71 Where in mm is the veneer length and r
f
in mm the final log radius. The integration domain depends on both the final and the initial log radius.
Indeed, logs are not peeled up to the log center, there remains a peeler core of r
f
radius. Initial and final log radii were chosen to be respectively 280 and 60 mm.
Affectation of mechanical properties
Using the above equations Equation 3 to 6, it is then possible to determine the wood’s modulus of elasticity along the peeling sheet by composing them
Equation 9 to obtain parametric coordinates on segmented ring basis : =
�
� , ,
�
=
�
� �
…………………..9 Where l in mm is the veneer length and E in MPa is the modulus of elasticity.
These values are parametrized by the segmented ring C
a
5.2.2 Calculation of Flatwise and Edgewise modulus of elasticity
Since each layer in the LVL sample has a different modulus of elasticity along its length, it is necessary to compute a specific modulus of elasticity for
each LVL samples. The value of SMOE depends on the loading direction, because the impact of each layer on the mechanical behavior of the LVL is
different in flatwise or edgewise configuration. In the model, SMOE of LVL is considered higher than that of SMOE solid wood. This is a prerogative in
European standard EN 14374 concerning LVL panels.
5.2.2.1 Edgewise
In an edgewise bending test, each layer can be considered as independent. Thus, according to Equation 10 the specific modulus of elasticity is determined by
calculating the sum of the SMOE of each layer, as in the following Equation �� � =
∑ �
� � ��
�= � �
………………………10 Where : SMO
�� = Specific modulus of elasticity of the i
th
layer � ; ��= Local
inertia of the i
th
layer
4
; � = �
� ℎ ℎ �
4
; n
l
= number of layers and ρt = density of the i
th
layer kg m
-3
5.2.2.2 Flatwise
In a flatwise bending test, the location of layers with different mechanical properties has more influence on global properties than in an edgewise bending
test, due to a different stress rate between surface and core layers. To calculate the specific modulus of elasticity in the flatwise direction, the
position of neutral axis has to be determined thanks to the following Equation 11 Figure 45:
=
∑
,�
�
�
�
� ��
�=
∑ �
�
�
� ��
�=
……………………11
72 In accordance with Figure 45: Z
= Distance from the neutral axis to an arbitrary reference
; �0= Distance from the i
th
layer netral axis to an arbitrary reference
; ��= Distance from the i
th
layer netral axis to the global neutral axis
Figure 45 Geometrical and mechanical properties of LVL in a specimen cross- section
Thus, the distance between each layer’s neutral axis and the global neutral axis, ��
is deduced by Equation 12: �
�
= �
,�
− � ………. 12 Then the specific modulus of elasticity can be determined by the following
Equation 13: �� � =
∑ �
� �
+�
�
�
� ��
�= � �
……………………13 Where: SMO
�� = Specific modulus of elasticity of the i
th
layer � ; �� =
Inertia of the i
th
layer
4
; � = �
� ℎ ℎ �
4
; �� =
Section of the i
th
layer
2
; �� = Distance from the i
th
layer neutral axis to the neutral axis
; and ρt = density of the i
th
layer kg m
-3
5.2.3 Stochastic approach
Since in virtual peeling process the layout is based on a randomized assembly process, the process is repeated a thousand times to identify and enhance
a tendency. Indeed, the peeling process always gives the same veneer for a given thickness value, but there are several combinations of veneer arrangements after
the primary cutting. These combinations are high due to the possibility to turn each veneer upside-down. The veneer position influences the flatwise SMOE
whereas it does not impact the edgewise SMOE. Consequently, a great number of processes have to be performed to obtain results close to experimental conditions.
The number of possible combinations for a single
layer LVL is given by Equation 14 :
=
� ��
…………… 14 Where,
is the number of possible combinations and is the number of
layers. Symmetric cases are considered in the number of combinations; therefore the number is divided by two.
