Research on design rules for composite laminate

1 Introduction

Because of the specific strength and stiffness, lightweight, anti-fatigue, and high temperature resistance, composite laminates have been widely used in various fields of engineering such as aerospace and automobile industries [1]. With the development of advanced composite material, design optimization of composite laminated plates has become particularly important. There are a number of studies associated with the optimal design of composite laminates.

Based on the research carried out on related topics, plenty of algorithms for design optimization of composite laminated plates have been proposed. Deng et al. [2] have used simulated annealing algorithms for stacking sequence of a composite laminate plate with constant thickness. Shafei and Kabir [3] have studied the dynamic stability optimization of laminated composite plates with the combined boundary loading. Falzon and Faggiani [4] have used the genetic algorithm (GA) to improve the secondary instability yield strength of laminated composite plate. They also studied the effects of the secondary instability for structural design. According to the classic theory of composite laminate, Carrera and Miglioretti [5] have studied the high-order theory of the different boundary conditions and the mechanical properties of the sheet’s shear deformation and multilayer theory with GA. Marín et al. [6] have introduced optimal composite laminate stiffness in the hydrothermal load using neural networks and GAs. Sadr and Ghashochi [7] have used the number of layers, ply angle, boundary conditions, and the aspect ratio of the laminate as design conditions to optimize the fundamental frequency of laminated composite plates using elitist-GA and finite-strip method. Jin et al. [8] and Tang and Li [9] have used the master-slave parallel GA to optimize the structure of the large wing box, where the optimum time is shorter and more efficient. Seresta et al. [10] have proposed a simple GA framework applicable to multilayered laminated composite plate design, which is effective to solve the problem of independent design of unmatching adjacent layers of composite laminate. Li et al. [11] have used an improved ant colony algorithm to optimize the minimum number of layers of laminated composite plates and ply angle; the improved ant colony algorithm is shown to be more efficient and faster. Tabakov and Walker [12] and Pelletier and Vel [13] have given a methodology to improve stiffness of composite plates by the optimal orientation of fibers placed using fiber steering techniques. Park et al. [14] have proposed an improved GA to realize the multidisciplinary design optimization of composites. Abouhamze and Shakeri [15] have employed GAs and neural networks to optimize the stacking, aiming at the composite cylindrical panels natural frequency, buckling load, quality. Almeida and Awruch [16] have realized the optimization of layer material, stacking sequence, and layer thickness using finite element analysis and specially designed genetic operators. Mian et al. [17] have optimized the weight of the composite pressure vessels using the finite element and Matlab. Several researches have been carried out on expert system of composite. Kim et al. [18–20] have expounded the development of an expert system for the optimal stacking sequence design of composite laminates subjected to the various constraints. In the expert system, GA is used for optimization, and stacking sequence was optimized to maximize the strength of composite with a given thickness. Fu et al. [21] have established a composite design expert system based on a network. The system contains a number of knowledge that can be used to design typical composite structures. Zhang et al. [22] have discussed the application of expert systems in the composite solidification formation process. Fairuz et al. [23] have used an expert system in the selection of composites, and it is suitable for the hull design.

In this article, an improved standard GA (SGA) using rule operators is suggested, which is particularly applicable for the stacking sequence optimization of composite laminates. The design rules with several methods such as replacement approach, swap approach, combined approach, and zero fitness approach are introduced. This is a new way with using lamination parameters and the proposed method to optimal stacking sequence of composite laminate.

2 Optimal design model

The lamination parameters V_i (V_A and V_D are values of in-plane and out-of-plane lamination parameters) are defined by the classic laminate theory. For the symmetric laminates, the lamination parameters can be expressed as in Eq. (1):

where θ_k (k=1, 2, 3, …, n) is fiber angle, k is an arbitrary ply number, and n is the number of plies.

The coefficients K₁,K₂ are defined as follows:

In the study of Fukuna and Shkine [24], the optimization problem including object function and constraint condition can be expressed as in Eq. (3). In practical use, the fiber angles in laminate are limited to small sets composed of 0°, ±45°, and 90°.

where the penalty factor σ=1 in order to simplify the optimization process.

3 Design rules for composite laminate

In the proposed fitness function, the lamination parameters are independent of stacking order. However, it is limited in a sense for dealing with interlaminar stress issues, which is intensively dependent on the adjacent layers’ lay-up directions. Considering the experimental result and other practical aspects, some standard design considerations have been proposed for the design of composite laminates [25]. A methodology has been proposed to upgrade the result of GA according to this concept; using the method, corresponding rule operators will be selected. These rule operators will upgrade the mutated individuals to conform to the design rules randomly. Hence, the rule operators in the algorithm have added a new dimension for optimization without any change in the fitness function. Nonconforming individuals will be transformed into other form of conforming configuration. In this process, the volume of individuals of searching the optimization results can be significantly reduced; however, the best effort has been considered to keep the random mutation, which is a key characteristic of GA. Figure 1 shows the flowchart of GA-associated rule operators in this article.

Figure 1

Flowchart of GA-associated rule operator.

Among many design rules, some of them are selected for prototyping this concept. These rules are as follows:

1. In order to avoid the matrix loading, the minimum amount of plies in each direction should be at least 10%.

2. In order to minimize interlaminar stress, grouping of -45° and 45° plies in adjacent layers should be avoided, separated by a 0° or 90° ply. Similarly, grouping 0° and 90° plies in adjacent layers should be avoided, separated by a -45° or 45° ply.

3. In order to minimize interlaminar stress, grouping of ±45° plies in adjacent layers should be avoided, separated by a 0° or 90° ply.

4. Grouping of a maximum of four plies in the same direction is allowed in a stack.

5. Avoid grouping of 90° plies in adjacent layers, which should be separated by 0°, -45°, or 45° plies.

6. The first ply and the last ply of the stack should be -45°, which can improve the local buckling strength and laminate strength.

Based on the purpose, we can classify the rules into three categories as Table 1.

3.1 Algorithm design for rules

Three basic methods are proposed in this article to perform the operation of design rules. Each of them is described in detail. A comparative analysis is presented in the following sections.

3.1.1 Replacement approach

In general, the replacing approach will let the operator to change a gene in a chromosome string; then, the nonconformance is solved. Therefore, in the action of rule operators, the modified individual will conform to the design rule. If a gene is randomly changed to conform to the design rule, the value may be better or worse (cannot be guaranteed) or unpredictable. Because of changing a gene in the string will affect both V^A and V^D, their relationship is not defined as mentioned before.

3.1.2 Swap approach

The swap approach uses the exchange of genes between two adjacent loci. If nonconformity to the design rule is detected, the values are exchanged. In this method, no new ply is introduced; only the stacking order is changed between adjacent layers. If an individual is modified by operator purely by swap approach, it is easy to know that V^A is unchanged and V^D is changed. It can provide us a more stable and meaningful result.

3.1.3 Zero fitness approach

The main theme of this approach is that if some individuals are not conforming to the requirements of the design rule, the fitness of the individuals is forced to be zero. Hence, the probability of these individuals to be selected for next crossover will be very small. In addition to that, their fitness will be less than the individuals that conform to the design rule.

In this method, searching over a wide range is confirmed compared with other approaches, and it can provide more precise results. The probability of getting a child cannot be ensured. Therefore, a considerable number will be discarded in each generation that might result in slower speed.

In Figure 2, it refers to a modified version of the standard rule operator algorithm. In this method, the operator is introduced after mutation. Therefore, the method appears to be simple in the algorithm.

Figure 2

General execution process of zero fitness approach.

3.2 Rules are selected for example

To explain the difference of these three methods, some rules are selected in following section.

1. Replacement: For rule 1, it is first to know that there is any ply angle that is <10%. The second question is which ply orientation is <10%. Then, the maximum appearing ply direction is determined; one of these plies is replaced with ply that is <10%. Figure 3 is the Execution process of replacement approach for rule 1.

   From the flowchart, the location of replacement is not randomly selected, and the first appearing value of θ₂ is replaced with θ₁. It is difficult to assume how many times θ₂ will appear in a string. Thus, to randomly select a locus that contains θ₂ is also difficult. In order to simplify, the randomization of locus selection for replacement is limited between the first appearing θ₂ and last appearing θ₂. However, the randomness is still available, because the x_i and x_j are not fixed locus on the chromosome string and either of them is randomly selected.

2. Swapping: For rule 5, a two-step scanning is suggested for this kind of rules. Both the first locus and the last locus can get the action of rule operator. From the flowchart, the concepts of swapping between adjacent loci are clear. At the same time, the limitations of this method are also prominent. This method is not effective for grouping, which is more than two layers. Figure 4 is the Execution process of replacement approach for rule 5.

Figure 3

Execution process of replacement approach for rule 1.

Figure 4

Execution process of swap approach for rule 5.

3.3 Comparison and suggestion for a combined approach

From the discussion in this section, the features of different approaches to execute rule operators can be analyzed. It is clear that none of these approaches can provide the best result alone. Among the replace and swap approaches that can be critically compared, but the zero fitness approach is different in nature and more likely to hold features of conventional GA. A contrast between replace and swap approaches is as Table 2.

Table 2

Comprehensive comparison between replacement approach and swap approach.

	Replacement approach	Swap approach
No. of genes affected	1	2
Change of VA and VD	VA and VD both changes	Independent of VA, only VD changes
Example	To separate 90°/90° in [45°/90°/90°/-45°]
	[45°/90°/90°/-45°]→[45°/45°/90°/-45°]	[45°/90°/90°/-45°]→[90°/45°/90°/-45°]
	[45°/90°/90°/-45°]→[45°/90°/45°/-45°]	[45°/90°/90°/-45°]→[45°/90°/-45°/90°]
	The change takes place at second or third locus, which changes only one gene. Adding a 45 or -45 increase number of respected ply, reduce number of 90° ply.	The change takes place at first and second or third and fourth locus, which changes two genes. Numbers of certain ply direction remain unchanged.

For the first two cases, the rate can be assumed to increase because of the limitation of the search zone of the GA. Initially, the algorithm will converge very fast, but later the speed will reduce because of less variation among the individuals while using a rule operator. To get a better result, a combination of replacement and swap approach is proposed for rule 5. The steps of the calculation process are as follows and Figure 5.

Figure 5

Execution process of combined approach for rule 5.

1. Check for the grouping of 90° plies in adjacent layers.

2. If the group consists at least three layers, replace middle layer’s value with randomly selected 0°, 45°, or -45°.

3. For a group of two plies, the second one from the left side of the group should be swapped with its next right adjacent value.

4. In this step, if grouping still persists, the left value of the group should be swapped with left adjacent value.

5. If grouping still persists, any elements of the group are randomly changed with 0°, 45°, or -45°.

6. If the last layer is 90°, replace with randomly selected 0°, 45°, or -45°.

4 Application of the proposed method

In order to compare the effect of the rule operator using different methods, the contrast among them is demonstrated in this chapter. The results are generated for the same value of GA parameter, such as Table 3:

where h is the thickness of laminate and h₀ is the thickness of layer. For the SGA and different methods with rule operators, the same example values are used. The results of optimization are demonstrated according to the design rules.

4.1 Result of rule 1

Recalling from the previous sections, rule 1 requires having at least 10% of ply in each direction. The results of optimization are obtained as Figure 6:

Figure 6

Analyses of maximum fitness of particular generation and global best fitness over number generation for rule 1.

The results can be summarized as Table 4:

Table 4

Optimal results for rule 1.

	Laminate stacking sequence	Design rule conformance	Fitness	Gen
SGA	[903/45/90/-452/45]S	✗	0.870262	23
ZFA	[903/0/45/-45/45/-45]S	✓	0.806440	3
RPA	[90/0/90/-45/90/452/-45]S	✓	0.798765	5

RPA, GA equipped with rule operator using replacement approach; ZFA, GA equipped with rule operator using zero fitness approach.

✗, The method is not fit for the design rule. ✓, The method is fit for the design rule.

4.2 Result of rule 5

Requirement of rule 5 is to avoid grouping of 90° plies in adjacent layers, separated by 0°, -45°, or 45° plies. The results of optimization are obtained as Figure 7 and Table 5.

Figure 7

Analyses of maximum fitness of particular generation and global best fitness over number generation for rule 5.

4.3 Result of rule 6

The requirement of rule 6 is to have -45 ply in the first and last layers. The results of optimization are obtained as Figure 8 and Table 6.

Figure 8

Analyses of maximum fitness of particular generation and global best fitness over number generation for rule 6.

4.4 Validation of results using numerical simulation

In case of rule operators, the design rules are suggested based on practical experience and experiments. For rule 1, avoiding the matrix loading may result in an increase of maximum stress in laminate for the same loading condition, which have been proven and provided in the handbook. The stress analysis of simulation in ABAQUS is demonstrated in detail for rule 6 about minimizing the local buckling effect. In Figure 9, the maximum and minimum values of laminate stress are compared with the proposed solutions by replacement approach, swap approach, combined approach, and zero fitness approach.

Figure 9

Maximum and minimum global laminate stress analysis using envelope absolute tool.

The detailed analysis of rule 6 is provided in Figures 10 and 11. From these figures, it is obvious that the maximum global laminate stress is minimized with proposed solution. The maximum stress is appearing on ply 1 with no rule operator, but the maximum stress is appearing on ply 2, which is used for rule operator, so the maximum stress value is also reduced as mentioned earlier. These superior performances are much more suitable for the design rules of composite laminate.

Figure 10

Maximum stress and subjected plies for SGA solution.

Figure 11

Maximum stress and subjected plies for rule operator-equipped GA solution.

5 Conclusion

Following the discussions of the above section, an improved model of the SGA is proposed using rule operator, which can provide faster convergence speed of stacking sequence optimization of composite laminates. Usually, the fitness of the proposed optimal stack is compromised to meet the requirement of design rule. However, it is worth to compromise in a sense that provides one more suitable solution for particular practical application. Similarly, the solution to minimize the interlaminar stress or avoid matrix loading by means of rule operator can incur the cost increasing in maximum stress. It is considered as a price of obtaining a better result. The minimum stress of optimal laminate using GA with rule operator is lower than that of SGA. The most optimistic result of numerical solution can be obtained from the optimal solution provided by using rule operator in GA. Minimizing local buckling can reduce the maximum stress on the first layer significantly and eventually can reduce the global maximum stress appearing in the laminate.