# Manufacturing cost-effective weight minimization of composite laminate using uniform thickness and variable thickness approaches considering different failure criteria

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## Abstract

The present study aims at providing a solution to manufacturer and designer for selecting manufacturing cost-effective uniform ply thickness and finding necessary optimum ply angle stacking sequence when ply thicknesses and ply angles are available in discrete form, in two stages using combined variable thickness (VTA) and uniform thickness approach (UTA). The manufacturing cost is indirectly included in the analysis by restricting maximum number of layers and resizing ply thicknesses accordingly, while minimizing weight. Maximum stress theory, Tsai Wu theory, and Tsai Hill theory are used as constraints in the analysis independently. The capability of the conventional genetic algorithm is improved using direct value-coded representation for chromosomes to handle multiple design variables of different nature in discrete form, simultaneously. The effectiveness of combined VTA–UTA strategy in this regard is demonstrated with multiple examples.

## Keywords

Composite laminate Weight minimization OptiComp Uniform thickness approach Variable thickness approach## List of symbols

*E*_{11}Elastic modulus in longitudinal direction in GPa

*G*_{12}In-plane shear modulus in GPa

*E*_{22}Elastic modulus in transverse direction in GPa

- \(\mu_{12}\)
Major Poisson’s ratio

*S*_{Lt}Tensile strength in longitudinal direction in MPa

*S*_{Lc}Compressive strength in longitudinal direction in MPa

*S*_{Tc}Compressive strength in transverse direction in MPa

*S*_{Tt}Tensile strength in transverse direction, MPa

*S*_{Lts}Shear strength, MPa

- \(\theta\)
Ply orientation angle

*t*Thickness of each ply, mm

- \(\rho\)
Mass density, kg/m

^{3}*a*Length of laminate, mm

*b*Width of laminate, mm

*T*Laminate thickness, mm

## 1 Introduction

Composite laminate plate is a structural element made up of multiple fiber reinforced polymer (FRP) laminas/plies connected together to provide required engineering properties for an application. The light weight associated with high stiffness and strength along the direction of the reinforcement is the main advantage of using composite laminates. The properties of composite laminate depend on various design variables such as total number of plies/laminas, stacking sequence, ply angles, thickness of each ply, etc. The composite laminate can be tailor-made using combination of these design variables to suit the application under consideration.

*N*

_{xx},

*N*

_{yy}and

*N*

_{xy}is shown in Fig. 1. In figure,

*X*,

*Y*and

*Z*denote global co-ordinate system of the composite laminate, while 1 and 2 represent local co-ordinate system for individual lamina. Axis 1 of the local co-ordinate system is along the length of the fiber and axis 2 is perpendicular to local axis 1.

Majority automotive applications demand maximum strength to weight ratio for achieving maximum material performance in terms of durability of structure, fuel economy, cost, etc., in any structural application. To satisfy this major design requirement, many researchers (Almeida and Awruch 2009; Tripathi and Kulkarni 2014; Pelletier and Vel 2006; Lopez et al. 2009; Akbulut and Sonmez 2008; Naik et al. 2008; Omkar et al. 2008; Fakhrabadi et al. 2013; Assie et al. 2011; Seresta et al. 2007; Badalló et al. 2013; Walker and Smith 2003; Farshi and Herasati 2006; Irisarri et al. 2009; Khosravi and Sedaghati 2008; Barroso et al. 2017; Gillet et al. 2010; Adali et al. 1995; Lopez et al. 2011; Vo-Duy et al. 2017; Javidrad et al. 2018; Rocha et al. 2014) have taken weight minimization objective during design optimization of composite laminate for different loading conditions.

A composite laminate can be designed using two approaches; uniform thickness approach (UTA) and variable thickness approach (VTA). All laminas in a laminate will have same thickness in UTA, while in VTA, the laminas in the laminate may have same or different thicknesses. The use of ply thickness as user-defined discrete variable is rarely observed (Almeida and Awruch 2009; Walker and Smith 2003; Vo-Duy et al. 2017) so far in the available literature because of manufacturing difficulty and mathematical complexity. The comparison of both the approaches yields that the number of design variables in VTA becomes more than the design variables in UTA. Moreover, the nature of variables, i.e., ply angle and ply thickness is different. The ply angle is an integer number, while ply thickness is a real number. Ghiasi et al. (2009) and Nikbakt et al. (2018) concluded that genetic algorithm is the most popular and suitable method for design optimization of composite laminates.

Walker and Smith (2003) carried out minimization of weighted sum of deflection and weight of composite laminate subjected to normal loading using Tsai Wu theory as constraint considering fiber angles and layer thicknesses, both in discrete form as design variables using binary-coded genetic algorithm. Pelletier and Vel (2006) proposed integer-coded genetic algorithm for design optimization of composite laminates after exposing limitations of binary-coded genetic algorithm in catching exact incremental value of design variables commonly used in engineering applications. Lopez et al. (2009) carried out weight minimization of composite laminates using UTA subjected to in-plane loading conditions using direct value-coded genetic algorithm. The number of layers and fiber angles are used as design variables while designing conventional 0°, ±45°, 90° composite laminates subjected to Tsai Wu, Puch and maximum stress theory as constraints. The concept of direct value-coded representation of chromosomes used by Lopez et al. (2009) is extended in this article considering ply thicknesses along with ply angles as design variables. These two design variables are of different nature also. The concept of crossover and mutation operators presented by Pelletier and Vel (2006) for integer coded genetic algorithm is modified and adopted in the current simulation. Farshi and Herasati (2006) and Khosravi and Sedaghati (2008) adopted two-level strategy for weight minimization of composite laminate plate considering fiber angle and thickness of each layer as design variables.

To check the failure of the laminate while minimizing its weight, researchers working in the field have proposed various criteria (theories) for predicting failure of composite laminates, which may have some minor or major weaknesses. Hinton et al. (2002) and Soden et al. (2004) compared and assessed different leading theories for predicting failure in composite laminates under complex states of stress against experimental evidence through few selected test cases. In the present study, first ply failure criterion is adopted to predict failure of laminate. The laminate is considered as failed when any single ply of the laminate fails. Among all these theories maximum stress theory, maximum strain theory, Tsai Hill theory and Tsai Wu theory are the most preferred theories for predicting failure of laminate and used by majority of researchers (Almeida and Awruch 2009; Tripathi and Kulkarni 2014; Pelletier and Vel 2006; Lopez et al. 2009; Akbulut and Sonmez 2008; Naik et al. 2008; Omkar et al. 2008; Fakhrabadi et al. 2013; Nicholas et al. 2012; Sebaey et al. 2011; Khosravi and Sedaghati 2008; Barroso et al. 2017; Gillet et al. 2010; Lopez et al. 2011) for in-plane loading conditions. The strength-based theories such as maximum stress theory (MS), Tsai Wu theory (TW) and Tsai Hill theory (TH) are considered and applied individually as constraints in the current study. A brief description of these theories follows (Mallick 2007).

### 1.1 Maximum stress theory

### 1.2 Tsai Wu theory

### 1.3 Tsai Hill theory

In this study, theories of failure are used as constraints to be satisfied for safe design. A procedure, based on first-order shear deformation theory, used to find the stresses for developing constraint equations is explained in the next section (Mallick 2007).

## 2 Analysis of composite laminate

*t*

_{1},

*t*

_{j}and

*t*

_{n}denote thicknesses of first,

*j*th and

*n*th lamina;

*h*

_{0}denotes distance from the laminate mid-plane to the top of the first lamina while \(h_{1 }\) is the distance from the laminate mid-plane to the bottom of the first lamina. \(h_{j - 1}\) is the distance from the laminate mid-plane to the top of the

*j*th lamina and \(h_{j }\) is the distance from the laminate mid-plane to the bottom of the

*j*th lamina. \(Z_{j}\) is the distance from the laminate mid-plane to the mid-plane of the

*j*th lamina. The stiffness matrix for individual lamina \(\left[ { \bar{Q}} \right]\) is calculated from the material properties as,

*A*, coupling stiffness matrix

*B*and bending stiffness matrix

*D*for laminate are calculated as,

*K*] as given as follows:

*j*th lamina in global co-ordinate system:

These local stress values obtained using Eqs. (9)–(11) for individual laminas are used for constructing all the constraint equations as mentioned earlier.

## 3 OptiComp and genetic algorithm (GA)

- (i)
Direct value-coded representation is used for chromosomes instead of binary-coded representation. This kind of representation facilitates handling of multiple design variables of different nature in discrete form.

- (ii)
Direct value-coded chromosome representation can exactly catch increment value for any design variable provided by user within the limit bounds. This is difficult in binary representation.

- (iii)
Single-point crossover function and mutation function are defined in such a manner so as to suit the chromosome representation.

*n*’ number of plies, then each chromosome will have 2 × ‘

*n*’ number of elements. The first ‘

*n*’ number of elements of the chromosome are the randomly selected ply angle values from the series (\(\theta_{\text{s}} )\) while the remaining elements are the randomly selected ply thickness values from the series (\(t_{\text{s}} ).\) The first and second half of the chromosome have given separate treatment during crossover and mutation as shown in Fig. 3. In UTA all the genes of the chromosomes of size ‘n’ are represented only by the allowable angle values selected randomly.

## 4 Combined VTA–UTA strategy and results

In UTA, ply angles and number of plies are treated as design variables, while in VTA, the number of plies, ply angles and ply thicknesses are treated as design variables. The lamina thicknesses available with the manufacturer are normally in discrete form and the manufacturers are interested in making laminate with uniform thickness laminas. The present study explores usefulness of VTA to find minimum required thickness of laminate to satisfy given strength constraints which will afterwards leads to value of cost-effective uniform thickness of lamina to be used in UTA.

Material properties for carbon/epoxy laminate

Property | | | | \(\mu_{12 }\) | | | | | | \(\rho\) |
---|---|---|---|---|---|---|---|---|---|---|

Value | 116.6 | 7.673 | 4.173 | 0.27 | 1701 | 2062 | 240 | 70 | 105 | 1605 |

Different load cases used in the study

Load case | Longitudinal force (N/mm) | Transverse force (N/mm) | Shear force (N/mm) |
---|---|---|---|

LC1 | 3000 | 3000 | 0 |

LC2 | 3000 | 3000 | 500 |

LC3 | 3000 | 3000 | 1000 |

LC4 | − 3000 | − 3000 | 0 |

LC5 | − 3000 | − 3000 | 500 |

LC6 | − 3000 | − 3000 | 1000 |

- 1.Find [\(\theta_{n}\), \(t_{n}\)],$${\text{to minimize weight }}W\, = \,\rho \cdot a \cdot b \cdot T ,\;\left( {T = \mathop \sum \limits_{i = 1}^{n} t_{i} } \right).$$

Subjected to satisfying:

Optimum results for LC2 and LC5 using VTA

Load case | Failure criteria ( | Stacking sequence | Average thickness mm ( | Reliability | Weight (kg) by OptiComp |
---|---|---|---|---|---|

LC2 | MS (68) | \(\theta_{n}\) = [75/60/15/60/− 60/90/− 60/15/45/75/− 60/75/15/30/0/60/− 75/45/75/45/0/− 15/30/15/90/− 30 \(t_{n}\) = [0.1 | 0.0963 (6.55) | 0.65 | 10.512 |

LC2 | TW (72) | \(\theta_{n}\) = [30/75/− 45/− 15/45/60/30/75/− 15/60/− 15/75/45/45/− 75/− 15/− 60/60/90/45/0/75/0/45 \(t_{n}\) = [0.125/0.075/0.125/0.15/0.05 | 0.0979 (7.05) | 0.6 | 11.315 |

LC2 | TH (72) | \(\theta_{n}\) = [− 45/0 \(t_{n}\) = [0.075/0.125/0.075/0.125/0.075/0.1/0.15/0.1 | 0.0979 (7.05) | 1 | 11.315 |

LC5 | MS (36) | \(\theta_{n}\) = [− 60/90/− 75/− 15/15/90/15/− 45/− 60/45/− 60/15/90/45/− 60/− 15/− 15/0] \(t_{n}\) = [0.125/0.1/0.125/0.05/0.075/0.1/0.075 | 0.0917 (3.3) | 0.65 | 5.296 |

LC5 | TW (32) | \(\theta_{n}\) = [45/− 15/− 60/0/− 45/− 30/30/− 75/− 60/30/60 \(t_{n}\) = [0.05 | 0.0953 (3.05) | 1 | 4.895 |

LC5 | TH (32) | \(\theta_{n}\) = [30/15/75/30/− 75/− 45 \(t_{n}\) = [0.125/0.15/0.125/0.1/0.15 | 0.1156 (3.7) | 0.7 | 5.938 |

*T*’ and optimum weights using VTA are provided in Table 4 for all the load cases.

Minimum laminate thickness and optimum weight obtained using VTA

Load case | Failure criteria | Number of plies used (Lopez et al. 2009) | Average thickness mm ( | Weight (kg) by OptiComp |
---|---|---|---|---|

LC1 | MS | 68 | 0.0963 (6.55) | 10.512 |

TW | 72 | 0.0979 (7.05) | 11.315 | |

TH | 72 | 0.0979 (7.05) | 11.315 | |

LC2 | MS | 68 | 0.0963 (6.55) | 10.512 |

TW | 72 | 0.0979 (7.05) | 11.315 | |

TH | 72 | 0.0979 (7.05) | 11.315 | |

LC3 | MS | 72 | 0.093 (6.7) | 10.757 |

TW | 76 | 0.0934 (7.1) | 11.395 | |

TH | 76 | 0.0934 (7.1) | 11.395 | |

LC4 | MS | 36 | 0.0917 (3.3) | 5.296 |

TW | 32 | 0.0953 (3.05) | 4.895 | |

TH | 32 | 0.1156 (3.7) | 5.938 | |

LC5 | MS | 36 | 0.0917 (3.3) | 5.296 |

TW | 32 | 0.0953 (3.05) | 4.895 | |

TH | 32 | 0.1156 (3.7) | 5.938 | |

LC6 | MS | 40 | 0.0838 (3.35) | 5.376 |

TW | 36 | 0.0848 (3.05) | 4.895 | |

TH | 36 | 0.1042 (3.75) | 6.018 |

*N*

_{max}’ has to be rounded as even number on higher side so as to obtain a balanced symmetric laminate. Detailed procedure of selection of cost-effective lamina thickness for LC5 using Tsai Hill theory as constraint is explained here. The minimum laminate thickness obtained for this case from VTA is 3.7 mm. The required number of layers and possible laminate thickness for each available lamina thickness are given in Table 5.

Possible configurations of LC5 laminate using different lamina thicknesses

Ply thickness (mm) | 0.05 | 0.075 | 0.1 | 0.125 | 0.15 |
---|---|---|---|---|---|

Required number of layers | 74 | 50 | 38 | 30 | 26 |

Possible laminate thickness ( | 3.7 | 3.75 | 3.8 | 3.75 | 3.9 |

The two major considerations while designing composite laminate plate as structural member are structural consideration and manufacturing consideration. As per first, the obtained laminate thickness should be very close to required minimum thickness so as to achieve maximum strength to weight ratio. The cost of the composite laminates consists of two main parts: material cost and manufacturing cost. The major part of the material cost includes the cost of the raw material for composite laminates. On the other hand, the manufacturing cost is the function of the manufacturing time including the time lost in press from preparation, layer cutting, layer sequencing and final working (Fakhrabadi et al. 2013). Manufacturing cost can be reduced by controlling the maximum number of layers. Here an attempt has been made to satisfy both the requirements.

- 2.Find [\(\theta_{n}\)],$${\text{to minimize weight }}W\, = \,\rho \cdot a \cdot b \cdot T ,\left( {T = \mathop \sum \nolimits_{i = 1}^{n} t_{i} } \right).$$

Subjected to satisfying:

Optimum results for LC2 and LC5 by UTA using selected ply thickness

Load case | Failure criteria ( | Optimum ply angle stacking sequence | Selected thickness mm | Reliability | Weight (kg) by OptiComp |
---|---|---|---|---|---|

LC2 | MS (44) | [0/90/0/75 | 0.15 | 0.65 | 10.593 |

TW (48) | [0/− 15/60/90/− 15/75/0 | 0.15 | 1 | 11.556 | |

TH (48) | [75/90/0 | 0.15 | 1 | 11.556 | |

LC5 | MS (22) | [30/− 30/75/− 45/− 30 | 0.15 | 0.75 | 5.296 |

TW (32) | [− 60/60/− 45/30/− 75/60/75/− 15/− 45 | 0.1 | 1 | 5.136 | |

TH (30) | [− 60/− 45 | 0.125 | 1 | 6.018 |

Cost-effective lamina thickness and laminate weight for all load cases

Load case | Failure criteria | Cost-effective lamina thickness mm | | Cost-effective laminate thickness mm | Minimum laminate thickness mm using VTA | Cost-effective weight (kg) by OptiComp |
---|---|---|---|---|---|---|

LC1 | MS | 0.15 | 44 | 6.6 | 6.55 | 10.593 |

TW | 0.15 | 48 | 7.2 | 7.05 | 11.556 | |

TH | 0.15 | 48 | 7.2 | 7.05 | 11.556 | |

LC2 | MS | 0.15 | 44 | 6.6 | 6.55 | 10.593 |

TW | 0.15 | 48 | 7.2 | 7.05 | 11.556 | |

TH | 0.15 | 48 | 7.2 | 7.05 | 11.556 | |

LC3 | MS | 0.125 | 54 | 6.75 | 6.7 | 10.833 |

TW | 0.15 | 48 | 7.2 | 7.1 | 11.556 | |

TH | 0.15 | 48 | 7.2 | 7.1 | 11.556 | |

LC4 | MS | 0.15 | 22 | 3.3 | 3.3 | 5.296 |

TW | 0.1 | 32 | 3.2 | 3.05 | 5.136 | |

TH | 0.125 | 30 | 3.75 | 3.7 | 6.018 | |

LC5 | MS | 0.15 | 22 | 3.3 | 3.3 | 5.296 |

TW | 0.1 | 32 | 3.2 | 3.05 | 5.136 | |

TH | 0.125 | 30 | 3.75 | 3.7 | 6.018 | |

LC6 | MS | 0.1 | 34 | 3.4 | 3.35 | 5.457 |

TW | 0.1 | 32 | 3.2 | 3.05 | 5.136 | |

TH | 0.125 | 30 | 3.75 | 3.75 | 6.018 |

### 4.1 Reliability analysis

Reliability in case of design optimization process of composite laminates can be defined as fraction of successful optimization process runs reached to the global optimum solution against total number of optimization process runs (Rocha et al. 2014). In the present analysis, it is observed that the reliability of VTA is less than the reliability of UTA. It is logical as VTA deals with more design variables of different nature. The unsuccessful attempts also predict close global optimum solutions, e.g., in LC5, the unsuccessful attempts predicted optimum weight 5.315 kg for maximum stress theory in comparison with global optimum weight 5.296 kg.

### 4.2 Comparison of optimum results obtained at different stages

Comparison of optimum weights obtained at different stages

Load case | Failure criteria | Conventional weight (kg) by Lopez et al. (2009) | Weight (kg) using VTA only | Weight (kg) by combined VTA–UTA strategy | % reduction in weight using combined VTA–UTA over conventional weight |
---|---|---|---|---|---|

LC1 | MS | 10.914 | 10.512 | 10.593 | 2.94 |

TW | 11.556 | 11.315 | 11.556 | 0 | |

TH | – | 11.315 | 11.556 | – | |

LC2 | MS | 10.914 | 10.512 | 10.593 | 2.94 |

TW | 11.556 | 11.315 | 11.556 | 0 | |

TH | – | 11.315 | 11.556 | – | |

LC3 | MS | 11.556 | 10.757 | 10.833 | 6.25 |

TW | 12.198 | 11.395 | 11.556 | 5.26 | |

TH | – | 11.326 | 11.556 | – | |

LC4 | MS | 5.778 | 5.296 | 5.296 | 8.34 |

TW | 5.136 | 4.895 | 5.136 | 0 | |

TH | – | 5.938 | 6.018 | – | |

LC5 | MS | 5.778 | 5.296 | 5.296 | 8.34 |

TW | 5.136 | 4.895 | 5.136 | 0 | |

TH | – | 5.938 | 6.018 | – | |

LC6 | MS | 6.42 | 5.376 | 5.457 | 15 |

TW | 5.778 | 4.895 | 5.136 | 11.11 | |

TH | – | 6.018 | 6.018 | – |

The optimum weight results obtained using maximum stress theory as constraint are less than the reference weights for all the load cases and maximum 15% weight reduction is observed for LC6 over reference weight. Hinton et al. (2002) stated that Tsai Wu theory predicts enhanced strength under compression–compression biaxial loading. In line with this statement, the optimum weight results obtained by this theory in compression–compression region are less than results predicted by other theories. As rightly mentioned by Pelletier and Vel (2006), Tsai hill theory over predicts optimum weights for all the load cases.

## 5 Combined VTA–UTA strategy with contiguity constraint

Optimum results for LC5 using VTA considering contiguity constraint

Load case | Failure theory ( | Optimum stacking sequence | Weight (kg) and laminate thickness |
---|---|---|---|

LC5 | MS (36) | \(\theta\) = [(− 75/60/− 45/− 15/0/− 30/60/15/− 75)
| 5.457 (3.4) |

TW (32) | \(\theta\) = [(0/60/− 60/0/− 60/− 15/− 45/75)
| 4.975 (3.1) | |

TH (32) | \(\theta\) = [(− 30/− 60/0/− 75/− 45/− 15/60/45)
| 6.099 (3.8) |

Cost-effective optimum results for LC5 using UTA considering contiguity constraint

Load case | Failure theory ( | Optimum stacking sequence | Weight (kg) and laminate thickness |
---|---|---|---|

LC5 | MS (28) | \(\theta\) = [(0/90/− 15/− 75/45/− 75/− 15) | 5.617 (0.125) |

TW (32) | \(\theta\) = [(− 60/60/− 30/45/− 30/− 60/− 45/30) | 5.136 (0.1) | |

TH (32) | \(\theta\) = [(15/75/− 30/75/− 30/− 60/15/− 60) | 6.420 (0.125) |

It is observed that inclusion of additional contiguity constraint along with strength-based constraints, increases the minimum required laminate thickness as well as manufacturing cost-effective laminate thickness.

## 6 Conclusion

In the present study, a new insight on utilization of VTA for design optimization of composite laminate has been given through combined VTA–UTA strategy. This strategy can become a helpful tool to designers and manufacturers for selecting uniform ply thickness and for finding necessary optimum stacking sequence when both these variables are available in discrete form. Laminate is confined to be made within predefined number of layers to control the manufacturing cost. The weight and manufacturing cost reduction of laminate are achieved using single objective function by controlling maximum number of layers with this strategy.

The obtained results show that the laminate weight not only depends on design variables but also on choice of the failure criteria. The maximum stress theory is more sensitive towards the design variables as it does not consider stress interaction. VTA alone can produce less optimum weights for all the theories of failure and can be effectively used for finding the minimum laminate thickness required to bear the applied load. Tsai Hill theory is more conservative in weight prediction of composite laminate. Inclusion of contiguity constraint along with strength-based constraints results in increased cost-effective laminate thickness. This combined optimization methodology has ability to become useful for designing manufacturing cost-effective laminated composite structures for majority applications. The combined VTA–UTA strategy can be used with any optimization algorithm which can handle multiple design variables of different nature simultaneously.

## Notes

### Acknowledgements

The authors are thankful to College of Engineering, Pune and Vishwakarma Institute of Technology, Pune for providing necessary infrastructure and computational facility for this work.

### Compliance with ethical standards

### Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

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