# Estimation of the Compressive Strength of Corrugated Cardboard Boxes with Various Openings

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Ultimate Compressive Strength of a Single Panel

#### 2.2. Buckling—McKee’s Formula

#### 2.3. Buckling—Finite Element Method

- discretize the panel with finite elements (FE), see Figure 2a,
- compute the stiffness matrix ${K}^{e}$ of each element,
- assemble the global stiffness matrix ${K}_{0}$ of a whole panel,
- compute nodal forces representing initial loading configuration ${P}^{*}$, i.e., for loading multiplier $\lambda =1$ (one-parameter loading assumed $P=\lambda {P}^{*}$),
- take boundary conditions into account,
- solve equation ${}_{0}{d}^{*}={P}^{*}$ to obtain nodal displacements in pre-buckling state:$${d}^{*}={K}_{0}^{-1}\xb7{P}^{*},$$
- extract element displacements ${d}^{\left(e\right)*}$ (from displacements of the system ${d}^{*}$) and compute in each element the generalized stresses ${s}^{\left(e\right)*}$.

- generate initial geometrical stress matrices for each element ${K}_{\sigma}^{e}\left({s}^{*e}\right)$ and the whole panel ${K}_{\sigma}\left({s}^{*}\right)$,
- formulate initial buckling problem:$$\left[{K}_{0}+\lambda {K}_{\sigma}\right]v=\mathbf{0},$$
- solve the eigenproblem to determine the pairs $\left({\lambda}_{1},{v}_{1}\right),\dots ,\left({\lambda}_{N},{v}_{N}\right)$, where $N$ is a number of degrees of freedom, ${\lambda}_{i}$ is $i$th eigenvalue, ${v}_{i}=\Delta {d}_{i}$ is $i$th eigenvector (post-buckling deformation mode).

#### 2.4. Box Compression Strength—McKee’s Formula

#### 2.5. Box Compression Strength—General Case

#### 2.6. Practical Considerations

- (a)
- Equation (4)—for orthotropic rectangular plates without holes (analytical),
- (b)
- Equation (8)—for orthotropic rectangular plates with various openings (numerical – FEM).

## 3. Results

#### 3.1. Box Compression Strength—Experiment vs. Estimation

^{2}. All those cardboards had B or E flute, their thickness was 1.49, 1.52, 1.59, 2.80 and 2.82 mm. All cases tested are presented in Table 1 and Table 2 for FEFCO type boxes of F200 and F201, respectively. In Table 1 and Table 2 “opening type” columns contain the information about the opening, for instance O1B1S, means that the opening was located in the corner (O1) of the shorter wall (B). “Cardboard quality” columns contain the code of cardboard, for instance 3B400-1, means that three layer cardboard and B flute was used with 400 g/m

^{2}, 1st material production.

#### 3.2. Reduction of the Estimation Error—Optimal Parameters

## 4. Discussion

^{2}) the calculated values are lower than for the cases 18–23 (400 g/m

^{2}), this is caused by the different papers (predominance of recycled or virgin fibers) used for the production of these corrugated boards, which is noticeable through measured material properties, see Table 3. Comparing the parameters of the first three cardboard qualities: 3B400A1-1, 3B400A1-3 and 3B480-1, one can observe that the values are very close to each other and the ECT values, used in the original McKee formula, are even greater for the cardboards of lower grammage.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Derivation of the Substitute Transverse Shear Strain Matrix

## References

- Sohrabpour, V.; Hellström, D. Models and software for corrugated board and box design. In Proceedings of the 18th International Conference on Engineering Design (ICED 11), Copenhagen, Denmark, 15–18 October 2011. [Google Scholar]
- Kellicutt, K.; Landt, E. Development of design data for corrugated fiberboard shipping containers. Tappi J.
**1952**, 35, 398–402. [Google Scholar] - Maltenfort, G. Compression strength of corrugated containers. Fibre Contain.
**1956**, 41, 106–121. [Google Scholar] - McKee, R.C.; Gander, J.W.; Wachuta, J.R. Compression strength formula for corrugated boxes. Paperboard Packag.
**1963**, 48, 149–159. [Google Scholar] - Allerby, I.M.; Laing, G.N.; Cardwell, R.D. Compressive strength—From components to corrugated containers. Appita Conf. Notes
**1985**, 1–11. [Google Scholar] - Schrampfer, K.E.; Whitsitt, W.J.; Baum, G.A. Combined Board Edge Crush (ECT) Technology; Institute of Paper Chemistry: Appleton, WI, USA, 1987. [Google Scholar]
- Batelka, J.J.; Smith, C.N. Package Compression Model; Institute of Paper Science and Technology: Atlanta, GA, USA, 1993. [Google Scholar]
- Urbanik, T.J.; Frank, B. Box compression analysis of world-wide data spanning 46 years. Wood Fiber Sci.
**2006**, 38, 399–416. [Google Scholar] - Garbowski, T.; Gajewski, T.; Grabski, J.K. The role of buckling in the estimation of compressive strength of corrugated cardboard boxes. Materials
**2020**, 13, 4578. [Google Scholar] [CrossRef] - Avilés, F.; Carlsson, L.A.; May-Pat, A. A shear-corrected formulation of the sandwich twist specimen. Exp. Mech.
**2012**, 52, 17–23. [Google Scholar] [CrossRef] - Garbowski, T.; Gajewski, T.; Grabski, J.K. Role of transverse shear modulus in the performance of corrugated materials. Materials
**2020**, 13, 3791. [Google Scholar] [CrossRef] - Garbowski, T.; Gajewski, T.; Grabski, J.K. Torsional and transversal stiffness of orthotropic sandwich panels. Materials
**2020**, 13, 5016. [Google Scholar] [CrossRef] - Urbanik, T.J.; Saliklis, E.P. Finite element corroboration of buckling phenomena observed in corrugated boxes. Wood Fiber Sci.
**2003**, 35, 322–333. [Google Scholar] - Nordstrand, T. Basic Testing and Strength Design of Corrugated Board and Containers. Ph.D. Thesis, Lund University, Lund, Sweden, 2003. [Google Scholar]
- Nordstrand, T.; Carlsson, L. Evaluation of transverse shear stiffness of structural core sandwich plates. Comp. Struct.
**1997**, 37, 145–153. [Google Scholar] [CrossRef] - Pozorski, Z.; Wojciechowski, S. The influence of symmetrical boundary conditions on the structural behaviour of sandwich panels subjected to torsion. Symmetry
**2020**, 12, 2093. [Google Scholar] [CrossRef] - Yan, J.; Wang, G.; Li, Q.; Zhang, L.; Yan, J.D.; Chen, C.; Fang, Z. A comparative study on damage mechanism of sandwich structures with different core materials under lightning strikes. Energies
**2017**, 10, 1594. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Jin, X.; Xie, G.; Yan, H. Thermo-fluidic comparison between sandwich panels with tetrahedral lattice cores fabricated by casting and metal sheet folding. Energies
**2017**, 10, 906. [Google Scholar] [CrossRef] [Green Version] - Duan, Y.; Zhang, H.; Sfarra, S.; Avdelidis, N.P.; Loutas, T.H.; Sotiriadis, G.; Kostopoulos, V.; Fernandes, H.; Petrescu, F.I.; Ibarra-Castanedo, C.; et al. On the use of infrared thermography and acousto-ultrasonics NDT techniques for ceramic-coated sandwich structures. Energies
**2019**, 12, 2537. [Google Scholar] [CrossRef] [Green Version] - Garbowski, T.; Jarmuszczak, M. Homogenization of corrugated paperboard. Part 1. Analytical homogenization. Pol. Pap. Rev.
**2014**, 70, 345–349. (In Polish) [Google Scholar] - Garbowski, T.; Jarmuszczak, M. Homogenization of corrugated paperboard. Part 2. Numerical homogenization. Pol. Pap. Rev.
**2014**, 70, 390–394. (In Polish) [Google Scholar] - Hohe, J. A direct homogenization approach for determination of the stiffness matrix for microheterogeneous plates with application to sandwich panels. Compos. Part B
**2003**, 34, 615–626. [Google Scholar] [CrossRef] - Buannic, N.; Cartraud, P.; Quesnel, T. Homogenization of corrugated core sandwich panels. Comp. Struct.
**2003**, 59, 299–312. [Google Scholar] [CrossRef] [Green Version] - Biancolini, M.E. Evaluation of equivalent stiffness properties of corrugated board. Comp. Struct.
**2005**, 69, 322–328. [Google Scholar] [CrossRef] - Abbès, B.; Guo, Y.Q. Analytic homogenization for torsion of orthotropic sandwich plates: Application. Comp. Struct.
**2010**, 92, 699–706. [Google Scholar] [CrossRef] - Marek, A.; Garbowski, T. Homogenization of sandwich panels. Comput. Assist. Methods Eng. Sci.
**2015**, 22, 39–50. [Google Scholar] - Garbowski, T.; Marek, A. Homogenization of corrugated boards through inverse analysis. In Proceedings of the 1st International Conference on Engineering and Applied Sciences Optimization, Kos Island, Greece, 4–6 June 2014; pp. 1751–1766. [Google Scholar]
- Singh, J.; Olsen, E.; Singh, S.P.; Manley, J.; Wallace, F. The effect of ventilation and hand holes on loss of compression strength in corrugated boxes. J. Appl. Packag. Res.
**2008**, 2, 227–238. [Google Scholar] - Fadiji, T.; Coetzee, C.J.; Opara, U.L. Compression strength of ventilated corrugated paperboard packages: Numerical modelling, experimental validation and effects of vent geometric design. Biosyst. Eng.
**2016**, 151, 231–247. [Google Scholar] [CrossRef] - Fadiji, T.; Ambaw, A.; Coetzee, C.J.; Berry, T.M.; Opara, U.L. Application of the finite element analysis to predict the mechanical strength of ventilated corrugated paperboard packaging for handling fresh produce. Biosyst. Eng.
**2018**, 174, 260–281. [Google Scholar] [CrossRef] - Heimerl, G.J. Determination of plate compressive strengths. In National Advisory Committee for Aeronautics; Technical Note Number 1480; National Advisory Committee for Aeronautics (NACA): Washington, DC, USA, 1947. [Google Scholar]
- FEMat Systems. Available online: http://www.fematsystems.pl/en/systems/bct/ (accessed on 26 March 2020).
- Ventsel, E.; Krauthammer, T. Thin Plates and Shells. Theory, Analysis, and Applications; Marcel Dekker, Inc.: New York, NY, USA; Basel, Switzerland, 2001. [Google Scholar]
- Norstrand, T. On buckling loads for edge-loaded orthotropic plates including transverse shear. Comp. Struct.
**2004**, 65, 1–6. [Google Scholar] [CrossRef] - Wang, C.M.; Wang, C.Y.; Reddy, J.N. Exact Solutions for Buckling of Structural Members; CRC Press: Boca Ratton, FL, USA, 2005. [Google Scholar]
- Zienkiewicz, O.C.; Taylor, R.L.; Papadopoulus, P.; Onate, E. Plate bending elements with discrete constraints: New Triangular Elements. Comput. Struct.
**1990**, 35, 505–522. [Google Scholar] [CrossRef] - Zienkiewicz, O.C.; Taylor, R.L. The Finite Element Method for Solid and Structural Mechanics, 6th ed.; Butterworth-Heinemann: Oxford, UK, 2005. [Google Scholar]
- Onate, E.; Castro, J. Derivation of plate elements based on assumed shear strain fields. In Recent Advances on Computational Structural Mechanics; Ladeveze, P., Zienkiewicz, O.C., Eds.; Elsevier Pub: Amsterdam, The Netherlands, 1991. [Google Scholar]
- Buchanan, J.S.; Draper, J.; Teague, G.W. Combined board characteristics that determine box performance. Paperboard Packag.
**1964**, 49, 74–85. [Google Scholar] - Shick, P.E.; Chari, N.C.S. Top-to-bottom compression for double wall corrugated boxes. Tappi J.
**1965**, 48, 423–430. [Google Scholar] - Wolf, M. New equation helps pin down box specifications. Package Eng.
**1972**, 17, 66–67. [Google Scholar] - FEMat Systems. Available online: http://www.fematsystems.pl/en/systems/bse/ (accessed on 26 September 2020).
- Garbowski, T.; Borysiewicz, A. The stability of corrugated board packages. Pol. Pap. Rev.
**2014**, 70, 452–458. (In Polish) [Google Scholar] - Hughes, T.R.J. The finite element method. In Linear Static and Dynamic Analysis; Prentice Hall: Upper Saddle River, NJ, USA, 1987. [Google Scholar]
- Onate, E. Structural Analysis with the Finite Element Method. Vol. 1: Basis and Solids; Springer-CIMNE: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Onate, E.; Zienkiewicz, O.C.; Suarez, B.; Taylor, R.L. A general methodology for deriving shear constrained Reissner-Mindlin plate elements. Int. J. Numer. Meth. Eng.
**1992**, 33, 345–367. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) A single panel of width $b$ and height $a$ separated from cardboard packaging as a supported plate under compression; (

**b**) cardboard packaging in press during box compression testing; (

**c**) box compression press used in laboratory tests [32].

**Figure 2.**(

**a**) triangular mesh on a single panel with circular opening; (

**b**) eighteen degrees of freedom triangular Reissner-Mindlin plate FE; (

**c**) isoperimetric formulation of 6-node triangle FE.

**Figure 3.**Designs of boxes without upper flaps with openings considered in the study—modifications of FEFCO F200 with box height of $300$ mm: (

**a**) O1L1S, (

**b**) O2L1S, (

**c**) O3L1S and box height of $200$ mm (

**d**) O1B1S, (

**e**) O2B1S, and (

**f**) O3B1S.

**Figure 4.**Designs of flap boxes with openings considered in the study—modifications of FEFCO F201 with box height of $300$ mm: (

**a**) O1L1S, (

**b**) O2L1S, (

**c**) O3L1S, (

**d**), O3B1S1a (

**e**) O3B1S1b, (

**f**) O3B1S1c, (

**g**) O3B1S1d, (

**h**) O3L1S1a, (

**i**) O3L1S1b, (

**j**) O3L1S1c, (

**k**) O3L1S1d, (

**l**) O3L1S1e and with box height of $200$ mm: (

**m**) O1B1S, (

**n**) O2B1S, and (

**o**) O3B1S.

**Figure 5.**Example measurements from a box compression test machine [32] for different boxes and holes: (

**a**) F0200—O1B1S, (

**b**) F201—O3B1S1c and (

**c**) F201O3L1S1d.

**Figure 6.**Measured box compression strengths (black line), computed by a method proposed with typical (blue line) and optimal (red line) parameters of $k$ and$r$, and McKee formula (dashed magenta line) for FEFCO type boxes of (

**a**) F200 and (

**b**) F201.

**Figure 7.**Contour plot of mean errors [%] obtained for presented method, see Equation (39), and different values of $k$ and $r$.

**Table 1.**Experimental data for FEFCO F200 boxes for different: opening type, box height, cardboard quality and box compression strength.

No. | Opening Type | Box Height [mm] | Cardboard Quality | Box Strength [N] |
---|---|---|---|---|

1 | O3B1S | 200 | 3E350-3 | 901.4 |

2 | O2B1S | 200 | 3E350-3 | 742.3 |

3 | O2B1S | 200 | 3B400A1-1 | 1777.9 |

4 | O1B1S | 200 | 3B400A1-1 | 1842.7 |

5 | O3B1S | 200 | 3B400A1-1 | 2101.6 |

6 | O1L1S | 300 | 3B400A1-3 | 1819.4 |

7 | O2L1S | 300 | 3B400A1-3 | 1903.0 |

8 | O3L1S | 300 | 3B400A1-3 | 2184.3 |

**Table 2.**Experimental data for FEFCO F201 boxes for different: opening type, box height, cardboard quality and box compression strength.

No. | Opening Type | Box Height [mm] | Cardboard Quality | Box Strength [N] |
---|---|---|---|---|

9 | O1B1S | 200 | 3E350-3 | 694.3 |

10 | O3B1S | 200 | 3E350-5 | 783.8 |

11 | O1L1S | 300 | 3E350-5 | 699.3 |

12 | O3L1S | 300 | 3E350-5 | 842.9 |

13 | O2B1S | 200 | 3E350-5 | 697.5 |

14 | O3B1S | 200 | 3E380A2-1 | 944.6 |

15 | O2L1S | 300 | 3E380A2-1 | 935.7 |

16 | O3L1S | 300 | 3E380A2-1 | 1085.3 |

17 | O1L1S | 300 | 3E380A2-2 | 854.8 |

18 | O1B1S | 200 | 3B400A1-1 | 1630.0 |

19 | O3B1S | 200 | 3B400A1-1 | 1902.9 |

20 | O2B1S | 200 | 3B400A1-1 | 1629.0 |

21 | O1L1S | 300 | 3B400A1-3 | 1647.5 |

22 | O2L1S | 300 | 3B400A1-3 | 1701.9 |

23 | O3L1S | 300 | 3B400A1-3 | 2133.6 |

24 | O3B1S1a | 300 | 3B480-1 | 1841.1 |

25 | O3B1S1b | 300 | 3B480-1 | 1717.9 |

26 | O3B1S1c | 300 | 3B480-1 | 1606.6 |

27 | O3L1S1e | 300 | 3B480-1 | 1362.9 |

28 | O3L1S1b | 300 | 3B480-1 | 1743.5 |

29 | O3L1S1c | 300 | 3B480-1 | 1772.4 |

30 | O3L1S1d | 300 | 3B480-1 | 1591.3 |

31 | O3L1S1a | 300 | 3B480-1 | 1782.3 |

No. | Cardboard Quality | Thickness [mm] | ECT | ${\mathit{D}}_{11}[\mathbf{N}/\mathbf{mm}]$ | ${\mathit{D}}_{22}[\mathbf{N}/\mathbf{mm}]$ | ${\mathit{D}}_{33}[\mathbf{N}/\mathbf{mm}]$ | ${\mathit{R}}_{44}[\mathbf{N}/\mathbf{mm}]$ | ${\mathit{R}}_{55}[\mathbf{N}/\mathbf{mm}]$ |
---|---|---|---|---|---|---|---|---|

1 | 3B400A1-1 | 2.82 | 5.79 | 3484.8 | 1789.8 | 2262.1 | 7.73 | 12.78 |

2 | 3B400A1-3 | 2.80 | 5.50 | 3443.5 | 1565.5 | 2115.5 | 6.09 | 11.30 |

3 | 3B480-1 | 2.82 | 5.29 | 3491.2 | 1820.1 | 2359.9 | 5.66 | 12.40 |

4 | 3E350-3 | 1.49 | 3.96 | 958.9 | 431.5 | 870.2 | 2.49 | 2.79 |

5 | 3E350-5 | 1.49 | 4.68 | 878.6 | 376.9 | 904.0 | 2.86 | 2.97 |

6 | 3E380A2-1 | 1.59 | 5.41 | 1272.1 | 505.4 | 1084.5 | 3.83 | 3.86 |

8 | 3E380A2-2 | 1.52 | 5.31 | 1042.9 | 445.1 | 963.2 | 3.39 | 3.70 |

**Table 4.**Mean error of McKee formula and the method proposed for typical and optimal set of $k$ and $r$ parameters.

Case | k [—] | r [—] | Mean Error [%] |
---|---|---|---|

McKee formula | 0.4215 | 0.746 | 15.5 |

method proposed (typical parameters) | 0.500 | 0.750 | 14.9 |

method proposed (optimal parameters) | 0.755 | 0.435 | 6.5 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Garbowski, T.; Gajewski, T.; Grabski, J.K.
Estimation of the Compressive Strength of Corrugated Cardboard Boxes with Various Openings. *Energies* **2021**, *14*, 155.
https://doi.org/10.3390/en14010155

**AMA Style**

Garbowski T, Gajewski T, Grabski JK.
Estimation of the Compressive Strength of Corrugated Cardboard Boxes with Various Openings. *Energies*. 2021; 14(1):155.
https://doi.org/10.3390/en14010155

**Chicago/Turabian Style**

Garbowski, Tomasz, Tomasz Gajewski, and Jakub Krzysztof Grabski.
2021. "Estimation of the Compressive Strength of Corrugated Cardboard Boxes with Various Openings" *Energies* 14, no. 1: 155.
https://doi.org/10.3390/en14010155