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Dynamical Analysis of Long Fiber-Reinforced Laminated Plates with Elastically Restrained Edges
This paper presents a variational formulation for the free vibration analysis of unsymmetrically laminated composite plates with elastically restrained edges. The study includes a micromechanics approach that allows starting the study considering each layer as constituted by long unidirectional fibers in a continuous matrix. The Mori-Tanaka method is used to predict the mechanical properties of each lamina as a function of the elastic properties of the components and of the fiber volume fraction. The resulting mechanical properties for each lamina are included in a general Ritz formulation developed to analyze the free vibration response of thick laminated anisotropic plates resting on elastic supports. Comprehensive numerical examples are computed to validate the present method, and the effects of the different mechanical and geometrical parameters on the dynamical behavior of different laminated plates are shown. New results for general unsymmetrical laminates with elastically restrained edges are also presented. The analytical approximate solution obtained in this paper can also be useful as a basis to deal with optimization problems under, for instance, frequency constraints.
Fiber-reinforced composite laminated plates are extensively used in many engineering applications. The free vibration analysis of these plates plays a very important role in the design of civil, aerospace, mechanical, and marine structures. In addition to the favorable high specific strength and high specific stiffness, fiber-reinforced composite laminates offer the possibility of optimal design through the variation of stacking pattern, angle of fiber orientation, fiber content, and so forth, known as composite tailoring. All these mechanical and geometrical characteristics, as well as the various coupling effects that take place, must be considered in the prediction of the laminates dynamical response to assure that this is reliable, accurate, and adequate to the design requirements.
It is well known that laminated composite plates have relatively low transverse shear stiffness, playing the shear deformation an important role in the global and local behavior of these structures. Among the numerous theories used for laminated plates that include the transverse shear strain, the first-order shear deformation theory (FSDT) [1, 2] is adequate for the computation of global responses (such as natural frequencies) and simultaneously has some advantages due to its simplicity and low computational cost. Many investigations have been reported for free vibration analysis of moderately thick composite laminates using the FSDT kinematics (see for instance [3–13]). However, the results are, in most cases, limited to certain lamination schemes and boundary conditions. As far as the study of thick plates with elastically restrained edges is concerned, most of the previous works are limited to isotropic ones ([14–19] among others). But, limited information is found for the case of thick anisotropic laminated plates resting on elastic supports. For instance, Setoodeh and Karami  implemented a layer-wise laminated plate theory linked with three-dimensional elasticity approach for vibration and buckling of symmetric and antisymmetric fiber-reinforced composite plates having elastically restraint edges support and results for cross-ply laminates are presented, whereas Karami et al.  applied the differential quadrature method for the free vibration analysis of moderately thick symmetric laminated plates with elastically restrained edges. For the same boundary conditions, semianalytical solutions for the free vibration of angle-ply symmetrically laminated plates were presented by Ashour . Nallim and Grossi  also studied the vibration of symmetric laminated plates resting on elastic support employing the Ritz method and beam orthogonal polynomials as approximated functions. These kind of approximate functions (in one or two variables) have been used by many authors to the free vibration analysis of, both homogeneous and nonhomogeneous, plates (Chakraverty et al. [24–26] and Chow et al. , among others).
In this paper, a general Ritz formulation for the free vibration analysis of anisotropic laminated plates is developed. All kind of boundary conditions including elastically restrained edges are considered enhancing the study. This feature allows a more realistic analysis of some structural problems. The analysis includes a micromechanical approach (according to the classification of Altenbach et al. ), where the average mechanical properties of each anisotropic lamina are estimated from the known characteristics of the fibers and the matrix materials taking into account the fiber volume ratio and the fiber-packing arrangement. At structural level, the dynamic response of the unsymmetrical laminated plate, with elastically restrained edges, is analyzed using the first-order shear deformation theory and the Ritz method with beam orthogonal polynomials as coordinate functions. The approximate analytical solution developed here is very useful to understand, both qualitatively and quantitatively, the behavior of complex laminated plates.
2.1. Effective Elastic Moduli of Long Fiber-Reinforced Laminae
The micromechanics-based Mori-Tanaka method  is used in this section to predict the elastic mechanical properties of the orthotropic unidirectional laminae. This method may be viewed as the simplest mean field approach for inhomogeneous materials that encompass the full physical range of phase volume fraction.
Eshelby’s results  show that if an elastic homogeneous ellipsoidal inclusion in an infinite linear elastic matrix is subjected to an eigenstrain , uniform strain states is induced, and it is related to the eigenstrain by the expression where is the Eshelby tensor, which depends on the reinforcement dimensions and the Poisson ratio of the matrix . The components of this tensor for a circular, cylindrical inclusion with an infinite length-to-diameter ratio parallel to the 1-axis (parallel to the fiber direction, Figure 1) are The transformations strains are obtained considering the equivalent homogeneous inclusion for inhomogeneous inclusions developed by Eshelby  together with the interaction effects of Mori-Tanaka . These transformations strains are used to equate the total stresses in the inhomogeneities and their equivalent inclusions, as described in the following equation: where and are the stiffness tensors of fiber and matrix, respectively, is the uniform far field strain applied to the domain at infinity, and is the average elastic strain defined by Mori-Tanaka which is given by where is the fiber volume fraction.
Using this method the mechanical properties of unidirectional carbon/epoxy laminae are found considering various fiber volume fractions, and they are depicted in Table 1. These properties, for each unidirectional lamina, are then used in the next section to obtain the reduced constitutive matrix.
2.2. General Laminated Plate Resting on Elastic Supports
Let us consider a rectangular fiber-reinforced composite laminated plate, of dimension and total thickness (, represents the thickness of a layer). The laminated plate is composed of an arbitrary number of orthotropic layers and fibre orientation as shown in Figure 1. A rectangular Cartesian coordinate system is used to represent the plate geometry and the strain–displacement relations. The - plane coincides with the middle plane of the plate. The displacement field of the first-order shear deformation theory is assumed to be of the form [1, 2] where is the time dimension , and denote the mid-surface translational displacements along the , , and axes, and , are the rotations about - and -axes, respectively.
The displacement model (6) yields the following kinematic relations: where The stress-strain relation of each layer is given by the generalized Hooke’s law as follows: where are the components of the plane-stress reduced constitutive matrix  which are function of the elastic constant determined in Section 2.1 and the ply angle .
2.3. Energy Functional Components
Taking into account (7) and (9), the strain energy due to the laminated plate deflection can be written as where is the mid-surface area (Figure 1) and the stiffness coefficients [35, 36] are given by , being the shear correction factors.
The strain energy corresponding to the elastic edge restraints is given by where ( and ) are the elastic translational coefficients and () are the elastic rotational coefficients.
The kinetic energy is expressed as being the mass inertias of the plate defined as  where is the material density of the -th layer.
3. Application of the Ritz Method for the Free Vibration Analysis
The Ritz method is applied to determine analytical approximate solutions for dynamical behavior of arbitrarily laminated plates resting on elastic supports. During free vibration, the displacements components are assumed split in the spatial and temporal parts, being the last one periodic in time; that is, where is the natural frequency in radian.
Putting these displacements into the energy functional components ((10) to (12)) the maximum values of the kinetic energy () and the strain energies () are derived. Then, the energy functional for free vibration of the laminated plate is given by which is to be minimized according to the Ritz principle.
3.1. Boundary Conditions and Approximating Functions
There are some options when choosing the unknown functions of displacement components to apply the Ritz method. Particularly, the use of orthogonal polynomials as coordinate functions has important advantages related to numerical stability and fast convergence as has been demonstrated in previous works [23, 37, 38], even for plates with complicated boundary conditions and high degree of anisotropy. For these reasons, in this work, the displacement components are expressed by sets of beam characteristic orthogonal polynomials , , , resulting in where , , , , are the unknown coefficients, and , are the numbers of polynomials in each coordinate.
The procedure for the construction of the orthogonal polynomials has been developed by Bhat . The first members of the sets, and are obtained as the simplest polynomials that satisfy all the geometrical boundary conditions of the plate in their respective and directions. The higher members of each set are constructed by employing the Gram-Schmidt orthogonalization procedure. The coefficients of the polynomials are chosen in such a way as to make the polynomials orthonormal. However, the functions and for are obtained from relative rotation conditions starting from polynomials of an order lower than the chosen for the transversal displacements and then applying the sequence of Gram-Schmidt orthogonalization procedure. This particular choice is made to avoid the overestimation of the rate of elastic energy due to the shear respect to the rate due to the bending. This concept has been applied by Auciello and Ercolano , to Timoshenko beams, to avoid the shear locking effect and is extended here for laminated plates.
The classical boundary conditions considered in this study are depicted in Table 2. By keeping in mind that in the Ritz method only the geometric boundary conditions need to be satisfied, it is possible to work with any sets of required edge boundary condition and also is very simple the consideration of elastically restrained edges where there are not essential boundary conditions to satisfy.
Upon inserting the displacement forms (16) into the energy functional of the system (15), the minimization with respect to the coefficients of the displacement functions is given by From (17) a set of algebraic simultaneous equations is obtained. The number of these equations becomes . The algebraic equations obtained are given as follows, in the form of the generalized eigenvalue problem: where and are stiffness and inertia matrices, respectively (their expressions are given in the Appendix, contains the unknown coefficients of (16).
For a nontrivial solution, the eigenvalues which make the determinant equal to zero, correspond to the free vibration frequencies.
4. Verification of the Formulation and Numerical Applications
4.1. General Description
The variational algorithm developed in this paper was programmed in Fortran language and used for the free vibration analysis of generally laminated thin and moderately thick laminated plates having different geometric parameters, stacking sequences, material properties, fiber volume fractions, and boundary conditions. The examples considered in this study are confined to laminates with layers of equal thickness, even though the procedure was formulated for plies with arbitrary thickness. In all cases the shear correction factor was taken a 5/6.
Let us introduce the terminology to be used throughout the remainder of the paper for describing the boundary conditions of the considered plates. The designation , for example, identifies a plate with edges (1) clamped, (2) simply supported, (3) free, and (4) simply supported (see Figure 1) the subscript () identifies the in-plane constraints according to Table 2. When the edges are elastically restrained against rotation or translation, the following nondimensional restraint parameters are used
The main purposes of the numerical applications presented in this section are twofold. One is to demonstrate the accuracy, the flexibility, and the efficiency of the proposed method and the other is to produce some results which may be regarded as benchmark solutions for other academic research workers and design engineers.
4.2. Validation and Convergence Studies
The accuracy and reliability of the results obtained with the present approach are next demonstrated by comparing them with some selected values published by Shi et al.  for moderately thick () and thin () arbitrarily clamped laminated plates. The comparison presented in Table 3 authenticates the validity of the present method for arbitrarily laminated plates. Very close agreement for the first sixth nondimensional frequencies is obtained for all cases and display monotonic convergence tendency to constant values. For thick plates, as shown in Table 3, as number of and is increased from 7 to 10, the frequency parameter decreases merely 0.002% for the first mode and 0.26% for the sixth. For thin plates the relative decreases of the frequency parameters are 0.004% for the first mode and 1.88% for the sixth as the numbers of polynomials , are increased from 7 to 10, exhibiting slower convergence rate than that of moderately thick plates. Consequently the number of beam characteristic polynomials used in the following computations for thin and thick plates is chosen as .
4.3. Numerical Results and Discussion
Several examples including new results for arbitrarily laminated plates with elastically restrained edges are presented in this section. The elastic properties of the composite materials used here are those shown in Table 1. The influence of different values of fiber volume ratios () is analyzed in several figures and tables.
Values of the first four frequency parameters for square thick () and thin () unsymmetric laminated plates are shown for increasing values of the translational restraint parameter , in Tables 5(a) and 5(b). Moreover, the influence of rotational restraint parameter in the free vibration frequency coefficients is shown in Tables 6(a) and 6(b).
In Figures 2–4 the fundamental frequency coefficients corresponding to two laminated square plates are plotted against the restraint parameters and . Figure 2 shows the variation of for various values of the rotational restraint , while Figure 3 shows the variation of for various values of the translational restraint . A major increase of frequency occurs when the elastic restraint values are in the interval 0.1–50. Figure 4 shows the variation of for various values of the rotational and translational restraint parameters: (a) , ; (b) , , and (c) . The obtained curves illustrate the restraint parameters intervals for which the frequency coefficient is sensitive to and .
To asses the influence of the aspects ratio in the laminated plate response, values of the first four frequency parameters for rectangular thick () unsymmetric laminated plates are shown, for increasing values of the translational restraint parameter (Table 7(a)) and the rotational restraint parameter (Table 7(b)) considering and .
Figure 5 shows the variation of for various values of the rotational restraint , while Figure 6 shows the variation of for various values of the translational restraint for rectangular laminated plates.
To evaluate the effect of different fiber orientation angles (β) and fiber volume fraction on the dynamic properties of the laminates, the variation of the first free vibration coefficient is plotted in Figures 7 and 8, considering two lamination stacking sequences, [β/−β] and [0/β]. Two boundary conditions have been included, in Figure 7 and in Figure 6. It is observed that the [β/−β] laminate is more sensitive to the fiber orientation angle than [0/β] lamination scheme. The adimensional frequency parameter is noticeable higher as the fiber volume fraction increases and as the boundary conditions become clamped.
Finally, the first four free vibration coefficients are presented in Table 8 to illustrate the influence of various fiber volume fractions and boundary conditions on the dynamical behavior of an unsymmetric [0°/45°] laminated plate.
5. Concluding Remarks
A Ritz approach for free vibration analysis of general laminated plates with edges elastically restrained against translation and rotation is presented in this work. The study includes the effective elastic moduli of each lamina obtained using the Mori-Tanaka mean field theory, which allows taking into account the influence of the fiber volume ratios and the elastic properties of the components (fiber and matrix) into the vibration behavior. The formulation is based on the first-order shear deformation theory, and the generalized displacements are approximate using sets of characteristic orthogonal polynomials generated by the Gram-Schmidt procedure. The consideration of all possible rotational and translational restraints allows generating any classical boundary condition, only approaching the corresponding spring parameter to zero or infinity. The algorithm is computationally efficient, and the solutions are stables and convergent. Close agreement with existing results in the literature is shown and new results are presented in tables and figures which could be useful for design and optimization problems of general long fiber-reinforced laminated plates.
The matrices and in (18) are given by where with
The authors wish to thank the economic support of CONICET (PIP no. 0105/2010) and CIUNSa.
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