Materials 2022 , 15 , 663
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Figure5. Compressed and stretched segments of a corrugated board cross-section during bending.
For a compressed i -th segment (see Figure 6) with a geometric imperfection, the longi- tudinal shortening of i -th segment, δ i can be computed by the classical equation: δ i = L 0 N i N i E i A i dx + L 0 M i M i E i I i dx , (4) where: N i = P i is the normal force; N i = 1 is a virtual normal force; A i = bt i is the cross- section area (with b —segment width and t —segment thickness); I i = bt 3 i /12 is the cross- section moment of inertia; M i = P i w i ( x ) is the bending moment; and M i = 1 w i ( x ) is a virtual bending moment. By inserting all the relationships described above into Equation (4) and taking all the constant values out of the integral, the longitudinal deflection takes the form: δ i = P i E i t i b L i 0 dx + 12 P i E i t 3 i b L i 0 ( w i ( x )) 2 dx , (5) where the deflection function w i can be described by e.g., the sine function: w i ( x )= f i sin 2 π x L i , (6) with the maximum deflection f i in the middle of the element span L i (see Figure 6), which was assumed here as a small fraction of the element length: f i = L i · 10 − k while the value of k is a quantity assumed between 2 and 4.
Figure6. Compression of a single segment with a geometric imperfection.
When measuring the bending stiffness of five-layer corrugated board one can place a corrugated board sample with the E wave facing upwards or vice versa. As the result, in one case, two liners are compressed on the E wave side (see Figure 7) while in the other case, a single liner on the B wave side (see Figure 8) is compressed. In a case where two liners on the E-flute side are compressed, a higher value of bending stiffness is usually obtained. This is because larger number and shorter (therefore, less slender) segments are compressed and even when imperfections are present, the effect is less pronounced.
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