An exact solution for free vibration of thin functionally graded rectangular plates

Abstract

The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

Free Vibration Analyses of FGM Thin Plates by Isogeometric Analysis Based on Classical Plate Theory and Physical Neutral Surface

“…The following features are evident: (1) results based on the CPT neu and FSDT match well with the analytical solutions [17,21]; (2) as the CPT, natural frequencies obtained by the CPT neu are larger than the analytical solutions; (3) the accuracy of the CPT neu is higher than that of the FSDT when the number of control points is small, whereas the accuracy of the CPT neu is almost the same as that of the FSDT with increasing number of control points. The computational time versus the number of control points is plotted in Figure 5.…”

“…The same Al/Al 2 O 3 material considered in the previous section is employed. [17,21]. Figure 4 shows the comparison of the first normalized natural frequencies from the proposed method and the reference solutions with different control points.…”

“…The effects of the gradient indices, length-to-width ratios ( / ), and boundary conditions of the FGM thin plates on the natural frequencies are investigated to demonstrate the validity of the present method. For this purpose, the normalized natural frequencies of two length-to-width ratio plates with four gradient indices and three boundary conditions are examined; the results are then compared with the analytical solutions [17] and the NURBS-based IGA based on the CPT and the FSDT. The three boundary conditions are as follows: (1) fully simply supported (SSSS); (2) left and right boundaries are simply supported, whereas bottom and upper boundaries are free (SFSF); (3) left and right boundaries are simply supported, whereas bottom and upper boundaries are clamped (SCSC).…”

Isogeometric locking-free plate element: A simple first order shear deformation theory for functionally graded plates

“…Table 9 IGA, CPT-neu-based IGA [42], and analytical solutions [43]. When the boundary conditions change from CCCC to SCSC, SSSS, and SFSF, the magnitude of the natural frequency gradually decreases.…”

A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates

“…Large applications of functionally graded (FG) structures have led the development of various plate theories to predict accurately their static, buckling and vibration behaviours. They are generally followed: classical plate theory (CPT) neglecting the transverse shear deformation effects ( [2]- [7]), first-shear deformation theory (FSDT) with linear variation of displacements ( [5], [8]- [14]), higher-order shear deformation theory (HSDT) with higher-order variations of displacements through the plate thickness such as thirdorder shear deformation plate theory (TSDT), sinusoidal shear deformation plate theory (SSDT), hyperbolic shear deformable plate theory (HDT) ( [15]- [25]), quasi-3D theories taking into account normal stretching effect ( [26]- [37]). Moreover, owing to smooth variations of material properties, FG sandwich plates have recently been developed to overcome interface problems between faces and core found in conventional sandwich structures.…”