Materials 2022 , 15 , 663
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Figure7. Corrugated board bending–compressing the layers of lower fluting.
Figure8. Corrugated board bending–compressing the layer of higher fluting.
In a four-point bending test, only the bending moment occurs in the center of the specimen, so the model simplifies to pure bending. In this model, the moment is balanced by the normal forces P i acting in the liners on the arms z i (see Figure 7) with respect to the neutral axis z 0 : z 0 = ∑ N i z i t i b ∑ N i t i b . (7) The starting point for determining the bending stiffness is the kinematic excitation in the form of rotation φ (see Figure 7), which causes elongation or contraction δ i of liners (see Figure 8). By taking a small value δ 1 (e.g., 10 − 2 mm) and using the known values of z i , the remaining values δ i can be determined (see Figure 8) and finally the rotation angle φ can be obtained: φ = atan δ 1 z 1 . (8) By solving the integrals in Equation (5), it is possible to determine the longitudinal deflection in liners under compression or tension:
2
6 P i L i f
P i L i E i t i b
i
.
(9)
δ i =
+
3 i b
E i t
For the known deflection δ i , the compressive force P i inthe i -th liner can be determined:
E i t i δ i b
,
(10)
P i =
L i 1 + 6 f
2 i
2 i t −
while the tensile force P i (for f i = 0) is:
E i t i δ i b L i
.
(11)
P i =
The i -th bending moment is:
N ∑ i = 1
P i z i ,
(12)
M i =
and the bending stiffness can be calculated as the sum of the integrals from the formula:
N ∑
N ∑ i = 1
L i
M i φ
M i L i φ
i
.
(13)
dx =
EI =
0
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