Free vibration analysis of composite laminated and sandwich plate with circular cutout

Abstract

This paper presents the free vibration analysis of a composite laminated and sandwich square plate with circular cutout. The problem formulation is based on the higher order shear deformation plate theory HDST C⁰ coupled with a curved quadrilateral p-element. The elements of the stiffness and mass matrices are calculated analytically. The curved edges are accurately represented using the blending function method. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as the cutout size and location, plate thickness, fiber orientation angle and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New results for the fundamentals frequencies of composite laminated and sandwich plates with circular cutout are presented.

Keywords

Free vibration laminated plate with circular cutout sandwich plate with circular cutout curved quadrilateral p-element

Introduction

The composite materials are an assembly of at least two components materials having, however, a high adhesion capacity and showing different mechanical and chemical properties. The laminated composite structures are generally manufactured by stacking orthotropic layers; each composite layer is characterized by a type of matrix material, the reinforced fibers, the thickness and the fiber orientation angle. The sandwich composite structures are composed of a thick core and two skins (or face sheets) which are relatively thin and made of composite materials. High strength-to-weight and stiffness-to-weight ratios can be achieved by using laminated and sandwich composite materials.

Cutouts are inevitable in structures and designed for many fields as aeronautical, mechanical and civil structures, etc. Cutouts with different shapes are widely used in plates as ventilation openings to facilitate heat dissipation, to reduce the weight and to provide access to different components.

Several researchers studied the free vibration analysis of isotropic and composite plates with and without cutouts, see for instance [1 –8]. However, the research of perforated laminates plates was very limited; these laminated structures lead to a more complicated analysis. Reddy [9] studied the large amplitude vibration of anisotropic rectangular laminated composite plates, which have a rectangular cutout, by using the finite element method and the first order plate theory (FSDT). He varied the side-to-thickness ratio and the cutout size of the plate to show the effect of these parameters on the natural frequencies.

The theoretical and the experimental studies of the free vibration of a symmetrical laminated composite plate with a central cutout are carried-out by Boay [10]. He determined the fundamental frequencies by varying the frequency parameters, the size of the cutout and the boundary conditions. Bicos and Springer [11] used the finite element method and the FSDT theory to study the damped free vibration of a rectangular laminated composite plate, with one or two circular cutouts. They performed a calculation code to determine the fundamental frequencies, modes shape and damping factors, and the results obtained are compared with the results of the experimental tests. Ramakrishna et al. [12] considered a composite laminated plate with a central circular cutout. They developed a computer program for predicting the natural frequencies of the plate by using a hybrid-stress finite element. They studied the effects of fiber orientation, width-to-thickness ratio and cutout size on the first four natural frequencies.

Kumar and Shrivastava [13] studied the free vibration of a composite rectangular plate with a rectangular cutout. They used the finite element method and the high order plate theory (HSDT), and the frequency parameters are determined by varying the size of the cutout, the plate thickness and the boundary conditions, and the results obtained are compared with the solution of the first-order plate theory (FSDT). Khaldoon [14] used the ANSYS software to determine the fundamental frequencies of the composite laminated thick plate, having a cutout; the results are determined by varying the size and shape of the cutout, the orientation angle of the fibers, the thickness of the plate and the boundary conditions. Sahoo [15] analyzed the vibratory behavior of composite laminated spherical panels by varying the size and location of the rectangular cutout. He used the finite element method and the FSDT theory. While Bhardwaj et al. [16] investigated the fundamental frequencies of a composite laminated plate having a triangular cutout, by using the ANSYS software.

The research clearly showed that there are very limited studies on the dynamic analysis of composite sandwich plates with cutout. The vibration response of sandwich composite plate with a circular cutout for both experimentally and numerically analysis is studied by the published article of Mondal et al. [17]. They examined the fundamental frequencies with various parameters such as the size and location of the cutout, the core and face sheets thicknesses and plate thickness. Mishra et al. [18] used the ANSYS software to study the free vibration of a composite sandwich plate with square cutout. They examined the fundamental frequencies by varying the size of cutout, the number of layers and the boundary conditions.

This paper deals with the free vibration analysis of composite laminated and sandwich plates with a circular cutout. The study is based on the higher order shear deformation theory (HSDT C⁰) and the p-version of the finite element method (FEM). By using the finite element method, the formulation of the interpolation functions in the C¹ space is much more constraining than those of the C⁰ space. In order to avoid the difficulties caused by C¹ continuity elements and the shear correction factors, we used the HSDT C⁰ model of Shankara and Iyengar [19,20] in this analysis. The p-version of FEM presented a great advantage in modeling of composite plate [21 –23] plates, since it founded on the enrichment of the element by increasing the degree p of the hierarchical shape function, to obtain the convergence results. A new C⁰ enriched curved quadrilateral p-element is developed and applied to free vibration analysis of composite laminated and sandwich square plates having a circular cutout. The plate is discretized into only four p-elements, in which the transformed reduced stiffness elements are computed exactly within stiffness matrix and the curved edges of element are modeled exactly by employing the blending function method. The vibratory response of composite plates with cutouts can be affected by several parameters. A calculation code is developed to find out the effects of the fiber orientation angles, the cutout size and location, the plate thickness, the core and face sheet thicknesses and the boundary condition on the fundamental frequencies. The obtained results of the present model are compared with available solutions in the literature.

Energy formulation of a composite laminated plate

The higher order shear deformation theory (HSDT C⁰) was developed by Shankara and Iyengar [19,20] for thin and moderately thick plates. This theory is characterized by the hyperbolic variation of the displacement field along the thickness of the plate, which allows taking into account a possible warping of the cross section of the plate during the deformation. In addition, this model does not require shear correction factors, because the distribution of deformations and transverse shear stresses is parabolic according to the plate thickness.

The laminate composite plate is considered in this study with a uniform thickness h, width α and length b. The field displacement of the higher-order shear deformation theory (HSDT C⁰) [19] can be expressed as functions of the mid-plane displacements $u_{0}$ , $v_{0}$ , $w_{0}$ , the independent rotations $β_{x}$ and $β_{y}$ and warping functions $ϕ_{x}$ and $ϕ_{y}$ along the x and y axis of the plate, respectively

\begin{array}{l} u (x, y, z) = u_{0} + z β_{x} - f (z) (β_{x} + ϕ_{x}) \\ v (x, y, z) = v_{0} + z β_{y} - f (z) (β_{y} + ϕ_{y}) \\ w (x, y, z) = w_{0} \end{array}

(1)

where

f (z) = \frac{4 z^{3}}{3 h^{2}}

(2)

The linear strain–displacement relationship is given by

\begin{array}{l} {ε} = {\begin{matrix} ε_{x x} \\ ε_{y y} \\ γ_{x y} \\ γ_{x z} \\ γ_{y z} \end{matrix}} = {\begin{matrix} \frac{\partial u_{0}}{\partial x} \\ \frac{\partial v_{0}}{\partial y} \\ \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} \\ \frac{\partial w_{0}}{\partial x} + β_{x} \\ \frac{\partial w_{0}}{\partial y} + β_{y} \end{matrix}} + z {\begin{matrix} \frac{\partial β_{x}}{\partial x} \\ \frac{\partial β_{y}}{\partial y} \\ \frac{\partial β_{x}}{\partial y} + \frac{\partial β_{y}}{\partial x} \\ 0 \\ 0 \end{matrix}} \\ - δ \frac{\partial f (z)}{\partial z} {\begin{matrix} 0 \\ 0 \\ 0 \\ β_{x} + ϕ_{x} \\ β_{y} + ϕ_{y} \end{matrix}} - δ f (z) {\begin{matrix} \frac{\partial β_{x}}{\partial x} + \frac{\partial ϕ_{x}}{\partial x} \\ \frac{\partial β_{y}}{\partial y} + \frac{\partial ϕ_{y}}{\partial y} \\ \frac{\partial β_{x}}{\partial y} + \frac{\partial ϕ_{x}}{\partial y} + \frac{\partial β_{y}}{\partial x} + \frac{\partial ϕ_{y}}{\partial x} \\ 0 \\ 0 \end{matrix}} \end{array}

(3)

The constitutive equations for a k^th layer, in the orthotropic local coordinate derived from Hook’s law for plane stress is given by

{\begin{matrix} σ_{1} \\ σ_{2} \\ σ_{12} \\ σ_{13} \\ σ_{12} \end{matrix}}^{k} = {[\begin{matrix} Q_{11} & Q_{12} & Q_{13} & 0 & 0 \\ Q_{21} & Q_{22} & Q_{23} & 0 & 0 \\ Q_{31} & Q_{32} & Q_{33} & 0 & 0 \\ 0 & 0 & 0 & Q_{44} & 0 \\ 0 & 0 & 0 & 0 & Q_{55} \end{matrix}]}^{k} {\begin{matrix} ε_{1} \\ ε_{2} \\ ε_{12} \\ ε_{13} \\ ε_{12} \end{matrix}}^{k}

(4)

The well-known engineering constants $Q_{i j}$ as functions of the longitudinal and transverse modulus of elasticity ( $E_{1}, E_{2}$ ), Poisson ratios ( $υ_{12}, υ_{21}$ ) and shear modulus ( $G_{1, 2},$ $G_{1, 3}$ , $G_{2, 3}$ ) are given by

Q_{11} = \frac{E_{1}}{1 - υ_{12} υ_{21}}

(5)

Q_{22} = \frac{E_{2}}{1 - υ_{12} υ_{21}}

(6)

Q_{12} = υ_{12} E_{12} / (1 - υ_{12} υ_{21})

(7)

and:

Q_{21} = Q_{12} Q_{33} = G_{1, 2} Q_{44} = G_{2, 3} Q_{55} = G_{1, 3}

where equations (5) to (7) represent the directions parallel and perpendicular to the fibers direction.

By performing a proper coordinate transformation, the stress–strain relationships of a single lamina in the o-xyz coordinate system can be obtained

{\begin{matrix} σ_{x} \\ σ_{y} \\ σ_{xy} \\ σ_{xz} \\ σ_{yz} \end{matrix}}^{k} = {[\begin{matrix} {\bar{Q}}_{11} & {\bar{Q}}_{12} & {\bar{Q}}_{13} & 0 & 0 \\ {\bar{Q}}_{21} & {\bar{Q}}_{22} & {\bar{Q}}_{23} & 0 & 0 \\ {\bar{Q}}_{31} & {\bar{Q}}_{32} & {\bar{Q}}_{33} & 0 & 0 \\ 0 & 0 & 0 & {\bar{Q}}_{44} & {\bar{Q}}_{45} \\ 0 & 0 & 0 & {\bar{Q}}_{54} & {\bar{Q}}_{55} \end{matrix}]}^{k} {\begin{matrix} ε_{x} \\ ε_{y} \\ ε_{xy} \\ ε_{xz} \\ ε_{yz} \end{matrix}}^{k}

(8)

where

[{\bar{Q}}^{k}]

is the transformed reduced stiffness matrix. The first elements of matrix

[{\bar{Q}}^{k}]

are written as

{\bar{Q}}_{11} = Q_{11} \cos^{4} θ + 2 (Q_{12} + 2 Q_{33}) \cos^{2} θ \sin^{2} θ + Q_{22} \sin^{4} θ

(9)

The other elements of matrices $[{\bar{Q}}^{k}]$ are defined in Appendix 1.

The strain energy and kinetic energy of composite laminate plate are given by

E_{d} = \frac{1}{2}_{\iint}^{x y} [σ_{x x} ε_{x x} + σ_{y y} ε_{y y} + τ_{x y} γ_{x y} + τ_{x z} γ_{x z} + τ_{y z} γ_{y z}] d x d y

(10)

E_{c} = \frac{1}{2} ρ_{\iint}^{x y} [{(\frac{\partial u}{\partial t})}^{2} + {(\frac{\partial v}{\partial t})}^{2} + {(\frac{\partial w}{\partial t})}^{2}] d x d y

(11)

By inserting the strain–displacement relationships (equation (2)), the strain and kinetic energy become in following form

E_{d} = E_{d}_{1} + E_{d}_{2} + E_{d}_{3} + E_{d}_{4} + E_{d}_{5} + E_{d}_{6} + E_{d}_{7} + E_{d}_{8}

(12)

where

\begin{array}{l} E_{d}_{1} = \frac{1}{2}_{\iint}^{x y} [A_{11} {(\frac{\partial u_{0}}{\partial x})}^{2} + A_{22} {(\frac{\partial v_{0}}{\partial y})}^{2} + 2 A_{12} (\frac{\partial u_{0}}{\partial x} \frac{\partial v_{0}}{\partial y}) \\ + A_{33} [{(\frac{\partial u_{0}}{\partial y})}^{2} + {(\frac{\partial v_{0}}{\partial x})}^{2} + 2 (\frac{\partial u_{0}}{\partial y} \frac{\partial v_{0}}{\partial x})] + 2 A_{13} [(\frac{\partial u_{0}}{\partial x} \frac{\partial u_{0}}{\partial y}) + (\frac{\partial u_{0}}{\partial x} \frac{\partial v_{0}}{\partial x})] \\ + 2 A_{23} [(\frac{\partial v_{0}}{\partial x} \frac{\partial v_{0}}{\partial y}) + (\frac{\partial u_{0}}{\partial y} \frac{\partial v_{0}}{\partial y})] + A_{44} {(\frac{\partial w_{0}}{\partial x})}^{2} + A_{55} {(\frac{\partial w_{0}}{\partial y})}^{2} + 2 A_{45} (\frac{\partial w_{0}}{\partial x} \frac{\partial w_{0}}{\partial y})] d x d y \\ E_{d}_{2} = \frac{1}{2}_{\iint}^{x y} [2 B_{11} (\frac{\partial u_{0}}{\partial x} \frac{\partial β_{x}}{\partial x}) + 2 B_{22} (\frac{\partial v_{0}}{\partial y} \frac{\partial β_{y}}{\partial y}) + 2 B_{12} [(\frac{\partial u_{0}}{\partial x} \frac{\partial β_{y}}{\partial y}) + (\frac{\partial v_{0}}{\partial y} \frac{\partial β_{x}}{\partial x})] \\ + 2 B_{33} [(\frac{\partial u_{0}}{\partial y} \frac{\partial β_{x}}{\partial y}) + (\frac{\partial v_{0}}{\partial x} \frac{\partial β_{y}}{\partial x}) + (\frac{\partial u_{0}}{\partial y} \frac{\partial β_{y}}{\partial x}) + (\frac{\partial v_{0}}{\partial x} \frac{\partial β_{x}}{\partial y})] \\ + 2 B_{13} [(\frac{\partial u_{0}}{\partial y} \frac{\partial β_{x}}{\partial x}) + (\frac{\partial u_{0}}{\partial x} \frac{\partial β_{x}}{\partial y}) + (\frac{\partial u_{0}}{\partial x} \frac{\partial β_{y}}{\partial x}) + (\frac{\partial v_{0}}{\partial x} \frac{\partial β_{x}}{\partial x})] \\ + 2 B_{23} [(\frac{\partial v_{0}}{\partial x} \frac{\partial β_{y}}{\partial y}) + (\frac{\partial v_{0}}{\partial y} \frac{\partial β_{y}}{\partial x}) + (\frac{\partial u_{0}}{\partial y} \frac{\partial β_{y}}{\partial y}) + (\frac{\partial v_{0}}{\partial y} \frac{\partial β_{x}}{\partial y})]] d x d y \\ E_{d}_{3} = \frac{1}{2}_{\iint}^{x y} [2 D_{11} (\frac{\partial u_{0}}{\partial x} \frac{\partial ϕ_{x}}{\partial x}) + 2 D_{22} (\frac{\partial v_{0}}{\partial y} \frac{\partial ϕ_{y}}{\partial y}) + 2 D_{12} [(\frac{\partial u_{0}}{\partial x} \frac{\partial ϕ_{y}}{\partial y}) + (\frac{\partial v_{0}}{\partial y} \frac{\partial ϕ_{x}}{\partial x})] \\ + [2 D_{33} [(\frac{\partial u_{0}}{\partial y} \frac{\partial ϕ_{x}}{\partial y}) + (\frac{\partial v_{0}}{\partial x} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial u_{0}}{\partial y} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial v_{0}}{\partial x} \frac{\partial ϕ_{x}}{\partial y})]] \\ + 2 D_{13} [(\frac{\partial u_{0}}{\partial y} \frac{\partial ϕ_{x}}{\partial x}) + (\frac{\partial u_{0}}{\partial x} \frac{\partial ϕ_{x}}{\partial y}) + (\frac{\partial u_{0}}{\partial x} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial v_{0}}{\partial x} \frac{\partial ϕ_{x}}{\partial x})] \end{array} \begin{array}{l} + 2 D_{23} [(\frac{\partial v_{0}}{\partial x} \frac{\partial ϕ_{y}}{\partial y}) + (\frac{\partial v_{0}}{\partial y} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial u_{0}}{\partial y} \frac{\partial ϕ_{y}}{\partial y}) + (\frac{\partial v_{0}}{\partial y} \frac{\partial ϕ_{x}}{\partial y})]] d x d y \\ E_{d}_{4} = \frac{1}{2}_{\iint}^{x y} [E_{11} {(\frac{\partial β_{x}}{\partial x})}^{2} + E_{22} {(\frac{\partial β_{y}}{\partial y})}^{2} + 2 E_{12} (\frac{\partial β_{x}}{\partial x} \frac{\partial β_{y}}{\partial y}) \\ + E_{33} [{(\frac{\partial β_{x}}{\partial y})}^{2} + {(\frac{\partial β_{y}}{\partial x})}^{2} + 2 (\frac{\partial β_{x}}{\partial y} \frac{\partial β_{y}}{\partial x})] + 2 E_{13} [(\frac{\partial β_{x}}{\partial x} \frac{\partial β_{x}}{\partial y}) + (\frac{\partial β_{x}}{\partial x} \frac{\partial β_{y}}{\partial x})] \\ + 2 E_{23} [(\frac{\partial β_{y}}{\partial x} \frac{\partial β_{y}}{\partial y}) + (\frac{\partial β_{x}}{\partial y} \frac{\partial β_{y}}{\partial y})] + F_{44} {(β_{x})}^{2} + F_{55} {(β_{y})}^{2} + 2 F_{45} (β_{x} β_{y})] d x d y \\ E_{d}_{5} = \frac{1}{2}_{\iint}^{x y} [G_{11} {(\frac{\partial ϕ_{x}}{\partial x})}^{2} + G_{22} {(\frac{\partial ϕ_{y}}{\partial y})}^{2} + 2 G_{12} (\frac{\partial ϕ_{x}}{\partial x} \frac{\partial ϕ_{y}}{\partial y}) \\ + G_{33} [{(\frac{\partial ϕ_{x}}{\partial y})}^{2} + {(\frac{\partial ϕ_{y}}{\partial x})}^{2} + 2 (\frac{\partial ϕ_{x}}{\partial y} \frac{\partial ϕ_{y}}{\partial x})] + 2 G_{13} [(\frac{\partial ϕ_{x}}{\partial x} \frac{\partial ϕ_{x}}{\partial y}) + (\frac{\partial ϕ_{x}}{\partial x} \frac{\partial ϕ_{y}}{\partial x})] \\ + 2 G_{23} [(\frac{\partial ϕ_{y}}{\partial x} \frac{\partial ϕ_{y}}{\partial y}) + (\frac{\partial ϕ_{x}}{\partial y} \frac{\partial ϕ_{y}}{\partial y})] + H_{44} {(ϕ_{x})}^{2} + H_{55} {(ϕ_{y})}^{2} + 2 H_{45} (ϕ_{x} ϕ_{y})] d x d y \\ E_{d}_{6} = \frac{1}{2}_{\iint}^{x y} [2 I_{11} (\frac{\partial β_{x}}{\partial x} \frac{\partial ϕ_{x}}{\partial x}) + 2 I_{22} (\frac{\partial β_{y}}{\partial y} \frac{\partial ϕ_{y}}{\partial y}) + 2 I_{12} [(\frac{\partial β_{x}}{\partial x} \frac{\partial ϕ_{y}}{\partial y}) + (\frac{\partial β_{y}}{\partial y} \frac{\partial ϕ_{x}}{\partial x})] \\ + 2 I_{33} [(\frac{\partial β_{x}}{\partial y} \frac{\partial ϕ_{x}}{\partial y}) + (\frac{\partial β_{y}}{\partial x} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial β_{x}}{\partial y} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial β_{y}}{\partial x} \frac{\partial ϕ_{x}}{\partial y})] \\ + 2 I_{13} [(\frac{\partial β_{x}}{\partial y} \frac{\partial ϕ_{x}}{\partial x}) + (\frac{\partial β_{x}}{\partial x} \frac{\partial ϕ_{x}}{\partial y}) + (\frac{\partial β_{x}}{\partial x} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial β_{y}}{\partial x} \frac{\partial ϕ_{x}}{\partial x})] \\ + 2 I_{23} [(\frac{\partial β_{y}}{\partial x} \frac{\partial ϕ_{y}}{\partial y}) + (\frac{\partial β_{y}}{\partial y} \frac{\partial ϕ_{y}}{\partial x}) + (\frac{\partial β_{x}}{\partial y} \frac{\partial ϕ_{y}}{\partial y}) + (\frac{\partial β_{y}}{\partial y} \frac{\partial ϕ_{x}}{\partial y})] \\ + 2 S_{44} (β_{x} ϕ_{x}) + 2 S_{55} (β_{y} ϕ_{y}) + 2 S_{45} [(β_{x} ϕ_{y}) + (β_{y} ϕ_{x})]] d x d y \\ E_{d}_{7} = \frac{1}{2}_{\iint}^{x y} [2 S 1_{44} (\frac{\partial w_{0}}{\partial x} β_{x}) + 2 S 1_{55} (\frac{\partial w_{0}}{\partial y} β_{y}) + 2 S 1_{45} [(\frac{\partial w_{0}}{\partial y} β_{x}) + (\frac{\partial w_{0}}{\partial x} β_{y})]] d x d y \\ E_{d}_{8} = \frac{1}{2}_{\iint}^{x y} [2 S 2_{44} (\frac{\partial w_{0}}{\partial x} ϕ_{x}) + 2 S 2_{55} (\frac{\partial w_{0}}{\partial y} ϕ_{y}) + 2 S 2_{45} [(\frac{\partial w_{0}}{\partial y} ϕ_{x}) + (\frac{\partial w_{0}}{\partial x} ϕ_{y})]] d x d y \end{array}

(13)

The functions $A_{i j}$ , $B_{i j}$ , $D_{i j}, E_{i j}$ , $F_{i j}$ , $G_{i j}$ , $H_{i j}$ , $I_{i j}$ , $S_{i j}$ , $S 1_{i j}$ et $S 2_{i j}$ are indicated as

A_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] d z

(14)

B_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (z - f (z)) d z

(15)

D_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (- f (z)) d z

(16)

E_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (z^{2} + {(f (z))}^{2} - 2 z (f (z))) d z

(17)

F_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (1 + {(\frac{\partial f (z)}{\partial z})}^{2} - 2 (\frac{\partial f (z)}{\partial z})) d z

(18)

G_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] {(f (z))}^{2} d z

(19)

H_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] {(\frac{\partial f (z)}{\partial z})}^{2} d z

(20)

I_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (({(f (z))}^{2} - z (f (z)))) d z

(21)

S_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (({(\frac{\partial f (z)}{\partial z})}^{2} - (\frac{\partial f (z)}{\partial z}))) d z

(22)

S 1_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (1 - (\frac{\partial f (z)}{\partial z})) d z

(23)

S 2_{i j} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} [{\bar{Q}}^{k}_{i j}] (- (\frac{\partial f (z)}{\partial z})) d z

(24)

and

E_{c} = E_{c}_{1} + E_{c}_{2} + E_{c}_{3} + E_{c}_{4} + E_{c}_{5} + E_{c}_{6}

(25)

where

\begin{array}{l} E_{c}_{1} = \frac{1}{2}_{\iint}^{x y} ρ_{A} [{(\frac{\partial u_{0}}{\partial t})}^{2} + {(\frac{\partial v_{0}}{\partial t})}^{2} + {(\frac{\partial w_{0}}{\partial t})}^{2}] d x d y \end{array} \begin{array}{l} E_{c}_{2} = \frac{1}{2}_{\iint}^{x y} 2 ρ_{B} [(\frac{\partial u_{0}}{\partial t} \frac{\partial β_{x}}{\partial t}) + (\frac{\partial v_{0}}{\partial t} \frac{\partial β_{y}}{\partial t})] d x d y \\ E_{c}_{3} = \frac{1}{2}_{\iint}^{x y} 2 ρ_{C} [(\frac{\partial u_{0}}{\partial t} \frac{\partial ϕ_{x}}{\partial t}) + (\frac{\partial v_{0}}{\partial t} \frac{\partial ϕ_{y}}{\partial t})] d x d y \\ E_{c}_{4} = \frac{1}{2}_{\iint}^{x y} ρ_{D} [{(\frac{\partial β_{x}}{\partial t})}^{2} + {(\frac{\partial β_{y}}{\partial t})}^{2}] d x d y \\ E_{c}_{5} = \frac{1}{2}_{\iint}^{x y} ρ_{E} [{(\frac{\partial ϕ_{x}}{\partial t})}^{2} + {(\frac{\partial ϕ_{y}}{\partial t})}^{2}] d x d y \\ E_{c}_{6} = \frac{1}{2}_{\iint}^{x y} 2 ρ_{F} [(\frac{\partial β_{x}}{\partial t} \frac{\partial ϕ_{x}}{\partial t}) + (\frac{\partial β_{y}}{\partial t} \frac{\partial ϕ_{y}}{\partial t})] d x d y \end{array}

(26)

and

ρ_{A}

ρ_{B}

ρ_{C}

ρ_{D}

ρ_{E}

ρ_{F}

are expressed as

ρ_{A} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} ρ^{k} d z

(27)

ρ_{B} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} ρ^{k} (z - f (z)) d z

(28)

ρ_{C} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} ρ^{k} (- f (z)) d z

(29)

ρ_{D} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} ρ^{k} ((z^{2} + {(f (z))}^{2} - 2 z (f (z)))) d z

(30)

ρ_{E} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} ρ^{k} ({(f (z))}^{2}) dz

(31)

ρ_{F} = \sum_{k = 1}^{N L} \int_{h_{k}}^{h_{k + 1}} ρ^{k} (({(f (z))}^{2} - z (f (z)))) d z

(32)

Here $N L$ is the number of layers.

Hierarchical finite element formulation

The plate with circular cutout is discretized by using the p-version of the finite element method. A new C⁰ enriched curved quadrilateral p-element is developed based on the higher order shear deformation theory, in which the plate is divided in only four curved quadrilateral p-elements (Figure 1). The blending function method is a well-suited approach to represent accurately the circular cutout. The insertion of the blending function is necessary to map the elements of the plate from the physical domain to the computational space, as shown in Figure 2. The width, length and radius of circular cutout are $a, b$ and r, respectively.

Figure 2 shows a general example of the coordinate transformation, for mapping the geometry to the computational space, and the blending function method is used to define the point coordinate and the boundary parametric curves of the original physical space.

The mapping by the blending function method [22] is given as

\begin{array}{l} X (ξ, η) = (\frac{1 - η}{2}) X_{1} (ξ, - 1) + (\frac{1 + ξ}{2}) X_{2} (1, η) + (\frac{1 + η}{2}) X_{3} (ξ, 1) \end{array} \begin{array}{l} + (\frac{1 - ξ}{2}) X_{4} (- 1, η) - [(\frac{(1 - ξ) (1 - η)}{4}) x_{1} + (\frac{(1 - ξ) (1 + η)}{4}) x_{2} \\ + (\frac{(1 + ξ) (1 + η)}{4}) x_{3} + (\frac{(1 + ξ) (1 - η)}{4}) x_{4}] \end{array}

(33)

\begin{array}{l} Y (ξ, η) = (\frac{1 - η}{2}) Y_{1} (ξ, - 1) + (\frac{1 + ξ}{2}) Y_{2} (1, η) + (\frac{1 + η}{2}) Y_{3} (ξ, 1) \\ + (\frac{1 - ξ}{2}) Y_{4} (- 1, η) - [(\frac{(1 - ξ) (1 - η)}{4}) y_{1} + (\frac{(1 - ξ) (1 + η)}{4}) y_{2} \\ + (\frac{(1 + ξ) (1 + η)}{4}) y_{3} + (\frac{(1 + ξ) (1 - η)}{4}) y_{4}] \end{array}

(34)

where the functions

X_{i} (ξ, η)

and

Y_{i} (ξ, η)

represent the four sides of the boundary parametric curves of the original physical domain and the function

x_{i} (ξ, η)

and

y_{i} (ξ, η)

signifies the x and y coordinates of the point corresponding to the coordinates

ξ

and

η

in the computational space.

The curved quadrilateral p-element (Figure 3) is defined by its four nodes in the vertices $(x_{i}, y_{i})$ and the four sides are $(X_{i}, Y_{i})$ , where $i = 1, 2, 3 and 4$ . By using the blending function method, the corresponding equations that represented the geometry of Figure 4 are expressed as

X (ξ, η) = \frac{x_{1}}{4} + x_{2} (\frac{ξ η}{4} - \frac{η}{4}) + x_{3} (\frac{1}{4} + \frac{η}{4} + \frac{ξ}{4} + \frac{ξ η}{4}) + \frac{r \cos (\frac{π η}{4})}{2} - \frac{ξ r \cos (\frac{π η}{4})}{2}

Y (ξ, η) = \frac{y_{1}}{4} + y_{2} (\frac{ξ η}{4} - \frac{η}{4}) + y_{3} (\frac{1}{4} + \frac{η}{4} + \frac{ξ}{4} + \frac{ξ η}{4}) + \frac{r \sin (\frac{π η}{4})}{2} - \frac{ξ r \sin (\frac{π η}{4})}{2}

(36)

The rule chain of transformation matrix from the Cartesian system to the local system can be performed by

{\begin{matrix} \frac{\partial ()}{\partial ξ} \\ \frac{\partial ()}{\partial η} \end{matrix}} = J {\begin{matrix} \frac{\partial ()}{\partial x} \\ \frac{\partial ()}{\partial y} \end{matrix}}

(37)

where J is called the Jacobian matrix of the transformation, and is expressed as

J = [\begin{matrix} J_{1, 1} & J_{1, 2} \\ J_{2, 1} & J_{2, 2} \end{matrix}] = [\begin{matrix} \frac{\partial X (ξ, η)}{\partial ξ} & \frac{\partial Y (ξ, η)}{\partial ξ} \\ \frac{\partial X (ξ, η)}{\partial η} & \frac{\partial Y (ξ, η)}{\partial η} \end{matrix}]

(38)

The inverse of Jacobian matrix becomes

J^{- 1} = \frac{1}{| J |} [\begin{matrix} \frac{\partial Y (ξ, η)}{\partial η} & - \frac{\partial Y (ξ, η)}{\partial ξ} \\ - \frac{\partial X (ξ, η)}{\partial η} & \frac{\partial X (ξ, η)}{\partial ξ} \end{matrix}]

(39)

The determinant of $J$ is evaluated as

| J | = J_{1, 1} (ξ, η) J_{2, 2} (ξ, η) - J_{1, 2} (ξ, η) J_{2, 1} (ξ, η)

(40)

A four-node quadrilateral p-element with seven degrees of freedom per node ( $u_{0}, v_{0}, w_{0}, β_{x}, β_{y}, ϕ_{x}, ϕ_{y}$ ) is developed on the basis of a higher-order plate theory. The model requires $C^{0}$ continuity for all variables. The displacements and rotations of the quadrilateral curved p-element are expressed as

u_{0} (ξ, η) = \sum_{i = 1}^{P} \sum_{j = 1}^{P} U_{i j} (t) g_{i} (ξ) g_{j} (η)

(41)

v_{0} (ξ, η) = \sum_{i = 1}^{P} \sum_{j = 1}^{P} V_{i j} (t) g_{i} (ξ) g_{j} (η)

(42)

w_{0} (ξ, η) = \sum_{i = 1}^{P} \sum_{j = 1}^{P} W_{i j} (t) g_{i} (ξ) g_{j} (η)

(43)

β_{x} (ξ, η) = \sum_{i = 1}^{P} \sum_{j = 1}^{P} β_{x}_{i j} (t) g_{i} (ξ) g_{j} (η)

(44)

β_{y} (ξ, η) = \sum_{i = 1}^{P} \sum_{j = 1}^{P} β_{y}_{i j} (t) g_{i} (ξ) g_{j} (η)

(45)

ϕ_{x} (ξ, η) = \sum_{i = 1}^{P} \sum_{j = 1}^{P} ϕ_{x}_{i j} (t) g_{i} (ξ) g_{j} (η)

(46)

ϕ_{y} (ξ, η) = \sum_{i = 1}^{P} \sum_{j = 1}^{P} ϕ_{y}_{i j} (t) g_{i} (ξ) g_{j} (η)

(47)

where

g (ξ) and g (η)

are the hierarchical shape function, and are the commonly used in hierarchical finite element method. The polynomial hierarchical shape functions

g_{i} (ξ) (i = 1, 2, \dots, p)

for C⁰ continuity are given by

{\begin{matrix} g_{1} (ξ) = \frac{1}{2} (1 - ξ) \\ g_{2} (ξ) = \frac{1}{2} (1 + ξ) \\ g_{i} (ξ) = \sqrt{\frac{2 i - 3}{2}} \int_{- 1}^{ξ} P_{i - 1} (t) d t; i = 3, 4, \dots p \end{matrix}

(48)

where

{\begin{matrix} P_{0} (x) = 1 \\ P_{n} (x) = \frac{1}{2^{n} n!} \frac{d^{n}}{d x^{n}} ({(x^{2} - 1)}^{n}) \end{matrix}

(49)

Equations (39) to (43) can be written in the matrix form as

{\begin{matrix} u_{0} \\ v_{0} \\ w_{0} \\ β_{x} \\ \begin{matrix} β_{y} \\ \begin{matrix} ϕ_{x} \\ ϕ_{y} \end{matrix} \end{matrix} \end{matrix}} = [N] {q}

(50)

where [N] is the matrix of shape functions and is given by

[N] = [\begin{matrix} [N_{u}] & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & [N_{v}] & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & [N_{w}] & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & [N_{β_{x}}] & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & [N_{β_{y}}] & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & [N_{ϕ_{x}}] & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & [N_{ϕ_{y}}] \end{matrix}]

(51)

and

{q} = {\begin{matrix} q_{u} \\ q_{v} \\ q_{w} \\ q_{β_{x}} \\ \begin{matrix} q_{β_{y}} \\ \begin{matrix} q_{ϕ_{x}} \\ q_{ϕ_{y}} \end{matrix} \end{matrix} \end{matrix}}

(52)

in which

q_{u}

q_{v}

q_{w}

q_{β_{x}}

, and

q_{β_{y}}

are the generalized displacements.

The Lagrange equations of free motion are given by

\frac{d}{d t} (\frac{\partial E_{c}}{\partial \dot{q}}) - \frac{\partial E_{c}}{\partial q} + \frac{\partial E_{d}}{\partial q} = 0

(53)

By inserting the strain energy and kinetic energy in the Lagrange equation, the equations of motion of free vibration can be expressed as

[M] {\ddot{q}} + [K] {q} = 0

(54)

where [K] is called the stiffness matrix and [M] is the mass matrix of the p-element, and they are shown as

[K] = [\begin{matrix} [K_{u u}] & [K_{u v}] & 0 & [K_{u β_{x}}] & [K_{u β_{y}}] & [K_{u ϕ_{x}}] & [K_{u ϕ_{y}}] \\ {[K_{u v}]}^{T} & [K_{v v}] & 0 & [K_{v β_{x}}] & [K_{v β_{y}}] & [K_{v ϕ_{x}}] & [K_{v ϕ_{y}}] \\ 0 & 0 & [K_{w w}] & [K_{w β_{x}}] & [K_{w β_{y}}] & [K_{w ϕ_{x}}] & [K_{w ϕ_{y}}] \\ {[K_{u β_{x}}]}^{T} & {[K_{v β_{x}}]}^{T} & {[K_{w β_{x}}]}^{T} & [K_{β_{x} β_{x}}] & [K_{β_{x} β_{y}}] & [K_{β_{x} ϕ_{x}}] & [K_{β_{x} ϕ_{y}}] \\ {[K_{u β_{y}}]}^{T} & {[K_{v β_{y}}]}^{T} & {[K_{w β_{y}}]}^{T} & {[K_{β_{x} β_{y}}]}^{T} & [K_{β_{y} β_{y}}] & [K_{β_{y} ϕ_{x}}] & [K_{β_{y} ϕ_{y}}] \\ {[K_{u ϕ_{x}}]}^{T} & {[K_{v ϕ_{x}}]}^{T} & {[K_{w ϕ_{y}}]}^{T} & {[K_{β_{x} ϕ_{x}}]}^{T} & {[K_{ϕ_{x} ϕ_{x}}]}^{T} & [K_{ϕ_{x} ϕ_{x}}] & [K_{ϕ_{x} ϕ_{y}}] \\ {[K_{u ϕ_{y}}]}^{T} & {[K_{v ϕ_{y}}]}^{T} & {[K_{w ϕ_{y}}]}^{T} & {[K_{β_{x} ϕ_{y}}]}^{T} & {[K_{β_{y} ϕ_{y}}]}^{T} & [K_{ϕ_{x} ϕ_{y}}] & [K_{ϕ_{y} ϕ_{y}}] \end{matrix}]

(55)

[M] = [\begin{matrix} [M_{u u}] & 0 & 0 & [M_{u β_{x}}] & 0 & [M_{u ϕ_{x}}] & 0 \\ 0 & [M_{v v}] & 0 & 0 & [M_{v β_{y}}] & 0 & [M_{v ϕ_{y}}] \\ 0 & 0 & [M_{w w}] & 0 & 0 & 0 & 0 \\ {[M_{u β_{x}}]}^{T} & 0 & 0 & [M_{β_{x} β_{x}}] & 0 & [M_{β_{x} ϕ_{x}}] & 0 \\ 0 & {[M_{v β_{y}}]}^{T} & 0 & 0 & [M_{β_{y} β_{y}}] & 0 & [M_{β_{y} ϕ_{y}}] \\ {[M_{u ϕ_{x}}]}^{T} & 0 & 0 & {[M_{β_{x} θ_{x}}]}^{T} & 0 & [M_{ϕ_{x} ϕ_{x}}] & 0 \\ 0 & {[M_{v ϕ_{y}}]}^{T} & 0 & 0 & {[M_{β_{y} ϕ_{y}}]}^{T} & 0 & [M_{ϕ_{y} ϕ_{y}}] \end{matrix}]

(56)

The matrices formulation of [K] and [M] are detailed in Appendix 1.

Results and discussions

In this section, solution accuracy and convergence studies of the free vibration of the composite laminated square plate ( $a = b$ ) without and with square cutout are analyzed. The plate with square cutout is discretized into four trapezoidal p-elements. In order to calculate the frequency parameter $Ω$ where $Ω = ω (a^{2} / h) (\sqrt{ρ / E_{2}})$ , the degree of the polynomial shape function p is increased. The comparison study is performed by varying the ratio of cutout to width $c / a$ , fiber orientations angle and boundary conditions. Two examples are treated to validate the model: the first example concerns a symmetric laminated plate with 40 layers [(±45°/0°₂)₃(90°/0°₂/90°)₂]_s and the second example concerns an anti-symmetric laminated plate with two layers [0°/90°]. The mechanical properties of materials are shown in Table 1, where materials I and II are used, respectively, for example 1 and example 2.

Table 1.

Mechanical properties of materials.

	$E_{1} (P a)$	$E_{2} (P a)$	$G_{12} = G_{13}$ $(P a)$	$G_{23} (P a)$	$ν_{12}$	$ρ (k g / m^{3})$
material I	$13 \times 10^{10}$	$10^{10}$	$0.5 \times 10^{10}$	$0.33 \times 10^{10}$	$0.35$	1500
material II	$25$	$1$	$0.5$	$0.2$	$0.25$	$1$
material III	$40$	$1$	$0.6$	$0.5$	$0.25$	$1$
material IV	24.51 × 10⁹	7.77 × 10⁹	$3.34 \times 10^{9}$	$3.34 \times 10^{9}$	0.0078	1800
material V	0.104 × 10⁹	0.104 × 10⁹	0.05 × 10⁹	0.05 × 10⁹	0.32	130
material VI	131 × 10⁹	10.34 × 10⁹	6.895 × 10⁹	6.205 × 10⁹	0.22	1627
material VII	6.89 × 10⁹	6.89 × 10⁹	3.45 × 10⁶	3.45 × 10⁶	0	97

The frequency parameters are calculated by varying the ratio c/a from 0 to 0.8 for example 1, and two ratios of length to thickness are adopted ( $a / h = 75$ and $a / h = 15$ ). While in example 2, the ratio $c / a$ is varied from 0.2 to 0.6 with a thickness of $a / h = 100$ . Two boundary conditions as CCCC, SSSS are imposed at the four external edges, where C and S signified the clamped and simply supported, respectively. The boundary condition in the p-version of FEM is applied by ignoring the lines and columns related with restrained degrees of freedom.

The results of example 1 are given in Tables 2 to 10 and are compared with FSDT and HSDT C¹ solutions of Kumar and Shrivastava [13]. The results of example 2 are indicated in Table 11 and are validated with solutions of the first-order shear deformation theory (FSDT) [24] and ANSYS software [14]. It can be noticed that the present results are in excellent agreement with available solutions in the literature.

Table 2.

Convergence and comparison of the first five frequency parameters Ω of a square laminated plate without cutout, SSSS, c/a = 0, a/h = 75, material I.

		Mode
Method	p	1	2	3	4	5
	5	13.707	29.002	38.496	54.215	55.169
	6	13.707	29.001	38.493	53.786	54.556
Present	7	13.707	28.997	38.487	53.715	54.528
	8	13.707	28.997	38.487	53.710	54.526
	9	13.707	28.997	38.487	53.710	54.526
FSDT [13]		13.59	29.11	37.79	53.93	55.07
HSDT C¹ [13]		13.71	29.50	38.30	54.85	55.97

Table 3.

Convergence and comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, SSSS, c/a = 0.2, a/h = 75, material I.

		Mode
Method	p	1	2	3	4	5
	5	13.318	28.903	37.102	54.031	55.451
	6	13.246	28.331	36.453	52.653	55.252
Present	7	13.223	28.292	36.314	52.545	54.254
	8	13.212	28.252	36.234	52.494	54.226
	9	13.203	28.231	36.174	52.477	54.195
FSDT [13]		13.15	28.39	35.39	52.40	55.30
HSDT C¹ [13]		13.40	29.06	36.90	53.42	56.27

Table 4.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, SSSS, c/a = 0.4, a/h = 75, material I.

	Mode
Method	1	2	3	4	5
Present	14.249	25.590	28.788	48.726	51.007
FSDT [13]	14.24	25.65	28.64	48.71	51.44
HSDT C¹ [13]	14.86	26.68	29.91	49.76	54.10

Table 5.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, SSSS, c/a = 0.6, a/h = 75, material I.

	Mode
Method	1	2	3	4	5
Present	19.377	27.783	28.962	43.965	51.215
FSDT [13]	19.52	28.20	29.37	45.22	51.65
HSDT C¹ [13]	21.06	30.06	31.41	49.42	53.47

Table 6.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, SSSS, c/a = 0.8, a/h = 75, material I.

	Mode
Method	1	2	3	4	5
Present	38.465	45.485	46.155	55.984	82.358
FSDT [13]	38.75	45.77	46.51	56.45	86.86
HSDT C¹ [13]	40.73	48.53	48.99	60.27	88.73

Table 7.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, CCCC, c/a = 0.2, a/h = 15, material I.

	Mode
Method	1	2	3	4	5
Present	20.972	32.182	36.248	49.710	53.562
HSDT C¹ [13]	21.50	32.89	37.83	50.47	–

Table 8.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, CCCC, c/a = 0.4, a/h = 15, material I.

	Mode
Method	1	2	3	4	5
Present	27.174	31.325	34.955	45.953	47.716
HSDT C¹ [13]	27.52	32.06	35.98	47.91	–

Table 9.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, CCCC, c/a = 0.6, a/h = 15, material I.

	Mode
Method	1	2	3	4	5
Present	44.578	44.796	52.901	55.334	58.524
HSDT C¹ [13]	45.48	45.72	54.90	57.77	–

Table 10.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, CCCC, c/a = 0.8, a/h = 15, material I.

	Mode
Method	1	2	3	4	5
Present	113.881	113.885	121.862	121.910	126.959
HSDT C¹ [13]	117.17	117.20	126.96	126.99	–

Table 11.

Comparison of the first five frequency parameters Ω of a square laminated plate with square cutout, SSSS, a/h = 100, material II.

		Mode
c/a	Method	1	2	3	4	5
	Present	9.116	25.505	25.505	38.032	54.020
0.2	FSDT [24]	9.116	25.500	25.515	38.045	54.033
	ANSYS [14]	9.11	25.63	25.80	38.11	53.154
	Present	9.101	20.421	20.421	35.479	44.580
0.4	FSDT [24]	9.093	20.409	20.429	35.483	44.599
	ANSYS [14]	9.12	20.25	20.34	35.67	44.76
	Present	11.146	18.589	18.589	32.996	34.355
0.6	FSDT [24]	11.113	18.536	18.549	32.941	34.271
	ANSYS [14]	11.31	18.69	18.71	31.56	34.34

The free vibration of the composite sandwich plate without cutout is realized in the next step. The fundamental frequency parameters are determined by using a uniform mesh of four p-elements and the degree p of the polynomial shape function is fixed at nine. The results shown in Table 12 and 13 concern a composite sandwich square plate with two configurations of cross ply layers; [0°/90°/core/0°/90°] and [0°/90°/0°/core/0°/90°/0°]. The frequency parameters Ω where $Ω = ω (a^{2} / h) (\sqrt{ρ_{c} / E_{2}_{c}})$ are computed with different boundary conditions. The ratio $a / h$ is taken equal to 10 and the ratio of thickness of the core to thickness of the face sheet $h_{c} / h_{f}$ is taken equal to 16. The mechanical properties of materials IV and V of Table 1 are used, respectively, for the face sheets and the core.

Table 12.

Comparison of the first five frequency parameters Ω of a square sandwich plate without cutout, a/h = 10, h _c /h _f = 16, [0°/90°/core/0°/90°].

BoundaryConditions	Method	Mode
BoundaryConditions	Method	1	2	3	4	5
	Present	19.431	30.932	31.742	40.425	45.372
CCCC	HSDT [8]	19.98	32.53	32.88	42.14	48.72
	HSDT C¹ [23]	20.13	32.74	32.74	42.29	48.23
	Present	15.076	28.528	29.299	38.248	40.307
SSSS	HSDT [8]	15.03	28.43	28.78	37.66	40.82
	HSDT C¹ [23]	15.30	29.39	29.39	38.88	40.83
	Present	17.159	30.003	30.073	39.251	40.307
CSCS	HSDT [8]	17.56	29.85	31.50	39.94	40.82
	HSDT C¹ [23]	17.78	30.39	31.75	40.60	40.83
	Present	13.317	14.887	26.161	27.142	29.378
CFCF	HSDT [8]	13.83	15.45	26.40	28.85	30.98
	HSDT C¹ [23]	14.07	15.63	26.59	29.03	31.18
	Present	3.809	6.411	15.406	15.434	19.279
CFFF	HSDT [8]	3.85	6.51	15.62	15.89	19.72
	HSDT C¹ [23]	3.94	6.59	15.63	16.20	20.02

Table 13.

Comparison of the first five frequency parameters Ω of a square sandwich plate without cutout, a/h = 10, h _c /h _f = 16, [0°/90°/0°/core/0°/90°/0°].

BoundaryConditions	Method	Mode
BoundaryConditions	Method	1	2	3	4	5
	Present	19.419	31.186	31.526	40.47	46.03
CCCC	HSDT [8]	20.01	32.23	33.34	42.27	48.16
	HSDT C¹ [23]	20.11	32.13	33.32	42.29	47.19
	Present	15.051	28.596	29.168	38.233	40.307
SSSS	HSDT [8]	15.04	28.10	29.20	37.76	40.82
	HSDT C¹ [23]	15.29	28.69	30.01	38.86	40.83
	Present	17.341	30.042	30.35	39.421	40.307
CSCS	HSDT [8]	17.54	30.23	31.14	39.96	40.82
	HSDT C¹ [23]	17.84	29.76	32.39	40.64	40.83
	Present	13.366	14.996	26.519	27.462	29.721
CFCF	HSDT [8]	13.52	15.21	26.83	28.32	30.55
	HSDT C¹ [23]	13.55	15.24	27.17	28.1	30.42
	Present	3.611	6.395	14.863	15.260	19.261
CFFF	HSDT [8]	3.63	6.43	15.07	15.39	19.43
	HSDT C¹ [23]	3.65	6.47	15.07	15.44	19.55

The simply supported SSSS sandwich square plate with symmetric cross ply [0°/90°/0°/core/0°/90°/0°] and anti-symmetric angle ply [45°/–45°/45°/core/–45°/45°/–45°] is considered in Table 14. The obtained results are in good agreement with those available solutions of HSDT [8], HSDT C¹ [23] and HSDT [25].

Table 14.

Comparison of the first four frequency parameters Ω of a SSSS square sandwich plate without cutout, $a / h = 10$ , ${h_{c} / h}_{f} = 16$ .

Fiber		Mode
Orientation	Method	1	2	3	4
	Present	15.051	28.596	29.168	38.233
[0°/90°/0°/core/0°/90°/0°]	HSDT^a [8]	15.04	28.10	29.20	37.76
	HSDT^b [8]	15.34	30.18	31.96	40.94
	HSDT [23]	15.29	28.89	30.14	39.14
	HSDT [25]	15.28	28.69	30.01	38.86
	Present	16.205	29.207	29.207	39.427
[45°/−45°/45°/core/−45°/45°/−45°]	HSDT^a [8]	16.09	28.93	28.93	38.76
	HSDT^b [8]	16.43	31.17	31.17	42.78
	HSDT [23]	16.44	29.84	29.84	40.34
	HSDT [25]	16.38	29.65	29.65	40.00

HSDT^a [8]: 4 node element; HSDT^b [8]: 9 node element.

The next comparison in Table 15 is carried out for an anti-symmetric [0°/90°/core/0°/90°] sandwich square plate with fully simply supported SSSS boundary condition. The first frequency parameters

Ω

where

Ω = ω (a^{2} / h) (\sqrt{ρ_{f} / E_{2}_{f}})

are given by varying the ratio a/h from 2 to 100 and the ratio

h_{c} / h_{f}

is taken equal to 10. The mechanical properties of materials VI and VII are adopted, respectively, for the face sheets and the core. The results are compared with the solutions of HSDT [26], FSDT [27], HSDT C¹ [23] and those HSDT^a (12 degrees of freedom) and HSDT^b (9 degrees of freedom) models [7]. It clearly seen that the present results are very close to the HSDT models [7].

Table 15.

Comparison of the first frequency parameters Ω of a SSSS square sandwich plate without cutout, ${h_{c} / h}_{f} = 10$ .

	Method
a/h	Present	HSDT^a [7]	HSDT^b [7]	HSDT C⁰ [23]	HSDT [26]	FSDT [27]
2	1.195	1.1941	1.1734	1.6260	1.6252	5.2017
4	2.137	2.1036	2.0913	3.1039	3.1013	9.0312
10	4.959	4.8594	4.8519	7.0529	7.0473	13.8694
20	8.740	8.5955	8.5838	11.2725	11.2664	15.5295
30	11.237	11.0981	11.0788	13.3723	13.6640	15.9155
40	12.800	12.6821	12.6555	14.4440	14.4390	16.0577
50	13.785	13.6899	13.6577	15.0366	15.0323	16.1264
60	14.427	14.3497	14.3133	15.3911	15.3868	16.1612
70	14.861	14.7977	14.7583	15.6174	15.6134	16.1845
80	15.164	15.119	15.0702	15.7699	15.7660	16.1991
90	15.383	15.3380	15.2946	15.8770	15.8724	16.2077
100	15.546	15.5093	15.4647	15.9551	15.9522	16.2175

HSDT^a [27]: 12 degrees of freedom; HSDT^b [27] : 9 degrees of freedom.

Due to the relative lack of vibration analysis studies of composite laminated and sandwich plates with circular cutout, it is interesting to investigate the effect of fibers orientation angle, the radius $r$ of the circular cutout and the boundary conditions on the natural frequencies of composite laminated and sandwich plates.

The composite laminated square plate with circular cutout is treated in first study through three different cases. In the first and second cases, the cutout is considered, respectively, as free and clamped and the third case concerns a plate with an offset cutout.

The non-dimensional frequency parameter used in this investigation is $Ω = ω (a^{2} / h) (\sqrt{ρ / E_{2}})$ . The composite laminated square plates ( $a = b$ ) with thickness ratio $a / h$ equal to 10 are considered. The mechanical properties selected are those of material III.

First case: plate with free cutout

In the first case, the five first frequency parameters are calculated by varying the radius r from 0.1 to 0.4. The circular cutout is free in this case. The composite laminated plates are adopted with configurations of [0°/90°/90°/0°], [0°/90°/0°/90°], [45°/−45°/−45°/45°] and [45°/−45°/45°/−45°]. The boundary conditions used in this case are fully clamped CCCC and simply supported SSSS. The results determined in this example are shown in Tables 16 to 19. From the tables listed below, it is noticed that the effect of the radius r on the frequency parameters is very significant, especially by the first modes. It can be observed that the first modes increase with the increase of the radius r from 0.1 to 0.4 with an average of 61%, 19%, 62% and 54% for the Tables 16 to 19, respectively (see Figure 4(a) and (b)). The same is observed for the other modes, and only in the case of the simply supported SSSS laminated plates in Table 17, the frequency parameters from modes 2 to 5 decrease with the increase of radius r.

Table 16.

The first five frequency parameters Ω of a CCCC composite laminated square plate.

		Mode
Fiber orientation		1	2	3	4	5
	0.1	23.625	31.622	37.713	47.556	47.711
[0°/90°/90°/0°]	0.2	28.462	31.534	36.342	44.479	47.806
	0.3	38.765	38.999	44.614	45.822	55.019
	0.4	59.284	59.358	59.694	59.772	84.020
	0.1	24.348	36.717	36.717	49.796	57.614
[0°/90°/0°/90°]	0.2	29.534	35.638	35.638	46.792	52.906
	0.3	42.027	44.316	44.316	48.205	57.739
	0.4	64.056	64.156	64.156	64.245	91.313

Table 17.

The first five frequency parameters Ω of a SSSS composite laminated square plate.

		Mode
Fiber orientation	$γ$	1	2	3	4	5
	0.1	13.983	24.682	33.097	42.322	43.253
[0°/90°/90°/0°]	0.2	12.872	21.448	24.946	38.722	40.635
	0.3	13.254	19.501	21.461	31.979	36.611
	0.4	17.372	21.206	22.596	28.230	39.719
	0.1	14.356	30.506	30.506	44.321	54.134
[0°/90°/0°/90°]	0.2	13.223	24.722	24.722	40.021	47.158
	0.3	13.561	21.565	21.565	33.250	40.820
	0.4	17.670	22.457	22.457	28.927	42.589

Table 18.

The first five frequency parameters Ω of a CCCC composite laminated square plate.

		Mode
Fiber orientation	γ	1	2	3	4	5
	0.1	21.960	32.904	34.645	46.237	52.727
[45°/−45°/−45°/45°]	0.2	26.510	32.305	32.765	43.897	48.453
	0.3	37.842	40.380	40.650	44.750	53.019
	0.4	58.688	58.757	58.849	58.919	84.564
	0.1	23.140	36.109	36.109	51.161	56.197
[45°/−45°/45°/−45°]	0.2	27.777	34.594	34.594	47.495	51.920
	0.3	39.709	42.595	42.595	47.714	56.530
	0.4	62.150	62.306	62.306	62.466	90.562

Table 19.

The first five frequency parameters Ω of a SSSS composite laminated square plate.

		Mode
Fiber orientation	γ	1	2	3	4	5
	0.1	17.118	29.121	31.708	43.414	50.318
[45°/−45°/−45°/45°]	0.2	19.009	26.968	27.784	41.337	45.213
	0.3	24.284	27.854	31.546	40.042	43.611
	0.4	35.370	35.804	46.104	48.808	55.241
	0.1	19.084	32.884	32.884	49.317	53.222
[45°/−45°/45°/−45°]	0.2	21.750	30.297	30.297	45.913	48.595
	0.3	28.834	33.959	33.959	44.601	48.138
	0.4	45.168	47.958	47.958	52.907	63.388

Second case: plate with clamped cutout

In order to examine the effect of boundary conditions on the fundamental frequency of laminated composite plates, the frequency parameters are determined in this case by constraining the degrees of freedom of circular cutout, in which the radius r is taken equal to 0.2. Two configurations of the fiber orientation angle are considered: symmetric [0°/90°/90°/0°] and antisymmetric [0°/90°/0°/90°]. The boundary conditions at the edges of plate are proposed as follows; FFFF, CFCF and SFSF where F indicate that the edge is free. The results of the first five modes are shown in Tables 20 and 21.

Table 20.

The first five frequency parameters Ω of a symmetric composite laminated square plate with clamped cutout.

	Boundarycondition	Mode
Fiber orientation	Boundarycondition	1	2	3	4	5
	FFFF	9.101	11.302	11.737	13.029	23.171
[0°/90°/90°/0°]	CFCF	28.184	29.395	32.038	33.863	47.825
	SFSF	25.377	26.864	28.268	30.154	44.804

Table 21.

The first five frequency parameters Ω of an antisymmetric composite laminated square plate with clamped cutout.

	Boundarycondition	Mode
Fiber orientation	Boundarycondition	1	2	3	4	5
	FFFF	9.48	13.349	13.349	16.045	23.093
[0°/90°/0°/90°]	CFCF	28.18	28.702	38.938	39.867	58.887
	SFSF	25.185	25.827	35.585	36.575	56.24

Third case: plate with an offset cutout

The effect of the position of the circular cutout on the free vibrations is examined in the last case, by shifting the position of the cutout along the y-axis of the plate (see Figure 5), in which the radius r of the circular cutout is equal to 0.2. The circular cutout is shifted by a value of α and the first frequency parameters are given in Tables 22 and 23. The investigation is carried out with a symmetric [0°/90°/90°/0°] and antisymmetric [0°/90°/0°/90°] laminated plate. The boundary conditions applied to the four edges are CCCC and SSSS, and the circular cutout remains free.

Table 22.

The first frequency parameters Ω of a symmetric composite laminated square plate with an offset cutout.

	Boundary	α
Fiber orientation	Condition	0	0.25	0.26	0.27	0.28	0.29
	CCCC	28.462	23.561	23.444	23.332	23.225	23.123
[0°/90°/90°/0°]	SSSS	12.872	13.776	13.828	13.879	13.928	13.977

Table 23.

The first frequency parameters Ω of an antisymmetric composite laminated square plate with an offset cutout.

	Boundary	α
Fiber orientation	Condition	0	0.25	0.26	0.27	0.28	0.29
	CCCC	29.534	23.218	23.059	22.908	22.765	22.63
[0°/90°/0°/90°]	SSSS	13.223	13.428	13.444	13.459	13.476	13.493

It clearly seen from the Tables 22 and 23 that the increase of a value α decreases the first frequency in case of CCCC laminated plate and the opposite occurs for SSSS laminated plate (see Figure 6(a) and (b). The variation is more sensitive in case of CCCC laminated plate.

The effect of cutout size with the ratio of the thickness of the core to thickness of the face sheets (h_c/h_f) on the frequency parameters $Ω = ω (a^{2} / h) (\sqrt{ρ_{c} / E_{2}_{c}})$ of a SSSS sandwich plate is examined in the next example, with symmetric cross ply [0°/90°/0°/core/0°/90°/0°] and antisymmetric angle ply [45°/−45°/45°/core/−45°/45°/−45°]. The first frequency parameters indicated in Tables 24 and 25 are given by varying the radius r of cutout from 0 to 0.4 and the ratio h_c/h_f from 4 to 16 with a step of 2, where a/h is taken equal to 10.

Table 24.

The first frequency parameters Ω of a symmetric [0°/90°/0°/core/0°/90°/0°] composite sandwich square plate with circular cutout, SSSS.

	h_c/h_f
γ	4	6	8	10	12	14	16
0	15.300	14.655	14.551	14.654	14.804	14.942	15.051
0.1	15.505	14.927	14.831	14.918	15.044	15.156	15.237
0.2	16.919	16.407	16.311	16.376	16.470	16.544	16.586
0.3	21.179	20.616	20.493	20.544	20.626	20.863	20.704
0.4	18.481	18.002	17.566	17.167	16.799	16.460	16.145

Table 25.

The first frequency parameters Ω of an antisymmetric [45°/−45°/45°/core/−45°/45°/−45°] composite sandwich square plate with circular cutout, SSSS.

	h_c/h_f
γ	4	6	8	10	12	14	16
0	16.763	15.777	15.570	15.671	15.857	16.045	16.205
0.1	17.323	16.362	16.156	16.250	16.425	16.599	16.743
0.2	19.765	18.742	18.512	18.603	18.779	18.950	19.088
0.3	25.770	24.440	24.129	24.235	24.454	24.669	24.841
0.4	16.068	15.661	15.292	14.956	14.647	14.362	14.098

The mechanical properties of materials IV (Face sheets) and V (Core) are used. It is clearly seen that the first frequency parameters influenced by both the radius r and the ratio of h_c/h_f. The first frequency increases with increasing the radius r up to 0.3 and deceases with increasing the ratio h_c/h_f until to 10. Figure 7(a) and (b) shows that the sandwich plate loses its mass with the cutout, but retains its rigidity by the core to face sheet thickness ratio, by respecting the maximum size of cutout.

The last investigation is made for a symmetric [90°/45°/60°/core/90°/45°/60°] and antisymmetric [90°/−45°/60°/core/−90°/45°/−60°] composite sandwich plate with circular cutout. The radius r of cutout is taken equal to 0.2. In order to determine the frequency parameters $Ω = ω (a^{2} / h) (\sqrt{ρ_{c} / E_{2}_{c}})$ , the ratio of width to thickness a/h is varied and the ratio hc/hf is taken equal to 14. The same materials used in the previous example are considered here. The first five frequency parameters are shown in Tables 26 to 28 with different boundary conditions. It is can be observed that the frequency parameters increased with increasing the ratio a/h. The sandwich plate with cutout can improve its rigidity by the size of the plate and the number of layers.

Table 26.

The first five frequency parameters Ω of a SSSS composite sandwich square plate with circular cutout, $γ = 0.2$ .

		Mode
Fiber orientation	a/h	1	2	3	4	5
	10	17.836	24.219	24.913	34.959	39.043
[90°/45°/60°/core/90°/45°/60°]	20	20.416	38.841	42.367	59.687	68.313
	40	22.405	47.941	56.948	78.376	89.452
	50	22.747	49.334	59.419	82.013	93.786
	10	18.085	24.627	24.948	35.218	39.236
[90°/−45°/60°/core/−90°/45°/−60°]	20	20.802	39.527	42.752	60.972	68.806
	40	22.869	49.318	57.149	81.742	89.238
	50	23.224	50.831	59.578	86.023	93.259

Table 27.

The first five frequency parameters Ω of a CCCC composite sandwich square plate with circular cutout, $γ = 0.2$ .

		Mode
Fiber orientation	a/h	1	2	3	4	5
	10	23.605	28.281	28.478	36.816	40.961
[90°/45°/60°/core/90°/45°/60°]	20	30.452	46.597	49.015	65.769	74.128
	40	37.409	64.683	74.123	95.635	107.310
	50	38.930	68.496	80.116	103.135	115.540
	10	23.738	28.433	28.613	37.025	41.147
[90°/−45°/60°/core/−90°/45°/−60°]	20	30.764	47.225	49.374	66.842	74.716
	40	37.841	66.014	74.429	98.618	107.456
	50	39.384	69.963	80.391	106.868	115.295

Table 28.

The first five frequency parameters Ω of a CFCF composite sandwich square plate with circular cutout, $γ = 0.2$ .

		Mode
Fiber orientation	a/h	1	2	3	4	5
	10	14.433	15.104	25.545	26.860	28.775
[90°/45°/60°/core/90°/45°/60°]	20	21.183	23.568	35.746	44.480	49.107
	40	26.525	30.225	44.521	65.386	70.265
	50	27.563	31.533	46.540	70.243	75.527
	10	14.523	15.244	25.688	27.355	28.800
[90°/−45°/60°/core/−90°/45°/−60°]	20	21.366	23.366	36.388	44.817	49.601
	40	26.751	30.875	45.610	65.918	71.275
	50	27.788	32.241	47.768	70.826	76.570

Conclusion

The p version of the finite element method based on a new C⁰ enriched curved quadrilateral p-element has been developed and applied to free vibration analysis of a composite laminated and sandwich plate having a circular cutout. The HSDT C⁰ model is used in this investigation, to avoid the difficulties associated with these C¹ continuity elements, in which the strain vector contains only derivatives of the first-order and the shear correction factor is not required. In order to reduce the error of the discretization caused by the classical method of FEM, the composite plate has been discretized into only four curved quadrilateral p-elements and the curved edges are represented accurately by using the blending function method.

The elements of the stiffness and mass matrices are calculated analytically. Monotonic and uniform convergence occurs when the number of polynomial shape functions is increased. High accuracy, stable numerical computation, and rapid convergence have been observed in the analysis. The results of the present model show a great agreement compared to the available solutions.

The fundamental frequencies are examined by considering the fiber orientation, radius and location of the circular cutout, the plate thickness and the boundary conditions. The obtained results calculated in this study show that the fundamental frequencies are influenced significantly by varying the radius of circular cutout of the composite plate. The increase of the radius of cutout increases the fundamental frequencies for both laminated and sandwich plate, and it is observed that the plate with cutout can increase its stiffness by the fiber orientation and number of layers. The plate loses its mass by the size of cutout, but it can increase its stiffness by the size of the sandwich plate and the number of layers. The fundamental frequencies of laminated plate are influenced by the boundary conditions imposed at its edges, when changing the location of the circular cutout.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix 1

References

Paramasivam

Free vibration of square plates with square openings. J Sound Vibrat 1973; 30: 173–178.

Pandit

Haldar

Mukhopadhyay

Free vibration analysis of laminated composite rectangular plate using finite element method. J Reinf Plastics Compos 2007; 27: 69–79.

Lee

Lim

Chow

ST.

Free vibration of composite rectangular plates with rectangular cutouts. J Compos Struct 1987; 8: 63–81.

Liewa

Kitipornchaib

Leungb

AYT

, et al. Analysis of the free vibration of rectangular plates with central cut-outs using the discrete Ritz method. Int J Mech Sci 2003; 45: 941–959.

Liu

Zhao

Dai

, et al. Static and free vibration analysis of laminated composite plates using the conforming radial point interpolation method. J Compos Sci Technol 2008; 68: 354–366.

Torabi

Azadi

AR.

Vibration analysis for rectangular plate having a circular central hole with point support by Rayleigh-Ritz method. J Solid Mech 2014; 6: 28–42.

Kant

Swaminathan

Analytical solutions for free vibration of laminated composite and sandwich plates based on a higher order refined theory. Compos Struct 2001; 53: 73–85.

Nayak

Moy

SSJ

Shenoi

RA.

Free vibration analysis of composite sandwich plates based on Reddy’s higher-order theory. Compos: Part B 2002; 33: 505–519.

Reddy

JN.

Large amplitude flexural vibration of layered composites plates with cutouts. J Sound Vibrat 1982; 83: 1–10.

10.

Boay

CG.

Free vibration of laminated plates with a central circular hole. Compos Struct 1996; 35: 357–368.

11.

Bicos

Springer

GS.

Vibrational characteristics of composite panels with cutouts. AIAA J 1989; 27: 1116–1122.

12.

Ramakrishna

Rao

NS.

Free Vibration analysis of laminates with circular cutout by hybrid-stress finite element. Compos Struct 1992; 21: 177–185.

13.

Kumar

Shrivastava

RP.

Free vibration of square laminates with delamination around a central cutout using HSDT. Compos Struct 2005; 3: 317–333.

14.

Khaldoon

FB.

Free vibration analysis of a symmetric and anti-symmetric laminated composite plate with a cutout at the center. Al-Qadisiya J Eng Sci 2009; 2: 324–334.

15.

Sahoo

Laminated composite stiffened shallow spherical panels with cutouts under free vibration. Eng Sci Technol Int J 2014; 4: 247–259.

16.

Bhardwaj

Jyoti

Study of free vibration analysis of laminated composite plates with triangular cutouts. Eng Solid Mech 2015; 3: 43–50.

17.

Mondal

Patra

Chakraborty

, et al. Dynamic performance of sandwich composite plates with circular hole/cut-out: a mixed experimental–numerical study. Composite Structures 2015; 131: 479–489.

18.

Mishra

Basa

Sarangi

SK.

Free vibration analysis of sandwich plates with cutout. IOP Conference Series: Materials Science and Engineering, Volume 1. 2016.

19.

Shankara

Iyengar

NGR.

Analysis of composite plates with higher order shear deformation theory. Mech Res Commun 1992; 4: 301–314.

20.

Shankara

Iyengar

NGR.

A C0 element for the free vibration analysis of laminated composite plates. J Sound Vibrat 1996; 5: 721–738.

21.

Hamza-Cherif

SM.

Free vibration analysis of rotating cantilever plates using the p-version of the finite element method. J Struct Eng Mech 2006; 22: 151–167.

22.

Hachemi

Hamza-Cherif

Houmat

Free vibration analysis of variable stiffness composite laminate plate with circular cutout.

Aus J Mech Eng 2020; 18: 63–79.

23.

Serdoun

SMN

Hamza-Cherif

SM.

Free vibration analysis of composite and sandwich plates by alternative hierarchical finite element method based on Reddys C1 HSDT. J Sandwich Struct Mater 2016; 18: 501–528.

24.

Sheikh

Haldar

Sengupta

Free flexural vibration of composite plates in different situations using a high precision triangular element. J Vibrat Control 2004; 10: 371–386.

25.

Meunier

Shenoi

RA.

Free vibration analysis of composite sandwich plates. Proc Inst Mech Eng Part C J Mech Eng Sci 1999; 213: 715–727.

26.

Senthilnathan

Lim

Lee

, et al. Buckling of shear deformable plates. AIAAJ 1987; 25: 1268–1271.

27.

Whitney

Pagano

NJ.

Shear deformation in heterogeneous anisotropic plates. J Appl Mech 1970; 37: 1031–1036.