Free vibration analysis of angle-ply laminate composite beams by mixed finite element formulation using the Gâteaux differential
-
Atilla Ozutok
,
Emrah Madenci
Abstract
Free vibration analyses of angle-ply laminated composite beams were investigated by the Gâteaux differential method in the present paper. With the use of the Gâteaux differential method, the functionals were obtained and the natural frequencies of the composite beams were computed using the mixed finite element formulation on the basis of the Euler-Bernoulli beam theory and Timoshenko beam theory. By using these functionals in the mixed-type finite element method, two beam elements, CLBT4 and FSDT8, were derived for the Euler-Bernoulli and Timoshenko beam theories, respectively. The CLBT4 element has 4 degrees of freedom (DOFs) containing the vertical displacement and bending moment as the unknowns at the nodes, whereas the FSDT8 element has 8 DOFs containing the vertical displacement, bending moment, shear force and rotation as unknowns. A computer program was developed to execute the analyses for the present study. The numerical results of free vibration analyses obtained for different boundary conditions were presented and compared with the results available in the literature, which indicates the reliability of the present approach.
1 Introduction
In recent years, the use of laminated composite beams is increasingly growing in many applications of civil engineering, aerospace, nuclear and automotive industries, because it has some properties favorable in practical design such as high stiffness-to-weight ratios, high modulus-to-weight ratios, corrosion resistance, temperature-dependent behavior and acoustical insulation. The behavior of laminated composite beams under the dynamic loads should be well known by considering its application areas, characteristic properties and structural behaviors.
For the free vibration analysis of the composite beams, researchers have applied various beam theories including Euler-Bernoulli beam theory (CLBT), first-order shear deformation theory (FSDT), higher order shear deformation theory and refined shear deformation theory. In order to solve the involved equations of dynamic analysis, many numerical techniques based on various beam theories have been developed and used by researchers. The most commonly used methods; Lagrange multiplier, analytical method, Ritz method, finite element (FE) method, etc. Krishnaswamy et al. [1] present dynamic equations governing the free vibration of the laminated composite beams by using Hamilton’s principle. Analytical solutions were obtained through the use of the Lagrange multiplier method for the free vibrations of the composite beams such that the effects of shear deformation and inertia were included in the formulation. Abramovich and Livshits [2] studied the free vibration analysis of cross-ply laminated composite beams based on first-order shear deformation theory. It was applied to obtain the equations of motion which include shear deformation, rotary inertia and bending stretching coupling terms, where six different combinations of boundary conditions were considered. Khedir and Reddy [3] developed the analytical solutions of various beam theories to study the free vibration behavior of cross-ply rectangular beams with arbitrary boundary conditions. Banerjee and Williams [4] derived explicit expressions for the exact dynamic stiffness matrix elements of a uniform bending-torsion-coupled Timoshenko composite beam. The theory includes the effects of shear deformation and rotary inertia. Kant et al. [5] studied an analytical solution for the dynamic analysis of laminated beams using higher order refined theory by applying Hamilton’s principle. The theory incorporates cubic axial, transverse shear and quadratic transverse normal strain components in the basic formulation. Aydoğdu [6, 7] investigated the vibration of cross-ply and angle-ply composite beams for six different boundary conditions by using the Ritz method. In the analyses, algebraic polynomial trial functions and a refined higher order shear deformation beam theory were used. In practical engineering, the FE method is very functional for solving the complicated problems of composite beams. Shi and Lam [8] presented a new FE formulation for the free vibration analysis of cross-ply composite beams based on third-order beam theory. Variational consistent stiffness and mass matrices were derived for FE modeling by using the Hamilton principle, where a two-noded higher order composite beam element was used. Bassiouni et al. [9] derived a displacement type of FE for investigation of the dynamic behavior of laminated composite beams which was used to obtain the natural frequencies and mode shapes. Their model included transverse shear deformation. In theoretical analysis, the FE technique was used to construct the mass and stiffness matrices. Chakraborty et al. [10] presented a new refined locking free first-order shear deformable FE model for solving the free vibration and wave propagation problems in laminated composite beams. The exact shape functions were derived arising from exact solutions to the governing equations; thus the exact stiffness matrix and consistent mass matrix and load matrix were obtained by using the shape functions. Jun et al. [11] obtained natural frequencies and mode shapes for cross-ply and angle-ply laminated beams by using dynamic FE method based on the first-order shear deformation. The influences of the Poisson effect, slender ratio, anisotropy, shear deformation and boundary condition were investigated. A dynamic stiffness matrix was formed to solve the free vibration of generally laminated composite beams. Ramos Loja et al. [12] developed a FE model to predict the static and free vibration behavior of anisotropic multi-laminated thick and thin beams with higher order shear deformation theory. The model was based on a single-layer Lagrangian four-node straight beam element with 14 degrees of freedom (DOFs) per node. Thuc and Thai [13] presented the vibration and buckling analyses of composite beams with arbitrary lay-ups using refined shear deformation theory. A two-noded C1 beam element of 5 DOFs per node was developed. By using Hamilton’s principle, three governing equations of motion were derived. A displacement-based FE method was carried out theoretically.
Several different element models are present in the FE method which are known as displacement model, force model, hybrid model and mixed model [14]. There is an apparent superiority of both displacement-type and mixed-type models in engineering solutions, but the mixed-type FE model has greater importance since the forces and moments can be calculated with less number of elements but with more sensitivity. The mixed-type FE method proved itself for static and dynamic analyses of various structural members and is extensively used by researchers [15–18]. Variational methods using the Gâteaux differential (GD) were also adapted to formulate the mixed-type FE method. Akoz et al. [19–26] employed GD to make it functional for various problems. The GD method was successfully applied in obtaining functionality for the static and dynamic analyses of beam, plate, shells, etc. This method was also carried out to solve composite beam problems by some researchers. Kadioglu and Iyidogan [27] investigated the free vibration analysis of cross-ply composite curved beam using the GD method. Natural frequencies were computed using the mixed-type FE formulation based on the Timoshenko beam theory. The FE method consists of a total of 6 DOFs, two displacements, two shear forces, one rotation and one bending moment. Ozutok and Madenci [28] applied the GD method to construct the functional for the free vibration analysis of cross-ply composite beams. The solutions were obtained by minimizing the functional allowing a mixed FE method. Two finite elements having two nodes were developed, one with 4 DOFs and the other with 8 DOFs.
In the present study, the free vibration analysis of angle-ply laminated composite beams was investigated considering different boundary conditions depending on the mixed-type FE formulation using the GD method. The field equations were written in operator form by including the dynamic and geometric boundary conditions. The GD approach was employed to construct the functional using variational methods. With the application of the mixed-type FE method, CLBT4 and FSDT8 beam elements were derived for CLBT and FSDT, respectively, where the CLBT4 element has a total of four unknowns of displacements and bending moments, and the FSDT8 element has a total of eight unknowns of displacements, bending moments, shear forces and rotations. Through the numerical examples, the natural frequencies of angle-ply laminated composite beams were investigated using CLBT4 and FSDT8 finite elements by applying various material properties and boundary conditions. The performance of the elements for free vibration analysis was verified with good accuracy by the solution of the numerical examples present in the literature.
2 Formulation
The laminated composite beam’s coordinate system is formed as in Figure 1. And the total thickness, the lamination thickness and the width properties of the beam with L length are h, hk and b, respectively.
Coordinate system and layer numbering used for a laminated composite beam.
The composite beam was made up of many unidirectional plies stacked up in different orientations with respect to a reference axis.
There are a number of theories used to represent the kinematics for the deflection of beams [29]. The most simple and commonly used theories are CLBT and FSDT, which are based on the displacement field.
For generality purposes, the displacement field in the beam may be assumed to be
where (u, v, w) are the displacements of a point (x, y, z) along the x, y and z coordinates, respectively. w0 and ϕ are the transverse displacement and the rotation of the normal of the mid-plane in the z- and y directions, respectively. The displacement fields in Eq. (1) can be specialized to various beam theories as in the following:
Euler-Bernoulli beam theory (CLBT): c0=-1 and c1=0
Timoshenko beam theory (FSDT): c0=0 and c1=1.
The linear strains associated with the displacement field are
where
2.1 Beam theories
The principle of virtual displacements will be used to derive the governing equations
where the total internal virtual work δU and the virtual work done by applied forces δW are given by
Where the transverse load of q along the length of the beam and the virtual strain δεx are related to the virtual displacement δw as in Eq. (6);
Substituting Eq. (4) into Eq. (3), Eq. (7) is obtained as
and substituting Eq. (6) into Eq.(7), the following Euler-Lagrange equations are obtained
where M and Q are the bending moment and shear force, respectively.
2.2 Stress-strain relations
The stress-strain relationships of composite laminate beams can be defined as being layer by layer. Note that, since the lamination is made up of several orthotropic layers with their material axes oriented arbitrarily with respect to the lamination coordinates, the constitutive equations of each layer must be transformed to the lamination coordinate (x, y, z) as given by Reddy [30]. After the transformation of the lamination coordinates, the relationship of the stresses with the strains in the laminate coordinates can be expressed as
where
The constitutive equations related to the shear force and bending moment can be expressed as
where the shear correction factor K is assumed to be 5/6 for a general lamination as commonly used in the literature. A55 and D11 are the axial stiffness and bending stiffness values, respectively.
The laminate stiffness coefficients A55 and D11 are defined by
where n is the total number of layers, and zk and zk+1 are the lower and upper z coordinates of the kth layer.
The field equations of the composite beams can be written as in the following:
The equilibrium equations related to Eq. (8) are
(12)The constitutive equations related to Eq. (10) are
(13)where
(14)The boundary conditions of the laminated composite beams can be written in symbolic form as follows:
The dynamic boundary conditions are
(15)and the geometric boundary conditions are
(16)
An explicit form of the boundary conditions operator can be obtained after some variational manipulations. The quantities with hat have known values on the boundary R, M, Ω and u, representing the force, moment rotation and deflection vectors, respectively.
2.3 The Gâteaux differential and functional
The field equations including the boundary conditions for composite beams can be written in the operator form as in the following:
Having obtained the field equations, one needs a method to obtain the functional. We believe that the Gâteaux differential method is suitable for this purpose. Since this method was extensively used and explained in other studies [21–28], for the sake of simplicity, the basic steps and definitions will be summarized briefly.
The Gateaux derivative of an operator is defined as
where τ is a scalar quantity. A required and sufficient condition for Q to be a potential is
where parentheses indicate the inner products. If the operator Q is a potential, then the functional corresponding to the field equations will be given as
where s is a scalar quantity. The clear form of the functional corresponding to the field equations is obtained as in the following:
The parentheses with the subscripts σ and ε indicate the dynamic and geometric boundary conditions, respectively. And the terms with “∧” are valid if the boundary conditions are linear; otherwise, they vanish.
2.4 The mixed finite element method
The free vibration behavior of orthotropic angle-ply laminated composite beams is based on the mixed-type FE formulation using the GD method. CLBT4 and FSDT8 beam elements based on Euler-Bernoulli (CLBT) and Timoshenko (FSDT) beam theories are derived by applying the mixed-type FE formulation. They have two nodes and were obtained using linear shape functions.
Initially, the interpolation function should be chosen to derive the mixed FE formulation. All the external known and internal unknown quantities expressed by these interpolation functions are inserted into Eq. (21) where the variables concerning CLBT4 and FSDT8 are w=w(x), M=M(x) and w=w(x), M=M(x), Q=Q(x) and ϕ=ϕ(x), respectively. Both CLBT4 and FSDT8 have first derivatives, and, as seen in Figure 2, a dimensionless and normalized coordinate system defined in local coordinates is used for the shape function. The shape functions in terms of ξ coordinates are given as
Coordinate system for shape functions.
Then, at the element nodes, the variables are expressed as
To give an explicit form of the element matrix for composite beams, the following submatrices are defined:
where i,j=1, 2. These submatrices can be written in expanded form as in the following:
The element matrices of the composite beams can be obtained as in the following by using the expressions in Eq. (24) for the expressions in Eq. (21):
2.5 Free vibration analysis
The problem of obtaining the natural vibration frequencies of a structural system reduces to the solution of a standard eigenvalue problem
where {F} denotes the resultants and {w} indicates the transverse displacement vector. The elimination of {F} from Eq. (27) yields the following equation
where
and [M] and [K*] are defined as the mass matrix and the condensed system matrix of the problem, respectively. The eigenvalue ω2, for which the determinant of the coefficient matrix of Eq. (28) is zero, leads to the free vibration frequencies.
3 Numerical results
Numerical examples were presented for the accuracy verification of the present method and the investigation of the element behavior of angle-ply composite beams in terms of vibration. In the examples, orthotropic, rectangular cross-section angle-ply composite beams were used. The boundary conditions of the beams were denoted by “S” for simply supported edge, “C” for clamped edge and “F” for free edge, and different combinations of these boundary conditions were also taken into consideration. The material properties and the natural frequencies computed in the used non-dimensional form are given as in the following:
Material I
E1=144.8 GPa, E2=9.65 GPa, L/h=15, v=0.3, G12=4.14 GPa and G23=3.45 GPa. ρ=13.89 kN/m3 and non-dimensional form
Material II
E1/E2=open, v=0.25, G12=0.5E2 and G23=0.2E2.ρ=1 and non-dimensional form
In the first example, a convergence study was carried out considering the various boundary conditions, and the effects of the number of finite elements for simply supported symmetric (30°/-30°)s angle-ply composite beams within FSDT8 elements are shown in Figure 3. The material properties were taken as those of material I, and, as seen in Figure 3, the frequency parameters were obtained after five finite elements and compared with the solution of Ref. [10].
Non-dimensional natural frequency convergence study for a symmetric (30°/-30°)s composite beam with SS boundary condition and material I.
In the second example, after a convergence study again using material I properties, the effects of angle orientation on the natural frequencies were investigated, and the corresponding mode shapes were considered. The free vibration analyses of symmetric lay-up (θ°/-θ°/-θ°/θ°) composite beam for various boundary conditions using the CLBT4 and FSDT8 elements were performed, and the results are presented in Table 1. Even here, the results were obtained in compatibility with the results reported using different solutions in the references. It is obvious that the natural frequencies decrease with the increasing angles from 0° to 90°. Since the Euler-Bernoulli beam theory ignores the shear effect, the natural frequencies obtained using the CLBT4 elements were greater than those obtained by the FSDT8 elements in each boundary condition. The first three natural frequency mode shapes were plotted for symmetric (45°/-45°)s FSDT8 beam element with various boundary conditions as shown in Figures 4–7. In general, good agreements were obtained in comparison to the values available in the references.
Comparison of the non-dimensional natural frequencies of symmetric lay-up (θ°/-θ°/-θ°/θ°) composite beams with various boundary conditions and material I.
| Boundary condition | Theory | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
|---|---|---|---|---|---|---|---|---|
| SS | CLBT4 | 2.859 | 2.672 | 2.167 | 1.504 | 0.937 | 0.730 | 0.738 |
| FSDT8 | 2.658 | 2.505 | 2.071 | 1.469 | 0.928 | 0.724 | 0.733 | |
| Ref. [31] | 2.656 | 2.510 | 2.103 | 1.536 | 1.012 | 0.761 | 0.732 | |
| Ref. [10] | 2.657 | 2.511 | 2.096 | 1.526 | – | – | – | |
| Ref. [7] | 2.651 | 1.896 | 1.141 | 0.804 | 0.736 | 0.725 | 0.729 | |
| Ref. [13] | 2.649 | 2.403 | 1.554 | 0.907 | 0.736 | 0.724 | 0.729 | |
| SC | CLBT4 | 4.468 | 4.175 | 3.385 | 2.350 | 1.465 | 1.139 | 1.153 |
| FSDT8 | 3.734 | 3.553 | 3.017 | 2.209 | 1.426 | 1.120 | 1.133 | |
| Ref. [31] | 3.731 | 3.559 | 3.057 | 2.303 | 1.551 | 1.175 | 1.136 | |
| Ref. [10] | 3.734 | 3.562 | 3.050 | 2.289 | – | – | – | |
| Ref. [7] | 3.775 | 2.960 | 1.671 | 1.178 | 1.150 | 1.122 | 1.129 | |
| Ref. [13] | 3.818 | 3.507 | 2.358 | 1.402 | 1.140 | 1.123 | 1.130 | |
| CC | CLBT4 | 6.483 | 6.059 | 4.913 | 3.410 | 2.126 | 1.653 | 1.674 |
| FSDT8 | 4.851 | 4.655 | 4.052 | 3.066 | 2.029 | 1.604 | 1.622 | |
| Ref. [31] | 4.848 | 4.663 | 4.098 | 3.184 | 2.198 | 1.681 | 1.620 | |
| Ref. [10] | 4.857 | 4.670 | 4.092 | 3.170 | – | – | – | |
| Ref. [7] | 4.973 | 4.294 | 2.195 | 1.929 | 1.669 | 1.612 | 1.619 | |
| Ref. [13] | 4.897 | 4.569 | 3.235 | 1.992 | 1.631 | 1.605 | 1.615 | |
| CF | CLBT4 | 1.018 | 0.951 | 0.771 | 0.535 | 0.334 | 0.260 | 0.263 |
| FSDT8 | 0.982 | 0.922 | 0.755 | 0.529 | 0.332 | 0.259 | 0.262 | |
| Ref. [31] | 0.982 | 0.924 | 0.767 | 0.555 | 0.363 | 0.272 | 0.261 | |
| Ref. [10] | 0.982 | 0.925 | 0.765 | 0.551 | – | – | – | |
| Ref. [7] | 0.981 | 0.676 | 0.414 | 0.288 | 0.262 | 0.258 | 0.260 | |
| Ref. [13] | 0.980 | 0.883 | 0.561 | 0.325 | 0.263 | 0.259 | 0.261 |
The first three mode shapes of symmetric (45°/-45°)s composite beam for the SS boundary condition.
The first three mode shapes of symmetric (45°/-45°)s composite beam for the CC boundary condition.
The first three mode shapes of symmetric (45°/-45°)s composite beam for the CF boundary condition.
The first three mode shapes of symmetric (45°/-45°)s composite beam for the SC boundary condition.
In the third example, the natural frequencies of a simply supported anti-symmetric angle-ply (θ°/-θ°) composite beam were investigated using various angle conditions and material I properties. The non-dimensional natural frequency parameters of the first four modes are shown in Table 2 for CLBT4 and FSDT8 beam elements, and compared with the results of Ref. [13]. Additionally, a clamped-clamp unsymmetric (0°/θ°) angle-ply composite beam was selected, and the non-dimensional natural frequency values of the first four modes are presented in Table 3 for CLBT4 and FSDT8 beam elements. And a good agreement was observed.
The first four non-dimensional natural frequencies of a simply supported anti-symmetric angle-ply (θ°/-θ°) composite beam with material I.
| Mode | Theory | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
|---|---|---|---|---|---|---|---|---|
| Mode 1 | CLBT4 | 2.8597 | 2.6724 | 2.1669 | 1.5042 | 0.9376 | 0.7292 | 0.7383 |
| FSDT8 | 2.6580 | 2.5037 | 2.0708 | 1.4689 | 0.9281 | 0.7242 | 0.7331 | |
| Ref. [13] | 2.6494 | 2.4039 | 1.5540 | 0.9078 | 0.7361 | 0.7247 | 0.7295 | |
| Mode 2 | CLBT4 | 11.463 | 10.713 | 8.6862 | 6.0296 | 3.7585 | 2.9231 | 2.9598 |
| FSDT8 | 8.9709 | 8.5780 | 7.3902 | 5.5157 | 3.6118 | 2.8468 | 2.8799 | |
| Ref. [13] | 8.9572 | 8.3223 | 5.7944 | 3.5255 | 2.8798 | 2.8352 | 2.8526 | |
| Mode 3 | CLBT4 | 25.884 | 24.189 | 19.614 | 13.615 | 8.4868 | 6.6003 | 6.6833 |
| FSDT8 | 16.5900 | 16.0610 | 14.3640 | 11.3380 | 7.7872 | 6.2291 | 6.2953 | |
| Ref. [13] | 16.6431 | 15.7685 | 11.8313 | 7.5850 | 6.2626 | 6.1639 | 6.1977 | |
| Mode 4 | CLBT4 | 46.246 | 43.217 | 35.042 | 24.325 | 15.163 | 11.792 | 11.941 |
| FSDT8 | 24.4570 | 23.8720 | 21.9330 | 18.162 | 13.1130 | 10.6750 | 10.7750 | |
| Ref. [13] | 24.7032 | 23.7045 | 18.8714 | 12.7587 | 10.6606 | 10.4930 | 10.5426 |
The first four non-dimensional natural frequencies of a clamped-clamp unsymmetric angle-ply (0°/θ°) composite beam with material I.
| Mode | Theory | 0° | 15° | 30° | 45° | 60° | 75° | 90° |
|---|---|---|---|---|---|---|---|---|
| Mode 1 | CLBT4 | 6.483 | 6.275 | 5.752 | 5.180 | 4.825 | 4.732 | 4.735 |
| FSDT8 | 4.842 | 4.748 | 4.496 | 4.191 | 3.982 | 3.920 | 3.918 | |
| Ref. [13] | 4.897 | 4.742 | 4.272 | 4.009 | 3.950 | 3.938 | 3.935 | |
| Mode 2 | CLBT4 | 17.916 | 17.340 | 15.895 | 14.314 | 13.332 | 13.074 | 13.084 |
| FSDT8 | 10.813 | 10.661 | 10.246 | 9.722 | 9.342 | 9.213 | 9.203 | |
| Ref. [13] | 11.493 | 11.212 | 10.330 | 9.802 | 9.665 | 9.625 | 9.615 | |
| Mode 3 | CLBT4 | 35.256 | 34.121 | 31.279 | 28.168 | 26.236 | 25.728 | 25.748 |
| FSDT8 | 17.797 | 17.594 | 17.039 | 16.327 | 15.793 | 15.596 | 15.571 | |
| Ref. [13] | 18.400 | 18.037 | 16.901 | 16.192 | 15.977 | 15.896 | 15.872 | |
| Mode 4 | CLBT4 | 58.584 | 56.698 | 51.974 | 46.806 | 43.595 | 42.751 | 42.784 |
| FSDT8 | 25.179 | 24.943 | 24.287 | 23.433 | 22.773 | 22.509 | 22.467 | |
| Ref. [13] | 26.448 | 26.011 | 24.637 | 23.743 | 23.437 | 23.306 | 23.264 |
In the last example, with the use of material II properties and a length-thickness ratio of L/h=10, the non-dimensional natural frequencies of eight-layer symmetric simply supported and cantilever angle-ply composite beams were obtained for two angle combinations. The results are listed as in Tables 4 and 5 and compared with Ref. [30] where the frequency parameters for the symmetric (0°/±45°/90°)s laminate were larger than those of laminate (90°/±45°/0°)s. Finally, the effects of modulus ratio (E1/E2) on the natural frequencies of anti-symmetric composite beams with L/h=10 based on FSDT were investigated depending on various boundary conditions. It was observed from Figure 8–10 that the natural frequencies also increase with the increasing orthotropy.
Comparison of non-dimensionalized natural frequencies of a simply supported symmetric angle-ply composite beam with material II (E1/E2=25; L/h=10).
| Angle | Theory | |||
|---|---|---|---|---|
| CLBT4 | Ref. [30] | FSDT8 | Ref. [30] | |
| (0°/±45°/90°)s | 11.826 | 11.236 | 9.696 | 9.367 |
| (90°/±45°/0°)s | 5.461 | 5.054 | 5.195 | 4.843 |
Comparison of non-dimensionalized natural frequencies of a cantilever symmetric angle-ply composite beam with material II (E1/E2=25; L/h=10).
| Angle | Theory | |||
|---|---|---|---|---|
| CLBT4 | Ref. [30] | FSDT8 | Ref. [30] | |
| (0°/±45°/90°)s | 4.209 | 4.002 | 3.793 | 3.721 |
| (90°/±45°/0°)s | 1.944 | 1.800 | 1.898 | 1.772 |
Non-dimensional natural frequencies of simply supported angle-ply composite beam depending on E1/E2 ratio changing with L/h=10.
Non-dimensional natural frequencies of clamped-clamp angle-ply composite beam depending on E1/E2 ratio changing with L/h=10.
Non-dimensional natural frequencies of cantilever angle-ply composite beam depending on E1/E2 ratio changing with L/h=10.
4 Conclusions
In this study, the Gâteaux differential method was employed in the mixed-type FE formulation for the free vibration analysis of laminated composite beams based on CLBT and FSDT. In order to model the laminated composite beams, two different beam elements with two nodes, namely, CLBT4 and FSDT8, were derived each with 4 and 8 DOFs, respectively. A computer program was developed using the FORTRAN computer programming language to carry out the analyses. To demonstrate the accuracy and validity of this study, numerical examples were presented in which the effects of angle orientation, corresponding mode shapes and boundary conditions on natural frequencies were investigated for angle-ply composite beams and compared with the results in the literature. The examples were observed, and the present results were in good agreement with the results of previous studies. It was shown that the Gâteaux differential approach provides an efficient and accurate means for the free vibration analysis of laminated composite beams.
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