Free vibration analysis of laminated FG-CNT reinforced composite beams using finite element method

Abstract

In the present study, the free vibration of laminated functionally graded carbon nanotube reinforced composite beams is analyzed. The laminated beam is made of perfectly bonded carbon nanotubes reinforced composite (CNTRC) layers. In each layer, single-walled carbon nanotubes are assumed to be uniformly distributed (UD) or functionally graded (FG) distributed along the thickness direction. Effective material properties of the two-phase composites, a mixture of carbon nanotubes (CNTs) and an isotropic polymer, are calculated using the extended rule of mixture. The first-order shear deformation theory is used to formulate a governing equation for predicting free vibration of laminated functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beams. The governing equation is solved by the finite element method with various boundary conditions. Several numerical tests are performed to investigate the influence of the CNTs volume fractions, CNTs distributions, CNTs orientation angles, boundary conditions, length-to-thickness ratios and the numbers of layers on the frequencies of the laminated FG-CNTRC beams. Moreover, a laminated composite beam combined by various distribution types of CNTs is also studied.

Keywords

free vibration analysis / laminated FG-CNTRC beam / finite element method / first-order shear deformation theory / composite material

Introduction

Carbon nanotube (CNT), a new advantaged material with exceptional electronic and mechanical properties over carbon fibers [¹], was considered as a newly excellent candidate for reinforcement of composite materials [²–¹⁵]. Moreover, the CNTs were successfully studied in regard to functionally graded (FG) distribution along the thickness direction [¹⁶], which significantly change the mechanical behavior of CNT reinforced composite structures. Owing to these CNT’s advantages, a large number of studies have recently focused on the mechanical analysis of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) structures. For example, recent works done on analysis structural behavior of FG-CNTRC structures (e.g., beams and plates) are reported in Refs. [⁴,¹⁷–³⁴]. A comprehensive review of the mechanical analysis of FG-CNTRC structures is reported by Liew et al. [³⁵].

Due to various exceptional mechanical properties like high strength-to-weight ratio, high stiffness-to-weight ratio, superior fatigue properties, high corrosion resistance, and flexibility in design, composite beams have been widely used in aircraft structures, space vehicles, turbo-machines and other engineering applications. It is well known that composite beams in these applications often operate in complex environmental conditions and are commonly exposed to a variety of dynamic excitations which may result in excessive vibration and fatigue damage [³⁶]. Therefore, to design composite beams working effectively in these applications, the accurate knowledge of the vibration behaviors of composite beams is very importance (see, for example, some references on frequency optimization of composite structures [³⁷–³⁹]). Moreover, for damage assessment of composite beam during their service life, the vibration characteristic of composite beams is really necessary [⁴⁰–⁴²]. In the last few decades, a lot of relevant papers [⁴³–⁵³] have fully addressed the free vibration of the traditional composite beams. Meanwhile, research works pertaining to the free vibration analysis of FG-CNTRC beams [¹⁷,²²,²³,³⁴] have been developed. There still exist some issues that need to be addressed adequately, e.g., the free vibration analysis of multi-layer FG-CNTRC beams and the effect of different stacking of FG-CNTRC lamina.

This paper hence makes an effort to fulfill in the above-mentioned research gaps by studying the free vibration analysis of laminated FG-CNTRC beams. Here, it was considered that the laminated beam is composed of perfectly bonded carbon nanotubes reinforced composite (CNTRC) layers and distinct distributions of single-walled carbon nanotubes (SWCNTs) through the thickness of the layers. Concerning the effective material properties of the laminated nanocomposite beams, the extended rule of mixture is used. In order to build the system of equations ruling the beam deformation, we used the linear two-node element combined with the first-order shear deformation theory. In this work, two types of multilayered composite beams are analyzed including: (1) The multilayered with the same CNT distributions; and (2) the multilayered with various CNT distributions, and the corresponding numerical results are presented. In addition, the effects of CNT distributions, CNT volume fractions, CNT orientation angles, number of layers, length-to-thickness ratios and boundary conditions on the free vibration response of laminated FG-CNTRC beams are also illustrated.

The remainder of the paper is organized as follows. Section 2 presents material properties of the FG-CNTRC materials. Section 3 describes the finite element method for free vibration analysis of the laminated FG-CNTRC beam based on the first-order deformation theory. Section 4 examines some numerical examples. Section 5 draws some conclusions.

Material properties of FG-CNTRC beams

In this study, laminated composite beams composed of many layers of CNTRC materials are considered. In each layer, it is assumed that CNTs are (10,10) armchair SWCNTs and the matrix is supposed to be isotropic and homogeneous. The effective material properties of the two-phase composites, mixture of CNTs and an isotropic polymer, can be defined according to the extended rule of mixtures [¹⁶],

(1)E_{\mathrm{11}}={\eta }_{\mathrm{1}}{V}_{\mathrm{CNT}}{E}_{\mathrm{11}}^{\mathrm{CNT}}+V_m{E}_m,

(2){\displaystyle \frac{{\eta }_{\mathrm{2}}}{E_{\mathrm{22}}}}={\displaystyle \frac{V_{\mathrm{CNT}}}{E_{\mathrm{22}}^{\mathrm{CNT}}}}+{\displaystyle \frac{V_m}{E_m}},

(3){\displaystyle \frac{{\eta }_{\mathrm{3}}}{G_{\mathrm{12}}}}={\displaystyle \frac{V_{\mathrm{CNT}}}{G_{\mathrm{12}}^{\mathrm{CNT}}}}+{\displaystyle \frac{V_m}{G_m}},

(4)v_{\mathrm{12}}=V_{\mathrm{CNT}}{v}_{\mathrm{12}}^{\mathrm{CNT}}+V_m{v}_m,

(5)\rho =V_{\mathrm{CNT}}{\rho }_{\mathrm{CNT}}+V_m{\rho }_m,

where E_{\mathrm{11}}^{\mathrm{CNT}}, E_{\mathrm{22}}^{\mathrm{CNT}} and G_{\mathrm{12}}^{\mathrm{CNT}} are Young’s moduli and shear modulus of the CNTs, E_m and G_m are Young’s modulus and shear modulus of the isotropic matrix, {\eta }_{\mathrm{1}}, {\eta }_{\mathrm{2}} and {\eta }_{\mathrm{3}} are CNT/matrix efficiency parameters which can be determined by matching the elastic modulus of CNTRCs observed from the molecular dynamics (MD) simulation results with the numerical ones obtained from the rule of mixture, v_{\mathrm{12}}^{\mathrm{CNT}} and v_m are Poisson’s ratios of CNTs and matrix, respectively, {\rho }_{\mathrm{CNT}} and {\rho }_m are mass densities of the CNTs and matrix, respectively, V_{\mathrm{CNT}}and V_m, related by V_{\mathrm{CNT}}+V_m=\mathrm{1}, are respectively the volume fractions for CNTs and matrix.

Currently, CNTRC materials have been developed to five distributions of CNTs along the thickness direction, including a uniformly distributed (UD) and four different FG, namely FG-V, FG-Λ, FG-X and FG-O. The volume fraction of CNTs, therefore, depends on its distribution as follows

where V_{\mathrm{CNT}}^{*} is the volume fraction of CNTs that is determined by

(6)V_{\mathrm{CNT}}^{*}={\displaystyle \frac{w_{\mathrm{CNT}}}{w_{\mathrm{CNT}}+{\displaystyle \frac{{\rho }_{\mathrm{CNT}}}{{\rho }_m}}-{\displaystyle \frac{{\rho }_{\mathrm{CNT}}}{{\rho }_m}}{w}_{\mathrm{CNT}}}},

in which w_{\mathrm{CNT}} is the mass fraction of the CNTs in composite beams.

Free vibration of laminated FG-CNTRC beams

In the literature, many papers have been published in recent decades and many mathematical models and solution techniques have been developed for structural analysis (see, for example, Refs. [⁴⁴,⁵⁴–⁷¹]). In these approaches, the finite element method is one of the most popular methods because it is simple to understand and to implement. Besides, the first-order shear deformation theory (FSDT) provides a balance between computational efficiency and accuracy for the global structural behavior of thin and moderately thick composite beams. Under such circumstances, in the present work, the finite element method based on the first order shear deformation theory is used for free vibration analysis of laminated FG-CNTRC beams. More detail on the formulation of this method is presented below.

Let us consider a laminated composite beam consisting of N layers. The size of the beam is characterized by the length L, the width b and the thickness h. A global coordinate Oxyz is attached at the center of the beam such that the x-axis is in the longitudinal direction, as depicted in Fig. 1. Here the bending of the beam on the yz-plane is not considered. In each layer, we denote the fiber orientation angles by q⁽¹⁾, q⁽²⁾, q⁽³⁾, ...., q^(N), the fiber volume fractions by V_{\mathrm{CNT}}^{\mathrm{1}}, V_{\mathrm{CNT}}^{\mathrm{2}}, ..., V_{\mathrm{CNT}}^N, and the vertical coordinate of layers by z₁, z₂, ..., z_N, z_N+1.

Based on FSDT where the influence of shear deformation and rotary inertia is considered, the displacement field of the laminated composite beam is given by

(7)\{\begin{array}{l}u(x,z)=u_{\mathrm{0}}+z{\beta }_x(x)\\ w(x,z)=w_{\mathrm{0}}(x)\end{array},

where u and w are x-direction and z-direction displacements of the beam, respectively, u_{\mathrm{0}} and w₀ are the x-direction and z-direction displacements of the beam neutral axis, respectively, and {\beta }_x is the rotation of the cross section.

According to Eq. (7), the strain field of the beam is expressed as follows

(8)\epsilon =\left[\begin{array}{l}{\epsilon }_x\\ {\gamma }_{xz}\end{array}\right]=\left[\begin{array}{l}{u}_{\mathrm{0},x}+z{\beta }_{x,x}\\ w_{\mathrm{0},x}+{\beta }_x\end{array}\right]\mathrm{.}

If transverse normal stresses are neglected, the constitutive relations for the kth layer of the beam take the form

(9){\sigma }^{(k)}={\overline{Q}}^{(k)}(z){\epsilon }^{(k)},

where

(10){\sigma }^{(k)}=\left[\begin{array}{l}{\sigma }_x^{(k)}\\ {\tau }_{xz}^{(k)}\end{array}\right],{ϵ}^{(k)}=\left[\begin{array}{l}{ϵ}_x^{(k)}\\ {\gamma }_{xz}^{(k)}\end{array}\right],{\overline{Q}}^{\left(k\right)}(z)=\left[\begin{array}{cc}{\overline{Q}}_{11}^{\left(k\right)}(z)& 0\\ 0& {\overline{Q}}_{55}^{\left(k\right)}(z)\end{array}\right],

where

(11){\overline{Q}}_{\mathrm{11}}^{(k)}(z)=Q_{\mathrm{11}}^{(k)}(z){\cos }^{\mathrm{4}}{\theta }^{(k)}+\mathrm{2}(Q_{\mathrm{12}}^{(k)}(z)+\mathrm{2}{Q}_{\mathrm{66}}^{(k)}(z)){\sin }^{\mathrm{2}}{\theta }^{(k)}{\cos }^{\mathrm{2}}{\theta }^{(k)}+Q_{\mathrm{22}}^{(k)}(z){\sin }^{\mathrm{4}}{\theta }^{(k)},

(12){\bar{Q}}_{\mathrm{55}}^{(k)}(z)=Q_{\mathrm{44}}^{(k)}(z){\sin }^{\mathrm{2}}{\theta }^{(k)}+Q_{\mathrm{55}}^{(k)}(z){\cos }^{\mathrm{2}}{\theta }^{(k)}

where the stiffness coefficients Q_{ij}^{(k)}(z) of the kth lamina in the material coordinate system are

(13)Q_{\mathrm{11}}^{(k)}(z)={\displaystyle \frac{E_{\mathrm{11}}^{(k)}(z)}{\mathrm{1}-v_{\mathrm{12}}^{(k)}(z)v_{\mathrm{21}}^{(k)}(z)}},Q_{12}^{(k)}={\displaystyle \frac{v_{12}^{(k)}{E}_{22}^{(k)}(z)}{\mathrm{1}-v_{\mathrm{12}}^{(k)}(z)v_{\mathrm{21}}^{(k)}(z)}},Q_{22}^{(k)}={\displaystyle \frac{E_{22}^{(k)}(z)}{\mathrm{1}-v_{\mathrm{12}}^{(k)}(z)v_{\mathrm{21}}^{(k)}(z)}},Q_{44}^{(k)}(z)=G_{23}^{(k)}(z),Q_{55}^{(k)}(z)=G_{13}^{(k)}(z),Q_{66}^{(k)}(z)=G_{12}^{(k)}(z),

where E_{\mathrm{11}}^{(k)} and E_{\mathrm{22}}^{(k)} are the effective Young’s moduli, G_{\mathrm{12}}^{(k)}, G_{\mathrm{13}}^{(k)} and G_{\mathrm{23}}^{(k)} are the shear moduli, v_{\mathrm{12}}^{(k)}and v_{\mathrm{21}}^{(k)}are Poisson’s ratios. These parameters depend on the CNT volume fraction of the kth layer. The five distributions of CNTs for the kth layer of the laminated CNTRC beam are rewritten by

(14)\{\begin{array}{l}\begin{array}{ll}{V}_{\mathrm{CNT}}(z)=V_{\mathrm{CNT}}^{*}& \mathrm{UD}\end{array}\\ \begin{array}{ll}{V}_{\mathrm{CNT}}(z)=\mathrm{2}{\displaystyle \frac{z-z_k}{z_{k+\mathrm{1}}-z_k}}{V}_{\mathrm{CNT}}^{*}& \mathrm{FG-V}\end{array}\\ \begin{array}{ll}{V}_{\mathrm{CNT}}(z)=-\mathrm{2}{\displaystyle \frac{z-z_{k+\mathrm{1}}}{z_{k+\mathrm{1}}-z_k}}{V}_{\mathrm{CNT}}^{*}& \mathrm{FG-}\Lambda \end{array}\\ \begin{array}{l}\begin{array}{ll}{V}_{\mathrm{CNT}}(z)=\mathrm{4}{\displaystyle \frac{\left|z-{\displaystyle \frac{z_{k+\mathrm{1}}+z_k}{\mathrm{2}}}\right|}{z_{k+\mathrm{1}}-z_k}}{V}_{\mathrm{CNT}}^{*}& \mathrm{FG-X}\end{array}\\ \begin{array}{ll}{V}_{\mathrm{CNT}}(z)=\left(\mathrm{2}-\mathrm{4}{\displaystyle \frac{\left|z-{\displaystyle \frac{z_{k+\mathrm{1}}+z_k}{\mathrm{2}}}\right|}{z_{k+\mathrm{1}}-z_k}}\right)V_{\mathrm{CNT}}^{*}& \mathrm{FG-O}\end{array}\end{array}\end{array}\mathrm{.}

The governing equation for free vibration of the laminated composite beam can be obtained by using the Hamilton’s principle

(15)\delta {\displaystyle \underset{t_{\mathrm{0}}}{\overset{t_{\mathrm{1}}}{\int }}\left[{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \underset{V}{\int }{({\sigma }^{(k)})}^T{\epsilon }^{(k)}dV-{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\displaystyle \underset{V}{\int }{\rho }^{(k)}\left({\dot{u}}^{\mathrm{2}}+{\dot{w}}^{\mathrm{2}}\right)dV}}\right]}dt=\mathrm{0},

where \delta is the notation of variation operator, {\rho }^{(k)} is the mass density of the kth layer and the over-dot represents a partial derivative with respect to time t.

Then the Galerkin weak form of free vibration analysis of the laminated composite beam is given by

(16){\displaystyle {\int }_L\delta {(\Lambda u)}^{T}D(\Lambda u)dx}+{\displaystyle {\int }_L\delta u^{T}m\ddot{u}dx}=\mathrm{0},

where

(17)\Lambda =\left[\begin{array}{lll}{\displaystyle \frac{\partial }{\partial x}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {\displaystyle \frac{\partial }{\partial x}}\\ \mathrm{0}& {\displaystyle \frac{\partial }{\partial x}}& \mathrm{1}\end{array}\right],u=\left[\begin{array}{l}{u}_{\mathrm{0}}\\ w_{\mathrm{0}}\\ {\beta }_x\end{array}\right],

and \ddot{u} is the second-order derivative with respect to time of u. The formulas for D and m are expressed as follows

(18)D=\left[\begin{array}{lll}{A}_{\mathrm{11}}& B_{\mathrm{11}}& \mathrm{0}\\ B_{\mathrm{11}}& D_{\mathrm{11}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \kappa A_{\mathrm{55}}\end{array}\right],m=\left[\begin{array}{lll}{I}_{\mathrm{0}}& \mathrm{0}& I_{\mathrm{1}}\\ \mathrm{0}& I_{\mathrm{0}}& \mathrm{0}\\ I_{\mathrm{1}}& \mathrm{0}& I_{\mathrm{2}}\end{array}\right],

where \kappa =\mathrm{5}/\mathrm{6} is the shear correction factor and A_{\mathrm{11}},B_{\mathrm{11}},D_{\mathrm{11}},A_{\mathrm{55}},I_{\mathrm{0}},I_{\mathrm{1}},I_{\mathrm{2}} are defined by

(19)(A_{\mathrm{11}},B_{\mathrm{11}},D_{\mathrm{11}})=b{\displaystyle \sum _{k=\mathrm{1}}^{nL}{\displaystyle {\int }_{z_k}^{z_{k+\mathrm{1}}}{\overline{Q}}_{\mathrm{11}}^{(k)}(z)(\mathrm{1},z,z^{\mathrm{2}})dz}},A_{\mathrm{55}}={\displaystyle \sum _{k=\mathrm{1}}^{nL}{\displaystyle {\int }_{z_k}^{z_{k+\mathrm{1}}}{\overline{Q}}_{\mathrm{55}}^{(k)}(z)dz}},(I_{\mathrm{0}},I_{\mathrm{1}},I_{\mathrm{2}})=b{\displaystyle \sum _{k=\mathrm{1}}^{nL}{\displaystyle {\int }_{z_k}^{z_{k+\mathrm{1}}}{\rho }^{(k)}(z)(\mathrm{1},z,z^{\mathrm{2}})dz}}\mathrm{.}

By using finite element method, the beam is divided into Ne elements and each element has two nodes. On each element, the displacement field of the beam, u^e, is approximated by linear shape functions in the natural coordinate, N_{\mathrm{1}}^e(\xi )={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}(\mathrm{1}-\xi ) and N_{\mathrm{2}}^e(\xi )={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}(\mathrm{1}+\xi ), as follows:

(20)u^e=N^e{d}^e

in which d^e={\left[\begin{array}{llllll}{u}_{\mathrm{01}}& w_{\mathrm{01}}& {\beta }_{x\mathrm{1}}& u_{\mathrm{02}}& w_{\mathrm{02}}& {\beta }_{x\mathrm{2}}\end{array}\right]}^T is the nodal displacement vector of element e and

(21)N^e=\left[\begin{array}{llllll}{N}_{\mathrm{1}}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2}}^e& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& N_{\mathrm{1}}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2}}^e& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& N_{\mathrm{1}}^e& \mathrm{0}& \mathrm{0}& N_{\mathrm{2}}^e\end{array}\right]\mathrm{.}

After substituting Eq. (20) into Eq. (16), the discrete equation for free vibration analysis of the laminated composite beam is represented by

(22)M\ddot{d}+Kd=\mathrm{0}

where d is the displacement vector, \ddot{d}is the second-order derivative with respect to time of the displacement, M and K are the global mass matrix and stiffness matrix which are the assemblies of elemental ones and given by

(23)\{\begin{array}{l}K={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{K}^e}={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{\displaystyle \underset{-\mathrm{1}}{\overset{\mathrm{1}}{\int }}{(B^e)}^{T}D{B}^e{\displaystyle \frac{l^e}{\mathrm{2}}}d\xi }}\\ M={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{M}^e}={\displaystyle \sum _{e=\mathrm{1}}^{Ne}{\displaystyle \underset{-\mathrm{1}}{\overset{\mathrm{1}}{\int }}{(N^e)}^{T}m{N}^e}{\displaystyle \frac{l^e}{\mathrm{2}}}d\xi }\end{array}

where l^e is the length of the eth element and B^e is defined by

(24)B^e=\left[\begin{array}{llllll}{N}_{\mathrm{1},\xi }^e\left({\displaystyle \frac{\mathrm{2}}{l^e}}\right)& \mathrm{0}& \mathrm{0}& N_{\mathrm{2},\xi }^e\left({\displaystyle \frac{\mathrm{2}}{l^e}}\right)& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& N_{\mathrm{1},\xi }^e\left({\displaystyle \frac{\mathrm{2}}{l^e}}\right)& \mathrm{0}& \mathrm{0}& N_{\mathrm{2},\xi }^e\left({\displaystyle \frac{\mathrm{2}}{l^e}}\right)\\ \mathrm{0}& N_{\mathrm{1},\xi }^e\left({\displaystyle \frac{\mathrm{2}}{l^e}}\right)& N_{\mathrm{1}}^e& \mathrm{0}& N_{\mathrm{2},\xi }^e\left({\displaystyle \frac{\mathrm{2}}{l^e}}\right)& N_{\mathrm{2}}^e\end{array}\right]\mathrm{.}

To ensure the well-behaved element performance that is devoid of shear locking in slender beams, a selective reduced integration scheme [⁷²,⁷³] is used. The idea of the reduced integration scheme is to split the strain energy into two parts: The bending- related term and the shear-related one. Two different integration rules are then used, respectively, for the bending strain energy and the shear strain energy. In this study, a single Gaussian point is used for calculating the bending stiffness (with respect to the bending strain energy) and transverse shear stiffness (with respect to shear strain energy). More detail on this technique, interesting readers can refer to Refs. [⁷²,⁷³].

The natural frequency (w) and mode shape (f) of the beam are obtained by solving the eigenvalue problem which is derived from Eq. (22) as follows

(25)(\mathit{K}-{\omega }^{\mathrm{2}}\mathit{M})\varphi =\mathrm{0.}

The support conditions for clamped (C), hinged (H) and free (F) at the ends x = 0, L are

(26)\{\begin{array}{l}(C)\hspace{1em}{u}_{\mathrm{0}}=w_{\mathrm{0}}={\beta }_x=\mathrm{0}\\ (H)\hspace{1em}{u}_{\mathrm{0}}=w_{\mathrm{0}}=\mathrm{0}\\ (F)\hspace{1em}\mathrm{no\; contraints}\end{array}\mathrm{.}

It should be mentioned that in the previous work by Lin and Xiang [³⁴], they analyzed the free vibration of FG-CNT beams. However, this study only limited to single layer-FG-CNTRC beams. Therefore, their finite element formulation is simpler than the present formulation while the effects of CNT orientation angles (Eq. (10)) and the change of CNT distributions in each layer (as in Eq. (14)) are neglected in their work.

Numerical examples

Comparison study

As mentioned before, to the best of our knowledge there is no work reported on free vibration analysis of laminated FG-CNTRC beams. Therefore, to assure the validity of free vibration analysis for single-layer FG-CNTRC beams and multilayered composite beams, two comparison studies are given in this section.

For the first comparison study, the fundamental frequency parameter of a single layer FG-CNTRC beam is evaluated and compared with the results of Yas and Samadi [²] using the generalized differential quadrature method (GDQM) and Lin and Xiang [³⁴] using the p-Ritz method. The material properties of the FG-CNTRC beam at room temperature (300 K) are the same as those used in study of Yas and Samadi [²] in which poly methyl methacrylate, referred to PMMA, is considered as the matrix and the armchair (10,10) SWCNTs are selected as reinforcements. Details of the parameters are listed in Table 1. It is also assumed that G_{\mathrm{23}}^{(k)}(z)=G_{\mathrm{13}}^{(k)}(z)=G_{\mathrm{12}}^{(k)}(z) and v_{\mathrm{21}}^{(k)}(z)=v_{\mathrm{12}}^{(k)}(z). Three different values of CNTs volume fraction and their corresponding CNT/matrix efficiency parameters are given as follows: V_{\mathrm{CNT}}^{*} = 0.12, h₁ = 1.2833 and h₂ = 1.0556; V_{\mathrm{CNT}}^{*} = 0.17, h₁ = 1.3414 and h₂ = 1.7101; and V_{\mathrm{CNT}}^{*} = 0.28, h₁ = 1.3238 and h₂ = 1.7380. In addition, it is assumed that h₃ = h₂. The beam for length to thickness ratio (L/h) of 15 is divided into 32 elements equal of lengths for finite element analysis. The obtained dimensionless frequencies with various CNT volume fractions, boundary conditions and types of distributions are provided in Tables 2 and 3. It can be seen that the present results are in good agreement with those solved in Refs. [²,³⁴]. This result demonstrates the reliability of the finite element analysis for the FG-CNTRCs beams.

For the second comparison study, the dimensionless frequency of a four-layered angle-ply [q/–q/–q/q] beam with four different types of boundary conditions and various CNTs orientation angles is calculated from the present approach, and the obtained results are compared with those by Chandrashekhara et al. [⁴³] and Nguyen et al. [⁵⁰]. The beam is made of AS/3501-6 graphite-epoxy materials whose properties are used the same as in Refs. [⁴³,⁵⁰]. For finite element analysis, the beam is divided into 32 elements equally. The frequencies for ratio L/h = 15 of the present study and the reference are shown in Table 4. It can be observed that the difference between the results is very small.

Parametric study

After performing comparison studies, parametric studies are done in this section to analyze the influences of involved parameters on the free vibration of a laminated FG-CNTRC beam. In this study, the material and geometrical parameters of the beam are the same as in the first comparison study of the previous examples. It should be noted that the length-to-thickness ratio is chosen to be 15 for all examinations except for the case of examination the effect of length-to-thickness ratio on the frequencies.

Table 5 presents the dimensionless frequencies of the cross-ply [0°/90°] FG-CNTRC beams for the case of

V_{\mathrm{CNT}}^{*}

= 0.12 with different boundary conditions. It can be seen that the trend of frequency looks very similar to the case of only one layer, as shown in Table 2, i.e., the FG-X, UD, FG-V and FG-O beams have decreasing frequency, respectively for all boundary conditions. In this case, however, there is a small difference when the first frequency of the FG-O beam is greater than that of the FG-V beam for boundary conditions of CH and HH.

Table 6 shows the dimensionless frequencies of the cross-ply [0°/90°/0°] FG-CNTRC beams for the case of

V_{\mathrm{CNT}}^{*}

= 0.12 with different boundary conditions. Similar to the results in Table 2, the FG-X beam yields the largest frequency while the FG-O beam has the smallest frequency for all boundary conditions.

Tables 7–10 provide the dimensionless frequency of four-layered angle-ply [q/–q/–q/q] FG-CNTRC beams for the case of V_{\mathrm{CNT}}^{*} = 0.12 with various CNTs orientation angles and different types of boundary conditions. In addition, the effect of CNTs orientation angle on frequency for other cases of CNT volume fraction is also performed and illustrated in Fig. 2 where the FG-X beam with boundary condition of CC is examined. As can be observed, the frequency decreases as the CNTs orientation angle q decreases from 90° to 0° for all distributions of CNT, boundary conditions and for three values of CNTs volume fraction.

Figure 3 shows the dimensionless frequency of the angle-ply [45°/–45°/–45°/45°] FG-CNTRC beam with different distributions and boundary conditions for the case of V_{\mathrm{CNT}}^{*} = 0.17. It can be seen from the figure that the frequencies of FG-X beam are the largest; however, the difference of frequencies between FG-X beam and the other beams is small for all boundary conditions.

Figure 4 shows the dimensionless frequency of the angle-ply [45°/–45°/–45°/45°] FG-X beam with clamped-clamped boundary condition and various length-to-thickness ratios. As can be seen from Fig. 4, the dimensionless frequency of the beam is reduced when the length-to-thickness ratio increases.

Table 11 shows the dimensionless frequency of a laminated FG-CNTRC beam for the case of V_{\mathrm{CNT}}^{*} = 0.12 with various types of distributions, boundary conditions and numbers of layers. The CNTs orientation angle of each lamina is 0°. It is seen from Table 11 that when the number of layers increases the frequency of the FG-V and FG-O beams increase while that of the FG-X beam decreases. Moreover, the frequency of these beams converges to that of the UD beam. The illustration of this comment can be seen in Fig. 5 that depicts the first dimensionless frequency of these beams for the boundary condition of CC.

Table 12 lists the dimensionless frequency of the three-layered [0°/0°/0°] beams that are combined by different types of CNTs distributions for the case of V_{\mathrm{CNT}}^{*} = 0.12. The combined beams are named by joining the letter of distributions of each layer. For examples, UD-X-UD means that the first and third layers have uniform distributions and the second layer has FG-X distribution. From the results in Table 12, it can be recognized that the Λ-X-V beam yields the largest frequency while the V-X-V beam produces the smallest frequency in comparison with X-UD-X and X-V-X beams. Also, the frequency of the X-UD-X and X-V-X beams is nearly the same. It can also be seen that the frequency of these combined beams is higher than that of the initial UD-UD-UD and V-V-V beams for all boundary conditions.

Conclusions

The paper presents the free vibration analysis of laminated functionally graded carbon nanotube reinforced composite beams. Here, it was considered that the laminated beam is composed of perfectly bonded carbon nanotubes reinforced composite (CNTRC) layers and distinct distributions of single-walled carbon nanotubes (SWCNTs) through the thickness of the layers. In order to build the system of equations ruling the beam deformation, the linear two-node element combined with the first-order shear deformation theory is used. The influence of the CNTs volume fractions, CNTs distributions, CNTs orientation angles, boundary conditions, the numbers of layers and length-to-thickness ratios on the free vibration of a laminated FG-CNTRC beam is investigated. Moreover, a laminated composite beam combined by various CNTs distributions is also studied. The numerical results indicate that

i) The FG-X distribution yields the largest frequency while the FG-O distribution yields the smallest frequency of laminated FG-CNTRC beams.

ii) The CNT orientation angle of 0° for each layer yields the largest frequency of the laminated FG-CNTRC beams.

iii) When the number of layers increases, the frequency of FG-X distribution decreases while the frequency of the FG-V and FG-O increases. In addition, the frequencies of other CNT distributions tend to approach to the frequency of the UD distribution.

iv) The stacking sequence of CNT distributions has little influence on the frequency of the FG-CNTRC beams. Also, the stacking sequence of FG-Λ-X-V gives the largest frequencies.

v) Boundary conditions, CNT volume fractions and length-to-thickness ratios strongly effect on the frequency of laminated FG-CNTRC beams.

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