A General Ritz Algorithm for Static Analysis of Arbitrary Laminated Composite Plates using First Order Shear Deformation Theory

This paper is concerned with the bending of laminated composite plates with arbitrary lay-up and general boundary conditions. The analysis is based on the small deflection, first-order shear deformation theory of composite plates, which utilizes the Reissner-Mindlin plate theory. In solving the aforementioned plate problems, a general algorithm based on the Ritz method and the use of beam orthogonal polynomials as coordinate functions is derived. This general algorithm provides an analytical approximate solution that can be applied to the static analysis of moderately thick laminated composite plates with any lamination scheme and combination of edge conditions. The convergence, accuracy, and flexibility of the obtained general algorithm are shown by computing several numerical examples and comparing some of them with results given in the literature. Some results, including general laminates and nonsymmetrical boundary conditions with free edges, are also presented.


Introduction
The advantages of composite materials over conventional ones have been well-established because of their high strength-to-density and stiffness-to-density ratios.These advantages are especially desirable in aerospace, aircraft, automotive, and other industrial applications.Among the different types of composite materials, fiber-reinforced laminates are very important as structural plate components.The classical laminate plate theory (CLPT) (Lekhnitskii 1968), which neglects the effects of out-of-plane strains, has been widely used for obtaining the mechanical response of several composite plates.However, composite laminated plates have relatively low transverse shear stiffness making, in consequence, the transverse shear strains more noticeable for laminated plates than for isotropic plates.CLPT predicts the response of thin isotropic plates with reasonable accuracy, but it usually fails to yield similar accuracy for composite plates of a similar configuration.Because of this, CLPT often under-predicts deflections and over-predicts natural frequencies and buckling loads, even in thin laminated plates.
There are numerous plate theories that include the transverse shear strains in the analysis eg. (Noor and Burton 1989;Maiti and Sinha 1996;Liew et al. 1996;Reddy 2003).The first-order shear-deformation theory (FSDT) proposed by (Reissner 1945 andMindlin 1951) assumes that planes normal to the midplane remain straight, but not necessarily normal to it after deformation.Since FSDT accounts for a constant state of transverse shear stresses, shear correction coefficients are needed to rectify the unrealistic variation of the shear strain/stress through the thickness and which ultimately define the shear strain energy.Some other plate theories, such as the higher order shear deformation theories (HSDT), include the effect of transverse shear strains (Khdeir and Reddy 1989;Reddy 2003;Xiao et al. 2008;Oktem and Chaudhuri 2008) and do not need the use of shear correction coefficients in computing the transverse shear stresses.
Notwithstanding the limitations of FSDT, from an engineering point of view it is still a very attractive approach due to its simplicity and low computational cost.It is well recognized that while FSDT is adequate for global structural behavior (eg.transverse deflections, fundamental vibration frequencies, critical buckling loads, force and moment resultants), it is not adequate for accurate prediction of local response parameters, such as the interlaminar stress distributions (Qi and Knight 1996).
For problems involving different common and refined plate theories, several numerical and analytical approximate solutions have been proposed (Tessler 1993;Nguyen et al. 2005;Daghia et al. 2008;Fares and Elmarghany 2008;Xiang 2009;Bodaghi and Saidi 2010).Most papers are limited to certain edges support and only to cross-ply or angle-ply laminated plates.Closed-form solutions for laminated composite plates using FSDT have been provided for some simple boundary conditions and some particular lamination schemes (Whitney 1987;Reddy 2003).Other analytical solutions obtained by the Ritz method employ trigonometric and hyperbolic beam functions to form the approximate shape functions.However, that approach for plates with general anisotropy leads to resultant moments that appear to be oscillating about a relative constant value, as has been shown by (Nallim and Grossi 2003).
In this study, FSDT was employed in conjunction with the Ritz method and sets of beam characteristic orthogonal polynomials for obtaining a generalized approximate analytical solution to study the static behavior of arbitrary composite laminated plates with all possible combinations of boundary conditions.All coupling effects including, stretching-bending, stretching-shear, and bending-twisting were considered in the formulation.The accuracy of the present method, is shown through the convergence tests for selected plate problems and selected results are compared with those published by other authors.Few selected cases are solved and the corresponding deflections and resultant moments are presented by means of plots and in tabular form.Results include laminated plates with general lamination sequences, non-symmetric boundary conditions and free edges.
The algorithm developed offers an interesting alternate from an engineering view point to perform design analysis.It can be applied to the analysis of a wide range of rectangular laminated plates with different boundary conditions, number of layers, stacking sequences, and fiber orientation.

Energy Functional Components
A rectangular fiber reinforced composite laminated plate of uniform thickness h is shown in Fig. 1.The plane x, y coincides with the middle surface along the plate thickness, while z remains normal to it.In each layer of the laminate, denotes the angle of fiber orientation and the major and minor principal material axes are denoted by 1 and 2, respectively.
In the first order shear deformation theory, the plate kinematics is governed by the midplane displacements u o , v o , w o and rotations x , y given by: (4)

Boundary Conditions and Approximating Functions
The situation for asymmetrically laminated plates is more complex than for symmetrical ones because the transverse and in-plane displacements are coupled.Actually, there are four types of boundary conditions that can be called simply supported (S), clamped (C), or free edges (F), and the combinations of these constraints are summarized in Table 1.
In the application of the Ritz method only the essential boundary conditions are required to be satisfied by the assumed functions.The use of beam orthogonal polynomials to study anisotropic plates is very satisfactory, as has been demonstrated by Nallim et al. (2003Nallim et al. ( , 2005Nallim et al. ( , 2008)), since the convergence of the solution is rapid and practically without oscillations.In the present work, the five displacement components are expressed by sets of beam characteristic orthogonal polynomials as follows: (5) where B k , C k are obtained using the orthogonality property of the p k (x) polynomials: The same procedure is used for y variable.

Application of the Ritz Method
The total energy functional for the static bending analysis of laminated plates is given by: (6) The Ritz procedure requires the minimization of the energy functional (6) with respect to each of the unknown coefficients: Table 1.Notations for various combinations of boundary conditions, in which n and s indicate the normal and tangential directions to the respective plate edges Substituting Eq. ( 5) into the expression of the functional Eq. ( 6), and subsequently applying Eq. ( 7) results in the following governing equation: where {f} is the load vector and [K] is the syammetric stiffness matrix given by: and the unknown coefficient vector is: (10) The elements of the stiffness matrix are given in Appendix A.

Numerical Results
A number of numerical examples are presented in this section to demonstrate the performance of the general algorithm developed for bending analysis of laminated plates with various boundary conditions, spanto-thickness ratios, fiber orientations, etc.The following terminology is used for describing the plate boundary conditions.For instance, S i S i C i C i identifies a plate with edges simply supported along x = 0, a, and edges clamped along y = 0, b.The subscript i (i=1,…,4) indicates the in-plane constraints according to Table 1.The results for square laminated composite plates are compared with (Moleiro et al. 2008).The gtraphite/epoxy material with properties are used.
The reference flexural stiffness is

Validation and Convergence Studies
Comparison and convergence studies were conducted to evaluate the accuracy of the present formulation.Table 2 shows results for square angle-ply anti-symmetric laminates ( / -) with a S 1 S 1 C 1 C 1 boundary conditions and thickness ratio.The results are given in the form of deflection, bending moments at the plate center, and axial forces at one of the corners of the plate.The number of polynomials (M, N ) are stepped steadily from 5 to 8 to demonstrate the convergence.In the same table the accuracy and reliability of the present results are demonstrated by comparison with (Sheikh et al. 2002) who used a high precision triangular element considering the effect of shear deformation by taking transverse displacement and bending rotations as independent field variables.The comparison shows a very good agreement.
Table 3 shows results of deflections, and moment and transverse force resultants for simply supported cross ply (0/90) square plates with uniform loading.The lamina properties are E 1 / E 2 = 25, G 12 / E 2 = G 13 / E 2 = 0.5, G 23 / E 2 = 0.2, v 12 = 0.25.The results for different thickness ratios were compared with those published by (Moleiro et al. 2008) and a very good agreement was observed.

Numerical Examples
The developed Ritz formulation is applied in this section to obtain the plate deflections and moment resultants of general laminated plates with several combinations of boundary conditions and stacking sequences.The capability and generality of the present approach is shown through selected examples depicted in Table 4 and Figs The effect of span to thickness ratio on center deflection for symmetrical and anti-symmetrical cross-ply and angle-ply laminates with S 1 S 1 C 2 C 2 boundary conditions is shown in Fig. 5.The nondimensional deflection decreased as b / h increased, but the decrease was small (beyond b / h = 40).The value of the non-dimensional deflection became, as expected, practically constant for thin plates.Also, the difference between the symmetrical and anti-symmetrical lay-up values was small for cross-ply laminates and much larger for angle-ply laminates.
Finally, antisymmetrical two layered laminates ( / -), with a fiber orientation varying between 0°a nd 45° were analyzed when simply supported (S 1 S 1 S 1 S 1 ), fully clamped (C 2 C 2 C 2 C 2 ), and when mixed (S 1 S 1 C 2 C 2 ) boundary conditions were used.Variations of non-dimensional deflection are shown in Fig. 6.A change in fiber orientation led to an increase in the non-dimensional deflection in case of fully clamped plates, while for simply supported and mixed boundary conditions, the non-dimensional deflection showed a convex shape.

Conclusions
A general algorithm for static analysis of thick, arbitrarily laminated composite plates with multiple combination of boundary conditions is presented in this    The five components of the displacement field were approximated by sets of characteristic orthogonal polynomials generated using the Gram-Schmidt procedure.The algorithm developed was very general and allowed the analysis of moderately thick plates with different materials, laminate lay-ups, span-to-thickness ratios, and boundary conditions.The computational implementation is also very simple, and the formulation is such that we could appreciate how the various geometrical and mechanical parameters involved influenced the static response.
Consequently, the developed formulation constitutes an efficient tool in design and optimization problems.Comparisons of the present results with a variety of other published results were made with very good agreement.Some typical results were given for crossply, angle-ply, and arbitrarily laminated plates with various thickness ratios ranging from moderately thick to thin, and for different boundary conditions.ite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method.Composites Part B: Engineering 39:414-427.
. 2-4.Two different combinations of boundary conditions, three thickness ratios, and four lamination schemes have been chosen as examples.

Figure 2 .
Figure 2. Contour plot of moment resultants for different laminated plates with S 1 S 1 C 3 C 3 boundary conditions

Figure 4 .Figure 5 .Figure 3 .
Figure 4. Contour plot of moment for different laminated plates with C 1 F 1 S 4 S 4 boundary

Table 4 .
Laminated plates with S 1 S 1 C 3 C 3 , S 1 S 1 F 4 F 4 , S 1 F 4 C 3 C 3 and C 1 F 1 S 4 S 4 boundary conditions