Numerical and experimental analyses of multiple-dowel steel-to-timber joints in tension perpendicular to grain

Engineering Structures 31 (2009) 2357–2367

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Numerical and experimental analyses of multiple-dowel steel-to-timber joints in tension perpendicular to grain B.H. Xu, A. Bouchaïr ∗ , M. Taazount, E.J. Vega LaMI, Polytech, Blaise Pascal University, B.P. 206, 63174 Aubière Cedex, France

article

info

Article history: Received 10 January 2009 Received in revised form 16 March 2009 Accepted 25 May 2009 Available online 11 June 2009 Keywords: Damage evolution Finite element Perpendicular to grain Plastic criterion Splitting strength Timber joint Three-dimensional model

abstract The behaviour of dowel-type steel-to-timber joints loaded in tension perpendicular to grain is analyzed experimentally and numerically. Two main types of failures are observed in the experiments such as wood splitting and embedding. The experimental results are used to validate a three-dimensional (3D) nonlinear finite element model. The non-linear model uses the Hill criterion to control the plastic yielding of wood material. The Hoffman failure criterion, controlling the damage evolution in wood, is used to take into account the brittle failure in shear and tension perpendicular to grain. The comparison with experimental results shows that the numerical results are in good agreement with them. The validated model is used to investigate the effect of some influential parameters on the splitting strength of the joints loaded in tension perpendicular to grain. Besides, the splitting strengths given by the numerical model are used to evaluate the accuracy of some analytical formulae available in the literature. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The resistance and the durability of timber structures are mainly dependent on the design of the joints between the elements. To ensure the structural performance of timber joints, the correct location of the fasteners with respect to the end and edge distances of the members is of high importance. As timber is orthotropic with dissymmetric strengths between tension and compression, the behaviour of the joints is not only controlled by the load-carrying capacity of the individual fasteners. It may also depend on the shape of the joint and on the interaction between the fasteners which influences the stress distribution in the joint area. Due to the anisotropy of timber, this leads in many cases to brittle failure related to the stresses in shear and in tension perpendicular to grain. However, dowelled joints can exhibit ductile behaviour if they are designed in order to obtain the large plastic deformations of the dowels before the occurrence of the brittle failure of timber in tension perpendicular to grain or shear. In real joints with dowels, timber is loaded in tension or compression parallel or perpendicular to grain combined with shear [1]. For example, in some truss nodes, the connections between the members have various angles between the tension or compression load and the wood grain [2] (Fig. 1). When timber

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Corresponding author. Tel.: +33 473407532; fax: +33 473407494. E-mail address: [email protected] (A. Bouchaïr).

0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.05.013

joint is loaded in tension perpendicular to grain, its failure may result either in yielding of the dowels, crushing of the wood under the dowels or splitting of the timber member. The first two failure modes are considered as a ductile failure, while splitting or fracture of wood is a brittle failure and may lead to catastrophic collapse [3]. In practice, it is not realistic to avoid this direction of loading although it may lead to brittle splitting failure at load levels lower than the bearing capacity defined by the ductile embedding of wood and the bending of dowels. This kind of failure has different effects not only on the load-carrying capacity of the joints but also on the ductility that influences the behaviour of the whole structure, mainly with load redistribution in the statically undetermined structures and under seismic loads. According to Eurocode 5 (EC5) [4], the load-carrying capacity of a single-dowelled timber joint is based on Johansen’s yield theory if the specifications of minimum distances are checked. Thus, the failure mode occurs by plastic hinges in the dowels combined to the wood embedding. When the loading is in the grain direction, the strength is well predicted by this approach [5]. In multi-dowelled joints, the block shear and plug shear failure have to be considered and in the joints loaded in the direction perpendicular to grain, the splitting has to be checked. Thus, when the loading is in the direction perpendicular to grain, the splitting may arise and it has to be considered in addition to the yield theory approach in order to take into account the failure due to the brittle fracture of wood. The joint design has to ensure that brittle failure does not happen prior to the yielding of dowels and wood. In some cases, to prevent splitting and to enhance the ductility of the joint, reinforcement of the

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Fig. 1. Connection loaded partially in the direction perpendicular to grain.

joint area can be applied on the timber surface or inside the timber elements in the vicinity of the fasteners [1,6]. In order to check and to develop a prediction formula of splitting, some experimental studies on the splitting strength of beams loaded in the direction perpendicular to grain by doweltype joints were carried out [3,5,7–13]. In these studies, tests were performed on timber beams, with different sizes and different joint fasteners (nails, dowels, bolts and ring connectors), and with steel plates or punched metal plates in a simply supported beam configuration with the joint at mid-span or cantilevered beams. The dowel-type joints are very common in timber structures mainly in combination with glulam members. Thus, this study is limited to this type of fastener. It may concern timber-to-timber joints or timber-to-steel joints with various configurations. Complying with the actual architectural trends and high strength requirements, most experimental studies have been focused on beams loaded by tension perpendicular to grain by single- or multiple-dowel steel-to-timber joints. Kasim et al. [3] have tested glulam beams loaded at mid-span by means of perpendicular to grain connections with four bolts. The beam dimensions were 80 mm × 304 mm and the diameter of the bolt was 19 mm. All specimens developed significant cracks shortly prior to failure or at failures. The cracks in all specimens passed through the bolt holes furthest from the loaded edge. Due to the small slenderness of the dowels, there were no signs of noticeable bearing of wood at all bolt hole locations or through the bearing length of the supports. The bolt holes in the steel plates did not show any sign of bearing failure either. The steel bolts themselves did not bend but remained straight during testing. Ballerini [9] has also presented experimental results of bending tests on simply supported glulam beams (40 mm thickness), loaded at mid-span by means of perpendicular to grain connections with one or two dowels (10 mm diameter). Three types of failure of the beams have been observed: the splitting failure, the bending failure and the embedding failure. All specimens experienced the splitting failure and stable crack propagations have been observed till the maximum load is reached. Also, some specimens have experienced embedding failure. These reported tests concern mostly joints with steel side members and rigid dowel-type fasteners. In this case, the splitting determines the load-carrying capacity of joint. However, when the connections with the slender dowel-type fasteners are placed near the loaded edge of the beams, the failure mode may occur by plastic hinges in the dowels combined with wood splitting. Moreover, in practice, joints with steel main member are frequently used in order to obtain better aesthetic appearance and greater fire safety. The number of experimental studies on steel-to-timber joints, with slotted-in steel plates, available in the literature is rather limited. Only two series of tests providing ultimate loads have been reported [5,13]. Thus, it is necessary to study the mixed failure mode in multiple-dowel steel-to-timber joints with steel main member in tension perpendicular to grain.

The paper summarizes the major available prediction formulae and the existing numerical models. These references are used as a basis for the development of a 3D numerical model considering the elasto-plastic behaviour of timber and steel with a failure criterion based on the damage evolution in timber. In order to validate the numerical model, an experimental study is carried out on a multiple-dowel timber joint with steel main member loaded in the direction perpendicular to grain. The ultimate loads characterized by splitting are analyzed and the types of failure are highlighted. The 3D non-linear finite element model is used to predict the behaviour of the joints. As the model uses a failure criterion, it predicts well the splitting strength of the joint. This numerical model has been successfully applied to predict the behaviour of the multiple-dowel timber joint with steel main member loaded parallel to grain [14] and timber joints with glued-in rods [15]. Using the validated model, the effects of main joint parameters are investigated. Finally, the values of splitting strength obtained from the models are used to evaluate the available analytical formulae. 2. Review of analytical and numerical studies 2.1. Analytical approaches According to EC5, the resistance of a single dowel in double shear steel-to-timber joint, is based on the load-carrying capacity formulae according to Johansen’s yield theory. This approach can be used whatever be the direction of load applied with regard to the wood grain. However, in the case of the load applied in the direction perpendicular to grain, an additional relationship is proposed to cover the brittle failure of timber (splitting). The failure modes considered are shown in Fig. 2 and the corresponding formulae are given in Eq. (1). The resistance Fv,Rk for each shear plan is the weakest among the values associated to the three failure modes. The parameters influencing the failure modes are the embedding strength of timber, the yield limit of steel and the dowel slenderness. In general, a large plasticity of the joint can be provided when relatively slender fasteners are used. In that case, the more ductile failure modes 2 and 3 are governing. Mode 1 concerns the timber joint with rigid fasteners (low slenderness) where timber embedding is dominant. The application to the tested joints shows that failure mode 2 is dominant. It is due to a combination of dowel yield failure in bending and timber failure in embedding.

Fv,Rk = min

 fh,α,k t1 d  "s    fh,α,k t1 d

   

mode 1 2+

4My,Rk fh,α,k dt12

# −1

2.3 My,Rk fh,α,k d

p

mode 2

(1)

mode 3

where: My,Rk = 0.3fu,k d2.6 ;

fh,α,k =

fh,0,k = 0.082(1 − 0.01d)ρk ;

fh,0,k

;

k90 sin α + cos2 α k90 = 1.35 + 0.015d. 2

To take account of the specific failure types of the joints loaded in the direction perpendicular to grain, various research works are dedicated to the analysis of their load-carrying capacity. These studies pointed out that the capacity of dowel-type joint loaded in the direction perpendicular to grain is rarely based on the embedding resistance of the single fastener, as for joints loaded parallel to grain. It is normally governed, at the final stage, by the splitting or fracture of wood. This splitting is induced by tensile stresses in the direction perpendicular to grain and shear stresses. So, some mechanical models have been derived on the basis of these two types of stresses. As the fracture in the joints is related

B.H. Xu et al. / Engineering Structures 31 (2009) 2357–2367

2359

F (kN) 2F1

2F1

2F1

Eq. 2

270

Eq. 3 Eq. 4

220

Eq. 5

170

F1

F1

F1

Mode 1

F1 Mode 2

F1

120

F1 Mode 3

70

Fig. 2. Plastic failure modes of steel-to-timber joints.

20 64

144

224

304

384

he (mm)

Fig. 4. Comparison of different analytical splitting strength formulae.

Fm

√

he

V1

hm

h

V2

h3 h2 h1 Ir Fig. 3. Joint with timber members loaded in the direction perpendicular to grain.

to the crack propagation in wood, other mechanical models based on fracture mechanics and fracture energy have been derived. Some experimental studies show that the failure of joint loaded in the direction perpendicular to grain is governed by the critical length where the crack propagation becomes unstable [11]. It is quite difficult to determine the critical crack length from the experiments as it is hard to measure the crack propagation during the test. The relation between the lateral force and the crack length can be easily calculated using the linear elastic fracture mechanics. The critical crack length has a correlation with the ratio between the distance (he ), of the innermost row of fasteners from the loaded edge, and the beam depth (h) (Fig. 3). Thus, the majority of the current prediction formulae are based on this ratio. Based on the experimental results, several prediction formulae of the splitting strength have been developed. In the CSA 2001 [16], the splitting strength may be determined on the basis of a fictitious shear design over the residual cross-section as follows: Vu =

2fv he t

(2)

3

with he > 0.5h, where, fv is the shear strength of wood; Vu is to be compared to the shear force [max(V1 , V2 )] produced in the wood member of thickness t by the fasteners (V1 + V2 = Fm ) (Fig. 3). Van der Put and Leijten [17] derived a theoretical prediction formula on the basis of an energetic approach in the framework of the Linear Elastic Fracture Mechanics (LEFM). The prediction formula [18] is simplified as follows:

r Vu = t

GGc 0.6

s

he 1−

he h

(3)

where GGc is the apparent fracture parameter (G is the shear modulus and Gc is the fracture energy). It is considered that the crack propagation is always a combination of fracture modes I and II. The value of this parameter depends on the specific conditions under which crack opening or unstable crack growth takes place, which among others depend on the fastener type. At this stage, it is impossible to estimate the influence of the failure mode of the fastener in relation with the apparent fracture parameter as test data taken from literature were not tailored to check this aspect. For this reason a lower bound approach √ is taken to derive the value of the apparent fracture parameter GGc by evaluation of test data. For a structural design code proposal, the apparent √ fracture parameter GGc = 12 N/mm1.5 is taken as the mean lower bound [18]. The EN 1995-1-1 [4] proposes a relationship for dowel-type timber joint (Eq. (4)), derived from the studies of Van der Put and Leijten [17]. This relationship √ uses a factor equal to 14 that correspond approximately to GGc = 10.84 N/mm1.5 in Eq. (3).

s Vu = 14t

he 1−

he h

.

(4)

Another relationship based on fracture mechanics is proposed by Larsen and Gustaffson [19]. The maximum capacity of the joint is expressed as follows: Fm = 2η · t

p

2he GGc

(5)

where η is an efficiency factor, suggested by the authors to be assumed equal to 0.66. According to the formulae presented here above (Eqs. (2)– (5)), the load-carrying capacity for joints depends only on the member thickness t and the distance he . Thus, the influence of the number and the distribution of fasteners is not considered. To illustrate the difference between these formulae, a comparison has been performed for the glulam beam used in this study (h = 486 mm, t = 150 mm, fv = 3.24 MPa). The value of he varies in the range between 64 and 438 mm according to the minimum geometry dimensions proposed by EC5 [4]. The evolution of strength versus he is shown in Fig. 4. It can be observed that Eq. (5) gives a linear evolution of strength and shows the lowest values. Eqs. (3) and (4), with an exponential evolution, give quasi-similar values and the highest values are given by the linear equation (Eq. (2)). In order to evaluate the various available prediction formulae, comparisons with experimental results are performed. It can be observed in the experiments that the splitting strength of the beams grows quite linearly with he [9]. Besides, with the same he , the presence of a second fastener in the joint increases the

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splitting strength [9,13]. Moreover, as the number of fasteners and he remain constant, the resistance of the connection increases with the increase of row spacing, up to a maximum value of spacing [3]. These results confirm that the joint geometry has a great influence on its strength. So, the capacity of multiple-fastener timber joint loaded perpendicular to grain depends not only on the ratio (he /h), but also on the distribution of fasteners in the joint (number of fastener rows and spacing of fasteners) and on the thickness of the beam [5]. Other prediction formulae (Eqs. (6) and (8)) consider the influence of the distribution of fasteners on the load-carrying capacity of a joint. Ehlbeck and Görlacher [20] proposed a relationship based on experimental and theoretical investigations: Fm = 13 ·

0.8 Aef ft ,90

(6)

λkr The factor λ considers that only part of the joint load causes tensile stresses. The factor kr takes into account the fact that the joint load is distributed over several rows of fasteners, so that only a reduced portion of tensile stresses is acting in the line of the farthest row of fasteners. The effective area Aef represents a fictitious area where uniform stresses perpendicular to grain are considered. These parameters are given by Eq. (7).

λ=1−3

Aef = t

q



he

2

 +2

h

he

3 ;

h

l2r + (c · h)2 ;

c=

kr = 4 3

s

he h

n 1X



h1

n i =1

 1−

he

2

hi

; (7)

3

h

where n is the number of rows of fasteners, lr is the distance between the outer fasteners in the innermost row and hi is the distance of the ith row of fasteners from the unloaded edge (Fig. 3). Ballerini [21] presented a semi-empirical prediction formula based on all the experimental data available from literature, and on the main results of theoretical and numerical works. The formula assumes the soundness of the Van der Put LEFM energetic approach but takes also into account the effect of joint geometry. From the semi-empirical prediction formula, a simple design formula [22] for the characteristic splitting strength of beams loaded by multiple-dowel joints can be given by Eq. (8).

s FR,k = 18t ·

he 1 − ( hhe )3

fw = 1 + 0.75 ·

lr h

· fw · fr ;

≤ 2.0;

fr = 1 + 1.75 ·

n · hm 1000 + n · hm

(8)

where the factor fw takes into account the effect of the joint width lr ; the factor fr is correlated with the joint height hm and the number of rows n (Fig. 3). The prediction of splitting strength of timber joints is a difficult task, since it is influenced by a large number of parameters. Thus, in order to evaluate the different prediction formulae, it is necessary to perform the comparisons with experimental results. Although some comparisons have been performed, these experimental results concern the connections with rigid fasteners where the failure arises in embedment (mode 1) or splitting. The cases of realistic connections with slender fasteners where the failure may arise in plastic yielding (modes 2 and 3) are not common in the literature. The tests realized in this study allow performing a comparison with the available prediction formulae considering a combination of splitting and failure mode 2. 2.2. Existing numerical models In the last decade, a great deal of numerical research based on 2D finite element models (FEM) was performed [22–25].

These models are developed in the framework of linear elastic fracture mechanics (LEFM). In the numerical analysis performed by Ballerini et al. [22,24], the splitting failure loads were derived using Wu’s fracture criterion. In the analyses on beams loaded by single-dowel joints, with the ratio (he /h) lower than 0.5, experimental and numerical results agreed well. For larger (he /h) values, the numerical results overestimate the splitting strength in comparison with experimental results. Yasumura and Daudeville [25] presented a finite element model to analyze the fracture of multiple-bolted joints under lateral force perpendicular to wood grain. The maximum loads are calculated by LEFM and the crack initiating loads are calculated by the average stress method (ASM). The maximum loads calculated by LEFM were 4% to 20% higher than the experimental results in the joints of two fasteners aligned with the grain. The maximum loads calculated by LEFM showed comparatively good agreement with the experimental results in joints of two or three fasteners aligned perpendicular to grain. Usually, the assumption of elastic bodies is adopted in the LEFM models presented here above. According to the experimental observations, this assumption seems valid for the brittle failure and these models based on LEFM can simulate well the splitting behaviour of timber joint with rigid fasteners loaded in the direction perpendicular to grain. For dowel-type joints with a ductile failure mode, the behaviour is characterized by the bending of the fastener and/or the embedment of the fastener into the wood. Furthermore, when the span to depth ratio of the beam exceeds about 5, in many cases the governing failure mode will not be splitting due to the joint but due to bending or shear of the loaded member itself [26]. In this case, the failure is the result of a complex stress interaction. LEFM model on the assumption of elastic bodies is not suitable for this case. It is known that FE nonlinear analysis can approximate the load–slip behaviour of doweltype joints considering the elasto-plastic behaviour of wood and dowels. Sawata and Yasumura [5] developed a non-linear 2D FE beam on foundation model to investigate the wood embedding and the dowel yielding in double shear bolted timber joints with steel side plates and a slotted-in steel plate under lateral loads parallel and perpendicular to grain. However, it has been recognized that these 2D FE models only give reasonable predictions for very specific situations such as very thin or very thick timber members. Typically, dowel-type connections are 3D problems (non-uniform stress distributions across the thickness of members) that must be accounted for a convenient modeling. Few 3D models are available in the literature to predict the mechanical behaviour of singlefastener joints. However, no FE models for multi-fastener joints are available in the literature [27]. Moreover, many modeling issues are important for the FE models, such as the choice of the appropriate constitutive models for wood [28] and the adequate failure criteria to judge failure [29]. 3. Experimental setup Two specimens of steel-to-timber joints were tested with the load applied in the direction perpendicular to grain. The configurations and geometry of the specimens are shown in Fig. 5 and Table 1. Each specimen consists of two lateral glulam GL24h (Douglas Fir) beams connected to a main steel plate using four 16 mm dowels. The load is applied by a displacement control process on the steel plate and two supports are set on the glulam beams. The slips between the steel plate and the timber were measured using two LVDT displacement transducers. Tests were carried out at a constant rate of 1 mm/min. The perpendicular to grain loading was applied in the tests according to the European Norm ‘‘NFEN 26891’’ [30] which is the technical protocol to determine the

B.H. Xu et al. / Engineering Structures 31 (2009) 2357–2367

2361

F e

b

LVDT

LVDT

he

h

hm h1 t

Ir a

I

a

Fig. 5. Test configuration of the steel-to-timber joints. Table 1 Geometry of the steel-to-timber joints (mm). d

t

a

b

lr

h1

hm

h

l

e

16

75

100

115

65

140

206

486

1800

8

F (kN)

90

60

30 Test 1 Test 2

0

0

5

10

15

Slip (mm)

Fig. 6. Load–slip curves.

stiffness and the strength of timber joints. Firstly, the joints were loaded until 40% of the maximum estimated load (Fest ), and the crosshead position held during 30 s. After this stage, specimens are unloaded until 0.1Fest and the crosshead position maintained again for 30 s. Finally, specimens are reloaded until failure. Fig. 6 shows the experimental load–slip curves for the two tested specimens. During tests, two failure modes are developed successively, the embedding failure and the splitting failure of wood. The first mode is associated to a large plastic deformation due to the yielding of the dowel and the embedding of wood (Fig. 7). This large deformation, similar to the failure mode 2 defined by Eq. (1), is followed by cracks propagating along grain, in the two side

members, near the dowels at the beam loaded edge. During test, the load increases as the slip increases. A non-linear deformation without failure is observed at a load value close to 40 kN. This load is followed by an increase of the stiffness and the load without failure. This seems to be due to the initial gap between the dowels and the steel plate. Thus, at the initial stage of loading, only a part of the dowels, in contact with the plate, is carrying the load. Then, a first splitting failure occurred in the line of the dowels, close to the loaded edge in one timber beam. The load continues to increase and the failure of the second beam arises and the length of the crack increases at the maximum load. At this ultimate load, with the increase of displacement, the beam continues to carry a decreasing load through the second row of dowels near the unloaded edge of the beam. The remaining resistance of the joint is due to two contributions. The first one is due to the part of the beam, between the dowel row near the loaded edge and the beam supports, which behaves as a simply supported beam. The second contribution is due to the other row of dowels which shows a large deformation of dowels without splitting. Finally, the embedding failure led the load to decrease. There were noticeable bearing of wood at all dowel holes associated to the large flexural deformation of dowels. Therefore, the splitting strength is defined by the first peak load. The dowel holes in the steel plates did not show any bearing deformation and an estimation of stresses shows that they remain in the elastic stage. To obtain a base for comparison with the numerical model, the stiffness and the strength are considered with a lesser focus on stiffness for the design purpose. Two values of stiffness (slip modulus) are defined: an initial stiffness (Ki ) and an elastic stiffness (Ke ). Ki is determined from a linear regression on the load–slip curves between 0.2 and 0.4Fest . Ke is determined from a linear regression during the unloading/reloading stages. Two characteristic values of load are defined: the failure load (Fu1 ) defines the first splitting failure in the wood member and the ultimate load (Fu ). The stiffness and failure load are summarized in Table 2.

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Fig. 7. Failure mode of specimen (during and after test).

Load Steel plate

Steel member

Steel plate

(a) Tested specimen (load and supports).

(b) Dowel zone.

Fig. 8. Meshing of the tested specimen.

F (kN)

Table 2 Experimental main values (strength and stiffness).

Test 1 Test 2

Fu1 (kN)

Fu (kN)

Ki (kN/mm)

Ke (kN/mm)

95.6 102.7

106.4 114

18.3 30.3

48.9 67.2

4. Finite element model

120

80

4.1. Meshing, boundary conditions and material properties To describe the behaviour of the timber–steel–timber joint under load perpendicular to grain, a 3D finite element model is developed using the MARC.MSC software package [31]. Considering the symmetry, only a half of the beam geometry is modeled. The geometry of the finite element model, based on 8-noded hexahedral elements, is shown in Fig. 8. The selected model consists of approximately 27 000 elements and 34 000 nodes with a computational time of ten hours. The model takes into account different sources of non-linearity such as the elasto-plastic behaviour of materials and the contact evolution between steel, dowels and timber. As the dowels do not develop the rope effect, the large non-linear geometric deformation is not considered in this study because it is time consuming. The comparison between the two approaches showed that the global behaviour and the failure load are very close. Thus, the FE model with small deformation theory can simulate well the behaviour of timber dowelled joint (Fig. 9). Besides, Nishiyama [32] developed FE models including both the non-linear geometric analysis and the small deformation theory for timber joint. In the timber dowelled joints, the calculated results by both non-linear geometric analysis and small deformation theory were similar to the double shearing tests results. In the bolted joints exhibiting the rope effect, the load calculated by small deformation theory was lower than the experimental value of the double shearing tests, but the calculated result by non-linear geometric analysis

Test 1 Test 2

40

Model 1 Model 2

0

0

5

10

15

Slip (mm)

Fig. 9. Comparison of experimental and numerical load–slip curves.

was similar to the experimental value. Therefore, non-linear geometric analysis is more important for bolted timber joints that exhibit the rope effect than for the timber dowelled joint. The loads are introduced by increasing controlled displacements applied on all nodes in the top of steel member with the conditions of supports reproducing the real conditions of the experimental setup. The displacement steps for the numerical calculation are defined in order to obtain a well defined load–slip curve and to obtain a convergent calculation process. Steel is considered isotropic (E = 210 000 MPa, ν = 0.3). The main mechanical characteristics, such as the yield strength (fy ) and the ultimate strength (fu ), were determined from tensile tests (Table 3). The stress–strain curve used in the numerical model is a non-linear one from tests with some simplifications (multilinear curve). The timber properties are orthotropic with identical properties in radial and tangential directions. This combined direction is referred to as perpendicular to grain (⊥) while the

B.H. Xu et al. / Engineering Structures 31 (2009) 2357–2367 Table 3 Steel properties used in the model (MPa). Steel plate

Steel dowel

fy

fu

fy

fu

309

434

366

543

Table 4 Wood properties used in the model. E0 (MPa)

E90 (MPa)

Gmean (MPa)

νTR = νLT

νRL

11 600

390

750

0.41

0.02

longitudinal direction is referred to as parallel to grain (//). The glued-laminated timber used for steel-to-timber joint corresponds to resistance class GL24h according to EN 1194. Its measured mean density ρmean is equal to 438 kg/m3 and the moisture content hmean is equal to 10%. The values of its mechanical elastic parameters are given in Table 4. Due to the anisotropic plastic behaviour of timber, the Hill criterion is chosen. The Hill criterion is a generalized version of the von Mises yield criterion to consider the anisotropy of the materials [31]. Its stress potential can be expressed as follows (Eq. (9)):

 2 2 2 σ = [a1 (σy − σz )2 + a2 (σ  √z − σx ) + a3 (σx − σy ) + 3a4 τzx   2 2 1/2 +3a5 τyz + 3a6 τxy ] / 2 (9) 2 1 1 2   a1 = 2 − 2 ; a2 = a3 = 2 ; a4 = a5 = a6 = 2 . fc ,90

f c ,0

3fv

fc ,0

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

∂ σ¯ ∂σ

(10)

where dλ is the plastic multiplier. In this study, perfect plasticity is considered for timber without strain-hardening. As the Hill criterion is easy to use and is available in many software tools, it was used in the first part of this study to model timber. However, this criterion is not taking into account the difference of strength between tension and compression in the direction perpendicular to grain. In order to evaluate the effect of this component of strength on the load-carrying capacity of the studied joints, a failure criterion, combined to the elasto-plastic Hill criterion, is used in the finite element model. The failure criterion is based on that of Hoffman [33] that is based on the Hill criterion with some modifications to allow unequal ultimate stresses in tension and compression which is representing the real behaviour of timber. In this criterion, 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. The Hoffman criterion is associated with the Hill criterion to control the plastic yielding combined with the damage evolution. It can be expressed as follows:

 C1 (σy − σz )2 + C2 (σz − σx )2 + C3 (σx − σy )2 + C4 σx + C5 σy    +C6 σz + C7 τ 2 + C8 τ 2 + C9 τ 2 = 1   yz xz xy   1 1 1   C1 = − ; C2 = C3 = ; ft ,90 fc ,90

2ft ,0 fc ,0

2ft ,0 fc ,0

1 1   C4 = − ;    ft ,0 f c ,0    1 1 1  C5 = C6 = − ; C7 = C8 = C9 = 2 ft ,90

fc ,90

fv

2363

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. In this study, timber tensile and shear strength values are considered: ft ,0 = 19.8 MPa, ft ,90 = 0.48 MPa and fv = 3.24 MPa. These mean values are obtained from the characteristic values using the coefficient 1.2. Considering a normal distribution, this equivalence corresponds approximately to a coefficient of variation equal to 10%. Mean values of the compressive strengths are determined by tests: fc ,0 = 39 MPa and fc ,90 = 3.24 MPa. 4.2. Contact description The interaction between the different members was always modeled with deformation contact elements. Contact occurs in the interface between: the glulam and the steel plate, the dowel and the glulam, the dowel and the steel plate and finally the stiff steel plates at the supports and the glulam. Contact was modeled using the direct constraint method in MSC.Marc. The method requires the definition of the ‘‘contact body’’ that potentially may come in contact with the other. Contact bodies can simply be the physical bodies themselves (e.g. timber, fasteners and steel plate). In this method, when a node of a body contacts another body, a multipoint constraint is imposed. This constraint allows the contacting node to slide on the contacted segment. In this way, a contacting node is forced to be on the contacted segment. During the iteration procedure, a node can slide from one segment to another, changing the retained nodes associated with the constraint. A node is considered sliding off a contacted segment if it passes the end of the segment over a distance more than the contact tolerance. 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 fasteners and the timber is set equal to 0.3. The friction coefficient between fasteners and steel plate is set equal to 0.001 because of the small contact zone between the two materials. The stiff steel plates used on the supports in the tests are modeled with a control of the contact conditions between the steel plates and the glulam. The friction coefficient between timber and steel plate is set equal to 0.2. To avoid the potential rigid body modes, ‘‘nodal ties’’ were applied to the components not fully constrained in the glulam. In order to evaluate the effect of the friction coefficient, several simulations with friction coefficients varying from 0.1 to 0.5 between fasteners and timber were carried out. The comparison showed that the influence of the friction coefficients on the failure loads can be neglected. Moreover, two simulations with friction coefficients 0.1 and 0.3 between fasteners and steel plate were carried out. No obvious difference is found. 5. Validation and application of the numerical model The 3D finite element model is validated by comparison of the global numerical and experimental load–slip curves. The validated model is used to illustrate the bending moment and load distributions among dowels and to understand the evolution of the failure positions. Then, the model is applied to evaluate the influence, of the joint geometry and of the span to depth ratio of the beam, on the behaviour of the joint zone. Finally, the various analytical prediction formulae are evaluated through the comparison with the numerical results.

(11) 5.1. Global load–slip behaviour To validate the numerical model, the load–slip curves given by the 3D finite element model are compared with those from

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Table 5 Comparison of results (test and model with the Hoffman criterion).

Test (mean) Model 2

M (Nm)

Ke (kN/mm)

Splitting load (kN)

Ultimate load (kN)

200

58.0 53.3

99.2 107.4

110 117.7

150

experiments (Fig. 9). Two models have been developed on the basis of the Hill and Hoffman criterion. The model 1 includes the Hill yield criterion considering the elasto-plastic behaviour of timber without failure or damage. The model 2 includes the Hill yield criterion in combination with the Hoffman failure criterion to model the plastic yielding combined with the damage evolution. The model 1 represents in an acceptable way the initial stiffness and it shows a non-linear character mobilizing the wood embedding in compression and dowels in bending. However, its curve is higher than the experimental one (Fig. 9) and it does not represent the strength because it does not include any damage evolution. The model 2 is in good agreement with the whole experimental load–slip curve. It predicts well the stiffness and the failure load of joint (Fig. 9 and Table 5). From the numerical model, the failure 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 splitting load, and the last decreasing load is defined as the ultimate load.

82 kN

D1 D3

50 kN

100

D1

D2

D3

D4

50 0

0

0.02

0.04

0.06

x (m)

Fig. 10. Approximated moment along the dowel and dowels’ positions.

5.2. Plastic moment of the dowels

5.3. Stress analysis During the test, the cracks parallel to grain appeared and propagated in the dowel line near the loaded edge of the beam. The crack is related to tensile stresses perpendicular to grain and shear stresses. The failure index of the Hoffman criterion based on the stresses’ interaction provided by the numerical model is used to evaluate the position of the first potential crack. Fig. 11 shows the zone where the Hoffman failure criterion is reached for different levels of loading (31, 42 and 50 kN). It can present the potential failure zone before the appearance of cracks. The region of a large failure index value is close to the hole line. The zone at the line of dowels close to the loaded edge has a greater potential than the other line, but the difference is not very high. This is consistent with the observation from perpendicular to grain

31 kN

42 kN

50 kN

Fig. 11. Contour plot of the Hoffman failure index.

5f t,90 Tensile stress perpendicular to grain

In the test, a plastic hinge appears in the mid-span of all the dowels (Fig. 7). The EC5 formulae (Eq. (1)) with (fu = 543 MPa) gives a yield moment of the dowel equal to My = 220 N m. If the elastic limit (fy = 366 MPa) is used, the yield moment of the cross-section, considering an elastic perfectly plastic law, is equal to Mpl = 250 N m. In order to find out numerically the elasto-plastic moment that appears in each dowel, an approximated procedure is used. The section is divided into ten rectangles. At the centre of each of these rectangles, the stress level is calculated using the finite element results. Each area of these ten rectangles was multiplied by this stress and by the distance between the centre of the total area and the centre of each rectangle. Using this approach, the moment at the position of the hinge at the centre of the joint is calculated. At 50 kN, the dowels reached their elastic limit moment and at 82 kN, the dowels reached their plastic moment. The procedure is also used to calculate the moment along the length of the dowel. The shape of moments for the dowels D1 and D3 at 50 kN and 82 kN are shown in Fig. 10. The maximum moment is in the mid-span of the dowel and at 82 kN, the moment in this zone is equal to the plastic moment of the cross-section. The difference between the dowels D1 and D3 is not significant when all the dowels are in perfect initial contact.

D1

4f t,90

D3

3f t,90

2f t,90

f t,90

0 –25

–15

–5 5 15 Distance from dowel hole (mm)

25

Fig. 12. Tensile stress perpendicular to grain near the dowel hole (at 50 kN).

tensile stress distribution in Fig. 12 which presents the tensile stress perpendicular to grain near the dowel hole along the length of the beam at 50 kN. These results show that tensile stresses near the dowel D1 hole are slightly higher than those near the dowel D3 hole. This can explain why the splitting occurs at the line of fasteners nearest to the loaded edge in this case. Fig. 13 shows the shear stress parallel to grain near the dowel hole along the length of the beam at 50 kN. Though shear stress near the dowel D1 hole was slightly less than that near the dowel D3 hole, the shear stress is lower than the shear strength. Thus, the tension perpendicular to grain seems more influential than the shear stress parallel to grain in the failure of the joint. As for the global behaviour of the joints, the density of meshing selected in this study gives a stable evolution of the Hoffman failure index. In fact, various meshing densities showed that the zones where the Hoffman failure criterion is reached are very similar.

B.H. Xu et al. / Engineering Structures 31 (2009) 2357–2367

Shear stress

0.5fv

2365

D1

Table 6 Geometry (mm) and strength (kN) of the steel-to-timber joints.

D3

Joint

n

A1 A2 A3 A4 A5 A6 A7 A8

1 2 2 4 4 4 4 4

0

lr

h1

65 65 65 65 96 128

140 140 140 140 140 140 140 140

F (kN) –0.5fv –25

–15

–5 5 15 Distance from dowel hole (mm)

25 100

hm

Fu1 40.7 81.2 77.5 107.4 83.7 90 108 114

206 206 64 128 206 206

A1

A2

A3

A4

A5

A6

A7

A8

Fig. 13. Shear stress parallel to grain near the dowel hole (at 50 kN).

75 F (kN) 50

100 80

C1

25

C2

60

C3

40

0

5

10

15

20

Slip (mm)

Fig. 15. Load–slip curves with different geometries.

C5

20

Model 2

0

0

C4

0

2

4

6

8

Slip (mm)

Fig. 14. Load–slip curves with different dowels–holes in contact.

5.4. Effect of hole clearance The hole clearance in the steel plate was not considered initially in the model. In reality, the dowel hole in the steel plate has a diameter equal to 17 mm. Thus, the clearance between the dowel and the steel plate used in the modeling is equal to 1 mm. In order to analyze the influence of hole clearances on the behaviour of the joint, five configurations are considered (C1 to C5). They represent various situations where different dowels are in initial contact with the holes. It is considered that two or three dowels are in initial contact. The configurations are: C1 (D1 + D2), C2 (D3 + D4), C3 (D2 + D4), C4 (D1 + D2 + D4) and C5 (D2 + D3 + D4). The results show that, the number of dowels in initial contact has a non-negligible effect on the load–slip curve mainly at the initial stage (Fig. 14). They also show that there is no significant difference between the configurations with the same number of dowels in contact. For the configurations with two (C1 to C3) or three (C4 and C5) dowels in contact, a plastic deformation appears at a level of load slightly higher than 30 kN and 40 kN, respectively. For the two configurations, the same tendency of the curve evolution is observed. It is similar to those observed during tests. Thus, the hole clearance can explain the shape of the experimental curves for the two specimens tested (Fig. 6). The configuration with the four dowels in contact corresponds to that called model 2 (Fig. 14). It will be noticed that all the configurations show practically the same ultimate strength. 5.5. Effect of the joint geometry Joints, loaded in the direction perpendicular to grain, with different geometries are modeled to evaluate the effect of this

parameter on their strength. Eight joints, named A1 to A8, are considered (Table 6). The dimensions of the beam, the steel plate and the dowels are the same as those used in the tests (Table 1) and the dimensions he (346 mm) and h1 remain constant. The tested joint corresponds to the A4 configuration. The splitting strength is defined, from the load–slip curve, when the load decreases with the increase of the displacement. This decrease of load is due to the partial damage evolution in timber according to the Hoffman criterion. It is observed that the ductility decreases as the number of dowels increases. The splitting strengths are summarized in Table 6 and the load–slip curves are given in Fig. 15. The comparison of the strength of different joints configurations such as A1, A2 and A3, shows that the presence of a second dowel in the joint increases the strength of the joint and does not weaken the splitting strength of the beams. The joints A2 and A3, with different positions of the second dowel, show a quasi-similar strength which is close to the double of the joint A1. The maximum strength is given by A8 which has the largest distance between the dowels in the direction parallel to grain. This study shows that many geometrical parameters influence the splitting strength of dowelled joint in tension perpendicular to grain. It has to be completed by a more extensive study considering a large number of dowels with different slenderness of dowels. 5.6. Evaluation of the available prediction formulae As stated here above, various formulae predicting the splitting strength, of multiple-dowel joint loaded in tension perpendicular to grain, are available in the literature. These analytical formulae are applied to the configurations A1 to A8 to obtain the splitting strength and their results are compared to those given by the numerical model. The results are summarized in Table 7. The results are also compared to the plastic strength (Eq. (1)). In these equations, the mechanical characteristics used are defined hereafter. In Eq. (1), the embedding strength is (fh = 18.97 MPa) and the dowel yielding moment is (My = 220 N m). In Eq. (2) the shear strength of wood is (fv = 3.24 MPa). In Eq. (3) the apparent

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B.H. Xu et al. / Engineering Structures 31 (2009) 2357–2367

Table 7 Comparison between numerical (FEM) and analytical strengths (kN). Joint

FEM

Eq. (1)

Eq. (2)

Eq. (3)

Eq. (4)

Eq. (5)

Eq. (6)

Eq. (8)

A1 A2 A3 A4 A5 A6 A7 A8

40.7 81.2 77.5 107.4 83.7 90 108 114

26.7 53.4 53.4 106.8 106.8 106.8 106.8 106.8

224.2 224.2 224.2 224.2 224.2 224.2 224.2 224.2

161.1 161.1 161.1 161.1 161.1 161.1 161.1 161.1

145.6 145.6 145.6 145.6 145.6 145.6 145.6 145.6

62.5 62.5 62.5 62.5 62.5 62.5 62.5 62.5

59.5 102.2 71.6 123 97.3 112.5 142.3 164.6

62.8 94.9 69.1 104.4 82.8 93.8 109 113.6

√

fracture parameter ( GGc ) is assumed equal to 12 N/mm1.5 . In Eq. (6) the tensile strength of wood perpendicular to the grain is (ft ,90 = 0.48 MPa). Table 7 shows significant differences among the various prediction formulae and a significant scatter with numerical results. For the joints with one or two dowels, the failure mode predicted by Eq. (1) is dominant. For the joints with four dowels the splitting failure becomes dominant in some cases but the plastic deformation is present. The predictions of the splitting strength by CSA 2001 [16] (Eq. (2)), Van der Put and Leijten [18] (Eq. (3)) and EN 1995 [4] (Eq. (4)), overestimate the capacity in all cases. The prediction proposed by Larsen and Gustaffson [19] (Eq. (5)) underestimates the capacity of joints with more than one dowel but it is the more conservative. The prediction proposed by Ehlbeck and Görlacher [20] (Eq. (6)) overestimates the capacities in all cases except A3. The prediction proposed by Ballerini and Rizzi [22] (Eq. (8)) overestimates the capacities of joint A1 and predicts well the effect of the joint geometry on the splitting strength of the joint. Eq. (8) seems to be the best formula. It has to be more explored for various geometrical and mechanical configurations. The influence of the interaction between the shear stress and the perpendicular to grain stress can be considered using an empirical criterion [34]. 5.7. Effect of the span to depth ratio of the beam The span to depth ratio of the beam used in the test configuration also affects the strength of the joint loaded in the direction perpendicular to grain [26]. To evaluate numerically this effect, four beams are analyzed considering different span to depth ratios but keeping constant the height of the beam and the joint geometry. The beams are named Bi with i representing the span to depth ratio. For example, B3.7 represents the tested beam that has a span to height ratio equal to 3.7. It can be observed (Fig. 16) that the splitting strength and the ductility decrease when the span to depth ratio increases. The span to depth ratios equal to 1, 3, 3.7 and 5 give a splitting strength equal to 133.5, 113.1, 107.4 and 96.6 kN, respectively. The strength of the joint becomes relatively stable for the span to height ratio larger than 3. It can represent a good compromise between the representativeness of the joint in tension perpendicular to grain, the influence of supports and the large bending of the beam. 6. Conclusion The experimental results of timber-to-steel with multiple dowels, loaded in the direction perpendicular to grain, showed that different failure modes arose (plastic deformation and splitting). The numerical model developed is based on the finite element method with contact and non-linear behaviour of materials. The plastic flow law based on the Hill criterion associated to the Hoffman criterion representing the evolution of damage in wood represents well the load–slip experimental results with failure. The results show that the numerical model is a good approach for the investigation of the splitting strength of timber joints.

F (kN) 120 100 80 B1

60

B3

40 B3.7

20 B5

0

0

2

4

6

8

10

Slip (mm)

Fig. 16. Load–slip curves with different span to depth ratios of the beams.

This approach can be profitably used to limit the expensive experiments. The model is used for the investigation of the effect of different parameters, such as the geometry and the span to depth ratio of the beam, on the joint splitting strength. The joint geometry affects considerably the strength of the beam. The numerical model is used to compare the various analytical formulae, predicting the splitting strength, available in the literature. This comparison shows that these analytical formulae overestimate or underestimate the splitting strength. One of the formulae is promising but it has to be confirmed by a more extensive study. So, more theoretical and experimental studies are needed to refine the available analytical models. References [1] Bouchaïr A, Racher P, Bocquet JF. Analysis of dowelled timber to timber moment-resisting joints. Mater Struct 2007;40:1127–41. [2] Silih S, Premrov M, Kravanja S. Optimum design of plane timber trusses considering joint flexibility. Eng Struct 2005;27:145–54. [3] Kasim M, Quenneville JHP. Effect of row spacing on the capacity of bolted timber connections loaded perpendicular-to-grain. In: Proceedings of CIBW18 conference. 2002. Paper 35-7-6. [4] EN 1995-1-1:2004. Eurocode 5 – Design of timber structures. Part 1-1: General rules and rules for buildings. CEN/TC 250/SC5, 2004-11-01. [5] Sawata K, Yasumura M. Estimation of yield and ultimate strengths of bolted timber joints by nonlinear analysis and yield theory. J Wood Sci 2003;49: 383–91. [6] Guan ZW, Rodd PD. Hollow steel dowels—a new application in semi-rigid timber connections. Eng Struct 2001;23:110–9. [7] Yasumura M, Murota T, Sakai H. Ultimate properties of bolted joints in gluedlaminated timber. In: Proceedings of CIB-W18 conference. 1987. Paper 20-7-3. [8] Reffolds A, Reynolds TN, Choo BS. An investigation into the tension strength of nail plate timber joints loaded perpendicular to the grain. J Inst Wood Sci 1995;15(1). [9] Ballerini M. A new set of experimental tests on beams loaded perpendicularto-grain by dowel-type joints. In: Proceedings of CIB-W18 conference. 1999. Paper 32-7-2. [10] Reshke RG, Mohammad M, Quenneville JPH. Influence of joint configuration parameters on strength of perpendicular-to-grain bolted timber connections. In: Proceedings of the world timber engineering conference. 2000.

B.H. Xu et al. / Engineering Structures 31 (2009) 2357–2367 [11] Yasumura M. Criteria for damage and failure of dowel-type joints subjected to force perpendicular to the grain. In: Proceedings of CIB-W18 conference. 2001. Paper 34-7-9. [12] Quenneville JPH, Mohammad M. A proposed Canadian design approach for bolted connections loaded perpendicular-to-grain. In: Proceedings of the International RILEM symposium on joints in timber structures. 2001. p. 61–70. [13] Gattesco N, Toffolo I. Experimental study on multiple-bolt steel-to-timber tension joints. Mater Struct 2004;37:129–38. [14] Xu BH, Taazount M, Bouchaïr A, Racher P. Numerical 3D finite element modelling and experimental tests for dowel-type timber joints. Constr Build Mater 2009. doi:10.1016/j.conbuildmat.2009.04.006. [15] Xu BH, Bouchaïr A, Racher P. Mechanical analysis of timber connections with glued-in rods in bending. Mater Struct 2008 (under revision). [16] CSA 086-01. Canadian Standard Association. Engineering Design of Wood. CSA, Toronto, Ontario, Canada, 2001. [17] Van der Put TACM, Leijten AJM. Evaluation of perpendicular to grain failure of beams caused by concentrated loads of joints. In: Proceedings of CIB-W18. 2000. Paper 33-7-7. [18] Van der Put TACM, Leijten AJM. Splitting strength of beams loaded perpendicular to grain by connections, a fracture mechanical approach. In: Proceedings of the world timber engineering conference. 2004. [19] Larsen HJ, Gustafsson PJ. Dowel joints loaded perpendicular to grain. In: Proceedings of CIB-W18 conference. 2001. Paper 34-7-3. [20] Ehlbeck J, Görlacher R. Tension perpendicular to the grain in joints. STEP 1, Lecture C2, Centrum Hout, The Netherlands, 1995. [21] Ballerini M. A new prediction formula for the splitting strength of beams loaded by dowel-type connections. In: Proceedings of CIB-W18 conference. 2004. Paper 37-7-5. [22] Ballerini M, Rizzi M. Numerical analyses for the prediction of the splitting strength of beams loaded perpendicular-to-grain by dowel-type connections. Mater Struct 2007;40:139–49.

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