# GA Optimization of Variable Angle Tow Composites in Buckling and Free Vibration Analysis through Layerwise Theory

## Abstract

**:**

## 1. Introduction

## 2. Carrera Unified Formulation for Beams

#### Preliminaries

## 3. Constitutive Equations for VAT Laminates

## 4. Optimization for VAT Composite Studies

#### 4.1. Direct Search Stochastic Methods

#### 4.2. Surrogate Model: Response Surface

#### 4.3. Modeling of VAT Composite in Buckling Analysis

#### 4.4. Modeling of VAT Composite in Free Vibration Analysis

## 5. Optimization Results and Discussion

#### 5.1. Buckling Results of VAT Laminate

^{16}. Therefore, the significant cost reduction from 91

^{16}to 1000 shows the efficiency of the GA method for stacking sequence optimization.

#### Reduction of Search Domain by Latin Hypercube Method

#### 5.2. Results of VAT Laminate in Free Vibration

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

1D | One-Dimensional |

CUF | Carrera Unified Formulation |

VAT | Variable Angle Tow |

LW | Layer Wise |

GA | Genetic Algorithm |

CLT | Classical Laminate Theory |

DQM | Differential Quadrature Method |

FEM | Finite Element Method |

RS | Response Surface |

LHS | Latin Hypercube Sampling |

FN | Fundamental Nucleus |

BC | Boundary Condition |

QI | Quasi-Isotropic |

CS | Constant Stiffness |

DOF | Degree Of Freedom |

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**Figure 1.**The modeling of a beam structure using a 1D model where y is the along the beam axis, and the cross-section lies on the x–z plane through thickness.

**Figure 5.**Flowchart of genetic algorithm procedure (MATLAB) in combination with CUF approach (FORTRAN).

**Figure 6.**1D CUF cross-section refinement with $10B3$ influence on buckling load in $VA{T}_{1}$ in terms of FEM and degrees of freedom. (

**a**) Cross-section refinement. (

**b**) DOF based on cross-section refinement.

**Figure 7.**Refinement of a 1D CUF beam element with a $160L9$ effect on the buckling load in $VA{T}_{1}$ in relation to the FEM and the number of degrees of freedom. (

**a**) Beam refinement. (

**b**) DOF based on beam refinement.

**Figure 9.**The first buckling modes in different laminates. (

**a**) $QI$: 13.79 kN. (

**b**) $C{S}_{1}$: 16.51 kN. (

**c**) $C{S}_{2}$: 10.12 kN. (

**d**) $VA{T}_{1}$: 13.77 kN. (

**e**) $VA{T}_{OPT}$: 17.38 kN.

**Figure 10.**The first five buckling modes of optimum VAT laminates. (

**a**) Mode 1: 17.38 kN. (

**b**) Mode 2: 27.38 kN. (

**c**) Mode 3: 49.77 kN. (

**d**) Mode 4: 57.36 kN. (

**e**) Mode 5: 71.72 kN.

**Figure 11.**Distribution of variables and first critical buckling load, ${T}_{0}$-${T}_{1}$; first buckling load—${T}_{0}$, and first buckling load—${T}_{1}$.

**Figure 12.**Contour plots and response surfaces based on GA and LHS for ${T}_{0}$ and ${T}_{1}$ as the variables and first critical buckling load as the response. (

**a**) GA contour plot of variables’ sample points (${F}_{cr}$ is represented by the colors). (

**b**) LHS contour plot of variables’ sample points (${F}_{cr}$ is represented by the colors).

**Figure 13.**First natural frequency modes in various layup designs. (

**a**) QI. (

**b**) CS1. (

**c**) CS2. (

**d**) $VA{T}_{1}$. (

**e**) $VA{T}_{OPT}$.

**Figure 14.**The first five natural frequency modes. (

**a**) Mode 1: 513.15 Hz (

**b**) Mode 2: 809.30 Hz. (

**c**) Mode 3: 1135.00 Hz. (

**d**) Mode 4: 1312.00 Hz (

**e**) Mode 5: 1400.00 Hz.

**Figure 15.**Distribution of variables and first natural frequency; first natural frequency—${T}_{0}$, first natural frequency—${T}_{1}$.

**Figure 16.**During implementation of the GA technique in a free vibration issue, the contour plot and response surface reveal the impacts of variables (fiber orientation angles). (

**a**) Colors represent the ${f}_{1}$ in this contour plot of variables. (

**b**) Response surface.

${\mathit{E}}_{1}$ (GPa) | ${\mathit{E}}_{2}={\mathit{E}}_{3}$ (GPa) | ${\mathit{G}}_{12}={\mathit{G}}_{13}$ (GPa) | ${\mathit{G}}_{23}$ (GPa) | ${\mathit{\nu}}_{12}$ | |
---|---|---|---|---|---|

Material | 181 | 10.270 | 7.170 | 3.780 | 0.28 |

Laminated Scheme | Layup Design | |
---|---|---|

$QI$ | ${[45,0,-45,90]}_{2S}$ | |

$C{S}_{1}$ | ${[\pm 45]}_{4S}$ | |

$C{S}_{2}$ | ${\left[0\right]}_{16T}$ | |

$VA{T}_{1}$ [44,51] | $[<{60}^{\circ}|{15}^{\circ}><-{60}^{\circ}|{15}^{\circ}>/<-{60}^{\circ}|{15}^{\circ}><{60}^{\circ}{|{15}^{\circ}>]}_{4}$ |

Description | Maximum Number of Generation | Population Size | Crossover Percentage | Mutation Rate |
---|---|---|---|---|

Value | 20 | 50 | 0.8 | 0.04 |

Laminated Scheme | First Natural Frequency (Hz) | |||
---|---|---|---|---|

$QI$ | 438.385 | |||

$C{S}_{1}$ | 433.064 | |||

$C{S}_{2}$ | 446.235 | |||

$VA{T}_{1}$ | 438.659 | |||

$VA{T}_{OPT}$ | 513.159 |

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**MDPI and ACS Style**

Fallahi, N.
GA Optimization of Variable Angle Tow Composites in Buckling and Free Vibration Analysis through Layerwise Theory. *Aerospace* **2021**, *8*, 376.
https://doi.org/10.3390/aerospace8120376

**AMA Style**

Fallahi N.
GA Optimization of Variable Angle Tow Composites in Buckling and Free Vibration Analysis through Layerwise Theory. *Aerospace*. 2021; 8(12):376.
https://doi.org/10.3390/aerospace8120376

**Chicago/Turabian Style**

Fallahi, Nasim.
2021. "GA Optimization of Variable Angle Tow Composites in Buckling and Free Vibration Analysis through Layerwise Theory" *Aerospace* 8, no. 12: 376.
https://doi.org/10.3390/aerospace8120376