Wednesday, January 6, 2010

ANALYSIS OF COMPOSITE MATERIAL

ANALYSIS OF COMPOSITE MATERIAL USING
FINITE ELEMENT METHOD

ABSTRACT:
The objective of the paper is the finite element analysis of pressurized laminated composite shell with elliptical cutout / inclusion with two pressure load mechanisms and comparison with different materials.
One is uniform shear force around the elliptical cutout boundary and the other is the same with the opening closed with an elastic cover (inclusion). The pressurized composite shell is a heterogeneous body consisting of finite number of layers bonded together, with an elliptical hole. The basic equations governing the elastic behavior of the composite structure can be derived but the analytical solution of these complex equations cannot be obtained easily. So FEA with 8-noded isoparametric quadrilateral shell element with 6 degree of freedom is employed in present study. The numerical results of this study with elliptical hole and with elastic cover can be presented. In addition to finite element model, material modeling, layerwise stress recovery and failure criteria are also considered.

NOTE: We have performed the above mentioned analysis at COMPOSITES SECTION, ARDC DIVISION OF HINDUSTAN AERONAUTICS LTD, BANGALORE and acquired results that were quite satisfactory to theoretical results.







INTRODUCTION
Multilayered composite materials are increasingly used in the construction of shell type structures for various applications, creating a need for structural analysis which considers the laminated characteristics of such materials. Individual laminae with different fibre orientations are to be bonded together called as laminates. Shells are advantageous over plates since the deformation and stresses are less in shells due to their geometry. Openings are necessary in such structures due to a variety of functional requirements. Stress concentration around holes in shells is of great importance in the design of shell structures. The basic equations governing the elastic behaviour of shells can be derived but analytical solution of these complex equations is not available. The shell structures are required to have structural rigidity so that they don’t fail if cut-outs are made. The problem of stresses around the holes in shells has been investigated extensively. Although circular holes are used widely in practice, an elliptical hole has certain advantage over the circular hole as it reduces stress concentration.
The application of finite elements to cut-out problems in shells is limited to two shell types. They are cylindrical & spherical and with two simple hole shapes which are circular and rectangular. The determination of stress around an elliptical cut-out in pressurized composite shells has not been given much attention. Also analysis of stresses around openings in composite cylindrical shells is still not available. Analytical methods are complex and it is impossible to compute cut-out problems in composite structures. Also the problem of stress concentration around elliptical cut-outs in composite shells cannot be computed. Hence FEM appears to be the only alternative. In FEM we have to develop elements, meshes, constraints, procedures & post-processing and variety of options. Thus without FEM it is doubtful whether these type of problems could be analysed and so the present effort is directed towards the stress problem of openings\inclusions.






METHODS AND PROCEDURES
FEM is one of the most effective methods for the numerical solution of field problems. It is capable of modelling and solving complex problems of multilayered composite structures. It provides elements required for the analysis of shell. Among these an eight-noded isoparametric quadrilateral shell element with six degrees of freedom at each node is employed in the present paper. Within element we can specify arbitrary number of layers, each with its own thickness, ply orientation and orthotropic elastic properties. The element output includes layer stresses in material coordinate system.
Unlike the stiffness parameters these strength parameters cannot be transformed directly for an angle lamina. Hence the failure theories are based on first finding the stress in the local axes and then using these strength parameters of a lamina to find whether a lamina has failed. Unidirectional lamina in laminae can fail in three distinct modes i.e. fibre fracture, matrix cracking and interface shear. The measured laminate strengths corresponding to these modes of failure are usually denoted by X,Y & S. a lamina is considered to fail after all these modes have occurred i.e. when (σ1\ X) > 1, (σ2\ Y) > 1 and (τ12 \ S) > 1. No work has been done on the pressurized composite cylindrical shell with an elliptical cut-out considering the assumption of uniform shear force along the boundary. In most of the earlier analyses, for the pressure loading case for various hole configurations including elliptic holes, the elastic behaviour of the cover has been neglected. This paper includes elastic behaviour of the cover too. The stress distribution is highly sensitive to the force transmission from the cover to the shell and in order to get accurate results, the elastic behaviour of the cover must also be taken into account. Thus the main objective of the paper is the analysis of pressurized laminated composite cylindrical shell with elliptical cut-out\ inclusion with two pressure load mechanisms. The two pressure load mechanisms are
(a)force around the elliptical cut-out
(b)Elliptical opening is closed with perfectly edge bonded elastic inclusion i.e. cover so that proper transmission takes place from the boundary of the shell.

ELLIPTICAL CUT-OUT IN A COMPOSITE CYLINDRICAL SHELL
The geometry and constructional details of the laminated composite shell with an elliptical cut-out used in this analysis is described as in fig.


The shell wall is made up of specified number of layers of equal thickness stacked with specified fibre orientation as shown in figure. [902/0]2­ lay-up scheme is considered in the present analysis of composite shells. The winding angle of the fibre 900 is in the circumferential direction. The axes Z and Y oriented along the longitudinal and circumferential directions of the shell respectively. Fibre direction 00 is measured with reference to longitudinal (Z) direction. Hence fibre direction 00 means that the fibres are along the longitudinal direction whereas the fibre direction is 900 means that the fibres are along the circumferential direction. The shell with an elliptical cut-out is loaded by unit normal pressure. The major axis of the ellipse is taken to be perpendicular to the axis of the shell. The elastic constants of the composite material systems are given in table below.


RESULTS AND DISCUSSION
The finite element solutions for the practically important problem of elliptical cut-outs in laminated composite shell are presented
The in-plane stresses σ1, σ2 and τ12 are evaluated on top, middle and bottom surfaces of each layer along the hole boundary resulting eighteen surfaces for six layers in a shell. The observed critical locations are at η is about 00 and 900 i.e. at the end of major axis and minor axis of the cut-out respectively. The stress τ12 is considerably less at 00 and 900. The out of plane stresses σ3; τ13 and τ23 are small compared to that of in-plane stresses and hence neglected. Typical numerical results showing the effect of composite material properties on the state of stress across the thickness at critical locations on the whole boundary in a shell are shown in figures below.

X-COMPONENT STRESS


Y-COMPONENT STRESS
The histogram showing the comparison of different materials are shown in figs below.


Stress distribution across thickness of the shell (Boron\epoxy)

a) At η = 900 b).at η = 00



Stress distribution across thickness of the shell (Graphite/epoxy)
a) At η = 900 b).at η = 00








Stress distribution across thickness of the shell (Glass/epoxy)

a) At η = 900 b).at η = 00


The following are observed in laminated composite shell with an elliptical cut-out under given loading conditions.
The normalized stresses (σ1\ X) is maximum on 900 layers and least on 00 layers at η = 900, whereas it is maximum on 00 layers and least on 900 layers at η = 00. The stress σ1 increases from top surface to bottom surface in 900 layers at η = 900 while it decreases in 00 layers.
(σ1\ X) is more on bottom surface of layer1, whereas it is less on bottom surface of the layer 3 at η = 900. (σ1\ X) is more on top surface of layer 4 at η = 00 in graphite\epoxy shell. Its magnitude is compressive in 00 fibre direction layers and tensile in 900 directions. Its magnitude decreases in 900 layers from bottom to top surface.
The normalized stress (σ2\ Y) is more on 900 plies, and least in 00 plies at η = 00, whereas it is more on 00 plies and least on 900 plies at η = 900. Its value increases from top surface to bottom surface in each layer at η = 900 and decreases at η = 00
(σ2\ Y) on top surface of the layer 6 is maximum in boron\epoxy shell, whereas it is minimum on top surface of the layer 3 at η = 00. (σ2\ Y) is more on bottom surface of layer 3, and less on top surface of the layer 6 (900) in graphite\epoxy shell at η = 900.
It was observed that the stress levels changes as the surface of the respective ply is approached. The stress σ1 is almost negligible on 900 plies, decreases from top surface to bottom surface on 00 plies at η = 00. The stress σ2 is compressive in layers 1, 2 and tensile in layers 3, 4, 5 and 6 at η = 00.
The stress σ2 gradually decreases from top to bottom surfaces in layer 4 and 5 while its value increases in layer 1 and 2 and is compressive at η = 00.
The normalized stress (τ12\ S) is more at η is about 250 and 1050 and less at η is about 1550 and 3350 in 900 plies. Similarly the normalized stress (τ12\ S) is less at η is about 250 and 1050 and more at η is about 1550 and 3350 in 00 plies.

EDGE BONDED COVER IN COMPOSITE CYLINDRICAL SHELL
The geometry and ply orientations sequence of the cross ply laminated cylindrical shell with an edge bonded elastic inclusion in the present investigation is shown in figure.
The analysis of shell structures with cut-outs filled by elastic covers was attempted by the present method. The cut-out could be filled by the elastic cover, which is perfectly edge-bonded along the hole boundary of the shell, where the windows are introduced in pressure cabins for the purpose of visibility, the problem can be treated as one of the inclusion problem. If the window is of low modulus material such as Perspex or Plexiglas with young’s modulus (Ec) = 2758 MPa and poisson ratio γc = 0.36 while the shell wall is made up of six layers of laminae stacked with [902\0] s lay-up. The unit normal internal pressure loading is considered.

RESULTS AND DISCUSSIONS
Here the results are presented for the analysis of an edge bonded elastic inclusion in composite shell. The inclusion tends to behave more like a shell than a plate. Proper force transmission takes place from the inclusion to the shell for the stated loading condition. This naturally leads to lower stress concentration due to smaller bending effects. The method developed in this work for satisfying boundary conditions at the interface between shell and the inclusions should be of particular interest. The in-plane stresses σ1, σ2 and τ12 are evaluated at the top, middle, bottom surface of each layer along the hole boundary in a shell. The following figure gives the stress distribution across the thickness at η = 00 & 900. The stress τ12 is less at 00 & 900 and hence ignored.

Stress distribution across thickness of the shell (Boron/epoxy)
With low modulus edge-bonded cover
a) At η = 900 b).at η = 00



Stress distribution across thickness of the shell (Graphite/epoxy)
a) At η = 900 b).at η = 00





Stress distribution across thickness of the shell (Glass/epoxy)
With low modulus edge-bonded cover

a) At η = 900 b).at η = 00

The following are observed in laminated composite shell with an elliptical cut-out filled with an elastic cover.
The normalized stress (σ1\ X) is more on 900 plies and least on 00 plies at η = 900 whereas it is more on 00 plies & least on 900 plies at η = 00. The trend is same is all the material system considered.
(σ1\ X) on top surface of layer 6 (900) is maximum in graphite\epoxy shell and minimum on middle surface of the layer 4 (00) in glass\epoxy shell at η = 900.
(σ1\ X) on bottom surface of layer 1 (900) is maximum in graphite\epoxy shell and minimum on middle surface of the layer 5 (00) in glass\epoxy shell at η = 00.
The normalized stresses (σ2\ Y) is more on 00 plies and least in 900 plies at η = 900, whereas it is more on 900 plies(layer 5& 6) and least on inner 900 plies (layer 1& 2) at η = 00
(σ2\ Y) on top surface of layer 6 (900) is maximum in glass\epoxy shell and minimum on middle surface of the layer 3 (00) in graphite\epoxy shell at η = 00.
(σ2\ Y) is more on top surface of layer (00) and less on top surface of the layer 2 (900) in boron\epoxy shell at η = 900.
The normalized stress (τ12 \ S) is more at η is about 250 and 1050 and less at η is about 1550 and 3350 in 900 plies. Similarly it is less at η is about 250 and 1050 and more at η is about 1550 and 3350 in 00 plies.
(τ12\ S) is maximum on top surface of layer 6 (900), whereas it is minimum on bottom surface of the layer 3 (00).
The magnitude of τ12 is more on top surface of layer 6 (900) in graphite\epoxy shell, whereas it is less on bottom surface of layer 3 (00) in glass\epoxy shell.
The magnitude of τ12 is gradually increases from bottom to top surface of the shell in all material system of the shell.
The stress σ1 of outer 900 plies (layer 5& 6) is tensile and reaches maximum value at η = 900 and its value approaches 0 steeply at η = 900 at layer 4.
The results show that the stress σ2 in layer 5 & 6 had compressive characteristics at η = 00

CONCLUSION
Accurate stress analysis of laminated composite shell structures with elliptical cut-out \ inclusion under internal pressure loading is presented in this paper. From the results, it was observed that the stress distribution across the thickness of the laminated composite shell is not uniform but at the same time varying in geometry and loading. However, this method can be extended to apply to many other problems without complexity. Some of these problems are reinforced cut-outs, cut-outs in the presence of stiffening elements or shell boundaries, multiple cut-outs, load diffusion problems and certain shell intersection problems.









REFERENCES
Lakshminarayana H.V and Viswanath.S. “APPLICATION OF FINITE ELEMENT METHOD TO THE ANALYSIS OF LAMINATED COMPOSITE SHELL STRUCTURES”
Ramesh Kumar.R., Belkune.R.M and Kant.T. “A FINITE ELEMENT METHOD FOR EVALUATING STRESSES AROUND CUT-OUT IN SHELLS”
Mark.W.Hilburger and James.H.Starnses “RESPONSE OF COMPOSITE SHELLS WITH CUT-OUTS TO INTERNAL PRESSURE AND COMPRESSION”
Murthy.M.V.V “ON THE STRESS PROBLEM OF ELLIPTICAL HOLES AND INCLUSIONS AND STRAIGHT LINE CRACKS IN CYLINDRICAL SHELLS”
Tsai.S.W and Hahn.H.T. “INTRODUCTION TO COMPOSITE MATERIALS”

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