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A comparison between two optimization methods on the stacking sequence of fiber-reinforced composite laminate
Conrpurm & Swucrum Vol. 55, No. 3. PP. 515-525, 1995 CopyrIght 1995 Els&er Science Ltd Printed m Great Britain. All rkhts reserved
Asa%
t
> Pergamon
A COMPARISON BETWEEN TWO OPTIMIZATION METHODS ON THE STACKING SEQUENCE OF FIBER-REINFORCED COMPOSITE LAMINATE Li-ren TsauP and Chyuan-huei Liu$ TDepartment
of Mechanical
$Chung-Shan
Institute
Engineering,
Chung Cheng Institute of Technology, Ta-Shi, Taiwan 33509, Republic of China of Science and Technology, Taichung, Taiwan, Republic of China (Received I4 November 1993)
Abstract-This article is dealing with the study of optimization for the stacking sequence of composite laminates by two optimization methods of simplex and interior penalty. In the simplex method, both the applicable domain of magnifying factors (MFs) and the differences of stacking sequence optimization between symmetrical and unsymmetrical laminate composites are studied. When the values of MFs are 50, 70, 85, the converging of optimization process is faster and its result is more accurate. The strength of the unsymmetrical laminate with fewer layers is larger than that of the symmetric one by 8% only. But when the layers are increased, the strength of symmetric laminate is higher. In the interior penalty method, the error value can be kept within 5%6%, while in the simplex method the more layers the larger the error value. To sum up, proved by progressive failure simulation of finite element analysis, the best strength of composite laminate with optimal stacking sequence can be obtained.
NOTATION
composite lamina cannot meet the strength requirement, composite laminates stacked by laminae are developed. There have been many researches about the stacking problem of composite laminates. Some used the factors of frequency, damping, displacement or material sizes as constraints to get the thickness and stacking sequence of a laminate under load without failure [I, 21. Some solved this problem by classic laminate theory (CLT) [3,4]. Flangan and Palazotto [5] used Tsai-Wu failure criterion and firstply failure to judge the failure in order to find suitable thickness under the limitation of strength. However, since CLT theory could not solve such problems as interlaminar stress and free edge effect nor deal with as many layers as needed [4, 61. it could not be applied to analyzing the complex structure in a composite laminate. A laminate is usually composed of pre-preg laminae, so when designing a suitable laminate. the number of laminate layers instead of its thickness is used as the object of analysis [7, 81. If stacking scquence only is used as a design variable. then the simplex method, with features of nonlinear, nonconstraint and direct search, can be used. in which the stress state analyzed by the finite element method is substituted into Hashin Failure criterion to get an objective function value. However, the range of magnifying factors (MFs) which influcncc the converging and accuracy was not discussed [9]. Neither did Nelder and Mead [IO] deal with the effect of MFs. though they made tests on reflection. expansion, contraction coefficients of the simplex method. Therefore, one of the purposes of this research is to study
stiffness matrix of whole structure displacement matrix of nodal points force distribution matrix of nodal points operator matrix for element the general stress-strain relation [I I] Jacobian transformation matrix local coordinate system range from - 1 to + I hygrothermal strains in each element residual strain in each element
laminar thermal coefficient of expansion laminar moisture coefficient of expansion temperature change in each element percent weight gained by moisture absorption tensile strength in fiber direction compressive strength in fiber direction tensile strength transverse to fiber direction compressive strength transverse to fiber direction transverse shear strength axial shear strength
1. INTRODUCTION
A composite material is composed of two or more kinds of materials and has its own properties. Unidirectional fiber-reinforced composite materials have been widely applied in the aerospace industry. This kind of material has a very good strength/weight ratio, whose structure strength is controlled by its anisotropic features. Reinforced fibers dominate the longitudinal property and sustain most load, while matrix materials, such as epoxy, dominate the transverse property and transfer load among fibers. Therefore, since the transverse property of a unidirectional 515
Li-ren Tsau and Chyuan-huei
516 Table
I. Properties
of graphite/epoxy
Property Ply thickness Longitudinal modulus E,? Transverse modulus E,, Shear modulus G,, Major Poisson’s ratio r’,: Minor Poisson‘s ratio rl?, Longitudinal tensile strength Longitudinal compress& strength Transverse tensile strength Transverse compressive strength In-plane shear strength
SP-286T300 Value 0.0051 in 21.90 x IO” psi 1.53 x IOhpsi 0.96 x lOh psi 0.31 0.014 203 ksi 164 ksi 7.8 ksi 30.6 ksi 10.5 ksi
the effects of MFs. Moreover, besides the factors of stacking sequence, there are problems of thickness, fiber volume, displacement and sizes to be considered. And it becomes a nonlinear constraint problem, and so the simplex method becomes unsuitable. Cauchy [I I ] developed a steepest descent method, which was prone to get a local minimum, whose optimization path appeared zigzag when approaching the minimum, and whose convergence in solving complex problems was very slow. Fletcher and Reeves 1121greatly modified it as the conjugate gradient method, which. better than steep descent method but worse than the quasi-Newton method, converged only after numerous calculations resulting from the errors accumulated in each calculating period. The converging of the quasi-Newton method was faster than that of any direct research method and could avoid approaching maximum or critical value. However, it had the demerit of being unable to calculate Jacobian matrix, when inverse matrix becomes inhnitc. especially in the problem of complex structure. Fletcher and Powell improved on this descent method by a variable metric method and developed one of the most efficient methods for solving the problem of unconditional optimization [I 3:. Therefore, when considering conditional problems, this optimal design method is applied to this research to discuss its feasibility. Hence, the objective of this article is to hnd the MFs suitable for the simplex method. to test the feasibility of the interior penalty method, and to study the differences between these two methods. Besides, this article also deals with the optimization of symmetrical and unsymmetrical composite laminate by analyzing the relation between error value and the number of variables (such as layer numbers). In this article. the macromechanics approach is used to study the stress and strain of a composite lamina under the assumption that on the individual ply level the fiber-reinforced material is continuous. linear elastic. homogeneous, orthotropic. The I6 x 16 in SP-286T300 graphite/epoxy plate laminates with a central hole and without one are used for analysis and the properties of single laminae are given in Table I. The residual stress is due to a curing temperature difference of (76 - 350 = -274 F) and a moisture content of 0.6%.
Liu
2. FAILURE
CRITERION
AND
PROGRESSIVE
FAILURE
In a very broad sense, the failure of a structural element can be stated to have taken place when it ceases to perform satisfactorily. Therefore, the definition of failure will change from one application to another. In the case of composite materials, internal material failure usually takes place much earlier, even before any change in the macromechanical appearance of structure. Owing to the features of heterogeneity and anisotropy of composite materials, their failure characteristics are completely different from those of isotropic materials. In metals, the appearing of detectable cracks is generally considered unsafe because a little damage can rapidly grow to final fracture. In composite materials, however, this is not always so; although internal damage may appear very early, its propagation is arrested by the internal structure of the composite. Therefore, composite laminates can still sustain much higher loads after the occurrence of localized damage such as matrix cracking or delamination [l4-161. Hashin developed a failure criterion in which the failure modes, tensile fiber, compressive fiber, tensile matrix and compressive matrix failure are dealt with by four separate criteria [l4]. And during the process the failure modes of composites can be identified, which are of significance to finite element analysis of progressive failure. These four criteria are: (I) tensile fiber mode oLL > 0
(2) tensile matrix
mode mrT + (T,, > 0
Cfl(q, +o,,Y= , (3) compressive
matrix
c7,,
(3)
:
mode oTT + gLL< 0
+:2(~i.,-“ir’T,,)+ili((ilT+(i;/)=
I
(2)
fiber mode oLL < 0 OLL =
(4) compressive
I;
h
I.
(4)
If any failure function value is greater than 1.0, then failure occurs. It is matrix failure. if the matrix failure function value is larger: it is fiber failure if its function value is larger. The elements that have failed are assumed to have reduced stiffness for subsequent
Stacking
sequence
of fiber-reinforced
in the progressive failure analysis. It has been proposed [17] that the cracked plies be replaced with a continuum of lower stiffness so that conventional stress analysis in the finite element method can be applied. Following the recommendation in the matrix degradation for the graphite/epoxy fiber-reinforced composite 45% of its original value, and that of the shear modules to 35% of its original value, the major Poisson’s ratio is reduced to 30% of its original value. The fiber direction stiffness decreases gradually with increasing strain after the first fiber fails. Therefore, all the terms in the stiffness matrix for the elements with failure fiber are conservatively assumed to be zero in the subsequent analysis. And, for reasonably saving the computing time of calculating the laminate with thousands of elements and that of getting the load increment for sufficiently accurate results, a 45% increment loading rate [I81 was used in this investigation. All the elements which have either a fiber or matrix failure function value equal to or larger than 1.0 are recorded as a basis to modify stiffness to recalculate the stress distribution. Then the laminate was loaded with the same load. If this load did not make further damage, 45% of the initial failure load was increased and the same process was repeated. When the load did not make any damage with a two-fold load increment, and fiber failure occurred in all of the elements lying in a particular radial direction out to about half the radius of the hole in almost all the layers of the laminae, then those elements in the same radial direction as the remaining laminae could not sustain a load perpendicular to the fiber direction. Therefore, this was considered to be catastrophic failure [6, 181. The current step was redone with a half load increment if a final catastrophic failure was found [ 191; that is, the failure function values of fiber failure for elements failing in this step are much larger than 1.O. The increment was reduced further until the values of the failure function were 1.0 or just a little larger than I.0 to obtain the ultimate strength of a laminate. If the failure function was less than 1.O, the load was increased by half the previous decrement. analysis
3. FINITE
ELEMENT
ANALYSIS
The three-dimensional finite element mesh consists of a family of eight-node elements. The element equation which links the nodal point displacement vector [U] and the nodal point force vector can be written as [20]:
composite
laminate
517
Changes in temperature or moisture will cause strains in materials. One way of calculating the residual strains in the finite element model is first to calculate the nodal forces necessary to keep the nodes unmovable and thus unstrained. If one then imposes equal and opposite nodal forces, the set of nodal forces will be zeros and the residual strains can be calculated by using the stiffness matrix and these artificial nodal forces. The hygrothermal strains in each element are given by:
kTll = aoWl, + P,[AW. The 24 components of the equivalent element can be computed by:
The total equivalent laminate is:
Then the residual from:
(6) forces of each
force at each nodal point of the
nodal displacement
can be found
[UT = j%‘[%>
(9)
where the stiffness matrix and the equivalent forces are known. After computing the residual nodal displacements, the residual strains are found from:
M = {B),([W),
(a = 1,.
, m).
(10)
Finally, the residual stress can be computed with the stress-strain relation for a linearly elastic material subjected to mechanical strain using the equivalent forces needed to hold the finite element in an unstrained state. Thus, assuming linear elastic behavior, the hygrothermal stresses of the material in each element in accordance with the generalized Hooke’s law are:
bl, = {~~,(k”L - kTL).
(11)
The problem investigated in this study is that of a square plate made of a layered composite material, with and without a centered circular hole subjected to in-plane loading. The applied load was uniformly distributed in the x direction on x = + L and/or in the y direction on y = + L, as shown in Fig. 1. 4. SIMPLEX
METHOD
where The simplex method is a direct search for solving nonconstraint problems. Each search only takes one calculation to get an objective function value. If the objective function value has no abrupt variation, then
Li-ren Tsau and Chyuan-huei
518
Liu
L
F
’ r
Fig. I. The mesh and loading the searching path is followed to get the optimum. Though fiber angles are usually - 90” < 0 d 90” when analyzing the problems of stacking sequences, yet the angles obtained are larger than k 90’, which in turn are divided by 90 to get the remainder, which are the angle values we need. Therefore, the nonconstrained simplex method can be applied. The process is as follows: first, calculate the objective function values of the n + 1 verticals in the simplex and compare them. Second, make the simplex gradually approach the smallest objective function value through the searching mechanism, which includes (1) reflection, (2) expansion, (3) contraction [19]. If the smaller objective function values cannot be obtained in the process of reflection or contraction, or when oscillation phenomenon happens, then discard the abovementioned searching method, and change the searching direction by way of contracting the whole simplex toward the smallest objective function value. The search mechanism in simplex method helps discard all the worst objective function values one by one until it finds the optimum. The selection of the searching direction is decided not only by the objective function value, but also the selection of the initial simplex. In order to cover the domain of all the direction, we set
*I
a2
(4 +P) (u,+y)
(a.?+Y)
(4 +4)
(a?+Y)
(az+P)
.’ ”
a, (ai + Y) (ai, + 4)
Do=
1
p=k;‘zi(k -
”
I)+ (k
6%+-PI ~
of a laminate
with a central
4=, &(k
hole.
+
I)‘?- I}.
In the D, matrix, each row represents a set of stacking sequences, in which D, is the initial simplex, k is the number of variables, and a,, a*, , a, are the initial points for coordinate transformation. If the number of laminate layers is k, then k + I sets of initial stacking sequences are obtained. Take a twolayer composite laminate for example: k = 2, p = 0.966, q = 0.259 and suppose (a,, u2) = (-0.408, -0.408), then the initial simplex will be
which is in the form of an equilateral triangle, and its three vertices are located in three different quadrants and so can expand the coverage of the possible searching direction. Since its edge length is 1.0, it is also called a unit simplex. If the vertice coordinates of the simplex are multiplied by a magnifying factor, a new simplex is formed, larger and having an original direction feature. Some researches found that, when the reflection coefficient was 1.0, the expansion coefficient was 2.0 and the contraction coefficient was 0.5, the accuracy of convergence was obtained [IO, 211. But they did not discuss the values of proper MFs. If the values of MFs and the initial simplex are too small, then the searching domain is also small and the obtaining of optimum is slow; on the contrary, if the initial simplex is too large, then it is not easy to converge quickly. Therefore it is necessary to study the optimal domain of the MFs to improve the speed of convergence and enhance the accuracy. In this article, for evaluating MFs, the fiber angle of each lamina is used as a variable, which is a means of studying the optimization of stacking sequence.
Stacking
sequence
of fiber-reinforced
composite
laminate
519
Table 2. Optimization of two-layer unsymmetrical plate laminate Set
1
2
3
4
5
6
7
8
9
5
15
30
50
70
85
100
115
130
90.9’ - 89.9”
90.27” 90.41
89.68” - 90.26”
91.02” 91.15”
-90.72” - 89.25”
- 89.87” -90.04
-90.35’ - 89.68”
-89.57. -90.33”
-90.38” - 89.68,
0.03
0.38
0.32
1.21
0.81
0.09
0.37
0.42
0.39
71
83
81
71
69
60
65
72
102
Factor Magnifying factor Optimal Stacking sequence
Error(%) Calculating times
MFs are judged by both convergence speed and the error between optimum angle combination and the best angle 90”. In testing the unsymmetrically laminated plate loaded by unixial tensile force on the y-axis, all the best fiber angles with the x-axis should be 90”. Since the range of fiber angles is f 90”, the vertices of the initial simplex are + 0.5, which when multiplied by MFs are treated as the values of fiber angles. Therefore, the range of MF is from 1.0 to 180.0. From a test, it is found that when the magnifying factor is more than 140.0 the converging proceeds very slowly, because the size of simplex becomes too large. Tables 2 and 3 are the tests for two and four-layer unsymmetrical composite laminates with different MFs: 5, 15, 30,50, 70, 85, 100, 115, 130. The caiculating angle errors can be obtained from the following equation: i ,=l
I(lVX,I -9O”)l n*90
’
where (opx), means the calculated optimal angle value and n means the layer number. Figure 2 is a diagram for Tables 2 and 3 and the convergence speed is presented by the calculating times of finite element analysis. It is found that when MFs are 50, 70 or 85, the convergence speed and accuracy of the optimization are better. Among others, 70 is the best. It is note-worthy that when MFs are used improperly the result of optimization cannot even approach k 90’. Take Table 4 for example, when the MF is 5 or 15, the optimization of a three-layer composite laminate cannot converge. Hence the importance of choosing proper MFs is obvious. If MFs are chosen casually, they are very liable to get an erroneous
result. Especially when the loading situation is complex or the shape of composite material is unusual, it is impossible to predict the optimum of stacking sequence of laminate or to infer its accuracy. Take for example a laminate with a central hole. When the load in the y-axis direction is twice that in the x-axis, it is not easy to predict the optimal stacking angles. To verify the result of optimization, this research applies the progressive failure method to analyze the stacking sequences obtained by various MFs to evaluate the strength of these laminates. It is found that the better the stacking sequence the higher the strength. As mentioned above, since MF 70 is the most ideal, a comparison, as shown in Table 5, between MFs 70 and 5 in the case of a laminate with a central hole indicates that both the strength and convergence speed in MF 70 are better, which also proves the previous conclusion. During the process of searching for suitable MFs, it is found that the simplex method is a kind of direct search, in which the objective function value is not always smaller than the one in the previous searching step. That is, when approaching the optimum, the variation of the objective function values presents oscillatory decline instead of monotonous, which is different from the general optimization process. Figure 3 is the optimization process of a two-layer unsymmetrical laminate. When MFs are 5 and 15, both the simplex itself and the variation of searching amplitude each time are small and so the variation of objective function values each time is also small. Therefore, the searching process converges monotonously. When the MF is big, such as 85, an oscillatory convergence phenomenon will appear. Its convergence speed and accuracy are also better than
Table 3. Optimization of a four-layer unsymmetrical plate laminate Factor
I
2
3
4
Set 5
6
7
8
9
Magnifying factor
5
15
30
50
70
85
100
115
130
-76.1 106.5. 74.2 - 104.2
- 119.9” -88.7’ 96.1 91.6’
-89.8 - 87.4’ -95.3’ -87.6
- 76. I 103.5’ 76.1 - 103.4‘
-9l.V -85.6” 86.6 -90.0’
-74.5‘ 74.3 ’ - 105.5‘ -74.4’
-87.8’ -91.1 - 90.8’ -23.1
-87.2 -97.2 -86.3 ’ -89.3
Optimal stacking sequence
Error(%) Calculating times
16.8 256
10.8 337
2.9 407
15.2 318
-88.3’ -91.5” 87.8” - 88.0’ 2.1 152
2.4 205
17.3 194
19.7 143
4 228
Li-ren Tsau and Chyuan-huei Liu
520
T 1.5
* \
0.81; I
/
0.7.01 0.6 2 0.57
d i
,?-
1
oj”I.-
-*
r25
l
/
L,
o.3et
-0-
0.2 /
+ 15
30
45
75
60
Magnifying
Fig. 2. The testing
Usually, a symmetrical laminate is used for the analysis of stacking sequences [l-6]. For instance, for a four-layer laminate, only half of the upper or lower two layers are analyzed to save computing time. Up to now there have been no specific researches about the influence of this simplified process on the optimal stacking sequence and its strength. In this section, an entire unsymmetrical laminate with a central hole is analyzed by the simplex method to obtain the stacking sequence, whose strength is evaluated by factors
factors.
progressive failure analysis. Then this strength is compared with that of same laminate evaluated in a symmetrical way. Table 6 is the result of four and six-layer laminates analyzed in an unsymmetrical way. Take the four-layer laminate for instance, the strength of laminate obtained in the unsymmetrical way is higher than that in a symmetrical way by 8%, while the computing time in the former way increases 13 times. On the contrary, the ultimate strength of a six-layer laminate obtained in the unsymmetrical way is lower than that in a symmetrical way. The reason is that the resultant errors will increase, along with the increase of design variables (layer number). As shown in Table 7, when analyzing the IO-layer laminate (the variable is 5) the error reaches 13%. The number of design variables of a six-layer laminate analyzed in a symmetrical way is 3, while that in an unsymmetrical way is 6. Consequently, the increase of errors in the latter results in the reduction of laminate strength and increasing computing time. Hence, it is proved that, to save time and to obtain optimal stacking sequence, the symmetrical way is
AND
Table 4. Magnifying
120
105
of magnifying
those in MFs 5 and 15. It is the same with the case of a four-layer unsymmetrical laminate, as shown in Fig. 4, with MFs 15, 30, and 70. To sum up, the convergence of objective function value may appear either monotonous or oscillatory, and the former is not always better than the latter, and this is one of the features of the simplex method in solving the problem of optimal stacking sequence of the laminate. 5. THE EVALUATION OF SYMMETRICAL UNSYMMETRICAL WAYS
90
Factor
for a three-layer
unsymmetrical
laminate
Set Factor Magnifying Error(%) Calculatina
factor times
I
2
3
4
5
6
7
8
9
5
15
30 8.2 129
50 1.5 94
70 6.1 94
85 31.2 97
100 1.5 111
115 1.9 175
130 16.7 158
-
Table 5. Ontimization
and its ultimate
streneth
of a four-laver
unsvmmetrical
laminate
with a central
Magnifying factor
Optimal stacking sequence ( )
Ultimate failure load (lb)
CPU time (Micro VAX 3500)
5
37.0 -52.2 70.5 84.3
9214.52
94 h 27 min
70
-35.5 92.6 96.8’ 47.3
14548.
I
58 h 23 min
hole
Stacking
sequence
of fiber-reinforced
Table 6. The comparison
of symmetrical
Initial Four-layer Six-layer
laminate laminate
failure
521
laminates
Ultimate
strength
Symmetric Unsymmetric
4,632.7 3,586.6
13.446.3 14,548. I
Symmetric Unsymmetric
5,217.6 5,000.6
23,058.5 20,658.7
6. INTERIOR PENALTY METHOD
The optimization problems with constrained ditions can be standardized as: find X,, minimize f (A’,) and j=l,Z
laminate
and unsymmetrical
load (lb)
better. The increase of 8% in the strength of a four-layer laminate analyzed in an unsymmetrical way does not make a significant difference in an engineering application, not to mention the reverse result when the layer number increases. Besides, when the simplex method is used to solve a problem with five or more variables, the errors tend to increase greatly. Therefore other optimization methods must be explored.
g,(x)<0
composite
con-
m;
,...,
g,(x) ,< 0 is the constrained
condition.
(13)
The solution to this kind of problem can be divided into two groups: direct and indirect methods. Since there is no direct relationship between design variables with constrained conditions and stacking sequence, or between a defined area and an objective function value, no equation can be derived. Therefore, indirect optimization is more suitable. The interior penalty method transforms the problem of eqn (13) into an unconstrained minimization problem:
cm?A)=f(x)
-
7*,g, --&
1
(14)
(lb)
Fig. 3. The searching
process
11 Searching
of the simplex
17h 32min 314h 47min
(1) select an initial point X, to satisfy all the constrained conditions, g,(x) ,< 0 and set an initial value yk > 0, k = 1; (2) use an unconstrained minimization method to minimize 4(x, yk) to get the X2; (3) test whether A’, is the best solution; if so, stop calculating; if not, go on to do the next step; (4) find the next penalty parameter yk+, = c x ykr c < 1; (5) go on to step 2 with a new initial point Xi = A’, , and then continue the cycling calculation.
When using the interior penalty several things worthy of note.
method,
there are
(1) The selected initial interior penalty constant y, must be tested. The calculating process of the unconstrained minimization of function 4(x, yk) is to decrease the yk value in the series. The advantage of selecting a very small y, value is that the calculating times of minimization 4(x, yr) will decrease, but it is not easy to approach the optimal point. On the other hand a better optimal point can be obtained and the
Factor 5
A--a
15
U-0
a5
J=w=-=-n---6
4h 27min 58 h 23 min
in which g,(x) is a constrained condition, and yL is a penalty parameter, a positive constant. The second term in the right side of this equation is a penalty term. The 4(x, yk) function, which is an unconstrained minimization problem after yk (k = 1,2, , n) has been substituted iteratively, will fill up the problem of eqn (13). When this method is used, the initial point coincident to all the constrains must first be found to search for the optimum gradually. The whole process is as follows:
Magnifying o-o
1
CPU time
16 21 Sequence
method
for a two-layer
26
plate laminate
Li-ren Tsau and Chyuan-huei Liu
522
Magnifying
Factor
o--o
15
A--A
30 7n
‘IO 6.0 ‘S p 5.0 5
4.0
0
Fig.
10
4. The searching
20
30
process
40 50 60 70 80% Searching Sequence
of the simplex
computing time increases when choosing a larger ;‘, value. Therefore, for being able to converge quickly, the y, value for every initial X, point can be got through eqn (15):
f (X,1
Y’=a x[-;,&]’
method
100110120:30
for a two-layer
plate laminate
(a) the difference of two sequential objective function values is smaller than the set tolerance t,; that is,
(15)
where 0.1 < a < 1.O. (2) Choose suitable product factor. After yk is selected, yk + , can be obtained by c x ykr here c <: 1 because of yk + , < yk. The value of c is generally set as 0.1, 0.2 or 0.5. (3) Choose suitable convergence criterion to get the optimal point. In the process of the unconstrained minimization of function 4(x, yk) by decreasing ;ak gradually, a convergence criterion must be used to ascertain the optimal point and to finish the process, in which the following conditions must be satisfied:
(b) the difference of two sequential optimal point X,* and X:_, is smaller than set value f2. The interior penalty method used in this research is first to transform the optimization with constrained conditions into unconstrained ones and then to do the calculating by way of variable metric method unconditioned descent optimization. Here, the laminated plate is first used for analysis. Choose an initial value of the design variables and calculate the interior penalty constant domain by eqn (15) through its objective function value. According to the initial values and interior penalty constants listed in Table 8, the finite element analysis and optimization of composite laminates with various layers are shown in
Table 7. The optimization of a symmetrical plate laminate Layer number
Magnifying factor
4
6
Error(%)
-90.1 90.0.
87
0.1
(-25.0’, -25.0 . -25.0 ) (22.1 . - 13.2’,-13.2 ) (- 13.2”, 22.1 , - 13.2 ) (- 13.2 . - 13.2 , 22.1 )
-90.2’ -88.0 -91.9
94
1-5
(-2.5 , -2.5 . -2.5‘. -2.5 ) (2.1..-1.4. -1.4’,-1.4) (-1.4 .2.1 ( - I .4’, - I .4 ‘) (-1.4’.-1.4,2.1’,-1.4) (-1.4. -14, -1.4 ,2.1“)
-97.0 83.3 96.9 -83.7
231
1.4
(-35’, -35, -35, -35, -35“) (29‘, -21 , -21 , -21 . -21’) (-21., 29 , -21.21, -21’) (-21 . -21’, -21 .29 , -21 ) (-21 ) -21 ) -21’, 29 , -21 ) (-21 , -21 , -21 , -21,29 1
-98.4 -69.2 -98.2 80.4 -79.5’
251
13
Optimization
5
(-2.04. -2.04’) (2.79‘, -0.75’) (-0.75’. 2.79’ )
50
8
IO
Calculating times
Initial simplex
70
Stacking
sequence
of fiber-reinforced
composite
Table 8. The initial values and penalty Layer
Objective Penalty
variables
(‘)
function
value
constant
6
8
10
70” 75” 80”
70 75.’ 80” 85”
60 65‘ 70’ 75’ 80’
0.68
0.34
0.12
0.0023
O.OI~.l
Penalty constant
constant
4
Table 9. Optimization Layer number
523
70” 80”
number
Design
laminate
0.005-0.05
0.0018~.018
of plate laminate
with interior
Optimal stacking
Calculating times
Initial stacking
0.00003-0.0003
penalty
method
CPU time/each
Error 6
(%)
4
0.03
70” 80”
97.3” 82.5”
2370
24s
6
0.01
70” 75” 80”
94.5” 91.6” 83.4”
1734
40 s
5.27
8
0.01
70” 75” 80” 85”
84.8” 89.2” 87.8” 94.0”
3495
63 s
4.26
10
0.0001
60” 65” 70” 75” 80”
84.0” 86.9” 87.7” 92.5” 95.3”
3780
91 s
5.18
Table 9. And its comparison with the simplex method is shown in Fig. 5, from which we know the error values increase along with the increase of the layer number, while the interior penalty method does not have such a situation. The accuracy of the interior penalty method varies with its penalty constants. When a symmetrical four- and six-layer laminate with a central hole are pulled in the y-axis direction by twice the tensile load of that in the x-axis, only a quarter of this structure is used to do the analyzing because of its symmetric structure and load. Since its loading is more complicated than the unidirectional tensile load on a laminate without a central hole, the stacking sequence of which can be verified by the 15 1 4 -13__
O-O
Simplex
Method
12 --
O--D
Interior
Penalty
known optimal angle combination, progressive failure analysis is applied to evaluating the ultimate strength of this laminate in order to verify the feasibility of this optimization model. First, select an initial angle combination and then use a suitable penalty constant to make an optimization analysis, whose process and results are shown in Table 9. Since the tensile load in they-axis is twice that in the x-axis, the fiber direction of each lamina in this laminate with a central hole is supposed to approach the y-axis direction, and this is in accordance with our optimization result. The optimal convergence process of fourand six-layer laminates with a central hole is shown in Fig. 6. The process is monotonously
0
Method
11 10
3 Layer
Fig. 5. The comparison
of analysis
error
4 no. of Laminate
between
5
6
(Symmetric)
the simplex
method
and the penalty
method.
Li-ren Tsau and Chyuan-huei
524
Liu
________.__
O--O
4-layer
o----o
6-layer
i 1
2
3
4
5
6 7 S Searching
9 10 11 12 13 1415161718 Sequence
Fig. 6. The searching process of interior penalty method for a laminate with a central hole
converging toward the minimum objective function, which is different from the oscillatory searching process of the simplex method. Hence, the major difference between the interior penalty method and the simplex method is that the searching direction of the former is obtained by calculation, while the latter is according to searching mechanism. The optimization result is verified by progressive failure analysis. Meanwhile, in order to verify that the searching of stacking sequence during the optimization process is gradually approaching the optimum, four groups of fiber angle combination are chosen and individually put under progressive failure analysis. The results are shown in Tables 10 and 11, from which we know that the initial failure strength and the ultimate strength are increasing along with the decrease of objective function value. Hence, the laminate strength is gradually increasing along with the convergence process. And the optimum stacking sequence can result in the highest strength, which thereby proves the feasibility of the interior penalty method. Besides, the strength of a six-layer laminate without optimization is lower than that of a four-layer one with optimization. Therefore, by optimization, the goal of less material
with higher strength is achieved, objective of this research.
which is also the
7. CONCLUSIONS
a more
a four-layer
2 3 4 5
200 265 416 Optimization
Table Set
Search
)
.
114.8 ) 117.5 )
I I. Results of progressive
failure
seauence
754.27 917.95 1231.1 1358.1
. 105.5 ) (51.2 (49.6
Stacking
analysis
seauence
.75.0 , 80.0) 2 3 4 5
452 689 862 Optimization
(65.7 , 78.7 ,95.3’) (61.1 , 78.7 , 101.1 ) (57.5 (75.9 . 107.9 ) (47.7 ,76.4 , 127.7 )
for a six-layer Initial failure load (lb) 734.5 977.4 1119.5 1207.1 1870. I
7114.1 7509.7 7826
laminate Ultimate failure load (lb) 6243.6 7721.5 7920.5 8985.9 10548.5
Stacking feasibility optimization, and
so can
of
this
optimization.
material be put
can
into
get
the best
sequence
That the
is,
highest
of fiber-reinforced with
the
strength
use.
REFERENCES
I. T. Y. Kam and M. D. Lai, Multilevel optimal design of laminated composite plate structures. Comput. Struct. 31, 197-202 (1989). of laminated composite plate 2. Y. Hirno, Optimization and shells. In Mechanics ofcomposite Materials, Recent Advances (Edited by Zvi Hashin and Carl T. Hearkovich), Proceedings of the IUTAM Symposium on Mechanics of Composite Materials, Virgmia Polytechnic Institute and State University, pp. 355-365 (1982). Optimal design of 3. T. Y. Kam and R. R. Chang, laminated composite plates with dynamic and static considerations. Corngut. Struct. 32, 387-393 (1988). weight design of 4. C. C. Lin and A. J. Yu, Optimum composite laminated plates. Comput. Struct. 38, 581-583 (1991). 5. G. Flanagan and A. N. Palazotto, Composite laminate optimization program suitable for microcomputers. Comput. Struct. 22, 995-1009 (1986). 6. L. R. Tsau and R. Plunkett, Finite element analysis of progressive failure for laminated FRP plates with inplane loading. Engng Fracture Mech. 45, 529-546, (1993). finite element analysis of 7. J. D. Lee, Three dimensional damage accumulation in composite laminate. Comput. Struct. 15, 335-350 (1980). laminates. 8. R. Y. Kim, Matrix damage in composite Presented at Eleventh Southeastern Conf: on Applied Mechanics, held at Huntsvil, AL (1982).
composite
laminate
525
9. L. R. Tsau, Y. H. Chang and F. L. Tsao, The design of optimal stacking sequence for laminated FRP plates with inplane loading, Comput. Struct.. in press. 10. J. A. Nelder and R. Mead, A simplex method for function minimization. Comput. J. 7, 308-313 (1965). 1I. A. L. Cauchy, Methods generale pour la resolution des system d’equation simulates, C. r. Acad. Sci. Paris 25, 536-538 (1847). 12. R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients. Comput. J. 7, 149-154 (1964). 13. R. Fletcher and M. J. D. Powell, A rapidly convergent descent method for minimization. Compuf. J. 6, 1633168 (1963). Failure criterion for unidirectional fiber 14. Z. Hashin, composites. ASME J. Appl. Mech. 47, 329-334 (1980). and L. J. Broutman, Analysk and 15. B. D. Agarwal Performance qf Fiber Composites. Wiley, New York (1980). 16. M. R. Piggott, Load Bearing Fibre Composites. Pergamon Press. Oxford (1980). Day17. S. W. Tsai, Composite Design. Think Composite, ton, OH (1986). 18. L. R. Tsau, Finite element analysis of composite laminates. Ph.D. Thesis, University of Minnesota. MN (1987). 19. C. F. Liu, The optimal stacking sequence design for fiber reinforced composite laminate with inplane loading. Master Thesis, Chung-Cheng Institute of Technology (1990). The Finite Element Method. Mc20. 0. C.. Zienkiewicz, Graw-Hill. New York (1982). 21. L. R. Tsau. C. Y. Huang and F. L. Tsao, The optimal stacking sequence design for laminated FRP plates with inplane loading. In 9th Nat1 ConJ: on Mechanical Engineering. CSME, Kaohsiung (1992).
Asa%
t
> Pergamon
A COMPARISON BETWEEN TWO OPTIMIZATION METHODS ON THE STACKING SEQUENCE OF FIBER-REINFORCED COMPOSITE LAMINATE Li-ren TsauP and Chyuan-huei Liu$ TDepartment
of Mechanical
$Chung-Shan
Institute
Engineering,
Chung Cheng Institute of Technology, Ta-Shi, Taiwan 33509, Republic of China of Science and Technology, Taichung, Taiwan, Republic of China (Received I4 November 1993)
Abstract-This article is dealing with the study of optimization for the stacking sequence of composite laminates by two optimization methods of simplex and interior penalty. In the simplex method, both the applicable domain of magnifying factors (MFs) and the differences of stacking sequence optimization between symmetrical and unsymmetrical laminate composites are studied. When the values of MFs are 50, 70, 85, the converging of optimization process is faster and its result is more accurate. The strength of the unsymmetrical laminate with fewer layers is larger than that of the symmetric one by 8% only. But when the layers are increased, the strength of symmetric laminate is higher. In the interior penalty method, the error value can be kept within 5%6%, while in the simplex method the more layers the larger the error value. To sum up, proved by progressive failure simulation of finite element analysis, the best strength of composite laminate with optimal stacking sequence can be obtained.
NOTATION
composite lamina cannot meet the strength requirement, composite laminates stacked by laminae are developed. There have been many researches about the stacking problem of composite laminates. Some used the factors of frequency, damping, displacement or material sizes as constraints to get the thickness and stacking sequence of a laminate under load without failure [I, 21. Some solved this problem by classic laminate theory (CLT) [3,4]. Flangan and Palazotto [5] used Tsai-Wu failure criterion and firstply failure to judge the failure in order to find suitable thickness under the limitation of strength. However, since CLT theory could not solve such problems as interlaminar stress and free edge effect nor deal with as many layers as needed [4, 61. it could not be applied to analyzing the complex structure in a composite laminate. A laminate is usually composed of pre-preg laminae, so when designing a suitable laminate. the number of laminate layers instead of its thickness is used as the object of analysis [7, 81. If stacking scquence only is used as a design variable. then the simplex method, with features of nonlinear, nonconstraint and direct search, can be used. in which the stress state analyzed by the finite element method is substituted into Hashin Failure criterion to get an objective function value. However, the range of magnifying factors (MFs) which influcncc the converging and accuracy was not discussed [9]. Neither did Nelder and Mead [IO] deal with the effect of MFs. though they made tests on reflection. expansion, contraction coefficients of the simplex method. Therefore, one of the purposes of this research is to study
stiffness matrix of whole structure displacement matrix of nodal points force distribution matrix of nodal points operator matrix for element the general stress-strain relation [I I] Jacobian transformation matrix local coordinate system range from - 1 to + I hygrothermal strains in each element residual strain in each element
laminar thermal coefficient of expansion laminar moisture coefficient of expansion temperature change in each element percent weight gained by moisture absorption tensile strength in fiber direction compressive strength in fiber direction tensile strength transverse to fiber direction compressive strength transverse to fiber direction transverse shear strength axial shear strength
1. INTRODUCTION
A composite material is composed of two or more kinds of materials and has its own properties. Unidirectional fiber-reinforced composite materials have been widely applied in the aerospace industry. This kind of material has a very good strength/weight ratio, whose structure strength is controlled by its anisotropic features. Reinforced fibers dominate the longitudinal property and sustain most load, while matrix materials, such as epoxy, dominate the transverse property and transfer load among fibers. Therefore, since the transverse property of a unidirectional 515
Li-ren Tsau and Chyuan-huei
516 Table
I. Properties
of graphite/epoxy
Property Ply thickness Longitudinal modulus E,? Transverse modulus E,, Shear modulus G,, Major Poisson’s ratio r’,: Minor Poisson‘s ratio rl?, Longitudinal tensile strength Longitudinal compress& strength Transverse tensile strength Transverse compressive strength In-plane shear strength
SP-286T300 Value 0.0051 in 21.90 x IO” psi 1.53 x IOhpsi 0.96 x lOh psi 0.31 0.014 203 ksi 164 ksi 7.8 ksi 30.6 ksi 10.5 ksi
the effects of MFs. Moreover, besides the factors of stacking sequence, there are problems of thickness, fiber volume, displacement and sizes to be considered. And it becomes a nonlinear constraint problem, and so the simplex method becomes unsuitable. Cauchy [I I ] developed a steepest descent method, which was prone to get a local minimum, whose optimization path appeared zigzag when approaching the minimum, and whose convergence in solving complex problems was very slow. Fletcher and Reeves 1121greatly modified it as the conjugate gradient method, which. better than steep descent method but worse than the quasi-Newton method, converged only after numerous calculations resulting from the errors accumulated in each calculating period. The converging of the quasi-Newton method was faster than that of any direct research method and could avoid approaching maximum or critical value. However, it had the demerit of being unable to calculate Jacobian matrix, when inverse matrix becomes inhnitc. especially in the problem of complex structure. Fletcher and Powell improved on this descent method by a variable metric method and developed one of the most efficient methods for solving the problem of unconditional optimization [I 3:. Therefore, when considering conditional problems, this optimal design method is applied to this research to discuss its feasibility. Hence, the objective of this article is to hnd the MFs suitable for the simplex method. to test the feasibility of the interior penalty method, and to study the differences between these two methods. Besides, this article also deals with the optimization of symmetrical and unsymmetrical composite laminate by analyzing the relation between error value and the number of variables (such as layer numbers). In this article. the macromechanics approach is used to study the stress and strain of a composite lamina under the assumption that on the individual ply level the fiber-reinforced material is continuous. linear elastic. homogeneous, orthotropic. The I6 x 16 in SP-286T300 graphite/epoxy plate laminates with a central hole and without one are used for analysis and the properties of single laminae are given in Table I. The residual stress is due to a curing temperature difference of (76 - 350 = -274 F) and a moisture content of 0.6%.
Liu
2. FAILURE
CRITERION
AND
PROGRESSIVE
FAILURE
In a very broad sense, the failure of a structural element can be stated to have taken place when it ceases to perform satisfactorily. Therefore, the definition of failure will change from one application to another. In the case of composite materials, internal material failure usually takes place much earlier, even before any change in the macromechanical appearance of structure. Owing to the features of heterogeneity and anisotropy of composite materials, their failure characteristics are completely different from those of isotropic materials. In metals, the appearing of detectable cracks is generally considered unsafe because a little damage can rapidly grow to final fracture. In composite materials, however, this is not always so; although internal damage may appear very early, its propagation is arrested by the internal structure of the composite. Therefore, composite laminates can still sustain much higher loads after the occurrence of localized damage such as matrix cracking or delamination [l4-161. Hashin developed a failure criterion in which the failure modes, tensile fiber, compressive fiber, tensile matrix and compressive matrix failure are dealt with by four separate criteria [l4]. And during the process the failure modes of composites can be identified, which are of significance to finite element analysis of progressive failure. These four criteria are: (I) tensile fiber mode oLL > 0
(2) tensile matrix
mode mrT + (T,, > 0
Cfl(q, +o,,Y= , (3) compressive
matrix
c7,,
(3)
:
mode oTT + gLL< 0
+:2(~i.,-“ir’T,,)+ili((ilT+(i;/)=
I
(2)
fiber mode oLL < 0 OLL =
(4) compressive
I;
h
I.
(4)
If any failure function value is greater than 1.0, then failure occurs. It is matrix failure. if the matrix failure function value is larger: it is fiber failure if its function value is larger. The elements that have failed are assumed to have reduced stiffness for subsequent
Stacking
sequence
of fiber-reinforced
in the progressive failure analysis. It has been proposed [17] that the cracked plies be replaced with a continuum of lower stiffness so that conventional stress analysis in the finite element method can be applied. Following the recommendation in the matrix degradation for the graphite/epoxy fiber-reinforced composite 45% of its original value, and that of the shear modules to 35% of its original value, the major Poisson’s ratio is reduced to 30% of its original value. The fiber direction stiffness decreases gradually with increasing strain after the first fiber fails. Therefore, all the terms in the stiffness matrix for the elements with failure fiber are conservatively assumed to be zero in the subsequent analysis. And, for reasonably saving the computing time of calculating the laminate with thousands of elements and that of getting the load increment for sufficiently accurate results, a 45% increment loading rate [I81 was used in this investigation. All the elements which have either a fiber or matrix failure function value equal to or larger than 1.0 are recorded as a basis to modify stiffness to recalculate the stress distribution. Then the laminate was loaded with the same load. If this load did not make further damage, 45% of the initial failure load was increased and the same process was repeated. When the load did not make any damage with a two-fold load increment, and fiber failure occurred in all of the elements lying in a particular radial direction out to about half the radius of the hole in almost all the layers of the laminae, then those elements in the same radial direction as the remaining laminae could not sustain a load perpendicular to the fiber direction. Therefore, this was considered to be catastrophic failure [6, 181. The current step was redone with a half load increment if a final catastrophic failure was found [ 191; that is, the failure function values of fiber failure for elements failing in this step are much larger than 1.O. The increment was reduced further until the values of the failure function were 1.0 or just a little larger than I.0 to obtain the ultimate strength of a laminate. If the failure function was less than 1.O, the load was increased by half the previous decrement. analysis
3. FINITE
ELEMENT
ANALYSIS
The three-dimensional finite element mesh consists of a family of eight-node elements. The element equation which links the nodal point displacement vector [U] and the nodal point force vector can be written as [20]:
composite
laminate
517
Changes in temperature or moisture will cause strains in materials. One way of calculating the residual strains in the finite element model is first to calculate the nodal forces necessary to keep the nodes unmovable and thus unstrained. If one then imposes equal and opposite nodal forces, the set of nodal forces will be zeros and the residual strains can be calculated by using the stiffness matrix and these artificial nodal forces. The hygrothermal strains in each element are given by:
kTll = aoWl, + P,[AW. The 24 components of the equivalent element can be computed by:
The total equivalent laminate is:
Then the residual from:
(6) forces of each
force at each nodal point of the
nodal displacement
can be found
[UT = j%‘[%>
(9)
where the stiffness matrix and the equivalent forces are known. After computing the residual nodal displacements, the residual strains are found from:
M = {B),([W),
(a = 1,.
, m).
(10)
Finally, the residual stress can be computed with the stress-strain relation for a linearly elastic material subjected to mechanical strain using the equivalent forces needed to hold the finite element in an unstrained state. Thus, assuming linear elastic behavior, the hygrothermal stresses of the material in each element in accordance with the generalized Hooke’s law are:
bl, = {~~,(k”L - kTL).
(11)
The problem investigated in this study is that of a square plate made of a layered composite material, with and without a centered circular hole subjected to in-plane loading. The applied load was uniformly distributed in the x direction on x = + L and/or in the y direction on y = + L, as shown in Fig. 1. 4. SIMPLEX
METHOD
where The simplex method is a direct search for solving nonconstraint problems. Each search only takes one calculation to get an objective function value. If the objective function value has no abrupt variation, then
Li-ren Tsau and Chyuan-huei
518
Liu
L
F
’ r
Fig. I. The mesh and loading the searching path is followed to get the optimum. Though fiber angles are usually - 90” < 0 d 90” when analyzing the problems of stacking sequences, yet the angles obtained are larger than k 90’, which in turn are divided by 90 to get the remainder, which are the angle values we need. Therefore, the nonconstrained simplex method can be applied. The process is as follows: first, calculate the objective function values of the n + 1 verticals in the simplex and compare them. Second, make the simplex gradually approach the smallest objective function value through the searching mechanism, which includes (1) reflection, (2) expansion, (3) contraction [19]. If the smaller objective function values cannot be obtained in the process of reflection or contraction, or when oscillation phenomenon happens, then discard the abovementioned searching method, and change the searching direction by way of contracting the whole simplex toward the smallest objective function value. The search mechanism in simplex method helps discard all the worst objective function values one by one until it finds the optimum. The selection of the searching direction is decided not only by the objective function value, but also the selection of the initial simplex. In order to cover the domain of all the direction, we set
*I
a2
(4 +P) (u,+y)
(a.?+Y)
(4 +4)
(a?+Y)
(az+P)
.’ ”
a, (ai + Y) (ai, + 4)
Do=
1
p=k;‘zi(k -
”
I)+ (k
6%+-PI ~
of a laminate
with a central
4=, &(k
hole.
+
I)‘?- I}.
In the D, matrix, each row represents a set of stacking sequences, in which D, is the initial simplex, k is the number of variables, and a,, a*, , a, are the initial points for coordinate transformation. If the number of laminate layers is k, then k + I sets of initial stacking sequences are obtained. Take a twolayer composite laminate for example: k = 2, p = 0.966, q = 0.259 and suppose (a,, u2) = (-0.408, -0.408), then the initial simplex will be
which is in the form of an equilateral triangle, and its three vertices are located in three different quadrants and so can expand the coverage of the possible searching direction. Since its edge length is 1.0, it is also called a unit simplex. If the vertice coordinates of the simplex are multiplied by a magnifying factor, a new simplex is formed, larger and having an original direction feature. Some researches found that, when the reflection coefficient was 1.0, the expansion coefficient was 2.0 and the contraction coefficient was 0.5, the accuracy of convergence was obtained [IO, 211. But they did not discuss the values of proper MFs. If the values of MFs and the initial simplex are too small, then the searching domain is also small and the obtaining of optimum is slow; on the contrary, if the initial simplex is too large, then it is not easy to converge quickly. Therefore it is necessary to study the optimal domain of the MFs to improve the speed of convergence and enhance the accuracy. In this article, for evaluating MFs, the fiber angle of each lamina is used as a variable, which is a means of studying the optimization of stacking sequence.
Stacking
sequence
of fiber-reinforced
composite
laminate
519
Table 2. Optimization of two-layer unsymmetrical plate laminate Set
1
2
3
4
5
6
7
8
9
5
15
30
50
70
85
100
115
130
90.9’ - 89.9”
90.27” 90.41
89.68” - 90.26”
91.02” 91.15”
-90.72” - 89.25”
- 89.87” -90.04
-90.35’ - 89.68”
-89.57. -90.33”
-90.38” - 89.68,
0.03
0.38
0.32
1.21
0.81
0.09
0.37
0.42
0.39
71
83
81
71
69
60
65
72
102
Factor Magnifying factor Optimal Stacking sequence
Error(%) Calculating times
MFs are judged by both convergence speed and the error between optimum angle combination and the best angle 90”. In testing the unsymmetrically laminated plate loaded by unixial tensile force on the y-axis, all the best fiber angles with the x-axis should be 90”. Since the range of fiber angles is f 90”, the vertices of the initial simplex are + 0.5, which when multiplied by MFs are treated as the values of fiber angles. Therefore, the range of MF is from 1.0 to 180.0. From a test, it is found that when the magnifying factor is more than 140.0 the converging proceeds very slowly, because the size of simplex becomes too large. Tables 2 and 3 are the tests for two and four-layer unsymmetrical composite laminates with different MFs: 5, 15, 30,50, 70, 85, 100, 115, 130. The caiculating angle errors can be obtained from the following equation: i ,=l
I(lVX,I -9O”)l n*90
’
where (opx), means the calculated optimal angle value and n means the layer number. Figure 2 is a diagram for Tables 2 and 3 and the convergence speed is presented by the calculating times of finite element analysis. It is found that when MFs are 50, 70 or 85, the convergence speed and accuracy of the optimization are better. Among others, 70 is the best. It is note-worthy that when MFs are used improperly the result of optimization cannot even approach k 90’. Take Table 4 for example, when the MF is 5 or 15, the optimization of a three-layer composite laminate cannot converge. Hence the importance of choosing proper MFs is obvious. If MFs are chosen casually, they are very liable to get an erroneous
result. Especially when the loading situation is complex or the shape of composite material is unusual, it is impossible to predict the optimum of stacking sequence of laminate or to infer its accuracy. Take for example a laminate with a central hole. When the load in the y-axis direction is twice that in the x-axis, it is not easy to predict the optimal stacking angles. To verify the result of optimization, this research applies the progressive failure method to analyze the stacking sequences obtained by various MFs to evaluate the strength of these laminates. It is found that the better the stacking sequence the higher the strength. As mentioned above, since MF 70 is the most ideal, a comparison, as shown in Table 5, between MFs 70 and 5 in the case of a laminate with a central hole indicates that both the strength and convergence speed in MF 70 are better, which also proves the previous conclusion. During the process of searching for suitable MFs, it is found that the simplex method is a kind of direct search, in which the objective function value is not always smaller than the one in the previous searching step. That is, when approaching the optimum, the variation of the objective function values presents oscillatory decline instead of monotonous, which is different from the general optimization process. Figure 3 is the optimization process of a two-layer unsymmetrical laminate. When MFs are 5 and 15, both the simplex itself and the variation of searching amplitude each time are small and so the variation of objective function values each time is also small. Therefore, the searching process converges monotonously. When the MF is big, such as 85, an oscillatory convergence phenomenon will appear. Its convergence speed and accuracy are also better than
Table 3. Optimization of a four-layer unsymmetrical plate laminate Factor
I
2
3
4
Set 5
6
7
8
9
Magnifying factor
5
15
30
50
70
85
100
115
130
-76.1 106.5. 74.2 - 104.2
- 119.9” -88.7’ 96.1 91.6’
-89.8 - 87.4’ -95.3’ -87.6
- 76. I 103.5’ 76.1 - 103.4‘
-9l.V -85.6” 86.6 -90.0’
-74.5‘ 74.3 ’ - 105.5‘ -74.4’
-87.8’ -91.1 - 90.8’ -23.1
-87.2 -97.2 -86.3 ’ -89.3
Optimal stacking sequence
Error(%) Calculating times
16.8 256
10.8 337
2.9 407
15.2 318
-88.3’ -91.5” 87.8” - 88.0’ 2.1 152
2.4 205
17.3 194
19.7 143
4 228
Li-ren Tsau and Chyuan-huei Liu
520
T 1.5
* \
0.81; I
/
0.7.01 0.6 2 0.57
d i
,?-
1
oj”I.-
-*
r25
l
/
L,
o.3et
-0-
0.2 /
+ 15
30
45
75
60
Magnifying
Fig. 2. The testing
Usually, a symmetrical laminate is used for the analysis of stacking sequences [l-6]. For instance, for a four-layer laminate, only half of the upper or lower two layers are analyzed to save computing time. Up to now there have been no specific researches about the influence of this simplified process on the optimal stacking sequence and its strength. In this section, an entire unsymmetrical laminate with a central hole is analyzed by the simplex method to obtain the stacking sequence, whose strength is evaluated by factors
factors.
progressive failure analysis. Then this strength is compared with that of same laminate evaluated in a symmetrical way. Table 6 is the result of four and six-layer laminates analyzed in an unsymmetrical way. Take the four-layer laminate for instance, the strength of laminate obtained in the unsymmetrical way is higher than that in a symmetrical way by 8%, while the computing time in the former way increases 13 times. On the contrary, the ultimate strength of a six-layer laminate obtained in the unsymmetrical way is lower than that in a symmetrical way. The reason is that the resultant errors will increase, along with the increase of design variables (layer number). As shown in Table 7, when analyzing the IO-layer laminate (the variable is 5) the error reaches 13%. The number of design variables of a six-layer laminate analyzed in a symmetrical way is 3, while that in an unsymmetrical way is 6. Consequently, the increase of errors in the latter results in the reduction of laminate strength and increasing computing time. Hence, it is proved that, to save time and to obtain optimal stacking sequence, the symmetrical way is
AND
Table 4. Magnifying
120
105
of magnifying
those in MFs 5 and 15. It is the same with the case of a four-layer unsymmetrical laminate, as shown in Fig. 4, with MFs 15, 30, and 70. To sum up, the convergence of objective function value may appear either monotonous or oscillatory, and the former is not always better than the latter, and this is one of the features of the simplex method in solving the problem of optimal stacking sequence of the laminate. 5. THE EVALUATION OF SYMMETRICAL UNSYMMETRICAL WAYS
90
Factor
for a three-layer
unsymmetrical
laminate
Set Factor Magnifying Error(%) Calculatina
factor times
I
2
3
4
5
6
7
8
9
5
15
30 8.2 129
50 1.5 94
70 6.1 94
85 31.2 97
100 1.5 111
115 1.9 175
130 16.7 158
-
Table 5. Ontimization
and its ultimate
streneth
of a four-laver
unsvmmetrical
laminate
with a central
Magnifying factor
Optimal stacking sequence ( )
Ultimate failure load (lb)
CPU time (Micro VAX 3500)
5
37.0 -52.2 70.5 84.3
9214.52
94 h 27 min
70
-35.5 92.6 96.8’ 47.3
14548.
I
58 h 23 min
hole
Stacking
sequence
of fiber-reinforced
Table 6. The comparison
of symmetrical
Initial Four-layer Six-layer
laminate laminate
failure
521
laminates
Ultimate
strength
Symmetric Unsymmetric
4,632.7 3,586.6
13.446.3 14,548. I
Symmetric Unsymmetric
5,217.6 5,000.6
23,058.5 20,658.7
6. INTERIOR PENALTY METHOD
The optimization problems with constrained ditions can be standardized as: find X,, minimize f (A’,) and j=l,Z
laminate
and unsymmetrical
load (lb)
better. The increase of 8% in the strength of a four-layer laminate analyzed in an unsymmetrical way does not make a significant difference in an engineering application, not to mention the reverse result when the layer number increases. Besides, when the simplex method is used to solve a problem with five or more variables, the errors tend to increase greatly. Therefore other optimization methods must be explored.
g,(x)<0
composite
con-
m;
,...,
g,(x) ,< 0 is the constrained
condition.
(13)
The solution to this kind of problem can be divided into two groups: direct and indirect methods. Since there is no direct relationship between design variables with constrained conditions and stacking sequence, or between a defined area and an objective function value, no equation can be derived. Therefore, indirect optimization is more suitable. The interior penalty method transforms the problem of eqn (13) into an unconstrained minimization problem:
cm?A)=f(x)
-
7*,g, --&
1
(14)
(lb)
Fig. 3. The searching
process
11 Searching
of the simplex
17h 32min 314h 47min
(1) select an initial point X, to satisfy all the constrained conditions, g,(x) ,< 0 and set an initial value yk > 0, k = 1; (2) use an unconstrained minimization method to minimize 4(x, yk) to get the X2; (3) test whether A’, is the best solution; if so, stop calculating; if not, go on to do the next step; (4) find the next penalty parameter yk+, = c x ykr c < 1; (5) go on to step 2 with a new initial point Xi = A’, , and then continue the cycling calculation.
When using the interior penalty several things worthy of note.
method,
there are
(1) The selected initial interior penalty constant y, must be tested. The calculating process of the unconstrained minimization of function 4(x, yk) is to decrease the yk value in the series. The advantage of selecting a very small y, value is that the calculating times of minimization 4(x, yr) will decrease, but it is not easy to approach the optimal point. On the other hand a better optimal point can be obtained and the
Factor 5
A--a
15
U-0
a5
J=w=-=-n---6
4h 27min 58 h 23 min
in which g,(x) is a constrained condition, and yL is a penalty parameter, a positive constant. The second term in the right side of this equation is a penalty term. The 4(x, yk) function, which is an unconstrained minimization problem after yk (k = 1,2, , n) has been substituted iteratively, will fill up the problem of eqn (13). When this method is used, the initial point coincident to all the constrains must first be found to search for the optimum gradually. The whole process is as follows:
Magnifying o-o
1
CPU time
16 21 Sequence
method
for a two-layer
26
plate laminate
Li-ren Tsau and Chyuan-huei Liu
522
Magnifying
Factor
o--o
15
A--A
30 7n
‘IO 6.0 ‘S p 5.0 5
4.0
0
Fig.
10
4. The searching
20
30
process
40 50 60 70 80% Searching Sequence
of the simplex
computing time increases when choosing a larger ;‘, value. Therefore, for being able to converge quickly, the y, value for every initial X, point can be got through eqn (15):
f (X,1
Y’=a x[-;,&]’
method
100110120:30
for a two-layer
plate laminate
(a) the difference of two sequential objective function values is smaller than the set tolerance t,; that is,
(15)
where 0.1 < a < 1.O. (2) Choose suitable product factor. After yk is selected, yk + , can be obtained by c x ykr here c <: 1 because of yk + , < yk. The value of c is generally set as 0.1, 0.2 or 0.5. (3) Choose suitable convergence criterion to get the optimal point. In the process of the unconstrained minimization of function 4(x, yk) by decreasing ;ak gradually, a convergence criterion must be used to ascertain the optimal point and to finish the process, in which the following conditions must be satisfied:
(b) the difference of two sequential optimal point X,* and X:_, is smaller than set value f2. The interior penalty method used in this research is first to transform the optimization with constrained conditions into unconstrained ones and then to do the calculating by way of variable metric method unconditioned descent optimization. Here, the laminated plate is first used for analysis. Choose an initial value of the design variables and calculate the interior penalty constant domain by eqn (15) through its objective function value. According to the initial values and interior penalty constants listed in Table 8, the finite element analysis and optimization of composite laminates with various layers are shown in
Table 7. The optimization of a symmetrical plate laminate Layer number
Magnifying factor
4
6
Error(%)
-90.1 90.0.
87
0.1
(-25.0’, -25.0 . -25.0 ) (22.1 . - 13.2’,-13.2 ) (- 13.2”, 22.1 , - 13.2 ) (- 13.2 . - 13.2 , 22.1 )
-90.2’ -88.0 -91.9
94
1-5
(-2.5 , -2.5 . -2.5‘. -2.5 ) (2.1..-1.4. -1.4’,-1.4) (-1.4 .2.1 ( - I .4’, - I .4 ‘) (-1.4’.-1.4,2.1’,-1.4) (-1.4. -14, -1.4 ,2.1“)
-97.0 83.3 96.9 -83.7
231
1.4
(-35’, -35, -35, -35, -35“) (29‘, -21 , -21 , -21 . -21’) (-21., 29 , -21.21, -21’) (-21 . -21’, -21 .29 , -21 ) (-21 ) -21 ) -21’, 29 , -21 ) (-21 , -21 , -21 , -21,29 1
-98.4 -69.2 -98.2 80.4 -79.5’
251
13
Optimization
5
(-2.04. -2.04’) (2.79‘, -0.75’) (-0.75’. 2.79’ )
50
8
IO
Calculating times
Initial simplex
70
Stacking
sequence
of fiber-reinforced
composite
Table 8. The initial values and penalty Layer
Objective Penalty
variables
(‘)
function
value
constant
6
8
10
70” 75” 80”
70 75.’ 80” 85”
60 65‘ 70’ 75’ 80’
0.68
0.34
0.12
0.0023
O.OI~.l
Penalty constant
constant
4
Table 9. Optimization Layer number
523
70” 80”
number
Design
laminate
0.005-0.05
0.0018~.018
of plate laminate
with interior
Optimal stacking
Calculating times
Initial stacking
0.00003-0.0003
penalty
method
CPU time/each
Error 6
(%)
4
0.03
70” 80”
97.3” 82.5”
2370
24s
6
0.01
70” 75” 80”
94.5” 91.6” 83.4”
1734
40 s
5.27
8
0.01
70” 75” 80” 85”
84.8” 89.2” 87.8” 94.0”
3495
63 s
4.26
10
0.0001
60” 65” 70” 75” 80”
84.0” 86.9” 87.7” 92.5” 95.3”
3780
91 s
5.18
Table 9. And its comparison with the simplex method is shown in Fig. 5, from which we know the error values increase along with the increase of the layer number, while the interior penalty method does not have such a situation. The accuracy of the interior penalty method varies with its penalty constants. When a symmetrical four- and six-layer laminate with a central hole are pulled in the y-axis direction by twice the tensile load of that in the x-axis, only a quarter of this structure is used to do the analyzing because of its symmetric structure and load. Since its loading is more complicated than the unidirectional tensile load on a laminate without a central hole, the stacking sequence of which can be verified by the 15 1 4 -13__
O-O
Simplex
Method
12 --
O--D
Interior
Penalty
known optimal angle combination, progressive failure analysis is applied to evaluating the ultimate strength of this laminate in order to verify the feasibility of this optimization model. First, select an initial angle combination and then use a suitable penalty constant to make an optimization analysis, whose process and results are shown in Table 9. Since the tensile load in they-axis is twice that in the x-axis, the fiber direction of each lamina in this laminate with a central hole is supposed to approach the y-axis direction, and this is in accordance with our optimization result. The optimal convergence process of fourand six-layer laminates with a central hole is shown in Fig. 6. The process is monotonously
0
Method
11 10
3 Layer
Fig. 5. The comparison
of analysis
error
4 no. of Laminate
between
5
6
(Symmetric)
the simplex
method
and the penalty
method.
Li-ren Tsau and Chyuan-huei
524
Liu
________.__
O--O
4-layer
o----o
6-layer
i 1
2
3
4
5
6 7 S Searching
9 10 11 12 13 1415161718 Sequence
Fig. 6. The searching process of interior penalty method for a laminate with a central hole
converging toward the minimum objective function, which is different from the oscillatory searching process of the simplex method. Hence, the major difference between the interior penalty method and the simplex method is that the searching direction of the former is obtained by calculation, while the latter is according to searching mechanism. The optimization result is verified by progressive failure analysis. Meanwhile, in order to verify that the searching of stacking sequence during the optimization process is gradually approaching the optimum, four groups of fiber angle combination are chosen and individually put under progressive failure analysis. The results are shown in Tables 10 and 11, from which we know that the initial failure strength and the ultimate strength are increasing along with the decrease of objective function value. Hence, the laminate strength is gradually increasing along with the convergence process. And the optimum stacking sequence can result in the highest strength, which thereby proves the feasibility of the interior penalty method. Besides, the strength of a six-layer laminate without optimization is lower than that of a four-layer one with optimization. Therefore, by optimization, the goal of less material
with higher strength is achieved, objective of this research.
which is also the
7. CONCLUSIONS
a more
a four-layer
2 3 4 5
200 265 416 Optimization
Table Set
Search
)
.
114.8 ) 117.5 )
I I. Results of progressive
failure
seauence
754.27 917.95 1231.1 1358.1
. 105.5 ) (51.2 (49.6
Stacking
analysis
seauence
.75.0 , 80.0) 2 3 4 5
452 689 862 Optimization
(65.7 , 78.7 ,95.3’) (61.1 , 78.7 , 101.1 ) (57.5 (75.9 . 107.9 ) (47.7 ,76.4 , 127.7 )
for a six-layer Initial failure load (lb) 734.5 977.4 1119.5 1207.1 1870. I
7114.1 7509.7 7826
laminate Ultimate failure load (lb) 6243.6 7721.5 7920.5 8985.9 10548.5
Stacking feasibility optimization, and
so can
of
this
optimization.
material be put
can
into
get
the best
sequence
That the
is,
highest
of fiber-reinforced with
the
strength
use.
REFERENCES
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laminate
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9. L. R. Tsau, Y. H. Chang and F. L. Tsao, The design of optimal stacking sequence for laminated FRP plates with inplane loading, Comput. Struct.. in press. 10. J. A. Nelder and R. Mead, A simplex method for function minimization. Comput. J. 7, 308-313 (1965). 1I. A. L. Cauchy, Methods generale pour la resolution des system d’equation simulates, C. r. Acad. Sci. Paris 25, 536-538 (1847). 12. R. Fletcher and C. M. Reeves, Function minimization by conjugate gradients. Comput. J. 7, 149-154 (1964). 13. R. Fletcher and M. J. D. Powell, A rapidly convergent descent method for minimization. Compuf. J. 6, 1633168 (1963). Failure criterion for unidirectional fiber 14. Z. Hashin, composites. ASME J. Appl. Mech. 47, 329-334 (1980). and L. J. Broutman, Analysk and 15. B. D. Agarwal Performance qf Fiber Composites. Wiley, New York (1980). 16. M. R. Piggott, Load Bearing Fibre Composites. Pergamon Press. Oxford (1980). Day17. S. W. Tsai, Composite Design. Think Composite, ton, OH (1986). 18. L. R. Tsau, Finite element analysis of composite laminates. Ph.D. Thesis, University of Minnesota. MN (1987). 19. C. F. Liu, The optimal stacking sequence design for fiber reinforced composite laminate with inplane loading. Master Thesis, Chung-Cheng Institute of Technology (1990). The Finite Element Method. Mc20. 0. C.. Zienkiewicz, Graw-Hill. New York (1982). 21. L. R. Tsau. C. Y. Huang and F. L. Tsao, The optimal stacking sequence design for laminated FRP plates with inplane loading. In 9th Nat1 ConJ: on Mechanical Engineering. CSME, Kaohsiung (1992).