Vibration Analysis of Laminated Composite Beams Using a Novel Two-Variable Model with Various Boundary Conditions

Abstract

The kinematics of the beam having only two variables are increased in a hybrid form under polynomial, and trigonometric series in thickness and axial directions, respectively. Lagrange's equations are then used to derive characteristic equations of the beams. Numerical results for laminated composite beams are equalled with previous studies and are used to investigate the effects of length-to-depth ratio, fibre angles and material anisotropy on the vibration of laminated composite beams.

1 Introduction

Laminated composite materials are created by assembling multiple layers of fibrous materials to achieve the superior engineering properties such as bending stiffness, strength to weight ratio and thermal performance.

As a result, laminate composite has been widely applied in aerospace engineering, mechanical engineering as well as construction technology. In order to maximise the potential advantage of this multilayered material, numerous studies and computation modelling have been conducted to fine-tune the static and dynamic behaviours of laminated composite beams. Various beam theories have been developed in order to predict accurately their structural responses and capture anisotropy of laminated composite materials.

Classical beam theory (CBT) is the simplest one in analysing responses of laminated composite beams. Nonetheless, this theory underestimates deflections and overestimates natural frequencies of the beams due to neglecting effects of transverse shear deformation.

In order to account for this effect, thanks to its simplicity in formulation and programming, the first-order shear deformation beam theory (FSBT) is commonly used by researchers and commercial soft wares for the analysis of laminated composite beams [1, 2]. However, in this theory, the inadequate distribution of transverse shear stress in the beam thickness requires a shear correction factor to calculate the shear force.

This adverse in practice could be overcome by using higher-order deformation beam theory (HSBT) [2, 3] or Quasi-3D beam theory (Quasi-3D) [4, 5] owing to the higher-order variation of axial displacement or both axial and transverse displacements, respectively. In such an approach, stresses of the beam can be directly computed from constitutive equations without shear coefficient requirement.

Many higher-order shear deformation theories have been developed with different approaches in which its kinematics could be expressed in terms of polynomial [6, 7], trigonometric [8, 9], exponential ones [10], hyperbolic [11, 12] and hybrid higher-order shear functions [13].

A literature review shows that a vast number of researches on development HSBT and Quasi-3D have been developed, however the accuracy of these theories strictly depends on the choice of shear functions and number of variables defining the problem. The development of new beam theories as well as suitable solution methods is a complicated problem and needs to be studied further.

The purpose of this paper is to develop a two-directional elasticity solution for vibration analysis of laminated composite beams. Based on the elasticity equations, the proposed theory only requires two unknowns in which the axial and transverse displacements are approximated in series terms in its two in-plane directions for different boundary conditions and Lagrange’s equations are used to derive characteristic equations.

Numerical results are presented to investigate the effects of length-to-depth ratio, material anisotropy, Poisson's ratio and fiber angles of laminated composite beams.

2 Theoretical Formulation

Considering a laminated composite beam with rectangular section b x h and length L as shown in Fig. 1, the beam is composed of n layers of orthotropic materials.

Fig. 1.

Geometry of laminated composite beams.

2.1 Kinematic, Strain and Stress Relations

Denoting $u$ and $w$ are axial and transverse displacements at location ($x,\,z$) of the beam. The linear displacement-strain relations of the beam are given by:

$$ \varepsilon_x = u_{,x} $$

(1)

$$ \varepsilon_z = w_{,z} $$

(2)

$$ \gamma_{xz} = u_{,z} + w_{,x} $$

(3)

where the comma indicates partial differentiation with respect to the coordinate subscript that follows. Based on an assumption of the plan stress in the plane ($x,\,z$) of the beam, i.e. $\sigma_y = \sigma_{yz} = \sigma_{xy} = 0$, the elastic constitutive equation in the global coordinate system is expressed by:

$$ \left\{ \begin{gathered} \sigma_x \hfill \\ \sigma_z \hfill \\ \sigma_{xz} \hfill \\ \end{gathered} \right\} = \left[ {\begin{array}{*{20}c} {\overline{\overline{C}}_{11} } & {\overline{\overline{C}}_{13} } & 0 \\ {\overline{\overline{C}}_{13} } & {\overline{\overline{C}}_{33} } & 0 \\ 0 & 0 & {\overline{\overline{C}}_{55} } \\ \end{array} } \right]\left\{ \begin{gathered} \varepsilon_x \hfill \\ \varepsilon_y \hfill \\ \gamma_{xz} \hfill \\ \end{gathered} \right\} $$

(4)

where $\overline{\overline{C}}_{11}$, $\overline{\overline{C}}_{13}$ and $\overline{\overline{C}}_{55}$ are the reduced in-plane and out of plane elastic stiffness coefficients of the laminated composite beam in the global coordinates (see [5] for more details).

2.2 Variational Formulation

Lagrangian function is used to derive the equations of motion:

$$ \Pi = U - K $$

(5)

where $U$, and $K$ denote the strain energy, and kinetic energy, respectively.

The strain energy $U$ of a system is given by:

$$ \begin{gathered} U = \frac{1}{2}\int_V {\left( {\sigma_x \varepsilon_x + \sigma_z \varepsilon_z + \sigma_{xz} \gamma_{xz} } \right)} dV \hfill \\ \,\,\,\,\,\, = \frac{1}{2}\int_V {\left[ {\overline{\overline{C}}_{11} u_{,x}^2 + 2\overline{\overline{C}}_{13} u_{,x} w_{,z} + \overline{\overline{C}}_{33} w_{,x}^2 + \overline{\overline{C}}_{55} \left( {u_{,z}^2 + 2u_{,z} w_{,x} + w_{,x}^2 } \right)} \right]} dV \hfill \\ \end{gathered} $$

(6)

The kinetic energy $K$ is obtained as:

$$ K{ = }\frac{{1}}{{2}}\int_V {\rho \left( {\dot{u}^2 + \dot{w}^2 } \right)} dV $$

(7)

where the differentiation with respect to the time t is denoted by dot-superscript convention; $\rho \left( z \right)$ is the mass density of each layer.

By substituting Eq. (6) and (7) into Eq. (5), Lagrangian function is explicitly expressed as:

$$ \begin{gathered} \Pi = \frac{1}{2}\int_V {\left[ {\overline{\overline{C}}_{11} u_{,x}^2 + 2\overline{\overline{C}}_{13} u_{,x} w_{,z} + \overline{\overline{C}}_{33} w_{,x}^2 + \overline{\overline{C}}_{55} \left( {u_{,z}^2 + 2u_{,z} w_{,x} + w_{,x}^2 } \right)} \right]} dV \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - \frac{{1}}{{2}}\int_V {\rho \left( {\dot{u}^2 + \dot{w}^2 } \right)} dV \hfill \\ \end{gathered} $$

(8)

2.3 Two-Directional Ritz Solution

Based on the Ritz method, the axial and transverse displacements at location ($x,\,z$) of the beam can be generally approximated in the following forms:

$$ u(x,z,t) = \sum_{r = 1}^R {\sum_{s = 1}^S {\psi_{rs} (x,z)u_{rs} } } $$

(9)

$$ w(x,z,t) = \sum_{r = 1}^R {\sum_{s = 1}^S {\varphi_{rs} (x,z)w_{rs} } } $$

(10)

where $u_{rs}$,$w_{rs}$ are unknown displacement values to be determined; $\psi_{rs} (x,z)$, $\varphi_{rs} (x,z)$ are the two-directional shape functions which are composed of admissible hybrid exponential-trigonometric function in the x-axis and polynomial function in the z-axis are given in Table 1.

Table 1. Shape functions and essentials BCs Beams.

The governing equations of motion can be obtained by substituting Eq. (9), (10) into Eq. (8) and using Lagrange’s equations:

$$ \frac{\partial \Pi }{{\partial q_{rs} }} - \frac{d}{dt}\frac{\partial \Pi }{{\partial \dot{q}_{rs} }} = 0 $$

(11)

with $q_{rs}$ representing the values of ($u_{rs}$, $w_{rs}$), that leads to:

$$ \left( {\left[ {\begin{array}{*{20}c} {{{\varvec{K}}}^{11} } & {{{\varvec{K}}}^{12} } \\ {^T {{\varvec{K}}}^{12} } & {{{\varvec{K}}}^{22} } \\ \end{array} } \right] - \omega^2 \left[ {\begin{array}{*{20}c} {{\text{M}}^{11} } & {0} \\ {0} & {{\text{M}}^{22} } \\ \end{array} } \right]} \right)\left\{ \begin{gathered} {{\varvec{u}}} \hfill \\ {{\varvec{w}}} \hfill \\ \end{gathered} \right\} = \left\{ \begin{gathered} {0} \hfill \\ {0} \hfill \\ \end{gathered} \right\} $$

(12)

where the components of the stiffness matrix K and the mass matrix M are given by:

$$ \begin{gathered} K_{rspq}^{11} = \int_0^L {\int_{ - h/2}^{h/2} {\overline{\overline{C}}_{11} \psi_{rs,x} \psi_{pq,x} bdxdz} } + \int_0^L {\int_{ - h/2}^{h/2} {\overline{\overline{C}}_{55} \psi_{rs,z} \psi_{pq,z} bdxdz} } \hfill \\ K_{rspq}^{12} = \int_0^L {\int_{ - h/2}^{h/2} {\overline{\overline{C}}_{13} \psi_{rs,x} \varphi_{pq,z} bdxdz} } + \int_0^L {\int_{ - h/2}^{h/2} {\overline{\overline{C}}_{55} \psi_{rs,z} \varphi_{pq,x} bdxdz} } , \hfill \\ K_{rspq}^{22} = \int_0^L {\int_{ - h/2}^{h/2} {\overline{\overline{C}}_{33} \varphi_{rs,z} \varphi_{pq,z} bdxdz} } + \int_0^L {\int_{ - h/2}^{h/2} {\overline{\overline{C}}_{55} \varphi_{rs,x} \varphi_{pq,x} bdxdz} } , \hfill \\ M_{rspq}^{11} = \int_0^L {\int_{ - h/2}^{h/2} {\rho \psi_{rs} \psi_{pq} bdxdz} } ,\,M_{rspq}^{22} = \int_0^L {\int_{ - h/2}^{h/2} {\rho \varphi_{rs} \varphi_{pq} bdxdz} } \hfill \\ \end{gathered} $$

(13)

Finally, the vibration responses of the laminated composite beams can be determined by solving Eq. (12). It should be noted that the Eq. (12) does not consider damping materials, the investigation of vibration frequencies of damping materials [14] is also a very interesting problem and will be the future development of this paper.

3 Numerical Results

In this section, convergence and verification studies are carried out to demonstrate the accuracy of the present study. For vibration analysis, laminates are assumed to be equal thicknesses and made of the same orthotropic materials whose properties are given in Table 2. For convenience, the following non-dimensional terms are used:

$$ \overline{\omega } = \frac{\omega L^2 }{h}\sqrt {\frac{\rho }{E_2 }} \,\,{\text{for Material I }}\left( {\text{MAT I}} \right){\text{ and}}\,\,\overline{\omega } = \frac{\omega L^2 }{h}\sqrt {\frac{\rho }{E_1 }} \,{\text{for MAT II}} $$

(14)

Table 2. Material properties of composite beams.

The composite beams (MAT I, 0⁰/90⁰/0⁰, L/h = 5, E₁/E₂ = 40) with different BCs are considered to evaluate the convergence. The non-dimensional fundamental frequencies with respect to the number of series in $x - {\text{direction}}$ $(R)$ and $z - {\text{direction}}$ are given in Table 3. It can be seen that the responses converge quickly in x-direction and number of series in this direction $R = 10$ can be the point of convergence of the fundamental frequencies for the boundary conditions, whereas the frequency of vibration tends to decrease with increasing number of series in z-direction, and the beam tends to become softer. As an example for further verification, $R = 10$ and $S = 4$ will be chosen in the following examples.

Table 3. Convergence studies for normalized fundamental frequencies of 0⁰/90⁰/0⁰ laminated composite beams (MAT I, L/h = 5, E₁/E₂ = 40).

Vibration behaviors of cross-ply laminated composite beams are investigated in Table 4 which presents changes of the non-dimensional fundamental frequencies with S-S, C-F and C-C boundary conditions, span-to-thickness ratio $L/h = 5, \, 10, \, 50$ of the 0°/90°/0° and 0°/90° laminated composite beams. The solutions are computed with MAT I and $E1/E2 = 40$. The accuracy of the solutions is tested by verification with those derived from HSBTs (Nguyen et al. [3], Nguyen et al. [5], Khdeir et al. [17], Vo, T.P et al. [18], Murthy et al. [19]) and Quasi-3Ds (Nguyen et al. [5], Mantari et al. [20], Matsunaga [21]). It can be seen that the present solutions comply with those from the Quasi-3Ds, however there are slight deviations between them for the thick beams $(L/h = 5)$ and for C-F and C-C boundary conditions. The softer characteristic of the present beam model is again found for all solutions in comparison with the HSBTs and Quasi-3Ds.

Table 4. Non-dimensional fundamental frequencies of 0⁰/90⁰/0⁰ and 0⁰/90⁰ laminated composite beams (MAT I,$E_1 /E_2 = 40$).

In order to verify the vibration behaviors of present theory further, TableS 5 and Table 6 report non-dimensional fundamental frequencies with respect to the fiber angle. The results are obtained with three boundary conditions, two lay-ups 0°/θ^o/0° and 0°/θ^o, material MAT I, $E1/E2 = 40$ and $L/h = 5$. It is worth noticing that the solutions are compared with those of Nguyen et al. [5] based on a Quasi-3D theory. It is observed that there are significant differences between two models for C-C boundary conditions, whereas the solution field of the theories is in agreement for S-S and C-F boundary conditions.

The unsymmetrical (45⁰/-45⁰/45⁰/-45⁰) and (30⁰/-50⁰/50⁰/-30⁰) composite beams (MAT II) with various BCs are considered. The results of fundamental frequencies are given in Fig. 2 and Fig. 3. A good agreement between the present solutions and previous studies (Nguyen et al. [5], Chen et al. [22], Chandrashekhara et al. [23]) is again found.

Table 5. Non-dimensional fundamental frequencies of 0⁰/θ /0⁰ and 0⁰/θ laminated composite beams (MAT I, $E_1 /E_2 = 40$, $L/h = 5$).

Table 6. Non-dimensional fundamental frequencies of 0⁰/θ/0⁰ and 0⁰/θ laminated composite beams (MAT I, $E_1 /E_2 = 40$, $L/h = 10$).

Fig. 2.

Non-dimensional fundamental frequencies of 45⁰/-45⁰/45⁰/-45⁰ laminated composite beams (MAT II).

Fig. 3.

Non-dimensional fundamental frequencies of 30⁰/-50⁰/50⁰/-30⁰ laminated composite beams (MAT II).

Finally, the symmetric $\left( {\theta / - \theta } \right)_s$ composite beams (MAT II) are considered. The effects of the fiber angle on the natural frequencies is illustrated in Table 7. It can be seen that the present natural frequencies are closer to those of HSBT (Nguyen et al. [5] and Aydogdu [24]) smaller than those of HSBT (Nguyen et al. [3]), which neglected the Poisson’s effect, especially for $10^0 \le \theta \le 60^0$. This phenomenon can be explained by the fact that Poisson’s effect is incorporated in the constitutive equations by assuming $\sigma_y = \sigma_{xy} = \sigma_{yz} = 0$. It means that the strains $\left( {\varepsilon_y ,\,\gamma_{yz} ,\gamma_{xy} } \right)$ are nonzero and this causes the beams more flexible. This indicated that the Poisson’s effect is quite significant to composite beams with arbitrary lay-ups, and neglecting this effect is only suitable for cross-ply composite beams.

Table 7. Non-dimensional fundamental frequencies of $\left( {\theta / -\theta } \right)_s$ laminated composite beams (MAT II).

4 Conclusions

The authors proposed a new two-unknown model for vibration analysis of laminated composite beams. The axial and transverse displacements of the beam are expanded in a hybrid form under polynomial, and trigonometric series. Lagrange's equations are used to derive characteristic equations of the beams. Numerical results for laminated composite beams with different boundary conditions are compared with previous studies and investigate effects of length-to-depth ratio, material anisotropy, Poisson's ratio and fiber angles on the natural frequencies of laminated composite beams. The obtained results show that the normal strain effects are significant for un-symmetric and thick beams. The Poisson's effect is also important for composite beams with arbitrary lay-ups, and thus omitting this effects is only suitable for the cross-ply ones. The present model is found to be appropriate for vibration analyses of laminated composite beams.