# Buckling Analysis of a Thin-Walled Structure Using Finite Element Method and Design of Experiments

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Graph of load applied versus deflection [9]. Reprinted under the Creative Commons (CC) License (CC BY 4.0).

_{27}orthogonal array where 27 indicates the number of runs (simulation) required. Response surface methodology (RSM) was used to analyse the obtained data through the main effect plot, contour plot, regression, and analysis of variance (ANOVA). Additionally, the present study focused on perforated composite plates, and the supports were fixed at both ends and the orientation of the fibre was as mentioned. The cross-sectional shape considered was a C section. The novel work of this study focused on optimizing the parameters to achieve the highest probable critical buckling load for perforated composites and thin-walled members, through simulation using MINITAB and Design Epxert tools.

## 2. Finite Element Method

#### 2.1. Designed Model

^{3}. Table 1 shows the GFRP strut’s strength properties.

#### 2.2. Loading and Boundary Conditions

#### 2.3. Laminate Model

## 3. Design of Experiment

#### 3.1. Plan of Experiments with FEM Results

_{27}orthogonal array provided by MINITAB-20 (Coventry, UK) and Design Expert-13 software (Minneapolis, MN, USA). If the orthogonal array degrees of freedom are greater than, or at least equal to the previous ones, then the test parameters and the array are selected. The current experiment’s parameters had a greater impact. An orthogonal array of 27 tests was used, and each row had a parameter assigned to it (each column had a row number). The simulations were carried out following the orthogonal array L

_{27}standard. A planned analysis should be able to examine how different composite laminate types’ critical buckling behaviour was influenced by the opening ratio (W/W

_{o}), spacing ratio (D/W

_{o}), and cutout shape (Table 2).

_{27}orthogonal array, we were able to perform a linear buckling analysis for a variety of parameter combinations. Included here is the response for each laminate type as requested by Minitab and design expert software (Table 3).

#### 3.2. Response Surface Methodology (RSM)

#### 3.3. Analysis of Variance (ANOVA)

## 4. Finite Element Results

#### 4.1. Model Validation

Laminate Types | Critical Buckling Load (N) | Percentage Error (%) | |
---|---|---|---|

Previous Study [12] | Present Simulation (ABAQUS 2020) | ||

Quasi-isotropic | 11,258 | 11,255 | 0.026 |

Angle-ply | 12,652 | 11,382 | 0.017 |

#### 4.2. Mesh Convergence Test

#### 4.3. Displacement of Laminate Structures

#### 4.3.1. Effect of Shape

#### 4.3.2. Effect of Opening Ratio

#### 4.3.3. Effect of Spacing Ratio

#### 4.4. Nonlinear Analysis

^{2}values indicate the accuracy of the approximation for a curved trend in a post-buckling state. Additionally, the critical buckling load for Figure 19, Figure 20 and Figure 21 was 12,667.85 N, 12,551.1 N, and 7878.47 N, respectively.

## 5. Optimization Results

#### 5.1. Response Surface Analysis

#### 5.1.1. Main Effect Plot

#### 5.1.2. Contour Plot

_{o}, circular holes showed a better response compared to differently shaped holes. For cross-ply, the plot indicates a more sensitive response as the slope of each contour was steeper compared to the quasi-isotropic laminate. This means a slight adjustment of the input variable will affect the response of the output. Both angle-ply and balanced laminates had an almost linear contour plot. According to this result, one can freely choose the spacing ratio despite the size of the holes to obtain optimum results.

#### 5.1.3. Analysis of Variance (ANOVA)

_{o}, which had the greatest F-value for all laminate types, indicates that it is the parameter that most influenced one-way interaction. The F-value indicates how much the parameter is associated with the response. This once again supports the previously stated hypothesis that the size of the hole affects the eigenvalue load significantly.

_{o}and S/W

_{o}had the highest F-value for quasi-isotropic, cross-ply, and balanced laminates. In other words, when it comes to creating a composite thin-wall structure, combining these two elements had the greatest impact. Angle-ply, on the other hand, can utilize a variety of different combinations. There was a strong correlation between the W/W

_{o}ratio and cutout shape, which affects buckling loads.

#### 5.1.4. Regression Equation

_{27}orthogonal array through MINITAB software was used to identify an equation-based statistical model for the current study. A linear polynomial model (regression equations) was used to represent each component. The equation was obtained separately for each cutout shape as a categorical factor and different laminate types. The linear regression equation included below can be used to calculate the expected results and percentage variance for all test cases.

_{o}*S/W

_{o}, W/W

_{o}*S/W

_{o}, and W/W

_{o}*W/W

_{o}. It is believed that the interaction between those parameters affects the critical buckling load of the strut.

- For angle-ply,
| |||

CIRCULAR | P_cr(N) | = | −10522 + 19542 W/W_{o} + 1964 S/W_{o}—3628 W/W_{o}*W/W_{o} + 902 S/W_{o}*S/W_{o}—2914 W/W_{o}*S/W_{o} |

HEXAGON | P_cr(N) | = | −11411 + 19690 W/W_{o} + 2186 S/W_{o}—3628 W/W_{o}*W/W_{o} + 902 S/W_{o}*S/W_{o}—2914 W/W_{o}*S/W_{o} |

SQUARE | P_cr(N) | = | −14156 + 20988 W/W_{o} + 2318 S/W_{o}—3628 W/W_{o}*W/W_{o} + 902 S/W_{o}*S/W_{o}—2914 W/W_{o}*S/W_{o} |

- For balanced laminate,
| |||

CIRCULAR | P_cr(N) | = | −8939 + 17806 W/W_{o} + 645 S/W_{o}—3638 W/W_{o}*W/W_{o} + 740 S/W_{o}*S/W_{o}—1717 W/W_{o}*S/W_{o} |

HEXAGON | P_cr(N) | = | −9856 + 18013 W/W_{o} + 818 S/W_{o}—3638 W/W_{o}*W/W_{o} + 740 S/W_{o}*S/W_{o}—1717 W/W_{o}*S/W_{o} |

SQUARE | P_cr(N) | = | −10769 + 18271 W/W_{o} + 904 S/W_{o}—3638 W/W_{o}*W/W_{o} + 740 S/W_{o}*S/W_{o}—1717 W/W_{o}*S/W_{o} |

- For cross-ply,
| |||

CIRCULAR | P_cr(N) | = | −4456 + 7037 W/W_{o} + 3088 S/W_{o}—1052 W/W_{o}*W/W_{o} + 166 S/W_{o}*S/W_{o}—1515 W/W_{o}*S/W_{o} |

HEXAGON | P_cr(N) | = | −5179 + 7266 W/W_{o} + 3162 S/W_{o}—1052 W/W_{o}*W/W_{o} + 166 S/W_{o}*S/W_{o}—1515 W/W_{o}*S/W_{o} |

SQUARE | P_cr(N) | = | −5442 + 7209 W/W_{o} + 3244 S/W_{o}—1052 W/W_{o}*W/W_{o} + 166 S/W_{o}*S/W_{o}—1515 W/W_{o}*S/W_{o} |

- For quasi-isotropic,
| |||

CIRCULAR | P_cr(N) | = | −4921 + 10941 W/W_{o} + 2626 S/W_{o}—1569 W/W_{o}*W/W_{o} + 659 S/W_{o}*S/W_{o}—2358 W/W_{o}*S/W_{o} |

HEXAGON | P_cr(N) | = | −5825 + 11167 W/W_{o} + 2786 S/W_{o}—1569 W/W_{o}*W/W_{o} + 659 S/W_{o}*S/W_{o}—2358 W/W_{o}*S/W_{o} |

SQUARE | P_cr(N) | = | −6162 + 11065 W/W_{o} + 2922 S/W_{o}—1569 W/W_{o}*W/W_{o} + 659 S/W_{o}*S/W_{o}—2358 W/W_{o}*S/W_{o} |

## 6. Conclusions

_{o}parameter had the highest F-value in all laminate types, indicating it is the most influential for one-way interaction. In quasi-isotropic, cross-ply, and balanced laminates, W/W

_{o}and S/W

_{o}had the highest F-value. The design of the composite thin-wall structure was influenced by these two combinations. Angle-ply can be combined in many ways. W/W

_{o}and the cutout shape had the greatest impact on the buckling load. This study also found the best parameter combinations for different laminates in terms of critical buckling. An optimization analysis based on parameters showed that range changes affect buckling load values.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 18.**Nonlinear analysis for critical buckling load for the quasi-isotropic model without holes.

**Figure 22.**All laminate types’ main effects plot for fitted means: (

**a**) angle-ply, (

**b**) balanced, (

**c**) quasi-isotropic, and (

**d**) cross-ply.

E1 (Mpa) | E2 (Mpa) | υ | G12 (Mpa) | G13 (Mpa) | G23 (Mpa) |
---|---|---|---|---|---|

38,000 | 8100 | 0.27 | 2000 | 2000 | 2000 |

Parameter | Levels | ||
---|---|---|---|

1 | 2 | 3 | |

W/W_{o} | 1.5 | 1.6 | 1.7 |

D/W_{o} | 1.3 | 1.4 | 1.5 |

Shapes | Circular | Square | Hexagonal |

Run Order | W/W_{o} | S/W_{o} | SHAPE | Critical Buckling Load (N) | |||
---|---|---|---|---|---|---|---|

Angle-Ply | Balanced | Quasi-Isotropic | Cross-Ply | ||||

1 | 1.7 | 1.5 | CIRCULAR | 9773.7 | 9077.5 | 8554 | 5611.1 |

2 | HEXAGON | 9464.4 | 8765.2 | 8271.3 | 5387.8 | ||

3 | SQUARE | 9108.1 | 8411.4 | 7963.3 | 5146.9 | ||

4 | 1.6 | 1.5 | CIRCULAR | 9430.4 | 8742 | 8330.1 | 5483 |

5 | HEXAGON | 9109 | 8411 | 8024.7 | 5233.9 | ||

6 | SQUARE | 8644.6 | 8064.3 | 7727.5 | 4998.8 | ||

7 | 1.5 | 1.5 | CIRCULAR | 9040.9 | 8351.7 | 8066.8 | 5323 |

8 | HEXAGON | 8716.2 | 8011.3 | 7746.6 | 5055.8 | ||

9 | SQUARE | 8129 | 7600.4 | 7471.9 | 4844.5 | ||

10 | 1.7 | 1.4 | CIRCULAR | 9798.9 | 9084.6 | 8495.7 | 5507.2 |

11 | HEXAGON | 9475 | 8761.3 | 8201.6 | 5280 | ||

12 | SQUARE | 9107.2 | 8397.8 | 7879.9 | 5031 | ||

13 | 1.6 | 1.4 | CIRCULAR | 9438.7 | 8738.6 | 8253.3 | 5366.6 |

14 | HEXAGON | 9100 | 8393 | 7937.1 | 5118.1 | ||

15 | SQUARE | 8639.1 | 8028.9 | 7618.4 | 4866 | ||

16 | 1.5 | 1.4 | CIRCULAR | 9029 | 8332.6 | 7974 | 5200.1 |

17 | HEXAGON | 8680.1 | 7970 | 7637 | 4927.7 | ||

18 | SQUARE | 8049.5 | 7559.5 | 7335.3 | 4688.2 | ||

19 | 1.7 | 1.3 | CIRCULAR | 9844.5 | 9107.3 | 8452 | 5410.4 |

20 | HEXAGON | 9505.9 | 8774.6 | 8145 | 5172.5 | ||

21 | SQUARE | 9131.3 | 8406.4 | 7822.2 | 4930.9 | ||

22 | 1.6 | 1.3 | CIRCULAR | 9468.9 | 8752.6 | 8190 | 5253.5 |

23 | HEXAGON | 9105.6 | 8387 | 7854.5 | 4993.4 | ||

24 | SQUARE | 8653 | 8011.6 | 7530.4 | 4744.7 | ||

25 | 1.5 | 1.3 | CIRCULAR | 9035.3 | 8329.7 | 7890.6 | 5076 |

26 | HEXAGON | 8648.2 | 7940.5 | 7528.6 | 4789.8 | ||

27 | SQUARE | 7988.4 | 7521.3 | 7213.7 | 4543.7 |

Element Size | Critical Buckling Load (N) | Number of Element | Percentage Error (%) |
---|---|---|---|

1 | 11,232 | 40,000 | 0.230947 |

2 | 11,255 | 10,000 | 0.026648 |

3 | 11,313 | 4399 | 0.488541 |

4 | 11,365 | 2520 | 0.950435 |

5 | 11,423 | 1600 | 1.465624 |

Source | DF | Adj SS | Adj MS | F-Value | p-Value |
---|---|---|---|---|---|

Model | 11 | 3,257,334 | 296,121 | 11,702.66 | 0.000 |

Linear | 4 | 3,244,740 | 811,185 | 32,057.89 | 0.000 |

W/W_{o} | 1 | 134,5073 | 1,345,073 | 53,157.07 | 0.000 |

S/W_{o} | 1 | 129,914 | 129,914 | 5134.18 | 0.000 |

SHAPE | 2 | 176,9753 | 884,876 | 34,970.16 | 0.000 |

Square | 2 | 1738 | 869 | 34.35 | 0.000 |

W/W_{o}*W/W_{o} | 1 | 1478 | 1478 | 58.41 | 0.000 |

S/W_{o}*S/W_{o} | 1 | 260 | 260 | 10.29 | 0.006 |

2-Way Interaction | 5 | 10,856 | 2171 | 85.80 | 0.000 |

W/W_{o}*S/W_{o} | 1 | 6674 | 6674 | 263.76 | 0.000 |

W/W_{o}*SHAPE | 2 | 1532 | 766 | 30.28 | 0.000 |

S/W_{o}*SHAPE | 2 | 2649 | 1324 | 52.34 | 0.000 |

Error | 15 | 380 | 25 | ||

Total | 26 |

Source | DF | Adj SS | Adj MS | F-Value | p-Value |
---|---|---|---|---|---|

Model | 11 | 6,640,780 | 603,707 | 2127.51 | 0.000 |

Linear | 4 | 6,542,370 | 1,635,592 | 5763.96 | 0.000 |

W/W_{o} | 1 | 3,460,554 | 3,460,554 | 12,195.28 | 0.000 |

S/W_{o} | 1 | 69 | 69 | 0.24 | 0.629 |

SHAPE | 2 | 3,081,747 | 1,540,873 | 5430.17 | 0.000 |

Square | 2 | 8385 | 4192 | 14.77 | 0.000 |

W/W_{o}*W/W_{o} | 1 | 7896 | 7896 | 27.83 | 0.000 |

S/W_{o}*S/W_{o} | 1 | 488 | 488 | 1.72 | 0.209 |

2-Way Interaction | 5 | 90,025 | 18,005 | 63.45 | 0.000 |

W/W_{o}*S/W_{o} | 1 | 10,191 | 10,191 | 35.91 | 0.000 |

W/W_{o}*SHAPE | 2 | 75,981 | 37,991 | 133.88 | 0.000 |

S/W_{o}*SHAPE | 2 | 3853 | 1926 | 6.79 | 0.008 |

Error | 15 | 4256 | 284 | ||

Total | 26 | 6,645,036 |

Source | D | Adj SS | Adj MS | F-Value | p-Value |
---|---|---|---|---|---|

Model | 11 | 1,870,705 | 170,064 | 6065.49 | 0.000 |

Linear | 4 | 1,864,835 | 466,209 | 16,627.76 | 0.000 |

W/W_{o} | 1 | 509,713 | 509,713 | 18,179.39 | 0.000 |

S/W_{o} | 1 | 261,581 | 261,581 | 9329.54 | 0.000 |

SHAPE | 2 | 1,093,541 | 546,770 | 19,501.06 | 0.000 |

Square | 2 | 681 | 340 | 12.14 | 0.001 |

W/W_{o}*W/W_{o} | 1 | 664 | 664 | 23.69 | 0.000 |

S/W_{o}*S/W_{o} | 1 | 17 | 17 | 0.59 | 0.454 |

2-Way Interaction | 5 | 5189 | 1038 | 37.01 | 0.000 |

W/W_{o}*S/W_{o} | 1 | 2754 | 2754 | 98.23 | 0.000 |

W/W_{o}*SHAPE | 2 | 1702 | 851 | 30.36 | 0.000 |

S/W_{o}*SHAPE | 2 | 732 | 366 | 13.06 | 0.001 |

Error | 15 | 421 | 28 | ||

Total | 26 |

Source | DF | Adj SS | Adj MS | F-Value | p-Value |
---|---|---|---|---|---|

Model | 11 | 5,237,892 | 476,172 | 7286.76 | 0.000 |

Linear | 4 | 5,217,484 | 1,304,371 | 19,960.51 | 0.000 |

W/W_{o} | 1 | 2,855,333 | 2,855,333 | 43,694.55 | 0.000 |

S/W_{o} | 1 | 2307 | 2307 | 35.31 | 0.000 |

SHAPE | 2 | 2,359,843 | 1,179,922 | 18,056.09 | 0.000 |

Square | 2 | 8271 | 4136 | 63.28 | 0.000 |

W/W_{o}*W/W_{o} | 1 | 7942 | 7942 | 121.54 | 0.000 |

S/W_{o}*S/W_{o} | 1 | 329 | 329 | 5.03 | 0.040 |

2-Way Interaction | 5 | 12,137 | 2427 | 37.15 | 0.000 |

W/W_{o}*S/W_{o} | 1 | 3540 | 3540 | 54.17 | 0.000 |

W/W_{o}*SHAPE | 2 | 6514 | 3257 | 49.84 | 0.000 |

S/W_{o}*SHAPE | 2 | 2084 | 1042 | 15.94 | 0.000 |

Error | 15 | 980 | 65 | ||

Total | 26 | 5,238,872 |

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**MDPI and ACS Style**

Bin Kamarudin, M.N.; Mohamed Ali, J.S.; Aabid, A.; Ibrahim, Y.E.
Buckling Analysis of a Thin-Walled Structure Using Finite Element Method and Design of Experiments. *Aerospace* **2022**, *9*, 541.
https://doi.org/10.3390/aerospace9100541

**AMA Style**

Bin Kamarudin MN, Mohamed Ali JS, Aabid A, Ibrahim YE.
Buckling Analysis of a Thin-Walled Structure Using Finite Element Method and Design of Experiments. *Aerospace*. 2022; 9(10):541.
https://doi.org/10.3390/aerospace9100541

**Chicago/Turabian Style**

Bin Kamarudin, Mohamad Norfaieqwan, Jaffar Syed Mohamed Ali, Abdul Aabid, and Yasser E. Ibrahim.
2022. "Buckling Analysis of a Thin-Walled Structure Using Finite Element Method and Design of Experiments" *Aerospace* 9, no. 10: 541.
https://doi.org/10.3390/aerospace9100541