ADAPTIVE FINITE STRIP ANALYSIS OF LAYERED FIBRE COMPOSITE PLATES
K P Beena1 and V Kalyanaraman2
1
2
PhD Scholar, Professor, Dept of Civil Engineering, IIT Madras, Chennai-600016, India
Introduction
Composites are a combination of two or more materials
to form a new material that contains the best features of
each constituent, so as to optimize certain parameters
like strength, stiffness, toughness, fatigue, corrosion
resistance, temperature dependent behaviour, thermal
insulation, thermal conductivity, acoustical insulation,
etc. Due to their light weight and ease of fabrication,
glass and carbon fiber composites are extensively used
in plates and shells encountered in aerospace
applications. They may be subjected to a variety of static
and dynamic loads causing membrane as well as shear
force and bending moment. Since the laminated fibers
are anisotropic, the analysis is more involved, even
though such composites are desirable functionally.
The Finite Element Method has been extensively used
for the analysis of plated structures. Here the whole
structure has to be discretised into a number of elements.
The computational requirements of the FEM, in terms of
storage space and time, are very high, especially in linear
prismatic members having elements of small width.
Although the classical finite strip method (CFSM)
(Cheung 1976) can be used efficiently for the static,
dynamic, stability and non-linear analysis of such linear
thin walled members of regular geometry, it fails to
effectively deal with complex boundary conditions and
partial and concentrated loads, since the trigonometric
functions are infinitely continuous. This is overcome in
the spline finite strip method (SFSM) by replacing the
trigonometric function in the longitudinal direction by a
spline function. SFSM using unequal splines are more
efficient when a member is subjected to concentrated
loads or reactions, when the support of members are
either isolated or at irregular locations and when cutouts
are present.
are extensions of the conventional, isotropic plate
theories which are based on assumed variation of
displacements through the plate thickness. On extending
the single layer theories of homogeneous plates to the
laminated composite plates, we have to account for the
varying layer thickness and material properties in the
evaluation of displacement field. It is often assumed that
the layers are perfectly bonded and the heterogeneous
laminate is represented by a statically equivalent single
layer whose stiffnesses are a weighted average of the
layer stiffnesses through the thickness.
But these theories do not ensure a continuous variation
of transverse shear stresses across layer interfaces
leading to less accurate results, especially for
asymmetric laminates. But these deficiencies can be
solved efficiently by superimposing a zigzag linearly
varying in-plane displacement on a cubic varying
displacement field as proposed by Cho and Parameter
(1993). The cubic variation accounts for the overall
parabolic distribution of transverse shear strains known
from Reddy’s theory, while the zigzag accounts for the
strain discontinuities required for stress continuity
conditions across interlayer surfaces. Hence this theory
is used here for the analysis of laminated plates which
will efficiently take into account the shear deformation
effects.
Zigzag Theory
Based on Cho’s higher order zigzag theory(1993) and
Shu’s(1994) derivation, displacements u, v and w at any
point (x,y,z) within a layer k of the laminate have the
following relationships with the midplane displacements
u  u0   z w'   F  z  k  
w  w0  x, y 
where
The efficiency and accuracy of the SFSM with unequal
knot spacing depends on the discretisation, characterized
by the number of strips and the number and spacing of
knots. No well established criterion exists for deciding
on the spacing of knots and choice of strip width. As in
adaptive FEM, a measure of the error in the SFSM can
help in these modeling issues. In this paper the adaptive
meshing approach used in FEM, based on error analysis,
is extended to the SFSM. Locally Weighted Projection
Regression (LWPR) has been adopted for the first time
in this study to obtain smoothed stress field. It is shown
through an example that a more rational SFSM model
and analysis results to the desired level of accuracy can
be obtained by this SFSM approach, in the case of fibre
composite plates loaded and supported out of plane.
Most of the composite plate theories like classical
laminate plate theory(CLPT), first order shear
deformation
theory(FSDT),
third
order
shear
deformation theory(TSDT) which are used for analysis
u  u, v  , w'    wx' , w'y  ,
 F  z     F1 k  z  F2 k  z 2  F3   z 3  F4 
k
T
 F  F  F 
F 
1 k
4
in which
, 2 k , 3 and
are determined
from the material properties and laminate thicknesses.
Here the unknowns of different planes are expressed in
terms of those of the midplane after imposing the interlaminar shear stress continuity conditions and ensuring
the plate surfaces are free of transverse shear stresses.
Thus, the number of unknowns remains the same, but a
continuous distribution of shear stresses is achieved. In
this way, the transverse shear stresses can be calculated
directly and correctly from the constitutive equations,
and an artificial shear correction factor used in the first
order shear theory is no longer needed. In addition, the
assumed displacement variation through plate thickness
has been expressed explicitly in terms of material
properties and laminate thicknesses. Consequently, little
Error Estimation
The discretisation error caused by the assumed
displacement field approximations, resulting in
discontinuity of stresses ̂ , can be reduced by adaptive
meshing. A convenient estimate is obtained by the
procedure of Zienkiewicz and Zhu (1987), who proposed
a simple error estimator and a refinement strategy in
1
terms of energy norms as

 2
T
|| e ||     ˆ  D 1   ˆ  d  


where
‘ ̂ ’
is
the
approximated value of stresses and ‘σ’ is the exact value.
Since the exact value is not known, best guessed stresses
provided by a smoothed stress field are used to obtain
stresses which are continuous across the strips and
section knots. In this study adaptive mesh in the SFSM
formulation is evaluated, using a new smoothening
technique called LWPR.
LWPR Algorithm
LWPR is an algorithm that is helpful to represent the
implicit relationship between an input and output
(Vijayakumar, 2005).
It uses the concept of
approximating the non-linear functional relationship in a
region by using the weighted sum of many locally linear
models in the neighborhood as
yˆ  w yˆ / w ,

k
where
k
k

k
k
ŷ k is the hyper plane representing the functional
parameter within a domain referred to as the kth receptive
field. More details can be obtained from Vijayakumar et
al (2005).
Refinement Strategy
The refinement ratio,  i , defined as the ratio between
actual error and allowable error. Which ever segment has
 i >1 is refined. The newly generated segment size
1
should be no larger than
hnew   i
p
hi , where p is
the degree of polynomial used for displacement
formulation.
Example A simply supported symmetric cross ply
square laminate of size 100mmX100mmX25mm with
lay up 0/90/90/0 is analysed for uniformly distributed
loading. E1/E2=25, ν12=0.25, G12= G12=0.5E2.Taking
advantage of the symmetry only one quarter of the plate
is analysed. Meshing is done to get an error less than
1%. Figure 1 shows the final adaptive mesh obtained
which incorporates the effects of shear deformation
using the above zigzag theory. It is observed that there is
a higher gradient of stress near the support in y direction
since the stiffness is less in that direction. Hence more
no of knots are required.
Conclusions
It was shown that the LWPR can be used effectively to
smoothen the stress field for error analysis. The adaptive
SFSM is able to adjust the strip width and knot spacing
efficiently, depending on the stress gradient to achieve
the desired level of accuracy in energy norm.
Acknowledgements
The author acknowledges the financial support by the
Aeronautical Research and Development Board.
References
Cheung, Y. K. (1976) , Finite Strip Method in Structural Analysis,
Pergamon Press, Oxford
Maenghyo Cho and R Reid Parmerter(1993). Efficient Higher order
Plate Theory for General Lamination Configurations. AIAAJournal
Vol 31,No 7
Vijayakumar, D,Souza, S A and Schaal`S, (2005) Incremental
Learning in High Dimensions, Neural Computation, Massachusetts
Institute of Technology ,17, .2602-2634.
Zienkiewicz, O.C. and J.Z. Zhu, (1987) A Simple Error Estimator and
Adaptive Procedure for Practical Engineering Analysis, International
Journal for Numerical Methods in Engineering,24, 337-357.
Y
SS
SS
Using appropriate refinement strategies the accuracy of
the solution can be improved, particularly in segments
exhibiting higher error. The different refinement
strategies used are h- refinement (where the element is
sub-divided into a number of smaller size elements), prefinement (where the degree of polynomial representing
the field variable is changed) and r-refinement (where
the elements are resized retaining the same number of
degrees of freedom). Typically the mesh density is
higher near the regions of steep gradients of field
variable. By r-adaption process only an improvement of
existing solution is possible. i.e an optimal mesh can be
obtained, the ultimate aim of achieving a specified
accuracy cannot be realized. This can be achieved only
by successive mesh enrichment (h-refinement). In mesh
enrichment the estimated error from the current solution
is used to predict the desired segment or strip size, which
may be used to reconstruct an entirely new discretization
(remeshing). This process helps in achieving an optimal
mesh in which, the number of degrees of freedom is
minimal for a specified accuracy. The properties of
mesh enrichment and r-adaption are complimentary.
This led to the idea that the advantageous properties of
the two schemes could be combined. It is observed that
an optimal mesh is obtained for a lesser degrees of
freedom since the error is redistributed and hence this
combined strategy is used here.
Symmetry
extra computational effort is required in the solution
process. Hence this theory is adopted here for analysis.
CL
X
Symmetry
CL
Figure 1. Quadrant Model of a simply supported square symmetric
cross ply laminate under uniform load.