# Experimental and Numerical Investigation of the In-Plane Compression of Corrugated Paperboard Panels

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Edge Crush Test

_{max}is the failure load.

#### 2.2. Bending Stiffness Test

#### 2.3. Panel Compression

#### 2.4. Finite Element Modeling

#### 2.4.1. Bending Test

#### 2.4.2. Panel Compression

_{i,j}is the bending stiffness, w is the panel width and d is the panel height. The failure load can be calculated as:

#### 2.5. Box Compression

## 3. Results and Discussion

#### 3.1. ECT

#### 3.2. Bending Stiffness

#### 3.3. Panel Compression

^{2}> 0.99. The DIC panel stiffness was 3.6 kN/m, compared to 1 kN/m for the crosshead displacement curve. The ECT strength was included for reference, but it should be noted that the ECT had different boundary conditions (clamped horizontal edges) than the panel compression (simply supported). Compared to the ECT, the 400 mm panels were about 49% weaker. The decrease in strength is due to buckling failure rather than pure material failure.

#### 3.4. Finite Element Modelling

#### 3.4.1. Bending

#### 3.4.2. Panel Compression

_{0}or R

_{90}. For R

_{0}, the parameters may be given as:

_{N}is the yield stress in the thickness direction and σ

_{av}is the initial yield stress:

_{45}were unknown, some assumptions had to be made. It was assumed that σ

_{45}was equal to σ

_{90}, and the strain at σ

_{45}was equal to the strain at σ

_{0}, based on the failure trends seen in [32]. It was also assumed that the R-values were large, since the strain in the thickness direction was assumed small compared to the in-plane directions, i.e., plane stress [16].

_{N}was large, and finally, since R and σ

_{N}were large, YRSHR (1) was approximated as 1. σ

_{0}and σ

_{90}were estimated from Figure 14 by fitting a linear slope to the high- and low-stiffness regions (see Figure 15) and taking the intersect as the yield strength (Table 9).

#### 3.5. Box Compression Test

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Board geometry [11].

_{f}is the flute height and P is the wavelength [35].

_{vc}is the vertical thickness of the fluted sheet, given as:

## References

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**Figure 6.**(

**a**) Speckle pattern on a panel with top flaps; (

**b**) speckle pattern on a panel with bottom flaps.

**Figure 8.**Force (average and standard deviation) versus displacement for 400 mm panel compression: (

**a**) in-plane; (

**b**) out-of-plane.

**Figure 9.**(

**a**) The 400 mm panel test 13 in CD, maximum displacement towards the bottom left, approximate locations used to obtain DIC curve indicated in green. (

**b**) Typical displacement pattern before failure.

**Figure 12.**Panel load versus in-plane displacement for the detailed model, homogeneous model and experiment.

**Figure 13.**Panel load versus out-of-plane displacement for the detailed model, homogeneous model and experiment.

**Figure 14.**Paper sheet tensile test data [11].

**Figure 20.**(

**a**) BCT typical out-of-plane displacement pattern just before failure; (

**b**) BCT out-of-plane displacement pattern just before failure for sample two.

**Table 1.**Initial material properties [11].

Material Property | 250 KL | 150C-SC | 150C-SC Homogenised |
---|---|---|---|

E_{1} (MPa) | 6695 | 4709 | 27 |

E_{2} (MPa) | 2310 | 2918 | 235 |

E_{3} (MPa) | 35 | 25 | 3000 |

ν_{12} | 0.5 | 0.37 | 0.02 |

ν_{13} | 0.01 | 0.01 | 0.01 |

ν_{23} | 0.01 | 0.01 | 0.01 |

G_{12} (MPa) | 1522 | 1435 | 21 |

G_{13} (MPa) | 122 | 86 | 3 |

G_{23} (MPa) | 66 | 83 | 14 |

**Table 2.**Measured paper sheet strength properties [11].

Sheet | σ_{1,t} (MPa) | σ_{1,c} (MPa) | σ_{2,t} (MPa) | σ_{2,c} (MPa) |
---|---|---|---|---|

150C-SC | 45.3 ± 2.4 | 30.7 ± 1.7 | 22.1 ± 1.4 | 17.5 ± 1.1 |

250 KL | 68.0 ± 2.9 | 23.3 ± 1.4 | 24.1 ± 1.7 | 12.9 ± 1.2 |

Direction | Number of Samples | Average Force (N) | ECT Strength (kN/m) |
---|---|---|---|

CD | 20 | 298.3 ± 11.6 | 11.75 ± 0.46 |

Direction | P_{max} (N) | Y_{max} (mm) | S_{b} (N⋅m) |
---|---|---|---|

Four-point CD | 58.6 ± 2.4 | 14.3 ± 0.5 | 8.0 ± 0.3 |

Four-point MD | 41.6 ± 3.5 | 3.5 ± 0.4 | 20.8 ± 1.1 |

Panel | P_{max} (kN) | Y_{max} (mm) | W_{max} (mm) | Stiffness (kN/mm) | P_{norm} (kN/m) |
---|---|---|---|---|---|

ECT CD | - | - | - | - | 11.74 |

400 mm CD CH | 2.07 ± 0.07 | 3.5 ± 0.3 | 13.9 | 1.00 | 5.18 |

400 mm CD DIC | 2.07 ± 0.07 | 1.6 ± 0.2 | 13.9 | 3.62 | 5.18 |

Material Property | 250 KL | 150C-SC | 150C-SC Homogenised |
---|---|---|---|

E_{1} (MPa) | 7699 | 4709 | 32 |

E_{2} (MPa) | 2541 | 2918 | 246 |

E_{3} (MPa) | 35 | 25 | 3000 |

ν_{12} | 0.5 | 0.37 | 0.03 |

ν_{13} | 0.01 | 0.01 | 0.01 |

ν_{23} | 0.01 | 0.01 | 0.01 |

G_{12} (MPa) | 1583 | 1492 | 24.6 |

G_{13} (MPa) | 122 | 86 | 2.7 |

G_{23} (MPa) | 66 | 83 | 13 |

**Table 7.**Bending stiffness comparison of experimental data and adjusted detailed- and homogeneous model.

Model | Four-Point CD | Four-Point MD | |
---|---|---|---|

Experimental | Stiffness (N·m) | 8.0 | 20.8 |

Adjusted detailed model | Stiffness (N·m) | 8.2 | 21.2 |

Error | 2.8% | 1.6% | |

Adjusted homogeneous model | Stiffness (N·m) | 7.7 | 21.2 |

Error | 3.5% | 1.8% |

Method | Buckling Load (N) | Failure Load (N) | In-Plane Displacement (Failure) (mm) | Out-of-Plane Displacement (Failure) (mm) |
---|---|---|---|---|

Experimental | N\A | 2071 | 3.5 (CH), 1.6 (DIC) | 13.9 |

FEA detail | 1310 | 2209 | 1.25 | 11.45 |

Error | - | 6.7% | 64.6% (CH), 21.9% (DIC) | 17.6% |

FEA homogeneous (0.001 mm perturbation) | 1352 | 2220 | 1.31 | 11.60 |

Error | - | 7.2% | 62.9% (CH), 18.1% (DIC) | 16.5% |

Paper Type | σ_{0} (MPa) | σ_{45} (MPa) | σ_{90} (MPa) | σ_{avg} (MPa) |
---|---|---|---|---|

150C-SC | 26.11 | 18.87 | 18.87 | 20.68 |

250KL | 32.65 | 14.02 | 14.02 | 18.68 |

**Table 10.**Comparison of the 400 mm × 400 mm (415 mm × 400 mm in brackets) panel behaviour for experimental, theoretical and FEA values.

Method | Buckling Load (N) | Failure Load (N) | In-Plane Displacement (Failure) (mm) | Out-of-Plane Displacement (Failure) (mm) |
---|---|---|---|---|

Experimental | N\A | 2071 | 3.5 (CH), 1.6 (DIC) | 13.9 |

Theoretical | 1351 (1281) | 1742 (1769) | N\A | N\A |

Error | N\A | 15.9% (14.6%) | N\A | N\A |

FEA detail | 1310 | 2209 | 1.25 | 11.45 |

Error | 3.0% (2.3%) | 6.7% | 64.6% (CH), 21.9% (DIC) | 17.6% |

FEA homogeneous (0.001 mm) | 1352 | 2220 | 1.31 | 11.60 |

Error | 0.1% (5.5%) | 7.2% | 62.9% (CH), 18.1% (DIC) | 16.5% |

BCT (N) | Y (mm) | W (mm) | Stiffness, High (N/mm) | Stiffness, Low (N/mm) |
---|---|---|---|---|

5267 ± 464 | 13.1 ± 0.6 (CH) 0.7 ± 0.1 (DIC) | 11.3 ± 1.6 | 865.4 (CH) 11,054 (DIC) | 163.0 (CH) |

Method | BCT Load (N) | Error (%) |
---|---|---|

Experimental | 5 267 | N\A |

McKee long | 5 321 | 1.0 |

McKee short | 5 632 | 6.9 |

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**MDPI and ACS Style**

Cillie, J.; Coetzee, C.
Experimental and Numerical Investigation of the In-Plane Compression of Corrugated Paperboard Panels. *Math. Comput. Appl.* **2022**, *27*, 108.
https://doi.org/10.3390/mca27060108

**AMA Style**

Cillie J, Coetzee C.
Experimental and Numerical Investigation of the In-Plane Compression of Corrugated Paperboard Panels. *Mathematical and Computational Applications*. 2022; 27(6):108.
https://doi.org/10.3390/mca27060108

**Chicago/Turabian Style**

Cillie, Johan, and Corné Coetzee.
2022. "Experimental and Numerical Investigation of the In-Plane Compression of Corrugated Paperboard Panels" *Mathematical and Computational Applications* 27, no. 6: 108.
https://doi.org/10.3390/mca27060108