Material properties of unidirectional composite
In the past, the model of thin plate on the elastic foundation was mainly used in structural applications. Currently, thin films of metal, ceramic or synthetic materials deposited on the surface of the structural parts of the electronic devices are used to improve their mechanical, thermal, electrical and tribological properties. These thin films of material are considered as thin plates and in these applications, the substrate of thin film can be simulated as an elastic foundation [1-2].
The laminated composite rectangular plate is very common in many engineering fields such as aerospace industries, civil engineering and marine engineering. The ability to conduct an accurate free vibration analysis of plates with variable thickness is absolutely essential if the designer is concerned with possible resonance between the plate and driving force .
Ungbhakorn and Singhatanadgid  investigated the buckling problem of rectangular laminated composite plates with various edge supports by using an extended Kantorovich method is employed.
Setoodeh, Karami  investigated A three-dimensional elasticity approach to develop a general free vibration and buckling analysis of composite plates with elastic restrained edges.
Luura and Gutierrez  studied the vibration of rectangular plates by a non-homogenous elastic foundation using the Rayleigh-Ritz method.
Ashour  investigated the vibration analysis of variable thickness plates in one direction with edges elastically restrained against both rotation and translation using the finite strip transition matrix technique.
Grossi, Nallim  investigated the free vibration of anisotropic plates of different geometrical shapes and generally restrained boundaries. An analytical formulation, based on the Ritz method and polynomial expressions as approximate functions for analyzing the free vibrations of laminated plates with smooth and non-smooth boundary with non classical edge supports is presented.
LU, et al  presented the exact analysis for free vibration of long-span continuous rectangular plates based on the classical Kirchhoff plate theory, using state space approach associated with joint coupling matrices.
Chopra  studied the free vibration of stepped plates by analytical method. Using the solutions to the differential equations for each region of the plate with uniform thickness, he formulated the overall Eigen value problem by introducing the boundary conditions and continuity conditions at the location of abrupt change of thickness. However this method suffers from the drawback of excessive continuity, as in theory the second and third derivatives of the deflection function at the locations of abrupt change of thickness should not be continuous.
Cortinez and Laura  computed the natural frequencies of stepped rectangular plates by means of the Kantorovich extended method, whereby the accuracy was improved by inclusion of an exponential optimization parameter in the formulation.
Bambill et al.  subsequently obtained the fundamental frequencies of simply supported stepped rectangular plates by the Rayleigh–Ritz method using a truncated double Fourier expansion.
Laura and Gutierrez  studied the free vibration problem of uniform rectangular plates supported on a non-homogeneous elastic foundation based on the Rayleigh–Ritz method using polynomial coordinate functions which identically satisfy the governing boundary conditions.
Harik and Andrade  used the “analytical strip method” to the stability analysis of uni-directionally stepped plates. In essence, the stepped plate is divided into rectangular regions of uniform thickness. The differential equations of stability for each region are solved and the continuity conditions at the junction lines as well as the boundary conditions are then imposed.
1.1. The chapter aims
This chapter presents the finite strip transition matrix technique (FSTM) and a semi-analytical method to obtain the natural frequencies and mode shapes of symmetric angle-ply laminated composite rectangular plate with classical boundary conditions (S-S-F-F). The plate has a uniform thickness in x direction and varying thickness h(y) in y direction, as shown in Figure 1. The boundary conditions in the variable thickness direction are simply supported and they are satisfied identically and the boundary conditions in the other direction are free and are approximated. Numerical results for simple-free (S-S-F-F) boundary conditions at the plate edges are presented. The illustrated results are in excellent agreement compared with solutions available in the literature, which validates the accuracy and reliability of the proposed technique.
The equation of motion governing the vibration of rectangular plate under the assumption of the classical deformation theory in terms of the plate deflection W(x, y, t) is given by:
Where is the transverse deflection, ρ = the density per unit area of the plate and h(y) is the plate thickness at any point. The bending and the twisting moments in terms of displacements are given by:
The flexural rigidities of the plate are given by:
Where is the distance from the middle-plane of the plate according to to the bottom of the layer as shown in Figure 1. And are the plane stress transformed reduced stiffness coefficients of the lamina in the laminate Cartesian coordinate system. They are related to reduced stiffness coefficients of the lamina in the material axes of lamina by proper coordinate relationships they can be expressed in terms of the engineering notations as:
Where , are the longitudinal and transverse young's moduli parallel and perpendicular to the fiber orientation, respectively and is the plane shear modulus of elasticity, and are the poisson's ratios. Thus, the governing partial differential equation of laminated composite rectangular plate with variable thickness as shown in Figure 1 is reduced to:
Or in contraction form:
Equation (7) may be written as:
The equation of motion (8) can be normalized using the non-Dimensional variables ξ and η as follows :
Where is the aspect ratio, , , , , and .
3. Method of solution
The displacement can be expressed in terms of the shape function, chosen a prior; and the unknown function as:
The most commonly used is the Eigen function obtained from the solution of beam free vibration under the prescribed boundary conditions at ξ=0 and ξ=1.
The free vibration of a beam of length a can be described by the non-Dimensional differential equation:
Where is the flexural rigidity of the beam. The boundary conditions for free edges beam as shown in Fig. 2 are:
at ξ=0 and ξ=1
In this paper, the beam shape function in ξ-direction is considered as a strip element of the plate and the flexural rigidity of the beam can replaced by and for , it can be just approximated by . The solution of the beam equation is given as:
One can obtain the following system of homogenous linear equations by satisfying the boundary conditions (12) at ξ=0 and ξ=1.
The different value of are the roots of equation:
The roots of equation (15) are represented in the recurrence form:
Where , , , ,
From the orthogonality of the beam Eigen function, for , this is true for all boundary conditions except for plates having free edges in the ξ-direction.
The system of fourth order partial differential equations in equation (17) can be reduced to a system of first order homogeneous ordinary differential equations:
And after some manipulation, the governing differential equation (17) will become:
Where the frame denotes differentiation with respect to η.
Where: , , , ,
i= 0, 1, 2, 3, ……….,N, j= 0, 1, 2, 3, ……….,M
where the coefficients of the matrix in equation (18), in general, are functions of η and the Eigen value parameter λ. The vector is given by:
Solving the above system of first order ordinary differential equations using the transition matrix technique yields, at any strip element (i) with boundaries (i-1) and (i) to,
Where is called the transition matrix of the strip element (i), which can be obtained using the method of system linear differential equations of the strip element (i) in equation (18) (the exact solution of (ODE)).
The boundary conditions at η=0 and η=1 can be expressed as:
Using the assumed solution, equation (10) the boundary conditions can be given by the following equations:
At η=0 and η=1
Or in contraction form:
Where , ,
The solution is found using 2N initial vectors at η=0. Equation (22) is applied across the stripped plate until the final end at η=1 is reached. Thus, 2N solutions , where i= 0, 1, 2, 3, ……….,N, can be obtained. The true solutions can be written as a linear combination of these solutions :
Where are arbitrary constants. These constants can be determined by satisfying 2N boundary conditions at η=1 . The matrix forms a standard Eigen value problem.
4. Numerical results and discussion
In this section, some numerical results are presented for symmetrically laminated, angle-ply variable thickness rectangular plate with simple support in the variable thickness direction and free in the other direction. The designation (S-S-F-F) means that the edges x=0, x=a, y=0, y=b are free, free, simple supported and simple supported respectively. The plates are made up of five laminates with the fiber orientations [θ, - θ, θ, - θ, θ] and the composite material is Graphite/Epoxy, of which mechanical properties are given in Table 1. The Eigen frequencies obtained are expressed in terms of non-dimensional frequency parameter . To illustrate the solution, a plate with linear variable thickness, is used (see Appendix A).
Where is the tapered ratio of plate given by , is the thickness of the plate at η=0 and is the thickness of the plate at η=1. A convergence investigation is carried out for a uniform plate and for plate of variable thickness with aspect ratio . By varying the harmonic numbers of the series solution in equation (10). The results are shown in Table 2. It is found that excellent agreement and stable and fast convergence can be achieved with only a few terms of series solution (N= 3 to 5).
|Material||E1, (GPa)||E2, (GPa)||G12, (GPa)||υ12||E2/ E1||G12/ E1|
In order to validate the proposed technique, a comparison of the results with some results available for other numerical methods  for uniform laminated plates with simple support in the y-direction and free in the other direction. The first six natural frequencies of such uniform laminated plates are depicted in Table 2.
|Δ = 0.0|
Table 3 and Table 4 shows a convergence analysis of the first six frequencies parameters of symmetrically angle-ply five laminates [45/-45/45/-45/45] variable thickness plate with tapered ratio and with aspect ratio with simple support in the y-direction and free in the other direction (S-S-F-F).
Figure 4 and Figure 5 show the mode shapes of the first six fundamental frequencies of the above plate. Figure 4 and Figure 5 both are divided into two graphics. The first one shows the mode shapes of the plate in surface form and the other shows the mode shapes of the plate in surface contour form. All simulation results and graphics were obtained using MATLAB software.
|Δ = 0.5|
|β = 0.5|
|Δ = 0.5|
|β = 1.0|
5. Concluding remarks
A semi-analytical solution of the free vibration of angle-ply symmetrically laminated variable thickness rectangular plate with classical boundary condition (S-S-F-F) is investigated using the finite strip transition matrix technique (FSTM). The numerical results for uniform angle-ply symmetrically square plate with classical boundary condition (S-S-F-F) is presented and compared with some available results. The results agree very closely with other results available in the literature. It can be observed from Tables 2 and 3 that rapid convergence is achieved with small numbers of N in the series solution. Comparing to other techniques, the finite strip transition matrix (FSTM) proves to be valid enough in this kind of application. In all cases the FSTM method is easily implemented in a computer program a yields a fast convergence and reliable results. Also, the effect of the tapered ratio and aspect ratio on the fundamental natural frequencies and the mode shapes for five layers angle-ply symmetrically laminated variable thickness plates has been investigated for two cases of tapered ratio (uniform and variable thickness) and two cases of aspect ratio (square and rectangular). In fact the varying of the thickness and the increase the length (b) about a length (a) tend to increase the natural frequencies and the mode shapes of the laminated plate. The results from this investigation have been illustrated in the three dimensional surface contours for two different aspect ratios.
Plate thickness function
In this appendix the derivation of the relation of the plate thickness h(y) in y-direction as shown in the Figure 6 is given.
By similarity between the triangles (ABG) and (ACF):
By similarity between the triangles (ABG) and (ADE):
From equations (28) and (29) the plate thickness relation is:
Where at ,
Using the assumed solution, equation (10) The relation between the thickness of the plate h(y) can be given by the following equation:
Composite coefficients (function programs)