3D NON-LINEAR FINITE ELEMENT MODELLING OF TRADITIONAL TIMBER CONNECTIONS

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1 3D NON-LINEAR FINITE ELEMENT MODELLING OF TRADITIONAL TIMBER CONNECTIONS Bo-Han Xu 1, Abdelhamid Bouchaïr 1, Mustapha Taaount 1 ABSTRACT: In this paper, 3D non-linear inite element models are developed to simulate the mechanical behaviour o the rounded dovetail connection. The non-linear models use the Hill criterion to manage the plastic ielding o the wood material. The modiied Hill ailure criteria, managing the damage evolution in wood, are used to take into account the brittle ailure in shear and tension. Besides, the models also incorporate the contact and the non-linearit geometric. The experimental results are used to validate the 3D non-linear inite 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 ailure position o the rounded dovetail connection. KEYWORDS: Rounded dovetail connection, Finite element model, Failure criterion, Stress analsis 1 INTRODUCTION 13 The dovetail joint is a common traditional timber connection in carpentr. However, because o the productivit, it has been slowl replaced b the joist hangers and nail plates. In the last decade, with the development o machine tool with numerical control, the dovetail mortises and tenons can be made quickl and precisel. Thus, it is re-used due to aesthetic. The rounded dovetail connection (RDC) is a relativel new dovetail connection which is named ater the rounded shape in the longitudinal direction o the joist, similar to a dovetail (Figure 1). The use o rounded dovetail connections in timber construction has signiicantl increased in Europe. Nowadas, it has become more and more important to know the realistic load carring capacit o traditional timber joints or reconstruction, rebuilding and improvement o structural behaviour. Thus, a number o experimental studies on RDC have been carried out [1-3]. Two ailure modes were observed: tension perpendicular to grain ailure at the mortise base or tenon ailure with splitting o the joist member [3]. The mechanical perormance o RDC is dependent upon a number o actors, such as depth o the beam, clearance in the bottom o the notch, lange angle and connection angle. However, the available experimental results in the 1 Bo-Han Xu, LaMI, Poltech Clermont-Ferrand, Université Blaise Pascal, BP 6, Aubière Cedex, France. bohanxu@gmail.com 1 Abdelhamid Bouchaïr, LaMI, Poltech Clermont-Ferrand, Université Blaise Pascal, BP 6, Aubière Cedex, France. bouchair@poltech.univ-bpclermont.r 1 Mustapha Taaount, LaMI, Poltech Clermont-Ferrand, Université Blaise Pascal, BP 6, Aubière Cedex, France. taaount@univ-bpclermont.r literature are rather limited. In order to show the inluence o these actors, it is necessar to develop and adjust 3D inite element models which can predict the mechanical behaviour o RDC. Figure 1: RDC timber members A linear elastic numerical model was been developed to predict the strength o RDC [4]. A probabilistic method is used. The method, rather than being stress-based, incorporates sie eect or the combined action o tension perpendicular to grain and shear parallel to grain stresses in timber b comparing computed stress volume integrals to unit volume strength thresholds. Thereore not onl the magnitude o the stress distributions is considered but also the volume over which the act. The capacities o RDC conigurations were predicted and successull validated with experimental tests. However, the model has onl considered the brittle linear elastic behaviour o timber. The linear elastic model can not

2 accuratel represent the plastic compression behaviour at the one o RDC. In this paper, 3D non-linear inite element models have been developed. The timber properties are regarded as transverse isotropic. The timber plasticit is considered in compression in the parallel and perpendicular to grain directions. The compression mainl concerns the local behaviour in the contact one between the connected timber beams. Besides, the modiied Hill ailure criteria are used to take into account the quasi brittle behaviour o timber in shear and tension. Besides, the models also incorporate the contact and the non-linearit geometric. EXPERIMENTAL RESULTS A stud based on experimental analses has been carried out at Laborator LaMI in France in order to investigate the behaviour o the rounded dovetail connection [5]. In this paper, onl part experimental results are reported. The can be used to validate 3D inite element models. The specimen consists o two glulam beams. The experimental setup was illustrated in Figure. 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 1 mm x 5 mm. The main beam o the specimen was supported on two steel plates; our other steel plates on the both sides o the specimen prevented these rom lateral moving. The ree end o the joist was simpl supported on a plate. The RDC was used to connect the joist to the main beam (Figure 3). The load is applied b a displacement control process on the steel plate placed on the joist. Tests were carried out at a constant rate o mm/min according to the European Norm NF-EN 6891 [6]. The delections o beams and the slip between two beams were measured using LVDT displacement transducers. Figure 3: Test coniguration o RDC Glulam members were used in the stud, with mean densit value o 453 kg/m 3 and mean moisture content value o 11%. The dovetail geometr was chosen as: width b 1 = 87 mm, height h 1 = mm, depth t = 8 mm, dovetail angle α = 15, and lange angle k = (Figure 4). b b 1 α k h 1 t h Figure 4: Geometric parameters o RDC Figures 5 and 6 show the experimental F-u curves and F- g curves. F is the load, u is the delection o joist under the point o load and g is the vertical slip between the joist and main beam at the one o RDC. F (kn) Figure : Setup o experimental test 1 Test u (mm) 4 Figure 5: Load-delection curves

3 F (kn) 4 Table 1: Experimental strength values (kn) Test g (mm) 1 Figure 6: Load-slip curves During tests, two ailures modes were developed successivel. The irst ailure was the splitting o joist member, occurred in the elastic range o the load deormation response, and initiated at the bottom o the dovetail o the joist member. At the ultimate load, the ailure was due to the bending o residual section o the joist (Figure 7). Failure load Ultimate load Test FINITE ELEMENT MODEL 3.1 MODELLING OF THE BEAM COMPONENTS To describe the behaviour o the rounded timber connection, 3D inite element models are developed using the MARC.MSC sotware package [7]. Considering the smmetr, onl a hal o the geometr is modelled. The geometr o the inite element model, based on 8-noded hexahedral elements, is shown in Figure 8. The model includes several parts: two timber members, sti steel plates at supports and loading points. Figure 8: Meshing o joist and main beam Figure 7: Failure mode Two characteristic values o load are deined: the ailure load which deines the irst splitting ailure in the wood member and the ultimate load. These are summaried in Table MATERIAL PROPERTIES The glued-laminated timber used in the tests corresponds to the resistance class GL8h in accordance with standard EN The material characteristics that have been used in the numerical models are given in Table [8]. The directions L, R and T are the usual orthotropic directions o wood. In this stud, E and E 9 are considered with E L = E and E R = E T = E 9.

4 Table : Timber properties used in the model E (MPa) 16 E 9 (MPa) 4 G mean (MPa) 78 ν TR = ν LT.41 ν RL. Due to the anisotropic behaviour o timber, the Hill s criterion is chosen. This criterion is a generalised version o the von Mises ield criterion to take into account the anisotrop o the materials. It can be expressed according to Equation (1) [7]. a 1( ) a( x ) = a3( x ) 3a4τ x 3a5τ 3a6τ x 1 1 a1 = ; a = a3 = ; a 4 = a c, 9 5 = a 6 c, = 3 v c,. 5 The hpothesis o associated plasticit is considered. Thus the Hill criterion is used as a plastic low law. The relationship between the plastic strain increment and the stress increment is given b: ε = dλ d p () / (1) where dλ is the plastic multiplier. In this stud, perect plasticit is considered or timber without strainhardening. However, the Hill ield criterion does not take into account the dierence o strength between tension and compression. Also, the brittle ailure o wood in shear and tension is not taken into account. In order to predict the strength o RDC, the modiied Hill ailure criteria, combined to the elasto-plastic Hill ield criterion, are used in the inite element models. The Hill ailure criterion was reormulated in such a wa that onl tension and shear stresses which lead to brittle ailure modes were considered. It can be expressed b Equation (3). x t, 1 t, x t, τ x τ x x v τ = 1 The Hill ailure criterion was also reormulated in such a wa that both tension perpendicular to grain stresses (vertical and horiontal) and both shear parallel to grain stresses (radial and tangential planes). It can be expressed b Equation (4). (3) τ x τ x v = 1 Moreover, it was also reormulated in such a wa that onl vertical tension perpendicular to grain stresses and radial shear parallel to grain stresses. It can be expressed b Equation (5). τ x v = 1 (4) (5) where i and τ ij are the stresses in timber, t, and t,9 are the tensile strengths parallel and perpendicular to grain, c, and c,9 are the compressive strengths parallel and perpendicular to grain, v is the shear strength o timber. Numericall, in these criteria, the progressive ailure is simulated through a reduction o 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 E is set equal to the modulus in the direction perpendicular to grain E 9, and the modulus E 9 is reduced to 1% o its initial value. Thus, a kind o damage evolution in timber is considered. In this stud, mean values o strength are considered or timber: t, = 5.3 MPa, t,9 =.585 MPa, c, = 5.64 MPa, c,9 = 3.14 MPa and v = 1.98 MPa in accordance to the literature [8]. With the large deormation o 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 b using controlled displacements applied on the nodes o the sti steel plates at the loading point and supports with a management o the contact conditions between the steel plates and the timber beam. Contact ma be developed with riction based on the Coulomb criterion. The method allows no movement until the riction orce is reached. Ater that, the movement is initiated and the riction orces remain constant. The riction coeicient between the timber member and sti steel plate was set equal to. [9]. The riction coeicient between the joist and the main beam was set equal to.5 [1]. 4 VALIDATION AND APPLICATION OF THE NUMERICAL MODEL The 3D inite element models are validated on the basis o a comparison between the load-delection curves and load-slip curves given b FEM models and those rom experiments. Ater that, the numerical models are applied to justi the ailure mode and ailure position.

5 4.1 VALIDATION OF THE NUMERICAL MODEL Three models have been developed on the basis o the Hill ield criterion and the modiied Hill ailure criteria. The models with application o Equations (3-5) are named as model 1, model, and model 3, respectivel. From the numerical model, the load is deined as the maximum value rom which the applied load starts to decrease with the increase o the displacement. The irst decreasing load is deined as the ailure load, and the last decreasing load is deined as the ultimate load. Figures 9 and 1 illustrate the comparisons o experimental curves and numerical ones o model 1. F (kn) F (kn) Test u (mm) 4 Figure 11: Comparison o experimental and shited numerical load-delection curves F (kn) Test u (mm) 4 Figure 9: Comparison o experimental and numerical load-delection curves Test g (mm) 1 F (kn) Test g (mm) 1 Figure 1: Comparison o experimental and shited numerical load-slip curves The comparison o numerical results between three models is illustrated in Figures 13 and 14. Table 3 shows the comparison o ailure loads. The load-deormation curves between three models are similar. However, the ailure load o model is greater than one o model 1. The ailure load o model 3 is greatest. F (kn) 5 4 Figure 1: Comparison o experimental and numerical load-slip curves Due to non-consideration o the clearance, the numerical delection and slip o model 1 are slightl greater than the experimental results. In the process o test, the contact area increases progressivel when the load increases. Figure 11 and 1 are obtained through shiting the curves o model 1 along the axis o delection u and slip g. The model 1 relects well the experimentall observed behaviour. Moreover, the experimentall observed ailure loads are close to the predicted ailure load. Thus, model 1 is possible to accuratel simulate the load deormation response o RDC. 3 1 Model Model u (mm) 1 Figure 13: Comparison o numerical load-delection curves

6 F (kn) Model Model g (mm) 1 Figure 14: Comparison o numerical load-slip curves Table 3: Comparison o ailure loads (kn) 3.4 Test Model 4.41 Model STRESS ANALYSIS During the test, the cracks parallel to grain appeared and propagated along the axis o the joist. The crack is related to tension and shear stresses. The ailure index o the modiied Hill ailure criteria based on stresses interaction provided b the numerical model can be used to evaluate the position o the irst potential crack. Figure 15 shows the one where the modiied Hill ailure criterion (model 1) is reached or dierent levels o loading (5.3, 3.5 and 37.8 kn). The region o a large ailure index is consistent with the experimental observation. Figure 15: Contour plot o the ailure index (model 1) Figure 16 shows the ailure index o three models at the ailure load, the potential ailure ones o 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 ailure.

7 also be the useul tools to investigate the eect o some inluent parameters on the behaviour o the RDC. Figure 16: Contour plot o the ailure index REFERENCES [1] Hochstrate M.: Untersuchungen um Tragverhalten von CNC-geertigten Schwalbenschwanverbindungen. Diplomarbeit, FH Hildesheim, German, (in German). [] Tixier V.: Comportement d assemblages traditionnels en bois. Engineer Diploma thesis (Master), CUST, Blaise Pascal Universit, Clermont-Ferrand, France 5 (in French). [3] Tannert T., Prion H., Lam F.: Structural perormance o rounded dovetail connections under dierent loading conditions. Can. J. Civ. Eng., 34(1):16-165, 7. [4] Tannert T., Lam F., Vallée T.: Strength prediction or rounded dovetail connections considering sie eects. Journal o Engineering Mechanics, 136(3): , 1. [5] Racher P, Bressolette P, Barrerra A.: Assemblages traditionnels-comportement d assemblages traditionnels en bois. Rapport d étude, LGC-CUST, France, 7 (in French). [6] European Committee or Standardiation. EN6891: Timber structures. Test methods. Determination o embedding strength and oundation values or dowel-tpe asteners. European Standard, [7] MSC.MARC. User s Manual, vol. A: theor and user inormation. MSC.Sotware Corporation, 5. [8] Xu B. H., Taaount M., Bouchaïr A., Racher P.: Numerical 3D inite element modelling and experimental tests or dowel-tpe timber joints. Construction and Building Materials, 3(9): , 9. [9] Xu B. H., Bouchaïr A., Taaount M., Vega E. J.: Numerical and experimental analses o multipledowel steel-to-timber joints in tension perpendicular to grain. Engineering Structures, 31(1): , 9. [1] Tannert T.: Structural perormance o rounded dovetail connections. PhD thesis, Universit o British Columbia, Vancouver, Canada, 8. 5 CONCLUSIONS The experimental results o RDC loaded in bending showed that the brittle ailure occurred in the elastic range o the load deormation response, and initiated at the bottom o dovetail o the joist member. The cracks parallel to grain appeared and propagated along the axis o the joist. The crack is related to tension and shear stresses. The 3D numerical models developed are based on inite element method considering contact, non-linearit geometric and non-linear behaviour o materials. The plastic low law based on the Hill s criterion is associated to the modiied Hill ailure criteria, which represent the evolution o damage in wood. These models simulate well the behaviour o RDC. The will

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