Volume 9

Issue 4

##### Wood Technology

JOURNAL OF

POLISH

AGRICULTURAL

UNIVERSITIES

Available Online: http://www.ejpau.media.pl/volume9/issue4/art-32.html

**
RIGIDITY-STRENGTH MODELS AND STRESS DISTRIBUTION IN HOUSED TENON JOINTS SUBJECTED TO TORSION
**

Tomasz Gawroński*
Department of Furniture Design,
The August Cieszkowski Agricultural University of Poznan, Poland*

The aim of this study was an analysis of rigidity-strength numerical models based on the finite elements method of housed tenon joints subjected to torsion. The experiments aimed at selecting models which would allow effective solution of various verification tasks associated with the strength and optimisation of constructions in which connections are subjected to significant torsional loads. The research concerned beam and brick model. The results of dislocation provided by brick model in conditions of pure torsion were verified using the results of identical laboratory investigations. Then the dislocations obtained using both models in a complex state of loads were compared. The research proved that the beam model of the housed tenon connection can be employed to simulate torsional deformations of this connection in strength verification and optimization of furniture skeletons. Whereas the observed similarity between the laboratory experiments and the results of numerical calculations arrived at using the brick model allows to believe that the this model can be used successfully for the stress analysis and dimension optimization of connections in skeleton furniture. The distribution of stresses in the constructional node was determined using brick model.

**Key words:**
furniture, joint, model, stiffness, torsion.

**INTRODUCTION**

The stiffness and strength of skeleton constructions depend, primarily, on the parameters of constructional nodes. Therefore, in order to be able to carry out optimisation of this type of furniture with the aim to minimise material consumption, it is necessary to ascertain the characteristics of the employed connections. Investigations in this area carried out so far focused on the determination of elastic and strength constants of bent connections both in tenon and dowel joints [2, 3, 6, 11, 16]. However, it is evident from the analysis of internal forces and strains that occur in elements of skeleton furniture for sitting [5, 14] that construction nodes are subjected to much more complex loads. Only few researchers [9, 10] made an attempt to determine mechanical properties of tenon joints subjected to torsion. One of the objections concerning the above-mentioned experiments refers to the fact that the axis of rotation of the constructional nodes did not coincide with the geometrical centre of the tenon cross section which may have contributed to the development of an additional bending moment of the tenon. This unfavourable factor was eliminated in the investigations carried out by Gawoński [5], who applied a special device with a grip with bearings which ensured loading conditions similar to pure bending. The performed experiments allowed to develop a finite element method based beam model of a torsion connection with a tenon joint for which the equivalent modulus of elasticity was determined in the laboratory. The simplicity of the above-mentioned model made it possible to utilise it for the optimisation of entire furniture skeletons where a significant number of iterations carried out by the algorithm required rapid verification of strength conditions. However, a wider application of the proposed methodology involves verification of the correctness of the determined deformations of the constructional node in complex load conditions.

**RESEARCH OBJECTIVE**

The aim of this study was an analysis of rigidity-strength numerical models based on the finite elements method of housed tenon joints subjected to torsion. The experiments aimed at selecting models which would allow effective solution of various verification tasks associated with the strength and optimisation of constructions in which connections are subjected to significant torsional loads. Another objective was to determine the distribution of stresses in the constructional node in the above state of pure torsion.

**RESEARCH MOTHODOLOGY**

The object of investigations was a connection with a stop housed tenon joint with the elliptical cross section of the tenon. The dimensions of the experimental joint presented in Figure 1 were selected on the basis of the analysis of typical constructional solutions employed in skeleton furniture. It was further assumed that the construction elements were made of beech wood and glued with a PVAC adhesive.

The following two numerical models of the above connection were applied in the performed investigations: beam and solid models. The beam model, which is significantly simpler, allows obtaining calculation results in a considerably shorter time and, therefore, can be used as an excellent tool for the constriction analysis and optimisation of skeleton furniture when it is not necessary to consider joint dimensions directly and the final connection rigidity is essential. Equivalent elasticity characteristics of the beam model are determined on the basis of laboratory investigations as well as mathematical correlations describing a given model which combine the wanted material constants with the values of force and dislocation registered in the course of loading samples of real connections. That is why the constructional nodes simulated in this way show deformations which are identical with those which characterised samples employed to determine model constants. Therefore, these models do not need verification if the considered loading conditions are in keeping with those which occurred during the investigations of model samples. Hence, the assessment of their suitability for the analysis and optimisation of real constructions should be conducted on the basis of the evaluation of deformations at different, more complex loading conditions.

Figure 1. Dimensions of considered joint |

Another considered model was the solid model which contains finite elements of the *brick* type and which represents the shape of the applied joints and the adhesive bond. Therefore, its potential applications include searching for the optimal dimensions of joints and the analysis of stresses in the connection itself. Its shortcoming is the large number of finite elements and nodes which increase the calculation complexity of the task. Bearing in mind the fact that the consistency of this model with the real conditions depends on the suitability of the assumptions adopted at the stage of its development, it is essential to verify the consistency of results obtained with its assistance with values obtained in the laboratory experiments carried out in the identical conditions of load and support.

Taking into account the above-presented considerations, the author decided to prepare a solid model of the analysed connection and later verify the results provided by this model in conditions of pure torsion using the results of identical laboratory investigations. The performed comparison should allow to assess the usefulness of this model for analyses in which it is essential to utilise directly constructional details of the connection, as, for example, optimisation of the joint dimensions. Next, the author decided to compare the dislocations obtained using both models in a complex state of loads.

The applied beam model of the torsion connection is presented in Figure 2. This model was made up of three elements in which the deformations of *AB* and *CD* elements corresponded to the deformations of wooden slats, whereas the deformations of the *BC* element simulated the work of the torsional construction node. The Kirchoff’s modulus *G _{w}* of the

*AB*and

*CD*beams was identical with the Kirchoff’s modulus of the applied wood, while the Kirchoff’s modulus

*G*of the

_{j}*BC*element constituted the equivalent elasticity modulus of the connection. The length of the

*AB*and

*CD*elements was equal to the real length of wooden slats of the sample or of the simulated construction. On the other hand, the

*l*length of the

_{j}*BC*element was constant and equalled 10 mm. This value was selected on the basis of earlier experience gathered in the course of experiments on connection modelling and optimisation [ (4, 5, 13]. On the one hand, the

*BC*element as the one simulating the angular deformations should be as short as possible, however, the excessive reduction of the

*l*value could lead to the necessity of application of very low values of equivalent rigidity moduli which would be unwelcome taking into account the numerical accuracy and stability of the calculation process.

_{j}Figure 2. Beam model of considered join |

In the case of the above-described model, conditions of pure torsion of the tenon joint can be achieved by fixing of the point *A* and application of the force *P* perpendicular to the *ACD* plane at point *D*. When the *AB* segment is sufficiently short, the impact of its torsional deformation on the *d* dislocation of the *D* point, as affected by force *P*, is negligible. On the other hand, the *d _{b}* dislocation associated with the deformations of the

*CD*segment constitutes a significant part of the

*d*dislocation and it can be described as:

where:

*l _{b}* – length of the bent part of wood rail,

*E*– Young modulus of wood,

_{w}*J*– axial moment of inertia of wood rail.

On the other hand, the angular deformation of the constructional node equals

Moreover, the correlation between the *P* force and the *d* deformation and the Kirchhoff—s equivalent modulus is expressed by the formula:

where:

*J _{0}* – polar moment of inertia of wood rail.

The value of the Kirchoff’s equivalent modulus of the *G _{j}* connection was adopted on the basis of Gawroński’s investigations [5] as 113 MPa. However, the scope of investigations carried earlier was expanded by the measurement of the breaking force

*P*. The appropriate tests were performed on the test machine Zwick 1445 with the assistance of the equipment presented in Figure 3. The arm of the sample corresponding to the

_{max}*AB*segment of the model was fixed in a permanent grip 1, while the

*P*force was exerted at the point designated on the model (Fig. 2) as

*D*. In order to eliminate the bending moment in the

*BC*element, node C was placed in a grip 2 equipped in a bearing.

The next considered numerical model of the discussed connection was the solid model (Fig. 4). When discretising the model, in accordance with the assumptions of the finite elements method, the author employed the *brick* type elements. The wooden elements were treated as anisotropic bodies for which, on the basis of the study by Wilczyński [15], the following elastic constants were assumed: *E _{L}* = 14010 MPa,

*E*= 1160 MPa,

_{T}*E*= 2280 MPa, ν

_{R}*= 0.448 MPa ν*

_{LR}*= 0.073, ν*

_{RL}*= 0.708,*

_{RT}*G*= 470 MPa,

_{TR}*G*= 1640 MPa,

_{RL}*G*= 1080 MPa. On the other hand, the applied adhesive bond was 0.1 mm thick and constituted an isotropic body of Young’s modulus and Poisson’s coefficient determined on the basis of experiments conducted by Smardzewski [12]:

_{LT}*E*=460 MPa, ν = 0.3,

*G*=177 MPa.

Figure 3. Equipment for testing of joints under torsion |

Figure 4. Brick model of tested joint |

The outline of the loading and support in the designed numerical model complied with the assumptions adopted in the laboratory investigations. In order to make the conditions of both tests as similar as possible, an additional body simulating the action of grip 2 from the device was modelled. This body was characterised by rigidity which was many times greater than that of the remaining elements and was equipped in an unmovable support capable of rotation simulating the work of bearings employed in the device. Two independent cases of loading with a concentrated force in the node corresponding to point *D* in Figure 1 were considered simultaneously. In the first of these cases, the value of force *P _{1}* was equal to 30% of the breaking load

*P*. This allowed to determine the

_{max}*d*dislocation at the point of force application in the linear interval with the aim to compare it with the results of laboratory experiments. On the other hand, in the second case, the

_{1}*P*load was equal to the

_{2}*P*force which allowed to determine the state of stresses at the moment of breaking of the connection.

_{max}In the course of further investigations, the rotational unmovable support at point *C* was excluded from the beam model which allowed to perform the analysis of the deformations at point *D* in the complex state of loading. Similarly, the body simulating the action of grip 2 together with the support representing the work of bearings was removed from the solid model. Both models were loaded with force *P _{1}* identically as in the above tests and at this place the

*d*dislocation was determined.

_{1}’**ANALYSIS OF RESULTS**

Values of the *P _{max}* force obtained in the course of laboratory experiments are shown in Table 1 and on this basis, the value of the

*P*force was adopted as 151 N.

_{1}The value of the dislocation *d _{1}* determined for the

*P*force during laboratory experiments and numerical investigations with the assistance of the solid model are presented in Table 2. The dislocation obtained from the numerical calculations is by 7% lower in comparison with the value obtained as a result of tests on the test machine. The observed differences can be attributed to the discrepancies between the properties of beech wood assumed in the numerical model and real traits of the material employed in investigations.

_{1}Table 1. Values of the P force obtained in the course of laboratory experiments_{max} |

Mean [mm] |
503 |

Standard deviation [mm] |
67 |

Coefficient of variation |
0.13 |

Table 2. Value of the d dislocation for 30% P_{max} |

Method of determination |
d [mm] |

At the laboratory |
3.2 |

Numerically |
3.0 |

Table 3 presents values of the *d _{1}’* dislocations obtained for both models in the combined state of loading. The result obtained for the brick model was by 9.8% lower in comparison with the beam model. The obtained results are similar and indicate the possibility of replacing the brick model with the beam one during numerical calculations. However there is a need for further research in order to determine which of the result is more close to the real conditions.

Table 3. Values of the d dislocations obtained for both models in the combined state of loading_{1}’ |

Type of model |
d’ [mm] |

Beam model |
4.38 |

Brick model |
3.95 |

In the next part, results of calculations of stresses occurring in the construction obtained from the solid model are presented. For the adopted design of loading and support, we can expect considerable normal stresses in the glue bond in the direction perpendicular to the side plane of the tenon. This stress is presented in Figure 5. The maximum stress amounted to 70 MPa and was concentrated in the elliptical part of the tenon, whereas on the greater part of the flat surface of the tenon this stress did not exceed 10 MPa. In his experiments on double-cut samples made from beech wood and bonded with PVAC adhesive, Krystofiak [8] obtained delamination strength at the level of 14 MPa. However, since this value was calculated on the basis of the delamination coefficient according to Bock [1], therefore it reflects only the interval of mean normal stresses in the glue bond and does not take into account places of stress concentrations at the end of overlaps in which extreme values may be many times higher. Therefore, it can be presumed that the normal stresses obtained in the glue bond were not the direct cause of the destruction of the connection.

Figure 5. Normal stress in glue line |

Figure 6 shows reduced stresses in tenon according to Tresca’s hypothesis. The highest value of this stress occurs in the elliptical part of the tenon and amounts to 74 MPa which constitutes 60% of the bending strength of beach wood reported by Kollman [7]. The tangential stress in the *xy* plane, which amounted to 33 MPa, made up a considerable proportion of this stress.

Figure 6. Tresca stress in tenon |

On the other hand, the maximum Tresca’s reduced stress in the element in which the mortise was made was 46 MPa (Fig. 7) which constitutes 62% of the value obtained for the tenon. However, the analysis of the character of the obtained breakages (Fig. 8) indicates a high proportion of normal stresses in the *Y* axis. The distribution of these stresses is presented in Figure 9 and their maximum value amounts to 9.6 MPa, so it exceeds by 37% the value of the tensile strength across fibers. Therefore, it can be said that, for the adopted dimensions of the joints, the mortise is the place which is most exposed to destruction as confirmed by the obtained pictures of breakages.

Figure 7. Tresca stress in mortise |

Figure 8. Joint destruction at performed test |

Figure 9. Normal stress in Y-axis direction in mortise |

**CONCLUSIONS**

The beam model of the housed tenon connection can be employed to simulate torsional deformations of this connection in strength verification and optimization of furniture skeletons. Its application allows to obtain accurate results both in the case of pure torsion and complex state of loading at a time which is several times shorter in comparison with the solid model.

The observed similarity between the dislocation at the point of force application obtained during the laboratory experiments and the results of numerical calculations arrived at using the solid model allows to believe that the research model was designed correctly and it can be used successfully for the stress analysis and dimension optimization of connections in skeleton furniture.

It is evident from the analysis of stresses occurring in the torsion tenon joint that the mortise is the place which is most exposed to destruction as confirmed by pictures of breakages obtained in the course of laboratory experiments.

**REFERENCES**

Bock E., 1951. Prüfung von Holzleimen. Die Bestimmung der statischen Festigkeitseigenschaften der Leimvarbindungen [Investigation of glue wood. Investigation of static strength of glue joints]. Holz a. Roh u. Werkst. Jg. 9 H. 2, 62-71 [in German].

Dzięgielewski S., Smardzewski J., 1992. Sztywnosc i wytrzymałosc połaczeń katowych wykonanych z płyt MDF [Stiffness and strength of corner joint made of MDF]. Przem. Drzew. 5, 6-8 [in Polish].

Eckelman C. A., Suddarth S. K., 1969. Analysis and design of furniture frames. Wood Sci. Technol. 3, 239-255.

Gawroński T., 2002. Optymalizacja mebli szkieletowych metoda gradientowa [Optimisation of skeleton furniture with a gradient metod]. Ph. D. thesis. The August Cieszkowski Agricultural University of Poznań [in Polish].

Gawroński T., 2005. Multiobjective optimisation of a skeleton furniture construction. Rocz. AR Pozn. Rozpr. Nauk. Zeszyt 368.

Gogolin R. M., Wilczyński A., Warmbier K., 1996. Naprężenia na powierzchni płaskiego połaczenia narożnikowego o złaczu kołkowym [Stress on the surface of flat corner connection with dowel joint]. 8 Ses. Nauk. Badania dla meblarstwa. 49-59 [in Polish].

Kollmann F., 1982. Technologie des Holzes und der Holzwerkstoffe [Technology of wood and wood compound materials]. Springer-Verlag, Berlin [in German].

Krystofiak T., 2002. Badania nad zastosowaniem srodków proadhezyjnych do klejenia wybranych materiałów w przemysle drzewnym [Research on the application of adhesive promoters for gluing of selected materials in wood industry]. Ph. D. thesis. The August Cieszkowski Agricultural University of Poznań [in Polish].

Nakai T., Takemura T., 1995. Torsional Properties of Tenon Joints with Ellipsoidal-like Tenons and Mortises. Mokuzai Gakkaishi Vol. 41. No 4, 387-392.

Nakai T., Takemura T., 1996. Stress Analysis of the throug-tenon joint of wood under torsion I. Measurements of shear stresses in the male by using Rosette gauges. Mokuzai Gakkaishi 42, 4, 387-392.

Smardzewski J., 1991. Analiza odkształceń połaczeń drewnianych [Analysis of deformation of wooden connections]. Rocz. AR. Pozn. 216, 83-103 [in Polish].

Smardzewski J. 1998. Wpływ niejednorodnosci drewna i spoiny klejowej na rozkład naprężeń stycznych w połaczeniach meblowych [The influence of heterogeneity of wood and glue line on the distribution of tangential stress in furniture connections]. Rocz. AR Pozn. Rozpr. Nauk. 282, [in Polish].

Smardzewski J., Gawroński T., 2003. Gradient optimisation of skeleton furniture with different connections. EJPAU, Wood Technol. 6, (1) www.ejpau.media.pl.

Smardzewski J., Papuga T., 2004. Stress distribution in angle joints of skeleton furniture. EJPAU, Wood Technol. 7, (1) www.ejpau.media.pl.

Wilczyński A., 1987. Model anizotropii własciwosci sprężystych drewna [Model of anisotropy of elastic properties of wood]. Zesz. Probl. Post. Nauk Roln. 334, 115-132 [in Polish].

Wilczyński A., Warmbier K., 1996. Badania metodyczne połaczenia narożnikowego płaskiego o złaczu dwukołkowym [Methodical research of flat corner connection with double dowel joint]. 9 Ses. Nauk. Badania dla meblarstwa, Poznań, 23-32 [in Polish].

Wilczyński A., Warmbier K., 1997. Wpływ wymiarów złącza na nosnosc i sztywnosc połaczenia katowego płaskiego o złaczu dwukołkowym [The influence of joint dimensions on the load capacity and stiffness of flat angle joint of double dowel joint]. 10 Ses. Nauk. Badania dla meblarstwa, Poznań, 23-32 [in Polish].

Accepted for print: 29.11.2006

Tomasz Gawroński

Department of Furniture Design,

The August Cieszkowski Agricultural University of Poznan, Poland

Wojska Polskiego 38/42, 60-627 Poznan, Poland

phone: +48 61 848 74 74

email: TGawronski@au.poznan.pl

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.