Composite materials are preferred in places where lighter materials are desired or required without sacrificing strength. They have even become essential for many applications. Composite materials are being used in a variety of applications such as the structural parts of aircrafts, automobiles, chemical equipment, transformer tubes, boats, etc. Some transmission gears make use of plastic materials in many different places such as watches, instruments, washing machines, gear pumps, etc. Composite materials are efficient in applications that required high strength to weight and stiffness to weight ratios. Brebbia and Connor [1] presented the finite element displacement formulation applicable to arbitrary plate and shell elements for geometrically nonlinear problems and also developed appropriate equation for Newton-Raphson iteration. Wood and zienkiewicz [2] studied the geometrically nonlinear analysis of beams frames and arches using total Lagrangian coordinate system by using modified isoparametric element and system nonlinear equation was solved using Newton-Raphson method. Bathe and Bolourchi [3] presented finite element method for linear, geometricand material nonlinear analysis of plates and shell. Pica et al. [4] presented a geometrically nonlinear analysis of plates using finite element Mindilin formulation. Reddy et.al [5] developed a finite element model based on the combined theory of the yang, Norris, Stavsky and Von Karman [6] that, it accounts for the transverse shear strain, large rotations. Puctha and Reddy [7] developed a mixed shear flexible finite element with relaxed continuity, geometrically linear and nonlinear analysis of laminated anisotropic plates. Chia [8] solved the nonlinear bending of an unsymmetrically laminated angle ply rectangular plate analytically under lateral load satisfying the von-karman type strain. Striz et al. [9] investigated the behavior of thin, circular isotropic elastic plates with the immovable edges and undergoing large deflections .they used Newton-Raphson technique to solve the nonlinear systems of equations. Z.G. Azizian and D. J. Dawe [10] studied geometrically nonlinear analysis of rectangular mindlin plates using the finite strip method. Singh et al. [11] investigated the large deflections bending analysis of anti-symmetric rectangular cross-ply plates based on von-karman plate theory, with one–term approximation for the in-plane & transverse displacements, under sinusoidal loading. Clarke et al. [12] described various incremental- iterative technique based on the Newton-Raphson approach to analyze the geometric nonlinear behavior.

The nonlinear finite element modelling and analysis is more complicated than linear one, as the stiffness matrix in nonlinear equilibrium equations is dependent on deformations. The first order shear deformation plate theory is considered in the finite element

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analysis of composite plates subjected to mechanical loading. The geometry and origin of the material coordinates are at the middle of the laminate as shown in Fig.1

Displacement Field The first order shear deformation theory is used to describe the kinematics of deformation for the present analysis. The displacements components u, v and w along x, y and z respectively are expressed as in given below, Finally, for the deformation theory to finding the displacement expressions are given below,

where, the strains of the middle surface for the displacements. Constitutive relations orientations and equation related fiber orientation are given as follow with respect to the principal axis as shown in below.

Stress strain relationship for a substrate lamina is given by The subscripts x, y, z represent the principle material axes. E12, E21, G12, G13 and G23 are elastic longitudinal, transverse and shear moduli of laminated plate respectively. Transformed reduced stiffness matrix is calculated by following formula.

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[A] = In plane stiffness matrix, [B]= Coupling stiffness matrix, [D]= Bending stiffness matrix and are calculated using following formula Elemental stiffness matrix for the plate element is given by, Elemental load vector is given by Where,

Geometrically nonlinear finite element is developed for the static analysis of laminated composite plate using first order shear deformation theory. An eight noded continuous element is employed. Isoparametric element In the present analysis eight node isoparametric elements is considered and the shape function of the element is given below. The independent field variables based on the [u, v, w, θx, θy] values. Here θx and θy are local coordinates of the element. For the isoparametric element concept, the element geometry, displacement vectors over each element are represented by the interpolation functions.

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Ni is the interpolation function,

Geometrically nonlinear analysis of square plate is formulated by using the finite element modeled for the bending behavior. The laminated square plate is discretized using the eight nodded element .The development of a program was successfully done .In the present work nonlinear problem/ equation is linearized using Newton-Rapson method, now based on the numerical analysis carried out for the non-linear analysis. The finite element model is converted into MatLab code. 1. Non dimensional zed center Deflection :

resulting nonlinear finite element equation is written as Here the nonlinear term [K {d}] in the above nonlinear finite element equation (16), depends on the unknown solution {d}. Thus the above equation needs to be solved iteratively till the solution convergence is achieved with the said tolerance. Here nonlinear finite element equation (17), is linearized by using Newton-Raphson iterative method. The tangent stiffness matrix [KT]n and the secant stiffness matrix [Ks]n are given in above the boundary conditions and material properties are respectively as shown in the table 1. Finite element model derived into investigate the nonlinear bending behavior of plate subjected to the mechanical load. A wide variety of numerical examples considering with the plate with the different support conditions and laminates loading conditions, validated formulation is further employed to study the nonlinear bending analysis of plates by varying the loading condition .The nonlinear linear equations are solved using Newton-Rapson iterative method. For the analysis carried out here, the boundary conditions considered are given in the table 1.The obtained result, unless otherwise mentioned are presented in the following non-dimensional form:

The present nonlinear FE model is converted to MatLab code and is validated with available results for its accuracy and correctness. The comparison of present results with available results is presented in below table. Nonlinear deflection for simply supported angle ply [θ/ -θ] plate with θ =45, 30, 15 are presented in table 4.2 with respect results are available in open literature. Present study are having the material properties

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available reference

In this section numerical analysis is carried out by considering different boundary conditions to know the geometric nonlinear behavior of anti-symmetric cross ply laminated plates. In the present analysis the effect of number of layers and the lamination scheme on the center deflection is examined for anti-symmetric cross-ply laminates. A square plate with clamped, simply supported, hinged boundary condition and material. The material properties of the laminates are taken as follows:

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Geometrically nonlinear analysis of square plate is formulated by using the finite element method for the bending behavior. The laminated square plate is discretized using the eight nodded element. The formulated FE model is successfully converted to MatLab code and is been validated. It is concluded that the number of layers, stacking sequence, boundary condition have significant effect on the nonlinear deflection of the plate. In the present work nonlinear problem/equation is linearized using Newton-Rapson method, now based on the numerical analysis carried out for the non-linear bending analysis conclusions are as follows

• The effect of fiber orientation and number of layers on geometric nonlinear behavior as the number of ply increase then the nonlinearity increases.

• The effect on same number of layers with having different fiber orientation then the deflection will increases.

Brebbia, C. and Connor, J. (1969). Geometrically nonlinear finite-element analysis, Journal of Engineering Mechanics Division 95: 463-483. Wood, R. D. and Zienkiewicz, O. C. (1997). Geometrically nonlinear finite element analysis of beams, frames, arches and axisymmetric shells, Computer & Structure 7(6): 725-735. Bathe, K. J. and Bolourchi, S. (1980). A geometric and material nonlinear plate and shell element, Computer & Structure 11(1): 23-48. plate behaviour using mindlin formulation. Computer & Structure 11: 203-215 J. N. Reddy and W. C. Chao (1981). Nonlinear bending of thick rectangular, laminated composite plates. Int. J. Nonlinear Mechanics 11(3/4):291-301. P. C. Yang, C. H. Norris and Y. Stavsky (1966). Elastic wave propagation in heterogeneous plates. Int. J. Solids and structure 2:665-685. N. S. Puctha and J. N. Reddy (1986). A refined mixed shear felexible finite element for the nonlinear analysis of laminated plates, Computer & Structure 22(4): 529-538. Chia CY. (1985). Large deflection of unsymmetrical laminates with mixed boundary conditions. Int J Non-Linear Mech 20(4):273-82. Striz AG, Jang SK, Bert CW. (1988). Nonlinear bending analysis of thin circular plates by differential quadrature. Thin-Walled Struct 6(1):51-62. Z. G. Azizian and D. J. Dawe (1992). Geometrically nonlinear analysis of rectangular Mindilin plates using the finite strip method. Computer & structure 21(3): 423-436. Gajbir Singh, G. Venkateswara Rao, N.G.R. Iyengar. (1991).Some observations on the large deflection bending of rectangular anti-symmetric cross-ply plates. Compos Struct 18:77-91. Murray J. Clarke, Gregory J. Hancock. (1990) A study of incremental-iterative strategies for non-linear analyses. Int J Numer Methods Eng. 29:1365-91. Madhulit Mukhopadhya,(2012). Mechanics of Composite materials and structures, Universites press, ISBN-978 81 7371 4771. Zaghloul SA, Kennedy JB. Nonlinear behavior of symmetrically laminatedplates.J.Appl.Mech,42:2.

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PG Student, Dept. of Mechanical Engg., BEC. Bagalkot, India