Frequency Response of Higher-Order Shear-Deformable Multilayered Angle-Ply Cylindrical Shells

Abstract

This research is based on the frequency response of angle-ply laminated cylindrical shells under higher-order shear deformation theory. The higher-order shear deformation theory is used to model the displacement and rotational functions, which are approximated by cubic and quintic splines. The eigenvalue problem is obtained with the simply supported boundary condition. The frequency of cylindrical shells is analyzed by varying the circumferential node number, length, number of layers, and layer alignment. The competence of the formulation is verified by comparing it with the available results of higher-order zigzag theory.

1. Introduction

Engineers are vastly using cylindrical shells in designing structures because they can carry applied loads effectively by means of their curvatures. Shells consisting of composite materials enable researchers to achieve a wide range of mechanical properties and optimum results. The initial impetus to use composite materials as an element of advanced engineering structures is based on their properties such as specific strength and specific stiffness in comparison to their monolithic counterparts. Moreover, composite structures provide excellent opportunities by offering benefits while designing advanced lightweight, high-temperature resistance, fatigue resistance, impact resistance, corrosion resistance, better damping, and shock absorbing characteristics. Presently, engineers are mainly focusing on structural weights while designing any aerospace or automotive structures, which could be achieved by selecting such composite materials that have high strength-to-weight ratios or stiffness-to-weight ratios. Keeping in mind the importance of composite materials, a number of studies have used functionally graded materials using the wave-based method, discrete singular convolution method, finite element model [1,2,3], functionally graded graphene-reinforced piezoelectric nanoplates using the Halpin–Tsai micromechanical model [4], and graphene nanoplatelets using the Halpin–Tsai micromechanical model [5]. So, the present study is unique in terms of material usage and was designed to investigate the frequency of composite cylindrical shells consisting of Kevlar-49/epoxy and E-glass epoxy materials and using the spline approximation method, which, in return, produced a unique result as compared to other studies. The main practical application of the present study is that cylindrical shells are found in many applications like the aerospace and naval construction industries. They are often used as load-bearing structures for missiles, airplanes, space craft, and submarines, in which a minimum weight design is required.

The simple laminated shear deformation theory for layered plates was proposed by Reddy [6] and that for layered shells by Reddy and Liu [7]. To avoid the discrepancies of the first-order shear deformation theory, the higher-order shear deformation theory (HSDT) was developed to accurately evaluate the transverse shear stresses that effectively exist in thick plates. In higher-order plate theories, displacements are expanded up to any desired degree in terms of thickness coordinates [8,9,10], which eliminate the need for a shear correction factor. Further, HSDT yields more accurate inter-laminar stress distributions and satisfies the conditions of zero shear stress at the top and bottom surfaces of the plate [11,12].

Researchers are continuously working on laminated shells to obtain the best possible outcomes as desired by engineers and architectures. Among them, Roy et al. [13] conducted an investigation on the free vibration of shells based on higher-order zigzag theory. Peng et al. [14] examined the dynamic stiffness formulation for the free-vibration analysis of combined elliptical–cylindrical–conical shells using first-order shear deformation theory, but the present study used HSDT. Shinde and Sayyad [15] studied the free-vibration analysis of laminated shells based on a new higher-order shear and normal deformation theory, whereas the present study neglected the transverse normal deformation. Lore et al. [16] investigated the nonlinear free-vibration analysis of laminated composite plates and shell panels using non-polynomial higher-order shear deformation theory, while the present formulation is linear. Tong et al. [17] examined the free-vibration analysis of fiber-reinforced composite multilayer cylindrical shells under hydrostatic pressure, but the current study did not consider hydrostatic pressure. Parvez and Khan [18] studied the influence of geometric imperfections on the nonlinear forced vibration characteristics and stability of laminated angle-ply composite conical shells. Bochkarev et al. [19] examined the natural vibrations of composite cylindrical shells partially filled with a fluid. Gulizzi et al. [20] studied the high-order accurate transient and free-vibration analysis of plates and shells. The effect of anisotropic stiffness degradation on the forced vibration of cylindrical shells was studied by Li et al. [21]. A closed-form solution for the asymmetric free-vibration analysis of composite cylindrical shells with a metamaterial honeycomb core layer based on shear deformation theory was investigated by Eipakchi and Mahboubi Nasrekani [22]. Reddy’s third-order shear deformation shell theory was used to conduct the free-vibration analysis of rotating stiffened advanced nanocomposite toroidal shell segments in thermal environments by Nguyen et al. [23]. The free-vibration analysis of a cross-ply laminated conical shell, a cylindrical shell, and an annular plate with variable thickness using the Haar wavelet discretization method was performed by Kim et al. [24]. The finite element method was used by Attia et al. [25] to analyze the free vibrations of thick laminated composite shells. Open cylindrical shells coupled with rectangular plates were examined by Chen et al. [26]. Skew cylindrical shells reinforced with graphene platelets were investigated by Jahanbazi et al. [27]. The free-vibration analysis of hybrid laminated thin-walled cylindrical shells containing multilayer functionally graded plies was performed by Ma and Gao [28]. Ghassabi et al. [29] investigated an exact analytical method for the free-vibration analysis of functionally graded-sector cylindrical shells under Levy-type boundary conditions. Guo et al. [30] studied the free vibration of coupled structures of laminated composite conical, cylindrical, and spherical shells based on the spectral-Tchebychev technique. The simple first-order shear deformation theory for free vibration of functionally graded spherical shell segments was examined by Nguyen et al. [31]. Garg and Li [32] analyzed the data-driven uncertainty quantification and sensitivity of the free vibration behavior of bio-inspired helicoidal laminated composite cylindrical shells. A meshfree method was used to study the free vibration of arbitrary laminated composite shells and spatial structures by Chen et al. [33]. Khorasani et al. [34] performed a buckling analysis of functionally graded material beams, which were predicted through an exponential-law and power-law theory under a hydrothermal environment. Additionally, Khorasani et al. [35] investigated the vibration of a functionally graded porous three-layered beam resting on Vlasov’s foundation. A refined vibrational analysis of functionally graded material, porous-type beams resting on a silica aerogel substrate was conducted by Khorasani et al. [36]. The thermo-elastic buckling of honeycomb micro plates integrated with functionally graded reinforced epoxy skins with stretching effect was investigated [37].

The spline method was not used by any of the above-mentioned researchers to analyze multilayered cylindrical shells. So, the novelty of the present study is that multilayered cylindrical shells are analyzed using the spline method. Moreover, each layer of the shell consists of a different material. Therefore, the purpose of this research is to examine the free vibration of symmetric angle-ply laminated cylindrical shells based on HSDT using spline approximation. The displacement functions are based on higher-order shear deformation theory and approximated by cubic and quintic splines. Collocation with these splines yields a set of field equations, which, along with the equations of boundary conditions, are reduced to a system of homogeneous simultaneous algebraic equations on the assumed spline coefficients. The eigen solution technique is used to obtain the frequency parameter. The eigenvectors are the spline coefficients through which the mode shapes are constructed. The variation in the frequency is examined by varying the shell length, circumferential node number, number of layers, and layer alignment for simply supported boundary conditions. The results are shown in graphs and tables.

2. Problem Formulation

Consider a composite laminated circular cylindrical shell of length $\ell$, thickness $h$, and radius $r$, as shown in Figure 1. The $x$ coordinate of the shell is taken along the meridional direction, the $\theta$ coordinate along the circumferential direction, and the $z$ coordinate along the thickness direction; the origin is at the mid-surface of the shell.

Figure 2 shows the layer alignment for 4-layered (30°/0°/0°/30°) and 6-layered (60°/30°/0°/0°/30°/60°) shells. Total thickness is represented by $h$.

2.1. Displacement Field

The displacement field considered according to higher-order shear deformation theory [38]:

$$\begin{gathered} u(x,\theta ,z,t)=u_0(x,\theta ,t)+z{\varphi }_x(x,\theta ,t)-{\displaystyle \frac{4z^3}{3h^2}}\left({\varphi }_x+w_{0,x}\right) \\ v(x,\theta ,z,t)=v_0(x,\theta ,t)+z{\varphi }_{\theta }(x,\theta ,t)-{\displaystyle \frac{4z^3}{3h^2}}\left({\varphi }_{\theta }+w_{0,\theta }\right) \\ w(x,\theta ,z,t)=w_0(x,\theta ,t) \end{gathered}$$

(1)

where $t$ is the time; $u,v$, and $w$ are the displacement components in the $x,\theta$, and $z$ directions, respectively; $u_0$, $v_0$, and $w_0$ are the in-plane displacements of the middle plane; and ${\varphi }_x$ and ${\varphi }_{\theta }$ are the shear rotations of any point on the middle surface.

2.2. Strains and Stress Components

In-plane strains are defined as [39]

$$\epsilon =\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_{\theta }\\ {\gamma }_{x\theta }\end{array}\right\}$$

(2)

$$\epsilon ={\epsilon }^0+z{\epsilon }^1+z^3{\epsilon }^3$$

where

$${\epsilon }^0=\left\{\begin{array}{l}{\epsilon }_x^0\\ {\epsilon }_{\theta }^0\\ {\gamma }_{x\theta }^0\end{array}\right\} ,{\epsilon }^1=\left\{\begin{array}{l}{\epsilon }_x^1\\ {\epsilon }_{\theta }^1\\ {\gamma }_{x\theta }^1\end{array}\right\} ,{\epsilon }^3=\left\{\begin{array}{l}{\epsilon }_x^3\\ {\epsilon }_{\theta }^3\\ {\gamma }_{x\theta }^3\end{array}\right\}$$

and the shear strain components are defined as [39]

$$\gamma =\left\{\begin{array}{l}{\gamma }_{\theta z}\\ {\gamma }_{xz}\end{array}\right\}$$

(3)

$$\gamma ={\gamma }^0+z^2{\gamma }^2$$

where

$${\gamma }^0=\left\{\begin{array}{l}{\gamma }_{\theta z}^0\\ {\gamma }_{xz}^0\end{array}\right\}, {\gamma }^2=\left\{\begin{array}{l}{\gamma }_{\theta z}^2\\ {\gamma }_{xz}^2\end{array}\right\}$$

The stress–strain relations for the k-th layer, after neglecting transverse normal strain and stress, are of the form

$$\left(\begin{array}{l}{{\sigma }_x}^{(k)}\\ {{\sigma }_{\theta }}^{(k)}\\ {{\tau }_{x\theta }}^{(k)}\\ {{\tau }_{\theta z}}^{(k)}\\ {{\tau }_{xz}}^{(k)}\end{array}\right) = \left(\begin{array}{lllll}{Q_{11}}^{(k)}& {Q_{12}}^{(k)}& {Q_{16}}^{(k)}& 0& 0\\ {Q_{12}}^{(k)}& {Q_{22}}^{(k)}& {Q_{26}}^{(k)}& 0& 0\\ {Q_{16}}^{(k)}& {Q_{26}}^{(k)}& {Q_{66}}^{(k)}& 0& 0\\ 0& 0& 0& {Q_{44}}^{(k)}& {Q_{45}}^{(k)}& \\ 0& 0& {Q_{45}}^{(k)}& {Q_{55}}^{(k)}& \end{array}\right) \left(\begin{array}{l}{{\epsilon }_x}^{(k)}\\ {{\epsilon }_{\theta }}^{(k)}\\ {{\gamma }_{x\theta }}^{(k)}\\ {{\gamma }_{\theta z}}^{(k)}\\ {{\gamma }_{xz}}^{(k)}\end{array}\right)$$

(4)

When the materials are oriented at an angle ${\theta }_1$ with the $x$-axis, the transformed stress–strain relations are

$$\left(\begin{array}{l}{{\sigma }_x}^{(k)}\\ {{\sigma }_{\theta }}^{(k)}\\ {{\tau }_{x\theta }}^{(k)}\\ {{\tau }_{\theta z}}^{(k)}\\ {{\tau }_{xz}}^{(k)}\end{array}\right) = \left(\begin{array}{lllll}{{\overline{Q}}_{11}}^{(k)}& {{\overline{Q}}_{12}}^{(k)}& {{\overline{Q}}_{16}}^{(k)}& 0& 0\\ {{\overline{Q}}_{12}}^{(k)}& {{\overline{Q}}_{22}}^{(k)}& {{\overline{Q}}_{26}}^{(k)}& 0& 0\\ {{\overline{Q}}_{16}}^{(k)}& {{\overline{Q}}_{26}}^{(k)}& {{\overline{Q}}_{66}}^{(k)}& 0& 0\\ 0& 0& 0& {{\overline{Q}}_{44}}^{(k)}& {{\overline{Q}}_{45}}^{(k)} \\ 0& 0& 0& {{\overline{Q}}_{45}}^{(k)}& {{\overline{Q}}_{55}}^{(k)}& \end{array}\right) \left(\begin{array}{l}{{\epsilon }_x}^{(k)}\\ {{\epsilon }_{\theta }}^{(k)}\\ {{\gamma }_{x\theta }}^{(k)}\\ {{\gamma }_{\theta z}}^{(k)}\\ {{\gamma }_{xz}}^{(k)}\end{array}\right)$$

(5)

where ${Q_{ij}}^{(k)}$, as functions of ${{\overline{Q}}_{ij}}^{(k)}$, are fully furnished in [40].

The stress resultants are defined as [39]

$$\left\{\begin{array}{l}{N}_i\\ M_i\\ P_i\end{array}\right\}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\sigma }_i}\left\{\begin{array}{l}1\\ z\\ z^3\end{array}\right\}dz, \left\{\begin{array}{l}{Q}_i\\ R_i\end{array}\right\}={\displaystyle \underset{-h/2}{\overset{h/2}{\int }}{\tau }_i}\left\{\begin{array}{l}1\\ z^2\end{array}\right\}dz$$

(6)

where $N_i$, $M_i$, and $Q_i$ are stress, moment, and shear resultants, respectively. $P_i$ and $R_i$ denote higher-order stress resultants; normal stress ${\sigma }_i,$ $i=x,\theta$; and shear stress ${\tau }_i,$ $i=x\theta ,\theta z,xz$, respectively.

The stress–strain relations are obtained as follows.

$$\left(\begin{array}{l}N\\ M\\ P\\ Q\\ R\end{array}\right) = \left(\begin{array}{lllll}A& B& E& 0& 0\\ B& D& F& 0& 0\\ E& F& H& 0& 0\\ 0& 0& 0& A^{\prime }& D^{\prime }\\ 0& 0& 0& D^{\prime }& F^{\prime }\end{array}\right) \left(\begin{array}{l}{\epsilon }^0\\ {\epsilon }^1\\ {\epsilon }^3\\ {\gamma }^0\\ {\gamma }^2\end{array}\right)$$

(7)

The stiffness coefficients are defined as [39]

$$\begin{array}{l}{A}_{ij}= {\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( z_k - z_{k-1} ) , B_{ij}= {\displaystyle \frac{1}{2}}{\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( {z^2}_k - {z^2}_{k-1} ) ,D_{ij}= {\displaystyle \frac{1}{3}}{\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( {z^3}_k - {z^3}_{k-1} ) \\ E_{ij}= {\displaystyle \frac{1}{4}}{\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( {z^4}_k - {z^4}_{k-1} ) , F_{ij}= {\displaystyle \frac{1}{5}}{\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( {z^5}_k - {z^5}_{k-1} ) ,H_{ij}= {\displaystyle \frac{1}{7}}{\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( {z^7}_k - {z^7}_{k-1} ) \mathrm{For} i,j = 1, 2, 6,\\ A_{ij}^{\prime }= {\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( z_k - z_{k-1} ) , D_{ij}^{\prime }= {\displaystyle \frac{1}{3}}{\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( {z^3}_k - {z^3}_{k-1} ) \mathrm{and} F_{ij}^{\prime }= {\displaystyle \frac{1}{5}}{\displaystyle \sum _k{\overline{Q}}_{ij}^{(k)}} ( {z^5}_k - {z^5}_{k-1} ) \\ \mathrm{for} i,j = 4, 5\end{array}$$

(8)

where the elastic coefficients $A_{ij}$, $B_{ij}$, and $D_{ij}$ (extensional, bending-extensional coupling, and bending stiffnesses) and $E_{ij}$, $F_{ij}$, and $H_{ij}$ are the higher-order stiffness coefficients.

In the case of symmetric angle-ply lamination, the laminate stiffnesses $A_{16}, A_{26}, A_{45}, D_{16}, D_{26}, D_{45}, F_{16}, F_{26}, F_{45}, H_{16}, H_{26}$, ${B_{ij}}^{\prime }s$, and ${E_{ij}}^{\prime }s$ are zero.

2.3. Equilibrium Equations

The equilibrium equations considered are as follows [38]:

$$\begin{array}{l}{N}_{x,x}+{\displaystyle \frac{1}{r}}{N}_{x\theta ,\theta }=I_0{u}_{tt} +\left(I_1-c_1{I}_3\right){\varphi }_{x,tt} -c_1{I}_3{w}_{xtt} \\ N_{x\theta ,x}+{\displaystyle \frac{1}{r}}{N}_{\theta ,\theta }+{\displaystyle \frac{1}{r}}{Q}_{\theta }=\left(I_0+{\displaystyle \frac{2}{r}}{I}_1\right)v_{tt} +\left(I_1+{\displaystyle \frac{1}{r}}{I}_2\right. \left.-c_1{I}_3-c_1{\displaystyle \frac{1}{r}}{I}_4\right){\varphi }_{\theta ,tt}-\left(c_1{I}_3+c_1{\displaystyle \frac{1}{r}}{I}_4\right)w_{\theta tt} \\ \begin{array}{l}{\overline{Q}}_{x,x}+{\displaystyle \frac{1}{r}}{\overline{Q}}_{\theta ,\theta }-{\displaystyle \frac{1}{r}}{N}_{\theta }+c_2\left(P_{x,xx}+2{\displaystyle \frac{1}{r}}{P}_{x\theta ,x\theta }+{\displaystyle \frac{1}{r^2}}{P}_{\theta ,\theta \theta }\right)\\ =c_1{I}_3{u}_{xtt} +\left(c_1{I}_3+{\displaystyle \frac{c_1}{r}}{I}_4\right)v_{\theta tt}+I_0{w}_{tt}- c_1^2{I}_6\left(w_{xxtt}+w_{\theta \theta tt}\right)\\ +\left(c_1{I}_4-c_1^2{I}_6\right){\varphi }_{x,xtt}+\left(c_1{I}_4-c_1^2{I}_6\right){\varphi }_{\theta ,\theta tt}\end{array}\\ {\overline{M}}_{x,x}+{\displaystyle \frac{1}{r}}{\overline{M}}_{x\theta ,\theta }-{\overline{Q}}_x=\left(I_1-c_1{I}_3\right)u_{tt}-\left(c_1{I}_4-c_1^2{I}_6\right)w_{xtt}+\left(I_2-2c_1{I}_4+c_1^2{I}_6\right){\varphi }_{x,tt} \\ \begin{array}{l}{\overline{M}}_{x\theta ,x}+{\displaystyle \frac{1}{r}}{\overline{M}}_{\theta ,\theta }-{\overline{Q}}_{\theta }=\left(I_1+{\displaystyle \frac{1}{r}}{I}_2-c_1{I}_3-c_1{\displaystyle \frac{1}{r}}{I}_4\right)v_{tt}-\left(c_1{I}_4-c_1^2{I}_6\right)w_{\theta tt} \\ +\left(I_2-2c_1{I}_4+c_1^2{I}_6\right){\varphi }_{\theta ,tt} \end{array}\end{array}$$

(9)

where

• $c_1={\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$h^2$}\right.}, c_2={\displaystyle \raisebox{1ex}{$c_1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$.
• $I_i={\displaystyle \underset{z}{\int }{\rho }^{(k)}(z}{)}^{i}dz (i = 0, 1, 2, 3,\dots \dots , 6)$, and $\rho$ is the material density of the $k$-th layer.

• And

  $$\begin{gathered} {\overline{M}}_{\alpha \beta }=M_{\alpha \beta }-c_1{P}_{\alpha \beta }, {\overline{Q}}_{\alpha }=Q_{\alpha }-c_2{R}_{\alpha } \\ {J}_i=I_i-c_1{I}_{i+2}\hspace{1em}{K}_2=I_2-2c_1{I}_4+c_1^2{I}_6 \end{gathered}$$

The displacements and rotational functions are assumed in the separable form for cylindrical shells as

$$\begin{array}{l}u(x,\theta ,t) = U(x) \cos n\theta e^{i{\omega }_{n}t}\\ v(x,\theta ,t) = V(x)sin n\theta e^{i{\omega }_{n}t}\\ w(x,\theta ,t) = W(x) \cos n\theta e^{i{\omega }_{n}t} \\ {\varphi }_x(x,\theta ,t ) = {\Phi }_x(x) \cos n\theta e^{i{\omega }_{n}t}\\ {\varphi }_{\theta }(x,\theta ,t) = {\Phi }_{\theta }(x)sin n\theta e^{i{\omega }_{n}t}\end{array}$$

(10)

where $U(X) ,$$V(X),$ and $W(X)$ are displacement functions and ${\Phi }_X\left(X\right), {\Phi }_{\theta } (X)$ are the rotational functions. ${\omega }_n$ is the angular frequency of vibration, and $n$ is the circumferential node number.

The non-dimensional parameters are as follows:

• $X={\displaystyle \frac{x}{l}}$, a distant coordinate, and $X\in [0,1]$; ${\lambda }_m={\omega }_n\mathcal{l}\sqrt{{\displaystyle \frac{I_0}{A_{11}}}}$, a frequency parameter;
• $L={\displaystyle \frac{\mathcal{l}}{r}}$, a length parameter; $H={\displaystyle \frac{h}{r}}$, ratio of total thickness to radius;
• ${\delta }_k={\displaystyle \frac{h_k}{h}}$, relative layer thickness of the $k$-th layer

• where $h_k$ is the thickness of the kth layer.

Using Equation (10) and introducing non-dimensional parameters, the obtained equations are approximated using spline as follows.

2.4. Spline Approximation

The differential equations in Equation (10) contain derivatives of the third order in $U (X)$, second order in $V (X)$, fourth order in $W (X)$, third order in ${\Phi }_X(X)$, and second order in ${\Phi }_{\theta } (X)$. These functions are approximated by using cubic and quintic spline functions in the range of $X \in \left[ 0 , 1 \right]$, since splines are relatively simple and elegant and use a series of lower-order approximations rather than global higher-order approximations, affording fast convergence and a high accuracy.

The displacement functions $U(X) ,$$V(X),$ and $W(X)$ and the rotational functions ${\Phi }_X\left(X\right), {\Phi }_{\theta } (X)$ are approximated respectively by the following splines:

$$\begin{array}{l}U(X) = {\displaystyle \sum _{i = 0}^4 a_i} X^i + {\displaystyle \sum _{j = 0}^{N-1} b_j} {( X-X_j )}^5 H ( X - X_j )\\ \\ V(X) = {\displaystyle \sum _{i = 0}^2 c_i} X^i + {\displaystyle \sum _{j = 0}^{N-1} d_j} {( X-X_j )}^3 H ( X - X_j )\\ \\ W(X) = {\displaystyle \sum _{i = 0}^4 e_i} X^i + {\displaystyle \sum _{j = 0}^{N-1} f_j} {( X-X_j )}^5 H ( X - X_j )\\ \\ {\Phi }_X (X) = {\displaystyle \sum _{i = 0}^4 g_i} X^i + {\displaystyle \sum _{j = 0}^{N-1}{p}_j} {( X-X_j )}^5 H ( X - X_j )\\ \\ {\Phi }_{\theta } (X) = {\displaystyle \sum _{i = 0}^2 l_i} X^i + {\displaystyle \sum _{j = 0}^{N-1} q_j} {( X-X_j )}^3 H ( X - X_j )\end{array}$$

(11)

Here $a_i$, $c_i$, $e_i$, $g_i$, $l_i$, $b_i$, $d_i$, $f_i$, $p_i$, and $q_i$ are unknown coefficients (i.e., spline coefficients); $H ( X- X_j )$ is the Heaviside step function; and $N$ is the number of intervals into which the range $[ 0 , 1 ]$ of $X$ is divided. The points $X = X_s = {\displaystyle \frac{s}{N}}$, $( s = 0 , 1 , 2 ,\dots \dots N )$ 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 Equation (10) at all values of $X_s$. The resulting expressions contain $( 5 N + 5)$ homogeneous systems of equations in the $( 5 N + 21 )$ spline coefficients.

The boundary condition considered on the edges $X=0$ and $X=1$ are

• (S-S): both ends are simply supported.

This boundary condition gives 13 more equations, thus making a total of $(5N+18)$ equations, which is the same as the number of unknowns. The resulting field and boundary condition equations may be written in the form

$$\left[ M \right] \left\{ q \right\} = {\lambda }^2 \left[ P \right] \left\{ q \right\}$$

(12)

where $\left[ M \right]$ and $\left[ P \right]$ are square matrices, and $\left\{ q \right\}$ is a column matrix. This is treated as a generalized eigenvalue problem in the eigen parameter $\lambda$ and the eigenvector $\left\{ q \right\}$ whose elements are the spline coefficients. FORTRAN was used to analyze the data.

3. Discussion on Result Outcomes

The frequency response of higher-order shear deformable multilayered composite symmetric angle-ply cylindrical shells under simply supported boundary conditions is analyzed. We consider cylindrical shells of three, four, five, and six layers consisting of two materials: Kevlar-49/epoxy and E-glass epoxy.

The convergence of the spline method for cylindrical shells is studied in the number of subintervals $N$ of the range $X\in [0,1]$. The value of $N$ starts from 2 and is finally fixed at $N=14$, since for the next value of $N$, the percentage changes in the values of $\lambda$ are very small, the maximum being 2%.

A comparative study of the fundamental frequency of Roy et al. [13] and the present study of angle-ply lamination of 3-, 4-, 5-, and 6-layered cylindrical shells for different curvature ratios under S-S boundary conditions is shown in

Table 1. Comparative study of fundamental frequency of Roy et al. [13] and the present study of angle-ply lamination of 3-, 4-, 5-, and 6-layered cylindrical shells under S-S boundary conditions.

a/h	Layers	Angle-Ply Lamination Scheme	Fundamental Frequency
			R/a = 1		R/a = 2		R/a = 3		R/a = 4		R/a = 5
10			Ref. Roy et al. [13]	Present	Ref. Roy et al. [13]	Present	Ref. Roy et al. [13]	Present	Ref. Roy et al. [13]	Present	Ref. Roy et al. [13]	Present
	3	−45°/45°,/−45°,	1411.56	1411.41	1172.47	1172.32	1123.44	1123. 32	1105.607	1105.54	1097.134	1097.02
	4	[−45°/45°]s	1062.66	1062.52	893.92	893.83	861.35	861.22	850.00	850.00	844.815	844.74
	5	−45°/45°/−45°/45°/−45°	839.27	839.11	703.85	703.71	677.96	677.84	669.05	669.04	665.033	665.02
	6	[−45°/45°/−45°]s	668.23	668.11	556.93	556.81	536.92	536.82	530.49	530.35	527.82	527.71

. The effect of the length of cylindrical shells on the angular frequency ${\omega }_n$ of 3-, 4-, 5-, and 6-layered shells is shown in

Table 2. Angular frequency ${\omega }_n$(Hz) of 3-, 4-, 5-, and 6-layered cylindrical shells.

$\mathit{L}$	60°/0°/60°	60°/0°/0°/60°	60°/30°/0°/30°/60°	60°/30°/0°/0°/30°/60°
0.5	2.99357	2.62157	1.91732	2.67609
1	1.19244	1.32572	0.999881	1.0431
1.5	0.365161	0.30223	0.431398	0.362887
2	0.174763	0.102356	0.195176	0.208656
2.5	0.091274	0.08912	0.093216	0.081422

. There is an inverse relation between the length $L$ of cylindrical shells and the angular frequency ${\omega }_n$. As the length increases, the angular frequency decreases. There is a strict decrease in the angular frequency between $L=0.5$ and $L=1$, whereas the decrement in the angular frequency is very small afterwards. Practically, it can be interpreted that if the frequency decreases, the rigidity of the structure also decreases, but the flexibility increases. It is evident from the results that the short shells are more rigid and less flexible than the long shells.

Table 3. Fundamental frequency ${\lambda }_m$ of 3-, 4-, 5-, and 6-layered cylindrical shells.

$\mathit{n}$	30°/0°/30°	30°/0°/0°/30°	60°/30°/0°/30°/60°	60°/30°/0°/0°/30°/60°
2	0.0004	0.0004	0.00048	0.0004
4	0.00028	0.00031	0.00035	0.00027
6	0.00019	0.00027	0.00027	0.0003
8	0.00027	0.00032	0.0003	0.0003
10	0.00039	0.00042	0.000435	0.0004

depicts the relation between the frequency parameter value ${\lambda }_m$ and the circumferential node number $n$ of cylindrical shells with different layer alignments of 3, 4, 5, and 6 layers. The frequency parameter value decreases until $n=6$ and steadily starts increasing afterwards.

Figure 3 and Figure 4 show the relation between the angular frequency ${\omega }_n$ and length $L$ of the cylindrical shells of 3-, 4-, 5-, and 6-layered cylindrical shells with different layer alignments. The results depict that as the length of the shell increases, the angular frequency deceases. There is a steep decease in angular frequency from $L=0.5$ to $L=1$ and very slight deceases afterwards. Figure 5 and Figure 6 depict the relation between the frequency parameter value ${\lambda }_m$ and circumferential node number $n$ of cylindrical shells with different layer alignments. The frequency parameter value decreases until $n=6$ and gradually starts increasing afterwards, showing that its rigidity also decreases but its flexibility increases until n = 6, whereas its rigidity gradually increases and its flexibility decreases afterwards.

4. Conclusions

The present study explores the frequency response of higher-order shear deformable multilayered angle-ply cylindrical shells under HSDT for S-S boundary conditions. The cylindrical shells’ frequency is examined by varying the shell length, circumferential node number, number of layers, and layer alignment. It is concluded that all these factors affect the frequency of the shells, which, in turn, shows that these factors play a plausible role in manufacturing multilayered cylindrical structures. The research will be helpful for engineers in the related fields.