Reliability Analysis of Normal, Lognormal, and Weibull Distributions on Mechanical Behavior of Wood Scrimber

Abstract

Reliability analysis of mechanical strength could be used for evaluation of wood scrimber properties in this study. Normal, lognormal, and Weibull distributions were used to determine and selected the optimal model for wood scrimber for the first time. The results of reliability analysis indicated that the bending and tensile strength were well fit for normal distribution. Weibull distribution could describe the probability distribution law of compression strength, and lognormal distribution could reflect the probability distribution law of shear strength, respectively. The standard value of each mechanical strength was determined and compared in accordance with two methods. This illustrated that a significant difference between these two methods is evident in the case of modulus of elasticity (MOE), compression strength (CS), and shear strength (SS), while modulus of rupture (MOR) and tensile strength (TS) yielded similar data. The improvement in mechanical strengths was remarkably affected by the increase in density. Moreover, the microstructure of wood scrimber has a good ratio of deformation with respect to density, which can be significantly explained by compressive densification. The results suggest that the deformation ratio increased from 49.75% to 78.67%, which might reflect the variation in macroscopic mechanical strength of wood scrimber.

1. Introduction

Wood scrimber has high strength and excellent durability, and has been used widely in applications such as in the field of outdoor floor, indoor decoration windows, and city guardrails. For both these cases, functional properties must be balanced with the mechanical requirements of the application. However, there is an inherent strength variability measured across seemingly identical samples, and the mean strength cannot represent an adequate predictor of performance. There are many factors that affect and determine the strength of wood scrimber created during processing, such as density, scale of unit, and chemical components, which could lead to the dispersion of mechanical parameters. Wood scrimber composed of resin and natural fiber is a type of anisotropic elastic material. Accordingly, the mechanical strength of the composite material is determined by a distribution rather than a single value, and mechanical reliability must be characterized using a probabilistic method.

Appropriate data analysis is one of the cornerstones of research. There are some probability distribution models being proposed to describe the mechanical strength of materials, which could evaluate the correctness of experimental results completely. In most cases, normal, lognormal or Weibull distribution as the main approach are used for the estimation of mechanical reliability. A normal distribution has a symmetrical shape with the highest frequency at the center of the distribution. It has certain characteristics that help researchers to make predictions based on only the mean and the standard deviation of the data [1]. Further, non-normal skewed distribution is common in biomass materials, especially in anisotropic material. The frequency distribution of a variable is said to be lognormal if the distribution of logarithm of the variable values follows a normal distribution [2]. As is well known, Weibull distribution has been successfully used to characterize the statistical variation in the fracture strength of brittle materials, such as ceramics, glass, and fiber composite [3,4,5,6]. Thus, previous studies have discussed the mechanical properties by statistical approach of distribution and reliability. It was mentioned that the normal distribution has been applied to describe the distribution law in the elastic modulus of warm frozen soil [7] and bending properties of structural recombinant bamboo [8]. Moreover, it is claimed that lognormal distribution was used to estimate the ultimate bearing capacity of reinforced concrete slab [9] and analyzed the microelement strength to reflect the fracture toughness of rock. Further, the Weibull distribution being proposed to describe the strength and reliability evaluation of strength were also reported, such as the compression strength and bending properties of lumber [10,11], and the fatigue life of rubber materials [12]. The tensile strength reliability of metallic alloy was also characterized and compared with normal and Weibull distribution statistical analysis [13]. Although different models have been proposed to describe the strength of materials, the most optimum model should be compared and selected. Characterizing the strength distribution of material and linking it to affecting parameters and microstructural features is, therefore, of primary importance for industrial applications. Hence, this is particularly vital for biocomposite, which is controlled by introducing density variation that enables the desired strength.

This study aims to determine the optimal distribution model for the mechanical strength of wood scrimber through hypothesis testing, followed by conducting a reliability analysis and making reasonable value selections based on the requirements for strength. Additionally, the distribution and reliability of the mechanical properties need to be characterized, which is of importance for the application of wood scrimber in the field of structural materials. Thus, three types of probability distributions were employed for hypothesis testing of each mechanical parameter. The reliability of density impacting the variation in strength of the composite material was ascertained. To estimate the macrostructural variation, the quantitative variation of microstructure was also determined at different densities.

2. Materials and Methods

2.1. Sampling

Wood scrimber, as the experimental material in this study, has been extensively employed in engineered structural materials, such as outdoor flooring and architecture. The raw materials of wood scrimber are poplar wood with a diameter of 25–30 cm as reinforced phase, and PF adhesive solution with 20% solid content as matrix phase. The manufacturing process of wood scrimber was accomplished with rotary cutting, fluffing, impregnation of phenolic formaldehyde resin, drying, cold pressing, and hot curing (Figure 1). The production parameters were cold press of 70–80 MPa, and hot cured temperature of 140 °C for 12 h. The experimental wood scrimber was sourced from Nantong Qinghua Wood Industrial Co., Ltd. (Nantong, China). The dimensions of wood scrimber in this study had an average length, width, and thickness of 2600 mm × 300 mm × 100 mm, respectively. The density of wood scrimber was determined by measuring its air-dry weight and volume after keeping it in a conditioning room with a relative humidity of 65% ± 5% at 20 ± 2 °C for 2 weeks. After measurement, the samples with the exact density of 0.85~1.05 g/cm³ were prepared for evaluating mechanical parameters.

2.2. Methods

2.2.1. Mechanical Characterization

Mechanical properties were conducted on a universal testing machine (WDW-W10, Time Co., Ltd., Jinan, China). Experimental samples were prepared for determination of bending, compression, tensile, and shear behaviors in accordance with standards [14,15], and the specific information of samples are shown in Figure 1.

2.2.2. Quantitative Anatomical Characteristics

Wood scrimber samples with dimension of 5 (length) × 5 (width) mm × 5 (thickness) mm were prepared, and smooth surfaces were polished using a sliding microtome. The obtained samples were coated with gold by a Gressington sputter coater (ULVAC G-50A, Chigasaki, Japan), and observed using scanning electron microscopy (SEM, Hitachi SU-70, Chiyoda, Japan). The quantitative characteristic of transverse in wood scrimber sample was conducted by the measurement function of an ultra-depth microscope. The transformation ratio of wood fiber with different density was calculated according to Equation (1) as follows:

Transformation ratio = D1/D2 × 100%

(1)

where D1 and D2 represent the longest and shortest diameter of wood fiber, respectively.

2.2.3. Distribution Analysis

The reliability distribution was analyzed by normal, lognormal, and two-parameter Weibull (2-P Weibull) distribution methods, respectively. The probability density function is defined as follows:

$$f\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{e}^{-{\displaystyle \frac{{\left(x-\mu \right)}^2}{2{\sigma }^2}}}$$

(2)

where the μ and σ values are unknown parameters. The variables of x₁, x₂, …, x_n as experimental values could be calculated by the maximum likelihood method as follows:

$$\mu ={\displaystyle \frac{1}{n}}\sum _{i=1}^n{x}_i$$

(3)

$${\sigma }^2={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(x_i-\mu \right)}^2$$

(4)

The n in Equations (3) and (4) is the total number of samples equal to the sum of n_i, which is the observed number located in the i interval, and the x_i is the value of the interval center. The probability function of normal distribution can be calculated with the substitution of Equations (3) and (4) into Equation (2), as follows:

$$f\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{\int }_{-\infty }^x{e}^{{\displaystyle \frac{{\left(t-\mu \right)}^2}{2{\sigma }^2}}}dt$$

(5)

The values were also assumed to be fitted for lognormal distribution. The probability density function f(x) of lognormal distribution is displayed as follows:

$$f\left(x\right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\sqrt{2\pi }\sigma x}}{e}^{{\displaystyle \frac{{\left(\mathit{ln}x-\mu \right)}^2}{2{\sigma }^2}}} \left(x>0\right)\\ 0 \left(x\le 0\right) \end{array}\right.$$

(6)

The μ and σ values can be also calculated into the maximum likelihood method as follows:

$$\mu ={\displaystyle \frac{1}{n}}\sum _{i=1}^n{x}_i$$

(7)

$${\sigma }^2={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(\mathit{ln}{x}_i-\mu \right)}^2$$

(8)

The n in Equations (7) and (8) is the total number of samples equal to the sum of n_i, which is the observed number located in the i interval, and the x_i is the value of the interval center. The probability function of lognormal distribution can be calculated with the substitution of Equations (7) and (8) into Equation (6), as follows:

$$f\left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}{\int }_{-\infty }^x{{\displaystyle \frac{1}{t}}e}^{{\displaystyle \frac{{\left(\mathit{ln}t-\mu \right)}^2}{2{\sigma }^2}}}dt$$

(9)

The 2-P-Weibull distribution function is defined as follows:

$$f\left(x\right)={\displaystyle \frac{\beta }{\eta }}{\left({\displaystyle \frac{\chi }{\eta }}\right)}^{{\beta }_{-1}}\mathit{exp}\left[-{\left({\displaystyle \frac{x}{\eta }}\right)}^{\beta }\right] (x>0)$$

(10)

where β is shape parameter, and η is scale parameter. These parameters are obtained by calculating the nonlinear equations according to Newton’s method, and the theoretical Weibull distribution of those mechanical properties can be settled as follows:

$$f\left(x\right)={\int }_{-\infty }^x{\displaystyle \frac{\beta }{\eta }}{\left({\displaystyle \frac{t}{\eta }}\right)}^{{\beta }_{-1}}\mathit{exp}\left[-{\left({\displaystyle \frac{t}{\eta }}\right)}^{\beta }\right] dt$$

(11)

Following the completion of the experiment, the Kolmogorov–Smirnov test (K–S test) was employed to evaluate the probability distribution of samples accurately, particularly when the sample size was limited. The selected theoretical cumulative probability frequency distribution was compared with the measured cumulative distribution, to indicate the difference between the calculated theoretical value and experimental value, which could identify the optimal distribution. The regression analysis was employed to determine the fitting degree of the regression equation, as indicated by the coefficient of R square. An F-distribution was utilized to evaluate whether a significant correlation existed between two variables upon the completion of the experiment, while a t-test was conducted to determine the significance of the regression coefficients.

Comparisons of mechanical strength in different densities were conducted using analysis of variance (ANOVA). The total data of all mechanical parameters were at least of 200 to ensure the accuracy of statistical analysis. The significance level was set as 5% in this study.

3. Results and Discussion

3.1. Mechanical Behavior of Wood Scrimber

The mechanical parameters of wood scrimber were evaluated in this study. The results of mean value (MV), standard deviation (SD), and coefficient of variance (COV) are summarized schematically in

Table 1. Statistic evaluation of mechanical strength in wood scrimber.

Mechanical Parameters	Density (g/cm3)	Mean Value (MPa)	Standard Deviation (MPa)	Coefficient of Variance (%)	Standard Value of Parameter (MPa)	Standard Value of Nonparameter (MPa)
MOR	0.85–1.05	116.34	18.71	16.08	83.74	83.08
MOE	0.85–1.05	17,661.88	1129	6.39	15,694.03	15,327.56
CS	0.85–1.05	75.63	19.11	25.27	40.72	45.17
TS	0.85–1.05	100.79	24.40	24.20	52.71	52.74
SS	0.85–1.05	14.68	2.80	19.04	5.78	6.55

. For exhibiting the variation of bending strength, modulus of rupture (MOR) and modulus of elasticity (MOE) were evaluated. The mean value of MOR was 116.34 MPa with a COV value of 16.08%, and MOE was calculated as 17661.88 MPa with a COV of 6.39%, respectively. Compression strength (CS) was determined to analyze the capacity of springback after compressive process, and the mean value was 75.63MPa, with a higher COV of 25.27%. To examine the bonding strength and peeling resistance of interface between fiber and resin, the tensile strength (TS) was conducted at 100.79 MPa with a COV of 24.20%. Further, the shear strength was also tested with a mean value of 14.68 MPa and a COV of 19.04%. Accordingly, the minimum COV value was found in the MOE and was not significantly affected by the change of density. This proved that the fiber properties and wood species were closely related on the MOE [14], whereas compressive strength displayed a distinct variance among all mechanical parameters, which was attributable to the change in fiber accumulation per unit volume as variation of density.

3.2. Analysis of Probability Distribution

The probabilistic distribution of various mechanical properties is assumed to follow a normal distribution, lognormal distribution, and 2-P Weibull distribution. Subsequently, the cumulative distribution function was computed using Equations (5), (9) and (11) and compared with experimental data. The disparity between experimental and theoretical data was thoroughly investigated and scrutinized using the Kolmogorov–Smirnov (K–S) method, which is employed for selecting the appropriate distribution. The results of distribution for each mechanical parameter are presented in Figure 2 and

Table 2. The probability distribution model of each mechanical parameter.

Parameters	Distribution	K-S Test	Decision at Level (0.05)	D0.05
	Normal	0.0554	Can’t reject Normal
MOR	Lognormal	0.0810	Can’t reject Lognormal	0.1182
	Weibull	0.0763	Can’t reject Weibull
	Normal	0.0638	Can’t reject Normal
MOE	Lognormal	0.0765	Can’t reject Lognormal	0.1191
	Weibull	0.0693	Can’t reject Weibull
	Normal	0.1771	Can’t reject Normal
CS	Lognormal	0.1853	Can’t reject Lognormal	0.2024
	Weibull	0.1575	Can’t reject Weibull
	Normal	0.0746	Can’t reject Normal
TS	Lognormal	0.1045	Can’t reject Lognormal	0.1921
	Weibull	0.0877	Can’t reject Weibull
	Normal	0.0867	Can’t reject Normal
SS	Lognormal	0.0660	Can’t reject Lognormal	0.1739
	Weibull	0.0988	Can’t reject Weibull

. It was indicated that each curve of the three distributions closely approximated the experimental data, and all assumed probability distributions successfully fitted each mechanical parameter. Therefore, the probability distribution was assessed based on the examination of the disparity between experimental and theoretical data, which were well suited for different mechanical properties. This disparity of normal, lognormal, and Weibull distribution is denoted as D_N, D_L, and D_W, respectively. As shown in Table 1, those D values for MOR were lower than the standard difference D (α = 0.05) of 0.2095, with D_N (0.0503) exhibiting the lowest value compared to D_L and D_W. Hence, the optimal distribution for MOR is the normal distribution, which closely matched the experimental data (Figure 2b). Meanwhile, all three D values of MOE were also lower than the standard difference D (α = 0.05) of 0.2095, and D_W demonstrated the lowest value of 0.0562, making it suitable for the Weibull distribution. Furthermore, the statistical analysis revealed that the Weibull distribution was fit for compression strength exactly, with Dw of 0.2024. Tensile strength was found to be most accurately represented by the normal distribution, with the lowest difference being 0.0746 in comparison to D (α = 0.05) of 0.1921. Similarly, shear strength was assessed through a comparison of experimental and theoretical data. All three D values (D_N, D_L, and D_W) were lower than the standard difference D (α = 0.05) of 0.1739, and D_L (0.0660) was the lowest among the three D values, indicating that shear strength exhibited greater consistency with the lognormal distribution.

3.3. Determination of Standard Values

The determination of standard values for mechanical properties can be achieved through two methods, including the parameter method and the nonparametric method. In the parameter method, the key lies in obtaining the coefficient of value for standard mechanical properties, determined by the number of specimens containing single samples. Conversely, the nonparametric method focuses on obtaining the sampling order number for the standard mechanical properties.

Previous research on standard values of mechanical properties predominantly concentrated on strength values, typically defining the lower 5th percentile value at the 75th confidence level as the strength standard value. Thus, in accordance with the standard [16], the method for obtaining the strength standard value can be expressed as follows:

fk = m − kS

(12)

where fk represents the strength standard value, m denotes the average strength of the specimen, S indicates the standard deviation of the strength of the specimen, and k is the characteristic coefficient. The latter is determined according to ASTM D2915 [17].

Further comparisons between two methods are presented in Table 1 for each of the strength parameters in detail. The results suggested that there are discernible variations in the standard values obtained from different calculation methods, with MOR and TS yielding similar data, while significant difference between these two methods is evident in the case of MOE, CS, and SS. The obtained mechanical standard values resulting from different calculation could be assessed and evaluated according to diverse application demands.

3.4. Analysis of Effect Factors on Mechanical Properties

The analysis of reliability distribution on mechanical strength was conducted through the comparison of various distributions. The impact of density on mechanical parameters varies, resulting in different probability distributions. Thus, the variation of density was discussed in this study, and the related results are shown in Figure 3 and Figure 4. It is declared that the MOR of specimens increased from 102.37 to 138.98 MPa with an increase in density (Figure 3). This result could be attributed to an increase in the fiber amount or compression ratio, which suggests that the addition of fiber might lead to a higher compression ratio. As the compression ratio increased, the bonding points would be increased to join the adhesive with fiber. A significant increase was observed in the MOE as the density of the wood scrimber increased. The range of data in the MOE was from 16,573 to 18,395 MPa. Regarding bending strength, density had a positive effect on those values. The remarkable improvement in the MOR and the MOE could be generalized as follows: the load-bearing properties of wood mainly depend on the wood fiber [18]. Due to the densification of wood cell and vessels, the fiber content per unit of volume increased. In general, the mechanical performances of wood scrimber mainly depended on the structure and strength of cellulosic fibers, which were increased through structural densification, resulting in high endurance capacity for these composites. Moreover, the fracture failure with the fibrous break mainly appeared in bending samples, while the failure of adhesive layer was present occasionally (Figure 3).

Figure 4 shows the variation of selected mechanical strengths of wood scrimber at different densities and the failure modes in wood scrimber. The results of compressive strength are shown in Figure 4a. The results show that the density significantly affected the compressive strength. The CS value varied from 61.0 to 98.0 MPa as density increased from 0.90 to 1.05 g/cm³, with an overall increase of 60%. During the compression process, there are two primary failure modes associated with longitudinal compression [19,20]. One is delamination failure, resulting from fiber buckling, primarily due to the buckling of fibers and matrix failure leading to the loss of load-carrying capacity. The other is shear failure, in which the fiber breaks and delamination occurs within this failure mode. Moreover, the delamination of adhesive layers might occur in wood scrimber in the case of lower density. The shear strength of wood scrimber in the vertical direction at different densities is exhibited in Figure 4c. The results increased from 10.03 MPa to 13.39 MPa with an increase in the density from 0.90 to 1.05 g/cm³, which increased by 33.50%. This could be attributable to the chemical bond force between resin and wood fibers, the layer thicknesses, and surface treatment of bonding materials [21]. Moreover, wood scrimber exhibited fracture failure, which mainly showed the shear failure of fibers (Figure 4). Wood scrimber has fibers as the matrix, which are stacked layer by layer. The fibers have extremely high intralaminar strength, but due to the interactions between the fibers, it is easy for damage to occur between the fibers, and delamination is the main failure mode of fiber-reinforced composite materials [22]. There was failure of the adhesive layer observed in a small quantity of samples with lower density of 0.85 and 0.90 g/cm³ due to the bonding issue. Figure 4e illustrates the relation between the tensile strength and the density of wood scrimber. A significant enhancement in tensile strength of 63.49% was observed with increasing density from 0.85 to 1.00 g/cm³. The longitudinal tensile load of composite materials is primarily carried by the fibers, and the strength mainly depends on both the strength and volume fraction of the fibers [23]. As the density increases in the process of increasing the volume fraction of fibers, this results in an upward trend in overall strength. The failure mode is fiber extraction and fracturing, and delamination occurs at low density. The extraction failure mode of fiber specimens is mainly influenced by the bonding performance between the specimen and the resin matrix, as well as the tensile strength of fiber. A previous study discovered that the fiber bonding stress reached peak bonding strength and then detached [24]. Subsequently, the friction on the fiber surface was damaged, and the extent of damage continued to expand with the appearance of the extraction process, leading to rapid fracture of some fiber specimens after the fiber detached and experienced overall slippage. Thus, even after the fiber is impregnated with resin, there are still defects along the direction of force, resulting in the phenomenon of stress concentration during the pull-out process. Upon the failure of matrix bonding after reaching the bonding strength peak, the friction on the fiber surface is damaged, and the fiber fractures rapidly following slippage. Consequently, appropriate density should be considered for the improvement in mechanical properties.

3.5. Micro-Structure Evaluations

The anatomical characteristics of wood scrimber and transformation ratio are summarized in Figure 5. The transverse diameter of wood fiber was distinctly transformed with increasing density, while radial diameter was reduced (Figure 5a,b). As a result, density has a positive relation with transformation ratio, which increased from 49.75% to 78.67% with an increase in density. The compressive deformation of cells can be separated into stages: elastic deformation, stationary deformation, and compression deformation. At the stage of elastic deformation, there was basically no change in wood cells, while the cell wall and cell lumen happened to bend and decrease, respectively, with the linear increase in load. Further, the cells happened to be notably compressed with increasing density of 1.10 g/cm³, as shown in Figure 5c, which was decreased by 36.76% totally. Therefore, designing density is an important factor affecting the mechanical performance. With increasing density, the microstructure gradually becomes compressed, resulting in the pores becoming smaller. The wood rays and fibers also exhibit noticeable nonuniform deformations and are gradually compressed. The shapes of the conduit cells in the wood scrimber are diverse, and they are distributed irregularly.

4. Conclusions

The experimental results underwent reliability analysis, comparing three distributions. The analysis illustrated that the bending and tensile strength were fit for normal distribution, and compression and shear strength followed Weibull and lognormal distribution, respectively. Specifically, density was proven to influence the mechanical properties, exhibiting the same distribution model within the overall result analysis. The standard value of each parameter was also compared with different methods, using optimal calculation to ascertain. Additionally, there was a notable increasing trend of mechanical parameters impacted by the increase in density, indicating the important role of density in enhancing mechanical properties. Furthermore, the variation in density demonstrated a positive correlation with the transformation ratio of the cell wall. In conclusion, the reliability of strength for different densities could provide a support for controlling the properties of wood scrimber and is beneficial for industrial manufacturing according to different engineered structural applications.