1 introduction

1.1

Delamination buckling and post buckling of composite

previous studies on the

delamination buckling and post buckling of laminated composite structures can

be classified within three general categories: experimental, analytical and

numerical methods. The experimental methods are usually used to confirm the

results produced by the other two methods; therefore, in here we will focus on

the analytical and numerical works and will also mention their experimental

validations. Later we will further subdivide the works based on different sub categories, such as: one-, two- or

three-dimension delamination modelling, single or multiple delamination, etc.

One of the earliest works in delamination buckling and growth analysis of beams

was coming out by Chai et al 1. They studied the behaviour of an isotropic

homogeneous beam-column under axial compression from a thin film model to the

general case (when the supporting base laminate buckles globally, so that the

zero-slope boundary condition for the thin sub laminate becomes invalid). Semites

et al 2 also employed a similar model to study delamination buckling. They studied

the effect of the location, length and thickness of delamination on the

buckling load of a beam with clamped and simply supported ends, having a single

across-the-width delamination. The perturbation method was used to solve the

buckling equation. Kardomateas and Shmueser 3 used perturbation technique to

analyses the compressive stability of a one-dimensional across-the-width

delaminated orthotropic homogeneous elastic beam. They also considered the

transverse shear effect on the buckling load and post-buckling behaviour of the

beam. Using a variation energy approach and a shear-deformation theory, Chen

4 formulated the same problem. According to his results, inclusion of the

shear deformation causes reduction in the buckling and ultimate strength of

delaminated composite plates. Kyoung and Kim 5 used the variation principle

to calculate the buckling load and delamination growth of an axially loaded

beam-plate with a non-symmetric (with respect to the centre-span of the beam)

delamination. Wang et al 6 used an analytical procedure to determine the

buckling load of beams having multiple single-delamination. Free and

constrained models based on the beam-column theory were used to model the

perfect and separated parts. Successive corrections made by removing the

overlaps lead to physically permissible buckling mode. Steinman et al 7

solved the differential equations of a delaminated composite beam under arbitrary loading and

boundary conditions with a finite difference method. Bending-stretching coupling

was taken into account which was show to significantly influence the buckling

loads. Steinman used the finite difference method to solve the post buckling

problem of an imperfect composite laminate having a through-the-thickness

delamination. They employed the commonly used one-dimensional beam model and

formulated the response of the beam by dividing the delaminated beam into four

regions. Using the Von Kaman kinematic approach, the resulting non-linear

differential equations were solved by the method of Newton-Raphson. Davidson 8

used the Rayleigh-Ritz method to compute the buckling strains of a composite

laminate containing an elliptical delamination. The influence of the bending

stretching coupling behaviour of the delaminated region and the Poisson’s ratio

mismatch between the delaminated and base regions were also investigated.

2 generalized differential quadrature method

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Generalized Differential Quadrature Method (GDQM),

introduced by Bellman et al 9, is Generalized based on the weighted sum of

function values as an approximation to the derivatives of that function.

Bellman stated that partial derivative of a function with respect to a space

variable could be approximated by a weighted linear combination of function

values evaluated at some intermediate points in the domain of that variable.

Compared to FEM or FDM, GDQM is relatively a new method used for solving a

system of differential equations. In addition to the Less complex algorithm, in

comparison to FEM, GDQM also offers increased efficiency of the solution by

demanding less number of grid points (hence, equations) to mode1 the problem.

Therefore, owing to the improved performances of GDQM, this method has gained

increasing popularity in solving a variety of engineering problems. Bert et al

10 used DQM for static and free vibration analysis of anisotropic plates,

while Laura and Gutierrez 18 used the method in vibration analysis of

rectangular plates with non-uniform boundary conditions. Sherboume and Padney 12

used DQM to analyse the buckling of composite beams and plates. They used

different number of grid spacing in their analysis. The same problem was

addressed by Wang 13, who also used different grid spacing. He found that

employing uniform grid spacing could result in an inaccurate solution;

therefore, caution should be exercised when using such spacing. Liew et al 14

used the method for the analysis of thick symmetric cross-ply laminates with

first order shear deformation while Kang et al 15 used it to address the vibration

and buckling analysis of circular arches. Bert et al 16 analysed the large

deflection problem of a thin orthotropic rectangular plate in bending. The

three nonlinear differential equations of equilibrium of the plate were

transformed into differential quadrature form and solved numerically using the

method of Newton-Raphson. Lin et al 17 used the same procedures to solve the

problem of large deflection of isotropic plates under thermal loading. In their

analysis they used the generalized differential quadrature of Shu and Richards

18. Bert also examined the equally spaced grids as well as unequally spaced

in several structural mechanic’s applications. The domain is divided into N

discrete points and cij are the weighting coefficients of the derivative.

i values are important factors control the quality of the approximation,

resulting from the application of GDQM.

(1)

(2)

2.1

Choice of the sampling points

In most cases, one can obtain

a more accurate solution by choosing a set of unequally spaced sampling points.

A common method is to select the zeros of orthogonal polynomials. M(x) is

defined in terms of the Legendre polynomials. M1(x) is the first

derivative of M(x). Here xi, xj, i,j=1, …, N are the

coordinates of the sampling points which may be chosen arbitrarily.

(3)

(4)

(5)

(6)

(7)

Also use of zeros of the shifted Legendre polynomials good results,

while some authors have chosen the grid points based on trial (Sherborne and

Pandey 19).

2.2

Boundary Conditions

Essential and natural

boundary conditions can be approximated by DQM; they are treated the same way

as the differential equations are. In the resulting system of algebraic

equations from GDQM, each boundary condition replaces the corresponding field

equation. Note that at each boundary point only one boundary condition be

satisfied. However, in the case of fourth order differential equations or the

higher order, one must satisfy more than one boundary condition at each

boundary. Wang proposed a method in which the weighting coefficient matrices

for each order derivative can be developed by incorporating the boundary

conditions in the GDQ discretization. This method has significant limitations

when dealing with the boundary conditions other than simply supported or

clamped. Malik and Bert also explored the benefits and the limitations of this

method for various types of boundary conditions. Shu and Du 19 proposed

another approach in which the derivative representing the two opposite edges

are coupled to provide two solutions at two neighbouring points to the edges. a

and b are dimensions and nx, ny are number of grid points

(test points) in direction of X and Y respectively where A, B and D are the

stiffness terms and W represent Transverse deflection.

For simply support

(8)

(9)

For clamped support

(10)

(11)

2.3

Domain Decomposition

For problems having complicated domains

such as those in delaminated plates or beams or plates with cutouts, the

concept of domain decomposition may be used for solving the problems. With this

concept, first the domain is divided into several subdomains. A local mesh can

be generated for each subdomain with more density near the boundaries. Then, GDQ

representation of the governing differential equations for each domain can be

formulated. In this approach, each region may have different number of sampling

points. Finally, the boundary conditions and the compatibility conditions at

the subdomain interfaces should be taken into consideration and satisfied.

3 GDQM Formulation of Delamination Buckling & POST

BUCKLING

The essence of the differential quadrature

method is that the partial (ordinary) derivatives of a function with respect to

a variable in governing equation are approximated by a weighted linear sum of

function values at all discrete points in that direction. Its weighting

coefficients do not relate to any special problem and only depend on the grid

space. Thus any partial differential equation can be easily reduced to a set of

algebraic equations. The Von Karman equations are then used to find an analytical

expression for the post buckling behaviour. After the derivation the results

are used to define the ‘effective width’ of a post buckled plate. The geometry

configuration factors, including number and cross-shape or profile of

stiffener, and lay-out or configuration of stiffeners, rib numbers, fibre angle

as well as the stacking sequences of CFRP, have a great influence on the

buckling behaviours of stiffened composite structures. The role of stiffeners

to increase the buckling capacity of plates without increasing the plate

thickness was researched, and it was concluded that by stiffening a flat

rectangular plate, its critical shear stress increases. The amount of this

increase depends on the aspect ratio and both the type and number of stiffeners.