Materials 2022 , 15 , 663
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In Figure 11 it can be clearly noticed that for the imperfection value at the level of 0.1% of the initial length of the compressed segments (which corresponds to k equal to 3), not only the difference between the EB and BE configurations is not noticeable, but also the difference between the BS values for both EB and EB do not differ from the reference value (dashed lines) computed while using the theoretical model (see Equation (15)). The difference between the bending stiffness in the case of EB and BE increases with the augmentation of the imperfection coefficient and for the value k = 2 (i.e., imperfection equals to L i · 10 − 2 ) it is between 12% and about 22% (see Figure 12). In this work the assumed imperfection coefficient is 2.3, which corresponds to the initial imperfections in the compressed elements at the level of 0.5% of the initial length of these segments. Table 5 summarizes all calculated and literature values of bending stiffness for six examples of five-layer corrugated board. It is clearly seen than in just two cases the theoretical BS is higher than the experimentally measured BS. It can be evidently noticed that only in two cases (Board 2 and Board 6) the theoretical BS is higher than experimentally measured BS. This is an alarming observation, because in the case of real structures made of corrugated board, the cross-section is rarely ideal (usually the corrugated board is slightly crushed [67,68]), which means that the measured bending stiffness values should rather be lower than theoretical. Not only the theoretical values of BS are lower than those measured experimentally. Virtually all the results presented in Table 5 follow a similar trend, both the results obtained with the use of analytical and numerical models, including the results from the literature (column 6) [29]. Due to this observation, the results of experimental research presented in [29] may contain some errors or are incorrectly ordered. Despite these doubts the results obtained while using the analytical model are very good for Examples 2 and 6 (marked in Table 6), for other Examples the results are not as good but still better than results presented in [29] (see Table 6). The mean absolute error generated by the analytical model is 11.7% for all cases while the mean absolute error of the results presented in [29] is 16.4%.
Table6. Percentage error between BS measured experimentally and computed BS.
FEM[29]
Analytical
Face-up
Title1
(%) 9.18
(%)
EB BE EB BE EB BE EB BE EB BE EB BE
16.69
Board1
11.74 11.03 18.04 26.16 20.85 48.99 4.73
8.04 1.61 0.60 1.40
Board2
Board3
21.02 25.69 39.10
Board4
9.98
5.23
Board5
25.07
12.59
2.95 8.57
7.45 1.62
Board6
Due to the relatively large discrepancies between the calculated and measured values of bending stiffness, and due to the suspected measurement error or incorrect compilation of results in [29], the sensitivity analysis of the analytical model was also carried out in this study. The graphs shown in Figure 13 clearly indicate that both the EB and BE models have the greatest sensitivity to the change in the stiffness modulus and thickness of the flat inner and outer layers (i.e., Liner-1 and Liner-3). The sensitivity of BS to changes in the height of the corrugated layer B, h 2 , is similarly high. Thus, even a small change of these parameters (just a few percent), can dramatically change the computational value of bending stiffness of the corrugated board.
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