mechanics of Advanced Materials and Structures, 15, 355–363 (2008)
Zhong, Z. and Cheng, Z. Q. Fracture analysis of a functionally graded strip with arbitrary distributed material properties. International Journal of Solids and Structures, 45, 3711–3725 (2008)
Chen, S. P. and Zhong, Z. Three-dimensional elastic solution of a power form functionally graded rectangular plate. Journal of Mechanics and MEMS, 1(2), 349–358 (2009)
Birsan, M., Altenbach, H., Sadowski, T., Eremeyev, V. A., and Pietras, D. Deformation analysis of functionally graded beams by the direct approach. Composites: Part B, 43, 1315–1328 (2012)
Thai, H. T. and Choi, D. H. An efficient and simple refined theory for buckling analysis of functionally graded plates. Applied Mathematical Modelling, 36, 1008–1022 (2012)
Jodaei, A., Jalal, M., and Yas, M. H. Free vibration analysis of functionally graded annular plates by state-space based differential quadrature method and comparative modeling by ANN. Composites: Part B, 43, 340–353 (2012)
Wen, P. H. and Aliabadi, M. H. Analysis of functionally graded plates by meshless method: a purely analytical formulation. Engineering Analysis with Boundary Elements, 36, 639–650 (2012)
Liew, K. M., Zhao, X., and Lee, Y. Y. Postbuckling responses of functionally graded cylindrical shells under axial compression and thermal loads. Composites: Part B, 43, 1621–1630 (2012)
Malekzadeh, P., Fiouz, A. R., and Sobhrouyan, M. Three-dimensional free vibration of functionally graded truncated conical shells subjected to thermal environment. International Journal of Pressure Vessels and Piping, 89, 210–221 (2012)
Sadeghi, H., Baghani, M., and Naghdabadi, R. Strain gradient elasticity solution for functionally graded micro-cylinders. International Journal of Engineering Science, 50, 22–30 (2012)
Zenkour, A. M. Dynamical bending analysis of functionally graded infinite cylinder with rigid core. Applied Mathematics and Computation, 218, 8997–9006 (2012)
Shang, X. C. Exact solution on a problem of bending of double-direction rectangular elastic plates of variable rigidity (in Chinese). Journal of Lanzhou University (Natural Sciences), 27(2), 24–32 (1991)
Yang, J. The structural analysis of plates with unidirectionally varying rigidity on Galerkin line method (in Chinese). Journal of Wuhan Institute of Chemical Technology, 18(1), 57–60 (1996)
Liu, D. Y., Wang, C. Y., and Chen, W. Q. Free vibration of FGM plates with in-plane material inhomogeneity. Composite Structures, 92, 1047–1051 (2010)
Uymaz, B., Aydogdu, M., and Filiz, S. Vibration analyses of FGM plates with in-plane material inhomogeneity by Ritz method. Composite Structures, 94, 1398–1405 (2012)
Bodaghi, M. and Saidi, A. R. Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory. Applied Mathematical Modelling, 34, 3659–3673 (2010)
Thai, H. T. and Kim, S. E. Levy-type solution for buckling analysis of orthotropic plates based on two variable refined plate theory. Composite Structures, 93, 1738–1746 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Foundation of China (No. 11072177)
Rights and permissions
About this article
Cite this article
Yu, Tc., Nie, Gj., Zhong, Z. et al. Analytical solution of rectangular plate with in-plane variable stiffness. Appl. Math. Mech.-Engl. Ed. 34, 395–404 (2013). https://doi.org/10.1007/s10483-013-1679-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-013-1679-x