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Layer optimisation for maximum fundamental frequency of laminated composite plates for different edge conditions
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COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 537–550 www.elsevier.com/locate/compscitech
Layer optimisation for maximum fundamental frequency of laminated composite plates for different edge conditions M. Kemal Apalak b
a,*
, Mustafa Yildirim b, Recep Ekici
a
a Department of Mechanical Engineering, Erciyes University, Kayseri 38039, Turkey Graduate School of Natural and Applied Sciences, Erciyes University, Kayseri 38039, Turkey
Received 19 January 2007; accepted 2 June 2007 Available online 2 August 2007
Abstract In this study the layer optimisation was carried out for maximum fundamental frequency of laminated composite plates under any combination of the three classical edge conditions. The optimal stacking sequences of laminated composite plates were searched by means of Genetic Algorithm. The first natural frequencies of the laminated composite plates with various stacking sequences were calculated using the finite element method. Genetic Algorithm maximizes the first natural frequency of the laminated composite plate defined as a fitness function (objective function). However, the finite element method needs a certain calculation time of the first natural frequency for each new lay-up sequence and plate edge condition. In order to reduce the searching time of the optimal lay-up sequence an artificial neural network model was proposed and trained with small training and testing data composed of the natural frequencies of the composite plates calculated for random lay-up sequences, layer number, edge conditions and plate length/width ratios using the finite element method. The outer layers of the composite plate had a stiffness increasing effect, and as the clamped plate edges were increased both stiffness and natural frequency of the plate increased. In addition the Genetic Algorithm predicted successfully the optimal layer sequences without yielding a local optimum on the contrary the Ritz-based layerwise optimisation method [Y. Narita, Layerwise optimization for the maximum fundamental frequency of laminated composite plates. J Sound Vib 2003;263:1005–16]. 2007 Elsevier Ltd. All rights reserved. Keywords: Composites; Laminated plate; Natural frequency; Optimisation; Genetic algorithm; Artificial neural network
1. Introduction In structural design the laminated fibrous composite plates and shells offer light-weight structures with high stiffness and strength. However, they exhibit material anisotropy and require more complex analysis and design methods. Since these composite structures operate dynamic loading conditions as well as the static loads their response against the structural resonance should be optimized by maximizing the natural frequency of the composite structures. The free vibration analysis of isotropic and composite plates has been carried out extensively [1–7]. However, these studies are limited to the optimization of vibration *
Corresponding author.Tel.: +90 352 437 4901; fax: +90 352 437 5784. E-mail address: [email protected] (M. Kemal Apalak).
0266-3538/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.06.031
characteristics of the composite structures only for some stacking sequences and boundary conditions since this optimization problem requires the combination of the complicated aspects of both the plate free vibration and the optimization techniques. The number of design variables affects directly the computation time. The design variables of this optimization problem are considered as fiber angle of each layer, thickness, length/width ratio of the composite plate. An optimal solution maximizes the natural frequency of the composite plate defined as the objective function with n design variables subjected to m constraints. As the layer number of the plate is increased an considerable increase in the searching time of the optimal solution is observed. Narita offered a Ritz-based layerwise optimization approach for the symmetrical composite plates [1]. His
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M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
assumption for the layerwise optimization approach is introduced as the optimal stacking sequence for the maximum natural frequency of laminated plate can be determined sequentially in the order from the outermost to the innermost layer. He reduced evidently the searching time and calculation for the optimal solution. However, his approach may yield a local optimum for some edge conditions. In this study the optimal layer sequences of the symmetrical composite plates were determined using the Genetic Algorithms so that their first natural frequency can be maximized. The finite element method requires a certain time for the calculation of the natural frequency of the composite plates. In order to reduce this calculation time the artificial neural networks were proposed. The Genetic algorithm combined with these artificial neural networks found the optimal layer sequences without yielding a local optimum for all cases. 2. Problem description Fig. 1 shows a laminated composite plate in the problem coordinate (x, y) and each layer in the material coordinate (x1, x2). The composite plate with a dimension of (a · b · t) has symmetrical layered sequences, and composed of 2 · n layers (n layers in each of the upper and lower half sections of the plates). Free (F), simply supported (S) and clamped (C) edge conditions are assumed along the plate edges (1, 2, 3, 4) in Fig. 1. Based on the classical plate theory the free vibration of a laminated (symmetrical) thin plate is governed by o4 w o4 w o4 w o4 w D11 4 þ 2ðD12 þ 2D66 Þ 2 2 þ D22 4 þ 4D16 3 ox ox oy oy ox oy þ 4D26
o4 w qx2 w ¼ 0 oxoy 3
ð1Þ
x1 Edge 4
O
E1 ; 1 t12 t21 Q66 ¼ G12
Q11 ¼
x
½Q
ðkÞ
b t
Fig. 1. The edge numbering of a layered composite plate in the material coordinate (x1, x2) and the problem coordinate (x, y).
E1 t21 ; 1 t12 t21
ðkÞ
T
¼ ð½T ½Q½T Þ
Q22 ¼
E2 ; 1 t12 t21 ð3Þ
ð4Þ
where [T] is transformation matrix considering the fiber angle h(k) of the kth lamina [7]. Normalized natural frequency is 1=2 q X ¼ xa2 ð5Þ D0 where the reference bending rigidity is 1 E2 t3 : 12 1 t12 t21
ð6Þ
The Ritz method is used for the linear free vibration of a thin plate subjected to simple full edge conditions whereas the finite element method is preferred for the thin plates with complex edge conditions. The dynamic equations of motion of a structure can be derived by using either Lagrange equations and Hamilton’s principle [8,9]. The Lagrange equations are given by 8 9 8 9 8 9 d < oL = < oL = < oR= þ ¼ f0g ð7Þ dt :o q_ ; :o q; :o q_ ;
where ð8Þ
L ¼ T pp
Edge 2
Q12 ¼
where E1 and E2 are longitudinal and transverse moduli of elasticity, G12 is shear modulus, t12 and t21 are major and minor Poisson’s ratios of a lamina in the material ðkÞ coordinate. The reduced stiffness Qij of the kth lamina ðkÞ can be transformed to Qij in the problem coordinate (x, y) as
Edge 3
Edge 1
θ
a
where zk is through-the-thickness distance from the middle ðkÞ surface. The Qij are reduced stiffness of a lamina in the material coordinate (x1, x2) as
D0 ¼
y
x2
where w is the displacement of the middle surface, q is mean mass per unit area and x is circular frequency. The bending stiffness of a symmetrical laminate is given by [5–7] n 2X ðkÞ ð2Þ Q z3 z3k1 ði; j ¼ 1; 2; . . . ; 6Þ Dij ¼ 3 k¼1 ij k
is called the Lagrange function, T is the kinetic energy, pp is the potential energy, R is dissipation function, q is the nodal displacement and q_ is the nodal velocity. The kinetic and potential energies of each element of the structure which was divided into E elements can be written as Z Z Z 1 T ðeÞ ¼ qu_ T u_ dV ð9Þ 2 V ðeÞ
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
pðeÞ p ¼
1 2
Z Z Z
e T r dV
Z Z
V ðeÞ
uT /1 dS 1
Dij ¼
ðeÞ
Z Z Z
S1
uT /2 dV
ð10Þ
V ðeÞ
where q is density, u and u_ are displacement and velocity vectors. The dissipation function of each element Z Z Z 1 RðeÞ ¼ lu_ T u_ dV ð11Þ 2 V ðeÞ
where l is damping coefficient. Using the finite element method the overall system equations of motion can be written as [8,9] € ðtÞ þ ½C Q_ ðtÞ þ ½K QðtÞ ¼ P c ðtÞ ½M Q
ð12Þ
€ ðtÞ is the vector of nodal accelerations, and where Q E Z Z Z X T mass matrix½M ¼ q½N ½N dV e¼1
V ðeÞ
element stiffness matrix½K ¼
E Z Z Z X e¼1
element damping matrix½C ¼
total load vector½P ¼
E Z Z Z X e¼1
½BT ½D½BdV
ð14Þ
T
ð15Þ
V ðeÞ
E Z Z Z X e¼1
ð13Þ
l½N ½N dV
V ðeÞ T
½N f/1 gdS 1
ðeÞ
S1
þ
Z Z Z
½N T f/2 gdV
ð16Þ
V ðeÞ
where [N] is the matrix of shape functions, [B] is the matrix of coordinate derivatives of shape functions, [D] is the matrix of elastic constants, {/1} is surface traction vector, and {/2} is body force vector [8,9]. The equation of motion for an undamped system without external force becomes € þ½K Q ¼ 0: ð17Þ ½M Q
The solution of this eigenvalue problem detð½K k½MÞ ¼ f0g
ð18Þ
Each eigenvalue ki is associated with an eigenvector Qi
which is called a natural mode. The block Lanczos eigenvalue extraction method was used for the calculation of eigenvalues and eigenvectors [8–10]. In the free vibration analysis of the composite plates a layered shell element with four nodes was used, and three integration points in each layer through the shell element thickness were considered [9]. The bending stiffness coefficient matrix [D] of the shell element was modified with
n 2X ðkÞ Qij z3k z3k1 ði; j ¼ 1; 2; . . . ; 6Þ: 3 k¼1
539
ð19Þ
The main objective in the optimization problem is to determine the optimal layer sequences which maximize the normalized frequency X for the first natural mode of the composite plate. The optimization problem can be defined as Maximize the objective function X ¼ Xðh1 ; h2 ; . . . ; hn Þ ! max
ð20Þ
for the design variables h ¼ ðh1 ; h2 ; . . . ; hn Þ
ð21Þ
subject to the constraints 90 6 hk 6 90
ð22Þ
where n is the layer number. It is not convenient to search directly the optimal layer sequences satisfying a maximal natural frequency condition since the iteration number increases with increasing layer number, thus, 181n possibilities appear for a fiber angle (90, 89 6 hk 6 90). An optimal layer sequences can be searched using the Genetic Algorithm, which is very fast in finding for values of the design variables satisfying the maximum fitness value of the objective function. 3. Genetic algorithm Genetic algorithms are a particular class of evolutionary algorithms and inspire some techniques from the evolutionary biology, such as inheritance, mutation, natural selection, and crossover [11–14]. The candidate solutions (individuals) to an optimization problem are defined as a population of abstract representations (chromosomes). The different encodings, such as binary or float, can be used for the representations of the parameters set. The evolution process is performed so that better solutions to the optimization problem can be obtained, and is implemented as follows: (i) A solution of the problem is represented by a list of parameters called chromosome or genome. An initial set of the individuals is generated randomly and the value of the fitness function is evaluated for each of this initial set. These results returned from the fitness function for each individual are sorted and the individuals with better fitness are kept. (ii) In order to generate a consecutive population a pair of parent organisms is selected for breeding by considering generations giving better fitness. Roulette wheel selection and tournament selection are well-defined organism selection methods. Later, the reproduction processes are applied to the individuals through the genetic operators, such as crossover and mutation (Fig. 2). (iii) The crossover (recombination) operation is performed upon the selected chromosomes. Two selected organisms breed and recombined with a crossover probability between 0.6 and 1.0. These two new child chromosomes are added to the next generation population. The crossover operation
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Generate initial random population
Evaluate the fitness value
Yes
Convergence criteria
Stop
No
Selection
Crossover
Mutation
Reproduction Fig. 2. Evolutionary stages of genetic algorithm.
Hidden Layers
{
θ1 → θ2 → … …
Fiber angles =
Input Layer
θ n − 1→ θn →
Output Layer
1 . . .
Natural frequency
n
Plate length / width ratio, a/b
Information processing in an artificial neural network unit.
a) tangent sigmoid and b) logarithmic sigmoid transfer functions.
Fig. 3. Architecture of an ANN model with (n + 1), 10, 3 and 1 neurons in input, hidden, output layers.
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
mixes the chromosomes of the parents by swapping a portion of the data structure, and can be repeated with different parents to create a sufficient number of candidate solutions in the next generation population (Fig. 2). (iv) This created offspring is randomly mutated by flipping bits in the chromosome data structure based on a fixed very small probability of mutation on the order of 0.001 or less. The next generation population of chromosomes is ultimately different from the initial generation. (v) The evaluation is interrupted when the highest ranking individual’s fitness is reached or successive iterations do no longer produce better results.
10
Performance is 7.51687e−006, Goal is 0
0
Train Validation Test
10
Performance
10
10
10
10
10
−1
−2
−3
−4
−5
−6
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5000 Epochs
Fig. 4. The variation of the mean square error between the predicted values (ANN) and the calculated values (FEM) of the natural frequencies during the training seasons.
541
Genetic algorithm can be used successfully to determine the optimal layer sequences such that the first natural frequency of the layered symmetrical composite plate becomes maximal. The genetic algorithm toolbox of MATLAB [15] was used to solve this optimization problem. Genetic algorithm requires the evaluation of the fitness function X = X (h1, h2, . . ., hn) for each set of design variables h = (h1, h2, . . ., hn). The fitness function is the natural frequency of the composite plate which was calculated using the finite element method. However, this is a timeconsuming stage, and can be reduced considerably using a well trained neural network model instead of the finite element method to predict the first natural frequency of the composite plate for different values of the design variables. 4. Artificial neural networks Linear and non-linear relationships for a set of data can be established by Artificial neural networks (ANN) which obtain the knowledge through a learning period which is stored within inter-neuron connections. Artificial neural networks are similar to biological nervous systems, in which the simple elements, called neurons (Fig. 3), operate in parallel. An artificial neural network is modeled in two stages: (i) the training patterns are generated, and (ii) the neural network model is tested with a set of test data which was not used in the training stage. As in nature, the network function is determined via the connections among processing elements. The neuron receives weighted activation from other neurons through its coming connections. First these weighted activations are added (summation function) and the result is transferred through an activation function to the outcome being the activation of the neuron.
Table 1 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 1, fibre angle increment 5) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[55/45/55/35]s [0/0/0/0]s [45/45/45/45]s [75/50/65/65]s [65/35/40/40]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [5/0/5/5]s [5/5/0/0]s [0/0/0/0]s [0/0/0/0]s [45/45/45/45]s [90/75/60/60]s [0/45/45/45]s [90/90/90/90]s [0/0/0/0]s [0/90/0/90]s
20.88 13.79 11.28 16.28 18.80 38.69 60.47 87.77 39.84 40.28 61.49 61.88 88.41 88.63 56.32 65.27 68.72 90.89 91.99 93.67
20.620 13.787 11.155 16.261 18.779 38.682 60.385 87.511 39.811 40.256 61.389 61.781 88.137 88.356 55.611 65.086 68.476 90.596 91.697 93.538
[50/40/40/40]s [0/0/0/0]s [45/45/45/45]s [70/45/75/90]s [55/10/55/50]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/5]s [0/0/0/10]s [5/5/0/0]s [5/5/5/5]s [0/0/0/5]s [0/0/0/0]s [45/45/45/45]s [65/60/65/90]s [45/45/45/45]s [90/90/90/90]s [0/0/0/0]s [90/90/90/70]s
21.479 13.600 10.846 16.119 18.695 38.377 59.856 86.910 39.546 39.491 61.146 61.358 87.845 88.251 55.418 65.907 71.289 90.483 91.250 93.529
20.954 13.787 11.155 16.303 18.695 38.682 60.385 87.511 39.808 40.251 61.379 61.773 88.128 88.356 55.611 66.238 71.656 90.596 91.697 93.319
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For each of the outgoing connections, this activation value is multiplied with the specific weight and transferred to the next neuron [16].
A training set is a group of matched input and output patterns, which is used for the training of the network. The outputs are the dependent variables produced by the
Fig. 5. The mode shapes and the optimal natural frequencies Xopt of a symmetric eight-layered square composite plate with optimal layered sequence for different edge conditions.
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
network for the present input. This network uses the input data to produce an output after each pattern is read. If there is a difference, the connection weights are altered in such a direction that the error is reduced. After the network has run through all the input patterns repeatedly until all errors remain within a specified tolerance the training process reaches a satisfactory level. Consequently, the neural network holds all weight constants and this trained network could be used to decide identify patterns, or define relationships between a new set of input data not used during the training season.
543
In order to establish the neural network models which can predict accurately the first natural frequency of the composite plate for different design variables the training and testing data should be generated using the finite element method. For this purpose the natural frequencies of the composite plates were calculated for random values of the design variables: the ply angle (90, 89 6 hk 6 90) corresponding to the layer number (2 6 n 6 20), the plate length/width ratio (1 6 a/b 6 5) and for twenty different plate edge conditions. An individual artificial neural network model was trained and tested for
Table 2 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 1, fibre angle increment 1) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[55/45/55/35]s [0/0/0/0]s [45/45/45/45]s [75/50/65/65]s [65/35/40/40]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [5/0/5/5]s [5/5/0/0]s [0/0/0/0]s [0/0/0/0]s [45/45/45/45]s [90/75/60/60]s [0/45/45/45]s [90/90/90/90]s [0/0/0/0]s [0/90/0/90]s
20.88 13.79 11.28 16.28 18.80 38.69 60.47 87.77 39.84 40.28 61.49 61.88 88.41 88.63 56.32 65.27 68.72 90.89 91.99 93.67
20.620 13.787 11.155 16.261 18.779 38.682 60.385 87.511 39.811 40.256 61.389 61.781 88.137 88.356 55.611 65.086 68.476 90.596 91.697 93.538
[46/43/43/43]s [0/0/2/0]s [45/46/47/43]s [67/50/53/75]s [55/10/52/48]s [0/0/0/0]s [0/0/1/2]s [0/0/0/0]s [1/1/1/7]s [0/0/2/9]s [3/3/3/3]s [4/4/4/6]s [0/0/0/4]s [0/1/1/1]s [45/45/45/45]s [64/59/60/60]s [45/45/45/44]s [88/88/90/88]s [0/0/0/0]s [0/0/0/0]s
21.479 13.601 10.847 16.042 18.695 38.378 59.858 86.621 39.548 39.493 61.176 61.360 87.845 88.251 55.418 66.323 71.540 90.567 91.697 93.338
21.479 13.784 11.150 16.195 18.695 38.682 60.382 87.511 39.802 40.249 61.457 61.826 88.131 88.347 55.611 66.323 71.540 90.567 91.697 93.338
Table 3 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 2, fibre angle increment 5) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[5/40/50/45]s [0/0/0/0]s [35/45/45/45]s [85/85/85/85]s [85/85/85/85]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/30/40/35]s [90/70/55/55]s [10/0/5/25]s [10/65/35/35]s [0/0/0/0]s [0/0/0/0]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
32.23 13.79 21.89 57.06 57.71 38.66 60.44 87.74 45.26 61.94 64.84 69.88 90.28 92.28 159.9 245.7 246.4 353.9 247.1 354.9
32.06 13.79 21.73 57.03 57.68 38.68 60.48 87.80 45.14 61.84 64.83 69.78 90.33 92.32 159.81 245.35 245.99 352.87 246.79 353.89
[35/25/40/50]s [0/0/0/0]s [40/50/40/50]s [85/90/90/90]s [80/90/85/80]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [35/35/35/35]s [65/70/50/80]s [10/5/0/5]s [60/35/35/35]s [0/0/0/0]s [0/0/0/0]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
36.467 13.769 21.937 56.900 57.430 38.680 60.478 87.804 47.803 63.541 64.778 69.734 90.197 92.245 159.571 245.159 245.744 352.471 246.548 352.471
37.988 13.790 21.986 56.888 57.577 38.680 60.479 87.804 48.140 63.162 64.802 69.753 90.325 92.321 159.809 245.354 245.988 352.867 246.786 353.889
544
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
each of the twenty plate edge conditions. The structure of the proposed neural networks, such as layer number, neuron number and transfer functions, is designed depending on the prediction success of that ANN model using the testing data. This study uses the Levenberg– Marquardt back-propagation learning algorithm [16] in
a feed-forward, two hidden layers network. Tangent sigmoid transfer function f ðxÞ ¼
2 1 1 þ e2x
ð23Þ
and logarithmic sigmoid transfer function
Fig. 6. The mode shapes and the optimal natural frequencies Xopt of a symmetric eight-layered rectangular composite plate with optimal layered sequence for different edge conditions.
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
f ðxÞ ¼
2 1 þ ex
ð24Þ
are used as the activation functions of the hidden layers and the output layer, respectively. The values of the training and test data were normalized to a range of 0–1. The neural network toolbox of MATLAB was used to develop a proposed neural network model for each plate edge condition [15]. The input data set includes 1100 patterns based on the free vibration analysis for a random set of design variables. The 1000 patterns of the input data were used for the training of the neural network models and the 100 patterns selected randomly were used for testing the neural network model. Fig. 3 shows the topology of ANN model which can predict fast and accurately the first natural frequency of the composite plate with various edge conditions. The layer configuration composed of (n + 1), 10, 3 and 1 neurons in each layer (1–4). The neural network was trained with values of input(s) and output(s) to adjust connection weights between input layer and hidden layer and between hidden layer and output layer, and then this procedure was repeated until an acceptable accuracy is reached for the outputs generated by the neural network. The performance of the neural network was measured with the mean square error (MSE) between the outputs of the neural network and the test data from the finite element method. The training phase elapsed a CPU time of 30–45 min. on a PIV processor having a 3.0 Ghz CPU speed and 1 Gbyte RAM for a training cycle of 5000 (epochs). A mean square error between 105 and 103 during the training season could be archived depending on the plate edge condition as shown in Fig. 4. This neural network model provides expected accuracy for the first natural frequency of the composite plate.
545
5. Results The free vibration analysis is carried out for the composite plates made of the graphite/epoxy system with the material constants as follows [6,7] E1 ¼ 138 GPa; E2 ¼ 8:96 GPa; G12 ¼ 7:1 GPa;
m12 ¼ 0:3
Narita presented the Ritz-based layerwise optimization method for maximizing the first natural frequency parameter X of the symmetric 8-layered composite square plates (a/b = 1) for various plate edge conditions [1]. First, the natural frequency parameters for the optimal layer sequences given by Narita are compared to those of the finite element method and similar results are obtained as shown in Table 1. In addition, the optimal solutions for the first natural frequency parameters of the symmetric 8-layered composite square plates (a/b = 1) for various plate edge conditions are searched using the Genetic algorithms (GAs) and the proposed neural networks (ANNs). The fiber angle of each ply in the composite plate is changed with a step of Dh = 5 between (90 6 hk 6 90). Table 1 also compares the predicted natural frequency parameters and corresponding optimal layer sequences with those given by Narita. It is evident that the Genetic Algorithms and the proposed neural networks are successful in the determination of the optimal layer sequences maximizing the first natural frequency. Fig. 5 shows the mode shapes corresponding to the first maximal natural frequency parameter Xopt of the square composite plates (a/b = 1) for various plate edge conditions. The natural frequency parameter is increased as the plate edges are clamped, such as X = 93.319 for the edge condition CCCC whereas the X = 11.155 for the edge condition SSFF (Fig. 5). The fiber
Table 4 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 2, fibre angle increment 1) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[5/40/50/45]s [0/0/0/0]s [35/45/45/45]s [85/85/85/85]s [85/85/85/85]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/30/40/35]s [90/70/55/55]s [10/0/5/25]s [10/65/35/35]s [0/0/0/0]s [0/0/0/0]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
32.23 13.79 21.89 57.06 57.71 38.66 60.44 87.74 45.26 61.94 64.84 69.88 90.28 92.28 159.9 245.7 246.4 353.9 247.1 354.9
32.06 13.79 21.73 57.03 57.68 38.68 60.48 87.80 45.14 61.84 64.83 69.78 90.33 92.32 159.81 245.35 245.99 352.87 246.79 353.89
[37/29/25/28]s [0/0/0/0]s [43/47/47/43]s [84/90/84/84]s [82/90/84/82]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [36/37/36/36]s [60/56/56/56]s [9/11/21/11]s [31/57/57/35]s [0/0/0/0]s [0/0/0/0]s [90/88/88/88]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
38.603 13.785 21.088 56.745 57.471 38.680 60.479 87.804 48.045 63.556 64.764 71.001 90.330 92.321 159.667 245.354 245.988 352.872 246.786 353.889
39.103 13.785 22.088 56.954 57.581 38.680 60.479 87.804 48.158 63.778 64.875 71.112 90.330 92.321 159.777 245.354 245.988 352.872 246.786 353.889
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550 100
opt
90
Normalized natural frequency Ω
angle of each ply in the square composite plates is also changed with a step of Dh = 1 between (90 6 hk 6 90), and the optimal solutions are compared in Table 2 with those given by Narita [1]. The GAs and ANNs are successful for determining the optimal layer sequences, especially the better frequency parameters are predicted for some of the clamped plates in which the Ritz method converges hardly. Similarly, the optimal solutions for the natural frequency parameters of the symmetric eight-layered composite rectangular plates (a/b = 2) with various plate edge conditions are searched using the Genetic algorithms and the proposed neural networks, and are compared in Table 3 with the optimal solutions given by Narita [1]. The fiber angle of each ply in the composite plate is changed with a step of Dh = 5 between (90 6 hk 6 90). The direct solutions using the finite element method agree with those of the Ritz-based layerwise optimization [1]. In addition, the Genetic algorithm and the proposed neural networks are very successful in the prediction of the optimal solutions. The higher natural frequencies than those predicted by the Ritz-based layerwise optimization method are also achieved for the composite plates (Table 3), such as SFFF, SSFF, SSSF and SCSF edge conditions. Fig. 6 shows the mode shapes corresponding to the first maximal natural frequency parameter of the rectangular plates (a/b = 2) for various plate edge conditions. The natural frequency parameters increase evidently in comparison with those of the square composite plates (Fig. 5) whereas the corresponding mode shapes become similar. The edge conditions also play an important role on the natural frequency parameter of the rectangular composite plates. The number of the clamped plate edges is increased an evident increase in the natural frequency parameters is observed, such as X = 13.79 for the edge condition CFFF whereas the X = 353.89 for the edge condition CCCC (Fig. 6). The fiber angle of each ply in the rectangular composite plates is also changed with a step of Dh = 1 between (90 6 hk 6 90), and the optimal solutions are compared in Table 4 with those given by Narita [1]. The GAs and ANNs predict successfully the optimal layer sequences inducing the maximal natural frequencies (Table 4), and can go the better solutions than the Ritz-based layerwise optimization method, such as the plate edge conditions: SFFF, SSFF, SSSF, SCSF, SSCF and SCCF. Figs. 7 and 8 compare the present optimal solutions of the symmetric eight-layered composite square and rectangular plates with those of the square and rectangular plates with other stacking sequences, respectively. The typical stacking sequences are chosen as the specially orthotropic [0/0/0/0]s, [0/90/0/90]s, the alternating angle-ply sequence [30/30/30/30]s, [45/45/45/45]s and the quasi-isotropic [0/45/45/90]s. All the present optimal solutions exhibit higher natural frequencies than those of the square plates (a/b = 1) with the present five typical stacking sequences (Fig. 7). The Ritz-based layerwise optimization method fails into a local optimum for the edge condition SSCC
80 70 60 50 40 30 [0/0/0/0] s [0/90/0/90] s [30/−30/30/−30] s [45/−45/45/−45] s [0/−45/45/90] s Optimal
20 10 0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Boundary conditions
Fig. 7. Comparison of the optimal natural frequency Xopt and the frequencies of symmetric eight-layered square plate for various stacking sequences (a/b = 1) (see Table 2), (1) SFFF, (2) CFFF, (3) SSFF, (4) SCFF, (5) CCFF, (6) SFSF, (7) SFCF, (8) CFCF, (9) SSSF, (10) SCSF, (11) SSCF, (12) SCCF, (13) CSCF, (14) CCCF, (15) SSSS, (16) SSSC, (17) SSCC, (18) SCSC, (19) CCCS, (20) CCCC.
400 [0/0/0/0] s [0/90/0/90] s [30/−30/30/−30] s [45/−45/45/−45] s [0/−45/45/90] s Optimal
350
Normalized natural frequency Ωopt
546
300
250
200
150
100
50
0 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Boundary conditions
Fig. 8. Comparison of the optimal natural frequency Xopt and the frequencies of symmetric eight-layered rectangular plate for various stacking sequences (a/b = 2) (see Table 4), (1) SFFF, (2) CFFF, (3) SSFF, (4) SCFF, (5) CCFF, (6) SFSF, (7) SFCF, (8) CFCF, (9) SSSF, (10) SCSF, (11) SSCF, (12) SCCF, (13) CSCF, (14) CCCF, (15) SSSS, (16) SSSC, (17) SSCC, (18) SCSC, (19) CCCS, (20) CCCC.
[1]; thus, it gives Xopt = 68.72 for a stacking sequence [0/ 45/45/45]s (Table 2) while an angle-ply sequence [45/ 45/45/45]s gives slightly higher value of X = 71.21. In the case of the edge condition SSCC the Genetic Algorithms and the proposed neural networks predict the maximal natural frequency parameter Xopt = 71.54 with a fiber angle increment of Dh = 1 for a stacking sequence [45/ 45/45/44]s (SSCC in Table 2 and (17, SSCC) in Fig. 7) whereas it finds the maximal natural frequency parameter
Table 5 Optimum solutions for symmetric multi-layered composite plates with various plate length/width ratios (a/b), layer number (n) and boundary conditions (fibre angle increment = 1) 4
BC
a/b
Stacking
8
SFFF
1 2 3 4 5
[52/37]s [36/28]s [10/36]s [32/48]s [30/44]s
8.546 13.546 18.198 36.410 43.718
[46/43/43/43]s [37/29/25/28]s [20/24/24/24]s [4/27/22/21]s [0/4/36/28]s
21.479 38.603 47.475 54.546 57.868
[44/45/46/45/44/ 46]s [40/36/16/26/39/52]s [32/8/16/33/0/47]s [16/16/0/24/14/14]s [0/0/0/48/31/23]s
32.464 58.857 73.465 82.645 86.350
CFFF
1 2 3 4 5
[0/0]s [40/39]s [24/35]s [58/54]s [33/55]s
6.904 19.063 25.265 33.101 48.898
[0/0/2/0]s [0/0/0/0]s [0/0/0/0]s [22/22/21/22]s [15/18/21/22]s
13.601 13.785 13.783 70.168 76.817
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [22/16/10/6/4/4]s
20.692 20.680 20.673 20.667 116.303
SSFF
1 2 3 4 5
[45/45]s [38/51]s [12/8]s [64/61]s [34/50]s
5.232 10.242 39.179 45.177 68.189
[45/46/47/43]s [43/47/47/43]s [42/47/47/42]s [42/47/47/42]s [40/49/40/50]s
10.847 21.088 32.901 43.585 53.553
[45/45/45/45/44/44]s [41/48/42/42/48/48]s [40/49/40/40/40/49]s [40/49/40/40/49/40]s [40/40/50/50/50/50]s
SCFF
1 2 3 4 5
[74/41]s [84/90]s [86/90]s [88/90]s [88/90]s
8.141 28.548 63.064 111.404 173.567
[67/50/53/75]s [84/90/84/84]s [86/90/88/84]s [86/90/88/88]s [88/90/88/88]s
16.042 56.745 126.022 222.611 346.941
[67/47/66/46/68/64]s [90/84/84/84/84/80]s [86/90/86/86/88/88]s [86/90/88/88/88/84]s [88/90/88/88/88/88]s
CCFF
1 2 3 4 5
[58/32]s [84/90]s [86/90]s [88/90]s [88/90]s
9.280 28.880 63.266 111.549 173.673
[55/10/52/48]s [82/90/84/82]s [86/90/84/88]s [86/90/88/88]s [88/90/88/88]s
18.695 57.471 126.399 222.902 347.123
[48/40/48/44/40/32]s [82/90/82/82/82/80]s [86/90/86/86/84/88]s [86/90/88/88/88/84]s [88/90/88/88/88/88]s
SFSF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [24/28]s [4/39]s
19.583 19.389 19.348 46.075 61.231
[0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s
38.378 38.680 38.693 38.663 38.648
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s
58.626 58.139 58.031 57.989 57.969
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
97.311 96.797 96.676 96.624 96.599
SFCF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [53/12]s [22/28]s
30.860 30.375 30.279 46.779 66.445
[0/0/1/2]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s
59.858 60.479 60.544 60.476 60.441
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s
92.062 90.998 90.779 90.695 90.649
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
151.763 151.250 151.120 151.046 151.015
CFCF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [0/0]s [48/11]s
45.225 44.199 44.007 43.936 61.892
[0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s
86.621 87.804 87.982 87.851 87.786
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s
134. 292 132. 273 131. 878 131. 725 131.646
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
219.428 219.370 219.311 219.272 219.235
Xopt
Stacking
12 Xopt
Stacking
20 Xopt
16.774 33.062 48.531 64.460 79.716
Stacking
Xopt
[44/45/44/44/44/44/44/44/44/44]s [32/64/48/32/32/0/16/32/46/28]s [32/42/12/0/0/0/12/12/15/6]s [29/0/0/18/14/10/8/12/4/4]s [0/0/0/16/32/0/1/12/10/10]s
53.435 94.530 122.528 136.318 145.228
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
34.420 34.450 34.446 34.440 34.434
[45/45/45/45/45/45/45/44/44/44]s [41/48/41/41/40/40/48/48/48/48]s [40/41/49/41/49/49/41/41/41/49]s [40/40/40/49/49/49/40/50/50/50]s [41/41/41/49/49/49/49/49/41/48]s
27.568 54.224 80.681 106.291 132.930
24.477 85.416 188.931 333.734 520.092
[68/46/68/46/68/68/68/68/48/64]s [84/84/90/88/84/84/84/84/84/80]s [86/90/86/88/86/88/88/88/88/80]s [86/90/88/88/88/88/88/88/88/84]s [88/90/88/88/88/88/88/88/88/80]s
40.569 142.094 314.273 555.251 865.233
28.540 86.441 189.497 334.171 520.388
[50/44/40/48/32/56/44/48/48/32]s [82/90/84/84/84/84/84/80/80/80]s [86/90/86/86/86/86/86/84/84/88]s [86/86/90/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s
47.244 143.847 315.298 555.863 865.757
547
(continued on next page)
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
Layer number (n)
548
Table 5 (continued) 4
8
BC
a/b
Stacking
SSSF
1 2 3 4 5
[0/0]s [10/36]s [37/43]s [38/46]s [38/48]s
20.153 22.158 28.928 37.407 46.172
SCSF
1 2 3 4 5
[0/0]s [81/56]s [90/88]s [90/90]s [88/90]s
20.375 30.768 64.812 113.106 175.091
SSCF
1 2 3 4 5
[3/3]s [10/20]s [24/45]s [29/49]s [36/47]s
31.408 32.608 37.870 45.712 48.073
SCCF
1 2 3 4 5
[4/4]s [16/56]s [90/90]s [90/90]s [88/90]s
31.600 35.082 65.344 113.484 175.400
CSCF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [17/43]s [27/49]s
45.535 45.464 46.833 50.493 57.795
CCCF
1 2 3 4 5
[0/0]s [0/0]s [90/86]s [90/88]s [90/90]s
SSSS
1 2 3 4 5
SSSC
1 2 3 4 5
Xopt
Stacking
12 Xopt
Stacking
20 Xopt
Stacking
Xopt
[1/1/1/7]s [36/37/36/36]s [42/42/42/42]s [43/44/44/44]s [43/45/45/45]s
39.548 48.045 67.587 88.082 108.987
[0/0/0/0/0/0]s [36/36/36/36/36/40]s [42/42/42/42/44/40]s [43/43/44/43/44/44]s [45/44/44/44/45/44]s
60.320 72.527 101.294 132.267 163.453
[0/0/0/0/0/0/0/0/0/0]s [36/36/36/36/36/36/36/36/36/32]s [42/42/42/42/42/42/42/42/40/40]s [44/44/44/44/44/44/44/44/44/48]s [44/44/44/44/44/44/44/44/48/48]s
100.082 119.577 167.073 218.003 270.025
[0/0/2/9]s [60/56/56/56]s [90/88/88/88]s [88/90/88/88]s [88/90/88/88]s
39.493 63.556 129.549 226.008 350.130
[0/0/0/0/0/0]s [58/58/58/60/58/56]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
60.989 95.886 194.172 338.808 524.899
[0/0/0/0/0/0/0/0/0/0]s [60/60/60/60/60/60/60/60/56/64]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/88/88/88/88/88/88/88/88]s
101.193 158.446 322.893 563.475 872.888
[3/3/3/3]s [9/11/21/11]s [33/40/40/33]s [37/43/44/37]s [42/42/42/42]s
61.176 64.764 78.556 97.842 115.989
[3/4/4/4/0/0]s [2/10/10/10/8/28]s [34/40/40/34/34/32]s [37/44/37/44/44/40]s [44/39/39/39/39/39]s
93.675 97.563 116.877 146.235 176.699
[2/3/4/4/4/4/4/4/0/0]s [8/8/8/9/8/8/24/24/16/0]s [30/31/32/42/40/40/40/40/40/40]s [36/44/44/36/36/36/36/36/44/32]s [45/38/38/38/40/44/44/44/48/48]s
154.387 162.103 194.352 241.581 291.783
[4/4/4/6]s [31/57/57/35]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s
61.360 71.001 130.615 226.781 350.746
[4/4/4/4/0/0]s [26/26/56/56/56/56]s [88/90/88/88/88/80]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
94.258 106.807 195.780 339.960 525.788
[3/4/4/4/4/4/4/4/0/0]s [30/60/32/32/58/56/56/56/56/64]s [88/90/88/88/88/88/88/88/88/80]s [88/90/88/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
155.371 176.589 325.591 565.414 874.506
[0/0/0/4]s [0/0/0/0]s [6/18/20/16]s [32/37/36/36]s [36/40/40/40]s
87.845 90.330 93.953 108.162 127.027
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/16/18/20/16/16]s [34/34/36/36/36/32]s [39/37/37/38/40/40]s
135.204 136.035 140.802 162.216 190.437
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/16/16/12/16/16/16/16/16/0]s [34/34/34/34/34/36/34/36/32/32]s [38/38/38/38/40/40/40/40/40/32]s
220.877 225.556 234.058 268.527 315.341
45.643 46.459 66.019 113.938 175.881
[0/1/1/1]s [0/0/0/0]s [89/90/88/88]s [88/90/88/88]s [88/90/88/88]s
88.251 92.321 131.993 227.700 351.429
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [88/88/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
135.523 139.024 197.838 341.345 526.813
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [88/88/88/90/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
221.415 230.539 329.028 567.718 876.237
[45/45]s [90/88]s [90/88]s [88/90]s [88/90]s
25.789 80.882 178.818 315.755 492.001
[45/45/45/45]s [90/88/88/88]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s
55.418 159.667 357.177 631.146 983.454
[45/45/45/45/45/44]s [90/90/90/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
84.380 242.098 535.068 945.501 1473.268
[45/44/44/44/44/44/44/44/44/48]s [89/88/88/88/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
138.193 401.638 887.939 1569.226 2445.080
[84/58]s [90/90]s [88/90]s [88/90]s [88/90]s
33.142 125.343 279.439 495.366 773.007
[64/59/60/60]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s
66.323 245.354 557.82 988.851 1543.052
[62/60/60/60/64/64]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
100.902 373.715 833.806 1478.093 2306.559
[64/63/64/64/64/64/64/64/64/64]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
165.715 615.909 1374.312 2436.368 3801.812
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
Layer number (n)
236.516 886.171 1981.745 3516.541 5490.266 [0/0/90/88/88/0/0/0/0/0]s [88/88/90/88/88/88/88/88/88/80]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/80]s [0/0/0/0]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s [0/0]s [88/90]s [88/90]s [88/90]s [88/90]s
48.169 182.629 408.381 724.670 1131.408
93.338 353.889 813.785 1444.046 2254.522
[0/0/0/0/0/0]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
143.06 542.42 1212.898 2152.327 3360.276
229.517 619.517 1376.444 2437.826 3802.952 [0/0/0/0/0/0/0/0/0/0]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [0/0/0/0]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s 47.325 126.04 279.857 495.658 773.235 [0/0]s [90/88]s [88/90]s [88/90]s [88/90]s
91.697 246.786 558.673 989.464 1543.531
[0/0/0/0/0/0]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
140.508 375.866 835.118 1478.968 2307.242
226.628 883.655 1980.187 3515.520 5489.355 [88/88/90/88/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/88/90/88]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s 46.761 182.13 408.086 724.451 1131.225 [90/88]s [88/90]s [88/90]s [88/90]s [88/90]s
90.567 352.872 813.186 1443.609 2254.181
[90/88/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
138.785 540.925 1211.996 2151.598 3359.821
178.792 617.476 1375.296 2437.097 3802.496 [45/45/45/44]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s [10/46]s [88/90]s [88/90]s [88/90]s [88/90]s
34.265 125.551 279.628 495.497 773.098
71.540 245.988 558.205 989.128 1543.280
[44/44/44/48/48/48]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
109.022 374.663 834.380 1478.531 2306.787
[44/44/44/44/44/44/48/48/44/48]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s
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549
Xopt = 71.289 with a fiber angle increment of Dh = 5 for a stacking sequence [45/45/45/45]s (SSCC in Table 1). Similarly, all the present optimal solutions exhibit higher natural frequencies than those of the rectangular plates (a/b = 2) with the present five typical stacking sequences (Fig. 8). Finally, the optimal solutions are searched for the composite plates for twenty different edge conditions, the plate length/width ratios (a/b = 1, 2, 3, 4, 5), the layer number (n = 4, 8, 12, 20), and are given in Table 5. As either the plate length/width ratio (a/b) or the layer number (n) is increased the optimal natural frequency parameter Xopt is increased. Some outer layers of the composite plates have a more stiffening effect on the optimal natural frequency parameter and the orientation of the layers becomes similar toward the mid-plane of the composite plate as the layer number is increased (Table 5). 6. Conclusions In this study the optimal layered sequences of symmetrically laminated square and rectangular plates were searched by means of the Genetic Algorithms and the proposed neural networks. The natural frequencies of the composite plates were calculated using the finite element technique for various plate edge conditions and random plate length/width ratios and layer number, and the proposed neural networks were trained and tested based on these data. The Genetic Algorithm and the proposed neural networks predicted successfully the natural frequencies of the composite plates and the predicted frequencies and optimal layered sequences were in good agreement with those of the Ritz-based layerwise optimization method [1]. The present method finds optimal design maximizing the natural frequency of the composite plates without yielding a local optimum for all edge conditions and design parameters whereas this possibility is experienced in the Ritz-based layerwise optimization method. In addition, the natural frequencies of the composite square and rectangular plates are increased with increasing layer number. The outer layers of the composite plates have an effect on the plate bending stiffness; thus, the fiber orientation of the layers close to the mid-plane of the composite plate. Acknowledgement Authors would like to thank for the financial support of the Scientific Research Project Division of Erciyes University under the contract: FBT-05-34.
1 2 3 4 5 CCCC
1 2 3 4 5 CCCS
1 2 3 4 5 SCSC
SSCC
1 2 3 4 5
References [1] Narita Y. Layerwise optimization for the maximum fundamental frequency of laminated composite plates. J Sound Vib 2003;263: 1005–16. [2] Leissa AW. Vibration of Plates, NASA-160. Washington, DC: US Government Printing Office; 1969.
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[3] Blevins RD. Formulas for natural frequency and mode shape. New York: Van Nostrand Reinhold; 1979. [4] Gorman DJ. Free vibration analysis of rectangular plates. New York: Elsevier; 1982. [5] Jones RM. Mechanics of composite materials. Washington, DC: Scripta; 1975. [6] Vinson JR, Sierakowski RL. The behavior of structures composed of composite materials. Dordrecht: Martinus Nijhoff; 1986. [7] Reedy JN. Mechanics of laminated composite plates: theory and analysis. London: CRC Press; 1996. [8] Rao SS. The finite element method in engineering. Oxford, UK: Pergamon Press; 1989. [9] Cook RD. Concepts and applications of finite element analysis. Second ed. New York: John Wiley and Sons; 1981. [10] Grimes RG, Lewis JG, Simon HD. SIAM J Matrix Anal Appl 1994;15:228–72.
[11] Michalewicz Z. Genetic algorithms + data structures = evolution programs. Springer-Verlag; 1999. [12] Goldberg DE. Genetic algorithms in search, optimization and machine learning. Boston, MA: Kluwer Academic Publishers; 1989. [13] Goldberg DE. The design of innovation: lessons from and for competent genetic algorithms. Reading, MA: Addison-Wesley; 2002. [14] Gurdal Z, Haftka RT, Hajela P. In: Design and optimization of laminated composite materials. New York: Wiley Interscience; 1999. p. 297–310. [15] MATLAB (ver 7.0), The MathWorks Inc., 3 Apple Hill Drive, Natick, MA 01760-2098. [16] Hagan MT, Menhaj M. Training feed-forward networks with the Marquardt algorithm. IEEE Trans Neural Networks 1994;5(6): 989–93.
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 537–550 www.elsevier.com/locate/compscitech
Layer optimisation for maximum fundamental frequency of laminated composite plates for different edge conditions M. Kemal Apalak b
a,*
, Mustafa Yildirim b, Recep Ekici
a
a Department of Mechanical Engineering, Erciyes University, Kayseri 38039, Turkey Graduate School of Natural and Applied Sciences, Erciyes University, Kayseri 38039, Turkey
Received 19 January 2007; accepted 2 June 2007 Available online 2 August 2007
Abstract In this study the layer optimisation was carried out for maximum fundamental frequency of laminated composite plates under any combination of the three classical edge conditions. The optimal stacking sequences of laminated composite plates were searched by means of Genetic Algorithm. The first natural frequencies of the laminated composite plates with various stacking sequences were calculated using the finite element method. Genetic Algorithm maximizes the first natural frequency of the laminated composite plate defined as a fitness function (objective function). However, the finite element method needs a certain calculation time of the first natural frequency for each new lay-up sequence and plate edge condition. In order to reduce the searching time of the optimal lay-up sequence an artificial neural network model was proposed and trained with small training and testing data composed of the natural frequencies of the composite plates calculated for random lay-up sequences, layer number, edge conditions and plate length/width ratios using the finite element method. The outer layers of the composite plate had a stiffness increasing effect, and as the clamped plate edges were increased both stiffness and natural frequency of the plate increased. In addition the Genetic Algorithm predicted successfully the optimal layer sequences without yielding a local optimum on the contrary the Ritz-based layerwise optimisation method [Y. Narita, Layerwise optimization for the maximum fundamental frequency of laminated composite plates. J Sound Vib 2003;263:1005–16]. 2007 Elsevier Ltd. All rights reserved. Keywords: Composites; Laminated plate; Natural frequency; Optimisation; Genetic algorithm; Artificial neural network
1. Introduction In structural design the laminated fibrous composite plates and shells offer light-weight structures with high stiffness and strength. However, they exhibit material anisotropy and require more complex analysis and design methods. Since these composite structures operate dynamic loading conditions as well as the static loads their response against the structural resonance should be optimized by maximizing the natural frequency of the composite structures. The free vibration analysis of isotropic and composite plates has been carried out extensively [1–7]. However, these studies are limited to the optimization of vibration *
Corresponding author.Tel.: +90 352 437 4901; fax: +90 352 437 5784. E-mail address: [email protected] (M. Kemal Apalak).
0266-3538/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.06.031
characteristics of the composite structures only for some stacking sequences and boundary conditions since this optimization problem requires the combination of the complicated aspects of both the plate free vibration and the optimization techniques. The number of design variables affects directly the computation time. The design variables of this optimization problem are considered as fiber angle of each layer, thickness, length/width ratio of the composite plate. An optimal solution maximizes the natural frequency of the composite plate defined as the objective function with n design variables subjected to m constraints. As the layer number of the plate is increased an considerable increase in the searching time of the optimal solution is observed. Narita offered a Ritz-based layerwise optimization approach for the symmetrical composite plates [1]. His
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assumption for the layerwise optimization approach is introduced as the optimal stacking sequence for the maximum natural frequency of laminated plate can be determined sequentially in the order from the outermost to the innermost layer. He reduced evidently the searching time and calculation for the optimal solution. However, his approach may yield a local optimum for some edge conditions. In this study the optimal layer sequences of the symmetrical composite plates were determined using the Genetic Algorithms so that their first natural frequency can be maximized. The finite element method requires a certain time for the calculation of the natural frequency of the composite plates. In order to reduce this calculation time the artificial neural networks were proposed. The Genetic algorithm combined with these artificial neural networks found the optimal layer sequences without yielding a local optimum for all cases. 2. Problem description Fig. 1 shows a laminated composite plate in the problem coordinate (x, y) and each layer in the material coordinate (x1, x2). The composite plate with a dimension of (a · b · t) has symmetrical layered sequences, and composed of 2 · n layers (n layers in each of the upper and lower half sections of the plates). Free (F), simply supported (S) and clamped (C) edge conditions are assumed along the plate edges (1, 2, 3, 4) in Fig. 1. Based on the classical plate theory the free vibration of a laminated (symmetrical) thin plate is governed by o4 w o4 w o4 w o4 w D11 4 þ 2ðD12 þ 2D66 Þ 2 2 þ D22 4 þ 4D16 3 ox ox oy oy ox oy þ 4D26
o4 w qx2 w ¼ 0 oxoy 3
ð1Þ
x1 Edge 4
O
E1 ; 1 t12 t21 Q66 ¼ G12
Q11 ¼
x
½Q
ðkÞ
b t
Fig. 1. The edge numbering of a layered composite plate in the material coordinate (x1, x2) and the problem coordinate (x, y).
E1 t21 ; 1 t12 t21
ðkÞ
T
¼ ð½T ½Q½T Þ
Q22 ¼
E2 ; 1 t12 t21 ð3Þ
ð4Þ
where [T] is transformation matrix considering the fiber angle h(k) of the kth lamina [7]. Normalized natural frequency is 1=2 q X ¼ xa2 ð5Þ D0 where the reference bending rigidity is 1 E2 t3 : 12 1 t12 t21
ð6Þ
The Ritz method is used for the linear free vibration of a thin plate subjected to simple full edge conditions whereas the finite element method is preferred for the thin plates with complex edge conditions. The dynamic equations of motion of a structure can be derived by using either Lagrange equations and Hamilton’s principle [8,9]. The Lagrange equations are given by 8 9 8 9 8 9 d < oL = < oL = < oR= þ ¼ f0g ð7Þ dt :o q_ ; :o q; :o q_ ;
where ð8Þ
L ¼ T pp
Edge 2
Q12 ¼
where E1 and E2 are longitudinal and transverse moduli of elasticity, G12 is shear modulus, t12 and t21 are major and minor Poisson’s ratios of a lamina in the material ðkÞ coordinate. The reduced stiffness Qij of the kth lamina ðkÞ can be transformed to Qij in the problem coordinate (x, y) as
Edge 3
Edge 1
θ
a
where zk is through-the-thickness distance from the middle ðkÞ surface. The Qij are reduced stiffness of a lamina in the material coordinate (x1, x2) as
D0 ¼
y
x2
where w is the displacement of the middle surface, q is mean mass per unit area and x is circular frequency. The bending stiffness of a symmetrical laminate is given by [5–7] n 2X ðkÞ ð2Þ Q z3 z3k1 ði; j ¼ 1; 2; . . . ; 6Þ Dij ¼ 3 k¼1 ij k
is called the Lagrange function, T is the kinetic energy, pp is the potential energy, R is dissipation function, q is the nodal displacement and q_ is the nodal velocity. The kinetic and potential energies of each element of the structure which was divided into E elements can be written as Z Z Z 1 T ðeÞ ¼ qu_ T u_ dV ð9Þ 2 V ðeÞ
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
pðeÞ p ¼
1 2
Z Z Z
e T r dV
Z Z
V ðeÞ
uT /1 dS 1
Dij ¼
ðeÞ
Z Z Z
S1
uT /2 dV
ð10Þ
V ðeÞ
where q is density, u and u_ are displacement and velocity vectors. The dissipation function of each element Z Z Z 1 RðeÞ ¼ lu_ T u_ dV ð11Þ 2 V ðeÞ
where l is damping coefficient. Using the finite element method the overall system equations of motion can be written as [8,9] € ðtÞ þ ½C Q_ ðtÞ þ ½K QðtÞ ¼ P c ðtÞ ½M Q
ð12Þ
€ ðtÞ is the vector of nodal accelerations, and where Q E Z Z Z X T mass matrix½M ¼ q½N ½N dV e¼1
V ðeÞ
element stiffness matrix½K ¼
E Z Z Z X e¼1
element damping matrix½C ¼
total load vector½P ¼
E Z Z Z X e¼1
½BT ½D½BdV
ð14Þ
T
ð15Þ
V ðeÞ
E Z Z Z X e¼1
ð13Þ
l½N ½N dV
V ðeÞ T
½N f/1 gdS 1
ðeÞ
S1
þ
Z Z Z
½N T f/2 gdV
ð16Þ
V ðeÞ
where [N] is the matrix of shape functions, [B] is the matrix of coordinate derivatives of shape functions, [D] is the matrix of elastic constants, {/1} is surface traction vector, and {/2} is body force vector [8,9]. The equation of motion for an undamped system without external force becomes € þ½K Q ¼ 0: ð17Þ ½M Q
The solution of this eigenvalue problem detð½K k½MÞ ¼ f0g
ð18Þ
Each eigenvalue ki is associated with an eigenvector Qi
which is called a natural mode. The block Lanczos eigenvalue extraction method was used for the calculation of eigenvalues and eigenvectors [8–10]. In the free vibration analysis of the composite plates a layered shell element with four nodes was used, and three integration points in each layer through the shell element thickness were considered [9]. The bending stiffness coefficient matrix [D] of the shell element was modified with
n 2X ðkÞ Qij z3k z3k1 ði; j ¼ 1; 2; . . . ; 6Þ: 3 k¼1
539
ð19Þ
The main objective in the optimization problem is to determine the optimal layer sequences which maximize the normalized frequency X for the first natural mode of the composite plate. The optimization problem can be defined as Maximize the objective function X ¼ Xðh1 ; h2 ; . . . ; hn Þ ! max
ð20Þ
for the design variables h ¼ ðh1 ; h2 ; . . . ; hn Þ
ð21Þ
subject to the constraints 90 6 hk 6 90
ð22Þ
where n is the layer number. It is not convenient to search directly the optimal layer sequences satisfying a maximal natural frequency condition since the iteration number increases with increasing layer number, thus, 181n possibilities appear for a fiber angle (90, 89 6 hk 6 90). An optimal layer sequences can be searched using the Genetic Algorithm, which is very fast in finding for values of the design variables satisfying the maximum fitness value of the objective function. 3. Genetic algorithm Genetic algorithms are a particular class of evolutionary algorithms and inspire some techniques from the evolutionary biology, such as inheritance, mutation, natural selection, and crossover [11–14]. The candidate solutions (individuals) to an optimization problem are defined as a population of abstract representations (chromosomes). The different encodings, such as binary or float, can be used for the representations of the parameters set. The evolution process is performed so that better solutions to the optimization problem can be obtained, and is implemented as follows: (i) A solution of the problem is represented by a list of parameters called chromosome or genome. An initial set of the individuals is generated randomly and the value of the fitness function is evaluated for each of this initial set. These results returned from the fitness function for each individual are sorted and the individuals with better fitness are kept. (ii) In order to generate a consecutive population a pair of parent organisms is selected for breeding by considering generations giving better fitness. Roulette wheel selection and tournament selection are well-defined organism selection methods. Later, the reproduction processes are applied to the individuals through the genetic operators, such as crossover and mutation (Fig. 2). (iii) The crossover (recombination) operation is performed upon the selected chromosomes. Two selected organisms breed and recombined with a crossover probability between 0.6 and 1.0. These two new child chromosomes are added to the next generation population. The crossover operation
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Generate initial random population
Evaluate the fitness value
Yes
Convergence criteria
Stop
No
Selection
Crossover
Mutation
Reproduction Fig. 2. Evolutionary stages of genetic algorithm.
Hidden Layers
{
θ1 → θ2 → … …
Fiber angles =
Input Layer
θ n − 1→ θn →
Output Layer
1 . . .
Natural frequency
n
Plate length / width ratio, a/b
Information processing in an artificial neural network unit.
a) tangent sigmoid and b) logarithmic sigmoid transfer functions.
Fig. 3. Architecture of an ANN model with (n + 1), 10, 3 and 1 neurons in input, hidden, output layers.
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
mixes the chromosomes of the parents by swapping a portion of the data structure, and can be repeated with different parents to create a sufficient number of candidate solutions in the next generation population (Fig. 2). (iv) This created offspring is randomly mutated by flipping bits in the chromosome data structure based on a fixed very small probability of mutation on the order of 0.001 or less. The next generation population of chromosomes is ultimately different from the initial generation. (v) The evaluation is interrupted when the highest ranking individual’s fitness is reached or successive iterations do no longer produce better results.
10
Performance is 7.51687e−006, Goal is 0
0
Train Validation Test
10
Performance
10
10
10
10
10
−1
−2
−3
−4
−5
−6
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5000 Epochs
Fig. 4. The variation of the mean square error between the predicted values (ANN) and the calculated values (FEM) of the natural frequencies during the training seasons.
541
Genetic algorithm can be used successfully to determine the optimal layer sequences such that the first natural frequency of the layered symmetrical composite plate becomes maximal. The genetic algorithm toolbox of MATLAB [15] was used to solve this optimization problem. Genetic algorithm requires the evaluation of the fitness function X = X (h1, h2, . . ., hn) for each set of design variables h = (h1, h2, . . ., hn). The fitness function is the natural frequency of the composite plate which was calculated using the finite element method. However, this is a timeconsuming stage, and can be reduced considerably using a well trained neural network model instead of the finite element method to predict the first natural frequency of the composite plate for different values of the design variables. 4. Artificial neural networks Linear and non-linear relationships for a set of data can be established by Artificial neural networks (ANN) which obtain the knowledge through a learning period which is stored within inter-neuron connections. Artificial neural networks are similar to biological nervous systems, in which the simple elements, called neurons (Fig. 3), operate in parallel. An artificial neural network is modeled in two stages: (i) the training patterns are generated, and (ii) the neural network model is tested with a set of test data which was not used in the training stage. As in nature, the network function is determined via the connections among processing elements. The neuron receives weighted activation from other neurons through its coming connections. First these weighted activations are added (summation function) and the result is transferred through an activation function to the outcome being the activation of the neuron.
Table 1 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 1, fibre angle increment 5) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[55/45/55/35]s [0/0/0/0]s [45/45/45/45]s [75/50/65/65]s [65/35/40/40]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [5/0/5/5]s [5/5/0/0]s [0/0/0/0]s [0/0/0/0]s [45/45/45/45]s [90/75/60/60]s [0/45/45/45]s [90/90/90/90]s [0/0/0/0]s [0/90/0/90]s
20.88 13.79 11.28 16.28 18.80 38.69 60.47 87.77 39.84 40.28 61.49 61.88 88.41 88.63 56.32 65.27 68.72 90.89 91.99 93.67
20.620 13.787 11.155 16.261 18.779 38.682 60.385 87.511 39.811 40.256 61.389 61.781 88.137 88.356 55.611 65.086 68.476 90.596 91.697 93.538
[50/40/40/40]s [0/0/0/0]s [45/45/45/45]s [70/45/75/90]s [55/10/55/50]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/5]s [0/0/0/10]s [5/5/0/0]s [5/5/5/5]s [0/0/0/5]s [0/0/0/0]s [45/45/45/45]s [65/60/65/90]s [45/45/45/45]s [90/90/90/90]s [0/0/0/0]s [90/90/90/70]s
21.479 13.600 10.846 16.119 18.695 38.377 59.856 86.910 39.546 39.491 61.146 61.358 87.845 88.251 55.418 65.907 71.289 90.483 91.250 93.529
20.954 13.787 11.155 16.303 18.695 38.682 60.385 87.511 39.808 40.251 61.379 61.773 88.128 88.356 55.611 66.238 71.656 90.596 91.697 93.319
542
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
For each of the outgoing connections, this activation value is multiplied with the specific weight and transferred to the next neuron [16].
A training set is a group of matched input and output patterns, which is used for the training of the network. The outputs are the dependent variables produced by the
Fig. 5. The mode shapes and the optimal natural frequencies Xopt of a symmetric eight-layered square composite plate with optimal layered sequence for different edge conditions.
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
network for the present input. This network uses the input data to produce an output after each pattern is read. If there is a difference, the connection weights are altered in such a direction that the error is reduced. After the network has run through all the input patterns repeatedly until all errors remain within a specified tolerance the training process reaches a satisfactory level. Consequently, the neural network holds all weight constants and this trained network could be used to decide identify patterns, or define relationships between a new set of input data not used during the training season.
543
In order to establish the neural network models which can predict accurately the first natural frequency of the composite plate for different design variables the training and testing data should be generated using the finite element method. For this purpose the natural frequencies of the composite plates were calculated for random values of the design variables: the ply angle (90, 89 6 hk 6 90) corresponding to the layer number (2 6 n 6 20), the plate length/width ratio (1 6 a/b 6 5) and for twenty different plate edge conditions. An individual artificial neural network model was trained and tested for
Table 2 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 1, fibre angle increment 1) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[55/45/55/35]s [0/0/0/0]s [45/45/45/45]s [75/50/65/65]s [65/35/40/40]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [5/0/5/5]s [5/5/0/0]s [0/0/0/0]s [0/0/0/0]s [45/45/45/45]s [90/75/60/60]s [0/45/45/45]s [90/90/90/90]s [0/0/0/0]s [0/90/0/90]s
20.88 13.79 11.28 16.28 18.80 38.69 60.47 87.77 39.84 40.28 61.49 61.88 88.41 88.63 56.32 65.27 68.72 90.89 91.99 93.67
20.620 13.787 11.155 16.261 18.779 38.682 60.385 87.511 39.811 40.256 61.389 61.781 88.137 88.356 55.611 65.086 68.476 90.596 91.697 93.538
[46/43/43/43]s [0/0/2/0]s [45/46/47/43]s [67/50/53/75]s [55/10/52/48]s [0/0/0/0]s [0/0/1/2]s [0/0/0/0]s [1/1/1/7]s [0/0/2/9]s [3/3/3/3]s [4/4/4/6]s [0/0/0/4]s [0/1/1/1]s [45/45/45/45]s [64/59/60/60]s [45/45/45/44]s [88/88/90/88]s [0/0/0/0]s [0/0/0/0]s
21.479 13.601 10.847 16.042 18.695 38.378 59.858 86.621 39.548 39.493 61.176 61.360 87.845 88.251 55.418 66.323 71.540 90.567 91.697 93.338
21.479 13.784 11.150 16.195 18.695 38.682 60.382 87.511 39.802 40.249 61.457 61.826 88.131 88.347 55.611 66.323 71.540 90.567 91.697 93.338
Table 3 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 2, fibre angle increment 5) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[5/40/50/45]s [0/0/0/0]s [35/45/45/45]s [85/85/85/85]s [85/85/85/85]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/30/40/35]s [90/70/55/55]s [10/0/5/25]s [10/65/35/35]s [0/0/0/0]s [0/0/0/0]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
32.23 13.79 21.89 57.06 57.71 38.66 60.44 87.74 45.26 61.94 64.84 69.88 90.28 92.28 159.9 245.7 246.4 353.9 247.1 354.9
32.06 13.79 21.73 57.03 57.68 38.68 60.48 87.80 45.14 61.84 64.83 69.78 90.33 92.32 159.81 245.35 245.99 352.87 246.79 353.89
[35/25/40/50]s [0/0/0/0]s [40/50/40/50]s [85/90/90/90]s [80/90/85/80]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [35/35/35/35]s [65/70/50/80]s [10/5/0/5]s [60/35/35/35]s [0/0/0/0]s [0/0/0/0]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
36.467 13.769 21.937 56.900 57.430 38.680 60.478 87.804 47.803 63.541 64.778 69.734 90.197 92.245 159.571 245.159 245.744 352.471 246.548 352.471
37.988 13.790 21.986 56.888 57.577 38.680 60.479 87.804 48.140 63.162 64.802 69.753 90.325 92.321 159.809 245.354 245.988 352.867 246.786 353.889
544
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
each of the twenty plate edge conditions. The structure of the proposed neural networks, such as layer number, neuron number and transfer functions, is designed depending on the prediction success of that ANN model using the testing data. This study uses the Levenberg– Marquardt back-propagation learning algorithm [16] in
a feed-forward, two hidden layers network. Tangent sigmoid transfer function f ðxÞ ¼
2 1 1 þ e2x
ð23Þ
and logarithmic sigmoid transfer function
Fig. 6. The mode shapes and the optimal natural frequencies Xopt of a symmetric eight-layered rectangular composite plate with optimal layered sequence for different edge conditions.
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
f ðxÞ ¼
2 1 þ ex
ð24Þ
are used as the activation functions of the hidden layers and the output layer, respectively. The values of the training and test data were normalized to a range of 0–1. The neural network toolbox of MATLAB was used to develop a proposed neural network model for each plate edge condition [15]. The input data set includes 1100 patterns based on the free vibration analysis for a random set of design variables. The 1000 patterns of the input data were used for the training of the neural network models and the 100 patterns selected randomly were used for testing the neural network model. Fig. 3 shows the topology of ANN model which can predict fast and accurately the first natural frequency of the composite plate with various edge conditions. The layer configuration composed of (n + 1), 10, 3 and 1 neurons in each layer (1–4). The neural network was trained with values of input(s) and output(s) to adjust connection weights between input layer and hidden layer and between hidden layer and output layer, and then this procedure was repeated until an acceptable accuracy is reached for the outputs generated by the neural network. The performance of the neural network was measured with the mean square error (MSE) between the outputs of the neural network and the test data from the finite element method. The training phase elapsed a CPU time of 30–45 min. on a PIV processor having a 3.0 Ghz CPU speed and 1 Gbyte RAM for a training cycle of 5000 (epochs). A mean square error between 105 and 103 during the training season could be archived depending on the plate edge condition as shown in Fig. 4. This neural network model provides expected accuracy for the first natural frequency of the composite plate.
545
5. Results The free vibration analysis is carried out for the composite plates made of the graphite/epoxy system with the material constants as follows [6,7] E1 ¼ 138 GPa; E2 ¼ 8:96 GPa; G12 ¼ 7:1 GPa;
m12 ¼ 0:3
Narita presented the Ritz-based layerwise optimization method for maximizing the first natural frequency parameter X of the symmetric 8-layered composite square plates (a/b = 1) for various plate edge conditions [1]. First, the natural frequency parameters for the optimal layer sequences given by Narita are compared to those of the finite element method and similar results are obtained as shown in Table 1. In addition, the optimal solutions for the first natural frequency parameters of the symmetric 8-layered composite square plates (a/b = 1) for various plate edge conditions are searched using the Genetic algorithms (GAs) and the proposed neural networks (ANNs). The fiber angle of each ply in the composite plate is changed with a step of Dh = 5 between (90 6 hk 6 90). Table 1 also compares the predicted natural frequency parameters and corresponding optimal layer sequences with those given by Narita. It is evident that the Genetic Algorithms and the proposed neural networks are successful in the determination of the optimal layer sequences maximizing the first natural frequency. Fig. 5 shows the mode shapes corresponding to the first maximal natural frequency parameter Xopt of the square composite plates (a/b = 1) for various plate edge conditions. The natural frequency parameter is increased as the plate edges are clamped, such as X = 93.319 for the edge condition CCCC whereas the X = 11.155 for the edge condition SSFF (Fig. 5). The fiber
Table 4 Optimum solutions for symmetric 8-layered square plates with various boundary conditions (a/b = 2, fibre angle increment 1) BC
Boundary conditions
Optimal stacking Ref. [1]
Xopt Ref. [1]
Xopt FEM
Optimal stacking (GA and ANN)
Xopt (GA and ANN)
Xopt FEM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
SFFF CFFF SSFF SCFF CCFF SFSF SFCF CFCF SSSF SCSF SSCF SCCF CSCF CCCF SSSS SSSC SSCC SCSC CCCS CCCC
[5/40/50/45]s [0/0/0/0]s [35/45/45/45]s [85/85/85/85]s [85/85/85/85]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/30/40/35]s [90/70/55/55]s [10/0/5/25]s [10/65/35/35]s [0/0/0/0]s [0/0/0/0]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
32.23 13.79 21.89 57.06 57.71 38.66 60.44 87.74 45.26 61.94 64.84 69.88 90.28 92.28 159.9 245.7 246.4 353.9 247.1 354.9
32.06 13.79 21.73 57.03 57.68 38.68 60.48 87.80 45.14 61.84 64.83 69.78 90.33 92.32 159.81 245.35 245.99 352.87 246.79 353.89
[37/29/25/28]s [0/0/0/0]s [43/47/47/43]s [84/90/84/84]s [82/90/84/82]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [36/37/36/36]s [60/56/56/56]s [9/11/21/11]s [31/57/57/35]s [0/0/0/0]s [0/0/0/0]s [90/88/88/88]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s [90/90/90/90]s
38.603 13.785 21.088 56.745 57.471 38.680 60.479 87.804 48.045 63.556 64.764 71.001 90.330 92.321 159.667 245.354 245.988 352.872 246.786 353.889
39.103 13.785 22.088 56.954 57.581 38.680 60.479 87.804 48.158 63.778 64.875 71.112 90.330 92.321 159.777 245.354 245.988 352.872 246.786 353.889
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550 100
opt
90
Normalized natural frequency Ω
angle of each ply in the square composite plates is also changed with a step of Dh = 1 between (90 6 hk 6 90), and the optimal solutions are compared in Table 2 with those given by Narita [1]. The GAs and ANNs are successful for determining the optimal layer sequences, especially the better frequency parameters are predicted for some of the clamped plates in which the Ritz method converges hardly. Similarly, the optimal solutions for the natural frequency parameters of the symmetric eight-layered composite rectangular plates (a/b = 2) with various plate edge conditions are searched using the Genetic algorithms and the proposed neural networks, and are compared in Table 3 with the optimal solutions given by Narita [1]. The fiber angle of each ply in the composite plate is changed with a step of Dh = 5 between (90 6 hk 6 90). The direct solutions using the finite element method agree with those of the Ritz-based layerwise optimization [1]. In addition, the Genetic algorithm and the proposed neural networks are very successful in the prediction of the optimal solutions. The higher natural frequencies than those predicted by the Ritz-based layerwise optimization method are also achieved for the composite plates (Table 3), such as SFFF, SSFF, SSSF and SCSF edge conditions. Fig. 6 shows the mode shapes corresponding to the first maximal natural frequency parameter of the rectangular plates (a/b = 2) for various plate edge conditions. The natural frequency parameters increase evidently in comparison with those of the square composite plates (Fig. 5) whereas the corresponding mode shapes become similar. The edge conditions also play an important role on the natural frequency parameter of the rectangular composite plates. The number of the clamped plate edges is increased an evident increase in the natural frequency parameters is observed, such as X = 13.79 for the edge condition CFFF whereas the X = 353.89 for the edge condition CCCC (Fig. 6). The fiber angle of each ply in the rectangular composite plates is also changed with a step of Dh = 1 between (90 6 hk 6 90), and the optimal solutions are compared in Table 4 with those given by Narita [1]. The GAs and ANNs predict successfully the optimal layer sequences inducing the maximal natural frequencies (Table 4), and can go the better solutions than the Ritz-based layerwise optimization method, such as the plate edge conditions: SFFF, SSFF, SSSF, SCSF, SSCF and SCCF. Figs. 7 and 8 compare the present optimal solutions of the symmetric eight-layered composite square and rectangular plates with those of the square and rectangular plates with other stacking sequences, respectively. The typical stacking sequences are chosen as the specially orthotropic [0/0/0/0]s, [0/90/0/90]s, the alternating angle-ply sequence [30/30/30/30]s, [45/45/45/45]s and the quasi-isotropic [0/45/45/90]s. All the present optimal solutions exhibit higher natural frequencies than those of the square plates (a/b = 1) with the present five typical stacking sequences (Fig. 7). The Ritz-based layerwise optimization method fails into a local optimum for the edge condition SSCC
80 70 60 50 40 30 [0/0/0/0] s [0/90/0/90] s [30/−30/30/−30] s [45/−45/45/−45] s [0/−45/45/90] s Optimal
20 10 0
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Boundary conditions
Fig. 7. Comparison of the optimal natural frequency Xopt and the frequencies of symmetric eight-layered square plate for various stacking sequences (a/b = 1) (see Table 2), (1) SFFF, (2) CFFF, (3) SSFF, (4) SCFF, (5) CCFF, (6) SFSF, (7) SFCF, (8) CFCF, (9) SSSF, (10) SCSF, (11) SSCF, (12) SCCF, (13) CSCF, (14) CCCF, (15) SSSS, (16) SSSC, (17) SSCC, (18) SCSC, (19) CCCS, (20) CCCC.
400 [0/0/0/0] s [0/90/0/90] s [30/−30/30/−30] s [45/−45/45/−45] s [0/−45/45/90] s Optimal
350
Normalized natural frequency Ωopt
546
300
250
200
150
100
50
0 0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Boundary conditions
Fig. 8. Comparison of the optimal natural frequency Xopt and the frequencies of symmetric eight-layered rectangular plate for various stacking sequences (a/b = 2) (see Table 4), (1) SFFF, (2) CFFF, (3) SSFF, (4) SCFF, (5) CCFF, (6) SFSF, (7) SFCF, (8) CFCF, (9) SSSF, (10) SCSF, (11) SSCF, (12) SCCF, (13) CSCF, (14) CCCF, (15) SSSS, (16) SSSC, (17) SSCC, (18) SCSC, (19) CCCS, (20) CCCC.
[1]; thus, it gives Xopt = 68.72 for a stacking sequence [0/ 45/45/45]s (Table 2) while an angle-ply sequence [45/ 45/45/45]s gives slightly higher value of X = 71.21. In the case of the edge condition SSCC the Genetic Algorithms and the proposed neural networks predict the maximal natural frequency parameter Xopt = 71.54 with a fiber angle increment of Dh = 1 for a stacking sequence [45/ 45/45/44]s (SSCC in Table 2 and (17, SSCC) in Fig. 7) whereas it finds the maximal natural frequency parameter
Table 5 Optimum solutions for symmetric multi-layered composite plates with various plate length/width ratios (a/b), layer number (n) and boundary conditions (fibre angle increment = 1) 4
BC
a/b
Stacking
8
SFFF
1 2 3 4 5
[52/37]s [36/28]s [10/36]s [32/48]s [30/44]s
8.546 13.546 18.198 36.410 43.718
[46/43/43/43]s [37/29/25/28]s [20/24/24/24]s [4/27/22/21]s [0/4/36/28]s
21.479 38.603 47.475 54.546 57.868
[44/45/46/45/44/ 46]s [40/36/16/26/39/52]s [32/8/16/33/0/47]s [16/16/0/24/14/14]s [0/0/0/48/31/23]s
32.464 58.857 73.465 82.645 86.350
CFFF
1 2 3 4 5
[0/0]s [40/39]s [24/35]s [58/54]s [33/55]s
6.904 19.063 25.265 33.101 48.898
[0/0/2/0]s [0/0/0/0]s [0/0/0/0]s [22/22/21/22]s [15/18/21/22]s
13.601 13.785 13.783 70.168 76.817
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [22/16/10/6/4/4]s
20.692 20.680 20.673 20.667 116.303
SSFF
1 2 3 4 5
[45/45]s [38/51]s [12/8]s [64/61]s [34/50]s
5.232 10.242 39.179 45.177 68.189
[45/46/47/43]s [43/47/47/43]s [42/47/47/42]s [42/47/47/42]s [40/49/40/50]s
10.847 21.088 32.901 43.585 53.553
[45/45/45/45/44/44]s [41/48/42/42/48/48]s [40/49/40/40/40/49]s [40/49/40/40/49/40]s [40/40/50/50/50/50]s
SCFF
1 2 3 4 5
[74/41]s [84/90]s [86/90]s [88/90]s [88/90]s
8.141 28.548 63.064 111.404 173.567
[67/50/53/75]s [84/90/84/84]s [86/90/88/84]s [86/90/88/88]s [88/90/88/88]s
16.042 56.745 126.022 222.611 346.941
[67/47/66/46/68/64]s [90/84/84/84/84/80]s [86/90/86/86/88/88]s [86/90/88/88/88/84]s [88/90/88/88/88/88]s
CCFF
1 2 3 4 5
[58/32]s [84/90]s [86/90]s [88/90]s [88/90]s
9.280 28.880 63.266 111.549 173.673
[55/10/52/48]s [82/90/84/82]s [86/90/84/88]s [86/90/88/88]s [88/90/88/88]s
18.695 57.471 126.399 222.902 347.123
[48/40/48/44/40/32]s [82/90/82/82/82/80]s [86/90/86/86/84/88]s [86/90/88/88/88/84]s [88/90/88/88/88/88]s
SFSF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [24/28]s [4/39]s
19.583 19.389 19.348 46.075 61.231
[0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s
38.378 38.680 38.693 38.663 38.648
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s
58.626 58.139 58.031 57.989 57.969
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
97.311 96.797 96.676 96.624 96.599
SFCF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [53/12]s [22/28]s
30.860 30.375 30.279 46.779 66.445
[0/0/1/2]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s
59.858 60.479 60.544 60.476 60.441
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s
92.062 90.998 90.779 90.695 90.649
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
151.763 151.250 151.120 151.046 151.015
CFCF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [0/0]s [48/11]s
45.225 44.199 44.007 43.936 61.892
[0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s [0/0/0/0]s
86.621 87.804 87.982 87.851 87.786
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/0/0/0/0/0]s
134. 292 132. 273 131. 878 131. 725 131.646
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
219.428 219.370 219.311 219.272 219.235
Xopt
Stacking
12 Xopt
Stacking
20 Xopt
16.774 33.062 48.531 64.460 79.716
Stacking
Xopt
[44/45/44/44/44/44/44/44/44/44]s [32/64/48/32/32/0/16/32/46/28]s [32/42/12/0/0/0/12/12/15/6]s [29/0/0/18/14/10/8/12/4/4]s [0/0/0/16/32/0/1/12/10/10]s
53.435 94.530 122.528 136.318 145.228
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s
34.420 34.450 34.446 34.440 34.434
[45/45/45/45/45/45/45/44/44/44]s [41/48/41/41/40/40/48/48/48/48]s [40/41/49/41/49/49/41/41/41/49]s [40/40/40/49/49/49/40/50/50/50]s [41/41/41/49/49/49/49/49/41/48]s
27.568 54.224 80.681 106.291 132.930
24.477 85.416 188.931 333.734 520.092
[68/46/68/46/68/68/68/68/48/64]s [84/84/90/88/84/84/84/84/84/80]s [86/90/86/88/86/88/88/88/88/80]s [86/90/88/88/88/88/88/88/88/84]s [88/90/88/88/88/88/88/88/88/80]s
40.569 142.094 314.273 555.251 865.233
28.540 86.441 189.497 334.171 520.388
[50/44/40/48/32/56/44/48/48/32]s [82/90/84/84/84/84/84/80/80/80]s [86/90/86/86/86/86/86/84/84/88]s [86/86/90/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s
47.244 143.847 315.298 555.863 865.757
547
(continued on next page)
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
Layer number (n)
548
Table 5 (continued) 4
8
BC
a/b
Stacking
SSSF
1 2 3 4 5
[0/0]s [10/36]s [37/43]s [38/46]s [38/48]s
20.153 22.158 28.928 37.407 46.172
SCSF
1 2 3 4 5
[0/0]s [81/56]s [90/88]s [90/90]s [88/90]s
20.375 30.768 64.812 113.106 175.091
SSCF
1 2 3 4 5
[3/3]s [10/20]s [24/45]s [29/49]s [36/47]s
31.408 32.608 37.870 45.712 48.073
SCCF
1 2 3 4 5
[4/4]s [16/56]s [90/90]s [90/90]s [88/90]s
31.600 35.082 65.344 113.484 175.400
CSCF
1 2 3 4 5
[0/0]s [0/0]s [0/0]s [17/43]s [27/49]s
45.535 45.464 46.833 50.493 57.795
CCCF
1 2 3 4 5
[0/0]s [0/0]s [90/86]s [90/88]s [90/90]s
SSSS
1 2 3 4 5
SSSC
1 2 3 4 5
Xopt
Stacking
12 Xopt
Stacking
20 Xopt
Stacking
Xopt
[1/1/1/7]s [36/37/36/36]s [42/42/42/42]s [43/44/44/44]s [43/45/45/45]s
39.548 48.045 67.587 88.082 108.987
[0/0/0/0/0/0]s [36/36/36/36/36/40]s [42/42/42/42/44/40]s [43/43/44/43/44/44]s [45/44/44/44/45/44]s
60.320 72.527 101.294 132.267 163.453
[0/0/0/0/0/0/0/0/0/0]s [36/36/36/36/36/36/36/36/36/32]s [42/42/42/42/42/42/42/42/40/40]s [44/44/44/44/44/44/44/44/44/48]s [44/44/44/44/44/44/44/44/48/48]s
100.082 119.577 167.073 218.003 270.025
[0/0/2/9]s [60/56/56/56]s [90/88/88/88]s [88/90/88/88]s [88/90/88/88]s
39.493 63.556 129.549 226.008 350.130
[0/0/0/0/0/0]s [58/58/58/60/58/56]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
60.989 95.886 194.172 338.808 524.899
[0/0/0/0/0/0/0/0/0/0]s [60/60/60/60/60/60/60/60/56/64]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/88/88/88/88/88/88/88/88]s
101.193 158.446 322.893 563.475 872.888
[3/3/3/3]s [9/11/21/11]s [33/40/40/33]s [37/43/44/37]s [42/42/42/42]s
61.176 64.764 78.556 97.842 115.989
[3/4/4/4/0/0]s [2/10/10/10/8/28]s [34/40/40/34/34/32]s [37/44/37/44/44/40]s [44/39/39/39/39/39]s
93.675 97.563 116.877 146.235 176.699
[2/3/4/4/4/4/4/4/0/0]s [8/8/8/9/8/8/24/24/16/0]s [30/31/32/42/40/40/40/40/40/40]s [36/44/44/36/36/36/36/36/44/32]s [45/38/38/38/40/44/44/44/48/48]s
154.387 162.103 194.352 241.581 291.783
[4/4/4/6]s [31/57/57/35]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s
61.360 71.001 130.615 226.781 350.746
[4/4/4/4/0/0]s [26/26/56/56/56/56]s [88/90/88/88/88/80]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
94.258 106.807 195.780 339.960 525.788
[3/4/4/4/4/4/4/4/0/0]s [30/60/32/32/58/56/56/56/56/64]s [88/90/88/88/88/88/88/88/88/80]s [88/90/88/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
155.371 176.589 325.591 565.414 874.506
[0/0/0/4]s [0/0/0/0]s [6/18/20/16]s [32/37/36/36]s [36/40/40/40]s
87.845 90.330 93.953 108.162 127.027
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [0/16/18/20/16/16]s [34/34/36/36/36/32]s [39/37/37/38/40/40]s
135.204 136.035 140.802 162.216 190.437
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [0/16/16/12/16/16/16/16/16/0]s [34/34/34/34/34/36/34/36/32/32]s [38/38/38/38/40/40/40/40/40/32]s
220.877 225.556 234.058 268.527 315.341
45.643 46.459 66.019 113.938 175.881
[0/1/1/1]s [0/0/0/0]s [89/90/88/88]s [88/90/88/88]s [88/90/88/88]s
88.251 92.321 131.993 227.700 351.429
[0/0/0/0/0/0]s [0/0/0/0/0/0]s [88/88/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
135.523 139.024 197.838 341.345 526.813
[0/0/0/0/0/0/0/0/0/0]s [0/0/0/0/0/0/0/0/0/0]s [88/88/88/90/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
221.415 230.539 329.028 567.718 876.237
[45/45]s [90/88]s [90/88]s [88/90]s [88/90]s
25.789 80.882 178.818 315.755 492.001
[45/45/45/45]s [90/88/88/88]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s
55.418 159.667 357.177 631.146 983.454
[45/45/45/45/45/44]s [90/90/90/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
84.380 242.098 535.068 945.501 1473.268
[45/44/44/44/44/44/44/44/44/48]s [89/88/88/88/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
138.193 401.638 887.939 1569.226 2445.080
[84/58]s [90/90]s [88/90]s [88/90]s [88/90]s
33.142 125.343 279.439 495.366 773.007
[64/59/60/60]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s
66.323 245.354 557.82 988.851 1543.052
[62/60/60/60/64/64]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
100.902 373.715 833.806 1478.093 2306.559
[64/63/64/64/64/64/64/64/64/64]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s
165.715 615.909 1374.312 2436.368 3801.812
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
Layer number (n)
236.516 886.171 1981.745 3516.541 5490.266 [0/0/90/88/88/0/0/0/0/0]s [88/88/90/88/88/88/88/88/88/80]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/80]s [0/0/0/0]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s [0/0]s [88/90]s [88/90]s [88/90]s [88/90]s
48.169 182.629 408.381 724.670 1131.408
93.338 353.889 813.785 1444.046 2254.522
[0/0/0/0/0/0]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
143.06 542.42 1212.898 2152.327 3360.276
229.517 619.517 1376.444 2437.826 3802.952 [0/0/0/0/0/0/0/0/0/0]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [0/0/0/0]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s 47.325 126.04 279.857 495.658 773.235 [0/0]s [90/88]s [88/90]s [88/90]s [88/90]s
91.697 246.786 558.673 989.464 1543.531
[0/0/0/0/0/0]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
140.508 375.866 835.118 1478.968 2307.242
226.628 883.655 1980.187 3515.520 5489.355 [88/88/90/88/88/88/88/88/88/80]s [88/88/90/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s [88/88/90/88]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s 46.761 182.13 408.086 724.451 1131.225 [90/88]s [88/90]s [88/90]s [88/90]s [88/90]s
90.567 352.872 813.186 1443.609 2254.181
[90/88/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
138.785 540.925 1211.996 2151.598 3359.821
178.792 617.476 1375.296 2437.097 3802.496 [45/45/45/44]s [90/90/90/90]s [88/90/88/88]s [88/90/88/88]s [88/90/88/88]s [10/46]s [88/90]s [88/90]s [88/90]s [88/90]s
34.265 125.551 279.628 495.497 773.098
71.540 245.988 558.205 989.128 1543.280
[44/44/44/48/48/48]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s [88/90/88/88/88/88]s
109.022 374.663 834.380 1478.531 2306.787
[44/44/44/44/44/44/48/48/44/48]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/88/90/88/88/88/88/88/88/88]s [88/90/88/88/88/88/88/88/88/88]s
M. Kemal Apalak et al. / Composites Science and Technology 68 (2008) 537–550
549
Xopt = 71.289 with a fiber angle increment of Dh = 5 for a stacking sequence [45/45/45/45]s (SSCC in Table 1). Similarly, all the present optimal solutions exhibit higher natural frequencies than those of the rectangular plates (a/b = 2) with the present five typical stacking sequences (Fig. 8). Finally, the optimal solutions are searched for the composite plates for twenty different edge conditions, the plate length/width ratios (a/b = 1, 2, 3, 4, 5), the layer number (n = 4, 8, 12, 20), and are given in Table 5. As either the plate length/width ratio (a/b) or the layer number (n) is increased the optimal natural frequency parameter Xopt is increased. Some outer layers of the composite plates have a more stiffening effect on the optimal natural frequency parameter and the orientation of the layers becomes similar toward the mid-plane of the composite plate as the layer number is increased (Table 5). 6. Conclusions In this study the optimal layered sequences of symmetrically laminated square and rectangular plates were searched by means of the Genetic Algorithms and the proposed neural networks. The natural frequencies of the composite plates were calculated using the finite element technique for various plate edge conditions and random plate length/width ratios and layer number, and the proposed neural networks were trained and tested based on these data. The Genetic Algorithm and the proposed neural networks predicted successfully the natural frequencies of the composite plates and the predicted frequencies and optimal layered sequences were in good agreement with those of the Ritz-based layerwise optimization method [1]. The present method finds optimal design maximizing the natural frequency of the composite plates without yielding a local optimum for all edge conditions and design parameters whereas this possibility is experienced in the Ritz-based layerwise optimization method. In addition, the natural frequencies of the composite square and rectangular plates are increased with increasing layer number. The outer layers of the composite plates have an effect on the plate bending stiffness; thus, the fiber orientation of the layers close to the mid-plane of the composite plate. Acknowledgement Authors would like to thank for the financial support of the Scientific Research Project Division of Erciyes University under the contract: FBT-05-34.
1 2 3 4 5 CCCC
1 2 3 4 5 CCCS
1 2 3 4 5 SCSC
SSCC
1 2 3 4 5
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