Estimation of the Compressive Strength of Corrugated Cardboard Boxes with Various Openings
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 of each element,
- assemble the global stiffness matrix of a whole panel,
- compute nodal forces representing initial loading configuration , i.e., for loading multiplier (one-parameter loading assumed ),
- take boundary conditions into account,
- solve equation to obtain nodal displacements in pre-buckling state:
- extract element displacements (from displacements of the system ) and compute in each element the generalized stresses .
- generate initial geometrical stress matrices for each element and the whole panel ,
- formulate initial buckling problem:
- solve the eigenproblem to determine the pairs , where is a number of degrees of freedom, is th eigenvalue, is 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
3.2. Reduction of the Estimation Error—Optimal Parameters
4. Discussion
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
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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 |
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 | |||||
---|---|---|---|---|---|---|---|---|
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 |
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 |
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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
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 StyleGarbowski, 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