Investigating the Influence of Tenon Dimensions on White Oak (Quercus alba) Mortise and Tenon Joint Strength

Abstract

The dimensions of tenons in solid wood furniture significantly influence the mechanical performance of mortise and tenon joints. While previous studies have primarily focused on tenon length, width, and thickness, they often overlooked the impact of clearance between the mortise and tenon. This study investigates the effects of tenon length, tenon width, and clearance on the mechanical performance of mortise and tenon joints, aiming to enhance their bending moment capacity (BMC) and stiffness. A three-factor, three-level orthogonal test was conducted, utilizing range analysis and variance analysis to assess the effects of each factor on BMC and stiffness. The LSD post hoc test was employed to identify significant differences between levels of the same factor, and nonlinear regression analysis was used to fit the experimental results. Based on orthogonal experiment outcomes, a grey relational theory-based evaluation system was developed to assess the comprehensive performance of joints, including both moment capacity and stiffness. Results indicate that tenon length has the most significant effect on BMC, followed by clearance and tenon width, while clearance has the greatest impact on stiffness, followed by tenon length and tenon width. These findings are consistent with those obtained from grey relational analysis. When considering both BMC and stiffness as a comprehensive evaluation, the optimal combination is a tenon length of 40 mm, a tenon width of 35 mm, and a clearance of −0.1 mm. This study offers valuable insights for the rational design of mortise and tenon joints, contributing to improved performance and reduced manufacturing costs.

1. Introduction

The mortise and tenon joint is commonly used in solid wood furniture [1]. Wooden structures that utilize this type of joint typically exhibit semi-rigid characteristics [2]. Bending moment capacity is one of the key factors determining the strength of wooden structures. Previous research has found that the type of wood, growth rings, type of adhesive, tenon size, and tenon shape can affect the bending moment capacity and stiffness of the joint [3,4,5]. The change in ambient temperature can also lead to variations in the equilibrium moisture content of wood, thereby affecting the fit between the mortise and tenon joints and their mechanical performance [6]. Smardzewski et al. [7] conducted experimental studies on the effects of wood type and adhesive type on joints. The results indicated that the bending of the tenon and shearing of the adhesive caused the contact between the tenon and mortise. The value of the linear elastic modulus of the adhesive had a decisive impact on the compressive stress between the tenon and mortise. Kasal et al. [8] studied the effects of wood type, adhesive type, tenon width, and length on the bending load capacity and stiffness of T-shaped mortise and tenon joints through a full-factorial experiment. The results showed that both the bending moment capacity and stiffness of the joint increased with the increase in tenon length and width. Moreover, the tenon length had a more significant impact on the bending moment capacity than the tenon width, while the tenon width had a more significant impact on the joint stiffness than the tenon length. Zaborsky et al. [9] studied the effects of wood type, joint type, tenon thickness, adhesive type, loading method, and grain orientation on the bending stiffness of joints. The results indicated that wood type, joint type, tenon size, and adhesive type have significant impacts on bending stiffness, but their interactions were not significant. Grain orientation and loading method did not have a significant effect on joint stiffness. Harada et al. [10] studied the impact of moisture content on the mechanical properties of mortise and tenon joints. The results showed that for pre-cut joints, moisture content had no significant effect on ultimate bending moment capacity and initial stiffness. The stiffness of joints assembled with kiln-dried wood decreased as the moisture content increased.

The aforementioned research results indicated that the geometric shape of the tenon could significantly affect the bending moment capacity and stiffness of the joint. Previous studies have investigated the effects of tenon shape and size on joint performance using various methods. Eckelman et al. conducted tests to determine the impact of a tightly fitting shoulder on the bending moment capacity of round mortise and tenon joints. The results indicated that a tightly fitting shoulder could significantly increase the strength of the joint, and a mathematical model was established to predict the bending moment capacity of the joint [11]. Oktaee et al. [12] conducted experimental research on the impact of tenon geometric shape on the bending moment capacity of joints. The study found that the length of the tenon had the greatest influence on the bending moment capacity of the joint, while the impact of the tenon width was much less significant. A tenon thickness of 10 mm provided greater load-bearing capacity compared to thicknesses of 7.5 mm or 15 mm. Derikvand et al. [13] further investigated the effects of shoulder width, tenon embedment depth, and tenon width on the bending moment capacity of joints. The results indicated that the bending moment capacity was directly related to the embedment depth of the tenon and closely related to the width of the shoulder, while the impact of tenon width was relatively minor. These studies mainly used full-factorial experiments, which involve a large number of trials and higher experimental costs.

In addition, many studies have utilized finite element analysis to evaluate the performance of joints, aiming to find more effective ways to calculate joint strength. Mackerle conducted a literature review on the application of finite element analysis in wood analysis, primarily focusing on its use in wooden structures and wood products [14]. Derikvand et al. used finite element analysis to study the effects of tenon size on stress and strain distribution in T-shaped mortise and tenon joints under uniaxial bending loads [15]. The results showed that the ultimate shear stress occurred in the middle of the tenon, and the highest shear elastic strain was estimated at the glue layer between the tenon and mortise. Kasal et al. used finite element analysis to study the impact of tenon dimensions on the bending moment capacity and stiffness of L-shaped and T-shaped mortise and tenon joints [16]. The results indicated that tenon length had a more significant effect on bending moment capacity, while tenon width had a more significant impact on the stiffness of the joint, consistent with earlier experimental research findings. Hu et al. [17] further combined finite element analysis with response surface methodology to study the bending moment capacity and stiffness of mortise and tenon joints. The results similarly indicated that the length of the tenon had a more significant effect on the bending moment capacity of the joint, while the width of the tenon had a more significant impact on the stiffness of the joint. The aforementioned studies demonstrate that finite element analysis is helpful in studying the mechanical properties of mortise and tenon joints. However, since wood is a natural heterogeneous anisotropic polymer material with complex mechanical properties, experimental validation is needed to confirm the effectiveness of the finite element analysis results.

Some studies have established mathematical models to estimate the performance of mortise and tenon joints. Sandberg et al. developed a joint stiffness model for analyzing traditional timber frames. The study found that the stiffness model underestimated the stiffness of typical fixed knee joints by 20%–30% [18]. In timber structure construction, the gap between the tenon and mortise affects the performance of the joint. Ogawa et al. derived a method for estimating the mechanical performance of mortise and tenon joints using gap size as a parameter and validated the estimation method through experimentation. The results showed that the estimated relationship between the bending moment capacity and deformation angle of the joint matched the experimental outcomes [19]. He et al. studied the mechanical performance of penetrated mortise and tenon joints involving gaps and proposed a theoretical model for the bending moment of the joint. The results showed that the theoretical model corresponded well with experimental results, making it a viable tool for estimating the bending moment of such joints [20]. Currently, mortise and tenon joints are machined using CNC machines. Due to the low processing efficiency and high cost of square mortise and tenon joints, circular or oval mortise and tenon joints are more commonly used in wooden structures and furniture. Previous mathematical models were primarily focused on specific research subjects and lacked general applicability. Moreover, most of these models were developed for square mortise and tenon joints, with limited research conducted on oval mortise and tenon joints.

Grey system theory can solve uncertainty issues through small datasets and poor information, which is different from the problems solved by probability, fuzzy mathematics, or rough set theory [21]. The grey relational analysis within it is widely applied in the field of natural sciences [22], Prakash et al. optimized the turning of aluminum/rock powder composite materials and their processing parameters through Taguchi and grey relational analysis (GRA) to improve surface finish and material removal rate (MRR) [23]. Loganathan et al. first assessed the impact of feed rate, spindle speed, and step depth on the surface roughness and wall thickness of the AA6061 alloy through Taguchi experiments and then determined the optimal forming parameters using grey relational analysis [24]. Pagar et al. used the same method to study the effects of the number of turns (N) and pitch diameter (dp) on the radial deflection stress of bellows [25]. Khan et al. employed grey relational analysis to perform multi-objective optimization of turning titanium-based alloys under various conditions [26]. Yu et al. proposed a grey relational method based on game theory for quantitatively analyzing the combustion performance of wood [27]. The results showed that the combustion performance of the wood types, from worst to best, was as follows: pine > cedar > oriented strand board > medium-density fiberboard > plywood > elm. Wang et al. used grey system theory to analyze and determine the effects of ultrasonic-assisted dyeing parameters on wood performance and optimized the parameters [28]. From previous studies, it can be seen that grey relational analysis is widely used in multi-objective optimization of structures and materials. The bending moment capacity and stiffness are important indicators of the performance of mortise and tenon joints. Grey relational analysis can be used to optimize the joint parameters to maximize the performance of the joints.

In solid wood furniture, mortise and tenon joints are critical structures for connecting different parts. They are used in connections such as table and chair legs to frames, beams to uprights in bookshelves, cabinets, and beds, where they bear bending loads. Therefore, it is necessary to study the bending resistance of these joints through static bending tests, focusing mainly on their bending moment capacity (BMC) and stiffness. Previous research has shown that both the length and width of the tenon significantly affect the bending moment capacity and stiffness of the joint. Full-factorial experiments provide high accuracy in research, but they involve large-scale testing and complex data processing. Finite element analysis and mathematical models are helpful for studying the mechanical properties of mortise and tenon joints; however, their accuracy and general applicability need to be improved. In addition, there has been limited research on the effect of the clearance between the tenon and the mortise on the mechanical performance of joints. Moreover, most studies have only investigated either the bending moment capacity or the stiffness of joints separately. It is necessary to establish an evaluation system that includes bending moment capacity and stiffness to study the comprehensive performance of the joint.

This study aims to investigate the relationship between tenon dimensions and the mechanical performance of mortise and tenon joints. Specifically, we focus on the following: (1) identifying the tenon dimensions and the clearance between the mortise and tenon that have the greatest impact on bending moment capacity (BMC) and stiffness; (2) developing predictive models for BMC and stiffness using regression analysis and evaluating the comprehensive performance of the joints considering both BMC and stiffness through grey relational analysis; (3) determining the optimal combination of tenon dimensions to achieve the best comprehensive performance.

2. Materials and Methods

2.1. Materials

The wood used in this study was 50-year-old white oak (Quercus alba) purchased from Zhejiang Yuhua Timber Co., Ltd., Jiaxing, China. As the tree ages, the cell walls of white oak become thicker and the lignin content increases, providing better mechanical performance. White oak within this age range has high structural strength and is resistant to deformation or fracture. It is also less likely to shrink or expand due to changes in temperature and humidity. Therefore, white oak is commonly used in furniture structures.

2.2. Description of Specimens

The wood used for processing the specimens was kiln-dried, straight-grained, and defect-free. After processing, the specimens were conditioned for 7 weeks in a controlled environment with a temperature of 20 °C and a relative humidity of 65% to achieve an equilibrium moisture content of approximately 11.6%. At the moment of the tests, the mean specific gravity was 0.68, measured according to ASTM D2395-17, and the mean moisture content was 9.95 ± 0.21%, measured according to ASTM D4442-20. All specimens used in this study were machined by CNC machines (balestrini pico om, SCM, Rimini, Italy). Figure 1 shows the configuration of the T-shaped mortise and tenon joint used in this study. Each joint consisted of two components, a post and a rail. Posts measured 200 mm long × 40 mm wide × 40 mm thick; rails measured 180 mm long × 40 mm wide × 40 mm thick. There were three variables in this study: tenon length (30 mm, 35 mm, 40 mm), tenon width (25 mm, 30 mm, 35 mm), and the clearance between the tenon width and mortise height. Different clearance values determine different fits. The fits were interference (the clearance of 0.1 mm), transition (the clearance of 0 mm), and clearance (the clearance of −0.1 mm). The fit in the direction of tenon thickness and mortise width was clearance (the clearance of −0.1 mm). A 16 mm diameter drill bit was used to machine the mortises in this study, resulting in a constant tenon thickness of 16 mm. To study the effects of the three variables on joint performance, no adhesive was applied between the mortise and tenon joint.

2.3. Bending Test

Since there are no existing standards to guide the bending tests of mortise and tenon joints, this study determined the testing method based on the actual stress conditions of the joints in practical applications and previous research. Static bending tests were performed on a universal mechanical testing machine (Instron5582, Instron, Norwood, MA, USA), as shown in Figure 2. The post was kept fixed by specimen holder, while the rail had no support in the direction of the applied force. The rail accepted the pressure (P) from the load head of the testing machine, which was located 110 mm from the edge of the post, and the load was applied until the joint completely failed. The loading rate was 5 mm/min, and 3 samples were tested for each joint configuration, with a total of 27 samples tested. The deflection (mm) and load (N) during loading were recorded.

After measurement, the bending moment capacity (kN·m) and stiffness (kN·m/rad) of the joints were calculated using the following formulas:

(1)$M=P\cdot L$

(2)$k={\displaystyle \frac{M}{\theta }}$

where $M$ is the bending moment capacity (kN·m), $P$ is the reaction force measured by the load head (kN), and $L$ is the moment arm (m), which is 0.11 m in this study. $k$ is the stiffness, and $\theta$ is the rotation angle of the rail.

2.4. Orthogonal Experiment Design

Orthogonal testing has been widely used in engineering analysis, saving a lot of time in obtaining the optimal set of levels. The key to this method is the reasonable selection of representative factors and levels. This method selects a small number of representative experiments by using orthogonal arrays, and the results of these experiments can be statistically analyzed to determine better parameters. In this study, there were 3 factors, each with 3 levels, as shown in Table 1. The most appropriate orthogonal design table L₉(3⁴) was selected, reducing the number of tests from 27 to 9.

Experiments were conducted to study the effects of different combinations of three factors: tenon length (factor A), tenon width (factor B), and clearance (factor C) on the bending moment capacity and stiffness of the tenon joint. According to the L₉(3⁴) orthogonal experiment design table, a total of 9 groups of experiments were designed, and each group of experiments was repeated 3 times. The experiments were conducted using the L₉ mixed orthogonal array, as shown in Table 2.

2.5. Analysis of Variance (ANOVA)

ANOVA analysis is a statistical method used to determine the individual interactions of all factors in an orthogonal experiment. In this study, ANOVA was used to analyze the effects of tenon length, tenon width, and clearance on the ultimate bending moment capacity and stiffness of mortise and tenon joints. The analysis was performed at a 5% significance level and a 95% confidence level. The significance of factors in the ANOVA is determined by comparing the F-values of each factor.

2.6. Grey Relational Coefficient and Grey Relational Grade

Grey system theory is used to solve issues of data scarcity and uncertainty [29]. Grey relational analysis measures the approximate correlation between sequences by the absolute values of data differences between them [30]. Due to different units and ranges in various data sequences, it is necessary to preprocess the data, converting the original data into comparable sequences [31,32]. The calculation formula is as follows:

(3)$y_i\left(k\right)={\displaystyle \frac{x_i\left(k\right)-\underset{k}{min}{x}_i\left(k\right)}{\underset{k}{max}{x}_i\left(k\right)-\underset{k}{min}{x}_i\left(k\right)}}$

where $y_i\left(k\right)$ is the comparable sequences, $x_i\left(k\right)$ is the measured results, $\underset{k}{min}{x}_i\left(k\right)$ is the minimum value of $x_i\left(k\right)$, $\underset{k}{max}{x}_i\left(k\right)$ is the ultimate value of $x_i\left(k\right)$, $i=1,2,\dots ,m$, is the number of objective functions, and $k=1,2,\dots ,n$, is the number of experiments.

The preprocessed reference sequence is $Y_0=\left\{y_0\left(k\right),k=1,2,\dots ,n\right\}$, and the sequences of the objective function are $Y_i=\left\{y_i\left(k\right),k=1,2,\dots ,n\right\},i=1,2,\dots ,m$. The grey relational coefficient of $Y_i$ with respect to $Y_0$ at the $k$ point can be calculated as follows:

(4)${\xi }_i\left(k\right)={\displaystyle \frac{\underset{i}{min}\underset{k}{min}\left|y_0\left(k\right)-y_i\left(k\right)\right|+\rho \underset{i}{max}\underset{k}{max}\left|y_0\left(k\right)-y_i\left(k\right)\right|}{\left|y_0\left(k\right)-y_i\left(k\right)\right|+\rho \underset{i}{max}\underset{k}{max}\left|y_0\left(k\right)-y_i\left(k\right)\right|}}$

where ${\xi }_i\left(k\right)$ is the grey relational coefficient, $\rho$ is the resolution coefficient which is generally 0.5, $\left|y_0\left(k\right)-y_i\left(k\right)\right|$ is the absolute difference between $y_0\left(k\right)$ and $y_i\left(k\right)$, $\underset{i}{min}\underset{k}{min}\left|y_0\left(k\right)-y_i\left(k\right)\right|$ is the smallest value of $\underset{k}{min}\left|y_0\left(k\right)-y_i\left(k\right)\right|,$ and $\underset{i}{max}\underset{k}{max}\left|y_0\left(k\right)-y_i\left(k\right)\right|$ is the largest value of $\underset{k}{max}\left|y_0\left(k\right)-y_i\left(k\right)\right|$.

The grey relational degree can be calculated as follows:

(5)$\gamma \left(Y_i,Y_0\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum }_{k=1}^n{\lambda }_k\xi \left[y_i\left(k\right),y_0\left(k\right)\right]$

where ${\lambda }_k$ is the weighting coefficient.

2.7. Statistical Analysis

All data were statistically analyzed using IBM SPSS Statistics 27 software. One-way analysis of variance (ANOVA) and LSD post hoc test were conducted, with p < 0.05 indicating statistical significance.

3. Results and Discussion

3.1. The Force–Displacement Curve

Figure 3 shows the M-θ curves for various joints, which reflect the ultimate bending moment capacity (BMC) and stiffness of the joints. It can be observed that after exceeding the yield point, the joint can continue to deform without immediately breaking, undergoing significant plastic deformation before failure. Therefore, the failure mode of the mortise and tenon joint is ductile failure. The curve can be divided into the sliding phase, elastic phase, plastic phase, and failure phase. In this study, the calculation of stiffness is based on ASTM D4761-19, selecting 10%–40% of the ultimate load to calculate the joint stiffness. After reaching the ultimate load-bearing capacity, the curve exhibits a step-like appearance because some fibers oriented perpendicular to the grain direction, despite being crushed, still retain a certain load-bearing capacity.

3.2. Failure Modes

Figure 4 displays the mortise and tenon joint and the tenon after failure. During the loading process, the tenon and mortise gradually tighten together, accompanied by a slight “squeaking” sound. As the load increased, the compression deformation at the contact area between the tenon and the mortise increased, causing the tenon to be pulled outward. After examining the specimen post-testing, it was observed that the mortise exhibited no noticeable deformation. However, the head of the tenon experienced substantial plastic deformation, and compression deformation was evident at the neck of the tenon. The head of the tenon primarily withstood pressure perpendicular to the grain and friction parallel to the grain. As the angle of rotation increased, the pressure at the tenon’s end intensified, and the friction between the tenon and the mortise also increased. After the head of the tenon was crushed, cracks perpendicular to the grain formed under the tensile forces parallel to the wood grain. As shown in Figure 4b, after the initiation of the crack, it did not propagate in the direction perpendicular to the grain. Instead, it turned and extended parallel to the grain direction. This is because the crack tends to propagate in the direction where the normal stress ratio around the crack tip is greatest, and at the crack tip, the normal stress ratio parallel to the grain direction is the highest [33].

3.3. Range Analysis of Experimental Results

In orthogonal experiments, range analysis allows for a quick assessment of the factors affecting bending moment capacity and stiffness without complex statistical calculations. This is crucial for understanding the importance of each factor and how to optimize joint design.

The results of the orthogonal experiment are shown in Table 3. By conducting range analysis on the data in Table 3, we obtain Table 4 and Table 5. Table 4 and Table 5 showed the mean values and ranges of the bending moment capacity and stiffness of the joints, respectively. K_i (i = 1, 2, 3) is the sum of the corresponding test results when the number of levels in any column is i. k_i (i = 1, 2, 3) is the arithmetic mean of the corresponding test results when the number of levels in any column is i. The range R is the difference between the ultimate and minimum k values, used to measure the impact of different levels of this factor on the dependent variable. The greater the range, the greater the change in the test results caused by the change in the factor. Therefore, the order of significance of these three factors on the bending moment capacity of mortise and tenon joints is A > C > B, among which the tenon length has the most significant influence, followed by the clearance and the tenon width. The order of significance for the stiffness of mortise and tenon joints is C > A > B, among which the clearance has the most significant influence, followed by the tenon length and tenon width.

3.4. ANOVA Method

Normality and homogeneity tests need to be conducted on the data before performing the analysis of variance. A P-P plot compares the observed cumulative probability of a data sample with the expected cumulative probability. The horizontal axis represents the observed cumulative probability, while the vertical axis represents the expected cumulative probability. When the data sample is approximately distributed along the 45° diagonal line, it indicates that the data sample follows a normal distribution. From Figure 5, it can be seen that all 27 data samples for bending moment capacity and stiffness are approximately distributed along the 45° diagonal line, indicating that both datasets follow a normal distribution.

Levene’s test for equality of variances assesses whether the variances among different groups are equal. If p > 0.05, it indicates that there is no significant difference in variances among the groups, thereby satisfying the homogeneity assumption. From Table 6, it can be seen that both bending moment capacity and stiffness satisfy the homogeneity assumption.

Table 7 shows the results of variance analysis of the bending moment capacity of mortise and tenon joints. Since p < 0.001, all three factors are significant. It can be seen from Table 7 that the F values of factors A, B, and C are 28.405, 11.596, and 22.494, respectively. Therefore, the most important factor affecting the bending moment capacity of the mortise and tenon joint is the tenon length (factor A), followed by the clearance (factor C) and the tenon width (factor B), which is consistent with the range analysis results. Table 8 shows the results of the ANOVA for the stiffness of the mortise and tenon joints. According to the ANOVA results, p < 0.001 for factors A, B, and C, indicating that all three factors are significant. The F values of factors A, B, and C are 20.756, 20.183, and 54.868, respectively. This indicates that the most significant factor affecting the joint stiffness is the clearance (factor C), followed by the tenon length (factor A) and the tenon width (factor B), which is consistent with the range analysis results.

To investigate the significance differences between different levels of each factor, post hoc tests were conducted using the LSD (least significant difference) method for pairwise comparisons. The LSD post hoc test is used to further identify whether there are significant differences between different levels of the same factor after a one-way ANOVA shows an overall significant difference. The analysis was conducted at a significance level of 5% and a confidence level of 95%. Figure 6a shows the post hoc test results for the effect of different tenon lengths on the bending moment capacity of the joints. It can be found that the bending moment capacity increases with the increase in tenon length, and there are significant differences between different levels. Figure 6b shows the post hoc test results for the impact of different tenon widths on the bending moment capacity of the joints. It can be found that the bending moment capacity tends to increase with the increase in tenon width. However, there is no significant difference in the impact on the bending moment between the tenon widths of 30 mm and 35 mm. Figure 6c shows the post hoc test results for the impact of different clearances on the bending moment capacity of the joints. It can be found that the bending moment capacity decreases with the increase in clearance, and there are significant differences between different levels. Figure 6d presents the post hoc test results for the effect of different tenon lengths on the stiffness of the joints. It can be found that the stiffness tends to increase with the increase in tenon length. However, there is no significant difference in the impact on the stiffness of the joints between the tenon lengths of 35 mm and 40 mm. Figure 6e shows the post hoc test results for the impact of different tenon widths on the stiffness of the joints. It can be observed that the stiffness tends to increase with the increase in tenon width. However, there is no significant difference in the impact on the stiffness of the joints between the tenon widths of 30 mm and 35 mm. Figure 6f presents the post hoc test results for the effect of different clearances on the stiffness of the joints. It can be found that the stiffness tends to decrease with the increase in clearance. However, there is no significant difference in the impact on the stiffness of the joints between the clearance of −0.1 mm and 0 mm.

3.5. Evaluation of Experimental Results

Figure 7 illustrates the variations in bending moment capacity and stiffness of the joints as obtained from the experimental results. The length, width, and clearance of the tenon directly influence the bending moment capacity of the joint. In the interaction diagram of tenon length and tenon width, as shown in Figure 7a, when the tenon width is constant, the bending moment capacity increases as the tenon length increases. This may be due to the enlargement of the compressed area of the tenon, resulting in an increase in its compressive strength. When the tenon length is constant, the growth trend of the bending moment capacity becomes gentle as the width of the tenon increases, which is consistent with the post hoc test results. This may be due to the fact that with the increase in tenon width, part of the tenon is crushed and then slips out of the mortise, while the main factor affecting the ultimate bending moment capacity of the joint is the transverse compressive strength of the tenon head. The interaction between tenon length, width, and clearance is shown in Figure 7b,c. When the clearance remains unchanged, the bending moment capacity increases with the increase in the tenon length. When the tenon length and tenon width remain constant, the bending moment capacity decreases as the clearance increases. During the loading process, part of the energy is converted into elastic–plastic strain energy, and the rest is dissipated through frictional energy dissipation. The contact stress between mortise and tenon increases with the clearance, and the friction force also increases, resulting in it being more difficult for the tenon to slip. Therefore, more energy is converted into elastic–plastic strain energy, and the tenon is more likely to be crushed, leading to joint failure. In the case of clearance fit, the tenon is easier to pull out, and the ultimate bending moment capacity of the joint is jointly determined by the cross-grain compressive strength of the tenon and the friction between the mortise and tenon. Therefore, the clearance fit joint has a larger bending moment capacity.

Tenon length, width, and clearance directly affect the stiffness of the joint. As shown in Figure 7d, the joint stiffness increases with the increase in tenon length and width. The interaction of tenon length and tenon width with clearance is shown in Figure 7e,f, respectively. When the tenon length or width remains unchanged, the stiffness of the joint first increases and then decreases with the increase in clearance, and the peak appears when the clearance is 0 mm. This may be because when the joint is a clearance fit, the tenon will have a certain degree of free rotation, resulting in stress concentration, making the tenon more susceptible to deformation. When the joint has a clearance, most of the energy is converted into elastic–plastic strain energy, making the joint more susceptible to deformation.

Therefore, appropriately increasing the length of the tenon can improve the bending moment capacity and stiffness of the joint. When the clearance in the tenon width direction is 0 mm, better joint stiffness is exhibited. Although the tenon width affects the bending moment and stiffness of the joint, its significance is lower than the other two factors. The influence diagram of the control factors obtained through orthogonal methods on the bending moment capacity and stiffness variation (Figure 8) also validates the results of the experimental study.

3.6. Regression Analysis of Bending Moment Capacity and Stiffness

Regression analysis is used to model and analyze multiple variables, where there is a relationship between a dependent variable and one or more independent variables. In this study, the dependent variables are bending capacity and stiffness, and the independent variables are tenon length, tenon width, and clearance of the joint. Regression analysis was used in obtaining the predictive equations for joint bending moment capacity and stiffness. Predictive equations are formulated through linear and nonlinear regression. The prediction equations for bending moment capacity and stiffness obtained through linear regression are as follows:

(6)$\begin{array}{c}{M}_l=-0.063+0.004T_l+0.002T_w-0.002A\\ R^2=0.746\end{array}$

(7)$\begin{array}{c}{K}_l=-0.806+0.028T_l+0.03T_w-2.259A\\ R^2=0.805\end{array}$

Here, $M_l$ and $K_l$ show the prediction equations for bending moment capacity and stiffness, respectively. $T_l$, $T_w$, and $A$ represent tenon length, tenon width, and clearance, respectively.

Figure 9 shows a comparison of the actual test results and the predicted values of the linear regression model. The $R^2$ values of the equations obtained by $M_l$ and $K_l$ are 0.746 and 0.805, respectively.

The prediction equations for bending moment capacity and stiffness obtained through nonlinear regression are as follows:

(8)$\begin{array}{c}{M}_{nl}=-\mathrm{12,636.642}-1263.67\times T_l+1263.686\times T_w+\mathrm{63,182.931}\times A+252.73\times {T_l}^2+252.73\times {T_w}^2\\ +\mathrm{631,824.998}\times A^2-505.46\times T_l\times T_w-\mathrm{25,273.034}\times T_l\times A+\mathrm{25,273.018}\times T_w\times A\\ R^2=0.885\end{array}$

(9)$\begin{array}{c}{K}_{nl}=-8970.049-896.918\times T_l+896.979\times T_w+\mathrm{44,841.808}\times A+179.385\times {T_l}^2+179.384\times {T_w}^2\\ +\mathrm{448,445.725}\times A^2-358.770\times T_l\times T_w-\mathrm{17,938.663}\times T_l\times A+\mathrm{17,938.763}\times T_w\times A\\ R^2=0.941\end{array}$

Here, $M_{nl}$ and $K_{nl}$ show the prediction equations for bending moment capacity and stiffness.

Figure 10 shows the test results and a comparison of the predicted values obtained by the nonlinear regression model. It can be seen that there is a good correlation between the predicted values and the test results. The $R^2$ values of the equations obtained by the nonlinear regression models of $M_{nl}$ and $K_{nl}$ are 0.885 and 0.941, respectively. Therefore, nonlinear regression models obtain more intensive predictive values compared to linear regression models. The results show that the nonlinear regression model can more successfully estimate joint bending moment capacity and stiffness.

3.7. Grey Relational Analysis Results

Due to the different units of measurement for bending moment capacity and stiffness, the experimental results from Table 3 were normalized using Equation (3). The normalized data and the series of differences are presented in Table A1.

By substituting the results into Equations (4) and (5), respectively, the grey correlation coefficient and grey correlation degree of the response can be obtained, as shown in Table A2.

Grey relational analysis was conducted separately for the bending moment capacity and stiffness of the joints, focusing on single-factor objectives from the experimental results. Based on the grey relational coefficients corresponding to different levels of each parameter in Table A2, the average grey relational coefficients for different objectives were calculated, as shown in Table 9 and Table 10.

According to the concept of grey relational degree, the larger the difference in the average grey relational coefficients for different levels of the same factor, the greater the impact on the target; the parameter combination with the highest average grey relational coefficient is the optimal performance combination. Therefore, the factors affecting the bending moment capacity of the joints are ranked as A > C > B, with tenon length having the most significant impact, followed by clearance and tenon width. The combination yielding the highest bending moment capacity is A3B3C1, as shown in Table 9. For the stiffness of the joints, the impact order is C > A > B, with clearance having the most significant impact, followed by tenon length and tenon width. The combination with the greatest joint stiffness is A3B3C1, as shown in Table 10. These results are consistent with those of the range analysis and variance analysis.

When designing mortise and tenon joints, it is necessary to consider both the ultimate strength of the joint and its ability to resist deformation. Therefore, a multi-objective optimization analysis is required for both the bending moment capacity and stiffness of the joints, with results presented in Table 11.

According to the properties of grey relational degree, the size of the grey correlation degree reflects the degree of influence of different levels of each factor on the comprehensive performance of the mortise and tenon joint. By comparing the different levels of each factor, it is evident that the level with the highest grey relational degree is the optimal level for the comprehensive performance of the joint. The following conclusions can be drawn: for tenon length, the grey relational degree ranking affecting joint performance is A3 > A2 > A1; for tenon width, it is B3 > B2 > B1; for clearance, it is C1 > C2 >C3. Thus, for the comprehensive performance of the joint, the optimal combination of factors is A3B3C1. Based on the differences in grey relational degrees across different levels, the order of influence on the joint’s comprehensive performance is tenon length > clearance > tenon width.

4. Conclusions

This study used orthogonal experiments and grey relational analysis to examine the effects of tenon length, tenon width, and clearance on the bending moment capacity (BMC) and stiffness of joints. The results showed that tenon length has the greatest influence on BMC, followed by clearance and tenon width, while clearance has the greatest impact on stiffness, followed by tenon length and tenon width. Variance analysis revealed that although tenon width significantly affects BMC and stiffness, its F values (11.596 for BMC and 20.183 for stiffness) are smaller than those of tenon length (28.405 for BMC, 20.756 for stiffness) and clearance (22.494 for BMC, 54.868 for stiffness). Therefore, longer tenons and transition (the clearance of 0 mm) or clearance (the clearance of −0.1 mm) are recommended, with smaller tenon widths being acceptable. BMC increases with tenon length and width but decreases with clearance, with no significant difference between tenon widths of 30 mm and 35 mm. Stiffness also increases with tenon length and width but decreases with clearance, with no significant differences between tenon lengths of 35 mm and 40 mm, tenon widths of 30 mm and 35 mm, and clearance values of −0.1 mm and 0 mm. The nonlinear models for tenon length, tenon width, and clearance effectively predict and optimize BMC and stiffness, with R² values of 0.885 and 0.941, respectively. The impact of tenon length on joint performance is greater than that of clearance, followed by tenon width. The optimal combination is a tenon length of 40 mm, a tenon width of 35 mm, and a clearance of −0.1 mm, resulting in an average moment capacity of 0.2 kN·m and an average stiffness of 1.5 kN·m/rad.

This study focuses only on specific wood types and does not consider other factors such as wood moisture content and density. Additionally, the applicability of the nonlinear regression model requires further validation. Future research should expand to different types of wood, consider additional physical and mechanical properties, conduct dynamic condition analyses, and test long-term durability to optimize the design and performance of mortise and tenon joints.

Author Contributions

Conceptualization, K.L.; Methodology, K.L.; Software, Y.D.; Formal analysis, X.H.; Investigation, H.Z.; Data curation, L.W.; Writing—original draft, K.L.; Writing—review & editing, L.L. and H.L.; Visualization, Y.D. and W.G.; Supervision, L.L. and H.L.; Project administration, B.L.; Funding acquisition, K.L. and B.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Acknowledgments

The authors are grateful for the support of MOE Key Laboratory of Wooden Material Science and Application, Beijing Key Laboratory of Wood Science and Engineering at Beijing Forestry University.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Configuration of (a) T-shaped joint specimen and (b) mortise and tenon (mm).

Figure 2. The universal mechanical testing machine for bending test.

Figure 3. (a–i) represent the typical M-θ curves of nine different joints under bending load, where L represents the tenon length, with 30, 35, and 40 indicating the tenon length dimensions. W represents the tenon width, with 25, 30, and 35 indicating the tenon width dimensions. The numbers 1, 2, and 3 represent the codes for three repeat samples of the same specification. The green lines and dots represent the average stiffness and yield point of the joints, respectively.

Figure 4. Typical failure modes of mortise and tenon joints (a) and the tenon after failure (b).

Figure 5. P-P plot of bending moment capacity (a) and stiffness (b).

Figure 6. (a–c) represent the post hoc test results of the impact of different levels of tenon length, tenon width, and clearance on the BMC of the joints. (d–f) show the post hoc test results of the influence of different levels of tenon length, tenon width, and clearance on the stiffness of the joints. Different letters above the bars indicate significant differences between levels, while the same letters indicate no significant differences between levels.

Figure 6. (a–c) represent the post hoc test results of the impact of different levels of tenon length, tenon width, and clearance on the BMC of the joints. (d–f) show the post hoc test results of the influence of different levels of tenon length, tenon width, and clearance on the stiffness of the joints. Different letters above the bars indicate significant differences between levels, while the same letters indicate no significant differences between levels.

Figure 7. (a–c) represent the interaction effects of tenon length and tenon width, tenon length and clearance, and tenon width and clearance on the Bending Moment Capacity (BMC) of the joints. (d–f) illustrate the interaction effects of tenon length and tenon width, tenon length and clearance, and tenon width and clearance on the stiffness of the joints.

Figure 8. Effect of factors on the average S/N ratio of bending moment capacity (a) and on the average S/N ratio of stiffness (b).

Figure 9. Comparison between the linear regression model and experimental results for bending moment capacity (a) and stiffness (b).

Figure 10. Comparison between the nonlinear regression model and experimental results for bending moment capacity (a) and stiffness (b).

Table 1

Orthogonal factors and levels.

Levels	Factors
	A	B	C
1	30	25	−0.1
2	35	30	0
3	40	35	0.1

Factors A, B, and C represent tenon length, tenon width, and clearance, respectively.

Table 2

L₉(3⁴) orthogonal array.

NO	Factor A	Factor B	Factor C	e
1	30	25	−0.1	1
2	30	35	0	2
3	30	30	0.1	3
4	35	35	0.1	1
5	35	30	−0.1	2
6	35	25	0	3
7	40	30	0	1
8	40	25	0.1	2
9	40	35	−0.1	3

Factors A, B, and C represent tenon length, tenon width, and clearance, respectively. e represents the error term.

Table 3

The results of the orthogonal experiment.

NO	TenonLength(mm)	TenonWidth(mm)	Clearance(mm)	e	BMC (kN·m)	Stiffness (kN·m/rad)
1	30.00	25.00	−0.1	1.00	0.13	0.95
2	30.00	35.00	0	2.00	0.13	1.09
3	30.00	30.00	0.1	3.00	0.11	0.64
4	35.00	35.00	0.1	1.00	0.11	1.04
5	35.00	30.00	−0.1	2.00	0.17	1.13
6	35.00	25.00	0	3.00	0.13	1.15
7	40.00	30.00	0	1.00	0.19	1.33
8	40.00	25.00	0.1	2.00	0.14	0.70
9	40.00	35.00	−0.1	3.00	0.19	1.50
10	30.00	25.00	−0.1	1.00	0.11	0.95
11	30.00	35.00	0	2.00	0.12	0.92
12	30.00	30.00	0.1	3.00	0.13	0.73
13	35.00	35.00	0.1	1.00	0.11	0.94
14	35.00	30.00	−0.1	2.00	0.15	1.25
15	35.00	25.00	0	3.00	0.13	1.08
16	40.00	30.00	0	1.00	0.17	1.37
17	40.00	25.00	0.1	2.00	0.13	0.64
18	40.00	35.00	−0.1	3.00	0.21	1.52
19	30.00	25.00	−0.1	1.00	0.14	1.03
20	30.00	35.00	0	2.00	0.13	1.13
21	30.00	30.00	0.1	3.00	0.13	0.66
22	35.00	35.00	0.1	1.00	0.14	1.06
23	35.00	30.00	−0.1	2.00	0.18	1.33
24	35.00	25.00	0	3.00	0.14	0.95
25	40.00	30.00	0	1.00	0.19	1.49
26	40.00	25.00	0.1	2.00	0.12	0.67
27	40.00	35.00	−0.1	3.00	0.20	1.49

BMC: bending moment capacity.

Table 4

Range analysis of bending moment capacity.

	Factor A	Factor B	Factor C
K1	1.13	1.17	1.48
K2	1.26	1.43	1.33
K3	1.53	1.33	1.12
k1	0.13	0.13	0.16
k2	0.14	0.16	0.15
k3	0.17	0.15	0.12
R	0.05	0.03	0.04

Factors A, B, and C represent the tenon length, tenon width, and clearance, respectively. Ki (i = 1, 2, 3) is the sum of the corresponding test results. ki (i = 1, 2, 3) is the arithmetic mean of the corresponding test results. The range R is the difference between the ultimate and minimum k values.

Table 5

Range analysis of stiffness.

	Factor A	Factor B	Factor C
K1	8.09	8.11	11.14
K2	9.94	9.94	10.52
K3	10.71	10.69	7.08
k1	0.90	0.90	1.24
k2	1.10	1.10	1.17
k3	1.19	1.19	0.79
R	0.29	0.28	0.45

Table 6

Results of Levene’s test of equality of error variances for BMC and stiffness.

Dependent Variable	F	df1	df2	p
BMC	0.85	8	18	0.57
Stiffness	1.90	8	18	0.12

Table 7

Results of ANOVA of the bending moment capacity.

Factors	Sum ofSquares ofDeviation	Degree ofFreedom	MeanSquareDeviation	F Value	p Value
A	9 × 10−3	2	5 × 10−3	28.41	1.43 × 10−6
B	4 × 10−3	2	2 × 10−3	11.59	4.53 × 10−4
C	7 × 10−3	2	4 × 10−3	22.49	7.62 × 10−6
Error	3 × 10−3	20	1.62 × 10−4
Total	2 × 10−3	26

Table 8

Results of ANOVA of the stiffness.

Factors	Sum ofSquares ofDeviation	Degree ofFreedom	MeanSquareDeviation	F Value	p Value
A	0.40	2	0.20	20.76	1.32 × 10−5
B	0.39	2	0.19	20.18	1.59 × 10−5
C	1.07	2	0.53	54.87	7.58 × 10−9
Error	0.19	20	0.01
Total	2.05	26

Table 9

Average grey correlation coefficients of different levels of BMC.

	Level 1	Level 2	Level 3	Difference
Factor A	0.374	0.434	0.632	0.259
Factor B	0.387	0.525	0.529	0.142
Factor C	0.591	0.477	0.372	0.219

Table 10

Average grey correlation coefficients of different levels of stiffness.

	Level 1	Level 2	Level 3	Difference
Factor A	0.434	0.525	0.700	0.266
Factor B	0.427	0.584	0.647	0.220
Factor C	0.676	0.599	0.384	0.293

Table 11

Average grey relational degree of joint performance.

	Level 1	Level 2	Level 3	Difference
Factor A	0.202	0.240	0.333	0.131
Factor B	0.203	0.277	0.294	0.090
Factor C	0.317	0.269	0.189	0.128

Appendix A

Table A1

Raw data normalization and data sequence differences.

NO	y 1	y 2	Δ1(k)	Δ2(k)
1	0.200	0.352	0.800	0.648
2	0.200	0.511	0.800	0.489
3	0.000	0.000	1.000	1.000
4	0.000	0.455	1.000	0.545
5	0.600	0.557	0.400	0.443
6	0.200	0.580	0.800	0.420
7	0.800	0.784	0.200	0.216
8	0.300	0.068	0.700	0.932
9	0.800	0.977	0.200	0.023
10	0.000	0.352	1.000	0.648
11	0.100	0.318	0.900	0.682
12	0.200	0.102	0.800	0.898
13	0.000	0.341	1.000	0.659
14	0.400	0.693	0.600	0.307
15	0.200	0.500	0.800	0.500
16	0.600	0.830	0.400	0.170
17	0.200	0.000	0.800	1.000
18	1.000	1.000	0.000	0.000
19	0.300	0.443	0.700	0.557
20	0.200	0.670	0.800	0.330
21	0.200	0.023	0.800	0.977
22	0.300	0.477	0.700	0.523
23	0.700	0.784	0.300	0.216
24	0.300	0.352	0.700	0.648
25	0.800	0.966	0.200	0.034
26	0.100	0.034	0.900	0.966
27	0.900	0.966	0.100	0.034

y₁ and y₂ represent the normalized values of BMC and stiffness, respectively. Δ₁(k) and Δ₂(k) represent the data difference sequences of the normalized values of BMC and stiffness, respectively.

Table A2

Grey correlation coefficients and grey correlation degree.

NO	ξ1(k)	ξ2(k)	γ
1	0.385	0.436	0.205
2	0.385	0.506	0.223
3	0.333	0.333	0.167
4	0.333	0.478	0.203
5	0.556	0.530	0.271
6	0.385	0.543	0.232
7	0.714	0.698	0.353
8	0.417	0.349	0.191
9	0.714	0.957	0.418
10	0.333	0.436	0.192
11	0.357	0.423	0.195
12	0.385	0.358	0.186
13	0.333	0.431	0.191
14	0.455	0.620	0.269
15	0.385	0.500	0.221
16	0.556	0.746	0.325
17	0.385	0.333	0.179
18	1.000	1.000	0.500
19	0.417	0.473	0.222
20	0.385	0.603	0.247
21	0.385	0.338	0.181
22	0.417	0.489	0.226
23	0.625	0.698	0.331
24	0.417	0.436	0.213
25	0.714	0.936	0.413
26	0.357	0.341	0.175
27	0.833	0.936	0.442

ξ₁(k) and ξ₂(k) are the grey relational coefficients for BMC and stiffness, respectively. γ is the grey correlation degree.

References

1. Xie, Q.; Zhang, L.; Li, S.; Zhou, W.; Wang, L. Cyclic Behavior of Chinese Ancient Wooden Frame with Mortise–Tenon Joints: Friction Constitutive Model and Finite Element Modelling. J. Wood Sci.; 2018; 64, pp. 40-51. [DOI: https://dx.doi.org/10.1007/s10086-017-1669-5]

2. Eckelman, C.A.; Suddarth, S.K. Analysis and Design of Furniture Frames. Wood Sci. Technol.; 1969; 3, pp. 239-255. [DOI: https://dx.doi.org/10.1007/BF00367215]

3. Erdil, Y.; Kasal, A.; Eckelman, C. Bending Moment Capacity of Rectangular Mortise and Tenon Furniture Joints. For. Prod. J.; 2005; 55, pp. 209-213.

4. Demirci, S.; Diler, H.; Kasal, A.; Erdil, Y.Z. Bending Moment Resistances of L-Shaped Furniture Frame Joints under Tension and Compression Loadings. Wood Res.; 2020; 65, pp. 975-988. [DOI: https://dx.doi.org/10.37763/wr.1336-4561/65.6.975988]

5. Likos, E.; Haviarova, E.; Eckelman, C.A.; Erdil, Y.Z.; Ozcifci, A. Effect of Tenon Geometry, Grain Orientation, and Shoulder on Bending Moment Capacity and Moment Rotation Characteristics of Mortise and Tenon Joints. Wood Fiber Sci.; 2012; 44, pp. 462-469.

6. Sydor, M.; Majka, J.; Langová, N. Effective Diameters of Drilled Holes in Pinewood in Response to Changes in Relative Humidity. BioResources; 2021; 16, pp. 5407-5421. [DOI: https://dx.doi.org/10.15376/biores.16.3.5407-5421]

7. Smardzewski, J. Effect of Wood Species and Glue Type on Contact Stresses in a Mortise and Tenon Joint. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci.; 2008; 222, pp. 2293-2299. [DOI: https://dx.doi.org/10.1243/09544062JMES1084]

8. Kasal, A.; Haviarova, E.; Efe, H.; Eckelman, C.A.; Erdil, Y.Z. Effect of Adhesive Type and Tenon Size on Bending Moment Capacity and Rigidity of T-Shaped Furniture Joints Constructed of Turkish Beech and Scots Pine. Wood Fiber Sci.; 2013; 45, pp. 287-293.

9. Zaborsky, V.; Boruvka, V.; Kasickova, V.; Ruman, D. Effect of Wood Species, Adhesive Type, and Annual Ring Directions on the Stiffness of Rail to Leg Mortise and Tenon Furniture Joints. BioResources; 2017; 12, pp. 7016-7031. [DOI: https://dx.doi.org/10.15376/biores.12.4.7016-7031]

10. Harada, M.; Hayashi, Y.; Hayashi, T.; Karube, M.; Ohgama, T. Effect of Moisture Content of Members on Mechanical Properties of Timber Joints. J. Wood Sci.; 2005; 51, pp. 282-285. [DOI: https://dx.doi.org/10.1007/s10086-004-0656-9]

11. Eckelman, C.; Erdil, Y.; Haviarova, E. Effect of Shoulders on Bending Moment Capacity of Round Mortise and Tenon Joints. For. Prod. J.; 2006; 56, pp. 82-86.

12. Oktaee, J.; Ebrahimi, G.; Layeghi, M.; Ghofrani, M.; Eckelman, C.A. Bending Moment Capacity of Simple and Haunched Mortise and Tenon Furniture Joints under Tension and Compression Loads. Turk. J. Agric. For.; 2014; 38, pp. 291-297. [DOI: https://dx.doi.org/10.3906/tar-1211-74]

13. Derikvand, M.; Ebrahimi, G.; Eckelman, C.A. Bending Moment Capacity of Mortise and Loose-Tenon Joints. Wood Fiber Sci.; 2014; 46, pp. 159-166.

14. Mackerle, J. Finite Element Analyses in Wood Research: A Bibliography. Wood Sci. Technol.; 2005; 39, pp. 579-600. [DOI: https://dx.doi.org/10.1007/s00226-005-0026-9]

15. Derikvand, M.; Ebrahimi, G. Finite Element Analysis of Stress and Strain Distributions in Mortise and Loose Tenon Furniture Joints. J. For. Res.; 2014; 25, pp. 677-681. [DOI: https://dx.doi.org/10.1007/s11676-014-0507-5]

16. Kasal, A.; Smardzewski, J.; Kuşkun, T.; Erdil, Y.Z. Numerical Analyses of Various Sizes of Mortise and Tenon Furniture Joints. BioResources; 2016; 11, pp. 6836-6853. [DOI: https://dx.doi.org/10.15376/biores.11.3.6836-6853]

17. Hu, W.; Liu, N. Numerical and Optimal Study on Bending Moment Capacity and Stiffness of Mortise-and-Tenon Joint for Wood Products. Forests; 2020; 11, 501. [DOI: https://dx.doi.org/10.3390/f11050501]

18. Sandberg, L.B.; Bulleit, W.M.; Reid, E.H. Strength and Stiffness of Oak Pegs in Traditional Timber-Frame Joints. J. Struct. Eng.; 2000; 126, pp. 717-723. [DOI: https://dx.doi.org/10.1061/(ASCE)0733-9445(2000)126:6(717)]

19. Ogawa, K.; Sasaki, Y.; Yamasaki, M. Theoretical Estimation of the Mechanical Performance of Traditional Mortise–Tenon Joint Involving a Gap. J. Wood Sci.; 2016; 62, pp. 242-250. [DOI: https://dx.doi.org/10.1007/s10086-016-1544-9]

20. He, J.; Yu, P.; Wang, J.; Yang, Q.; Han, M.; Xie, L. Theoretical Model of Bending Moment for the Penetrated Mortise-Tenon Joint Involving Gaps in Traditional Timber Structure. J. Build. Eng.; 2021; 42, 103102. [DOI: https://dx.doi.org/10.1016/j.jobe.2021.103102]

21. Liu, S.; Lin, Y. On Measures of Information Content of Grey Numbers. Kybernetes; 2006; 35, pp. 899-904. [DOI: https://dx.doi.org/10.1108/03684920610662692]

22. Liu, S.; Tao, Y.; Xie, N.; Tao, L.; Hu, M. Advance in Grey System Theory and Applications in Science and Engineering. Grey Syst. Theory Appl.; 2022; 12, pp. 804-823. [DOI: https://dx.doi.org/10.1108/GS-09-2021-0141]

23. Soorya Prakash, K.; Gopal, P.M.; Karthik, S. Multi-Objective Optimization Using Taguchi Based Grey Relational Analysis in Turning of Rock Dust Reinforced Aluminum MMC. Measurement; 2020; 157, 107664. [DOI: https://dx.doi.org/10.1016/j.measurement.2020.107664]

24. Loganathan, D.; Satish Kumar, S.; Ramadoss, R. Grey Relational Analysis-Based Optimisation of Input Parameters of Incremental Forming Process Applied to the AA6061 Alloy. Trans. Famena; 2020; 44, pp. 93-104. [DOI: https://dx.doi.org/10.21278/TOF.44108]

25. Pagar, N.D.; Gawande, S.H. Parametric Design Analysis of Meridional Deflection Stresses in Metal Expansion Bellows Using Gray Relational Grade. J. Braz. Soc. Mech. Sci. Eng.; 2020; 42, 256. [DOI: https://dx.doi.org/10.1007/s40430-020-02327-0]

26. Khan, M.A.; Jaffery, S.H.I.; Khan, M.; Younas, M.; Butt, S.I.; Ahmad, R.; Warsi, S.S. Multi-Objective Optimization of Turning Titanium-Based Alloy Ti-6Al-4V under Dry, Wet, and Cryogenic Conditions Using Gray Relational Analysis (GRA). Int. J. Adv. Manuf. Technol.; 2020; 106, pp. 3897-3911. [DOI: https://dx.doi.org/10.1007/s00170-019-04913-6]

27. Yu, Z.; Song, J.; Xu, L.; Zhang, H. Evaluating the Combustion Performance of the Usual Timbers in Furniture Using a Grey Correlation Method Based on Thermolysis, Ignition, and Flame Spread. Fire; 2024; 7, 218. [DOI: https://dx.doi.org/10.3390/fire7070218]

28. Wang, X.; Yu, Z.; Zhang, Y.; Qi, C.; Chang, W. Evaluation of Ultrasonic-Assisted Dyeing Properties of Fast-Growing Poplar Wood Treated by Reactive Dye Based on Grey System Theory Analysis. J. Wood Sci.; 2018; 64, pp. 861-871. [DOI: https://dx.doi.org/10.1007/s10086-018-1768-y]

29. Deng, J. Introduction to Grey System Theory. J. Grey Syst.; 1989; 1, pp. 1-24.

30. Xie, Y.; Yu, H.; Chen, J.; Ruan, X. Application of Grey Relational Analysis in Sheet Metal Forming for Multi-Response Quality Characteristics. J. Zhejiang Univ.-Sci. A; 2007; 8, pp. 805-811. [DOI: https://dx.doi.org/10.1631/jzus.2007.A0805]

31. Tosun, N. Determination of Optimum Parameters for Multi-Performance Characteristics in Drilling by Using Grey Relational Analysis. Int. J. Adv. Manuf. Technol.; 2006; 28, pp. 450-455. [DOI: https://dx.doi.org/10.1007/s00170-004-2386-y]

32. Yang, C.-B.; Lin, C.-G.; Chiang, H.-L.; Chen, C.-C. Single and Multiobjective Optimization of Inconel 718 Nickel-Based Superalloy in the Wire Electrical Discharge Machining. Int. J. Adv. Manuf. Technol.; 2017; 93, pp. 3075-3084. [DOI: https://dx.doi.org/10.1007/s00170-017-0758-3]

33. Gregory, M.A.; Herakovich, C.T. Predicting Crack Growth Direction in Unidirectional Composites. J. Compos. Mater.; 1986; 20, pp. 67-85. [DOI: https://dx.doi.org/10.1177/002199838602000105]