A General Ritz Algorithm for Static Analysis of Arbitrary ...

TJER Vol. 10, No. 2, 1-12

__________________________________________*Corresponding author’s e-mail: [email protected]

A General Ritz Algorithm for Static Analysis of ArbitraryLaminated Composite Plates using First Order Shear

Deformation TheoryRF Rango*, FJ Bellomo and LG Nallim

INIQUI-CONICET, Faculted de Ingenierla, Universidad Nacional de salta, 4400-Salta, Argentina

Received 16 April 2011; accepted 3 July 2012

Abstract: This paper is concerned with the bending of laminated composite plates with arbitrary lay-upand general boundary conditions. The analysis is based on the small deflection, first-order shear defor-mation theory of composite plates, which utilizes the Reissner-Mindlin plate theory. In solving the afore-mentioned plate problems, a general algorithm based on the Ritz method and the use of beam orthogo-nal polynomials as coordinate functions is derived. This general algorithm provides an analytical approx-imate solution that can be applied to the static analysis of moderately thick laminated composite plateswith any lamination scheme and combination of edge conditions. The convergence, accuracy, and flexi-bility of the obtained general algorithm are shown by computing several numerical examples and com-paring some of them with results given in the literature. Some results, including general laminates andnonsymmetrical boundary conditions with free edges, are also presented.

Keywords: Plate bending, General laminated composite plates, First order shear deformation theory

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RF Rango, FJ Bellomo and LG Nallim

1. Introduction

The advantages of composite materials over con-ventional ones have been well-established because oftheir high strength-to-density and stiffness-to-densityratios. These advantages are especially desirable inaerospace, aircraft, automotive, and other industrialapplications. Among the different types of compositematerials, fiber-reinforced laminates are very impor-tant as structural plate components. The classical lam-inate plate theory (CLPT) (Lekhnitskii 1968), whichneglects the effects of out-of-plane strains, has beenwidely used for obtaining the mechanical response ofseveral composite plates. However, composite lami-nated plates have relatively low transverse shear stiff-ness making, in consequence, the transverse shearstrains more noticeable for laminated plates than forisotropic plates. CLPT predicts the response of thinisotropic plates with reasonable accuracy, but it usual-ly fails to yield similar accuracy for composite platesof a similar configuration. Because of this, CLPT oftenunder-predicts deflections and over-predicts naturalfrequencies and buckling loads, even in thin laminatedplates.

There are numerous plate theories that include thetransverse shear strains in the analysis eg. (Noor andBurton 1989; Maiti and Sinha 1996; Liew et al. 1996;Reddy 2003). The first-order shear-deformation theo-ry (FSDT) proposed by (Reissner 1945 and Mindlin1951) assumes that planes normal to the midplaneremain straight, but not necessarily normal to it afterdeformation. Since FSDT accounts for a constant stateof transverse shear stresses, shear correction coeffi-cients are needed to rectify the unrealistic variation ofthe shear strain/stress through the thickness and whichultimately define the shear strain energy. Some otherplate theories, such as the higher order shear deforma-tion theories (HSDT), include the effect of transverseshear strains (Khdeir and Reddy 1989; Reddy 2003;Xiao et al. 2008; Oktem and Chaudhuri 2008) and donot need the use of shear correction coefficients incomputing the transverse shear stresses.

Notwithstanding the limitations of FSDT, from anengineering point of view it is still a very attractiveapproach due to its simplicity and low computationalcost. It is well recognized that while FSDT is adequatefor global structural behavior (eg. transverse deflec-tions, fundamental vibration frequencies, criticalbuckling loads, force and moment resultants), it is notadequate for accurate prediction of local responseparameters, such as the interlaminar stress distribu-tions (Qi and Knight 1996).

For problems involving different common andrefined plate theories, several numerical and analytical

approximate solutions have been proposed (Tessler 1993; Nguyen et al. 2005; Daghia et al. 2008; Faresand Elmarghany 2008; Xiang 2009; Bodaghi and Saidi2010). Most papers are limited to certain edges sup-port and only to cross-ply or angle-ply laminatedplates. Closed-form solutions for laminated compositeplates using FSDT have been provided for some sim-ple boundary conditions and some particular lamina-tion schemes (Whitney 1987; Reddy 2003). Otheranalytical solutions obtained by the Ritz methodemploy trigonometric and hyperbolic beam functionsto form the approximate shape functions. However,that approach for plates with general anisotropy leadsto resultant moments that appear to be oscillatingabout a relative constant value, as has been shown by(Nallim and Grossi 2003).

In this study, FSDT was employed in conjunctionwith the Ritz method and sets of beam characteristicorthogonal polynomials for obtaining a generalizedapproximate analytical solution to study the staticbehavior of arbitrary composite laminated plates withall possible combinations of boundary conditions. Allcoupling effects including, stretching-bending,stretching-shear, and bending-twisting were consid-ered in the formulation. The accuracy of the presentmethod, is shown through the convergence tests forselected plate problems and selected results are com-pared with those published by other authors. Fewselected cases are solved and the correspondingdeflections and resultant moments are presented bymeans of plots and in tabular form. Results includelaminated plates with general lamination sequences,non-symmetric boundary conditions and free edges.

The algorithm developed offers an interesting alter-nate from an engineering view point to perform designanalysis. It can be applied to the analysis of a widerange of rectangular laminated plates with differentboundary conditions, number of layers, stackingsequences, and fiber orientation.

2. Formulation

2.1 Energy Functional ComponentsA rectangular fiber reinforced composite laminated

plate of uniform thickness h is shown in Fig. 1. Theplane x, y coincides with the middle surface along theplate thickness, while z remains normal to it. In eachlayer of the laminate, denotes the angle of fiber ori-entation and the major and minor principal materialaxes are denoted by 1 and 2, respectively.

In the first order shear deformation theory, the platekinematics is governed by the midplane displacementsuo, vo, wo and rotations x, y given by:

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A General Ritz Algorithm for the Static Analysis of Arbitrarily Laminated Composite Plates ...

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(4)

2.2 Boundary Conditions and Approximating Functions

The situation for asymmetrically laminated platesis more complex than for symmetrical ones becausethe transverse and in-plane displacements are coupled.Actually, there are four types of boundary conditionsthat can be called simply supported (S), clamped (C),or free edges (F), and the combinations of these con-straints are summarized in Table 1.

In the application of the Ritz method only the essen-tial boundary conditions are required to be satisfied bythe assumed functions. The use of beam orthogonalpolynomials to study anisotropic plates is very satis-factory, as has been demonstrated by Nallim et al.(2003, 2005, 2008), since the convergence of the solu-tion is rapid and practically without oscillations. In thepresent work, the five displacement components areexpressed by sets of beam characteristic orthogonalpolynomials as follows:

(5)

where Bk , Ck are obtained using the orthogonalityproperty of the pk (x) polynomials:

The same procedure is used for y variable.

3. Application of the Ritz Method

The total energy functional for the static bendinganalysis of laminated plates is given by:

(6)

The Ritz procedure requires the minimization of theenergy functional (6) with respect to each of theunknown coefficients:

Table 1. Notations for various combinations of boundary conditions, in which n and s indicate the normaland tangential directions to the respective plate edges

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Substituting Eq. (5) into the expression of the func-tional Eq. (6), and subsequently applying Eq. (7)results in the following governing equation:

[K] {C} = {f} (8)

where {f} is the load vector and [K] is the syammetricstiffness matrix given by:

and the unknown coefficient vector is:

(10)

The elements of the stiffness matrix are given inAppendix A.

4. Numerical Results

A number of numerical examples are presented inthis section to demonstrate the performance of the gen-eral algorithm developed for bending analysis of lam-inated plates with various boundary conditions, span-to-thickness ratios, fiber orientations, etc. The follow-ing terminology is used for describing the plate bound-ary conditions. For instance, SiSiCiCi identifies a platewith edges simply supported along x = 0, a, and edgesclamped 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 arecompared with (Moleiro et al. 2008). Thegtraphite/epoxy material with properties are used.The reference flexural stiffness is Do = E1h3 / 12 (1 -v12v21).

4.1 Validation and Convergence StudiesComparison and convergence studies were conduct-

ed to evaluate the accuracy of the present formulation.Table 2 shows results for square angle-ply anti-sym-metric laminates ( / - ) with a S1S1C1C1 boundaryconditions and thickness ratio. The results are given inthe form of deflection, bending moments at the platecenter, and axial forces at one of the corners of theplate. The number of polynomials (M, N ) are steppedsteadily from 5 to 8 to demonstrate the convergence. In

the same table the accuracy and reliability of the pres-ent results are demonstrated by comparison with(Sheikh et al. 2002) who used a high precision trian-gular element considering the effect of shear deforma-tion by taking transverse displacement and bendingrotations as independent field variables. The compari-son shows a very good agreement.

Table 3 shows results of deflections, and momentand transverse force resultants for simply supportedcross ply (0/90) square plates with uniform loading.The lamina properties are E1 / E2 = 25, G12 / E2 = G13/ E2 = 0.5, G23 / E2 = 0.2, v12 = 0.25. The results fordifferent thickness ratios were compared with thosepublished by (Moleiro et al. 2008) and a very goodagreement was observed.

4.2 Numerical ExamplesThe developed Ritz formulation is applied in this

section to obtain the plate deflections and momentresultants of general laminated plates with severalcombinations of boundary conditions and stackingsequences. The capability and generality of the presentapproach is shown through selected examples depict-ed in Table 4 and Figs. 2-4. Two different combina-tions of boundary conditions, three thickness ratios,and four lamination schemes have been chosen asexamples.

The effect of span to thickness ratio on centerdeflection for symmetrical and anti-symmetricalcross-ply and angle-ply laminates with S1S1C2C2boundary conditions is shown in Fig. 5. The non-dimensional deflection decreased as b / h increased,but the decrease was small (beyond b / h = 40). Thevalue of the non-dimensional deflection became, asexpected, practically constant for thin plates. Also, thedifference between the symmetrical and anti-symmet-rical lay-up values was small for cross-ply laminatesand much larger for angle-ply laminates.

Finally, antisymmetrical two layered laminates( / - ), with a fiber orientation varying between 0°and 45° were analyzed when simply supported(S1S1S1S1), fully clamped (C2C2C2C2), and whenmixed (S1S1C2C2) boundary conditions were used.Variations of non-dimensional deflection are shown inFig. 6. A change in fiber orientation led to an increasein the non-dimensional deflection in case of fullyclamped plates, while for simply supported and mixedboundary conditions, the non-dimensional deflectionshowed a convex shape.

5. Conclusions

A general algorithm for static analysis of thick, arbi-trarily laminated composite plates with multiple com-bination of boundary conditions is presented in this

(9)

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Table 2. Convergence and validation results for angle-ply antisymmetric laminates

Table 3. Convergence and validation results for S2S2S2S2, (cross-ply laminates)*

(*) Results at A, (x = a / 2, y = b / 2); B, (x = 0, y = 0); and C, (x = a / 2, y = 0).

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Table 4. Laminated plates with S1S1C3C3, S1S1F4F4, S1F4C3C3 and C1F1S4S4 boundary conditions

Figure 2. Contour plot of moment resultants for different laminated plates with S1S1C3C3 boundary conditions

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Figure 4. Contour plot of moment resultants for different laminated plates with C1F1S4S4 boundary

Figure 5. Variation of non-dimensional static deflec-tion at the plate centre with b / h ratio Figure 6. Variation of non-dimensional deflection

with fiber orientation angle (b / h = 20)

S1S1S1S1 C2C2C2C2 S1S1C2C2

Figure 3. Contour plot of moment resultants for different laminated plates with S1F4C3C3 boundaryconditions

work. The algorithm is based on the Ritz variationalmethod and the first order shear deformation theory.The five components of the displacement field wereapproximated by sets of characteristic orthogonalpolynomials generated using the Gram-Schmidt pro-cedure. The algorithm developed was very general andallowed the analysis of moderately thick plates withdifferent materials, laminate lay-ups, span-to-thick-ness ratios, and boundary conditions. The computa-tional implementation is also very simple, and the for-mulation is such that we could appreciate how the var-ious geometrical and mechanical parameters involvedinfluenced the static response.

Consequently, the developed formulation consti-tutes an efficient tool in design and optimization prob-lems. Comparisons of the present results with a varietyof other published results were made with very goodagreement. Some typical results were given for cross-ply, angle-ply, and arbitrarily laminated plates withvarious thickness ratios ranging from moderately thickto thin, and for different boundary conditions.

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Appenxid A

Elements in Eq. (9)

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