Abstract
The bending problem of a transverse load acting on an isotropic inhomogeneous rectangular plate using both two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions is considered. In the present 2-D solution, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The effects due to transverse shear and normal deformations are both included. The form of the assumed 2-D displacements is simplified by enforcing traction-free boundary conditions at the faces of the plate. No transverse shear correction factors are needed because a correct representation of the transverse shearing strain is given. The plate material is exponentially graded, meaning that Lamé’s coefficients vary exponentially in a given fixed direction (the thickness direction). A wide variety of results for the displacements and stresses of an exponentially graded rectangular plate are presented. The validity of the present 2-D trigonometric solution is demonstrated by comparison with the 3-D elasticity solution. The influence of aspect ratio, side-to-thickness ratio and the exponentially graded parameter on the bending response are investigated.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yamanouchi M., Koizumi M., Hirai T., Shiota I. (1990) Proceeding of the First International Symposium on Functionally Gradient Materials. Sendai, Japan
Koizumi M. (1993) The concept of FGM. Ceram. Trans. Funct. Gradient Mater. 34, 3–10
Praveen G.N., Reddy J.N. (1998) Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int. J. Solids Struct. 35, 4457–4476
Reddy J.N., Chin C.D. (1998) Thermomechanical analysis of functionally graded cylinders and plates. J. Ther. Stresses 21, 593–626
Loy C.T., Lam K.Y., Reddy J.N. (1999) Vibration of functionally graded cylindrical shells. Int. J. Mech. Sci. 41, 309–324
Reddy J.N. (2000) Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47, 663–684
Cheng Z.Q., Batra R.C. (2000) Deflection relationships between the homogeneous Kirchhoff plate theory and different functionally graded plate theories. Arch. Mech. 52, 143–158
Cheng Z.Q., Batra R.C. (2000) Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates. J. Sound Vib. 229, 879–895
Cheng Z.Q., Batra R.C. (2000) Three-dimensional thermoelastic deformations of a functionally graded elliptic plate. Composites B 31, 97–106
Vel S.S., Batra R.C. (2002) Exact solution for thermoelastic deformations of functionally graded thick rectangular plates. AIAA J. 40, 1421–1433
Whitney J.M., Pagano N.J. (1970) Shear deformation in heterogeneous anisotropic plates. ASME J. Appl. Mech. 37, 1031–1036
Bert C. W. (1973) Simplified analysis of elastic shear factors for beams of nonhomogeneous cross-section. J. Compos. Mater. 7, 525–529
Librescu L. (1975) Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Noordhoff, Leiden, The Netherlands
Reissner E. (1991) A mixed variational equation for a twelfth-order theory of bending of nonhomogeneous transversely isotropic plates. Comput. Mech. 7, 355–360
Reissner E. (1994) On the equations of an eighth-order theory of nonhomogeneous transversely isotropic plates. Int. J. Solids Struct. 31, 89–96
Fares M.E., Zenkour A.M. (1999) Buckling and free vibration of non-homogeneous composite cross-ply laminated plates with various plate theories. Compos. Struct. 44, 279–287
Zenkour A.M., Fares M.E. (1999) Non-homogeneous response of cross-ply laminated elastic plates using a higher-order theory. Compos. Struct. 44, 297–305
Zenkour A.M., Fares M.E. (2001) Bending, buckling and free vibration of non-homogeneous cross-ply laminated cylindrical shells using a refined first-order theory. Composites B 32, 237–247
Lo K.H., Christensen R.M., Wu E.M. (1977) A higher-order theory of plate deformation, Part 1: homogeneous plates. ASME J. Appl. Mech. 44, 669–676
Reddy J.N. (1984) A simple higher-order theory for laminated composite plates. ASME J. Appl. Mech. 51, 745–752
Reddy J.N. (1987) A generalization of two-dimensional theories of laminated composite plates. Commun. Appl. Numer. Meth. 3, 173–180
Noor A.K., Burton W.S. (1989) Assessment of shear deformation theories for multilayered composite plates. Appl. Mech. Rev. 42, 1–12
Reddy J.N. (1990) A general third-order nonlinear theory of plates with moderate thickness. Int. J. Nonlin, Mech. 25, 677–686
Reddy J.N. (1990) A review of refined theories of laminated composite plates. Shock Vib. Digest 22, 3–17
Zenkour A.M. (2004) Analytical solution for bending of cross-ply laminated plates under thermo-mechanical loading. Compos. Struct. 65, 367–379
Zenkour A.M. (2004) Buckling of fiber-reinforced viscoelastic composite plates using various plate theories. J. Eng. Math. 50, 75–93
Zenkour A.M. (2004) Thermal effects on the bending response of fiber-reinforced viscoelastic composite plates using a sinusoidal shear deformation theory. Acta. Mech. 171, 171–187
Zenkour A.M. (2005) A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses, Part 2—Buckling and free vibration. Int. J. Solids Struct. 42, 5224–5258
Zenkour A.M. (2005) On vibration of functionally graded plates according to a refined trigonometric plate theory. Int. J. Struct. Stab. Dynam. 5, 279–297
Zenkour A.M. (2006) Generalized shear deformation theory for bending analysis of functionally graded plates. Appl. Math. Model. 30, 67–84
Zenkour A.M.: Three-dimensional elasticity solutions for uniformly loaded cross-ply laminates and sandwich plates. J. Sand. Struct. Mater. (in press)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zenkour, A.M. Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate. Arch Appl Mech 77, 197–214 (2007). https://doi.org/10.1007/s00419-006-0084-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-006-0084-y