Frequency optimization of laminated annular circular plates

1 Introduction

Composite laminated structures are widely used in the fields of aerospace, automotive, and other engineering industries as they have considerably more stiffness and strength when compared to the structures formed using a single material. Present-day engineers are mainly focusing on the design of composite structures due to their high specific stiffness, better damping and absorbing characteristics.

Laminated annular circular plates have found wide applications as structural members in aerospace, marine, and other industries. In the literature, free vibrations of laminated annular circular plates have been extensively studied. For example, Lin and Tseng [1] studied the free vibration analysis of polar orthotropic laminated circular and annular plates. The effects of material property, stacking sequence, hole size, plate thickness to radius ratio, and boundary conditions on natural frequencies were investigated. Viswanathan and Sheen [2] investigated the free vibration of layered annular circular plates of variable thicknesses, made up of isotropic or specially orthotropic materials using spline function approximation by applying the point collocation method. Malekzadeh et al. [3] presented the free vibration of thick laminated circular and annular plates supported on the elastic foundation using a three-dimensional layerwise-finite element method. Viswanathan et al. [4] studied the asymmetric free vibrations of annular cross-ply circular plates using spline function approximation. Wang and Chen [5] considered the natural frequencies and modal loss factors of the three-layered annular plate with a viscoelastic core layer and two polar orthotropic laminated face layers. Ding and Xu [6] established the state-space equation of the axisymmetric vibration of laminated annular plates composed of transversely isotropic layers based on the basic equations of the three-dimensional theory of elasticity. Liu et al. [7] presented a finite element to analyze the three-dimensional vibration of piezoelectric laminated circular and annular plates. Yeh et al. [8] investigated the finite element analysis of polar orthotropic annular plate with an electrorheological fluid core and constraining layer. Greenberg and Lavan [9] presented the free vibrational response of homogeneous or symmetrically laminated polar orthotropic annular plates subjected to non-uniform boundary conditions on the inner and outer rims using a finite-difference method. Chen and Chen [10] considered the non-axisymmetric vibration and stability problem of the rotating sandwich plate by using the finite element method.

To the best of author’s knowledge, the frequency optimization of laminated annular circular plates has not been investigated yet. Therefore, in this study, the frequency optimization of laminated annular circular plates is investigated to fill this gap. The design objective is the maximization of the fundamental frequency, and the design variable is the fiber orientation in the layers. The first-order shear deformation theory and nine-node isoparametric finite element model in the polar coordinates are used in finding the natural frequencies. The modified feasible direction (MFD) method is used for the optimization routine. For this purpose, a program based on FORTRAN is used. Finally, the numerical analysis is carried out to investigate the effects of radius ratios, boundary conditions, and uniform external or internal pressure loads on the optimal designs, and the results are shown in tabular and graphical forms.

2 Basic equations

Consider a composite laminated annular circular plate with an arbitrary number of layers, which are perfectly bonded together. Figure 1 shows the geometry of a laminated annular circular plate. In the figure, a is the inner radius, b is the outer radius of the annular circular plate, and h is the thickness of the plate.

Figure 1

Geometry of a laminated annular circular plate.

The displacement field of the plate based on the first-order shear deformation theory is given by the following expressions

where u_o,v_o, and w are the displacements of the midsurface, and Ψ_r and Ψ_θ are the shear rotations of any point on the middle surface of the plate.

The stress-strain relations for a single lamina in the annular circular plate are given by

where

The kinematics relations in terms of the polar coordinates r, θ, and z can be expressed as

The stress resultants {N}, stress couples {M}, and transverse shear stress resultants {Q} are

In Eq. (5), K is the shear correction factor. In this study, the shear correction factor is taken as 5/6.

Using Eqs. (2), (3), and (4) into Eq. (5), one can obtain the equations of stress-resultants and displacements in the form

where A_ij, B_ij, and D_ij are extensional, coupling, and bending stiffnesses, respectively, which are defined in terms of the lamina stiffness

where z_k and z_k+1 denote the distances from the plate reference surface to the outer and inner surfaces of the kth layer.

3 Finite element formulation

In this study, a nine-node Lagrangian rectangular plate element, which is 5 degrees of freedom (u, v, w, Ψ_r, Ψ_θ), is used for the finite element solution of the laminates. The interpolation function of the displacement field is defined as

where d_i and N_i are the nodal variables and the interpolation function, respectively. The problem of free undamped vibrations of a finite element discretized linearly elastic structure is defined by the eigenvalue problem

where [K], [M], and λ=ω² are the stiffness matrice, mass matrice, and eigenvalue, respectively. Eq. (9) is a set of homogeneous linear equations in the unknown displacements {u}. For the non-trivial solution, the determinant is equal to zero, and the eigenvalues correspond to the natural frequencies of the laminated plates. The subspace iteration method is used for the frequency analysis. The obtained smallest natural frequency (fundamental frequency) is used as an objective function and will be designed to maximize its value in the present optimization problem.

4 Modified feasible direction method

The MFD method is one of the most powerful methods in optimization problems. This method takes into account not only the gradients of objective function and constraints but also the search direction in the former iteration. In this study, there is not any constraint. Figure 2 shows the iterative process within each optimization process. The details of the modified feasible direction method can be found in Topal [11] and Topal and Uzman [12].

Figure 2

Flow chart of the modified feasible direction method.

5 Optimization problem

In this study, the optimization problem is the maximization of the fundamental frequency by designing the fiber orientations in the layers. The optimal design problem can be stated mathematically as follows:

The fundamental frequency for a given fiber orientation is determined from the finite element solution of the eigenvalue problems given by Eq. (9). The optimization procedure involves the stages of evaluating the fundamental frequency and improving the fiber orientation θ to maximize ω. Thus, the computational solution consists of successive stages of analysis and optimization until a convergence is obtained, and the optimal angle θ_opt is determined within a specified accuracy.

6 Numerical results and discussion

6.1 Validation and convergence of the present study

In this section, the convergence behavior of the present study is investigated, and comparisons with other available solutions are made to verify the accuracy of the results. First, the first example is chosen to examine the finite element solution of the present study on the free vibration problems of isotropic annular plates. The convergence of the results for the annular circular plates with the outer edge is simply supported and the inner edge free boundary conditions (S–F) for different h/b ratios and a/b=0.2. The dimensionless frequency parameter is given below:

It can be shown from Table 1 that excellent solution agreements can be observed between the presented method and those of the other methods.

Table 1

Comparison of the present finite element solutions with the literature results for the isotropic annular plates.

h/b	Han and Liew [13]	Civalek and Gürses [14]	Present study
0.05	4.7087	4.7103	4.7088
0.1	4.6819	4.6823	4.6823
0.2	4.5805	4.5806	4.5817

In the second example, the free vibration of cross-ply (0°/90°) annular plates is investigated for different boundary conditions (a/b=0.5, h/a=0.05). The following boundary conditions are used to analyze the problem:

• Clamped-Clamped (C-C) (both ends are clamped),

• Simply supported-Simply supported (S-S) (both ends are simply supported),

• Clamped-Free (C-F) (outer edge clamped and inner edge free).

In the second example, E-glass/epoxy material is considered for the numerical results. The material properties are given below:

E₁=38.6 GPa, E₂=8.27 GPa, G₁₂=4.14 GPa, ν₁₂=0.26, ρ=1800 kg/m³

The dimensionless frequency parameter is given below:

where I₁ is the normal inertia coefficient. As seen from Table 2, the present study can yield results that are in close agreement with the literature results.

Table 2

Convergence study of the present finite element solutions with the literature results for cross-ply (0°/90°) annular plates.

Boundary conditions	Viswanathan et al. [4]	Present study
(C-C)	0.866186	0.861378
(S-S)	0.468485	0.461955
(C-F)	0.166496	0.165008

6.2 Optimization problem

In this study, the optimization problem is solved for the four-layered angle ply symmetric (θ/-θ/-θ/θ) annular plates for different parameters. Each of the lamina is assumed to be of the same thickness. The optimization results are given for T300/5208 graphite/epoxy material. The material properties are given below:

E₁=181 GPa, E₂=10.3 GPa, G₁₂=G₁₃=7.17 GPa, G₂₃=2.39 GPA, ν₁₂=0.28, ρ=1600 kg/m³

In this study, the effect of the radius ratios (a/b) on the optimum results is investigated for different boundary conditions (h/b=0.1). Six different boundary conditions are used to analyze the optimization problem given below:

• Clamped-Clamped (C-C) (both ends are clamped),

• Simply supported-Simply supported (S-S) (both ends are simply supported),

• Clamped-Simply supported (C-S) (outer edge clamped and inner edge simply supported),

• Simply supported-Clamped (S-C) (outer edge simply supported and inner edge clamped),

• Clamped-Free (C-F) (outer edge clamped and inner edge free),

• Simply supported-Free (S-F) (outer edge simply supported and inner edge free).

The non-dimensional fundamental frequency is defined as follows:

As seen from Figure 3, as the radius ratio (or hole size) increases, the fundamental frequency increases. This phenomenon that the fundamental frequencies increase with an increase in the hole size may seem strange. However, previous research showed that introducing a hole into a composite structure does not always reduce the fundamental frequency and, in some instances, may increase its fundamental frequency. This is because the fundamental frequency of a composite structure is not only influenced by the hole but also by the material orthotropy, boundary condition, structural geometry, and their interactions. On the other hand, the maximum and minimum fundamental frequencies occur for (C-C) and (S-F) boundary conditions, respectively.

Figure 3

Effect of the radius ratio on the fundamental frequency for laminated annular circular plate.

In Table 3, the effect of the radius ratio on the optimum fiber orientation for the laminated annular circular plate for different boundary conditions is illustrated. As seen, the radius ratio has no effect on the optimum fiber orientations for (C-C), (C-S), (C-F), and (S-F) boundary conditions.

Table 3

Effect of the radius ratio on the optimum fiber orientation for the laminated annular circular plate.

Boundary conditions	θopt(°)
	a/b
	0.2	0.4	0.6	0.8
(C-C)	90	90	90	90
(S-S)	55	72	69	69
(C-S)	90	90	90	90
(S-C)	62	66	66	68
(C-F)	90	90	90	90
(S-F)	0	0	0	0

In Figures 4–7, the mode shapes are given for the different a/b ratios and boundary conditions for the laminated annular circular plates (h/b=0.1).

Figure 4

Mode shapes for the laminated annular circular plates (a/b=0.2).

Figure 5

Mode shapes for the laminated annular circular plates (a/b=0.4).

Figure 6

Mode shapes for the laminated annular circular plates (a/b=0.6).

Figure 7

Mode shapes for the laminated annular circular plates (a/b=0.8).

In this study, the effect of the uniform external or internal pressure loads on the optimum design is investigated for both the ends of the clamped laminated annular circular plates (a/b=0.2, h/b=0.1). First, the critical buckling loads are calculated for uniform external and internal pressure loads, respectively. The generalized form of the buckling eigenvalue problem using finite element discretization can be written as

where [K] and [K_g] are the stiffness matrice and geometric matrice, respectively. For a non-trivial solution, the eigenvalues (λ_b), which make the determinant to be equal to zero, correspond to the critical buckling loads. The subspace iteration technique is used for the buckling solution of the laminates. After that, the fundamental frequencies are calculated by considering uniform external or internal pressure loads. The stiffness matrice [K] in Eq. (9) can be separated into two matrices as

In Figure 8, the effect of the different uniform external or internal pressure loads (N=0, N=0.2N_cr, N=0.4N_cr, N=0.6N_cr) on the fundamental frequency is given. As seen from Figure 8, as the uniform pressure load increases, the fundamental frequencies decrease. On the other hand, the fundamental frequencies of the uniform internal pressure load are higher than those of the uniform external pressure load. However, the optimum fiber orientations are obtained θ_opt=90° for all the different uniform external or internal pressure loads.

Figure 8

Effect of the different uniform external or internal pressure loads on the fundamental frequency.

7 Conclusions

This paper deals with the frequency optimization of symmetrically laminated angle-ply annular circular plates. The design objective is the maximization of the fundamental frequency, and the design variable is the fiber orientation in the layers. Initially, preliminary studies have been carried out by solving problems given in the literature for free vibration, and the results have been compared with the best known results available in the literature. The numerical results show that as the radius ratio increases, the fundamental frequency increases. This is because the fundamental frequency of a composite structure is not only influenced by the hole but also by the material orthotropy, boundary condition, structural geometry, and their interactions. The maximum and minimum fundamental frequencies occur for (C-C) and (S-F) boundary conditions, respectively. The radius ratio has no effect on the optimum fiber orientations for (C-C), (C-S), (C-F), and (S-F) boundary conditions. As the uniform pressure load increases, the fundamental frequencies decrease. On the other hand, the fundamental frequencies of the uniform internal pressure load are higher than those of the uniform external pressure load.