Article Numerical and Optimal Study on Bending Moment Capacity and Stiffness of Mortise-and-Tenon Joint for Wood Products

Wengang Hu 1,2,* and Na Liu 2 1 Co-Innovation Center of Efficient Processing and Utilization of Forest Resources, Nanjing Forestry University, Nanjing 210037, China 2 College of Furnishings and Industrial Design, Nanjing Forestry University, Nanjing 210037, China; [email protected] * Correspondence: [email protected]; Tel.: +86-150-7787-2795



Received: 14 April 2020; Accepted: 29 April 2020; Published: 1 May 2020


Abstract: 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 stiffness 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 stiffness were significantly affected by tenon size, and tenon length had a more significant effect on BMC than tenon width, while the tenon width affected the bending stiffness 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.

Keywords: bending moment capacity; mortise-and-tenon joint; FEM; structure optimization

1. 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,

Forests 2020, 11, 501; doi:10.3390/f11050501 www.mdpi.com/journal/forests Forests 2020, 11, 501 2 of 12 considering the effects of wood species, adhesive type, and tenon size (width and length), which indicated that the tenon length had a greater effect on the BMC than tenon width. Záborský et al. [7] further studied the effects of wood species, adhesive type, joint type, tenon thickness, loading type, annual rings on bending elastic stiffness of samples. The results suggested that wood species, joint type, and tenon thickness had a significant effect on the bending elastic stiffness, but the loading type and annual ring were not significant. Some of these studies also derived equations used to predict the BMC of M–T joint based on the regression method. Erdil et al. [3] proposed an empirical regression equation (Equation (1)) considering adhesive type, rail width, tenon width, and shear strength of wood to predict the BMC of M–T joints under compression load. Kasal et al. [6] derived an empirical expression to estimate the BMC of M–T joints under diagonal compression and tension states as functions of wood species, adhesive type, and tenon size shown as Equation (2). Wilczy´nskiand Warmbier [8] calculated regression functions (Equation (3)) of the BMC and bending stiffness under diagonal tension state considering the tenon length, tenon width, and tenon thickness. However, these equations were only effective in the situation in which they were derived. M = a(0.25W + 0.5D)Sl0.8 (1) 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). 0.42 M = 0.00227(wl)(0.229w + d)S k1k2 (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); k1 is the loading type: k1(tension) = 1, and k1(compression) = 1.006; k2 refers to adhesive type: k2 is equal to 1 and 0.85 for polyurethane (PU) and PVA, respectively.

M = 1.328t0.167l0.743w0.648 (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 [9–11]. 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 [14–16]. 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. Forests 2020, 11, 501 3 of 12

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.

2. Experimental Design

2.1. 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%. Table1 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

EL * ER ET υLR υLT υRT υTR υTL υRL 12,205 1858 774 0.502 0.705 0.526 0.373 0.038 0.078 Shear Modulus (MPa) Yield Strength (MPa) Ultimate Strength (MPa)

GLR GLT GRT LRTLRT 899 595 195 53.62 12 6.23 59.20 48.88 23.82 * E is elastic modulus, υ is Poisson’s ratio, G is shear modulus; L, R, and T refer to longitudinal, radial, and tangential directions of the wood, respectively.

2.2. Description of Specimens Figure1 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 (Figure1). The mortise and tenon were bonded by 65% solid content PVA. Forests 2020, 11, 501 4 of 12 Forests 2020, 11, x FOR PEER REVIEW 4 of 12

(a) (b)

Figure 1. DDimensionsimensions of of ( (aa)) T T-shaped-shaped specimens and ( b) mortise mortise-and-tenon-and-tenon joint ( (unitunit mm).mm).

2.3. Finite Finite E Elementlement M Modelodel and and S Simulationimulation M Methodethod Figure2 2shows shows a 3-D a 3 orthotropic-D orthotropic finite finite element element model ofmod anel M–T of jointan M T-shaped–T joint sample T-shaped established sample establishedusing finite using element finite software element (ABAQUS software 6.14-1, (ABAQUS Dassult, 6.14 USA).-1, Dassult, The mechanical USA). The mechanical properties used properties in this usedmodel in were this model determined were d frometermined the beechwood from the beechwood shown in Table shown1. Ductile in Table damage 1. Ductile was damage used as was a stress used ascriterion a stress for cri woodterion membersfor wood members in the model, in the and model the parameters, and the parameters needed were needed fracture were strainfracture (0.0083), strain (0.0083),stress triaxiality stress triaxial (0.333),ity strain (0.333) rate, strain (0.01), rate and (0.01), failure and displacement failure displacement (0.6452 mm). (0.6452 The grain mm) orientations. The grain orientationsof post leg and of poststretcher leg andwere stretcher differentiated were bydifferentiat local coordinates,ed by local i.e., coordinates, x, y, and z corresponded i.e., x, y, and to z correspondedlongitudinal, radial, to longitudi and tangentialnal, radial, directions and of tangential wood, respectively. directions Theof interactions wood, respectively. of mortise andThe interactionstenon were surface-to-surface of mortise and tenon contact, were for surface the width-to-surface direction contact, of the for tenon, the wid theth penalty direction contact of the property tenon, wasthe penalty specified contact with a pr frictionoperty coe wasffi specifiedcient of 0.54 with [22 a] friction to simulate coefficient the friction of 0.54 behavior [22] to with simulate 0.2-mm the frictioninterference behavior fit. In with addition, 0.2-mm for interference the thickness fit. direction In addition, of tenon, for cohesivethe thick behaviorness direction was applied of tenon, to cohesivesimulate behavior the cohesive was bondingapplied to behavior. simulate The the parameterscohesive bonding needed behavior. in ABAQUS The parameters software Knn, needed Kss, inand ABAQUS Ktt were so 1.23,ftware 3.49, Knn, and Kss 2.45, and N/ Kttmm, were respectively, 1.23, 3.49,measured and 2.45 N/mm using, therespectively, method described measured in using [23]. theThe method loading described conditions in referenced [23]. The loading to a previous conditions study referenced [24], specifically, to a previous the boundary study [24 conditions], specifically were, thethat boundary six-degree cond freedomsitions werewere constrained that six-degree at two freedoms parts, 20 were mm fromconstrained the two at ends two of parts the post, 20 mm leg, andfrom a thedisplacement two ends of load the was post imposed leg, and on a thedisplac loade headment atload the was point imposed of 30 mm on from the load the end head of stretcherat the point to get of 30the mm ultimate from reactionthe end forceof stretcher of load head,to get whichthe ultimate was equal reaction to the force ultimate of load force head (F) applied, which atwas the equal loading to thepoint ultimate (Figure force2). The (F) meshapplied of at the the model loading was point also ( shownFigure in2). FigureThe mesh2, the of sizesthe model of all elementswas also shown were inapproximate Figure 2, the 5 mm, size whiles of all for elements contact parts,were theapproximate sizes of elements 5 mm, werewhile approximate for contact parts, 2 mm the to make sizesthe of elementsmodel more were accurate. approximate The element 2 mm typeto make assigned the model to the more T-shaped accurate sample. The was element C3D8, type an 8-node assigned linear to thebrick T element.-shaped sampleA 4-node was 3-D C3D8, bilinear an rigid 8-node quadrilateral linear brick element element (R3D4). Awas 4-node assigned 3-D to bilinear load head. rigid quadrilateralIn this study, element the ( eR3D4ffects) was of tenon assigned width to (15,load 20, head 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 Table2. The BMC and bending sti ffness were evaluated by Equations (4) and (5), respectively.

M = F L (4) × K = ∆F/∆d (5) 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). Forests 2020, 11, 501 5 of 12

Table 2. Combinations of tenon width and tenon length evaluated in this study.

Dimensions of Tenon (mm) Width (w) Length (l) 15 20 25 10 l10-w15 l10-w20 l10-w25 20 l20-w15 l20-w20 l20-w25 30 l30-w15 l30-w20 l30-w25 Forests 2020, 11, x FOR PEER REVIEW40 l40-w15 l40-w20 l40-w25 5 of 12

Figure 2. The finite element model (FEM) of T-shaped mortise-and-tenon joint specimen with load and Figure 2. The finite element model (FEM) of T-shaped mortise-and-tenon joint specimen with load constraint conditions. and constraint conditions. 2.4. Verification of Finite Element Model In this study, the effects of tenon width (15, 20, and 25 mm) and tenon length (10, 20, 30, and 40 The finite element model proposed in this study was verified before conducting the numerical mm) on BMC of M–T joints were studied, aiming to obtain the optimal tenon size with maximum and optimal studies. A finite element model was built using the method described above and applied BMC. Therefore, a complete 3 × 4 factorial numerical simulation was conducted using 12 M–T joint to simulate the experimental tests [24] to get the BMC and compare with the experimental results. finite element models with different tenon sizes, as shown in Table 2. The BMC and bending stiffness 2.5.were Statistical evaluated Analysis by Equations (4) and (5), respectively. All data were analyzed by Design ExpertM (Version= F × L 8.06, Stat-Ease, Inc. Minneapolis, MN, USA)(4) using the response surface method. The response surface model was built by the modified cubic regression method, and the model terms wereK statistically= ΔF/Δd analyzed by analysis of variance (ANOVA)(5) at a 5% significant level. Model numerical optimization was also conducted by Design Expert using where M is bending moment capacity (Nm); F is the ultimate force (N); L is moment arm (mm); K is the nonlinear programming method. bending stiffness (N/mm); ΔF is the change of force in the linear portion of bending load and 3.deflection Results andcurve Discussions (mm), and Δd is the change of deflection corresponding to ΔF (mm).

3.1. Results of FiniteTable Element2. Combinations Model Verification of tenon width and tenon length evaluated in this study.

Table3 shows theDimensions results of the of BMC Tenon values (mm) of the experimentWidth [ 24 (w]) and those of the finite element model proposed in this study,Length indicating (l) that the results15 of FEM20 were well25 consistent with those of experimental tests with the ratios of10 FEM to observedl10 ranging-w15 froml10-w 0.85220 tol10 1.072.-w25 In other words, this finite element model was capable of20 replacing a largel20 number-w15 ofl20 experimental-w20 l20-w25 tests to study the effect of tenon size on BMC of M–T joints.30 l30-w15 l30-w20 l30-w25 40 l40-w15 l40-w20 l40-w25

2.4. 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.

2.5. 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. Forests 2020, 11, x FOR PEER REVIEW 6 of 12 Forests 2020, 11, x FOR PEER REVIEW 6 of 12 3. Results and Discussions 3. Results and Discussions 3.1. Results of Finite Element Model Verification 3.1. Results of Finite Element Model Verification Table 3 shows the results of the BMC values of the experiment [24] and those of the finite element modelTable proposed 3 shows in the this results study, of indicating the BMC values that the of resultsthe experiment of FEM were[24] and well those consistent of the finitewith elementthose of modelexperimental proposed tests in withthis study,the ratios indicating of FEM that to observed the results ranging of FEM from were 0.852 well to consistent 1.072. In withother those words, of experimentalthis finite element tests modelwith the was ratios capable of FEM of replacing to observed a large ranging number from of experimental0.852 to 1.072. tests In other to study words, the Foreststhiseffect finite2020 of tenon, 11element, 501 size model on BMC was of capable M–T joints. of replacing a large number of experimental tests to study6 of the 12 effect of tenon size on BMC of M–T joints. Table 3. Comparisons of bending moment capacities (BMCs) between experiment and FEM. Table 3.3. ComparisonsComparisons of bending moment capacities (BMCs)(BMCs) betweenbetween experimentexperiment andand FEM.FEM. Dimensions of Tenon (mm) Bending Moment Capacity (Nm) DimensionsDimensions of Tenon of T (mm)enon (mm) B Bendingending MomentMoment Capacity Capacity (Nm) (Nm) Width 30 WidthWidth 3030 Length Observed FEM Ratio Length Observed FEM Ratio Length 30 Observed219 (9) * FEM215 0.982 Ratio 303040 219 (9)219 *297 (9) (7) * 215215253 0.9820.852 0.982 404050 297 (7)279297 (10)(7) 253253264 0.8520.946 0.852 50 279 (10) 264 0.946 5060 279250 (10) (8) 264268 0.9461.072 6060 250 (8)250 (8) 268268 1.072 1.072 * The values in parentheses are coefficient of variance. * The* The values values in in parentheses parentheses areare coe coefficientfficient of variance.of variance. 3.2. Bending Behavior of M–T Joints During Loading Process 3.2. Bending BehaviorBehavior of M–TM–T JointsJoints DuringDuring LoadingLoading ProcessProcess Figure 3 shows a typical bending load and deflection curve of the M–T joint (30 mm long × 15 Figure3 3 shows shows a a typical typical bending bending load load and and deflection deflection curve curve of the of M–Tthe M joint–T joint (30 mm (30 longmm long15 mm× 15 mm wide × 10 mm thick) with four critical load points and a definition of stiffness (K) of ×the M–T wide 10 mm thick) with four critical load points and a definition of stiffness (K) of the M–T joint. mmjoint .wide × Fig ure× 10 4 mm shows thick the) with stress four distributionscritical load points of this and M a– Tdefin jointition during of stiffness the loading (K) of the process M–T Figure4 shows the stress distributions of this M–T joint during the loading process corresponding to jointcorresponding. Figure 4 to shows four critical the stress load points distributions shown in of Figure this 3M. –T joint during the loading process fourcorresponding critical load to points four critical shown load in Figure points3. shown in Figure 3.

900 900 F

) F (b) N ) (b) N 600 600 (c) (c)

300 (a) (d) (a) Bending load ( 300 K F (d)

Bending load ( K F d d 0 5 10 15 20 0 5 Deflection10 (mm)15 20 Deflection (mm) Figure 3. A typical bending load and deflection curve of the M–T joint measured 30 mm long by 15 Figure 3. A typical bending load and deflection curve of the M–T joint measured 30 mm long by 15 mm Figuremm wide 3. A with typical four bending critical loading load and points deflection, (a) 364 curve N in of linear the Mstage,–T joint (b) themeasured maximum 30 mm load long 778 byN, 15(c) wide with four critical loading points, (a) 364 N in linear stage, (b) the maximum load 778 N, (c) 594 N mm594 Nwide at failure with four stage, critical and ( loadingd) 366 N points at failure, (a) stage.364 N in linear stage, (b) the maximum load 778 N, (c) at594 failure N at failure stage, andstage, (d )and 366 ( Nd) at366 failure N at failure stage. stage.

(a) (b) (a) (b) Figure 4. Cont. Forests 2020, 11, 501 7 of 12 Forests 2020, 11, x FOR PEER REVIEW 7 of 12

(c) (d)

FigureFigure 4. 4.Stress Stress distributions distributions of the of M–T the joint M– T-shapedT joint T sample-shaped at di samplefferent loading at different stages loading corresponding stages tocorresponding the four points to shown the four in points Figure 3shown;( a) 364 in NFigure in linear 3; (a stage,) 364 (Nb) in the linear maximum stage, load(b) the 778 maximum N, (c) 594 Nload at failure778 N, stage,(c) 594 and N at (d failure) 366 N stage, at failure and stage.(d) 366 (unit N at MPa).failure stage. (unit MPa).

3.3.3.3. BendingBending MomentMoment CapacityCapacity andand StiStiffnessffness ofof thethe M–TM–T JointJoint TableTable4 4shows shows the the BMCs BMCs and and bending bending sti stiffnessffness ofof allall combinationscombinations ofof tenontenon lengthlength andand tenontenon width,width, which which indicated indicated that that the the BMC BMC increased increa assed either as either tenon tenon width widthor tenon or length tenon increased. length increased. Further comparingFurther comparing the effects the of effects tenon width of tenon and width tenon and length tenon on BMC,length the on tenon BMC, length the tenon influenced length BMCinfluenced more significantlyBMC more significantly when the tenon when length the tenon was relatively length was short, relatively ranging short from, ranging 10 to 30 from mm. 10 While to 30 the mm eff. Wecthile of tenonthe effect length of tenon on BMC length decreased on BMC when decreased the tenon when length the tenon was longerlength thanwas longer 30 mm than and 30 tenon mm width and tenon was biggerwidth thanwas bigger 15 mm. than The 15 bending mm. The stiff bendingness of the stiffness M–T joint of the increased M–T joint with increased the increase with ofthe tenon incre widthase of andtenon tenon width length and tenon in the length evaluated in the ranges. evaluated The tenonranges. width The tenon had a width more significanthad a more e ffsignificantect on bending effect stionff bendingness, whereas stiffness, this whereas effect of th tenonis effect length of teno reducedn length when reduced the tenon when length the tenon was longer length than was 30 longer mm. Thesethan 30 results mm. agreeThese withresults those agree of previouswith those studies of previous [5,24]. studies [5,24].

TableTable 4.4. BendingBending momentmoment capacitiescapacities (BMCs)(BMCs) andand stistiffnessffness ofof M–TM–Tjoint jointbased basedon onFEM. FEM.

DimensionsDimensions of of Tenon Tenon (mm) (mm) BendingBending Load Load Resistance Resistance (Nm) (Nm) BendingBending Sti Sfftiffnessness (N/ mm)(N/mm) Width Width 1515 2020 2525 1515 2020 2525 Length Length 1010 63.4263.42 64.5464.54 70.0470.04 173.64173.64 200.24200.24 222.76222.76 2020 97.2297.22 119.67119.67 151.17151.17 199.19199.19 240.00240.00 279.71279.71 30 101.12 128.03 158.35 206.14 251.58 302.09 30 101.12 128.03 158.35 206.14 251.58 302.09 40 114.99 129.11 158.62 211.06 257.85 329.82 40 114.99 129.11 158.62 211.06 257.85 329.82

3.4. Failure modes of M–T joints 3.4. Failure modes of M–T joints Figure5 summarized the failure modes of all finite element models with di fferent tenon sizes. 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, Two typical failure modes were observed. When the tenon length was shorter than tenon width, the the failure occurred at the front end of the tenon, indicating that the tenon was completely pulled out 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 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 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 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 to the grain. In other words, the tenon withdrew completely from the mortise when the ratio of tenon widthwidth toto tenontenon lengthlength waswas higherhigher thanthan 1,1, otherwise,otherwise, thethe M–TM–T jointjoint failedfailed becausebecause ofof thethe fracturefracture ofof tenon at the rear end of the tenon. Therefore, it was recommended to make the tenon length longer tenon at the rear end of the tenon. Therefore, it was recommended to make the tenon length longer thanthan tenontenon widthwidth toto getget thethe higherhigher BMCBMC inin thethe designdesign ofof thethe M–TM–T joints.joints. InIn general,general, thethe stressstress atat thethe failure part should reach the peak value. While in the finite element model, once the stress exceeds failure part should reach the peak value. While in the finite element model, once the stress exceeds thethe ultimateultimate strengthstrength ofof thethe material,material, thethe elementselements atat thethe placesplaces ofof damagedamage areare degradeddegraded andand deleted, which make the stress at damage part smaller than other undamaged parts. which make the stress at damage part smaller than other undamaged parts. Forests 2020, 11, 501 8 of 12 Forests 2020, 11, x FOR PEER REVIEW 8 of 12

l10-w15 l10-w20 l10-w25

l20-w15 l20-w20 l20-w25

l30-w15 l30-w20 l30-w25

l40-w15 l40-w20 l40-w25

FigureFigure 5. 5.The The failure failure modes modes of of tenons tenons with with di differentfferent sizes sizes based based on on FEM. FEM. (unit (unit MPa). MPa).

3.5.3.5. Response Response Surface Surface Model Model FigureFigure6 6shows shows the the response response surface surface model model used used to to predict predict the the BMC BMC of of M–T M–T joints joints relating relating to to the the tenontenon width width and and tenon tenon length. length. The The response response surface surface reduced reduced cubic cubic model model for for predicting predicting the the BMC BMC of of M–TM–T joints joints was was derived, derived as, as shown shown in in Equation Equation (6). (6) Additionally,. Additionally, the the ANOVA ANOVA result result of modelof model terms terms is shownis shown in Tablein Table5. 5.

M = 39.072 + 4.627A 6.4384B + 0.876AB 0.434A2 0.015A2B + 0.008A3 (6) − − − 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, A2,A2B, and A3 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 Forestsmaximum 2020, 1 BMC1, x FOR is PEER 160.18 REVIEW Nm, with the tenon length of 28.81 mm and tenon width 25 mm. 9 of 12

Figure 6. ResponseResponse surface surface model model of of bending bending moment relating to tenon width and tenon length. Table 5. ANOVA of response surface reduced cubic model.

2 2 3 M = 39.072 + 4.627ASource − 6.4384B + 0.876ABF-Value − 0.434Ap-Value − 0.015A B + 0.008A (6) Model 142.10 <0.05 Table 5. ANOVALength (A) of response 1.52surface reduced 0.27 cubic model. Width (B) 179.31 <0.05 ABSource F-V 21.27alue p-Value<0.05 2 AModel 142.10130.09 <0.0<50.05 2 LengthA B (A) 1.5230.22 0.27< 0.05 A3 23.26 <0.05 Width (B) 179.31 <0.05 AB 21.27 <0.05 3.6. Verification of the Optimal Solution A2 130.09 <0.05 Another M–T joint T-shaped finiteA element2B model30.22 was established<0.05 using the optimized tenon size, measuring 28.81 mm long by 25 mm thick.A3 Figure23.267 shows < the0.05 results of the finite element model, indicating that the maximum bending load is 1226.75 N (Figure7b). According to Equation (4), the BMCThe of optimizedmodel F-value M–T of joints 142.10 is 159.48 implie Nm,d that which the model is well wa consistents significant. with Values the result of p of less the than response 0.05 indicatesurface modelthat the (160.18 model Nm) terms and are higher significant. than other In this combinations case, B, AB, of A tenon2, A2B, widths and A and3 were tenon significant lengths modelevaluated terms in. this The study. “pred Therefore,icted R-squared the method” of 0.949 of combining4 was in reasonable FEM and theagreement response with surface the method“adjusted is Rcapable-squared of” optimizing of 0.9872. Therefore, the BMC of this M–T model joint i T-shapeds acceptable samples. 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.

3.6. 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. Forests 2020, 11, 501 10 of 12 Forests 2020, 11, x FOR PEER REVIEW 10 of 12

1500 1226.75 1200 ) N 900

600

Bending load ( 300

0 5 10 15 20 Deflection (mm) (a) (b)

FigureFigure 7.7. ResultsResults ofof thethe finitefinite elementelement modelmodel withwith optimizedoptimized tenontenon size,size, ((aa)) stressstress distributionsdistributions (unit(unit MPa),MPa), andand ((bb)) bendingbending load-deflectionload-deflection curvecurve ofof M–TM–T jointsjointswith with maximummaximum bendingbending loadload values.values.

4.4. Conclusions InIn thisthis study,study, bendingbending momentmoment capacitycapacity andand bendingbending stistiffnessffness werewere studiedstudied usingusing FEMFEM andand thethe responseresponse surfacesurface methodmethod toto optimizeoptimize thethe tenontenon size.size. TheThe followingfollowing conclusionsconclusions werewere drawn:drawn: 1.1. The proposed finitefinite element model using contactcontact behaviorbehavior toto simulatesimulate thethe gluedglued M–TM–T jointjoint waswas verifiedverified through predictingpredicting thethe bendingbending momentmoment capacitycapacity ofof thethe M–TM–T joint.joint. 2.2. The BMC and stistiffnessffness were significantlysignificantly affffectedected byby tenontenon size,size, andand tenontenon lengthlength hadhad aa moremore significantsignificant e ffectffect on bending bending moment than tenon width, width, while thethe tenon tenon width width aaffffectedected the the bending stistiffnessffness more significantly.significantly. 3.3. TThehe proposed proposed method method combining combining FEM FEM and andresponse response surface surface method method used to used predict to and predict optimize and optimizethe BMC the of M–TBMC jointsof M–T relating joints relating to tenon to lengthtenon length and tenon and tenon width width is capable is capable of replacing of replacing the theexperiments experiment ands and providing providing reasonable reasonable results results without without experimental experimental errors. errors. 4.4. It isis recommended to make the ratio of tenontenon lengthlength toto tenontenon widthwidth greatergreater thanthan 11 toto getget higherhigher BMC in the design of M–TM–T joints.

InIn conclusion,conclusion, thethe proposedproposed methodmethod combiningcombining FEMFEM andand thethe responseresponse surfacesurface methodmethod isis verifiedverified asas an efficient efficient way way to to optimize optimize the the M M–T–T joints, joints, which which will will contribute contribute to the to design the design of M– ofT joint M–T wood joint woodproducts. products. Further Further research research will focus will on focus the following on the following points: (1) points: additional (1) additionaltests will be tests repeated willbe to repeatedmake the to conclusions make the conclusions more reliable more; reliable; (2) the relat (2) theionship relationship between between bending bending moment moment capacity capacity and andfurniture furniture structure structure will willbe studied be studied based based on FEM on FEMto better to better instruct instruct the design the design of M– ofT joint M–T wood joint woodproducts products..

AuthorAuthor Contributions: Conceptualization,Conceptualization, W.H. W.H. and and N.L N.L.;.; methodology, methodology, W.H.; W.H.; software, software, W.H.; W.H.; validation, validation, W.H. W.H.and N and.L.; formal N.L.; formal analysis, analysis, W.H.; investigation, W.H.; investigation, W.H.; resources, W.H.; resources, W.H.; data W.H.; curation, data curation,W.H. and W.H.N.L.; andwriting N.L.;— writing—originaloriginal draft preparation, draft preparation, W.H.; W.H.; writing writing—review—review and and editing, editing, W.H. W.H. and and N N.L.;.L; supervision, supervision, W W.H.;.H.; funding funding acquisition, W.H. All authors have read and agreed to the published version of the manuscript. acquisition, W.H. Funding: This work was supported by the Scientific Research Foundation of Nanjing Forestry University (GXL2019074),Funding: This A work Project was Funded supported by the by National the Scientific First-class Research Disciplines Foundation (PNFD). of Nanjing Forestry University (GXL2019074), A Project Funded by the National First-class Disciplines (PNFD). Acknowledgments: The authors would like to express their heartfelt gratitude to the Priority Academic Program DevelopmentAcknowledgments: of Jiangsu The Higherauthors Educationwould like Institutions. to express their We heartfelt also would gratitude like to to thank the Priority the anonymous Academic reviewers Program andDevelopmen editor fort theirof Jiangsu valuable Higher comments Education and Institutions. suggestions forWe improvingalso would the like quality to thank of this the paper.anonymous reviewers Conflictsand editor of for Interest: their valuableThe authors comments declare and no suggestions conflict of interest. for improving the quality of this paper.

Conflicts of Interest: The authors declare no conflict of interest.

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