A three-dimensional model for bolted connections in wood

A three-dimensional model for bolted connections in wood D.M. Moses and H.G.L. Prion 555 Abstract: Recent criticism of the bolted connection requirements in the Canadian wood design code CSA Standard O86 indicates that the code lacks consideration of the different modes of failure, particularly as they relate to multiplebolt connections. A finite element model is proposed to predict load displacement behaviour, stress distributions, ultimate strength, and mode of failure in single- and multiple-bolt connections. The three-dimensional (3-D) model uses anisotropic plasticity for the wood member and elastoplasticity for the bolt. The Weibull weakest link theory is used to predict failure at given levels of probability. Predictions for connection behaviour in Douglas-fir and laminated strand lumber (LSL) correspond to experimentally observed behaviour. The output from the 3-D model is used for a multiplebolt connection spring model to illustrate many of the phenomena described in the literature. Key words: bolt, Douglas-fir, connection, model, plasticity, weakest link, wood. Résumé : Des critiques récentes concernant les exigences des raccordements boulonnés du code canadien de conception des charpentes en bois dans la norme CSA O86 indiquent que le code ne considère pas les différents modes de défaillance, plus particulièrement concernant les raccordements à boulons multiples. Un modèle à éléments finis est suggéré afin de prédire le déplacement en charge, les distributions de contraintes, la résistance à la rupture et le mode de défaillance dans les raccordements à boulon simple et à boulons multiples. Le modèle tridimensionnel utilise la plasticité anisotrope de la membrure de bois et l élastoplasticité pour le boulon. La théorie du lien le plus faible de Weibull est utilisée pour prédire la défaillance à certains niveaux de probabilité. Les prévisions concernant le comportement des raccordements en Douglas taxifolié et en bois de longs copeaux lamellés «Laminated Strand Lumber : LSL» correspondent au comportement observé lors des expériences. La sortie du modèle tridimensionnel est utilisée dans un modèle de ressort de raccordements à boulons multiples afin d illustrer plusieurs phénomènes décrits dans la littérature. Mots clés : boulon, Douglas taxifolié, connexion, modèle, plasticité, lien le plus faible, bois. [Traduit par la Rédaction] Moses and Prion 567 Introduction The Canadian wood design code CSA Standard O86 (CSA 1994) has recently been criticized for being overly conservative or inconsistent in its design requirements for multiple-bolt connections, particularly with regard to the lack of adequate consideration of brittle modes of connection failure (Quenneville and Mohammad 2000). The Canadian code predicts the ductile behaviour of single-bolt connections such that failure of the connection is governed by crushing of wood and bolt yielding. This is based on the simple European yield model (EYM) and empirical approximations of the ductile modes of failure (Smith and Foliente Received 11 February 2002. Revision accepted 3 February 2003. Published on the NRC Research Press Web site at http://cjce.nrc.ca on 6 June 2003. D.M. Moses. Equilibrium Consulting Inc., 1585 W. 4th Avenue, Vancouver, BC V6J 1L6, Canada. H.G.L. Prion. 1 Department of Civil Engineering, The University of British Columbia, 2324 Main Mall, Vancouver, BC V6T 1Z4, Canada. Written discussion of this article is welcomed and will be received by the Editor until 31 October 2003. 1 Corresponding author (e-mail: prion@civil.ubc.ca). 2002). Ductile behaviour is assumed to occur in all connections provided that the code-stipulated minimums for end distance and edge distance are followed. The strength of multiple-bolt connections is based on these single-bolt values multiplied by group-effect reduction factors that account for the number of rows of bolts, number of bolts within a row, bolt spacing, and bolt slenderness ratio (l/d, where l is the thickness of the wood member and d is the bolt diameter). These reduction factors were developed empirically following the publication of results by Yasumura et al. (1987) and Massé et al. (1988). The group-effect reduction factors neglect the complex distribution of stresses within a connection, however. Massé et al. and Quenneville and Mohammad (2000) showed that for multiple-bolt connections, connection failure was not ductile in many cases, thereby indicating that the current group factors must not only address uneven load distribution between bolts and fabrication defects but also account for the effect of brittle modes of connection failure. Quenneville and Mohammad (2000, 2001) showed that CSA Standard O86 is not capable of predicting connection strength in many cases for loading parallel to grain and perpendicular to grain because the code neglects the occurrence of brittle failure modes in multiple-bolt connections. The Canadian code also severely penalizes connections with multiple rows of bolts by neglecting the positive effect of adequate spacing between the rows (Mischler and Gehri Can. J. Civ. Eng. 30: 555 567 (2003) doi: 10.1139/L03-009

556 Can. J. Civ. Eng. Vol. 30, 2003 1999); this too results in overly conservative strength predictions. Each of these deficiencies in the code is tied directly to a poor understanding of the stress distributions in bolted connections. Clearly, the design and behaviour of bolted connections must be readdressed by moving away from empirical studies of laboratory data and placing new emphasis on analytical modelling of the stress distributions that lead to failure of bolted connections. Previous two-dimensional models Patton-Mallory et al. (1997b) provide an excellent review of past modelling work and the significant variables related to connection modelling in wood. Among the models are two-dimensional (2-D) plane stress finite element analyses for thin laminates, Johansen s (1949) yield model, and nonlinear beam on foundation models. The variables that must be considered include connection geometry, non-uniform stresses throughout wood member thickness, nonlinear orthotropic material properties, bolt bending and yielding, and bolt-hole clearance. Models in two dimensions must have built-in assumptions to address each of these variables. Designers of fibre composites have been using 2-D plane stress finite element models to predict bolted connection failure in brittle composites, such as carbon fibre, for many years with varying degrees of success (for example, Chang et al. 1982; Rowlands et al. 1982; Marshall et al. 1989). A contact interface between the bolt and a layered orthotropic member is used to predict the stress distribution around a bolt-loaded hole. The variables considered in these models include friction between the bolt and member, bolt-hole clearance, multiple-bolt connections, nonlinear compression properties, connection geometry, and stacking sequence of composite layers. Chen et al. (1995) used a threedimensional (3-D) model to show that delamination failure of fibre-reinforced plastic composites with l/d 1 can occur as a result of interlaminar (out-of-plane) shear stresses. It has also been shown that out-of-plane normal stresses are responsible for delamination, leading to failure in composites (Camanho and Matthews 1997), thereby justifying the use of a 3-D finite element model on connections with small slenderness ratios. Similar finite element models have been developed for bolted connections in wood (for example, Wilkinson et al. 1981; Hyer et al. 1987; Rahman et al. 1991). One main difference between bolted connections in nonwood composites and those in wood is the difference in slenderness ratio: nonwood composites are typically very thin, with l/d <1, whereas wood connections typically have l/d > 2. The earlier models assumed uniform stress distribution throughout the wood member. In reality, however, stresses are not uniform, as in the case of a connection where the bolt yields in bending. Again, 3-D models are better suited to this case, but approximations have been made using some 2-D models. The EYM and similar approximations that are used in CSA Standard O86 assume that both the wood member and steel dowel behave as ideal plastic materials. More details on this model can be found in Smith and Foliente (2002), for example. The difficulty with the EYM and similar embedment-based code equations is the assumption that embedment strength is a material property when in fact this strength is really a combination of many geometric and material factors. To account for some of these variables, many embedment tests are conducted for different bolt sizes and for parallel-to-grain and perpendicular-to-grain loading. Since the model cannot account for stress concentrations that lead to brittle failure, the EYM (and, hence, CSA Standard O86) strength predictions are restricted to ductile behaviour. However, brittle modes of failure can occur in multiple-bolt connections, in relatively thin members, or in connections with insufficient end distance, edge distance, and (or) bolt spacing. Jorissen (1998) attempted to account for brittle fracture using the EYM by calculating stress distributions along potentially critical load paths within the wood member. The average stresses for tension perpendicular to grain and for shear stresses were compared with those from a fracture mechanics model to predict ultimate strength. Jorissen found, however, that perpendicular-to-grain tension stresses were underestimated and, to allow crack initiation to be detected by the fracture theory, added an assumed peak stress perpendicular to grain to the bolt-hole location. This assumption limits the robustness of the model. In addition, this model works only for connections with small slenderness ratios where, it is assumed, the stresses are uniform throughout the wood member thickness. Two-dimensional finite element beam on foundation models using nonlinear behaviour have been developed to predict ductile load displacement behaviour of a single-bolt connection. Variables such as end restraint, bolt pretensioning, and friction have been included. One model has been developed that predicts ultimate load due to wood splitting by calculating perpendicular-to-grain tension forces (see Werner (1993), as noted in Patton-Mallory et al. (1997b)). Beam on foundation models are growing incrementally closer to solving the complete behaviour of a bolted connection, but they have never entirely considered all variables. For example, although tension perpendicular-to-grain stresses are included in one model, the other stresses that can lead to brittle failure are neglected. For this reason, a 3-D model is more robust. Previous three-dimensional models A number of 3-D finite element models for bolted connections exist. Guan and Rodd (2000) have developed a multiple-dowel moment connection model for hollow dowel connections using densified wood side plates. The model has orthotropic elastic material properties for wood, elastoplastic properties for the steel dowel, and a contact interface between the two materials. As was their intention, Guan and Rodd were able to shift the failure of these connections from the wood member to the hollow steel connector where ductile deformation could occur. Unlike traditional bolted connections, this hollow dowel connection eliminates any requirement for nonlinear behaviour of wood because dowel deformations dominate and far exceed any deformations in the wood. Patton-Mallory et al. (1997a) developed a 3-D finite element model of a single-bolt connection, shown in Fig. 1.

Moses and Prion 557 Fig. 1. (a) Connection test setup (based on Patton-Mallory 1996). (b) Finite element model geometry shown upside-down. Solid brick elements were used for the steel bolt and the wood, and contact elements were placed between the bolt and wood. Using symmetry, only one quarter of the geometry was modelled. The geometry of the model could be modified to accommodate changes in end distance, edge distance, member thickness, bolt diameter, hole clearance, and end restraint. To isolate the behaviour of the connector, side members were not included in the model. Nonlinear elasticity (i.e., reversible nonlinear stress strain with no permanent deformation) was assumed for compression parallel to grain and shear. Linear elastic properties were assumed for all other stress strain relationships, including all normal stresses in tension. The bolt was modelled as elastoplastic. The nonlinear material model is crucial to the development of accurate load displacement behaviour for connections in wood. Crushing in compression results in nonlinear stress strain behaviour in the three orthotropic material directions. Patton-Mallory et al. (1997a) assumed nonlinearity in the critical direction, parallel to loading, and found excellent correspondence with experimental load displacement curves for a range of end distances and slenderness ratios in Douglas-fir. This essentially elastic model did not, however, account for the energy dissipation associated with nonrecoverable deformations due to crushing, thereby ignoring the conservation of energy assumptions made in the development of the elastic theory. As a result, limitations existed in the nonlinear material model and were remedied by adding fictitious nonlinearity in the shear stress strain behaviour in two directions to prevent overprediction of these stresses. It was also shown that the elastic stiffness matrix would develop negative terms on the main diagonal if nonlinear behaviour was applied to the other two orthotropic material directions in compression (Patton-Mallory et al. 1997a). This material model can also result in poor estimates of the state of stress and failure strength. Failure prediction Studies of the stress field in this 3-D model provided evidence of stress concentrations in the vicinity of the bolt hole (Patton-Mallory et al. 1998a, 1998b). Tension stress perpendicular to grain and shear stresses in the wood member adjacent to the bolt were reviewed; however, predictions of ultimate strength due to brittle failure were not performed. Although several failure criteria exist for wood and orthotropic materials, most are difficult to apply to 3-D stress fields. Fracture mechanics models for mixed-stress crack propagation exist; however, there is some question over the accurate determination of the fracture toughness constants for wood (Fonselius and Riipola 1992). Smith and Hu (1994) considered a fracture model for a single-bolt connection loaded perpendicular to grain with a small slenderness ratio (i.e., stress uniform throughout member thickness). This model addresses only one particular state of stress and connection configuration. As mentioned earlier, Jorissen (1998) used fracture models with the EYM with limited application. In practice, Eurocode 5 (CEN 1995) uses a fracture mechanics approach to predict member strength in connections loaded at an angle to grain. There is consensus in Europe that fracture mechanics is the preferred approach to predicting strength because of perpendicular-to-grain splitting; however, improvements to the governing prediction equations continue to be debated (for example, Leijten 2002). This is a developing area of research. Stress interaction criteria, such as the Hankinson (1921) formula used in CSA Standard O86 or the postulation by Tsai and Wu (1971), relate stresses in different directions using polynomial equations. Interaction coefficients for the more complex equations are not well defined and are difficult to determine experimentally, particularly for wood products in 2-D applications (Clouston 1995). Patton-Mallory et al. (1998a, 1998b) analysed the results of their 3-D finite element model using the maximum stress and Tsai Wu criteria. Because of the difficulty of determining the interaction coefficients, they limited their analysis to a qualitative study. The size-effect (weakest link) failure criterion postulates that for brittle materials larger specimens are more likely to

558 Can. J. Civ. Eng. Vol. 30, 2003 fail at lower stresses because of the increased probability of a flaw in the specimen with a larger volume (Weibull 1939). Barrett (1974) showed this to be the case for tension stress perpendicular to grain in Douglas-fir, and CSA Standard O86 uses this as the basis for timber strength according to work by, for example, Madsen and Buchanan (1986). It can be shown that failure will occur when [1] σk V σ k d > * V* V where tension or shear stresses, σ, are integrated throughout the model volume, V, and compared with a reference stress, σ*, at a given probability of failure. The critical values for tension and shear stresses were determined from earlier tests along with the shape parameter, exponent k, according to the two-parameter Weibull probability distribution. 2 The benefits of this failure criterion are as follows: (i) non-uniform stresses can be analysed, while the highest concentrations are amplified by the shape parameter; (ii) failure can be predicted for a given probability; (iii) material variability is captured; and (iv) the location of failure and mode of failure can be isolated. Proposed material model Table 1. Anisotropic plasticity material properties for Douglas-fir. Uniaxial constant Value For the proposed model, wood is assumed to be a homogeneous continuum where material properties are directional but averaged throughout the member volume. Although wood does not exhibit the same microstructural plastic behaviour as that associated with metals, macroscopically wood exhibits plastic behaviour. In compression, wood has a linear elastic stress strain curve and will unload on the same path. On loading, once stresses exceed a critical level, permanent deformation occurs along with a drop in modulus. On unloading, the stress strain curve follows the initial elastic modulus. This was shown to be the case for laminated strand lumber (LSL) (Moses et al. 2003) and is similar for solid wood. This behaviour occurs in each of the orthogonal material directions, with different moduli and yield stresses in each direction. The anisotropic plasticity model was, therefore, chosen to model this behaviour. The anisotropic plasticity material model has been used to predict failure of nonwood composites using 2-D finite element models (Vaziri et al. 1991). In the current application, the anisotropic plasticity model was applied in three dimensions using the TB,ANISO option in ANSYS v.5.3 (Swanson Analysis Systems Inc. 1996). Unlike the nonlinear elastic models, this material model does not require modification of the elastic stiffness matrix. Instead, it accounts for permanent deformation and energy dissipation in three orthogonal planes. Details on this model can be found in Moses (2000), Hill (1947), Valliappan et al. (1976), and Shih and Lee (1978). Bilinear stress strain curves are assumed for tension and compression in each material direction. A yield stress and tangent modulus are required for each curve. As a consequence, 18 additional constants are required in addition to the 9 normally needed for orthotropic elastic materials. These constants are readily determined from standard materials tests, however (Moses et al. 2003). Brittle modes of failure are predicted according to the size-effect eq. [1]. Foschi and Longworth (1975) used this technique to determine the strength of timber rivets in Douglas-fir, and this formed the basis of the design requirements in CSA Standard O86. The maximum stress criterion is assumed, meaning that at each load increment, the three tension stresses and three shear stresses are independently compared against critical values. It is assumed that little or no interaction between these stresses exists. The evaluation of eq. [1] was carried out using a short user-programmable subroutine in ANSYS v.5.3 (Moses 2000). Results for Douglas-fir Shear constant Value E x 827 G xy 276 E y 13 780 G yz 276 E z 827 G xz 28 E Tx 4.0 G Txy 4.6 E Ty 140.0 G Tyz 4.6 E Tz 4.00 G Txz 0.05 σ x 7.6 σ ±xy 8.1 σ y 45.0 σ ±yz 8.1 σ z 7.6 σ ±xz 0.8 σ * a +x 3.2 σ * b ±xy 17.0 Note: E i, modulus of elasticity; E T, tangent modulus; G ij, shear modulus; G T, target modulus in shear; σ +i and σ i, yield stress in tension or compression, respectively; σ ij, shear stress in material coordinate system. a Tension perpendicular to the main strand axis (X direction) (V =1,p = 0.5). b Shear in the plane of the panel (XY) (V =1,p = 0.5). The 3-D finite element model, shown in Fig. 1, provided the starting point for the current study in which stresses are analysed, load displacement and ultimate strength are predicted, and the results are used to predict the behaviour of multiple-bolt connections. Based on the known behaviour of Douglas-fir from the literature (for example, Patton-Mallory 1996; Kollmann and Cote 1968), constants were chosen as listed in Table 1 for the anisotropic plasticity model. These constants can be determined from standard uniaxial and shear tests on small wood specimens, as described in Moses et al. (2003). The nine experimental single-bolt connection groups listed in Table 2 from Patton-Mallory (1996) were not originally analysed to predict failure. They are reanalysed here using both the original nonlinear elastic model and the proposed anisotropic plasticity model. Failure was predicted for both material models using eq. [1] with a 50% probability of failure (i.e., average). Load displacement behaviour was found 2 The theoretical three-parameter Weibull distribution is not necessary for material strengths because the third location parameter is assumed to be zero, i.e., physically, material strength cannot be less than zero (Barrett 1974).

Moses and Prion 559 Table 2. Single 12 mm diameter dowel connections in Douglas-fir. Geometry Experimental results (Patton-Mallory 1996) a (p =0.5) Nonlinear elastic model l/d e/d Edge/d Load at 1.0 mm (kn) Ultimate load (kn) Ultimate displacement (mm) Ultimate load (kn) Ultimate displacement (mm) Anisotropic plasticity model (p =0.5) Ultimate load (kn) Ultimate displacement (mm) 2 4 1.5 13.6 13.6 1.3 10.0 0.5 11.4 1.0 2 7 1.5 13.6 15.0 1.8 10.1 0.5 >11.7 >1.0 2 10 1.5 13.6 14.5 1.8 10.3 0.5 >11.9 >1.0 5 4 1.5 14.2 >17.6 >3.3 11.5 0.5 >12.3 >1.0 5 7 1.5 15.6 >19.1 >3.3 11.6 0.5 >12.3 >1.0 5 10 1.5 15.6 >17.6 >3.3 11.6 0.5 >12.4 >1.0 7 4 1.5 14.7 >17.0 >3.3 11.8 0.8 >12.1 >1.0 7 7 1.5 15.8 >17.4 >3.3 12.0 0.8 >12.2 >1.0 7 10 1.5 15.8 >17.1 >3.3 12.0 0.8 >12.2 >1.0 Note: The greater than symbol (>) in the experimental results indicates the test was stopped prior to any brittle failure, and in the model predictions indicates the analysis was stopped prior to the prediction of brittle failure. a Average of 10 specimens. Fig. 2. Tension stress concentrations perpendicular to grain (i.e., perpendicular to the direction of loading) at failure in Douglas-fir for l/d = 5 and e/d =4:(a) proposed plasticity model at 12.3 kn, and (b) earlier nonlinear elastic model at 11.5 kn. to be very similar to the behaviour of the experiments because a nonlinear stress strain model in compression in the direction of loading, coupled with elastoplastic steel properties, will provide good estimates of load displacement. A modest improvement over the nonlinear elastic model was achieved for predictions of ultimate strength and displacement using anisotropic plasticity. Table 2 shows that the anisotropic plasticity model predicts failure loads that are closer to and more consistent with those observed in experiments. Brittle failure was predicted to occur with small slenderness ratios, l/d, whereas ductile behaviour was predicted in all other cases and was consistent with experiments. Whereas the model of Patton-Mallory was not capable of predicting strength, the proposed model predicts strength and load displacement behaviour to failure. The improved behaviour using anisotropic plasticity (rather than nonlinear elasticity) is explained by the improved prediction of stresses in the other material directions. The perpendicular-to-grain stresses, shown in Fig. 2 with the steel dowel removed, were overpredicted by the nonlinear elastic model in areas under the steel dowel at ultimate load. A zone of high tension stress develops at the point of contact between the dowel and wood (noting that the hole is larger than the dowel diameter), with peak stresses predicted by the nonlinear model to be more than four times greater than those predicted using the anisotropic plasticity model. The nonlinear elastic model also predicted stresses in the compression zone to the side of the tension region to be more than one and a half times greater than the stresses predicted using the plasticity model. Stress overprediction leads to

560 Can. J. Civ. Eng. Vol. 30, 2003 premature failure predictions. Also note that the plasticity model predicts the tension zone at the end of the specimen; cracks are known to develop in this region (Jorissen 1998). In practice, the ultimate strength would be predicted for limit states design using a fifth percentile strength based on the material properties of Douglas-fir. Results for laminated strand lumber Fig. 3. Laminated strand lumber (LSL) panel stacking sequences: (a) A, fully oriented; (b) B, randomly oriented; and (c) C, 0 /R/0 ; (d) D, R/0 /R; and (e) E, 0 /+45 / 45 /0 /0 / 45 /+45 /0. R, randomly oriented layer. Laminated strand lumber (LSL) is a structural composite lumber made of wood strands up to 30 cm long, approximately 2.5 cm wide, and roughly 0.94 mm thick. Laminated strand lumber is made in large panels up to 75 mm thick and then cut into standard lumber sizes. Strand orientation in the plane of the panel can be controlled to increase axial and bending stiffness and strength. Limitations in the manufacturing process result in a significant percentage of crossaligned strands, although most are oriented in one direction. The cross-aligned strands reduce stiffness and strength in the direction parallel to the strands but increase stiffness and strength in the orthogonal direction in the plane of the strands, a potential benefit for the performance of connections. It is, therefore, possible to control the behaviour of bolted connections in LSL by controlling the strand orientation in a manner similar to the construction of composites using man-made fibres. Five different stacking sequences of strands were studied as shown in Fig. 3 (panel layup types A E), and each contains a different percentage of oriented strands per unit thickness as follows: (A) fully oriented (100%); (B) fully random (0%); (C) surfaces oriented, core random (66%); (D) surfaces random, core oriented (33%); and (E) eight oriented layers aligned at angles 0 and ±45 (50%). Results from single-bolt connection tests are listed in Table 3. End and edge distances were chosen to ensure that both brittle and ductile behaviour could be observed. The material properties used for the anisotropic plasticity model are listed in Table 4. The tension and shear results were obtained from material tests on a variety of specimen sizes to illustrate the validity of size effect and the Weibull probability distribution, particularly in tension (see Moses and Prion 2002 for details). The continuum approach to material properties was found to be valid for LSL as it was with solid wood. Panel layup types C, D, and E were modelled using layers of panel layup types A and B in the same 3-D connection model: this is equivalent to the global material properties of the entire composite panel (Moses et al. 2003). One analysis was performed for each connection configuration and compared with the average results listed in Table 3. Three sample plots of experimental load displacement curves are shown in Fig. 4 for specimens in the fully oriented, type A material loaded parallel to the main strand axis. These plots are shown here to illustrate the difference in ductility among specimen groups. Specimens in Fig. 4a had a low slenderness ratio (l/d = 2) with end distance 2d, and all failed suddenly. Though not shown, the same specimens with end distance 4d were all ductile and showed no brittle behaviour. In contrast, the specimen groups in Figs. 4b and 4c had a high slenderness ratio (l/d =4)and followed very similar load displacement paths to one another. Specimens with small end distance 2d (Fig. 4b), however, failed earlier than those with end distance 4d (Fig. 4c). The lower failure load is associated with the geometric effect of small end distance and the mode of failure for this stacking sequence. Note, however, that Fig. 4b indicates a transition between brittle and ductile behaviour. Geometry and stacking sequence were found to influence the average ultimate loads, displacements, and modes of failure for each of the test groups listed in Table 3. In particular, the transition from brittle to ductile behaviour is apparent. In addition, load displacement curves from the finite element analysis are superimposed on the experimental curves shown in Fig. 4, and these too show the transition from brittle to ductile behaviour. The predicted curves end at the lesser of (i) the point of predicted failure based on a probability of failure p = 0.5 (i.e., average values) or (ii) 2.5 mm (the point at which the analysis was stopped). Ultimate loads were always predicted conservatively to be within 50 84% of the experimental averages for all material types, configurations, and loading orientations. The reason for the conservative estimates is that failure of the entire connection was assumed to occur at the first instance that the governing stresses (tension or shear) reached capacity. Thus, if delamination (i.e., tension in the Z direction) failure was predicted to start in the specimen at a particular load, then the analysis was stopped: in reality, the specimen would continue to carry more load until it would either fail under the initial failure condition or fail in an entirely different mode that could become critical at a later stage. In addition, the two-parameter Weibull distribution is known to predict failure conservatively (Holmberg 1995). Ultimate displacements were found to be within 17 88% of the experimental averages. Ductile behaviour was predicted to occur when the load displacement curve exhibited nonlinearity and when the failure criterion was not satisfied up to the 2.5 mm displacement level. The large discrepancy in predicted ultimate displacements occurs because the analysis is terminated once the ultimate load is reached: in reality, the load may drop off while displacements continue. The predicted modes of failure from stresses matched the experimental observations, as indicated in Table 3. A sample

Moses and Prion 561 Table 3. Single-dowel connection geometry, experimental results, and predicted results. Ultimate Geometry Ultimate load (kn) displacement (mm) Failure mode Type a d (mm) e/d Edge/d l/d Expt. Predicted Expt. Predicted Expt. b Predicted c A-Pa 9.5 2 1.5 4 8.0 6.4 2.8 1.3 B Z A-Pa 9.5 3 1.5 4 >11.1 >7.3 >7.1 >2.5 D D A-Pa 9.5 4 1.0 4 >10.7 >7.3 >5.8 >2.5 D D A-Pa 9.5 4 1.5 4 >10.4 >7.3 >9.9 >2.5 D D A-Pa 13.0 2 1.5 3 12.0 12.2 2.0 1.0 B Z A-Pa 13.0 3 1.0 3 >19.0 >13.3 >8.4 >1.8 D D A-Pa 13.0 4 1.5 3 >19.4 >13.3 >6.6 >1.8 D D A-Pa 19.0 2 1.5 2 19.8 16.6 1.3 1.0 B X, Z A-Pa 19.0 3 1.5 2 33.0 23.6 2.3 1.5 B Z A-Pa 19.0 4 1.0 2 >27.5 17.1 >1.5 1.0 d Z A-Pa 19.0 4 1.5 2 >32.3 26.2 >2.8 1.5 d Z A-Pe 9.5 2 3.0 4 6.3 5.0 2.0 1.3 B X A-Pe 9.5 4 3.0 4 7.0 5.1 3.3 1.3 B X A-Pe 19.0 2 3.0 2 15.5 7.5 2.3 1.0 B X A-Pe 19.0 4 3.0 2 15.2 7.7 2.0 1.0 B X A-AN 9.5 2 3.0 4 8.6 6.4 3.3 1.8 B X A-AN 9.5 4 3.0 4 >10.7 >6.5 >9.9 >1.8 D D A-AN 19.0 2 3.0 2 18.1 9.1 1.8 0.8 B X A-AN 19.0 4 3.0 2 27.3 14.8 6.1 1.0 B X B-Pa 9.5 2 1.5 4 9.8 6.1 6.1 1.5 B, D Z B-Pa 9.5 3 1.5 4 >10.1 >6.2 >9.4 >1.5 D D B-Pa 9.5 4 1.0 4 5.8 5.3 1.0 1.0 B Z B-Pa 9.5 4 1.5 4 >10.0 6.2 >8.4 1.8 D Z B-Pa 13.0 2 1.5 3 14.3 11.1 3.6 1.0 B Z B-Pa 13.0 3 1.0 3 >17.9 12.0 >8.1 1.3 B, D Z B-Pa 13.0 4 1.5 3 >17.0 12.0 >6.9 1.3 D Z B-Pa 19.0 2 1.5 2 24.4 16.3 6.1 1.0 B Z B-Pa 19.0 3 1.5 2 31.4 17.4 4.3 1.3 B Z B-Pa 19.0 4 1.0 2 14.9 10.3 1.3 0.8 B Z B-Pa 19.0 4 1.5 2 30.9 17.4 6.4 1.3 B Z C-Pa 9.5 2 1.5 4 8.7 7.2 5.3 2.8 B, D Z, 0 C-Pa 9.5 3 1.5 4 >10.2 >7.0 >7.6 >2.5 D D C-Pa 9.5 4 1.0 4 9.6 >7.1 6.6 >2.5 B, D D C-Pa 9.5 4 1.5 4 >11.0 >7.0 >9.9 >2.5 D D C-Pa 13.0 2 1.5 3 15.6 12.2 5.1 1.0 B Z, 0 C-Pa 13.0 4 1.5 3 >20.4 >13.6 >11.4 >1.8 D D C-Pa 19.0 2 1.5 2 22.6 18.1 2.0 1.0 B Z, 0 C-Pa 19.0 3 1.5 2 32.8 23.0 5.3 1.0 B Z, 0, R C-Pa 19.0 4 1.0 2 30.0 14.7 2.3 0.8 B Z, R C-Pa 19.0 4 1.5 2 >35.8 >24.4 8.4 1.3 B, D Z, 0 C-Pe 9.5 2 3.0 4 8.8 6.0 5.1 1.5 B Z, 0 C-Pe 9.5 4 3.0 4 >11.0 6.1 >13.2 1.8 B, D Z, 0 D-Pa 9.5 2 1.5 4 9.4 6.4 5.8 1.3 B Z, R D-Pa 9.5 3 1.5 4 >10.7 >5.5 >9.9 >1.0 D D D-Pa 9.5 4 1.0 4 8.2 5.5 3.6 1.0 D Z, R D-Pa 9.5 4 1.5 4 >10.7 >6.3 >11.9 >1.8 D D D-Pa 13.0 2 1.5 3 13.8 11.8 4.1 1.0 B Z, R D-Pa 13.0 4 1.5 3 >20.6 12.6 >11.7 1.3 B, D Z, R D-Pa 19.0 2 1.5 2 27.0 17.8 2.0 1.0 B Z, R D-Pa 19.0 3 1.5 2 34.9 20.6 5.6 1.0 B, D Z, R D-Pa 19.0 4 1.0 2 24.7 12.5 2.0 0.8 B Z, R D-Pa 19.0 4 1.5 2 32.6 20.7 7.1 1.0 B Z, R D-Pe 9.5 2 3.0 4 >10.4 >6.0 >9.1 >1.3 D D D-Pe 9.5 4 3.0 4 >11.0 >6.2 >13.7 >1.3 D D

562 Can. J. Civ. Eng. Vol. 30, 2003 Table 3 (concluded). Ultimate Geometry Ultimate load (kn) displacement (mm) Failure mode Type a d (mm) e/d Edge/d l/d Expt. Predicted Expt. Predicted Expt. b Predicted c E-Pa 9.5 2 1.5 4 8.6 6.5 3.3 1.5 B Z, 0 E-Pa 9.5 3 1.5 4 >10.4 >7.3 >10.9 >2.5 B, D D E-Pa 9.5 4 1.0 4 9.7 6.5 5.1 1.5 B Z, +45 E-Pa 9.5 4 1.5 4 >11.3 >7.3 >10.7 >2.5 D D E-Pa 13.0 2 1.5 3 13.6 13.0 3.6 1.8 B Z, 0 E-Pa 13.0 4 1.5 3 >21.5 >12.8 >11.2 >1.5 D D E-Pa 19.0 2 1.5 2 24.4 19.8 2.0 1.3 B Z, 0 E-Pa 19.0 3 1.5 2 37.8 23.1 6.9 1.5 B XY, +45 E-Pa 19.0 4 1.0 2 24.5 15.0 1.8 1.0 B Z, 0, ±45 E-Pa 19.0 4 1.5 2 >37.1 22.1 >7.9 1.3 D XY, +45 E-Pe 9.5 2 3.0 4 >10.0 5.1 >8.9 1.0 B, D X, 0 E-Pe 9.5 4 3.0 4 >10.9 >5.6 >10.7 >1.8 B, D D a A E, panel layup types; AN, specimens cut at 45 to main strand axis; Pa, specimens cut parallel to main strand axis; Pe, specimens cut perpendicular to main strand axis. b B, brittle fracture; D, no fracture and ductile. c X and Z, tension failure in the X or Z direction, respectively; XY, shear failure in the XY plane; R, layer that failed with a random orientation; 0 and ±45, layer that failed oriented at 0 or ±45. d No failure achieved due to grip limitations. stress plot at ultimate load, shown in Fig. 5, shows that stresses are not uniform throughout the thickness. The model was found to be well behaved for predicting the failure modes in all loading orientations. For panel layup types C, D, and E the model was able to detect the layer in which failure initiated. Shear stress concentrations in Fig. 6 show the distinct differences between fully oriented specimens and eight-layer specimens. Figure 6a has a more or less uniform distribution throughout its thickness, whereas Fig. 6b has stress concentrations at the interfaces of the layers. Multiple-bolt connections In most practical designs, multiple-bolt connections are necessary. Rather than developing cumbersome 3-D finite element models of many bolted connections, and given the highly localized effects of stress concentrations around bolt holes, a simplified one-dimensional (1-D) model is proposed for multiple-bolt connections. This model requires output from the 3-D model. Isyumov (1967) proposed a 1-D multiple-spring model, shown in Fig. 7, to analyse bolted connections. The springs represent the main and side members and the interaction of each bolt with wood. This model simulates the load redistribution among the bolts as load is applied. The interaction stiffness can be nonlinear to capture the ductile behaviour of a single-bolt connection until failure. Jorissen (1998) used this model to predict brittle failure using fracture mechanics and assumed stress distributions based on the load level in each spring. Tan and Smith (1999) used a similar model with reasonable success using fitted load slip curves for single-bolt connections. A different approach, which is based on the relationship between the load level and the Weibull weakest link prediction, will be used here. The load displacement curves developed using the finite element model for single-dowel connections have distinct behaviours that depend on, for example, edge distance and slenderness ratio. These curves were found to be either roughly linear or nonlinear (depending on the situation), and this behaviour can be incorporated into the spring model through the wood dowel interaction stiffness, k b, shown in Fig. 7. The load displacement curves were found to be very similar if only end distance was changed; the only difference between these curves, as shown in Fig. 8 for LSL, was the end point (ultimate load) as a result of brittle failure. Thus, for this particular configuration, the spring stiffness, k b,is the same for both, regardless of end distance. The point of failure, on the other hand, must be predicted using the relationship between stress intensity and load level. At each load level prior to failure, the volume integral on the left side of eq. [1] was calculated for each of the connections listed in Table 3. In Fig. 9, the relationship between volume integral and load level is shown for one connection geometry for each of the three brittle stresses that were found to typically govern in these specimens, i.e., tension perpendicular to the main strand axis (X direction), tension perpendicular to the panel surface (delamination in the Z direction), and shear in the plane of the panel (XY). In general, a logarithmic relationship appears to exist, though this relationship could be further refined. These lines were fitted using linear regression, as were a corresponding set of curves for 19 mm dowels with end distance 4d. Finite element software was used to create a spring model similar to that in Fig. 7 for each of the four cases shown in Fig. 10. For each case, a single central LSL member and two 6.4 mm steel side plates with four dowels were analysed. The stiffness constants k m and k s, corresponding to the main wood member and the steel side plates, respectively, were also included. The end distance and spacing between dowels were fixed at 4d. The single-bolt 3-D finite element model was used to determine k b and the volume integral load relationship for

Moses and Prion 563 Table 4. Material properties for uniaxial and shear behaviour of LSL with anisotropic plasticity model. (A) Material properties for uniaxial behaviour of LSL Uniaxial constant Type A fully oriented panels E x 655 5516 E y 11 700 5516 E z 90 103 E Tx 31 57 E Ty 345 57 E Tz 25 23 σ x 6.6 16.0 σ y 24 16 σ z 5.6 9.0 Tension values Yield A σ +i Ultimate A σ i * (V = 16.4, p =0.5) Type B randomly oriented panels Yield B σ +i X 6.6 4.8 16.0 22.7 Y 24.0 52.7 16.0 22.7 Z 5.6 1.3 9.0 1.2 (B) Material properties for shear behaviour of LSL Shear constant Type A fully oriented panels G xy 1379 2068 G yz 421 345 G xz 179 345 G Txy 3.4 3.4 G Tyz 3.4 3.4 G Txz 3.4 3.4 σ ±xy 34 55 σ ±yz 55 55 σ ±xz 55 55 Note: Poisson s ratios not shown. Type B randomly oriented panels Ultimate B σ i * (V = 16.4, p =0.5) cases (a) and (b), and a new 3-D finite element model of two rows with one dowel per row (to allow for possible stress interaction between rows) was used to determine these relationships for cases (c) and (d). These relationships could then be used in the 1-D spring model by assuming independence between connector groups. The load was applied to the spring model in increments. At each load step, the load in each connector group was determined according to the relationships determined from the 3-D models. No interaction in stresses between neighbouring dowels in a row was assumed; only the redistribution in load (simulated by the spring model) was used to determine the current state of stress. The volume integral for each critical stress was determined for each connector group. Then, the volume integrals for each critical stress were summed for all connectors to provide a measure of the total stress state in the entire LSL member and to check for failure using eq. [1]. The predicted load displacement behaviour of these four connections is shown in Fig. 11. The curve for case (a) exhibits the same basic shape as that of a single dowel with l/d = 4 prior to brittle failure. However, the single dowel with this slenderness ratio and end distance was not found to have brittle failure either experimentally or by 3-D model predictions. Shear stresses were the cause of this brittle failure. As a result, the ultimate load per connector for the fourdowel connection is only 94% of the single-dowel connection strength. Case (b) load displacement was found to be linear as a result of the linear behaviour observed and predicted for the single-dowel connections with l/d = 2 with the same end distance. Brittle tension perpendicular-to-grain failure governed, resulting in the average load per connector of only 76% of the load expected for a single-dowel connection. Non-uniform load distribution between dowels resulted in the end dowel carrying the highest load and led to the lower efficiency of this connection. Case (c) has two rows of closely spaced dowels. The 3-D model predicted shear failure with load per connector of the

564 Can. J. Civ. Eng. Vol. 30, 2003 Fig. 4. Load displacement curves for type A LSL loaded parallel to grain: (a) e/d =2,l/d =2;(b) e/d =2,l/d =4;(c) e/d =4, l/d = 4., brittle splitting failure;, predictions. Fig. 5. Stress contours of normal stress, σ x, in type A LSL at peak load (7.3 kn) for e/d =4,l/d = 4, edge distance 1.5d, d = 9.5 mm, and loading parallel to grain. The reduction in efficiency of multiple-dowel connections is pronounced because of a combination of unequal load distribution and the logarithmic relationship between load and volume integral: at higher load levels, the end connectors were found to carry a much greater amount of load, while the stress intensity increased exponentially. We note that although some single-bolt connections were ductile, the same connector-units became brittle in a multiple-bolt connection. This corresponds to experimental observations by Massé et al. (1988) and Quenneville and Mohammad (2000). two-dowel connection at only 73% of the load expected for a single-dowel connection. Using the spring model for the two rows of four dowels, the efficiency dropped to 53%. In case (d), the spacing between rows was increased to 3d. As a result, the 3-D model predicted no loss in connection efficiency (i.e., 100% efficient or independence between connectors). Using the spring model, the efficiency dropped to 68%; however, this is greater than the 53% efficiency for case (c). This corresponds with experimental findings by Mischler and Gehri (1999). In both of these eight-dowel connections, it was found that the end connectors carried roughly twice as much load as each of the others. Conclusions In Canada, only the design of timber rivet connections is based on stress and failure analysis using the finite element method (Foschi and Longworth 1975). The analysis provides designers of timber rivet connections with a sense of the governing mode of behaviour for a connection. Given the acceptance of the timber rivet model, why then has the design community only recently started to address the behaviour of bolted connections with computer models? It is likely that increased computing speed and the development of sophisticated material models now make such analyses less prohibitive. Although stress analysis of bolted connections can be performed using a variety of analytical models, a 3-D finite element method is the most accommodating tool among them. This model can accommodate any species, bolt properties, and connection geometry, and it could even be used to simulate the embedment test. It provides load displacement behaviour and is well suited to evaluate the uneven stress distributions that lead to failure in bolted connections. These models and procedures describe the stress field, ultimate strength, and mode of failure in single- and multiple-

Moses and Prion 565 Fig. 6. Shear stress concentrations at failure (loading parallel to grain) for e/d =4,l/d = 2, edge distance 1.5d, and d =19mm: (a) type A LSL at 26.2 kn; and (b) type E LSL at 22.1 kn. Fig. 7. Multiple-bolt connection spring model (after Isyumov 1967). k b, stiffness of the bolt member interaction; k m, stiffness of the main member; k s, stiffness of the side members. Fig. 9. Stress volume integral variation in LSL for e/d =4,l/d = 4, edge distance 1.5d, and d = 9.5 mm. Fig. 8. Load displacement curves for e/d =2,3,and4andd = 9.5 mm. bolt connections in solid wood and in wood composites. They provide a rational approach to bolted connection design and, unlike the current European yield model design practice with code-stipulated group factors, the proposed stress-based models can be used to explain many of the phenomena described in earlier experimental studies by others. In practice, these models can be used for parametric studies to determine the transition between brittle and ductile failure as a result of changes in geometry. As with any model, the material properties and failure criterion can undoubtedly be fine-tuned; however, the concept and procedure described herein can be used, with engineering judgement, to develop new design standards for bolted connections in wood.

566 Can. J. Civ. Eng. Vol. 30, 2003 Fig. 10. Multiple-dowel connections used in 1-D spring model study. Central LSL member shown only. Fig. 11. Multiple-dowel connection load displacement curves (see Fig. 10). Acknowledgements Financial support for this project was provided by the Science Council of British Columbia, Trus Joist A Weyerhaeuser Business, and Forest Renewal British Columbia. References Barrett, J.D. 1974. Effect of size on tension perpendicular-to-grain strength of Douglas-fir. Wood and Fiber, 6: 126 143. Camanho, P.P., and Matthews, F.L. 1997. Stress analysis and strength prediction of mechanically fastened joints in FRP: a review. Composites Part A, 28A: 529 547. CEN. 1995. Eurocode 5: Part 1.1. Design of timber structures: general rules and rules for buildings. European Committee for Standardization (CEN), Brussels, Belgium. Chang, F.-K., Scott, R.A., and Springer, G.S. 1982. Strength of mechanically fastened composite joints. Journal of Composite Materials, 16: 470 494. Chen, W.-H., Lee, S., and Yeh, J.-T. 1995. Three-dimensional contact stress analysis of a composite laminate with bolted joint. Composite Structures, 30: 287 297. Clouston, P. 1995. The Tsai-Wu strength theory for Douglas-fir laminated veneer. M.Sc. thesis, The University of British Columbia, Vancouver, B.C. CSA. 1994. Engineering design in wood (limit states design). Standard O86.1-94, Canadian Standards Association, Rexdale, Ont. Fonselius, M., and Riipola, K. 1992. Determination of fracture toughness for wood. ASCE Journal of Structural Engineering, 118: 1727 1740. Foschi, R.O., and Longworth, J. 1975. Analysis and design of griplam nailed connections. ASCE Journal of the Structural Division, 101(ST12): 2537 2555. Guan, Z.W., and Rodd, P.D. 2000. A three-dimensional finite element model for locally reinforced timber joints made with hollow dowel fasteners. Canadian Journal of Civil Engineering, 27: 785 797. Hankinson, R.L. 1921. Investigation of crushing strength of spruce at varying angles of grain. Air Service Information Circular v.3 n.259, Chief of Air Service, Washington, D.C. Hill, R. 1947. A theory of the yielding and plastic flow of anisotropic metals. Proceedings of the Royal Society, Series A, 193: 281 297. Holmberg, J.A. 1995. Application of Weibull theory to random-fibre composites. Composites Science and Technology, 54: 75 85. Hyer, M.W., Klang, E.C., and Cooper, D.E. 1987. The effects of pin elasticity, clearance and friction on the stresses in a pinloaded orthotropic plate. Journal of Composite Materials, 21: 190 206. Isyumov, N. 1967. Load distribution in multiple shear-plate joints in timber. Canada Department of Forestry and Rural Development, Publication 1203. Johansen, K.W. 1949. Theory of timber connectors. International Association of Bridges and Structural Engineering (IABSE), Bern, Switzerland, Publication 9, pp. 249 262. Jorissen, A. 1998. Double shear timber connections with dowel type fasteners. Ph.D. thesis, Delft University, Delft, The Netherlands. Kollmann, F.F.P., and Cote, W.A. 1968. Principles of wood science and technology I: solid wood. Springer-Verlag, New York. Leijten, A.J.M. 2002. Splitting strength of beams loaded by connections, model comparison. In Proceedings of the International Council for Research and Innovation in Building and Construction (CIB), Kyoto, Japan, September. Lehrstuhl fuer Ingenieurhdzbau und Baukonstruktionen, University of Karlsruhe, Germany, CIB- W18, Paper 35-7-7. Madsen, B., and Buchanan, A.H. 1986. Size effects in timber explained by modified weakest link theory. Canadian Journal of Civil Engineering, 13: 218 232.

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Report to the Working Commission W18 Timber Structures, Dublin, Ireland. List of symbols d bolt or dowel diameter (mm, in.) e end distance (mm, in.) E i modulus of elasticity (MPa, psi) E T tangent modulus (MPa, psi) G ij shear modulus (MPa, psi) G T target modulus in shear k shape parameter for Weibull distribution k b stiffness of the bolt member interaction k i spring stiffness k m stiffness of the main member k s stiffness of the side members l thickness of wood member (mm, in.) p probability of failure V model volume V i specimen volume V* reference volume (m 3, in. 3 ) X, Y, Z principal axes of orthotropy x, y, z material and global coordinate system θ angle ( ) σ tension or shear stresses σ +i, σ i yield stress in tension or compression (MPa, psi) σ ij shear stress in material coordinate system (MPa, psi) σ* reference stress for Weibull distribution (MPa, psi)