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 Ibraheem Hssn 
 Postgraduate student (PhD), Department of Structural Engineering, Faculty of civil engineering, Tishreen University, Lattakia, Syria
ibhassan@tishreen.edu.sy

Sulaiman Abo Diab
Associate Professor, Department of Structural Engineering, Faculty of civil engineering, Tishreen University, Lattakia, Syria
‫sabodiab@tishreen.edu.sy‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬‬

Bassam Hwaija

Professor, Department of Structural Engineering, Faculty of civil engineering, Tishreen  University, Lattakia, Syria
h.bassam65@yahoo.com

Abstract

A displacement-based modified finite element is applied in this paper to analyze multilayered composite plates in the geometrical nonlinear domain. The proposed plate element is based on the first-order shear deformation theory (FSDT) and von Karman's large deflection theory while the displacement approach is based on the incremental form of the principle of virtual displacement. The displacement function is modified and new terms have been added to consider the effect of the external load. As a result, shape functions are classified into two categories the first one is the homogenous shape functions relevant to degrees of freedom and the second is the nonhomogeneous shape functions related to the element loading. Lastly, some examples are computed and the results of displacements are compared with analytical/experimental and other numerical solutions available in the literature. Numerical examples showed that results from applying the present element were in good agreement with other solutions.

 Introduction

Plates are very common elements used in all engineering fields such as transportation, aerospace and civil engineering. One of the most recent and currently used plates is the multilayered composite plates (Laminates), which are plates consists of number of layers placed over each other. In each layer we can use different materials also we can orient them in various directions. This structure of composite plates gives better engineering properties than ordinary plates with one material in one direction. Figure (1) shows a composite plate made of several layers of Carbon fibers connected with each other using Epoxy.

 Modified finite element

In the modified finite element method, the shape functions supposed to achieve the differential equation accurately, but because of the difficulty of finding exact solutions for the von Karman equation which expresses the geometrical nonlinear behavior of plates and since we study the nonlinear behavior as a series of small linear steps we will use the shape functions of the standard 4 node quadrilateral isoparametric  plane element as the homogeneous part of shape functions.

N_1 (θ^1,θ^2 )=(1/4)(1-θ^1 )(1-θ^2 )      ;     N_2 (θ^1,θ^2 )=(1/4)(1+θ^1 )(1-θ^2 )                (1)
N_3 (θ^1,θ^2 )=(1/4)(1+θ^1 )(1+θ^2 )      ;     N_4 (θ^1,θ^2 )=(1/4)(1-θ^1 )(1+θ^2 )                (2)

For nonhomogeneous part we will consider the displacement function for ACM element and extend it to contain fifteen parameters instead of twelve then match both sides of the differential equation governing the behavior of plates in linear state. These steps are explained in detail in references [17-21].
In final form we will obtain the displacement function as follows :

w_i=N_i^m(e) .u_m(e) +N ̅_ij.p ̅^j                                (3)

N_i^m(e)  : Homogeneous part of shape functions, u_m(e)  : Nodal degrees of freedom.
N ̅_ij : Nonhomogeneous part of shape functions which given in detail in [21] .
p ̅^j : Distributed load intensity at the nodal points of the element.

 Finite element formulation

 
δ∆π=0                                                                     (4)

Using incremental procedure to proceed from a given state of equilibrium to unknown state then the corresponding u_i^0 displacements are replaced by a new displacement u_i and the body moves to new position its displacement is u ̃_i :
u ̃_i=u_i^0+u_i                                                               (5)

u_i : Small increment in displacement.
σ_ij=σ_ij^0+〖∆σ〗_ij    ;   ε_ij=ε_ij^0+〖∆ε〗_ij                     (6)

ε_ij^0  ,σ_ij^0 : Strain and stress tensors.
〖∆ε〗_ij  ,〖∆σ〗_ij : Small increments in strain and stress respectively which resulted from the increment in displacement.
According to the principle of virtual displacement the variation of total energy is :

Linear stiffness matrix

Numerical results and discussion

    Clamped square plate under uniformly distributed load
This example is widely considered as benchmark problem to check and verify geometrical nonlinear behavior of thin plate formulations [16]. Plate dimensions are (L=300in ,t=3in). Material properties, Young modulus in x and y directions : (E_1=E_2=30×〖10〗^6 Psi), Possion ratio (ϑ_12=ϑ_21=0.316)
The plate is subjected to uniform distributed load, the thickness of all individual layers within the plate are equal, shear correction factor is taken (5/6),  20 incremental load steps are applied and the convergence tolerance used here is(0.001). The results of the nondimensional central deflection (w/t) and nondimensional normal stress (σ_xx.t^2/L^2.q) from modified element, Reddy and Urthaler [16] and analytical solution [5] are presented in table 1.
 

Conclusions

•    Modified method is able to deal with the incremental form of geometrical nonlinear analysis of multilayered composite plates, and even though it requires to use displacement functions satisfies the differential equation at the element level, it is possible to roughly use the solution of the differential equation for the linear case as an initial approximation to the nonlinear case.

•    The proposed modified element gave great convergence to the analytical solution, despite it can be considered simple in terms of mathematical formulation and compared to other proposed elements.

•    For the practical applications (especially composite plates in aerospace) the method presented in this article give fast and easy initial results for displacements since the shape functions aren't complicated from mathematical point of view and the nonhomogeneous part takes into account the type of loading on the plate.

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