Numerical and Optimal Study on Bending Moment Capacity and Sti ﬀ ness of Mortise-and-Tenon Joint for Wood Products

: Mortise-and-tenon (M–T) joint is a traditional joint type commonly used in wood constructions and wood products. Bending moment capacity (BMC) is a critical criterion to evaluate the strength of the M–T joint. In order to design the M–T joint structure more rationally, many researchers have been devoted to studying on this topic. However, the factors inﬂuencing the BMC are too many to conduct comprehensive studies using experimental tests, especially for tenon size. In this study, the BMC and bending sti ﬀ ness of the M–T joint were studied using a combination of ﬁnite element method (FEM) and response surface method to optimize the tenon size of the M–T joint. The results showed that (1) the proposed ﬁnite element model was capable of predicting BMC of M–T joints with the ratios of FEM to observed, ranging from 0.852 to 1.072; (2) the BMC and sti ﬀ ness were signiﬁcantly a ﬀ ected by tenon size, and tenon length had a more signiﬁcant e ﬀ ect on BMC than tenon width, while the tenon width a ﬀ ected the bending sti ﬀ ness more signiﬁcantly; (3) the response surface model proposed to predict and optimize the BMC of the M–T joint relating to tenon length and tenon width was capable of providing an optimal solution; (4) it was recommended to make the ratio of tenon length to tenon width higher than 1 to get higher BMC of M–T joints. In conclusion, this study will contribute to reducing the cost of a huge amount of experimental tests by applying FEM and the response surface method to design M–T joint wood products.


Introduction
Previous studies have found several factors influencing the bending moment capacity (BMC) and bending stiffness of mortise-and-tenon (M-T) joints, such as wood species, annual ring, adhesive type, load type, tenon size, tenon geometry [1,2]. Among these studies, Erdil et al. [3] studied the effects of wood species, adhesive type and tenon size on the bending strength and flexibility of M-T joint T-shaped samples, indicating that the stiffness of M-T joints increased with the increase of tenon length and tenon width, and the tenon width had a more significant effect on BMC than tenon length. Likos et al. [4] evaluated the BMC and moment rotation characteristics of M-T joints as a function of tenon geometry, grain orientation, length, and shoulder fit. The results indicated that the BMC of all joints in which tenons were fully inserted into mortise was 54% greater than joints that were not fully inserted. Oktaee et al. [5] examined the effect of tenon geometry (width, thickness, and length) on the BMC of simple and haunched M-T joints under compression and tension states. Tenon length was found to have the greatest effect on BMC of M-T joints, followed by tenon width. Kasal et al. [6] studied the BMCs of L-shaped mortise and tenon furniture joints under both compression and tension loadings, where M refers to the bending moment capacity (in-lb); a is adhesive type, which is equal to 1, 0.74, 0.79, and 0.83 for polyvinyl acetate (PVA), phenol resorcinol (PR), animal glue, and urea-formaldehyde (UF) adhesives, respectively; W is rail width; D is tenon depth; l is tenon length (in); S is the shear strength of wood (psi). M = 0.00227(wl)(0.229w + d)S 0.42 k 1 k 2 (2) where M refers to the bending moment capacity (Nm), w is the tenon width, l is the tenon length, and d is the width of shoulder (mm); S is the shear strength of wood (N/mm); k 1 is the loading type: k 1 (tension) = 1, and k 1 (compression) = 1.006; k 2 refers to adhesive type: k 2 is equal to 1 and 0.85 for polyurethane (PU) and PVA, respectively. M = 1.328t 0.167 l 0.743 w 0.648 (3) where M refers to the bending moment capacity (Nm); t, l, and w are thickness, length, and width of the tenon, respectively (mm). Beyond that, many studies have evaluated the M-T joint strength using the finite element method (FEM), aiming to search a more general method to study the M-T joint. At the early stage, most of these studies mainly focused on the stress distributions of wood products and wood constructions from a qualitative perspective [9][10][11]. Some studies regarded the M-T joint as a rigid joint without considering the contact and bonding behaviors between mortise and tenon [12,13] when establishing the finite element model of the M-T joint. There is no doubt it is unreasonable to assume the M-T joint is a rigid joint, which will much increase the stiffness of the M-T joint and make wood structures more solid. Furthermore, other studies have proposed semi-rigid finite element models to simulate the mechanical behaviors of wood M-T joints using FEM, considering the glue line between mortise and tenon. Among these studies, the glue line was built as a thin layer between mortise and tenon. Some researchers assumed the glue line was 0.1 mm thick around the entire surfaces of tenon [14][15][16]. Prekrat and Smardzewski [17] studied the effect of the glue line shape on the strength of oval M-T joints using FEM, assuming that the glue shape was rectangular and just bonding the flat faces of the joint, and the glue was ellipse-shape and bonding the entire face of the tenon, respectively. Hu et al. [18] determined the thickness of the glue line between oval mortise and tenon using a fluorescence microscope, observing that the thickness of the glue line decreased with the increase of the interference fit between curve surfaces of mortise and tenon, and little glue was observed between curve surfaces. Based on this finding, Hu et al. [19,20] established a semi-rigid M-T joint finite element model and studied the bending and withdrawal resistance of M-T joints. The above studies confirmed that factors such as wood species, annual ring, adhesive type, load type, tenon size, and tenon geometry significantly affected bending strength by conducting a huge number of experimental tests. The FEM was also proven to be a valid method to evaluate mechanical behaviors of wood M-T joints in a qualitative angle by comparing stress distributions of wood structures. However, until now, a general, efficient, accurate, and low-cost method has not been proposed to replace the huge number of experimental tests in optimizing wood M-T joints.
In this study, the main objective is to evaluate the BMC of M-T joint with various tenon sizes using FEM and response surface method to get the optimal solution of tenon size with maximum BMC from a quantitative perspective. The specific objectives of this study are to (1) verify the proposed finite element model through comparing with the previous experimental results; (2) numerically study the effects of tenon size on BMC and stiffness of M-T joints, and further study the effect of the tenon size on failure modes of M-T joints; (3) optimize the tenon size (length and width) to get the maximum BMC of the M-T joint through response surface method; (4) verify the optimal solution of tenon size based on FEM.

Materials
The wood used in this study was beech (Fagus orientalis. Lipsky), bought from a local commercial supplier (Nanjing, China). The specific gravity was 0.69, and the moisture content averaged 10.8%. Table 1 shows the mechanical parameters used in the finite element model, which were determined in the previous study [21]. Table 1. Summary of mechanical properties of solid beech wood.

Modulus of Elasticity (MPa)
Poisson's Ratio  Figure 1 shows the configurations of mortise-and-tenon joint T-shaped samples used in this study. The post leg measured 200 mm long × 40 mm wide × 40 mm thick, and the dimensions of stretcher were 160 mm long × 40 mm wide × 40 mm thick. A previous study [8] showed that the greatest effect on the joint strength was the tenon length, the influence of the tenon width was less significant, and the effect of the tenon thickness was slight. Therefore, in this study, the tenon length and tenon width were selected as two factors to evaluate the BMC of the M-T joint. The tenon length varied from 10 to 40 mm with an increment of 10 mm, and the tenon width ranged from 15 to 25 mm with an increment of 5 mm, and the tenon thickness was a constant of 10 mm. The interaction between tenon width and mortise height was a 0.2-mm interference fit, and a 0.2-mm clearance fit was applied between tenon thickness and mortise width ( Figure 1). The mortise and tenon were bonded by 65% solid content PVA.  Figure 2 shows a 3-D orthotropic finite element model of an M-T joint T-shaped sample established using finite element software (ABAQUS 6.14-1, Dassult, USA). The mechanical properties used in this model were determined from the beechwood shown in Table 1. Ductile damage was used as a stress criterion for wood members in the model, and the parameters needed were fracture strain (0.0083), stress triaxiality (0.333), strain rate (0.01), and failure displacement (0.6452 mm). The grain orientations of post leg and stretcher were differentiated by local coordinates, i.e., x, y, and z corresponded to longitudinal, radial, and tangential directions of wood, respectively. The interactions of mortise and tenon were surface-to-surface contact, for the width direction of the tenon, the penalty contact property was specified with a friction coefficient of 0.54 [22] to simulate the friction behavior with 0.2-mm interference fit. In addition, for the thickness direction of tenon, cohesive behavior was applied to simulate the cohesive bonding behavior. The parameters needed in ABAQUS software Knn, Kss, and Ktt were 1.23, 3.49, and 2.45 N/mm, respectively, measured using the method described in [23]. The loading conditions referenced to a previous study [24], specifically, the boundary conditions were that six-degree freedoms were constrained at two parts, 20 mm from the two ends of the post leg, and a displacement load was imposed on the load head at the point of 30 mm from the end of stretcher to get the ultimate reaction force of load head, which was equal to the ultimate force (F) applied at the loading point ( Figure 2). The mesh of the model was also shown in Figure 2, the sizes of all elements were approximate 5 mm, while for contact parts, the sizes of elements were approximate 2 mm to make the model more accurate. The element type assigned to the T-shaped sample was C3D8, an 8-node linear brick element. A 4-node 3-D bilinear rigid quadrilateral element (R3D4) was assigned to load head.  Figure 2 shows a 3-D orthotropic finite element model of an M-T joint T-shaped sample established using finite element software (ABAQUS 6.14-1, Dassult, USA). The mechanical properties used in this model were determined from the beechwood shown in Table 1. Ductile damage was used as a stress criterion for wood members in the model, and the parameters needed were fracture strain (0.0083), stress triaxiality (0.333), strain rate (0.01), and failure displacement (0.6452 mm). The grain orientations of post leg and stretcher were differentiated by local coordinates, i.e., x, y, and z corresponded to longitudinal, radial, and tangential directions of wood, respectively. The interactions of mortise and tenon were surface-to-surface contact, for the width direction of the tenon, the penalty contact property was specified with a friction coefficient of 0.54 [22] to simulate the friction behavior with 0.2-mm interference fit. In addition, for the thickness direction of tenon, cohesive behavior was applied to simulate the cohesive bonding behavior. The parameters needed in ABAQUS software Knn, Kss, and Ktt were 1.23, 3.49, and 2.45 N/mm, respectively, measured using the method described in [23]. The loading conditions referenced to a previous study [24], specifically, the boundary conditions were that six-degree freedoms were constrained at two parts, 20 mm from the two ends of the post leg, and a displacement load was imposed on the load head at the point of 30 mm from the end of stretcher to get the ultimate reaction force of load head, which was equal to the ultimate force (F) applied at the loading point ( Figure 2). The mesh of the model was also shown in Figure 2, the sizes of all elements were approximate 5 mm, while for contact parts, the sizes of elements were approximate 2 mm to make the model more accurate. The element type assigned to the T-shaped sample was C3D8, an 8-node linear brick element. A 4-node 3-D bilinear rigid quadrilateral element (R3D4) was assigned to load head.

Finite Element Model and Simulation Method
In this study, the effects of tenon width (15,20, and 25 mm) and tenon length (10, 20, 30, and 40 mm) on BMC of M-T joints were studied, aiming to obtain the optimal tenon size with maximum BMC. Therefore, a complete 3 × 4 factorial numerical simulation was conducted using 12 M-T joint finite element models with different tenon sizes, as shown in Table 2. The BMC and bending stiffness were evaluated by Equations (4) and (5), respectively.
where M is bending moment capacity (Nm); F is the ultimate force (N); L is moment arm (mm); K is bending stiffness (N/mm); ∆F is the change of force in the linear portion of bending load and deflection curve (mm), and ∆d is the change of deflection corresponding to ∆F (mm). Table 2. Combinations of tenon width and tenon length evaluated in this study.

Dimensions of Tenon (mm) Width (w)
Length (l) 15 20 Forests 2020, 11, x FOR PEER REVIEW 5 of 12 In this study, the effects of tenon width (15,20, and 25 mm) and tenon length (10, 20, 30, and 40 mm) on BMC of M-T joints were studied, aiming to obtain the optimal tenon size with maximum BMC. Therefore, a complete 3 × 4 factorial numerical simulation was conducted using 12 M-T joint finite element models with different tenon sizes, as shown in Table 2. The BMC and bending stiffness were evaluated by Equations (4) and (5), respectively.
where M is bending moment capacity (Nm); F is the ultimate force (N); L is moment arm (mm); K is bending stiffness (N/mm); ΔF is the change of force in the linear portion of bending load and deflection curve (mm), and Δd is the change of deflection corresponding to ΔF (mm). Table 2. Combinations of tenon width and tenon length evaluated in this study.

Verification of Finite Element Model
The finite element model proposed in this study was verified before conducting the numerical and optimal studies. A finite element model was built using the method described above and applied to simulate the experimental tests [24] to get the BMC and compare with the experimental results.

Verification of Finite Element Model
The finite element model proposed in this study was verified before conducting the numerical and optimal studies. A finite element model was built using the method described above and applied to simulate the experimental tests [24] to get the BMC and compare with the experimental results.

Statistical Analysis
All data were analyzed by Design Expert (Version 8.06, Stat-Ease, Inc. Minneapolis, MN, USA) using the response surface method. The response surface model was built by the modified cubic regression method, and the model terms were statistically analyzed by analysis of variance (ANOVA) at a 5% significant level. Model numerical optimization was also conducted by Design Expert using the nonlinear programming method. Table 3 shows the results of the BMC values of the experiment [24] and those of the finite element model proposed in this study, indicating that the results of FEM were well consistent with those of experimental tests with the ratios of FEM to observed ranging from 0.852 to 1.072. In other words, this finite element model was capable of replacing a large number of experimental tests to study the effect of tenon size on BMC of M-T joints.   Figure 4 shows the stress distributions of this M-T joint during the loading process corresponding to four critical load points shown in Figure 3. Table 3 shows the results of the BMC values of the experiment [24] and those of the finite element model proposed in this study, indicating that the results of FEM were well consistent with those of experimental tests with the ratios of FEM to observed ranging from 0.852 to 1.072. In other words, this finite element model was capable of replacing a large number of experimental tests to study the effect of tenon size on BMC of M-T joints.

Dimensions of Tenon (mm) Bending Moment Capacity (Nm)
Width   Figure 4 shows the stress distributions of this M-T joint during the loading process corresponding to four critical load points shown in Figure 3.    Table 3 shows the results of the BMC values of the experiment [24] and those of the finite element model proposed in this study, indicating that the results of FEM were well consistent with those of experimental tests with the ratios of FEM to observed ranging from 0.852 to 1.072. In other words, this finite element model was capable of replacing a large number of experimental tests to study the effect of tenon size on BMC of M-T joints.

Dimensions of Tenon (mm) Bending Moment Capacity (Nm)
Width Length   Figure 4 shows the stress distributions of this M-T joint during the loading process corresponding to four critical load points shown in Figure 3.    Table 4 shows the BMCs and bending stiffness of all combinations of tenon length and tenon width, which indicated that the BMC increased as either tenon width or tenon length increased. Further comparing the effects of tenon width and tenon length on BMC, the tenon length influenced BMC more significantly when the tenon length was relatively short, ranging from 10 to 30 mm. While the effect of tenon length on BMC decreased when the tenon length was longer than 30 mm and tenon width was bigger than 15 mm. The bending stiffness of the M-T joint increased with the increase of tenon width and tenon length in the evaluated ranges. The tenon width had a more significant effect on bending stiffness, whereas this effect of tenon length reduced when the tenon length was longer than 30 mm. These results agree with those of previous studies [5,24]. Table 4. Bending moment capacities (BMCs) and stiffness of M-T joint based on FEM.  Figure 5 summarized the failure modes of all finite element models with different tenon sizes. Two typical failure modes were observed. When the tenon length was shorter than tenon width, the failure occurred at the front end of the tenon, indicating that the tenon was completely pulled out from the mortise because of fracture of glue without damage of tenon, and the glue cannot provide enough strength to hold the tenon. In case of the tenon length being longer than or equal to the tenon width, the tenons fractured at the rear end of tenon because of tenon fracture in tension perpendicular to the grain. In other words, the tenon withdrew completely from the mortise when the ratio of tenon width to tenon length was higher than 1, otherwise, the M-T joint failed because of the fracture of tenon at the rear end of the tenon. Therefore, it was recommended to make the tenon length longer than tenon width to get the higher BMC in the design of the M-T joints. In general, the stress at the failure part should reach the peak value. While in the finite element model, once the stress exceeds the ultimate strength of the material, the elements at the places of damage are degraded and deleted, which make the stress at damage part smaller than other undamaged parts.  Table 4 shows the BMCs and bending stiffness of all combinations of tenon length and tenon width, which indicated that the BMC increased as either tenon width or tenon length increased. Further comparing the effects of tenon width and tenon length on BMC, the tenon length influenced BMC more significantly when the tenon length was relatively short, ranging from 10 to 30 mm. While the effect of tenon length on BMC decreased when the tenon length was longer than 30 mm and tenon width was bigger than 15 mm. The bending stiffness of the M-T joint increased with the increase of tenon width and tenon length in the evaluated ranges. The tenon width had a more significant effect on bending stiffness, whereas this effect of tenon length reduced when the tenon length was longer than 30 mm. These results agree with those of previous studies [5,24].  Figure 5 summarized the failure modes of all finite element models with different tenon sizes. Two typical failure modes were observed. When the tenon length was shorter than tenon width, the failure occurred at the front end of the tenon, indicating that the tenon was completely pulled out from the mortise because of fracture of glue without damage of tenon, and the glue cannot provide enough strength to hold the tenon. In case of the tenon length being longer than or equal to the tenon width, the tenons fractured at the rear end of tenon because of tenon fracture in tension perpendicular to the grain. In other words, the tenon withdrew completely from the mortise when the ratio of tenon width to tenon length was higher than 1, otherwise, the M-T joint failed because of the fracture of tenon at the rear end of the tenon. Therefore, it was recommended to make the tenon length longer than tenon width to get the higher BMC in the design of the M-T joints. In general, the stress at the failure part should reach the peak value. While in the finite element model, once the stress exceeds the ultimate strength of the material, the elements at the places of damage are degraded and deleted, which make the stress at damage part smaller than other undamaged parts. l10-w15 l10-w20 l10-w25 l20-w15 l20-w20 l20-w25

Failure modes of M-T joints
l30-w15 l30-w20 l30-w25 l40-w15 l40-w20 l40-w25  Figure 6 shows the response surface model used to predict the BMC of M-T joints relating to the tenon width and tenon length. The response surface reduced cubic model for predicting the BMC of M-T joints was derived, as shown in Equation (6). Additionally, the ANOVA result of model terms is shown in Table 5.  Figure 6 shows the response surface model used to predict the BMC of M-T joints relating to the tenon width and tenon length. The response surface reduced cubic model for predicting the BMC of M-T joints was derived, as shown in Equation (6). Additionally, the ANOVA result of model terms is shown in Table 5.

Response Surface Model
The model F-value of 142.10 implied that the model was significant. Values of p less than 0.05 indicate that the model terms are significant. In this case, B, AB, A 2 , A 2 B, and A 3 were significant model terms. The "predicted R-squared" of 0.9494 was in reasonable agreement with the "adjusted R-squared" of 0.9872. Therefore, this model is acceptable to predict the BMC of M-T joint T-shaped samples evaluated in this study.
Furthermore, numerical optimization was conducted, aiming to obtain the maximum BMC of M-T joints using a nonlinear programming method based on Design Expert software. The objective function was the response surface model shown in Equation (6). The constraints were that the tenon length ranged from 10 to 40 mm, and the tenon width varied between 15 and 25 mm. The final goal Forests 2020, 11, 501 9 of 12 was to get the maximum BMC. Considering the above conditions, the optimal solution was that the maximum BMC is 160.18 Nm, with the tenon length of 28.81 mm and tenon width 25 mm.  The model F-value of 142.10 implied that the model was significant. Values of p less than 0.05 indicate that the model terms are significant. In this case, B, AB, A 2 , A 2 B, and A 3 were significant model terms. The "predicted R-squared" of 0.9494 was in reasonable agreement with the "adjusted R-squared" of 0.9872. Therefore, this model is acceptable to predict the BMC of M-T joint T-shaped samples evaluated in this study.
Furthermore, numerical optimization was conducted, aiming to obtain the maximum BMC of M-T joints using a nonlinear programming method based on Design Expert software. The objective function was the response surface model shown in Equation (6). The constraints were that the tenon length ranged from 10 to 40 mm, and the tenon width varied between 15 to 25 mm. The final goal was to get the maximum BMC. Considering the above conditions, the optimal solution was that the maximum BMC is 160.18 Nm, with the tenon length of 28.81 mm and tenon width 25 mm.

Verification of the Optimal Solution
Another M-T joint T-shaped finite element model was established using the optimized tenon size, measuring 28.81 mm long by 25 mm thick. Figure 7 shows the results of the finite element model, indicating that the maximum bending load is 1226.75 N (Figure 7b). According to Equation (4), the BMC of optimized M-T joints is 159.48 Nm, which is well consistent with the result of the response surface model (160.18 Nm) and higher than other combinations of tenon widths and tenon lengths evaluated in this study. Therefore, the method of combining FEM and the response surface method is capable of optimizing the BMC of M-T joint T-shaped samples.

Verification of the Optimal Solution
Another M-T joint T-shaped finite element model was established using the optimized tenon size, measuring 28.81 mm long by 25 mm thick. Figure 7 shows the results of the finite element model, indicating that the maximum bending load is 1226.75 N (Figure 7b). According to Equation (4), the BMC of optimized M-T joints is 159.48 Nm, which is well consistent with the result of the response surface model (160.18 Nm) and higher than other combinations of tenon widths and tenon lengths evaluated in this study. Therefore, the method of combining FEM and the response surface method is capable of optimizing the BMC of M-T joint T-shaped samples.

Conclusions
In this study, bending moment capacity and bending stiffness were studied using FEM and the response surface method to optimize the tenon size. The following conclusions were drawn: 1. The proposed finite element model using contact behavior to simulate the glued M-T joint was verified through predicting the bending moment capacity of the M-T joint. 2. The BMC and stiffness were significantly affected by tenon size, and tenon length had a more significant effect on bending moment than tenon width, while the tenon width affected the bending stiffness more significantly. 3. The proposed method combining FEM and response surface method used to predict and optimize the BMC of M-T joints relating to tenon length and tenon width is capable of replacing the experiments and providing reasonable results without experimental errors. 4. It is recommended to make the ratio of tenon length to tenon width greater than 1 to get higher BMC in the design of M-T joints.
In conclusion, the proposed method combining FEM and the response surface method is verified as an efficient way to optimize the M-T joints, which will contribute to the design of M-T joint wood products. Further research will focus on the following points: (1) additional tests will be repeated to make the conclusions more reliable; (2) the relationship between bending moment capacity and furniture structure will be studied based on FEM to better instruct the design of M-T joint wood products.