3D NON-LINEAR FINITE ELEMENT MODELLING OF TRADITIONAL TIMBER CONNECTIONS Bo-Han Xu1, Abdelhamid Bouchaïr1, Mustapha Taazount1 ABSTRACT: In this paper, 3D non-linear finite element models are developed to simulate the mechanical behaviour of the rounded dovetail connection. The non-linear models use the Hill criterion to manage the plastic yielding of the wood material. The modified Hill failure criteria, managing the damage evolution in wood, are used to take into account the brittle failure in shear and tension. Besides, the models also incorporate the contact and the non-linearity geometric. The experimental results are used to validate the 3D non-linear finite element models. The comparison with experimental results shows that the numerical models are in good agreement with them. The validated models can be used to predict the strength and failure position of the rounded dovetail connection.

KEYWORDS: Rounded dovetail connection, Finite element model, Failure criterion, Stress analysis 1 INTRODUCTION 123 The dovetail joint is a common traditional timber connection in carpentry. However, because of the productivity, it has been slowly replaced by the joist hangers and nail plates. In the last decade, with the development of machine tool with numerical control, the dovetail mortises and tenons can be made quickly and precisely. Thus, it is re-used due to aesthetic. The rounded dovetail connection (RDC) is a relatively new dovetail connection which is named after the rounded shape in the longitudinal direction of the joist, similar to a dovetail (Figure 1). The use of rounded dovetail connections in timber construction has significantly increased in Europe. Nowadays, it has become more and more important to know the realistic load carrying capacity of traditional timber joints for reconstruction, rebuilding and improvement of structural behaviour. Thus, a number of experimental studies on RDC have been carried out [1-3]. Two failure modes were observed: tension perpendicular to grain failure at the mortise base or tenon failure with splitting of the joist member [3]. The mechanical performance of RDC is dependent upon a number of factors, such as depth of the beam, clearance in the bottom of the notch, flange angle and connection angle. However, the available experimental results in the 1 Bo-Han Xu, LaMI, Polytech’Clermont-Ferrand, Université Blaise Pascal, BP 206, 63174 Aubière Cedex, France. Email: bohanxu@gmail.com 1 Abdelhamid Bouchaïr, LaMI, Polytech’Clermont-Ferrand, Université Blaise Pascal, BP 206, 63174 Aubière Cedex, France. Email: bouchair@polytech.univ-bpclermont.fr 1 Mustapha Taazount, LaMI, Polytech’Clermont-Ferrand, Université Blaise Pascal, BP 206, 63174 Aubière Cedex, France. Email: taazount@univ-bpclermont.fr

literature are rather limited. In order to show the influence of these factors, it is necessary to develop and adjust 3D finite element models which can predict the mechanical behaviour of RDC.

Figure 1: RDC timber members

A linear elastic numerical model was been developed to predict the strength of RDC [4]. A probabilistic method is used. The method, rather than being stress-based, incorporates size effect for the combined action of tension perpendicular to grain and shear parallel to grain stresses in timber by comparing computed stress volume integrals to unit volume strength thresholds. Therefore not only the magnitude of the stress distributions is considered but also the volume over which they act. The capacities of RDC configurations were predicted and successfully validated with experimental tests. However, the model has only considered the brittle linear elastic behaviour of timber. The linear elastic model can not

accurately represent the plastic compression behaviour at the zone of RDC. In this paper, 3D non-linear finite element models have been developed. The timber properties are regarded as transverse isotropic. The timber plasticity is considered in compression in the parallel and perpendicular to grain directions. The compression mainly concerns the local behaviour in the contact zone between the connected timber beams. Besides, the modified Hill failure criteria are used to take into account the quasi brittle behaviour of timber in shear and tension. Besides, the models also incorporate the contact and the non-linearity geometric. 2 EXPERIMENTAL RESULTS A study based on experimental analyses has been carried out at Laboratory “LaMI” in France in order to investigate the behaviour of the rounded dovetail connection [5]. In this paper, only part experimental results are reported. They can be used to validate 3D finite element models. The specimen consists of two glulam beams. The experimental setup was illustrated in Figure 2. Two specimens were tested. The main timber dimension is the same as the joist member dimension. The member width b and member height h were chosen as 102 mm x 205 mm. The main beam of the specimen was supported on two steel plates; four other steel plates on the both sides of the specimen prevented these from lateral moving. The free end of the joist was simply supported on a plate. The RDC was used to connect the joist to the main beam (Figure 3). The load is applied by a displacement control process on the steel plate placed on the joist. Tests were carried out at a constant rate of 2 mm/min according to the European Norm “NF-EN 26891” [6]. The deflections of beams and the slip between two beams were measured using LVDT displacement transducers.

Figure 2: Setup of experimental test

Figure 3: Test configuration of RDC

Glulam members were used in the study, with mean density value of 453 kg/m3 and mean moisture content value of 11%. The dovetail geometry was chosen as: width b1 = 87 mm, height h1 = 141.5 mm, depth t = 28 mm, dovetail angle α = 15°, and flange angle k = 20° (Figure 4).

t

k

bb1

h1h

α

Figure 4: Geometric parameters of RDC

Figures 5 and 6 show the experimental F-u curves and F-g curves. F is the load, u is the deflection of joist under the point of load and g is the vertical slip between the joist and main beam at the zone of RDC.

0

10

20

30

40

50

60

0 8 16 24 32 40

Test 1

Test 2

F (kN)

u (mm)

Figure 5: Load-deflection curves

0

8

16

24

32

40

0 2 4 6 8 10

Test 1

Test 2

F (kN)

g (mm)

Figure 6: Load-slip curves

During tests, two failures modes were developed successively. The first failure was the splitting of joist member, occurred in the elastic range of the load deformation response, and initiated at the bottom of the dovetail of the joist member. At the ultimate load, the failure was due to the bending of residual section of the joist (Figure 7).

Figure 7: Failure mode

Two characteristic values of load are defined: the failure load which defines the first splitting failure in the wood member and the ultimate load. These are summarized in Table 1.

Table 1: Experimental strength values (kN)

Failure load Ultimate load Test 1 30.42 52.38 Test 2 36.44 52.45

3 FINITE ELEMENT MODEL 3.1 MODELLING OF THE BEAM

COMPONENTS To describe the behaviour of the rounded timber connection, 3D finite element models are developed using the MARC.MSC software package [7]. Considering the symmetry, only a half of the geometry is modelled. The geometry of the finite element model, based on 8-noded hexahedral elements, is shown in Figure 8. The model includes several parts: two timber members, stiff steel plates at supports and loading points.

Figure 8: Meshing of joist and main beam

3.2 MATERIAL PROPERTIES The glued-laminated timber used in the tests corresponds to the resistance class GL28h in accordance with standard “EN 1194”. The material characteristics that have been used in the numerical models are given in Table 2 [8]. The directions L, R and T are the usual orthotropic directions of wood. In this study, E0 and E90 are considered with EL = E0 and ER = ET = E90.

Table 2: Timber properties used in the model

E0 (MPa) 12600 E90 (MPa) 420 Gmean (MPa) 780 νTR = νLT 0.41 νRL 0.02

Due to the anisotropic behaviour of timber, the Hill’s criterion is chosen. This criterion is a generalised version of the von Mises yield criterion to take into account the anisotropy of the materials. It can be expressed according to Equation (1) [7].

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

===

==−=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++

+−+

−+−

=

2654

20

3220

290

1

50

26

25

24

23

22

21

32

112

2

33

3

v

,c,c,c

.

xyyz

zxyx

xzzy

faaa

;f

aa;ff

a

/

aa

a)(a

)(a)(a

ττ

τσσ

σσσσ

σ

(1)

The hypothesis of associated plasticity is considered. Thus the Hill criterion is used as a plastic flow law. The relationship between the plastic strain increment and the stress increment is given by:

σσλε∂∂

= dd p (2)

where dλ is the plastic multiplier. In this study, perfect plasticity is considered for timber without strain-hardening. However, the Hill yield criterion does not take into account the difference of strength between tension and compression. Also, the brittle failure of wood in shear and tension is not taken into account. In order to predict the strength of RDC, the modified Hill failure criteria, combined to the elasto-plastic Hill yield criterion, are used in the finite element models. The Hill failure criterion was reformulated in such a way that only tension and shear stresses which lead to brittle failure modes were considered. It can be expressed by Equation (3).

1122

222

20

290

290

20

290

2

290

2

20

2

=++

+⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−−++

v

yzxzxyzy

,t,t

,t

zx

,t

yx

,t

z

,t

y

,t

x

fff

fffff

τττσσ

σσσσσσσ

(3)

The Hill failure criterion was also reformulated in such a way that both tension perpendicular to grain stresses (vertical and horizontal) and both shear parallel to grain stresses (radial and tangential planes). It can be expressed by Equation (4).

12

2

22

290

290

2

290

2

=+

+−+v

xzxy

,t

zy

,t

z

,t

y

ffffττσσσσ

(4)

Moreover, it was also reformulated in such a way that only vertical tension perpendicular to grain stresses and radial shear parallel to grain stresses. It can be expressed by Equation (5).

12

2

290

2

=+v

xy

,t

y

ffτσ

(5)

where σi and τij are the stresses in timber, ft,0 and ft,90 are the tensile strengths parallel and perpendicular to grain, fc,0 and fc,90 are the compressive strengths parallel and perpendicular to grain, fv is the shear strength of timber. Numerically, in these criteria, the progressive failure is simulated through a reduction of the elastic modulus in both parallel and perpendicular to grain directions to represent the damage evolution in timber. Thus, when the criterion is reached at an integration point, the elastic modulus in the direction parallel to grain E0 is set equal to the modulus in the direction perpendicular to grain E90, and the modulus E90 is reduced to 10% of its initial value. Thus, a kind of damage evolution in timber is considered. In this study, mean values of strength are considered for timber: ft,0 = 25.3 MPa, ft,90 = 0.585 MPa, fc,0 = 50.64 MPa, fc,90 = 3.14 MPa and fv = 10.98 MPa in accordance to the literature [8]. With the large deformation of joist it is considered essential that both geometrical and material nonlinearities were included in the models. 3.3 LOAD, BOUNDARY AND CONTACT

CONDITIONS The loads are introduced by using controlled displacements applied on the nodes of the stiff steel plates at the loading point and supports with a management of the contact conditions between the steel plates and the timber beam. Contact may be developed with friction based on the Coulomb criterion. The method allows no movement until the friction force is reached. After that, the movement is initiated and the friction forces remain constant. The friction coefficient between the timber member and stiff steel plate was set equal to 0.2 [9]. The friction coefficient between the joist and the main beam was set equal to 0.25 [10]. 4 VALIDATION AND APPLICATION

OF THE NUMERICAL MODEL The 3D finite element models are validated on the basis of a comparison between the load-deflection curves and load-slip curves given by FEM models and those from experiments. After that, the numerical models are applied to justify the failure mode and failure position.

4.1 VALIDATION OF THE NUMERICAL MODEL

Three models have been developed on the basis of the Hill yield criterion and the modified Hill failure criteria. The models with application of Equations (3-5) are named as model 1, model 2, and model 3, respectively. From the numerical model, the load is defined as the maximum value from which the applied load starts to decrease with the increase of the displacement. The first decreasing load is defined as the failure load, and the last decreasing load is defined as the ultimate load. Figures 9 and 10 illustrate the comparisons of experimental curves and numerical ones of model 1.

0

10

20

30

40

50

60

0 8 16 24 32 40

Test 1Test 2Model 1

u (mm)

F (kN)

Figure 9: Comparison of experimental and numerical load-deflection curves

0

8

16

24

32

40

0 2 4 6 8 10

Test 1

Test 2

Model 1

g (mm)

F (kN)

Figure 10: Comparison of experimental and numerical load-slip curves

Due to non-consideration of the clearance, the numerical deflection and slip of model 1 are slightly greater than the experimental results. In the process of test, the contact area increases progressively when the load increases. Figure 11 and 12 are obtained through shifting the curves of model 1 along the axis of deflection “u” and slip “g”. The model 1 reflects well the experimentally observed behaviour. Moreover, the experimentally observed failure loads are close to the predicted failure load. Thus, model 1 is possible to accurately simulate the load deformation response of RDC.

0

10

20

30

40

50

60

0 8 16 24 32 40

Test 1Test 2Model 1

u (mm)

F (kN)

Figure 11: Comparison of experimental and shifted numerical load-deflection curves

0

8

16

24

32

40

0 2 4 6 8 10

Test 1

Test 2

Model 1

g (mm)

F (kN)

Figure 12: Comparison of experimental and shifted numerical load-slip curves

The comparison of numerical results between three models is illustrated in Figures 13 and 14. Table 3 shows the comparison of failure loads. The load-deformation curves between three models are similar. However, the failure load of model 2 is greater than one of model 1. The failure load of model 3 is greatest.

0

10

20

30

40

50

0 2 4 6 8 10 12

Model 1Model 2Model 3

u (mm)

F (kN)

Figure 13: Comparison of numerical load-deflection curves

0

10

20

30

40

50

0 2 4 6 8 10

Model 1Model 2Model 3

g (mm)

F (kN)

Figure 14: Comparison of numerical load-slip curves

Table 3: Comparison of failure loads (kN)

Test 1 30.42 Test 2 36.44 Model 1 37.71 Model 2 42.41 Model 3 43.00

4.2 STRESS ANALYSIS During the test, the cracks parallel to grain appeared and propagated along the axis of the joist. The crack is related to tension and shear stresses. The failure index of the modified Hill failure criteria based on stresses’ interaction provided by the numerical model can be used to evaluate the position of the first potential crack. Figure 15 shows the zone where the modified Hill failure criterion (model 1) is reached for different levels of loading (25.3, 30.5 and 37.8 kN). The region of a large failure index is consistent with the experimental observation.

Figure 15: Contour plot of the failure index (model 1)

Figure 16 shows the failure index of three models at the failure load, the potential failure zones of three models are similar. Thus, it can be proven that the vertical tension perpendicular to grain and radial shear parallel to grain contribute to RDC failure.

Figure 16: Contour plot of the failure index

5 CONCLUSIONS The experimental results of RDC loaded in bending showed that the brittle failure occurred in the elastic range of the load deformation response, and initiated at the bottom of dovetail of the joist member. The cracks parallel to grain appeared and propagated along the axis of the joist. The crack is related to tension and shear stresses. The 3D numerical models developed are based on finite element method considering contact, non-linearity geometric and non-linear behaviour of materials. The plastic flow law based on the Hill’s criterion is associated to the modified Hill failure criteria, which represent the evolution of damage in wood. These models simulate well the behaviour of RDC. They will

also be the useful tools to investigate the effect of some influent parameters on the behaviour of the RDC. REFERENCES [1] Hochstrate M.: Untersuchungen zum Tragverhalten

von CNC-gefertigten Schwalbenschwan-zverbindungen. Diplomarbeit, FH Hildesheim, Germany, 2000 (in German).

[2] Tixier V.: Comportement d’assemblages traditionnels en bois. Engineer Diploma thesis (Master), CUST, Blaise Pascal University, Clermont-Ferrand, France 2005 (in French).

[3] Tannert T., Prion H., Lam F.: Structural performance of rounded dovetail connections under different loading conditions. Can. J. Civ. Eng., 34(12):1600-1605, 2007.

[4] Tannert T., Lam F., Vallée T.: Strength prediction for rounded dovetail connections considering size effects. Journal of Engineering Mechanics, 136(3):358-366, 2010.

[5] Racher P, Bressolette P, Barrerra A.: Assemblages traditionnels-Comportement d’assemblages traditionnels en bois. Rapport d’étude, LGC-CUST, France, 2007 (in French).

[6] European Committee for Standardization. EN26891: Timber structures. Test methods. Determination of embedding strength and foundation values for dowel-type fasteners. European Standard, 1993.

[7] MSC.MARC. User’s Manual, vol. A: theory and user information. MSC.Software Corporation, 2005.

[8] Xu B. H., Taazount M., Bouchaïr A., Racher P.: Numerical 3D finite element modelling and experimental tests for dowel-type timber joints. Construction and Building Materials, 23(9): 3043-3052, 2009.

[9] Xu B. H., Bouchaïr A., Taazount M., Vega E. J.: Numerical and experimental analyses of multiple-dowel steel-to-timber joints in tension perpendicular to grain. Engineering Structures, 31(10): 2357-2367, 2009.

[10] Tannert T.: Structural performance of rounded dovetail connections. PhD thesis, University of British Columbia, Vancouver, Canada, 2008.