1 buckling and post buckling of laminated composite

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

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.