114
5.4 Shear Lag Model
As shown previously, HROM cannot account for the interface. But shear lag
methods have been used in composites to analyze fiber aspect ratio and interface effects
(Nairn 2005). Here we adapted the shear-lag approach
to account for interfacial, strand
length, and strand gap effects. The shear-lag equation for MOE of load in tension parallel
to direction of the grain in all strands (i.e. LVL geometry) is:
m
m
w
w
t
w
x
V
E
V
E
AR
D
E
+
=
)
,
(
η
(5.1)
where
D
t
is interfacial stiffness parameter;
V
w
and
V
m
are volume of wood
and glue;
E
w
and
E
m
are elastic moduli of wood and glue;
η
is an efficiency factor that describes the
ability of the interface to transfer load into the reinforcing phase. The efficiency factor
depends on interfacial D
t
and aspect ratio. The efficiency factor can be modeled by stress
transfer analysis, such as shear-lag analysis, resulting in (Hull and Clyne 1996):
γ
γ
β
β
η
tanh
1
2
2
tanh
1
−
=
−
=
r
l
t
t
l
t
w
(5.2)
where
2
)
(
*
AR
β
γ
=
,
t
β
β =
*
and
t
l
AR
=
Here,
β
is the
shear-lag parameter; and
l
and
t
is length and thickness of the strands. The
preferred shear-lag parameter has the form of (Nairn 2005):
+
+
+
=
t
m
m
w
w
w
m
w
m
w
w
m
m
D
t
G
V
G
V
V
V
E
E
V
E
V
E
*
2
*
1
3
3
β
(5.3)
where
G
w
and
G
m
are the shear moduli for wood and glue and
t
*
is strand-to-strand
spacing. Early shear-lag analysis always assumed a “perfect” interface implying
continuous displacement between phases (Cox 1952). Thus
the early models could not
115
account for interfacial stiffness effects and the modulus depended only on the mechanical
and geometric properties of the phases. The new shear-lag parameter in Equation 5.3
adds interfacial stiffness effect through the term in D
t
. Canceling terms and rearranging,
γ
becomes:
4
*
1
3
3
1
1
2
*
2
l
D
t
G
V
G
V
V
E
V
E
t
m
m
w
w
w
w
m
m
+
+
+
=
γ
(5.4)
To account for the gap spacing we used:
L
G
G
V
m
+
=
and
L
G
L
V
w
+
=
(5.5)
where G and L is the length and gap spacing. These volume fractions were derived from
the geometry in Figure 5.7. With shear lag model and length and gap values, Eq 5.1 gives
MOE of LVL specimens for face layers of OSB. The
set MOE for OSB, the HROM can
be modified by using E
x
in eq 5.1 for surface layers initial modulus and use prior
approach for core layer. For OSB with 50% core layer (as in all simulation), the new
modulus is
)
1
(
2
C
E
E
E
C
x
−
+
=
(5.6)
t
Matrix
Fiber
L
G/2
G/2
Figure 5.7. Embedded fiber (strand) into matrix (glue).
As shows in Figure 5.8, the results of MOE as the function of aspect ratio (strand
length over strand thickness) in
OSB for different D
t
. Here the thickness of the strand is
116
fixed and is equal to 0.8mm and MOE of individual strands is E
w
=9936 MPa, the MOE
of glue is 1500 MPa, shear modulus of wood is 743MPa, shear modulus of glue is
500MPa, volume of wood is 0.83% and volume of glue is 0.17% with
strand length of
150 mm and strand gap of 30 mm. Figure 5.8 are the results of equation 5.5 with
different D
t
. As aspect ratio increases, modulus of the composites increase and level off.
For optimum of stress transfer efficiency, the AR is around 200.
Figure 5.8. The results of MOE as the function of aspect ratio (strand length over strand
thickness) in
OSB for different
D
t
(where D
t
=100 [1/D
t
=0.01 is 25% glue coverage; D
t
=
60 [1/D
t
=0.016] is 1% glue coverage).
Figure 5.9 is shows MOE as a function of AR at constant gap size. The results are
compared to HROM modified by shear lag analysis for 1/D
t
=0.05. MOE increased as AR
increased. The numerical results are lower than the analytical results of combine HROM
with equation 5.5. Numerical values of MOE approached to analytical results as AR
300
1300
2300
3300
4300
0
50
100
150
200
250
300
350
400
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