Numerical and Experimental Study of Five-Layer Non-Symmetrical Paperboard Panel Stiffness
Abstract
:1. Introduction
2. Problem Formulation
2.1. Object of Analysis
2.2. Material Properties
2.3. Test Stand
2.4. Bending Stiffness
2.5. FE Models
3. Results
3.1. Stiffness Analysis
3.2. Deformations Maps
3.3. Full Experimental Curves
4. Conclusions
- The assumption of nominal thicknesses of the paper provides the lower stiffnesses in reference to those obtained in the experiment. Moreover, implementations of perfect structures with corrected thicknesses of paperboard also do not correlate well.
- The numerical results for perfect structures do not show the differences in BS between compressed B-wave and compressed E-wave.
- The measure results indicate a slight difference between the values of BS obtained at different signs of moments when bending paperboard in the machine direction. When an upper layer glued to the E-wave is compressed, the BS is higher in comparison to the BS of a compressed layer at the B-wave. This can be explained by a local deflection between joints (connection between waves and flat layers) that is greater if wave pitches are greater as well.
- The course of the curve seen in Figure 13 is characterized by a mild change between the initial part of chart (where compression dominates) and the field where buckling of the compressed plate occurs.
- We showed that numerically analysing of the behaviour of multilayer paperboard panels by including the aforementioned imperfections can reveal distinct differences in the BS for different signs of moment (in the case of compressed B-wave and E-wave). The lack of visible border in the results of the numerical simulation might just be caused by preliminary deflections.
- Depending on the analysed variant and arrangement of the panel, the mean values from the experiment were slightly higher than in the simulation but the discrepancies ranged from 3% up to almost 33%, at most (based on variant: CORRECT_GEOM_2). Firstly, it can be justified because in general, a realistic shape of paperboard can differ in the details from the numerical (idealized) model. Secondly, in the present simulation, the influence of the adhesive connecting all the layers of paper was not taken into account. This effect cannot be so significant, but the mentioned factors might have had an influence on the final scores.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Model Number | Layer Number | Thickness (mm) | EMD (GPa) | ECD (GPa) | nMD (-) | nCD (-) | GMD-CD (GPa) |
---|---|---|---|---|---|---|---|
Model_1 | 1-flat (t1) | 0.142 | 5.70 | 2.24 | 0.118 | 0.3 | 2.06 |
E-wave (tE) | 0.164 | 5.73 | 1.56 | 0.082 | 0.3 | 1.73 | |
3-flat (t2) | 0.126 | 6.46 | 1.97 | 0.091 | 0.3 | 2.06 | |
B-wave (tB) | 0.164 | 5.73 | 1.56 | 0.082 | 0.3 | 1.73 | |
5-flat (t3) | 0.146 | 5.65 | 2.47 | 0.131 | 0.3 | 2.16 | |
Model_2 | 1-flat (t1) | 0.185 | 6.69 | 2.50 | 0.112 | 0.3 | 2.36 |
E-wave (tE) | 0.227 | 5.49 | 1.87 | 0.102 | 0.3 | 1.85 | |
3-flat (t2) | 0.177 | 5.20 | 1.80 | 0.104 | 0.3 | 1.77 | |
B-wave (tB) | 0.199 | 6.64 | 1.95 | 0.088 | 0.3 | 2.08 | |
5-flat (t3) | 0.186 | 5.52 | 1.86 | 0.101 | 0.3 | 1.85 | |
Model_3 | 1-flat (t1) | 0.142 | 5.7 | 2.24 | 0.118 | 0.3 | 2.06 |
E-wave (tE) | 0.199 | 6.64 | 1.95 | 0.088 | 0.3 | 2.08 | |
3-flat (t2) | 0.126 | 6.46 | 1.97 | 0.091 | 0.3 | 2.06 | |
B-wave (tB) | 0.139 | 5.72 | 2.13 | 0.112 | 0.3 | 2.02 | |
5-flat (t3) | 0.146 | 5.65 | 2.47 | 0.131 | 0.3 | 2.16 | |
Model_4 | 1-flat (t1) | 0.142 | 5.70 | 2.24 | 0.118 | 0.3 | 2.06 |
E-wave (tE) | 0.177 | 5.20 | 1.80 | 0.104 | 0.3 | 1.77 | |
3-flat (t2) | 0.139 | 5.72 | 2.13 | 0.112 | 0.3 | 2.02 | |
B-wave (tB) | 0.177 | 5.20 | 1.80 | 0.104 | 0.3 | 1.77 | |
5-flat (t3) | 0.146 | 5.65 | 2.47 | 0.131 | 0.3 | 2.16 | |
Model_5 | 1-flat (t1) | 0.185 | 6.69 | 2.50 | 0.112 | 0.3 | 2.36 |
E-wave (tE) | 0.199 | 6.64 | 1.95 | 0.088 | 0.3 | 2.08 | |
3-flat (t2) | 0.177 | 5.20 | 1.80 | 0.104 | 0.3 | 1.77 | |
B-wave (tB) | 0.199 | 6.64 | 1.95 | 0.088 | 0.3 | 2.08 | |
5-flat (t3) | 0.186 | 5.52 | 1.86 | 0.101 | 0.3 | 1.85 | |
Model_6 | 1-flat (t1) | 0.142 | 5.70 | 2.24 | 0.118 | 0.3 | 2.06 |
E-wave (tE) | 0.177 | 5.20 | 1.80 | 0.104 | 0.3 | 1.77 | |
3-flat (t2) | 0.164 | 5.73 | 1.56 | 0.082 | 0.3 | 1.73 | |
B-wave (tB) | 0.177 | 5.20 | 1.80 | 0.104 | 0.3 | 1.77 | |
5-flat (t3) | 0.186 | 5.52 | 1.86 | 0.101 | 0.3 | 1.85 |
Variant | EXP | FEM_1 NOMINAL_GEOM_1 (Nm) | FEM_1 CORRECT_GEOM_1 (Nm) | FEM_2 CORRECT_GEOM_2 (Nm) | FEM_1 NOMINAL Decrease (+)/ Increase (−) with Respect Mean Value (%) | FEM_1 CORRECT Decrease (+)/ Increase (−) with Respect Mean Value (%) | FEM_2 CORRECT Decrease (+)/ Increase (−) with Respect Mean Value (%) | |||
---|---|---|---|---|---|---|---|---|---|---|
1 (Nm) | 2 (Nm) | 3 (Nm) | Mean Value (Nm) | |||||||
Model_1_B | 8.50 | 8.20 | 8.25 | 8.32 | 5.80 | 6.98 | 7.62 | 30.29 | 16.11 | 8.41 |
Model_1_E | 8.43 | 8.48 | 8.50 | 8.47 | 5.70 | 6.89 | 7.58 | 32.70 | 18.65 | 10.51 |
Model_2_B | 11.10 | 11.00 | 10.80 | 10.97 | 7.47 | 9.04 | 9.88 | 31.91 | 17.59 | 9.94 |
Model_2_E | 11.45 | 11.50 | 11.80 | 11.58 | 7.43 | 8.96 | 9.81 | 35.84 | 22.63 | 15.28 |
Model_3_B | 8.67 | 6.96 | 6.13 | 7.25 | 5.76 | 7.01 | 7.61 | 20.55 | 3.31 | −4.97 |
Model_3_E | 9.65 | 9.40 | 9.45 | 9.50 | 5.72 | 6.89 | 7.53 | 39.79 | 27.47 | 20.74 |
Model_4_B | 9.01 | 9.17 | 9.13 | 9.10 | 5.68 | 6.88 | 7.53 | 37.58 | 24.40 | 17.25 |
Model_4_E | 10.95 | 11.10 | 11.25 | 11.10 | 5.63 | 6.80 | 7.45 | 49.28 | 38.74 | 32.88 |
Model_5_B | 11.47 | 11.51 | 11.43 | 11.46 | 7.88 | 9.52 | 10.42 | 31.24 | 16.93 | 9.08 |
Model_5_E | 12.70 | 13.25 | 12.95 | 12.97 | 7.83 | 9.46 | 10.37 | 39.63 | 27.06 | 20.05 |
Model_6_B | 8.20 | 8.10 | 8.30 | 8.20 | 6.39 | 7.73 | 8.45 | 22.07 | 5.73 | −3.05 |
Model_6_E | 9.25 | 9.00 | 9.10 | 9.12 | 6.35 | 7.67 | 8.40 | 30.37 | 15.90 | 7.89 |
Force F (N) | FEM_1 CORRECT_GEOM_1 | FEM_2 CORRECT_GEOM_2 |
---|---|---|
1 | ||
2 | ||
4 | ||
8 |
Force F (N) | FEM_1 CORRECT_GEOM_1 | FEM_2 CORRECT_GEOM_2 |
---|---|---|
1 | ||
2 | ||
4 | ||
8 |
Point/ Force (N) | View | Magnified View of Bent Paperboard |
---|---|---|
B1 (2) | ||
B2 (3) | ||
B3 (5) | ||
B4 (6) | ||
B5 (7) |
Point/ Force (N) | View | Magnified View of Paperboard |
---|---|---|
E1 (2) | ||
E2 (3) | ||
E3 (5) | ||
E4 (6) | ||
E5 (10) | ||
E6 (16.5) |
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Czechowski, L.; Kmita-Fudalej, G.; Szewczyk, W.; Gralewski, J.; Bienkowska, M. Numerical and Experimental Study of Five-Layer Non-Symmetrical Paperboard Panel Stiffness. Materials 2021, 14, 7453. https://doi.org/10.3390/ma14237453
Czechowski L, Kmita-Fudalej G, Szewczyk W, Gralewski J, Bienkowska M. Numerical and Experimental Study of Five-Layer Non-Symmetrical Paperboard Panel Stiffness. Materials. 2021; 14(23):7453. https://doi.org/10.3390/ma14237453
Chicago/Turabian StyleCzechowski, Leszek, Gabriela Kmita-Fudalej, Włodzimierz Szewczyk, Jacek Gralewski, and Maria Bienkowska. 2021. "Numerical and Experimental Study of Five-Layer Non-Symmetrical Paperboard Panel Stiffness" Materials 14, no. 23: 7453. https://doi.org/10.3390/ma14237453