7 90
M. Nygårds
Δ
l 3 3
The aim of this work would be to implement analytical models that can be used to predict the performance of paper- boards. It will be based on how it is constructed in the thick- ness direction using the properties of three plies in the paper- board structures. This will be done by utilizing laminate theory Fellers and Carlsson [12] and Fellers [13], and Timoshenko beam bending [14] to construct a model that can predict the maximal bending moment for paperboards based on paper- making variables such as density, fiber orientation, and fiber length. Theoretical background In the analysis, we will assume that in-plane tensile and out- of-plane shear stresses will contribute to failure during fold- ing. With laminate theory, it is possible to calculate how a multilayer structure respond to external forces, which helps us to develop a ZD structure with accurate bending stiffness.
Δ F b
(3)
S b =
.
If the angle, θ , is measured in degrees the bending stiff- ness is given by
Δ
Δ F b
60 l 2
(4)
S b =
.
The two-point bending is commonly used as a quality control in industry. Then w = 38 mm wide samples are bent, and L = 50 mm is used. For this test, the force, P Sb , that is read on the testing device is often interpreted as the bending moment of the paperboard
15 wS b 60 L 2
(5)
P Sb =
.
Multi-ply paperboards consist of several plies, which can represent layers in a laminate. With a multi-ply model, it would be possible to calculate the bending stiffness. The model can be used to make predictions, and hence extend the design space for multi-ply paperboards. The ability to make calculations is beneficial for optimization, since it can give an indication of the choice of raw material, grammage, and ply layout. For multi-ply paperboard the same strategy as for homogenous plies are applied. However, it needs to be accounted for that the plies are not located along the neutral line. Then bending stiffness for a multi-ply structure can then be expressed as
Elastic bending
Bending stiffness of paperboard is normally measured by two- point bending, when the bending arm is L =50 mm [15]. With this configuration the shear stress component is negligible. Bending of a homogenous cantilever beam with a rectangular cross section will be performed with a bending moment M b per unit width. A tensile stress will arise on the convex side of the beam, and compressive stresses on the concave side. The theoretical framework for evaluation of the bending stiffness for multiply paperboards have been derived by Fell- ers and Carlsson [12] and Fellers [13]. By using the definition of bending moment, integrating the elastic stress in the beam and by utilizing strain compatibility the bending moment for a homogenous beam with thickness, t , and rectangular cross section the bending moment becomes
B 2 A
(6)
S b = D −
,
where
A = E k × z k − z k − 1 ,
(7)
1 2 1 3
E k × z E k × z
k − 1 k − 1
h ∕ 2
Et 3 12
S b =
(8)
2 k
− z 2
B =
,
(1)
Ez 2 d z =
.
− h ∕ 2
In the two-point bending setup, one end of the paperboard is clamped, and the other end is loaded by a concentrated load. The bending moment then decreases linearly as the distance x from the clamped end increase. If small deformations are assumed during the bending stiffness measurements, then the maximum deflection δ becomes
(9)
3 k
− z 3
D =
,
where E k is the elastic modulus of ply k , which might be in either MD or CD. The ply coordinates z k (Fig. 9) ( k = 0, 1, 2, …., N ) where N is the number of plies, are calculated as
(10)
z k = z k − 1 + t k
for k = 1,2, … , N ,
F b l 3 3 S b
(2)
=
.
t 2
(11)
for k = 0,
z 0 =−
Then bending stiffness which is evaluated as the steepest slope of a F − δ plot, becomes
where t is the total thickness of the paperboard. The bending stiffness is dependent of the thickness and grammage of the different plies.
1 3
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