# A finite element model of semi-rigid mortise-and-tenon joint considering glue line and friction coefficient

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## Abstract

The aim of this study is to build a new numerical model of a mortise-and-tenon joint based on the finite element method (FEM) considering glue line and friction coefficient to analyze the semi-rigid wood joint. Firstly, the friction coefficient, glue distributions and strengths of the mortise-and-tenon joint were determined by experimental methods. Secondly, these parameters were specified to a T-shaped mortise-and-tenon joint model to build a finite element model of joint by FEM. Finally, both withdrawal and bending load capacity of T-shaped specimens were investigated by experimental and numerical methods. The results showed that the testing methods used to determine the coefficient, distributions and strengths of the glued mortise-and-tenon joint were all effective enough to determine the mechanical properties of the wood mortise-and-tenon joint, and the finite element model of joint can be applied to analyze the semi-rigid mortise-and-tenon joint with consistency beyond 85%. These methods and finite element models will contribute to the analysis of wood products and wood constructions.

## Keywords

Finite element model Semi-rigid Wood framework Glue line Friction coefficient## Abbreviations

- FEM
finite element method

- PVAc
polyvinyl acetate

- L
longitudinal

- R
radial

- T
tangential

- RP
reference point

- COV
coefficient of variance

- ANOVA
analysis of variation

- LSD
least significant difference

## Introduction

Mortise-and-tenon joint is widely used in wood products and wooden constructions engineering [1, 2]. It is well known that the joints are critical part of wood frameworks [3, 4, 5]. Besides, it is a typical semi-rigid joint manufactured by wood. It was first introduced into stiffness evaluation of furniture joint by Eckelman [6]. Generally, it is difficult to evaluate the strength of a semi-rigid joint accurately by normal knowledge of mechanics. However, with the development of computer technique, the finite element method (FEM) has been popular with engineers and applied to structure design of wooden constructions and wood products. Although a number of studies have focused on this topic, the methods of modeling a reasonable semi-rigid mortise-and-tenon joint by FEM has not been figured out.

The mortise-and-tenon joint was regarded as a rigid joint in some studies [7, 8] using FEM to analyze the skeletal furniture. Obviously, although this can simplify the model, the results were not accurate enough to analyze the whole frame of furniture. Also, others considered it as semi-rigid by taking glue line into consideration when building finite element model of joint by FEM. However, the joint is usually assumed as a clearance fit and the gap is equal to the clearance [9, 10, 11, 12]. In addition, the mortise-and-tenon joint was seen as a semi-rigid connection by adding a spring which was predefined by spring constant value [13]. These studies contributed to analyze semi-rigid joint by FEM, but further studies must been conducted to make it more accurate. Džinčić and Živanić [14] studied the distributions of glue in the joint with entire clearance and total interference fit by metallographic microscope. The results showed that, with 0.1 mm clearance fit, the glue line was clear and not interrupted along the tenon, and the average thickness of joint was 0.095 mm. By contrast, with interference fit, no glue was observed but only the compressed wood, because the glue in the interference fit area was squeezed out. However, in factual industrial practice, the real-time manufacturing condition of the mortise-and-tenon joint is both with interference fit and clearance fit at the same time (i.e., the interference fit in wide direction of tenon, and clearance fit in thick direction of tenon). Thus, the distributions of glue in real-time condition should be determined, and a reasonable model of the mortise-and-tenon joint must take these points into consideration.

In this study, the concept of the semi-rigid mortise-and-tenon joint can be explanted as the effects of glue line, friction coefficient and the strength of wood itself. So it is vitally important to know more about the distributions of glue, the friction coefficient of joint and mechanical properties of wood before building the model of joint. In addition, the mechanical properties of joint inputted into the finite element model must be determined and assigned to a model with appropriate finite elements.

The aim of this study was to further investigate the distributions of glue and friction coefficient of joint considering the real-time condition of joint by experimental methods, and build a finite element model of mortise-and-tenon joint based on the results determined by experiments. In addition, a comparison was made between the results of FEM and experiment to verify the validity of the finite element model through determining withdrawal and bending resistance loading capacity of T-shaped mortise-and-tenon joint specimens.

## Materials and methods

### Materials

All of the specimens were made with beech (*Fagus orientalis.* Lipsky), bought from local wood commercial supplier (Nanjing, China). According to the ASTM D 4442 [15], the average density was 0.692 g/cm^{3}, and the moisture content of beech was conditioned to and held at 10.8% before and during the experiment. Besides, the joint was glued with polyvinyl acetate (PVAc), which was produced by Pattex, and the solid content was 52%. In addition, the temperature was controlled at 22 °C, and the relative humidity was 48% during the whole process of experiment.

### Specimen preparation

*G*

_{I}), and Fig. 3e shows the specimen to determine the shear strength of the joint perpendicular to the withdrawal direction (

*G*

_{II}). These parameters will be used in finite element model.

### Testing methods

#### Friction coefficient of the joint

*F*refers to the maximum force measured by machine,

*N*.

*p*is the efficiency of the pulley (0.94).

*m*

_{1}and

*m*

_{2}refer to the weight of counterweight (2.5) and upper mortise (0.02), respectively, kg.

*g*is acceleration of gravity, N/kg.

#### Shear and internal bond strength of glued joint

*G*

_{I}and

*G*

_{II}. It is worth paying attention not to fix the sample tightly by clamping the adjusting rod, but to make sure it can move smoothly when loaded (

*P*). The internal bond strength of glued joint in the flat area (Fig. 3c) was measured according to the procedure described in ASTM D1037-12 [16] and the setup is shown in Fig. 5b. All shear tests were performed on a 20 kN capacity universal testing machine at a loading rate of 1 mm/min according to the procedures described in GB/T 1935–2009 [17]. The shear strength

*G*

_{I}and

*G*

_{II}and internal bond strength were measured in 20 replications, respectively.

#### Distribution of glue in the joint

Fluorescence microscope (DM 50008B, Leica, Shanghai, China) was used to determine the actual distributions of glue in the joints. Toluidine blue solution with 0.5% solid content was used to stain all slices cut from block specimens (Fig. 3b, c). The glue line thickness of each specimen was measured 20 times by using Imagine J2x software.

#### Withdrawal and bending loading capacities

### Modeling

*G*

_{I}and

*G*

_{II}) of glue bonding. The elastic properties of beech wood were measured by strain gauges in compression tests in prophase study shown in Table 1 [18, 19], whereas the elastic properties of PVAc glue line were from literature [12], i.e., the modulus of elasticity was 460 MPa and Poisson’s ratio was 0.3. In general, 3D element, C3D8, was used to model beech wood materials; specifically, the element size of 1.4 mm × 1.4 mm × 1.4 mm was used for the material close to the contact surfaces of mortise-and-tenon and the element size of 5 mm × 5 mm × 5 mm was used for the rest of the joint member materials. Curve surfaces of the mortise-and-tenon joints were modeled using 3D element, C3D8, and surface-to-surface contact property was friction with their surface friction coefficient determined in this study. The reason for using nonbonding element for modeling curved surface was because in the actual experiment, it was observed that there was lack of adhesive in the contact surfaces of the curved area.

Modulus of elasticity (MPa) | Poisson’s ratio | Shear modulus (MPa) | Yield strength (MPa) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

| | | | | | | | | | | | L | T | R |

12,205 | 1858 | 774 | 0.502 | 0.705 | 0.526 | 0.373 | 0.038 | 0.078 | 899 | 595 | 195 | 42.51 | 4.49 | 9.83 |

A 30 mm displacement load was imposed on the stretcher to produce withdrawal (Fig. 6a) and bend (Fig. 6b) effects on the joint model, respectively. The total reaction force obtained at the reference points (RP) were the withdrawal and bending loading capacities of the modeled joint.

## Results and discussions

### Friction coefficient of joint

According to the testing method shown in Fig. 4, the friction coefficient was figured out by Eq. 1. The average friction coefficient of the mortise-and-tenon joint is 0.54 with coefficient of variance (COV) 7.62%. Besides, the results were analyzed by analysis of variance (ANOVA) and all mean comparisons were performed at the 1% significance level using the protected least significant difference (LSD) multiple comparisons procedure. The results shows that the effect of grain orientations of tenon on friction coefficient is not statistically significant with *p* value far more than 0.01. Besides, Hu and guan [20] studied the influences of pressures and grain orientations on the friction coefficient of beech wood, and the results showed that the effects of pressure on the friction coefficient of wood were not considered to be statistically significant.

### Distributions of glue in joint

### Shear strengths of the glue joint

*G*

_{I}and

*G*

_{II}) of the joint, which suggests that the shear strength

*G*

_{I}is a little bigger than that of

*G*

_{II}. However, the results of ANOVA suggest that the difference between

*G*

_{I}and

*G*

_{II}is not statistically significant with

*p*value > 0.01. The internal bond strength of the joint in the flat area is 1.63 MPa with COV 17.8%.

### Comparison between the experiment and FEM

Comparison of withdrawal and bending load capacities of experiment and FEM

Loading types | Experiment | COV | FEM | Ratio |
---|---|---|---|---|

Withdrawal ( | 5133 | 4.35 | 4998 | 0.97 |

Bending ( | 1130 | 8.24 | 960 | 0.85 |

## Conclusions

- 1.
The method used to measure the friction coefficient of the joint is effective, and the average friction coefficient of beech wood mortise-and-tenon joint is 0.54.

- 2.
The distributions of glue in the mortise-and-tenon joint can be determined by fluorescence microscope clearly. Specifically, the thickness of the glue line in the flat surface is 54.54 μm, while glue was absent in the curve surface of the joint.

- 3.
The procedures applied to process the specimens and measure the shear strengths of the mortise-and-tenon joint are valid, and the difference of shear strength

*G*_{I}and*G*_{II}is not statistically significant. - 4.
The results of the finite element model are well consistent with those of the experiments in withdrawal and bending loading capacities of the mortise-and-tenon joint T-shaped specimens.

In conclusion, the methods used in this paper can be applied to measure the distribution of glue, friction coefficient and strengths of the joint. In addition, the finite element model of the mortise-and-tenon joint can be applied to the structure design of wood furniture and wooden constructions.

## Notes

### Authors’ contributions

All authors materially participated in the research and article preparation. Both authors read and approved the final manuscript.

### Acknowledgements

The authors thank the support of a project funded by A Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

### Competing interests

This manuscript has not been submitted to any of the other journals and has not been published previously (partly or in full). Its publication is approved by all authors and tacitly or explicitly by the responsible authorities where the work was carried out. There are no competing interests in this manuscript.

### Availability of data and materials

The data are available if needed.

### Funding

This work was supported by a project funded by A Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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