Abstract

Free vibration of antisymmetric angle-ply laminated plates with variable thickness is studied. Higher-order shear deformation plate theory (HSDT) is introduced in the present method to remove the shear correction factors and improve the accuracy of transverse shear stresses. The thickness variations are assumed to be linear, exponential, and sinusoidal. The coupled differential equations are obtained in terms of displacement and rotational functions and approximated using cubic and quantic spline. A generalized eigenvalue problem is obtained and solved numerically by employing the eigensolution techniques with eigenvectors as spline coefficients to obtain the required frequencies. The results of numerical calculations are presented for laminated plates with simply supported boundary conditions. Comparisons of the current solutions and those reported in literature are provided to verify the accuracy of the proposed method. The effects of aspect ratio, number of layers, ply-angles, side-to-thickness ratio, and materials on the free vibration of cylindrical plates are discussed in detail.

1. Introduction

A composite is a material that consists of two or more constituent materials. Composite structures are widely used in structural applications because of their advantage properties such as having high strength-to-weight ratios, being lightweight, and having higher specific stiffness-to-weight ratio. Classical theory was developed originally based on the Love-Kirchoff assumption for thin elastic shells where transverse effects are neglected [1]. Therefore, refinements to Love’s first approximation theory of thin elastic shell were developed by Reissner [2] and Mindlin [3]. The most significant difference between the classical and shear deformation theories is the effect of including transverse shear deformation on the predicted deflections and frequencies [4]. As for the first-order shear deformation theory (FSDT), the theory extends the kinematics of the classical shell theory (CST) by relaxing the normality restriction and allowing for arbitrary but constant rotation of transverse normals. However FSDT requires the usage of shear correction factor in order to satisfy zero shear conditions on the top and bottom surfaces of the plate, which is considered a shortcoming of this theory. To overcome the limitation of FSDT, higher-order deformation theory which considers shear effect without any necessity to use shear correction factors was proposed [5]. HSDT further relaxes the kinematic hypothesis by removing the straightness assumption where the straight normal to the middle plane before deformation may become cubic curves after deformation [6]. According to Reddy [4], higher-order theories can represent the kinematics better, may not require shear correction factors, and can yield more accurate interlaminar stress distributions.

The study on HSDT for vibration analysis of shells had been conducted by a few researchers. Xiaoping and Liangxin [7] used Fourier series method to investigate vibration of symmetric cross-ply plates. A Navier solution was developed by Adim et al. [8] and Kumar et al. [9] to analyse vibration of symmetric and antisymmetric cross-ply laminates with simply supported boundary conditions. Ritz method was proposed by Hanna and Leissa [10] to study the vibration of completely free rectangular plate. Zuo et al. [11] presented the method of wavelet finite element method for vibration analysis of symmetric and antisymmetric cross-ply/angle-ply laminated composite plates under simply supported boundary conditions. The B-spline wavelet scaling functions were employed as approximating functions of the method of wavelet finite element. Dynamic analysis of multilayer symmetric composite plates was conducted by [12] Mallikarjuna and Kant [12, 13] by using a simple isoparametric finite element formulation based on a higher-order displacement model with an explicit time marching scheme. Using the same method, they studied the vibration of unsymmetrically laminated composite and sandwich plates. Simply supported and clamped boundary conditions were considered in both studies. Zhang and Selim [14] studied the vibration of carbon nanotube (CNT) reinforced functionally graded thick laminated composite plates. The element-free IMLS-Ritz method was employed for the solution of the problem and further utilizing the Mori–Tanaka approach for estimating the effective material properties of the composite. Saira et al. [15] investigated free vibration of antisymmetric angle-ply laminated plates using higher-order theory under simply supported boundary conditions. Constant thickness was considered in the study.

Bouazza et al. [16] have presented vibration behaviour of laminated composite simply supported rectangular plates using a closed form solution of Navier method. [17] D. B. Singh and B. N. Singh [17] have proposed Trigonometric Deformation Theory (TDT) and Trigonometric-Hyperbolic Deformation Theory (THDT) for the analysis of laminated and three-dimensional braided composite plate using FEM solution. The results were obtained with two types of boundary condition, that is, all edges simply supported and all edges clamped boundary conditions. Zamani et al. [18] developed direct viscoelastic approach to investigate the dynamic behaviour of thick polymer-matrix composite plates on visco-Pasternak foundation. Both symmetry and antisymmetric plates are considered in the research. The study on vibration of simply supported functionally graded plates using HSDT has been conducted by Thai and Kim [19] and Thinh et al. [20]. The solutions were assumed using Navier approach. Vibration analysis on antisymmetric cracked FGM plates for simply supported boundary conditions has been investigated by Tran et al. [21] using a formulation that combines the eXtended IsoGeometric Approach (XIGA) and HSDT. IsoGeometric Analysis (IGA) utilizing the Non-Uniform Rational B-Spline (NURBS) functions is incorporated with enrichment functions through the partition of unity method in order to stimulate the cracked FGM plates. Using generalized differential quadrature (GDQ) method, Asadi and Fariborz [22] analysed the vibration of laminated composite plates with mixed boundary conditions. Another study on FGM plates has been carried out by Selim et al. [23]. In the study, element-free IMLS-Ritz method together with Reddy’s HSDT was implemented to the solution of FGM plates with piezoelectric layers.

Although plenty of studies on the vibration of plates using higher-order theory have been conducted, the free vibration analysis of antisymmetric angle-ply laminated plates with variable thickness using third-order shear deformation theory has not been investigated yet. In this paper, the laminated plates of linear, exponential, and sinusoidal thickness variation with simply supported boundary conditions are considered. Governing equations are derived from Hamilton’s principle. Spline method is used to approximate the solution. Further, variations of frequency of laminated plates are discussed at various parameters such as aspect ratio, number of layers, ply-angles, side-to-thickness ratio, and materials. Numerical examples are presented to illustrate the accuracy and efficiency of the present theory in predicting the natural frequencies of laminates by comparing the present results with the results obtained in the literatures.

2. Formulation

Consider an antisymmetric angle-ply laminated rectangular plate in the Cartesian coordinates , , and where is taken as the normal to the plate and is the constant thickness as shown in Figure 1. According to third-order shear deformation theory (TSDT), the displacement field is assumed as [4] where , and are the displacement components in the , , and directions, respectively, and and are the in-plane displacements of the middle plane, and and are the shear rotations of any point on the middle surface of the plate.

Figure 1 

Geometry of the layered rectangular plate.

In-plane strains are defined aswhereand the shear strain components are defined aswhereThe stress-strain relations for the kth layer, after neglecting transverse normal strain and stress, are of the form When the materials are oriented at an angle with the -axis, the transformed stress-strain relations arewhere , as functions of , are expressed as where

The stress resultants are defined as where , , and are stress, moment, and shear resultants, respectively. and denote higher-order stress resultants. The stress-strain relations are obtained as follows:Stiffness coefficients are defined as where the elastic coefficients , , and (extensional, bending-extensional coupling, and bending stiffness) and , , and are the higher-order stiffness coefficients. The boundaries of the kth layer are represented by and . The thickness of the kth layer can be considered as where , in which and are the variable thickness and constant thickness of the th layer, respectively, and , , and are the coefficients of variable thickness. It should be noted that when , the thickness of the plate becomes constant.

The equilibrium equations considered are as follows:where I_i are the normal and rotary inertia coefficients defined bywhere is the material density of the th layer and Substitute (2) and (4) in (11) and then substitute in (14). Further, equate the following laminate stiffness equal to zero for antisymmetric angle-ply laminates [14]: The displacements and rotational functions are assumed in the separable form for antisymmetric angle-ply plates as where is the angular frequency of vibration and is the time and is the circumferential mode number. The nondimensional parameters introduced are as follows:Using (18) and introducing nondimensional parameters, modified equations in the matrix form are obtained aswhere are shown in Appendix.

3. Method of Solution

The differential equations in (20) contain derivatives of second order in , third order in , fourth order in , third order in , and second order in . These functions are approximated by using cubic and quantic spline functions, in the range of , since splines are relatively simple and elegant and use series of lower order approximations rather than global higher-order approximations, affording fast convergence and high accuracy.

The displacement functions , and and the rotational functions and are approximated, respectively, by the splines

Here is the Heaviside step function and is the number of intervals into which the range of is divided. The points are chosen as the knots of the splines, as well as the collocation points. Thus the splines are assumed to satisfy the differential equations given by (19), at all . The resulting expressions contain homogeneous system of equations in the spline coefficients.

The boundary condition considered is (S-S): both the ends simply supported where at and [27]. This boundary condition gives 13 more equations, thus making a total of equations, in the same number of unknowns. The resulting field and boundary condition equations may be written in the form where and are square matrixes and is a column vector. This is treated as a generalized eigenvalue problem in the eigenparameter and the eigenvector whose elements are the spline coefficients.

4. Results and Discussion

4.1. Validation

In order to demonstrate the accuracy of the present analysis, some illustrative examples are solved and the numerical results are compared with data available in the literature. The comparison of fundamental frequencies of four-layered antisymmetric angle-ply plates for side-to-thickness ratio fixed as 2 and 4 is tabulated in Table 1. The present results obtained for plate with S-S boundary conditions are compared with the results from Swaminathan and Patil [24] and Reddy [25]. The nondimensional frequency parameter is defined as . In Table 2, comparison study is presented for present results with FSDT results of Reddy [26] where he used FEM. The present parameter is converted into the nondimensional frequency parameter given in Reddy [26], where . The material properties used for both comparison in Tables 1 and 2 are , , , and . It can be observed from Tables 1 and 2 that the present results agree well with results of Swaminathan and Patil [24] and Reddy [25, 26].