Energies 2021 , 14 , 1095
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Figure 5. Box compression strength obtained within different approach for shelf-ready boxes considered in the study with various perforations, i.e., 25 × 75, 50 × 50 and75 × 25.
The comparison between the measured and estimated values of ultimate loads of the boxes with perforations may be represented by the mean error. It was computed for all three cases according to the following formula: error = BCT exp − BCT est BCT exp × 100% , (30) where BCT exp is an experimental value of the box compression strength averaged for four or five samples with peculiar perforation, and BCT est is its counterpart computed by the analytical–numerical approach proposed in this paper. The average error, while using the analytical–numerical approach for the boxes analyzed in this paper was 3.5%. A similar comparison with the experimental values may be considered for the esti- mation via McKee formula—the approach, which is easy, and thus in common use in the packaging industry. The results were presented by red bars in Figure 5, see Equation (30). Similarly, the mean error for the McKee formula was computed, it was equal to 19.5%. The summary of the mean errors is presented in Table 4.
Table4. The mean error for estimations of McKee formula and the method proposed.
Case
k (-)
r (-)
Mean Error (%)
McKee formula method proposed
0.4215
0.746
19.5
0.4
0.75
3.5
4. Discussion The analytical–numerical method proposed here shows the correct trend while com- paring it with the experimental results. Namely, with decreasing stiffness of the perforation, the strength of the box also decreases. Experimental results show that, if the box with 50 × 50 perforation would be the reference, the strength increase for 25 × 75 is 2.0%, while for 75 × 25 the decrease equals 6.3%. For the analytical–numerical approach proposed, the counterpart values are 8.4% and 6.5%, respectively. For the McKee formula, the values are constant for all three boxes. Moreover, note that the average error computed for the method proposed here was more than five times smaller than for the McKee formula, i.e., 3.5% vs. 19.5%, respectively. This is the most important conclusion from our study. The proposed approach is much more accurate than a simplified method. In the study presented, all designs of the boxes, due to the same overall geometry, had 1000 mm of the in-plane circumference. This caused the strength of the boxes to be the same for all three box designs with perforations, while considering the McKee formula approach, see Figure 5. The McKee formula does not include the information about perforations in the box design.
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