Experimental and Statistical Evaluation of the Size Effect on the Bending Strength of Dimension Lumber of Northeast China Larch

This study investigated the size effect on the bending strength (modulus of rupture—MOR) of dimension lumber of Northeast China larch (Larix gmelinii); providing a basis for the further application in light wood frame construction. Experimental and statistical evaluations were conducted on the bending strength. A total of 2409 full-size dimension lumber samples were tested by static bending tests; which included three different sizes: 2 × 3; 2 × 4; and 2 × 6. Results indicate that the size has a significant effect on the MOR. Both the chi-square (χ2) and Kolmogorov-Smirnov (K-S) test results show that the lognormal distribution generally fits to the MOR better than to the normal distribution. Additionally; the effects of partial safety factor (γR) and live-to-dead load ratio (ρ) were studied by reliability analysis. Reliability analysis results indicate that the reliability index increases nonlinearly with the decrease of γR and the rise of ρ. Finally; the design value of bending strength and its adjusting factor of size effect of 2 × 3; 2 × 4; and 2 × 6 larch dimension lumber were obtained according to the Chinese National Standards’ requirements of the reliability index.


Introduction
Chinese larch (Larix gmelinii), an abundant wood resource, comprises 55% of the wooded areas and 75% of forest stocks in China's frigid temperate zone, and can be used quite extensively to fabricate dimension lumber, glued lumber, and wood-based composites due to its favorable mechanical properties [1]. Due to the general lack of research regarding product fabrication processes and technologies, however, dimension lumber products manufactured from China's own tree species remain limited.
The mechanical performance of dimension lumber is generally determined by one of two common methods: full-size testing, or small clear wood specimen testing. Considering the acute necessity of accurate test results and the structural safety requirements in practice, the full-size test method substituting the small clear wood specimens test method, has been adopted widely in most developed countries, such as the United States, Canada, and Japan [2,3]. However, the design value of structural wooden materials, fabricated from China's native tree species, is still determined by the traditional small clear wood specimens test method.
Thus in an effort to ensure the safe use of dimension lumber and to promote the development of light wood frame construction, the mechanical performance of dimension lumber using China's own Larch forestry resources was studied by the full-size test method during the 11th Five-Year Plan of China. The Chinese government and a handful of research institutions also began promoting the use of larch in light wood frame construction around this time. Several works [4][5][6][7] have shown that mechanical strength is reduced with the increase of the size and the decrease of strength grade. In addition, the effects of various testing methods on the mechanical properties of dimension lumber have been investigated [8] as well as the relationships between the visual defects, such as knots, slope of grain, checks, and splits, and the mechanical strength of larch dimension lumber [9][10][11]. Research has shown that analysis via the lognormal distribution can provide a good fit for strength properties [12]. Based on reliability analysis, the design value of the compression strength parallel to grain and of the bending strength for larch 2ˆ4 lumber has been effectively determined [13,14]. Both the design value and its adjusting factor, as far as mechanical performance, are crucial considerations for the application of wood material in building construction; unfortunately, these have yet to be fully realized for China's lumber resources. To this effect, it remains necessary to determine the adjusting factor of the mechanical properties and the related design value based on full-size test results.
Therefore, the objective of this study was to investigate the size effects on the bending strength (MOR) of dimension lumber of Northeast China larch (Larix gmelinii), providing a basis for its further application in light wood frame construction. An extensive experimental study was conducted. Besides, the statistical analyses of MOR were also performed.

Materials
Larch (Larix gmelinii) was collected from the Cuigang and Pangu forest farms of Heilongjiang Province, China. The diameter range of logs was 160-340 mm, and the average tree age was 35 years. The material was cut into 2ˆ3, 2ˆ4, and 2ˆ6 dimension lumber samples. As specified in the Chinese National Standard [15], any single digit of lumber size, measured in millimeters (mm), was rounded to zero or five; unlike other countries' standards, then, the actual size of the 2ˆ3, 2ˆ4, and 2ˆ6 lumber specimens were 40ˆ65, 40ˆ90, and 40ˆ140 mm, respectively. The length of each specimen was 4000 mm. The strength grade of dimension lumber is determined by its visual defects, such as knots, slope of grain, checks, splits and so on, according to the NLGA standard [16]. Besides, grades SS, No. 1, No. 2, and No. 3 in the NLGA standard, which are referred to in this paper, can be equated to grades Ic, IIc, IIIc, and IVc, respectively, in the Chinese National Code [15].
The number of specimens, the mean value, and coefficient of variation (COV) of density for each size of dimension lumber are shown in Table 1. Due to lack of a sufficient number of specimens to obtain accurate results, grades No. 1 and No. 3 of 2ˆ6 lumber were not analyzed in this paper. According to ASTM D245 [17], an assumed minimum grade quality index (GQI) of each test sample was determined by the above visual defects.
Before bending testing, all specimens were conditioned at 20˝C and 65% relative humidity (RH) in a standard room, to arrive at the equilibrium moisture content (EMC). The measured average moisture content was 11.3% with a standard deviation of 1.11% [18].

Static Testing Method
According to ASTM D4761 [19], the third-point edgewise bending tests were carried out on an MTS universal testing machine. Specimens were loaded at a rate of 5 mm/min, which continued until failure. The span to depth ratio was 18 to 1. The MOR of each specimen was calculated as follows: where a is the distance between a loading position and the nearest support (mm), F max is the maximum load (N), and W is the section modulus (mm 3 ).
The MOR of each specimen, adjusted to 15% MC (MOR 15 ) in accordance with ASTM D1990 [20], can be represented as follows: where M 1 is the moisture content of the specimen (%).

One-Way Anova Analysis
The graphical analysis was conducted with Origin 9 software (OriginLab Corporation, Northampton, MA, USA). The differences in bending strength between different sizes were analyzed by one-way analysis of variance (ANOVA); multiple comparisons for different sizes were calculated by the Least Significant Difference (LSD) method using the ANOVA test results using SPSS 19.0 (IBM SPSS Corporation, Chicago, IL, USA). Significance level was set to 0.05.

Distribution Model
Both chi-square (χ 2 ) and Kolmogorov-Smirnov (K-S) tests were used to determine the probability distribution of MOR 15 .
For the chi-square (χ 2 ) test method, the probability distribution of MOR 15 was assumed to be either a normal distribution (N), or a lognormal distribution (L). The cumulative distribution function is defined as follows: where x and t are random variables of either MOR 15 or logarithmic MOR 15 obtained by the static bending test, and µ and σ are unknown parameters determined in the following analysis. First, the µ and σ parameters were calculated by the maximum likelihood method in which the likelihood function can be expressed: 2πσ e´p where x i is the random variable of the ith sample and n is the total number of samples. After a logarithmic transformation, the likelihood function can be rewritten as follows: By calculating the partial derivatives of Equation (5), the likelihood equations can be obtained, which are defined as: The maximum likelihood estimation of the µ and σ parameters is as follows: where k is the sum of interval numbers of observed MOR 15 , N is the total number of specimens equal to the sum of n i , n i is the observed number of specimens located in the ith interval, and x i is the value of the interval center. By substituting Equation (7) into Equation (3), F 0 (x) can be rewritten as: F(x) is the cumulative distribution function of MOR 15 obtained by the static bending test. Whether or not the F(x) is suitable for the assumed F 0 (x) at the setting level, can be judged by the chi-square (χ 2 ) test [21]. The calculation formula for χ 2 is as follows: wherep i and Np i are the cumulative distribution probability and predicted numbers of random variables on the ith interval, respectively. The formula for the Kolmogorov-Smirnov (K-S) test method is as follows: where D is the maximum absolute difference between the empirical distribution function and the theoretical distribution function, and S n pxq is the empirical distribution function of x.

Reliability Analysis
As reported in the authors' previous research [13,14], the characteristic values of MOR 15 for larch lumber can be determined by the fifth percentile value of probability distribution. As a lognormal distribution, the characteristic value (f k ) and design value (f d ) are as follows [22,23]: where µ f is the mean value of logarithmic MOR 15 , and δ f is the coefficient of variance (COV) ( Table 2). γ f is the partial safety factor of material property, µ k3 is the mean value of the adjusting factor for the effect of long-term load and is equal to 0.72, and γ R is the partial safety factor of bending resistance. In order to determine the design value of the bending strength and its adjusting factor of size effect of larch lumber, first-order second-moment reliability analyses were performed for all data cells and simulation load cases: dead load plus office occupancy load (G + L O ), dead load plus residential occupancy load (G + L R ), dead load plus wind load (G + L W ), and dead load plus snow load (G + L S ) [22,24]. A calculation program for reliability index (β) was developed in Matlab 7 software (MathWorks Corporation, Natick, MA, USA).
The performance function can be expressed as follows [25][26][27]: where R, D, and L are random variables representing bending resistance, dead load (G), and live load (L O , L R , L W , or L S ), respectively.

Results of the Bending Test
Each test sample met grade quality index (GQI) requirements, under which the difference between the observed GQI of the samples and the assumed GQI of the grade must be 5% or less of the total range of possible GQI [20]. An observed GQI for each test sample can be calculated for all specimens that did not fail in clear wood [17,20]. The difference between the observed GQI of the samples and the assumed GQI of the grade was less than 5%. Thus, the observed GQI of test data could be used to represent the grades of the samples.  [4,28]. The symbols ** and *** represent p-values less than 0.01 and 0.001, respectively.

Probability Distribution
The cumulative probability distributions of MOR 15 for different sizes are presented in Figures 1-3. The unknown parameters µ and σ were calculated according to Equations (4)- (8). For example, the estimated µ and σ of MOR 15      Both the χ 2 (Equation (9)) and K-S (Equation (10)) test values of different sizes of larch lumber are shown in Table 5. Results indicated that the lognormal distribution fitted the bending test data much better than the normal distribution for all sizes except the SS grade of 2ˆ3, SS grade of 2ˆ4, and No. 2 grade of 2ˆ4. Findings presented by Dahlen et al. [29] similarly showed that the lognormal distribution model provides a better fit for Douglas fir and Southern pine 2ˆ4 lumber.
The assumed F 0 (x) can be considered a good fit for the F(x) obtained by the static bending test when the critical value, which is dependent on probability levels, is greater than the test value [21]. At a probability level of 0.05, the χ 2 test results showed that normal distribution fitting did not provide accurate results for most sizes, the exceptions being grade No. 1 (Table 5). The K-S test results showed that both the normal and lognormal distribution can provide a good fit at probability levels of 0.05 and 0.01. In this paper, lognormal distribution was thus selected for the following reliability analysis.  Table 6 shows the characteristic values of MOR 15 (f k ) calculated by Equation (11). The f k value generally decreased alongside the reduction in strength of size. For instance, the f k for grade SS of 2ˆ3 was 38.2 MPa, 0.061, 0.345 times higher than those of 2ˆ4 and 2ˆ6, respectively. The γ R values were determined according to the reliability analysis described below. According to the authors' previous research [13,14], random variables of bending resistance (R) are also distributed lognormally. The calculated mean value and COV of R for larch lumber are shown in Table 7 based on statistical theory. Reliability analysis results indicated that β increased nonlinearly as live-to-dead load ratio (ρ) increased in all the simulation load cases. As an example, the relationship between β and ρ of 2ˆ4 larch lumber, under dead load (G) plus live office load (L O ), are shown in Figure 4. Several previous studies [13,[25][26][27]30] achieved a similar result regarding the relationship between ρ and β. The reliability level must meet the target level (β 0 = 3.2) to effectively determine the MOR 15 design value [24]. In order to evaluate the reliability level, the live-to-dead load ratio (ρ) was specified as 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0, 2.25, 2.5, 2.75, and 3.0 for each load combination (G + L O , G + L R , G + L W , and G + L S ), respectively, according to the Chinese National Standard [22], then the average of reliability index was calculated under different load ratios. The relationship between the average reliability index and the partial safety factor is shown in Figure 5.

Reliability Analysis
As shown in Figure 5, β increased nonlinearly with the increase of the partial safety factor (γ R ) in all simulation load cases, G + L O , G + L R , G + L W , and G + L S . For the same ratio of live-to-dead load (ρ), the simulated load cases of the maximum and minimum β were G + L O and G + L S , respectively. This is consistent with the findings of previous researchers [13,[25][26][27]. The partial safety factor (γ R ) of all sizes of larch lumber was obtained, as mentioned above, by taking the average of all simulation load cases. In order to obtain β, the linear, cubic, logarithmic, and inverse models were utilized to predict the relationship between β and γ R . For example, the predicted result of grade SS of 2ˆ4 larch lumber is shown in Figure 6.  Fitting results indicated that the linear, cubic, logarithmic, and inverse models precisely predicted β as a function of γ R for all sizes of larch lumber, though the logarithmic model (which was thus used for the reliability index calculation) fitted the data slightly better than other models.  According to the above reliability analysis, plus the requirements for the minimum reliability index (β > β 0 = 3.2) [22], the suggested design values of MOR 15 and partial safety factors are shown in Table 9. For instance, the partial safety factors which satisfied the conditions γ R ě 1.  Table 9 indicates that the size of larch lumber had significant impact on the design value of MOR 15 . For each grade of lumber, the bigger the size, the smaller the design value of MOR 15 . As an example, the design value of grade SS of 2ˆ3 was 20.8 MPa, which was 1.08 and 1.40 times that of 2ˆ4 and 2ˆ6, respectively. It is important to note that, for convenience in practical engineering application, the design value specified in the national standard requires multiplying different adjusting factors by the same reference value for different sizes of the same grade of lumber. The design value of 2ˆ3 larch lumber was selected as the reference value. The adjusting factors of 2ˆ4 were 0.925, 0.900, 0.900, and 0.942 for grades SS, No. 1, No. 2 and No. 3, and those of 2ˆ6 were 0.713 and 0.762 for grades SS and No. 2, respectively. These results indicate that the adjusting factors were not the same among different grades, though most countries do set the same adjusting factors for different grades in one dimension. To ensure conservative estimates and safe design, it was reasonable to set adjusting factors to 0.900, 0.713 for 2ˆ4 and 2ˆ6 lumber, as shown in Figure 7. In the United States Code [2], adjusting factors are equal to 1.0 if the section height is not more than 90 mm, and grade and adjusting factors are uncorrelated. The adjusting factors of 2ˆ4 and 2ˆ6 are 1.00 and 0.867 for any tree species-significantly higher than those for larch lumber.

Conclusions
In this work, a series of static bending tests and statistical analyses were performed for Northeast China larch (Larix gmelinii) dimension lumber in order to investigate the size effect on the bending strength of dimension lumber. Frequency histograms of bending strength were fitted by normal and lognormal distribution models. Additionally, the effects of partial safety factor and live-to-dead load ratio were studied by reliability analysis. Based on the analysis of the test data, the following conclusions can be drawn: ‚ MOR 15 is affected by the size of larch dimension lumber; the bigger the size, the smaller the MOR 15 .
‚ Design value of bending strength and adjusting factor of size effect of 2ˆ3, 2ˆ4, and 2ˆ6 larch dimension lumber were successfully obtained according to the Chinese National Standards' requirements.
‚ Further developments will be mainly devoted to carry out wider experimental activity particularly on more varied sizes and species, in order to validate the adjusting factor of size effect obtained in this work.