ijaerv10n19spl_31

OPTIMIZATION OF COMPOSITE LAMINATES USING SIMULATED
ANNEALING ALGORITHM USING DIFFERENT FAILURE CRITERIA
Vignesh Chellappan. N a
,Surajit Das b
Department of Civil Engineering, National Institute of Technology, Agartala, India.
a- M.Tech Scholar (email id: vigneshchellappan_1992@yahoo.co.in).
b- Assistant Professor,Department of Civil Engineering, NIT Agartala.
ABSTRACT
In this study composite laminates subjected to an in-plane loadingunder plane stress condition is
optimized. Fiber orientation and number of plies are used as design variables.The failure criterions chosen
for analysis are themaximum stress criterion, Tsai-Wu criterion and Tsai-Hill criterion,in order to evaluate
the safety factor. Modified simulated annealing (SA) algorithm is used to search the global optimal designs
within the design space.Numerical results demonstrate the influence of failure criterion chosen to analyze
the model when subjected to different loadings and also suggest that Tsai-Hill and maximum stress criterion
produceeconomical design.
Keywords:Composite laminates, optimal design, classical lamination theory, SA, failure criterions.
1. INTRODUCTION
The major reasons for increased usage of composite laminates are their high stiffness to weight ratio
which is essential for light weight structures. Light weight structures are designed byoptimizing the
structure. Optimized design of composite laminates is achievedby considering their stacking sequence as
discussed by Hossien Ghiasi [1], fiber orientation and number of plies in the laminate. Many researches
attempted to optimize the usage of material either by reducing the weight through minimization of the
laminate thickness or by maximizing thestatic strength of composite laminates for a given thickness as
discussed by Albert. T.Groenwold[2].
The design variables arefiber orientation and thickness which could be treated as continuous variables as
well as discrete variables. In practice, thecomposite laminates are manufactured with specific thickness
(prepegs) and the fiber orientations are chosen from a finite set of angles due to the limitations like
arranging the fibers exactly in that orientation, during the process of design. As theypossess major
limitations, the variables are taken as discrete variables as proposed by Mustafa Ali Akubult [3].The failure
criterions also play a major role in optimization process. Generally the Tsai-Wu failure criterion is used,
butapplication ofthis criterion alone will yield false optimal designsas demonstrated by R.H.Lopez [4]. To
obtain optimal design atleast two or more failure criterions are required. In the present work Tsai-Hill (TH)
failure criterion and maximum stress criterion (MS) are used in addition to Tsai-Wu (TW) criterion in
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 19 (2015)
© Research India Publications ::: http://www.ripublication.com
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pairs(MS&TW, MS&TH) and altogether; in order to study the influence of the failure criterion in
optimization and the variation with the optimization results.
For evaluating discrete variables the local search algorithms are to be used.Ali et. Al [5]has discussed
that as opposed to local search algorithms the stochastic optimization would be a suitable tool as it avoids
the local optimum points.In this study, stochastic search algorithm called ‘Direct Search simulated
Annealing algorithm (DSA)’ proposed by Ali et. Al [5]is adopted.DSA is not sensitive to starting point and
can search a large design space for global optimal designs by eluding the local optimum points as they allow
occasional uphill moves.
2. PROBLEM FORMULATION
2.1 Problem Statement
. The composite laminate considered for this study is balanced, symmetric and multilayered 2D structure.
According to the classical lamination theory, when only in plane loading Nxx, Nyy,and shear loading Nxy in
XX, YY and XY direction are considered to be applied over laminates as shown in fig.1 neither bending nor
twisting are to be taken into account for analysis of its mechanical behavior.
Each lamina having a constant thickness ‘to’ and of any fiber orientation makes up the laminateto
atotal thickness‘t’. The interval between the consecutive angles is taken as 15°. The fiber orientation of a
lamina is confined in between -90° ≤ 0° ≤ 90°. The objective is to find out the minimum thickness by
applying the failure criterions. Finally on comparing the results, the optimal design is found out for the
given loading. Reliability of the algorithm is evaluated by the ratio number of global optimums obtained
with respect to the number of trial runs executed.
Figure 1.Composite laminate subjected to in-plane loading
The total number of distinct angle of plies is denoted by m. While nk the number of plies in the kth
lamina, where the fiber orientation of plies is k .The factor 2 appears in the equation (1) is due to the
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symmetry condition of the laminate with respect to middle plane.Since the plies are made up of same
material, minimizing the thickness may lead to the same optimum configuration as the minimization of
weight. The laminate thickness is expressed as
1
2
m
o k
k
t t n

  (1)
2.2.Static or Material failure criteria
Weight Maximization and weight minimization of laminates necessarily involves strength constraints, as
variation inthe number of load carrying plies would result in uneconomical design or failure. Therequired
optimum thickness needs to be evaluated against the strength, bucklingand delamination parameters as
proposed by Ozgur Erdal [6].Static failure criterions such as Tsai-Wu, Tsai-Hill criterions incorporates the
interactive effects of the laminate, while the Maximum stress criterion predicts the failure of laminate if a
principal stress exceeds in any direction under plane stress condition.Other failure criterion’s like Hashin,
Feng’s, Puckas discussed by R.H.Lopez [4] maybe considered for generating the optimal configurations as
they deal with fibre or matrix failure.
3. METHODOLOGY
3.1 Objective Function
Failure of any ply signals inception of failure of the whole structure, even though its ultimate load
bearing capacity may not be exceeded. For this reason, this is considered as a design limit.Accordingly, the
first-ply failure approach is adopted in the design optimization, and during the optimization process safety
of each lamina in a laminate is checked using the Tsai–Wu, Tsai-Hill and Maximum Stress failure
criterion.Penalty functions are included as the failure is expected if one of the inequalities in Tsai-Wu, Tsai-
Hill and Maximum stress failure criterions is not satisfied for one of the lamina. The objective function is
presented using failure criterions which will be appropriate, Albert. T.Groenwold[2], Mustafa Ali Akubult
[3] and S.S.Rao [7].The objective function (F) is,
0 1 2 3 1 2 3
1
2
m
k MS TW TH MS TW TH
i
F t n C P C P C P C SF C SF C SF

       (2)
Where, the first term in the equation (2) represents the total thickness. MSP TWP & THP are the penalty values
of maximum stress, Tsai-Wu and Tsai-Hill criterion respectively.The second, third and fourth terms of the
equation (2) is used to increase the value of objective function for the designs which are predicted to fail and
thus restrict the search within feasible design space. MSSF TWSF & THSF are the safety factor of maximum
stress, Tsai-Wu and Tsai-Hill criterion respectively if they are greater 1.0, else these terms are zero. Ci are
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the suitable coefficients, for which the values are taken based on the in-plane loads or out-plane loads.Ci is
used minimize the failure criterion in order to formulate the objective function.The reason that the objective
function is reduced for safe designs is that there may be possible design with the minimum thickness. From
the designs obtained, the optimum design would be corresponding to the design that carries the maximum
load. The safety factor of the laminate according to the maximum stress criterion is calculated by the
following equations; first the principal stress 11 22 12, ,k k k
   in each lamina is calculated, then the safety factor
for each failure mode is calculated. Then the minimum of them is denoted as the maximum stress criterion.
Where the subscript t, c denotes the tensile and compressive strength; strength of fiber in the axial and
transverse direction are denoted by X and Y, S is the ultimate in-plane shear strength of the laminate under
pure shear loading.The safety factor for the maximum stress criterion SFMSis equal to the minimum of the
factor of safety ( k
MSSF ) for all the laminas.
SF MS= min of k
MSSF where K= 1, 2 … m-1, m. (4)
The penalty value of the maximum stress criterion is calculated when the respective configuration violates
its safety factor.The total penalty value of the laminate due to violation of the maximum stress criterion is
the summation of penalty values in each lamina as,
1
m
k k k
MS x y s
k
P P P P

  
(5)
Safety factor of kth
lamina according to Tsai-Wu stress Criterion as the multiplier of stress components at
lamina k, is
a( k
TwSF )2
+b( k
TwSF )=1 (6)
 
11 11
11 11
22 22
22 22
12
/ 0
/ 0
/ 0
min
/ 0
/
tk
x
c
tk k
MS y
c
k
s
X if
SF
X if
Y if
SF of SF
Y if
SF S
 
 
 
 

    
   
   
    
   
   
 
 
  
0 SF 1.0
((1/SF ) 1) SF 1.0
0 SF 1.0
((1/SF ) 1) SF 1.0
0 SF 1.0
((1/SF ) 1) SF 1.0
k
k x
x k k
x x
k
yk
y k k
y y
k
k s
s k k
s s
if
P
if
if
P
if
if
P
if
         
  
     
 
          
  
      
 
                 
(3)
a=
     
2 2 2
11 22 12 11 22
2
k k k k k
t c t c t c t c
X X Y Y S X X YY
     
   
 
 
b=    11 22
1 1 1 1k k
t c t cX X Y Y
 
   
        
   
(7)
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Equation (7)is used to obtain the safety factor of the laminate by employing Tsai-Wu stress criterion.
Negative value obtained from the root of the equation is not considered while the positive value obtained
from the root equation is considered for each lamina.
Then, the minimum of k
TwSF would be chosen as the safety factor of laminate,
SF TW = min of k
TwSF where K= 1, 2… m-1, m. (8)
The penalty k
TwP and the total penalty value TwP of Tsai-Wu criterion due tothe violationsafety factor is
obtained by the following equation,
0 SF 1.0
((1/ SF ) 1) SF 1.0
k
k Tw
Tw k k
Tw Tw
if
P
if
        
  
    
1
m
k
Tw Tw
k
P P

 
(9)
Similarly for Tsai-Hill failure criterion the safety factor and the penalty equation are as follows.
a( k
THSF )2
=1 (10)
a =
     
2 2 2
11 22 12 11 22
2
k k k k k
t c t c t c t c
X X Y Y S X X YY
     
   
 
 
(11)
Where Xt=Xc, Yt=Yc while using Tsai-Hill criterion.The minimum of k
THSF would be chosen as the safety
factor of laminate,
SF TH= min of k
THSF where K= 1, 2… m-1, m. (12)
The penalty value of Tsai-Hill criterion due to theviolation of the safety factor is calculated in a
similar manner to Tsai-Wu criterion and the total penalty value is obtained by the following equation,
0 SF 1.0
((1/ SF ) 1) SF 1.0
k
k TH
TH k k
TH TH
if
P
if
        
  
    
1
m
k
TH TH
k
P P

 
(13)
From these equations, the objective function for the problem to obtain the minimum thickness is formulated.
These equations are obtained from the classical lamination theory and indirect methods of optimization
techniques.
3.2 Optimization Procedure
3.2.1 Generation of new configurations
The objective function formulated in previous stepsis executed in DSA to obtain the cost of different
set of configurations and by using the cost function the laminates are optimized. Simulated annealing
proposed by Kirkpatrik et. Al [8] is modified to DSA by two steps i.e. at the start of theoptimization the
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procedure unlike SA, DSA starts with a set of current configurations rather than a single set configuration.
Then it searches the neighborhood of these current N configurations in DSA instead of only one point in
SA. The number of these configurations depends upon the dimension of the problem,
No. of initial configurations N = X(2m+1) (14)
‘X’ can have any value from 1 to 10 normally.Consideringthis it is started with a set of configurations at the
same time.‘X’ is taken as 7 and‘2m ’ is the number design variables adopted.
3.2.2 Acceptance Criteria
To accept the newly generated configurations, the DSA is used. Acceptance of the generated designs
is based on the value of cost function which relies on the objective function, as
j
1 ( )
exp(f )/ T ( )
t h
t
h t t h
if f f
A
f if f f
  
  
   
1 1j j jT T  (15)
Where fh is the highest cost in the current set. This means that every new design (trial configuration
ft) having a cost lower than the cost of the worst design or best configuration is accepted. On the other hand
if the cost is higher, the trial configuration may be accepted depending on the value of At. If it is greater than
a randomly generated number, Pr[-1,1], the trial configuration is accepted,otherwise it is rejected. If the trial
design is accepted, it replaces the worst configuration. fh and fl are updated in each iteration. At high
temperatures it is unlikely to from a dense cluster, which means that the current configurations are scattered
around the solution domain. At low temperatures the chance of accepting the worst configurations are
low.As annealing is a process of melting and freezing,the temperature (Tj) is a control parameter of
algorithm while it has no physical meaning.Tj is kept constant for certain iterations in the jth
Markov chain,
after which it is reduced by 1j  . The range of 1j  lies within [ min , max ] which is usually 0.85 for a
reliable convergence. Thus 1jT  is reduced linearly by small or large variations depending on the slow
convergence or quick convergence of the algorithm to yield the result depending on the assigned value of
1j  .
4. RESULTS AND DISCUSSION
A graphite/epoxy material is considered for optimization lay-up sequence for analyzing and
designing. T300/5308 with the material properties, E11= 40.91GPa, E22= 9.88GPa, G12= 2.84GPa, ѵ 12=
0.292GPa, Xt= 779 Mpa, Xc =-1134MPa, Yt= 19MPa, S = 75MPa, Yc= -131MPa, to =0.127mm is
considered.The laminates of any configuration with in-plane uniaxial or biaxial and shear loading are
evaluated.
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The result usually expectedon uniaxial loading when all thefibers of laminate isaligned along the
load direction while thefactor of safetymustbe more than one indicating that laminate is much stiffer. None
of the designsshowed higher factor of safety while the three failure criterions are used separately. On
comparing these results it may be explicitly said that the failure criteria chosen plays a major role in
searching the optimum design and it is also suggested that to ascertain the optimum design at least two
failure criterions are required.
Table 1 Failure criterion used and their effects (variations in ϴ=1°)
Failure Criteria
used to check
feasibility
Optimum
layup sequence
Loading
Nxx/Nyy/Nxy
Mpa m
Half
laminate
thickness
S.F. of
Tsai-Wu
S.F. of Max
stress
S.F. of Tsai-
Hill
Only Tsai-Wu [-925/1022]s
100/0/0 47 1.0007 0.9142 0.3639
Only
Max Stress
[551/ϴ0]s
100/0/0 51 0.6688 1.0168 0.6420
Only Tsai-Hill [051/190]s
100/0/0 51 0.6147 0.9463 1.0183
Both (MS & TW) [025/228]s
100/0/0 53 1.0441 1.0137 1.0480
Both (MS & TH) [325/026]s
100/0/0 51 1.0040 0.9392 1.0111
All (MS, TH & TW) [051/051-j]s
100/0/0 51 1.0091 1.0091 1.0183
Generally, Tsai-Wu criterion with maximum stress criterionis employed to calculate the failure
stress of any composite material since Tsai-Hill predicts the values only based on orientation. On the other
hand, on optimizationthe Tsai-Hill criterion proves to be economical with maximum stress criterion by
reducing the ply thickness. For example, in the table 1, when Tsai-Wu and Maximum stress criterion is used
to find the optimum design, Tsai-Hill criterion also demonstrates that the design is safe but the design has
more number of plies in comparison to another design for same loading when Tsai-Hill and maximum stress
criterion is used.It is unclear as it is tested only for limited practical values. Use of all the three failure
criterions to obtain the optimum designs proves to be overly conservative and also requires considerable
computational effort.
Table 2 Variations of thickness and Safety Factor with respect to failure criterion and loading conditions.
Loading
Nxx/Nyy/Nxy
Mpa m
Optimum layup
sequence
Half
laminate
thickness
S.F. of
Tsai-Wu
S.F. of
Max stress
S.F. of
Tsai-Hill
100/0/0 [0j/051-j]s(j<=51) 51 1.0091 1.0091 1.0183
10/0/0 [04/02]s 6 1.1872 1.1872 1.4094
10/10/0 [9047/047]s, [6047/-3047]s 94 1.0009 1.0050 1.0939
10/10/10 [4511/450]s
11 1.0883 1.0883 1.1843
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The increased complexity in computational process due to the coefficients ofthree failure criterions
needs to be assigned properly, to find optimal design by calculating the factor of safety and penalty values
else it will again yield incorrect optimal designs.A minimum of 94 plies are required when two different
fibre orientations of 0, 90 are used to obtain a balanced lay-up configurations [047/9047], [6047/-3047] as
shown in table 2 for biaxial loading cases. The results demonstrate that for a symmetric laminate layup or
balanced layup sequence there is no effect of stacking sequence, if the laminate is subjected to in-plane
loadings. One interesting load case is shear loading, on applying Nxx/Nyy/Nxy = 10/10/10 Mpa m the
laminate thickness reduced from 94 to 11 as half laminate thickness since the laminate fibers are arranged in
45°, else the laminate thickness was much higher than 11, so that the principal and transverse stresses might
be reduced.The reliability of simulated annealing algorithm is found to be satisfactory as we can see S.F. of
different failure criterions is nearly equal.
5. CONCLUSION
The results are obtained for T300/5308 graphite epoxy material for in-plane loading. Generally,
Tsai-Wu and maximum stress criterion is used for optimization under plane stress condition for in-plane
loading but it results inless economical design for some configurations when compared with Tsai-Hill and
maximum stress criterion. As each failure criterion used to find out the optimal design has their
ownlimitations,the failure criterions must be used in combination with other theories and not individually to
obtain the correct optimal design. The results should be verified further with other composite laminate
materials and also needs to be experimented. This study could furtherbe extended by including the effects of
matrix or fibre failure for in-plane or out-plane loading of the laminate.Modified simulated annealing
algorithm proves to be more reliable computationally.
REFERENCES
1. Hossein Ghiasi, Damino Pasini, Larry Lassard (2009), ‘Optimum Stacking sequence design of composite materials
part-I: Constant Stiffness design ’, Composite Structures, Elseveir, 90, 1-11.
2. Albert. T.Groenwold and Raphael T.Haftka (2006) ‘Optimization with non-homogeneous failure criteria like Tsai–
Wufor composite laminates’, springer, 32, 183-190.
3. Mustafa Akbulut, Fazil O.Sonmez (2008), ‘optimum design of composite laminates for minimum thickness’,Journal of
computers and structures, Elsevier, 86,1974-82.
4. R.H.Lopez, M.A.Leursen, E.S.Cursi (2011), ‘Optimization of laminated composites considering different failure criteria’,
journal on Composites Part B, 40, 731-40.
5. M.M.Ali, A.Torn, S.Viitanen (2002), ‘A direct search variant of the simulated annealing algorithm for optimization
involving continuous variables’, journal of computers and operations research, PERGAMON, 29, 87-102.
6. Ozgur Erdal, Fazil O.Sonmez (2005), ‘Optimum design of composite laminates for maximum buckling load capacity using
simulated annealing’, journal of composite structures, ELSEVIER, 71, 45-52.
7. S.S. Rao, ‘Engineering Optimization’, ISBN 978-0-470-18352-6, John Wiley & Sons, 2009, Fourth edition.
8. Kirkpatrick S, Gelatt Cd,Vecchi MP. (1983) ‘Optimization by Simulated Annealing ’, Science; 220, 671-80.
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