In the shifted Legendre polynomials good results, while some

     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 boundary
conditions maybe estimated by DQM; similar to the differential equations are.
From the algebraic equations, each boundary condition substitutes the
equivalent equation, that at each point only a single boundary condition is
satisfied. However, in the case of fourth order differential equations or the
higher order, one must satisfy more than one boundary condition at an specific
boundary. Wang suggested a method base on weighting coefficient matrices by
incorporating the boundary conditions in the GDQ discretization, by significant
limitations in boundary conditions for simply supported or clamped with more
than one boundary condition. Malik and Bert also explored the benefits and the
limitations of this method for various types of boundary conditions. Shu and Du
19 suggested different approach in which two opposite edges are coupled to
provide two solutions for two neighbouring points to the edges.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. 1      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.

 

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