Forest Stufee
Forest Stufee
Forest Stufee
1 Research Institute of Wood Industry, Chinese Academy of Forestry, Beijing 100091, China;
tianzhaopeng@caf.ac.cn (Z.T.); xu_junhua@163.com (J.X.); limingyue@caf.ac.cn (M.L.);
zhwang@caf.ac.cn (Z.W.); gongyingchun@caf.ac.cn (Y.G.)
2 Division of Structural Mechanics, Lund University, SE 221 00 Lund, Sweden; erik.serrano@construction.lth.se
* Correspondence: renhq@caf.ac.cn; Tel.: +86-010-6288-9460
Abstract: The density and modulus of elasticity (MOE) distribution can provide information on the
effectiveness of parametric and non-parametric methods in calculating the characteristic value of
MOE. In this study, we aim to determine the optimal distribution model of the actual measured
data of the lumber. We also estimate the lumber’s MOE characteristic value and compare the
difference in density and MOE between natural and planted larch. Approximately 1200 pieces of
dimension lumber of 4 m × 140 mm × 40 mm in size, made from larch and planted larch, were
obtained, tested, and the corresponding standard MOE value was calculated. Results revealed the
3-parameter Weibull distribution to be optimal in fitting the natural and planted larch distributions.
The parametric method proved effective in calculating the characteristic value of both larch groups,
with characteristic MOE values of 9.73 kN/mm2 and 8.84 kN/mm2 , and characteristic density values
Citation: Tian, Z.; Xu, J.; Li, M.;
of 530 kg/m3 and 460 kg/m3 for natural and planted larch, respectively. Moreover, the MOE and
Wang, Z.; Serrano, E.; Gong, Y.; Ren, density values followed grades C40 and C35. Thus, the conclusion is that the parametric method
H. Characteristic Value of the should be used to determine these characteristic values for natural and planted larch.
Modulus of Elasticity (MOE) for
Natural and Planted Larch in Keywords: MOE; planted larch; characteristic value; dimension lumber; probability distribution;
Northeast China. Forests 2021, 12, 883. non-destructive testing
https://doi.org/10.3390/f12070883
Zhong [6,7], Jiang [3], Lou [8], and Wang [9] et al., promoting the utilization of natural larch
in China. However, studies on the full-size mechanical properties of planted larch lumber
are limited, leading to a lack of test data for timber design applications. Thus, in order to
use planted larch to build frame structures and grade the lumber by stress, experimental
tests (particularly those based on full-size lumber) with extensive samples are urgently
required to provide basic data support for the necessary calculations in timber design.
Dimension lumber is the lumber processed to a specified size according to a standard
or code [10]. The modulus of elasticity (MOE) of the dimension lumber is key for the grad-
ing of lumber, the design of wood structures [11], and, in particular, to determine the level
of deflection that meets the serviceability limit states under bending [12]. The MOE of di-
mension lumber is commonly investigated using static test methods (denoted as MOEstatic ).
However, such test procedures are time-consuming, labor-intensive, and can damage the
lumber. Therefore, non-destructive tests (NDTs) have been developed to replace static test
methods and to determine the MOEDynamic of lumber [9]. NDTs are convenient, save time,
and most importantly they do not damage the lumber. Commonly used NDT techniques
include the transverse vibration method, the longitudinal fundamental frequency vibration
(FFV) test method, and the stress wave method [13–15]. The FFV method is commercially
employed for stress grading and developments and can be combined with laser scanning.
Previous work has reported the accurate and robust MOEDynamic measurements via the
FFV method [16–18]. Experimental tests reveal the correlation coefficients (R2 ) between the
dynamic elastic modulus (MOEDynamic ) and static elastic modulus (MOEstatic ) determined
by the FFV and static test method to exceed 0.8 [14,19,20]. Therefore, the MOEstatic can
potentially be replaced by MOEDynamic to grade the lumber and estimate the deflection.
In the current study, we focused on obtaining more data and knowledge on natural and
planted larch for designing and grading the wood and lumber. In particular, the MOEDynamic
and density differences between natural and planted larch lumbers were analyzed and
compared, and we also extensively measured the MOEstatic numbers of natural and planted
larch lumber via the FFV test method. Moreover, the density and MOEDynamic distributions
were evaluated by fitting them with normal, lognormal, and Weibull distribution functions.
The density and MOEDynamic differences were compared between natural and planted larch,
and the characteristic values of the density and MOEDynamic of natural and planted larch
dimension lumber were evaluated based on the density and MOEDynamic distributions. In
the end, parameter and non-parameter methods were used to calculate the characteristic
values, to compare which method was more suitable, and to determine the characteristic
values for natural and planted larch.
(a) (b)
(a) (b)
Figure 2. Test set-up (a) and screenshot of the software to determine the resonance frequency (f m ) (b).
The mean and standard deviation (SD) of the density and MOEDynamic values for
natural and planted larch lumber were determined in R language (Version 3.6.3, R Core
Team). Based on the Three Sigma Rule [22], if the absolute difference between the test value
Figure 2. Test set-up (a) and screenshot of the software to determine the resonance frequency (fm)
(b).
Forests 2021, 12, 883 The mean and standard deviation (SD) of the density and MOEDynamic values for4 of nat-
13
ural and planted larch lumber were determined in R language (Version 3.6.3, R Core
Team). Based on the Three Sigma Rule [22], if the absolute difference between the test value
and mean value of all data was greater than 3 times the standard deviation (|𝑥𝑖 − 𝑥̅ |>3𝜎),
and mean
the test value
value of rejected
was all data was greater than
to minimize 3 times the standard deviation (| xi − x | > 3σ),
errors.
the test value was rejected to minimize errors.
2.3. Testing the Lumber Static Bending MOE
2.3. Testing the Lumber Static Bending MOE
In order to verify the accuracy of the MOEDynamic determined via the FFV test method,
In order to verify the accuracy of the MOE determined via the FFV test method,
80 pieces of dimension lumber were randomlyDynamic selected from the natural larch dimension
80 pieces of dimension lumber were randomly selected from the natural larch dimension
lumber and their static bending elastic modulus (MOEstatic) was tested using the universal
lumber and their static bending elastic modulus (MOEstatic ) was tested using the universal
test machine. Linear regression analysis with the dynamic MOE data (MOEDynamic) was
test machine. Linear regression analysis with the dynamic MOE data (MOEDynamic ) was
then performed
then performed using using the
the MOE
MOEstatic values.
static values.
The static bending MOE(MOE
The static bending MOE (MOEstatic ) of the dimension lumber was tested based on the
static ) of the dimension lumber was tested based on
the ASTM D198-15 Standard Test MethodStatic
ASTM D198-15 Standard Test Method of Tests
of Static of Lumber
Tests of Lumberin Structure Sizes
in Structure [23][23]
Sizes andand the
Chinese standard GB/T 28993-2012 Standard test methods for
the Chinese standard GB/T 28993-2012 Standard test methods for mechanical properties ofmechanical properties of struc-
tural lumber
structural [24].[24].
lumber More specifically,
More specifically,wewe employed
employed the
thethird
thirdpoint
pointflatwise
flatwise bending
bending testtest
method (Figure 3) with a span-to-thickness ratio of 21 and loading
method (Figure 3) with a span-to-thickness ratio of 21 and loading distance at the center distance at the center
of 280
of 280 mmmm for for the
the MOE
MOEstatic tests. The load and deflection were determined by a load cell
static tests. The load and deflection were determined by a load cell
and electronic indicator, respectively,
and electronic indicator, respectively, and and measurement
measurement values values were
were collected
collected with
with aa data
data
logger at a 1 Hz sampling rate. In order to ensure accuracy, each
logger at a 1 Hz sampling rate. In order to ensure accuracy, each lumber was tested three lumber was tested three
times on
times on load
load data
dataranging
rangingbetween
between1.8 1.8kN kNandand3.33.3kN,
kN,asas these
these values
values were
were below
below 40% 40%of
of the
the estimated
estimated failure
failure load.
load. TheThe average
average MOE MOE
value value
of theoflast
thetwolast measurements
two measurements was
was used
used
as theas the MOE
static static forMOE thefor the lumber
lumber samplesampledue to thedueslight
to the slight difference
difference in valuesin values the
between be-
tween the first and
first and last two tests. last two tests.
Load
40mm
7h 7h 7h
parameters were
The 2- and similarly Weibull
3-parameter determined by calculating
distributions were alsothe logarithmand
determined, of the
thegoodness
individual
of
measured
fit data. Table 1 reports the basic expressions for the 3-parameter Weibull distribution.
was evaluated.
Forests 2021, 12, 883 5 of 13
Probability Density Function (PDF) Cumulative Distribution Function (CDF) Mean Variance
k−1 −[ x−xu ]k k
2
−[ x−λxu ]
f ( x ) = λk x−λxu e λ F(x) = 1 − e λΓ 1 + 1k λ2 Γ 1 + 2k − Γ 1 + 1k
i ρi k
= 1 − e−( λ ) (2)
( n + 1)
where i is the data point number in ascending order; and n is the number of samples.
The logarithm of both the sides of Equation (3) was then taken:
n+1 lge
lg(lg ) = klgρi + lg( k ) (3)
n+1−i λ
Equation (4) can be interpreted as a linear equation with slope k and intercept
lg(lge/λˆk). Taking the larch density data as an example, parameter k and scale parameter
λ were calculated as 9.03615 and 0.69555, respectively.
In order to estimate the 3-parameter Weibull distribution parameters, location param-
eter xu was subtracted from each data point. The other parameters were determined as
with the 2-parameter Weibull distribution.
MOEDynamic / GPa
From
From Figure
Figure 4,4, it
it was
was also
also found
found that
that the
the mean
mean MOEMOEDynamic
Dynamic was
was very
very close
close to to that
that of
of MOE , with values of 15.81 GPa and 15.25 GPa, respectively, and
MOEStatic, with values of 15.81 GPa and 15.25 GPa, respectively, and a 3.7% difference be-
Static a 3.7% difference
between them.There
tween them. Therewaswasaastrong
strong linear
linear correlation
correlationbetween
betweenthe theMOE
MOE and MOEStatic
Dynamic and MOE
Dynamic Static
values 2
(R ==0.758),
0.758),indicating
indicatingthethe ability MOE
values (R 2 ability of of
thethe FFV-determined
FFV-determined MOE values
Dynamic
Dynamic values to
to eval-
evaluate the lumber
uate the lumber mechanical
mechanical properties,
properties, potentially
potentially replacing
replacing the static
the static method.
method. AsFFV
As the the
FFV test easy
test was was easy and fast,
and fast, whilewhile the full-size
the full-size MOEMOE
staticstatic test method
test method was was
timetime consuming
consuming and
and based on a complex mechanical test machine, the FFV method
based on a complex mechanical test machine, the FFV method would reduce operation would reduce operation
complexity
complexity and and costs
costs compared
compared withwith the static MOE
the static MOE testtest processing,
processing, and and is is thus
thus more
more
suitable for large-scale factory production.
suitable for large-scale factory production.
3.2. Differences in the Density and MOEDynamic between Natural and Planted Larch
3.2. Differences in the Density and MOEDynamic between Natural and Planted Larch
3.2.1. Statistical Results of the Density and MOEDynamic Tests
3.2.1. Statistical Results of the Density and MOEDynamic Tests
By using the Three Sigma Rule, one natural larch sample and four planted larch spec-
By using the Three Sigma Rule, one natural larch sample and four planted larch spec-
imens were rejected; the results are shown in Table 2 and Figure 5. The violin plots in
imens were rejected; the results are shown in Table 2 and Figure 5. The violin plots in
Figure 5 contain the distribution information on the density and MOEDynamic . The box plots
Figure 5 contain the distribution information on the density and MOEDynamic. The box plots
inside the violin plot present the 5th, 25th, 50th, 75th, and 95th percentile of the data from
inside the violin plot present the 5th, 25th, 50th, 75th, and 95th percentile of the data from
the bottom to top. The density and MOEDynamic means are also represented as the mean
the bottom diamond
confidence to top. The
plotdensity andbox
inside the MOE Dynamic means are also represented as the mean
plot.
confidence diamond plot inside the box plot.
Table 2. Density and MOE statistics of the two-dimensional larch lumber types.
Table 2. Density and MOE statistics of the two-dimensional larch lumber types.
Density MOEDynamic MOEstatic
Group N Density MOEDynamic MOEstatic
3 ) SD (g/cm3 ) CV Mean (GPa) SD (GPa) CV Mean
Group Mean
N (g/cm
Mean Mean SD Mean SDSD
(GPa) (GPa) CV
Natural larch 599 0.66(g/cm )
SD (g/cm
0.08
3)
12.00%
CV 15.52
CV CV
3 (GPa) 2.97 (GPa) 19.12% 15.02
(GPa) (GPa) 2.35 15.62%
Planted larch 596 0.57 0.06 11.12% 13.10 2.78 21.20% 13.11 2.20 16.75%
Natural larch 599 0.66 0.08 12.00% 15.52 2.97 19.12% 15.02 2.35 15.62%
Planted larch Note:596
N is the number
0.57 of samples,
0.06SD denotes the standard
11.12% deviation, 2.78
13.10 and CV is the coefficient of
21.20% variation. 2.20
13.11 16.75%
Note: N is the number of samples, SD denotes the standard deviation, and CV is the coefficient of variation.
Forests 2021, 12,
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x FOR PEER REVIEW 77 of
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13
MOEDynamic /GPa
MOEStatic /GPa
ρ (g/cm3)
5. Density
Figure 5. Density and
andMOE
MOEtest
testresults
resultsofofthe
thenatural
naturaland planted
and larch
planted dimension
larch lumber.
dimension Note:
lumber. TheThe
Note: horizontal lineline
horizontal in the
in
boxplots are the 5th, 25th, 50th, 75th, and 95th percentile; inside the boxplot is the confidence diamond plot of
the boxplots are the 5th, 25th, 50th, 75th, and 95th percentile; inside the boxplot is the confidence diamond plot of the the mean
mean value,
value, and outside
and outside the boxplot
the boxplot is the plot.
is the violin violin plot.
Table 3. t-Test and Wilcoxon test results of the density and MOE.
Table 3. t-Test and Wilcoxon test results of the density and MOE.
Shapiro–Wilk Test F-Test t-Test Wilcoxon Test
Experimental Conditions Shapiro–Wilk Test F-Test t-Test Wilcoxon Test
Experimental Conditions W Prob < W F Ratio p-Value t Ratio Prob > |t| Z Prob > |Z|
W Prob < W F Ratio p-Value t Ratio Prob > |t| Z Prob > |Z|
Natural
Natural larch
larch 0.9976 0.5481
0.9976 0.5481
Density
Density 1.5745 <0.0001
1.5745 <0.0001**** 22.2776
22.2776 <0.0001
<0.0001**** 19.0425
19.0425 <0.0001
<0.0001 **
**
Planted
Planted larch
larch 0.9853 <0.0001
0.9853 <0.0001
** **
Natural
Naturallarch
larch 0.9977
0.9977 0.5863
0.5863
MOEDynamic
MOE Dynamic 1.1420
1.1420 0.1051
0.1051 14.5581
14.5581 <0.0001
<0.0001**** 13.5729
13.5729 <0.0001
<0.0001 **
**
Planted larch
Planted larch 0.9824
0.9824 <0.0001 ** **
<0.0001
Note: ** denotes
Note: ** denotessignificant differencesatat
significant differences the
the 0.01
0.01 level.
level.
The results reveal that the data failed the Shapiro–Wilk and F-tests, and thus the t-
The results reveal that the data failed the Shapiro–Wilk and F-tests, and thus the t-test
test prerequisites for density and MOEDynamic were not met. Therefore, the subsequent anal-
prerequisites for density and MOEDynamic were not met. Therefore, the subsequent analysis
ysis was based just on the Wilcoxon test. Significant differences were observed in the den-
was based just on the Wilcoxon test. Significant differences were observed in the density
sity and MOEDynamic between planted and natural larch dimension lumber. In particular,
and MOEDynamic between planted and natural larch dimension lumber. In particular, the
the natural larch’s MOEDynamic values exceeded those of planted larch after combining with
natural larch’s MOEDynamic values exceeded those of planted larch after combining with
the mean value of Figure 5. The faster the growth of the planted larch under favorable
the mean value of Figure 5. The faster the growth of the planted larch under favorable
silviculture
silviculture measures,
measures, the
the lower
lower the
the density,
density, the
the greater
greater the
the ratio
ratio value
value of
of the
the spring
springwood,
wood,
and the wider the width of the growth ring, resulting in lower MOE
and the wider the width of the growth ring, resulting in lower MOEDynamic values for Dynamic values for
planted
planted larch.
larch.Although
AlthoughthetheMOE
MOE Dynamic of the planted larch lumber was smaller than that
Dynamic of the planted larch lumber was smaller than
of natural larch, once converted to MOE
that of natural larch, once converted tostatic MOE, it was ,still obviously greater than the MOE of
static it was still obviously greater than the
SPF (Spruce–Pine–Fir from Canada), which is commonly
MOE of SPF (Spruce–Pine–Fir from Canada), which is commonly used in light-frame buildings.
used in light-frame
For example, the average MOE of SPF with a No. 1 grade was approximately
buildings. For example, the average MOE of SPF with a No. 1 grade was approximately 10 ± 1.82
GPa [25]; thus, the average MOE static of the planted larch lumber was 30% greater than that
10 ± 1.82 GPa [25]; thus, the average MOEstatic of the planted larch lumber was 30% greater
of SPF.
than Therefore,
that planted larch
of SPF. Therefore, can larch
planted effectively replace natural
can effectively replacelarch andlarch
natural is moreandresistant
is more
under bending
resistant under compared to SPF. to SPF.
bending compared
3.3.
3.3. Distribution
Distribution Parameter
Parameter of
of Density
Density and
and MOE,
MOE, and
and Their
Their K-S
K-S Test
Test
The
The K-S
K-Stest
testwas
wasadopted
adoptedtotocompare
comparethe thegoodness
goodness of of
fit fit
between
betweenthethe
different distri-
different dis-
tribution
bution models
models (Figure
(Figure 6 and
6 and Tables
Tables 4 and
4 and 5).5). Here,
Here, for for
thethe
normalnormal
andand lognormal
lognormal dis-
distri-
√
tributions,thethecritical
butions, criticalvalues
values were
were calculated
calculated as as D𝐷599,0.05
599,0.05 == 0.886/
0.886/√599 599== 0.03620
0.03620 and
√
D596,0.05
𝐷 =0.886/√596
596,0.05 = 0.886/ 596==0.03629,
0.03629 ,while
while the
thecorresponding
corresponding Weibull
Weibulldistribution values
distribution were
values
√ √
determined as D
were determined599,0.05 = 0.888/ 599 = 0.03628 and
as 𝐷599,0.05 = 0.888/√599 = 0.03628 596,0.05D = =
and 𝐷596,0.05 = 0.886/√596 =.
0.886/ 596 0.03637
Only the D
0.03637. Only is smaller than D , so the Weibull distribution
than 𝐷𝑛,0.05 , so the Weibull distribution
n the Dn is smallern,0.05 with three
withparameters
three param- is
accepted
eters after the
is accepted K-Sthe
after test.
K-SFigure 6 and6Tables
test. Figure 4 and45and
and Tables depict the K-S
5 depict the test
K-S results and
test results
distribution
and parameters.
distribution parameters.
Table 4. K-S tests of the normal and lognormal distributions based on the estimated parameters.
Table 4. K-S tests of the normal and lognormal distributions based on the estimated parameters.
Possible Distribution Parameters Critical Value
Samples Possible Distribution Parameters Value of Dn Reject or Accept
Samples Type Mean Variance Value of Dn (𝑫Value
Critical 𝒏,𝟎.𝟎𝟓 )(Dn,0.05 ) Reject or Accept
Type Mean Variance
ρ of natural larch Normal 0.660 48 0.006 28 0.031 16 Accept
Normal 0.66015
48 0.014 0.006 28 0.031 16 0.036 20 Accept
n = 599
ρ of natural larch n = 599 Lognormal −0.422 99 0.054 24 0.036 20 Reject
Lognormal −0.422 15 0.014 99 0.054 24 Reject
E of natural larch Normal 15.023 27 5.507 48 0.016 94 Accept
Normal 15.023 27 5.507 48 0.016 94 0.036 20 Accept
n = 599
E of natural larch n = 599 Lognormal
Lognormal
2.696 95
2.696 95
0.0260.026
02 02 0.036 55
0.036 55
0.036 20 Reject
Reject
ρ of planted larch Normal 0.568 21 0.003 99 0.039 20 Reject
ρ of planted
Normal
larch n = 596Lognormal −0.571 06
0.568 21 0.003 99 0.039 20 0.036 29 Reject
n = 596 0.036 29
Lognormal −0.571 06 0.012 13 13
0.012 0.021 6363
0.021 Accept
Accept
E of planted larch Normal
Normal 13.109
13.1092323 4.8224.822
48 48 0.064 6363
0.064 Reject
Reject
E of planted larch n = 596 0.036
0.036 29
29
n = 596 Lognormal
Lognormal 2.559 4343 0.0270.027
2.559 81 81 0.044 5454
0.044 Reject
Reject
Notation: ρ is density, E is the dynamic MOE determined via FFV, D n is the maximum difference value between the assumed
Notation: ρ is density, E is the dynamic MOE determined via FFV, Dn is the maximum difference value between the assumed distribution
distribution
functionfunction and statistics,
and the order the orderand
statistics,
Dn,0.05 isand 𝐷𝑛,0.05value
the critical is the
forcritical
the K-Svalue
test. for the K-S test
Table 5. K-S tests of the Weibull distribution with two and three parameters.
Table 5. K-S tests of the Weibull distribution with two and three parameters.
Possible Weibull Distribution Parameters Critical
Possible Weibull Distribution Parameters Value of
Samples Location Shape Scale Value Reject or Accept
Samples Mean Variance 𝑫𝒏 of Dn Critical Value Reject or
Value
𝒙𝒖Location 𝒌 Shape 𝝀 Scale Mean Variance (𝑫𝒏,𝟎.𝟎𝟓(D
) n,0.05 ) Accept
xu k λ
ρ of natural larch 0 9.036 15 0.695 55 0.66 0.007 60 0.048 70 Reject
0 0.048 70 0.036 28
n = 599
ρ of natural larch n = 5990.39 3.796 9.036
89 150.2990.695
16 550.660.66 0.006 0.007 60
32 0.024 44 0.036 28 Accept
Reject
0.39 3.796 89 0.299 16 0.66 0.006 32 0.024 44 Accept
E of natural larch 0 6.948 94 16.024 67 14.98 6.425 59 0.042 22 Reject
0 6.948 94 16.024 67 14.98 6.425 59 0.042 22 0.036 28 Reject
n = 599larch n = 5997.10 7.10 3.724 3.724
E of natural 02 028.7668.766
39 3915.01 15.01 5.608 9999
5.608 0.022 60
0.022 60
0.036 28 Accept
Accept
ρ of planted larch 0 9.042 21 0.597 05 0.57 0.005 60 0.075 71 Reject
0 9.042 21 0.597 05 0.57 0.005 60 0.075 71 0.036 37 Reject
n = 596larch n = 5960.41 0.41 2.707 2.707
ρ of planted 17 170.1790.179
90 900.570.57 0.004 0606
0.004 0.031 80
0.031 80
0.036 37 Accept
Accept
ρ of planted larch 0 0 6.252 6.252
06 06 14.05214.052
77 7713.07 13.07 5.942 0808
5.942 0.067 66 66
0.067 Reject
Reject
ρ of planted larch n = 596 0.036 370.036 37
n = 596 8.34 8.34 2.310 2.310
85 855.3845.384
09 0913.11 13.11 4.796 8282
4.796 0.035 16 16
0.035 Accept
Accept
Notation: ρ is density,
Notation: and
ρ is density, E is
and the
E is thedynamic
dynamic MOE determined
MOE determined viavia FFV.
FFV. OnlyOnly the Weibull
the Weibull distribution
distribution with
with two two parameters,
parameters, with
with a location
a location of 0,
of 0, and theand the Weibull
Weibull distribution
distribution with three with three with
parameters parameters withDthe
the smallest smallest
n value is shown𝐷𝑛invalue
Table is
5. shown in Table 5.
The figures
The figuresand
andtable
tableresults
resultsrevealed
revealedthat
thatthe
thenatural
naturallarch
larchlumber
lumberdensity
density exhibited
exhibited a
a good fit with the normal and 3-parameter Weibull distributions, with the
good fit with the normal and 3-parameter Weibull distributions, with the highest goodness highest good-
ness
of fit of fit observed
observed forthree-parameter
for the the three-parameter Weibull
Weibull distribution
distribution with location
with location parameter
parameter xu =
𝑥𝑢 = The
0.39. 0.39.MOE
TheDynamic
MOEDynamic of natural
of natural larch larch
lumber lumber fittedwith
fitted well wellthe
with the normal
normal and 3-pa-
and 3-parameter
rameter distributions,
Weibull Weibull distributions, with theexhibiting
with the former former exhibiting the best goodness
the best goodness of fit6A).
of fit (Figure (Figure
The
planted larch lumber density exhibited a good fit with the lognormal and 3-parameter
Forests 2021, 12, x FOR PEER REVIEW 10 of 13
6A). The planted larch lumber density exhibited a good fit with the lognormal and 3-pa-
rameter Weibull distributions, with the former presenting the highest goodness of fit. The
Weibull
MOE of distributions,
planted larch with
lumberthewas
former presenting
observed to fitthe highest
well with thegoodness of fit. Weibull
3-parameter The MOEdis- of
planted larch
tribution lumber was
at a location observed
parameter to fit
equal towell with theof3-parameter
97.6–98.9% the minimum Weibull distribution
MOEDynamic at
, with the
a location of
goodness parameter equal at
fit optimized to the
97.6–98.9%
locationof the minimum
parameter MOE(𝑥
8.34 GPa =
Dynamic
𝑢 , with
8.34) the
(Figuregoodness
7D). In
of fit optimized
order to simplifyatthethedistribution
location parameter
model, the 8.34 GPa (xu = Weibull
3-parameter 8.34) (Figure 7D). Inwas
distribution order to
sug-
simplify the distribution model, the 3-parameter Weibull distribution was
gested to fit the density and MOEDynamic of natural and planted larch as it passed the K-S suggested to fit
the density
test for the four MOE
and test data of natural
types.
Dynamic Besides,and planted
Figure 6 alsolarch as it passed
revealed that thethe K-S test for
distribution of the
four test data
measured datatypes. Besides,
of planted Figure
larch was6left-biased,
also revealed that thethat
meaning distribution
there were of more
the measured
samples
data of
with planted
a low and larch wasdensity
medium left-biased,
andmeaning that thereThus,
elastic modulus. werethe
more samples
Weibull with a lowwith
distribution and
medium density and elastic modulus. Thus, the Weibull distribution with three
three parameters was a better fit of the density and MOE distribution of planted larch full- parameters
was lumber.
size a better fit of the density and MOE distribution of planted larch full-size lumber.
(A) Distribution of density for natural larch (B) Distribution of MOEDynamic for natural larch
(C) Distribution of density for planted larch (D) Distribution of MOEDynamic for planted larch
Figure 7. Distribution
Figure 7. Distribution histograms and fitting
histograms and fitting curves
curves of
of larch
larch lumber
lumber MOE.
MOE.
3.4.
3.4. Determine
Determine the the Characteristic
Characteristic Based
Based on
on Distribution
Distribution of
of Density
Density andand MOE
MOE
The characteristicvalue
The characteristic valueis is
thethe 5th-percentile
5th-percentile valuevalue
at 75%at confidence
75% confidence level,
level, also also
known
known as the standard value in China according to GB 5005 [2],
as the standard value in China according to GB 5005 [2], forms the basis of the design forms the basis of the
design values and is crucial for the safety and reliability of wood constructions.
values and is crucial for the safety and reliability of wood constructions. According to According
to ASTM
ASTM D2915-10
D2915-10 [29]
[29] andand Zhong
Zhong Yong
Yong [30],
[30], bothboth parametric
parametric andand non-parametric
non-parametric meth-
methods
ods
can can be used
be used to calculate
to calculate thethe characteristic
characteristic value
value ofofthe
thelumber
lumberdensity
densityand
and MOE.
MOE. TheThe
parametric method was
parametric method was used
used to to calculate
calculate the
the characteristic value according
characteristic value according to to the
the formula
formula
𝐸 = 𝑥̅x − 𝑘k 𝑠,
k,e =
E𝑘,𝑒 s, where the k𝑘value
valueisisdecided
decidedby bythe
thenumber
numberof of samples
samples andand the confidence
level. IfIf the average value
the average value x 𝑥̅ and standard variation 𝑠 were known,
variation s were known, the the characteristic
characteristic
value E 𝐸𝑘,𝑒 could be calculated.
k,e could calculated. InInturn,
turn,the
thenon-parametric
non-parametric method
method was based
was on on
based the the
or-
der
order number
number of the testtest
of the value to estimate
value the characteristic
to estimate value,value,
the characteristic and theandorder was based
the order was
on
basedthe on
sample numbernumber
the sample and confidence level. Generally,
and confidence the parametric
level. Generally, method ismethod
the parametric consid-
ered as more effective
is considered as more when thewhen
effective data distribution model is known
the data distribution model is in known
advance, in while the
advance,
while the non-parametric method is associated with smaller errors. Table 6 reports the
determined characteristic values.
Forests 2021, 12, 883 11 of 13
Table 6. Characteristic values of density and MOE for the larch dimension lumber.
The characteristic values of the density and MOEDynamic for natural and planted larch
lumber via the parametric and non-parametric methods where highly similar, with dif-
ferences of 1.92%, 0.01%, 2.13%, and 2.90%, respectively. The differences between the
characteristic values calculated by the parametric method and the 5% quantile under the
best distribution were 0, −0.99%, −2.13%, and −4.48%, respectively. The corresponding
non-parametric method differences were −1.92%, −0.90%, 0, and −1.63%. This indicates
that the characteristics of the density and MOEDynamic for natural and planted larch cal-
culated by the parametric and non-parametric methods were almost equal. The planted
larch MOEDynamic calculated by the parametric method was slightly smaller than that of
the non-parametric method and the 5% quantile under the best distribution. This may
be attributed to the MOEDynamic distribution model (Figure 6D), with the median mean
value exceeding the mean value of the data. More specifically, with the exception of
the natural larch lumber density, the characteristic values calculated by the parametric
method were slightly lower than the non-parametric values. Thus, the parametric method
can more effectively estimate the characteristic value for larch and should be the pre-
ferred approach to determine the characteristic values of density and MOE for natural
and planted larch. The characteristic density values of natural and planted larch were
determined as 0.53 g/cm3 and 0.46 g/cm3 ; the characteristic MOEDynamic values were
11.05 GPa and 9.39 GPa; and the characteristic MOEstatic values in flatwise samples were
11.05 × 0.791 + 2.749 = 11.49 GPa and 9.39 × 0.791 + 2.749 = 10.17 GPa, respectively. By
taking into account the horizontal adjustment factor, the characteristic MOEstatic in edge-
wise values for natural and planted larch were 11.19/1.15 = 9.73 GPa (9730 MPa) and
10.17/1.15 = 8.84 GPa (8840 MPa); and the characteristic values of density were 0.53 g/cm3
(530 kg/m3 ) and 0.46 g/cm3 (460 kg/m3 ), respectively.
By comparing the value with the requirement of EN 338:2016, Structural timber.
Strength classes [31], the standard MOE value of natural and planted larch was able to
meet the requirements of the standard modulus with grades C40 and C35. The standard
MOE of both larch groups exceeded the standard modulus of elasticity for visual grade
larch with an Ic grade in GB 5005, where the standard MOE should greater than 8.6 GPa [2].
Thus, the number of MOE tests for visual natural and planted larch lumber can potentially
be reduced during factory processing. The grading of natural and planted larch should
be performed using the FFV method as it is able to increase the characteristic values for
high density and MOE lumber via increasing the mean value or decreasing the variation
in larch.
4. Conclusions
The results presented in the current paper can serve as a point of reference to promote
the application of natural and planted larch for wooden-based buildings. Based on the
experimental tests and analysis, we determined the following key conclusions as follows:
1. A relatively strong linear relationship was observed between the dynamic and static
MOE of the larch lumber, proving the FFV method as reliable for the testing of the
dynamic and static MOE estimations of larch dimension lumber based on the equation
MOEstatic = 0.791 × MOEDynamic + 2.749 (R2 = 0.758).
Forests 2021, 12, 883 12 of 13
2. According to statistical analysis and non-parametric testing, the density and MOEstatic
of the planted larch lumber were significantly lower (p = 0.01) than those of natural
larch lumber, and the average density and MOEstatic of planted larch were 13.6% and
12.7% lower those that of natural larch, respectively.
3. The density determined from clear samples could at times be used to evaluate the
average density of lumber; however, this was not the case for the MOEstatic . This
was because the average density obtained from full size testing was very close to the
average density of the small clear samples, yet the average MOEstatic obtained from
the full-size tests was significantly lower than that of the small clear specimens.
4. The 3-parameter Weibull distribution model optimally fits the density and MOE of
natural and planted larch, as it was the only distribution to pass the K-S test. In
particular, the distribution for the measured data was left-biased, and thus there were
more samples with a low and medium density and elastic modulus.
5. The parametric method was demonstrated to be more effective in calculating the
characteristic values of natural and planted larch compared to the non-parametric
method. The standard value of MOEstatic for natural and planted larch were 9.73 GPa
and 8.84 GPa, and hence the MOEstatic met lumber grades C35 and C30.
Author Contributions: Conceptualization, Z.T., Z.W. and H.R.; funding acquisition, H.R.; investi-
gation, Z.T.; methodology, Z.T., J.X. and M.L.; resources, Y.G.; supervision, E.S. and H.R.; writing—
original draft, Z.T.; writing—review and editing, E.S. All authors have read and agreed to the
published version of the manuscript.
Funding: This research was funded by National Natural Science Foundation of China, grant number
31971596 and the Chinese Scholarship Council grant number 20190320022.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Acknowledgments: The authors are grateful for the financial support provided by the National
Natural Science Foundation of China (Research on fracture mechanism of cross layer in cross-
laminated timbe (CLT) under bending stress. Grant number 31971596) and the Chinese Scholarship
Council (No. 20190320022).
Conflicts of Interest: The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported in this paper.
References
1. Pajchrowski, G.; Noskowiak, A.; Lewandowska, A.; Strykowski, W. Wood as a Building Material in the Light of Environmental
Assessment of Full Life Cycle of Four Buildings. Constr. Build. Mater. 2014, 52, 428–436. [CrossRef]
2. GB 5005, Code for Design of Timber Structures; Ministry of Housing and Urban Rural Construction of the People’s Republic of
China: Beijing, China, 2017.
3. Jiang, J.; Lu, J.; Ren, H. Study on Characteristic Values for Strength Properties of Chinese Larch Dimension Lumber. J. Build. Mater.
2012, 15, 361–365.
4. Tian, Z.; Wang, Z.; Wang, J.; Zhang, Z.; Ge, P.; Lv, Y.; Ren, H. Classification in Modulus of Elasticity and Mechanical Properties of
Larch Lumber in Mohe. J. Northwest For. Univ. 2017, 32, 211–215.
5. Zhong, Y.; Wu, G.; Ren, H. Design Value of Tension of Domestic Dimension Lumber for Structural Use. Sci. Silvae Sin. 2018, 54,
100–112.
6. Zhong, Y.; Ren, H.-Q.; Jiang, Z.-H. Experimental and Statistical Evaluation of the Size Effect on the Bending Strength of Dimension
Lumber of Northeast China Larch. Materials 2016, 9, 89. [CrossRef]
7. Zhong, Y.; Ren, H. Reliability Analysis for the Bending Strength of Larch 2x4 Lumber. Bioresources 2014, 9, 6914–6923. [CrossRef]
8. Lou, W.; Wang, Z.; Luo, X.; Guo, W.; Ren, H. Full Size Bending Mechanical Properties of Dahurian Larch Dimension Lumber. J.
Anhui Agric. Univ. 2011, 2, 009.
9. Wang, Z.; Ren, H.; Luo, X.; Zhou, H. Mechanical Stress Grading of Larch Dimension Lumber from Northeastern China. China
Wood Ind. 2009, 3, 002.
10. Porteous, J.; Kermani, A. Structural Timber Design to Eurocode 5; John Wiley & Sons: Hoboken, NJ, USA, 2013.
11. Lin, Y.; Wang, Y.; Jiang, S. GB 50005-2003 Code for Design of Timber Structures; China Architecture & Building Press: Beijing, China,
2003.
Forests 2021, 12, 883 13 of 13
12. Rosowsky, D.; Ellingwood, B. Limit-State Interactions in Reliability-Based Design for Wood Structures. J. Struct. Eng. 1992, 118,
813–827. [CrossRef]
13. Ross, R.J.; Brashaw, B.K.; Pellerin, R.F. Nondestructive Evaluation of Wood. For. Prod. J. 1998, 48, 14.
14. Sales, A.; Candian, M.; de Salles Cardin, V. Evaluation of the Mechanical Properties of Brazilian Lumber (Goupia Glabra) by
Nondestructive Techniques. Constr. Build. Mater. 2011, 25, 1450–1454. [CrossRef]
15. Wang, Z.; Li, L.; Gong, M. Measurement of Dynamic Modulus of Elasticity and Damping Ratio of Wood-Based Composites Using
the Cantilever Beam Vibration Technique. Constr. Build. Mater. 2012, 28, 831–834. [CrossRef]
16. Olsson, A.; Oscarsson, J.; Serrano, E.; Källsner, B.; Johansson, M.; Enquist, B. Prediction of Timber Bending Strength and
In-Member Cross-Sectional Stiffness Variation on the Basis of Local Wood Fibre Orientation. Eur. J. Wood Wood Prod. 2013, 71,
319–333. [CrossRef]
17. Olsson, A.; Oscarsson, J. Strength Grading on the Basis of High Resolution Laser Scanning and Dynamic Excitation: A Full Scale
Investigation of Performance. Eur. J. Wood Wood Prod. 2017, 75, 17–31. [CrossRef]
18. Brunetti, M.; Burato, P.; Cremonini, C.; Negro, F.; Nocetti, M.; Zanuttini, R. Visual and Machine Grading of Larch (Larix Decidua
Mill.) Structural Timber from the Italian Alps. Mater. Struct. 2015, 49, 2681–2688. [CrossRef]
19. Zhang, X.; Yin, Y.; Jiang, X. Evaluation of Bending Properties of Chinese Fir Plantation by Two Nondestructive Testing Methods. J.
Build. Mater. 2010, 13, 836–840.
20. Jiang, J.; Lv, J.; Ren, H.; Luo, X.; Long, C.; Guo, W. Evaluation of Modulus of Elasticity for Dimension Lumber by Three
Nondestructive Techniques. J. Zhejiang AF Univ. 2008, 25, 277–281.
21. Arriaga, F.; Monton, J.; Segues, E.; Íñiguez-Gonzalez, G. Determination of the Mechanical Properties of Radiata Pine Timber by
Means of Longitudinal and Transverse Vibration Methods. Holzforschung 2014, 68, 299–305. [CrossRef]
22. Pukelsheim, F. The Three Sigma Rule. Am. Stat. 1994, 48, 88–91.
23. ASTM. Standard Test Method of Static Tests of Lumber in Structure Sizes; ASTM International: West Conshohocken, PA, USA, 2015.
24. Lv, J.; Jiang, J.; Ren, H.; Luo, X. GB/T 28993-2012 Standard Test Methods for Mechanical Properties of Structural Lumber; Standardization
Administration of the People’s Republic of China: Beijing, China, 2012.
25. Wang, Z.; Wang, Z.; Wang, B.J.; Wang, Y.; Liu, B.; Rao, X.; Wei, P.; Yang, Y. Dynamic Testing and Evaluation of Modulus of
Elasticity (MOE) of SPF Dimensional Lumber. BioResources 2014, 9, 3869–3882. [CrossRef]
26. Pellicane, P. Goodness-of-Fit Analysis for Lumber Data. Wood Sci. Technol. 1985, 19, 117–129. [CrossRef]
27. Wu, S.; Ye, J. Critical Value Analysis of K-S Method Test with Unknown Parameters. Port Eng. Technol. 1990, 1, 6–20.
28. Zhou, X.; Zhang, J.; Zhou, H.; Sun, X.; Ren, H.; Zhao, R. Tree Age’s Effects on Physical and Mechanical Properties of Larix
Kaempferi Wood. China For. Sci. Technol. 2014, 28, 54.
29. ASTM. Standard Practice for Sampling and Data-Analysis for Structural Wood and Wood-Based Products; ASTM International: West
Conshohocken, PA, USA, 2017.
30. Zhong, Y.; Wu, G.; Ren, H.; Sun, Z.; Jiang, Z. Determination of Characteristic Strength for Structural Wood Materials by
Nonparametric Method. J. Build. Struct. 2018, 42, 142–150.
31. BS EN 338:2016. Structural Timber–Strength Classes, European Committee for Standardization; BSI standard Publucation: London, UK,
2016.