FINITE ELEMENT ANALYSIS OF THIN FRP SKEW LAMINATES WITH ELLIPTICAL CUTOUT
The present research work deals with the prediction of static behaviour of a thin five-layered symmetric cross-ply Fiber Reinforced Plastic (FRP) skew laminated composite plate with elliptical cutout subjected to uniform pressure loading using Classical Laminate Theory (CLT) based finite element method. The finite element software ANSYS has been successfully executed for the validation of the finite element model and evaluation of the transverse deflection and in-plane stresses in the skew laminate. The effect of size of the elliptical cutout for two different positions of the cutout on the deflection and stresses is discussed. The present analysis is useful for the safe and efficient design of skew laminates with elliptical cutouts.
Keywords: Skew laminates, Cutouts, FRP, FEM
Fiber reinforced composite material consists of fibers of high strength and modulus embedded in or bonded to a matrix with distinct interface between them. In this form, both fibers and matrix retain their physical and chemical identities, yet they produce a combination of properties that can not be achieved with either of the constituents acting alone. In general, fibers are the principal load-carrying members, while the surrounding matrix keeps them in the desired locations and orientation, acts as a load transfer medium between them, and protects them from environmental damages due to elevated temperatures and humidity.
The most common form in which fiber-reinforced composites are used in structural applications is called a laminate. It is obtained by stacking a number of thin layers of fibers and matrix and consolidating them in to the desired thickness. Fiber orientation in each layer as well as stacking sequence of various layers can be controlled to generate a wide range of physical and mechanical properties for the composite laminates. The properties that can be emphasized by lamination are strength, stiffness, low weight, corrosion resistance, wear resistance, attractiveness, thermal insulation, acoustical insulation, etc. Examples of laminated fiber-reinforced composites include Missile cases, Fiberglass boat hulls, Aircraft wing panels and body sections, Tennis rackets, Golf club shafts, etc.
The design of modern high-speed aircraft and missile structures, swept wing and tail surfaces are extensively employed. It is therefore necessary that we have good understanding of the behavior under multiple loading of such surfaces, which are in the form of skew plates. The heterogeneity and anisotropy of the laminates make the classical solution tools to fail, and hence one has to resort to numerical methods.
A lot of literature is available for in-plane loading problems and very few solutions are found for the bending case. Based on thin plate theory, Lekhnitskii  and Savin  gave the formulation of hole problems for the anisotropic case. Lekhnitskii  gave the results for a circular hole in plywood plate. In addition to the results for various shapes of holes in isotropic plate, Savin  gave detailed equations for an elliptical hole problem. Hwu  gave the solution for in-plane loading and in-plane bending of anisotropic plates with holes. Ukadgaonker et al,  gave a general solution to determine the stress resultants and moments around holes of any shape with simple mapping function under arbitrary bi-axial loading condition. Ukadgaonker et al,  gave a general solution for bending of symmetric laminates with holes based on the formulations of Lekhnitskii and Savin that considers any shape of hole in symmetric laminates subjected to remotely applied bending or twisting moments.
The problems of skew, or swept structures are invariably difficult and complex, but their solution is of considerable importance in enabling the construction of safe and efficient structures like skew bridges and swept wings. Morley et al,  developed an elementary bending theory for the small displacements of initially flat isotropic skew plates. Karami et al,  has applied Differential Quadrature Method (DQM) for static, free vibration, and stability analysis of skewed and trapezoidal composite thin plates.
From the review of literature, it is observed that there is a need for the analysis of Skew Laminates with cutouts. In the present work, finite element method is used to analyze a five-layered thin CFRP skew, cross-ply laminated plate subjected to uniform transverse pressure loading with elliptical cutouts.
2. PROBLEM MODELING
2.1. Geometric Modeling
The geometry of the problem is shown in Fig.1. The side of the plate l is taken as 20 mm and five layers are considered with total thickness (h) of 1mm so that the length to thickness ratio becomes s=20. The skew angle a is taken as 300. An elliptical hole is placed at the geometric centre of the plate with major (2a) and minor axis (2b) with a=2b. The size of the cutout is varied as per the ratio d/l (=2a/l) ranging from 0.1 to 0.6. Two different positions of the cutout are considered, i) major axis horizontal and ii) major axis vertical.
2. FE mesh on Skew plate
2.2 Finite Element Modeling
A laminated composite general shell element (SHELL99) is used for meshing geometry of the problem. This element is suited for modeling moderately thick to thin laminated composite shells. As shown in Fig. 3, the element consists of number of layers of perfectly bonded orthotropic materials. The element is quadratic and has six degrees of freedom per node namely, translations in x, y and z directions respectively, and rotations about x, y and z axes respectively. SHELL99 allows up to 250 layers.
This element gives results of high accuracy and discretization involves fewer elements. As shown in Fig. 3, the lamination sequence is between the bottom and top faces of the element with the layer setup starting from the bottom face. This element is used to model the present problem with 00/900/00/900/00 layer sequence. The individual layer thickness are taken so that the thickness of each 00 layer is equal to h/6 and the thickness of each 900 layer is equal to h/4.
2.3 Material Properties 
Each layer is unidirectional carbon fiber reinforced plastic possessing the following engineering constants.
Elastic modulus in the longitudinal direction of the fiber, EL = 175 GPa.
Elastic modulus in the transverse direction of the fiber, ET = 7 GPa.
Shear modulus in the longitudinal plane of the fiber, GLT = 3.5 GPa.
Shear modulus in the transverse plane of the fiber, GTT = 1.4 GPa.
Poissons ratio ?LT = ?TT = 0.25
2.4 Boundary Conditions
All the four sides of the skew plate are clamped i.e. all the six degrees of freedom of the nodes placed along the edges of the plate are constrained.
A uniformly distributed transverse pressure of intensity Po = 1 MPa is applied.
3. ANALYSIS OF RESULTS
For validation of the finite element method, the stresses are obtained for a simply supported square cross-ply (00/900/00) laminate (laminae thickness ratio equal to 1:2:1) for the static analysis of the laminates. The results obtained from the present work are in close agreement (Table 1) with exact elasticity solutions . Later the method is extended to study the static behavior of a skew laminated plate with an elliptical cutout. The maximum values of in-plane stresses and deformations by changing the ratio 2a/l are determined and are presented in Figs. 4 to 9.
The deflection for both horizontal and vertical ellipse increases up to d/l=0.2 and then decreases. When the size of the cut out increases, the resisting area of the plate decreases causing decrease in stiffness of the plate resulting in the increase in the deflection of the plate. In this process the resultant force acting on the plate decreases causing reduction in deflection. Therefore between d/l=0.1 to 0.2 the first factor is influencing and later deflection decreases due to the influence of second factor. The deflection of horizontal ellipse is less compared to vertical ellipse (Fig.4). The reason for more deflection of vertical ellipse is due to the small span of the plate in y direction than in x-direction.
For both the cases sx increases up to d/l=0.2 and later decreases. The reasons given for the variation of w are valid for sx also (Fig.5).
For both the cases, sy increases up to d/l=0.2 and later decreases. This is due to the variation of the stiffness as discussed in case of w. When the ellipse position is horizontal the resisting area normal to y-direction is less causing for the more value of sy for horizontal ellipse. For certain range of d/l it is observed that sy for vertical ellipse is more. This is due to the smaller value of span length in y-direction.
In both the cases txy decreases with increases in d/l which is due to influence of second factor.When compared to sx and sy the rate of decrease in txy is more. From the fig.8 it is observed that the sx varies linearly within the layers and the stress in middle and outer layers is more when compared to the stress in 2nd and 4th layers.
sx is more in the layers with fibre angle 0,this is due to the reaction force developed in the x-direction is more in these layers.This stress is maximum at the top and bottom surfaces of the plate and at the interface of the adjacent layers sx is not same which is due to the mismatch of poisons ratios at the junction of the adjacent layers.
sy along thickness is also varying linearly within the layers.The stress is more in 2nd and 4th layers than in other layers. sy is more in the layers with fibre angle 90,this is due to the reaction force developed in the y-direction is more in these layers.This stress is maximum at the interface of outer layers and adjacent layers. At interface this stress is also varying between the layers due to mismatching of Youngs Modulus.
. Lekhnitskii.S.G, " Anisotropic plates", Gordon and Breach, New York, 1968.
. Savin.G.N, "Stress concentration around holes", Pergamon Press, New York, 1961.
. Hwu. C, "Anisotropic plates with various openings under uniform loading or pure bending", J Appl. Mech., Trans ASME, 1990, Vol. 57, pp. 700 706.
. Ukadgaonker. V.G and Rao. D.K.N, "A general solution for stress resultants and moments around holes in unsymmetric laminates", Comp. Struct., 2000, Vol. 49, pp. 27-39.
. Ukadgaonker. V.G and Rao. D.K.N, "A general solution for moments around holes in symmetric laminates", Comp. Struct., 2000, Vol. 49, pp. 41-54.
. Morley.L.S.D,"Skew Plates and Structures", Division I: Solid and Structural Mechanics, Int. Series of Monographs on Aeronautics and Astronautics, Pergamon Press, New York, 1963.
. Karami. G, Shahpari. S.A and Malekzadeh. P, "DQM analysis of skewed and trapezoidal laminated plates", Comp. Struct., 2003, Vol.59, pp. 393-402.
. Paul.T.K and Rao.K.M, "Stress Analysis around circular holes in FRP Laminates under transverse load", Computers & Structures, 1989, Vol. 33, pp. 929-937.
. Pagano, N.J. and Hatfield, S.J. Elastic behavior of multilayered bidirectional composites. AIAA Journal, 1972, Vol.10, pp.931 33.
. ANSYS reference manuals, 2007.