a three-dimensional model for bolted connections in wooda three-dimensional model for bolted...

A three-dimensional model for bolted connectionsin wood

D.M. Moses and H.G.L. Prion

Abstract: Recent criticism of the bolted connection requirements in the Canadian wood design code CSA StandardO86 indicates that the code lacks consideration of the different modes of failure, particularly as they relate to multiple-bolt connections. A finite element model is proposed to predict load–displacement behaviour, stress distributions, ulti-mate strength, and mode of failure in single- and multiple-bolt connections. The three-dimensional (3-D) model usesanisotropic plasticity for the wood member and elastoplasticity for the bolt. The Weibull weakest link theory is used topredict failure at given levels of probability. Predictions for connection behaviour in Douglas-fir and laminated strandlumber (LSL) correspond to experimentally observed behaviour. The output from the 3-D model is used for a multiple-bolt 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 concep-tion des charpentes en bois dans la norme CSA O86 indiquent que le code ne considère pas les différents modes dedéfaillance, plus particulièrement concernant les raccordements à boulons multiples. Un modèle à éléments finis estsuggéré afin de prédire le déplacement en charge, les distributions de contraintes, la résistance à la rupture et le modede défaillance dans les raccordements à boulon simple et à boulons multiples. Le modèle tridimensionnel utilise laplasticité anisotrope de la membrure de bois et l’élastoplasticité pour le boulon. La théorie du lien le plus faible deWeibull est utilisée pour prédire la défaillance à certains niveaux de probabilité. Les prévisions concernant le comporte-ment 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 unmodèle 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 overlyconservative or inconsistent in its design requirements formultiple-bolt connections, particularly with regard to thelack of adequate consideration of brittle modes of connec-tion failure (Quenneville and Mohammad 2000). The Cana-dian code predicts the ductile behaviour of single-boltconnections such that failure of the connection is governedby crushing of wood and bolt yielding. This is based on thesimple European yield model (EYM) and empirical approxi-mations of the ductile modes of failure (Smith and Foliente

2002). Ductile behaviour is assumed to occur in all connec-tions provided that the code-stipulated minimums for enddistance and edge distance are followed. The strength ofmultiple-bolt connections is based on these single-bolt val-ues multiplied by group-effect reduction factors that accountfor the number of rows of bolts, number of bolts within arow, bolt spacing, and bolt slenderness ratio (l/d, where l isthe thickness of the wood member and d is the bolt diame-ter). These reduction factors were developed empirically fol-lowing the publication of results by Yasumura et al. (1987)and Massé et al. (1988). The group-effect reduction factorsneglect the complex distribution of stresses within a connec-tion, however. Massé et al. and Quenneville and Mohammad(2000) showed that for multiple-bolt connections, connec-tion failure was not ductile in many cases, thereby indicatingthat the current group factors must not only address unevenload distribution between bolts and fabrication defects butalso account for the effect of brittle modes of connectionfailure. Quenneville and Mohammad (2000, 2001) showedthat CSA Standard O86 is not capable of predicting connec-tion strength in many cases for loading parallel to grain andperpendicular to grain because the code neglects the occur-rence of brittle failure modes in multiple-bolt connections.The Canadian code also severely penalizes connections withmultiple rows of bolts by neglecting the positive effect ofadequate spacing between the rows (Mischler and Gehri

Can. J. Civ. Eng. 30: 555–567 (2003) doi: 10.1139/L03-009 © 2003 NRC Canada

555

Received 11 February 2002. Revision accepted 3 February2003. Published on the NRC Research Press Web site athttp://cjce.nrc.ca on 6 June 2003.

D.M. Moses. Equilibrium Consulting Inc., 1585 W. 4thAvenue, Vancouver, BC V6J 1L6, Canada.H.G.L. Prion.1 Department of Civil Engineering, TheUniversity of British Columbia, 2324 Main Mall, Vancouver,BC V6T 1Z4, Canada.

Written discussion of this article is welcomed and will bereceived by the Editor until 31 October 2003.

1Corresponding author (e-mail: [email protected]).

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1999); this too results in overly conservative strength predic-tions. Each of these deficiencies in the code is tied directlyto a poor understanding of the stress distributions in boltedconnections. Clearly, the design and behaviour of boltedconnections must be readdressed by moving away from em-pirical studies of laboratory data and placing new emphasison analytical modelling of the stress distributions that leadto failure of bolted connections.

Previous two-dimensional models

Patton-Mallory et al. (1997b) provide an excellent reviewof past modelling work and the significant variables relatedto connection modelling in wood. Among the models aretwo-dimensional (2-D) plane stress finite element analysesfor thin laminates, Johansen’s (1949) yield model, and non-linear beam on foundation models. The variables that mustbe considered include connection geometry, non-uniformstresses throughout wood member thickness, nonlinearorthotropic material properties, bolt bending and yielding,and bolt-hole clearance. Models in two dimensions musthave built-in assumptions to address each of these variables.

Designers of fibre composites have been using 2-D planestress finite element models to predict bolted connectionfailure in brittle composites, such as carbon fibre, for manyyears with varying degrees of success (for example, Changet al. 1982; Rowlands et al. 1982; Marshall et al. 1989). Acontact interface between the bolt and a layered orthotropicmember is used to predict the stress distribution around abolt-loaded hole. The variables considered in these modelsinclude friction between the bolt and member, bolt-holeclearance, multiple-bolt connections, nonlinear compressionproperties, connection geometry, and stacking sequence ofcomposite layers. Chen et al. (1995) used a three-dimensional (3-D) model to show that delamination failureof fibre-reinforced plastic composites with l/d ≤ 1 can occuras a result of interlaminar (out-of-plane) shear stresses. Ithas also been shown that out-of-plane normal stresses are re-sponsible for delamination, leading to failure in composites(Camanho and Matthews 1997), thereby justifying the use ofa 3-D finite element model on connections with small slen-derness ratios.

Similar finite element models have been developed forbolted connections in wood (for example, Wilkinson et al.1981; Hyer et al. 1987; Rahman et al. 1991). One main dif-ference between bolted connections in nonwood compositesand 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 earliermodels assumed uniform stress distribution throughout thewood member. In reality, however, stresses are not uniform,as in the case of a connection where the bolt yields in bend-ing. Again, 3-D models are better suited to this case, but ap-proximations have been made using some 2-D models.

The EYM and similar approximations that are used inCSA Standard O86 assume that both the wood member andsteel dowel behave as ideal plastic materials. More details onthis model can be found in Smith and Foliente (2002), forexample. The difficulty with the EYM and similarembedment-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 andmaterial factors. To account for some of these variables,many embedment tests are conducted for different bolt sizesand for parallel-to-grain and perpendicular-to-grain loading.Since the model cannot account for stress concentrationsthat lead to brittle failure, the EYM (and, hence, CSA Stan-dard O86) strength predictions are restricted to ductile be-haviour. However, brittle modes of failure can occur inmultiple-bolt connections, in relatively thin members, or inconnections with insufficient end distance, edge distance,and (or) bolt spacing.

Jorissen (1998) attempted to account for brittle fractureusing the EYM by calculating stress distributions along po-tentially critical load paths within the wood member. The av-erage stresses for tension perpendicular to grain and forshear stresses were compared with those from a fracture me-chanics model to predict ultimate strength. Jorissen found,however, that perpendicular-to-grain tension stresses wereunderestimated and, to allow crack initiation to be detectedby the fracture theory, added an assumed peak stress perpen-dicular to grain to the bolt-hole location. This assumptionlimits the robustness of the model. In addition, this modelworks only for connections with small slenderness ratioswhere, it is assumed, the stresses are uniform throughout thewood member thickness.

Two-dimensional finite element beam on foundation mod-els using nonlinear behaviour have been developed to predictductile load–displacement behaviour of a single-bolt connec-tion. Variables such as end restraint, bolt pretensioning, andfriction have been included. One model has been developedthat predicts ultimate load due to wood splitting by calcu-lating perpendicular-to-grain tension forces (see Werner(1993), as noted in Patton-Mallory et al. (1997b)). Beam onfoundation models are growing incrementally closer to solv-ing the complete behaviour of a bolted connection, but theyhave never entirely considered all variables. For example, al-though tension perpendicular-to-grain stresses are includedin one model, the other stresses that can lead to brittle fail-ure are neglected. For this reason, a 3-D model is more ro-bust.

Previous three-dimensional models

A number of 3-D finite element models for bolted con-nections exist. Guan and Rodd (2000) have developed amultiple-dowel moment connection model for hollow dowelconnections using densified wood side plates. The model hasorthotropic elastic material properties for wood, elastoplasticproperties for the steel dowel, and a contact interface be-tween the two materials. As was their intention, Guan andRodd were able to shift the failure of these connections fromthe wood member to the hollow steel connector where duc-tile deformation could occur. Unlike traditional bolted con-nections, this hollow dowel connection eliminates anyrequirement for nonlinear behaviour of wood because doweldeformations dominate and far exceed any deformations inthe wood.

Patton-Mallory et al. (1997a) developed a 3-D finite ele-ment model of a single-bolt connection, shown in Fig. 1.

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Moses and Prion 557

Solid brick elements were used for the steel bolt and thewood, and contact elements were placed between the boltand wood. Using symmetry, only one quarter of the geome-try was modelled. The geometry of the model could be mod-ified to accommodate changes in end distance, edgedistance, member thickness, bolt diameter, hole clearance,and end restraint. To isolate the behaviour of the connector,side members were not included in the model. Nonlinearelasticity (i.e., reversible nonlinear stress–strain with no per-manent deformation) was assumed for compression parallelto grain and shear. Linear elastic properties were assumedfor all other stress–strain relationships, including all normalstresses in tension. The bolt was modelled as elastoplastic.

The nonlinear material model is crucial to the develop-ment of accurate load–displacement behaviour for connec-tions in wood. Crushing in compression results in nonlinearstress–strain behaviour in the three orthotropic material di-rections. Patton-Mallory et al. (1997a) assumed nonlinearityin the critical direction, parallel to loading, and found excel-lent correspondence with experimental load–displacementcurves for a range of end distances and slenderness ratios inDouglas-fir. This essentially elastic model did not, however,account for the energy dissipation associated with non-recoverable deformations due to crushing, thereby ignoringthe conservation of energy assumptions made in the develop-ment of the elastic theory. As a result, limitations existed inthe nonlinear material model and were remedied by addingfictitious nonlinearity in the shear stress–strain behaviour intwo directions to prevent overprediction of these stresses. Itwas also shown that the elastic stiffness matrix would de-velop negative terms on the main diagonal if nonlinear be-haviour was applied to the other two orthotropic materialdirections in compression (Patton-Mallory et al. 1997a).This material model can also result in poor estimates of thestate of stress and failure strength.

Failure prediction

Studies of the stress field in this 3-D model provided evi-

dence of stress concentrations in the vicinity of the bolt hole(Patton-Mallory et al. 1998a, 1998b). Tension stress perpen-dicular to grain and shear stresses in the wood member adja-cent to the bolt were reviewed; however, predictions ofultimate strength due to brittle failure were not performed.Although several failure criteria exist for wood and ortho-tropic materials, most are difficult to apply to 3-D stressfields.

Fracture mechanics models for mixed-stress crack propa-gation exist; however, there is some question over the accu-rate determination of the fracture toughness constants forwood (Fonselius and Riipola 1992). Smith and Hu (1994)considered a fracture model for a single-bolt connectionloaded perpendicular to grain with a small slenderness ratio(i.e., stress uniform throughout member thickness). Thismodel addresses only one particular state of stress and con-nection configuration. As mentioned earlier, Jorissen (1998)used fracture models with the EYM with limited application.In practice, Eurocode 5 (CEN 1995) uses a fracture mechan-ics approach to predict member strength in connectionsloaded at an angle to grain. There is consensus in Europethat fracture mechanics is the preferred approach to predict-ing strength because of perpendicular-to-grain splitting;however, improvements to the governing prediction equa-tions 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 byTsai and Wu (1971), relate stresses in different directions us-ing polynomial equations. Interaction coefficients for themore complex equations are not well defined and are diffi-cult to determine experimentally, particularly for wood prod-ucts in 2-D applications (Clouston 1995). Patton-Mallory etal. (1998a, 1998b) analysed the results of their 3-D finite el-ement model using the maximum stress and Tsai–Wu crite-ria. Because of the difficulty of determining the interactioncoefficients, they limited their analysis to a qualitative study.

The size-effect (weakest link) failure criterion postulatesthat for brittle materials larger specimens are more likely to

Fig. 1. (a) Connection test setup (based on Patton-Mallory 1996). (b) Finite element model geometry shown upside-down.



fail at lower stresses because of the increased probability ofa flaw in the specimen with a larger volume (Weibull 1939).Barrett (1974) showed this to be the case for tension stressperpendicular to grain in Douglas-fir, and CSA StandardO86 uses this as the basis for timber strength according towork by, for example, Madsen and Buchanan (1986). It canbe shown that failure will occur when

[1] σ σk

V

kV V∫ >d * *

where tension or shear stresses, σ, are integrated throughoutthe model volume, V, and compared with a reference stress,σ*, at a given probability of failure. The critical values fortension and shear stresses were determined from earlier testsalong with the shape parameter, exponent k, according to thetwo-parameter Weibull probability distribution.2 The benefitsof this failure criterion are as follows: (i) non-uniformstresses can be analysed, while the highest concentrationsare amplified by the shape parameter; (ii) failure can be pre-dicted for a given probability; (iii) material variability iscaptured; and (iv) the location of failure and mode of failurecan be isolated.

Proposed material model

For the proposed model, wood is assumed to be a homo-geneous continuum where material properties are directionalbut averaged throughout the member volume. Althoughwood does not exhibit the same microstructural plastic be-haviour as that associated with metals, macroscopicallywood exhibits plastic behaviour. In compression, wood has alinear elastic stress–strain curve and will unload on the samepath. On loading, once stresses exceed a critical level, per-manent deformation occurs along with a drop in modulus.On unloading, the stress–strain curve follows the initial elas-tic modulus. This was shown to be the case for laminatedstrand lumber (LSL) (Moses et al. 2003) and is similar forsolid wood. This behaviour occurs in each of the orthogonalmaterial directions, with different moduli and “yield”stresses in each direction. The anisotropic plasticity modelwas, therefore, chosen to model this behaviour.

The anisotropic plasticity material model has been used topredict failure of nonwood composites using 2-D finite ele-ment models (Vaziri et al. 1991). In the current application,the anisotropic plasticity model was applied in three dimen-sions using the TB,ANISO option in ANSYS® v.5.3(Swanson Analysis Systems Inc. 1996). Unlike the nonlinearelastic models, this material model does not require modifi-cation of the elastic stiffness matrix. Instead, it accounts forpermanent deformation and energy dissipation in three or-thogonal planes. Details on this model can be found in Mo-ses (2000), Hill (1947), Valliappan et al. (1976), and Shihand Lee (1978). Bilinear stress–strain curves are assumedfor tension and compression in each material direction. Ayield stress and tangent modulus are required for each curve.As a consequence, 18 additional constants are required inaddition to the 9 normally needed for orthotropic elastic ma-

terials. These constants are readily determined from standardmaterials tests, however (Moses et al. 2003).

Brittle modes of failure are predicted according to thesize-effect eq. [1]. Foschi and Longworth (1975) used thistechnique to determine the strength of timber rivets inDouglas-fir, and this formed the basis of the design require-ments in CSA Standard O86. The maximum stress criterionis assumed, meaning that at each load increment, the threetension stresses and three shear stresses are independentlycompared against critical values. It is assumed that little orno interaction between these stresses exists. The evaluationof eq. [1] was carried out using a short user-programmablesubroutine in ANSYS® v.5.3 (Moses 2000).

Results for Douglas-fir

The 3-D finite element model, shown in Fig. 1, providedthe starting point for the current study in which stresses areanalysed, load–displacement and ultimate strength are pre-dicted, and the results are used to predict the behaviour ofmultiple-bolt connections.

Based on the known behaviour of Douglas-fir from the lit-erature (for example, Patton-Mallory 1996; Kollmann andCote 1968), constants were chosen as listed in Table 1 forthe anisotropic plasticity model. These constants can be de-termined from standard uniaxial and shear tests on smallwood specimens, as described in Moses et al. (2003). Thenine experimental single-bolt connection groups listed in Ta-ble 2 from Patton-Mallory (1996) were not originally ana-lysed to predict failure. They are reanalysed here using boththe original nonlinear elastic model and the proposedanisotropic plasticity model. Failure was predicted for bothmaterial models using eq. [1] with a 50% probability of fail-ure (i.e., average). Load–displacement behaviour was found

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558 Can. J. Civ. Eng. Vol. 30, 2003

Uniaxialconstant

Value(MPa)

Shearconstant

Value(MPa)

Ex 827 Gxy 276

Ey 13 780 Gyz 276

Ez 827 Gxz 28

ETx 4.0 GTxy 4.6

ETy 140.0 GTyz 4.6

ETz 4.00 GTxz 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+x* a 3.2 σ±xy* b 17.0

Note: Ei, modulus of elasticity; ET, tangent modulus; Gij,shear modulus; GT, target modulus in shear; σ+i and σ–i,yield stress in tension or compression, respectively; σij,shear stress in material coordinate system.

aTension perpendicular to the main strand axis (Xdirection) (V = 1, p = 0.5).

bShear in the plane of the panel (XY) (V = 1, p = 0.5).

Table 1. Anisotropic plasticity material propertiesfor Douglas-fir.

2 The theoretical three-parameter Weibull distribution is not necessary for material strengths because the third “location” parameter is as-sumed to be zero, i.e., physically, material strength cannot be less than zero (Barrett 1974).



to be very similar to the behaviour of the experiments be-cause a nonlinear stress–strain model in compression in thedirection of loading, coupled with elastoplastic steel proper-ties, will provide good estimates of load–displacement. Amodest improvement over the nonlinear elastic model wasachieved for predictions of ultimate strength and displace-ment using anisotropic plasticity. Table 2 shows that theanisotropic plasticity model predicts failure loads that arecloser to and more consistent with those observed in experi-ments. Brittle failure was predicted to occur with small slen-derness ratios, l/d, whereas ductile behaviour was predictedin all other cases and was consistent with experiments.Whereas the model of Patton-Mallory was not capable ofpredicting strength, the proposed model predicts strengthand load–displacement behaviour to failure.

The improved behaviour using anisotropic plasticity(rather than nonlinear elasticity) is explained by the im-proved prediction of stresses in the other material directions.The perpendicular-to-grain stresses, shown in Fig. 2 with thesteel dowel removed, were overpredicted by the nonlinearelastic model in areas under the steel dowel at ultimate load.A zone of high tension stress develops at the point of contactbetween the dowel and wood (noting that the hole is largerthan the dowel diameter), with peak stresses predicted by thenonlinear model to be more than four times greater thanthose predicted using the anisotropic plasticity model. Thenonlinear elastic model also predicted stresses in the com-pression zone to the side of the tension region to be morethan one and a half times greater than the stresses predictedusing the plasticity model. Stress overprediction leads to

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Geometry Experimental results (Patton-Mallory 1996)aNonlinear elastic model(p = 0.5)

Anisotropic plasticity model(p = 0.5)

l/d e/d Edge/dLoad at 1.0 mm(kN)

Ultimate load(kN)

Ultimatedisplacement(mm)

Ultimateload(kN)


Ultimate load(kN)


2 4 1.5 13.6 13.6 1.3 10.0 0.5 11.4 1.02 7 1.5 13.6 15.0 1.8 10.1 0.5 >11.7 >1.02 10 1.5 13.6 14.5 1.8 10.3 0.5 >11.9 >1.05 4 1.5 14.2 >17.6 >3.3 11.5 0.5 >12.3 >1.05 7 1.5 15.6 >19.1 >3.3 11.6 0.5 >12.3 >1.05 10 1.5 15.6 >17.6 >3.3 11.6 0.5 >12.4 >1.07 4 1.5 14.7 >17.0 >3.3 11.8 0.8 >12.1 >1.07 7 1.5 15.8 >17.4 >3.3 12.0 0.8 >12.2 >1.07 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 predictionsindicates the analysis was stopped prior to the prediction of brittle failure.

aAverage of 10 specimens.

Table 2. Single 12 mm diameter dowel connections in Douglas-fir.

Fig. 2. Tension stress concentrations perpendicular to grain (i.e., perpendicular to the direction of loading) at failure in Douglas-fir forl/d = 5 and e/d = 4: (a) proposed plasticity model at 12.3 kN, and (b) earlier nonlinear elastic model at 11.5 kN.



premature failure predictions. Also note that the plasticitymodel 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 forlimit states design using a fifth percentile strength based onthe material properties of Douglas-fir.

Results for laminated strand lumber

Laminated strand lumber (LSL) is a structural compositelumber made of wood strands up to 30 cm long, approxi-mately 2.5 cm wide, and roughly 0.94 mm thick. Laminatedstrand lumber is made in large panels up to 75 mm thick andthen cut into standard lumber sizes. Strand orientation in theplane of the panel can be controlled to increase axial andbending stiffness and strength. Limitations in the manufac-turing process result in a significant percentage of cross-aligned strands, although most are oriented in one direction.The cross-aligned strands reduce stiffness and strength in thedirection parallel to the strands but increase stiffness andstrength in the orthogonal direction in the plane of thestrands, a potential benefit for the performance of connec-tions. It is, therefore, possible to control the behaviour ofbolted connections in LSL by controlling the strand orienta-tion in a manner similar to the construction of compositesusing man-made fibres.

Five different stacking sequences of strands were studiedas shown in Fig. 3 (panel layup types A–E), and each con-tains a different percentage of oriented strands per unitthickness as follows: (A) fully oriented (100%); (B) fullyrandom (0%); (C) surfaces oriented, core random (66%);(D) surfaces random, core oriented (33%); and (E) eight ori-ented layers aligned at angles 0° and ±45° (50%). Resultsfrom single-bolt connection tests are listed in Table 3. Endand edge distances were chosen to ensure that both brittleand ductile behaviour could be observed.

The material properties used for the anisotropic plasticitymodel are listed in Table 4. The tension and shear resultswere obtained from material tests on a variety of specimensizes to illustrate the validity of size effect and the Weibullprobability distribution, particularly in tension (see Mosesand Prion 2002 for details). The continuum approach to ma-terial properties was found to be valid for LSL as it was withsolid wood. Panel layup types C, D, and E were modelledusing layers of panel layup types A and B in the same 3-Dconnection model: this is equivalent to the global materialproperties of the entire composite panel (Moses et al. 2003).

One analysis was performed for each connection configu-ration and compared with the average results listed in Ta-ble 3. Three sample plots of experimental load–displacementcurves are shown in Fig. 4 for specimens in the fully ori-ented, type A material loaded parallel to the main strandaxis. These plots are shown here to illustrate the differencein ductility among specimen groups. Specimens in Fig. 4ahad a low slenderness ratio (l/d = 2) with end distance 2d,and all failed suddenly. Though not shown, the same speci-mens with end distance 4d were all ductile and showed nobrittle behaviour. In contrast, the specimen groups inFigs. 4b and 4c had a high slenderness ratio (l/d = 4) andfollowed very similar load–displacement paths to one an-other. Specimens with small end distance 2d (Fig. 4b), how-

ever, failed earlier than those with end distance 4d (Fig. 4c).The lower failure load is associated with the geometric ef-fect of small end distance and the mode of failure for thisstacking sequence. Note, however, that Fig. 4b indicates atransition between brittle and ductile behaviour. Geometryand stacking sequence were found to influence the averageultimate loads, displacements, and modes of failure for eachof the test groups listed in Table 3. In particular, the transi-tion from brittle to ductile behaviour is apparent. In addition,load–displacement curves from the finite element analysisare superimposed on the experimental curves shown inFig. 4, and these too show the transition from brittle to duc-tile behaviour. The predicted curves end at the lesser of(i) the point of predicted failure based on a probability offailure p = 0.5 (i.e., average values) or (ii) 2.5 mm (the pointat which the analysis was stopped).

Ultimate loads were always predicted conservatively to bewithin 50–84% of the experimental averages for all materialtypes, configurations, and loading orientations. The reasonfor the conservative estimates is that failure of the entireconnection was assumed to occur at the first instance thatthe governing stresses (tension or shear) reached capacity.Thus, if delamination (i.e., tension in the Z direction) failurewas predicted to start in the specimen at a particular load,then the analysis was stopped: in reality, the specimen wouldcontinue to carry more load until it would either fail underthe initial failure condition or fail in an entirely differentmode that could become critical at a later stage. In addition,the two-parameter Weibull distribution is known to predictfailure conservatively (Holmberg 1995).

Ultimate displacements were found to be within 17–88%of the experimental averages. Ductile behaviour was pre-dicted to occur when the load–displacement curve exhibitednonlinearity and when the failure criterion was not satisfiedup to the 2.5 mm displacement level. The large discrepancyin predicted ultimate displacements occurs because the anal-ysis is terminated once the ultimate load is reached: in real-ity, the load may drop off while displacements continue.

The predicted modes of failure from stresses matched theexperimental observations, as indicated in Table 3. A sample

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560 Can. J. Civ. Eng. Vol. 30, 2003

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.



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Moses and Prion 561

GeometryUltimate load(kN)

Ultimatedisplacement(mm) Failure mode

Typea d (mm) e/d Edge/d l/d Expt. Predicted Expt. Predicted Expt.b Predictedc

A-Pa 9.5 2 1.5 4 8.0 6.4 2.8 1.3 B ZA-Pa 9.5 3 1.5 4 >11.1 >7.3 >7.1 >2.5 D DA-Pa 9.5 4 1.0 4 >10.7 >7.3 >5.8 >2.5 D DA-Pa 9.5 4 1.5 4 >10.4 >7.3 >9.9 >2.5 D DA-Pa 13.0 2 1.5 3 12.0 12.2 2.0 1.0 B ZA-Pa 13.0 3 1.0 3 >19.0 >13.3 >8.4 >1.8 D DA-Pa 13.0 4 1.5 3 >19.4 >13.3 >6.6 >1.8 D DA-Pa 19.0 2 1.5 2 19.8 16.6 1.3 1.0 B X, ZA-Pa 19.0 3 1.5 2 33.0 23.6 2.3 1.5 B ZA-Pa 19.0 4 1.0 2 >27.5 17.1 >1.5 1.0 —d ZA-Pa 19.0 4 1.5 2 >32.3 26.2 >2.8 1.5 —d ZA-Pe 9.5 2 3.0 4 6.3 5.0 2.0 1.3 B XA-Pe 9.5 4 3.0 4 7.0 5.1 3.3 1.3 B XA-Pe 19.0 2 3.0 2 15.5 7.5 2.3 1.0 B XA-Pe 19.0 4 3.0 2 15.2 7.7 2.0 1.0 B XA-AN 9.5 2 3.0 4 8.6 6.4 3.3 1.8 B XA-AN 9.5 4 3.0 4 >10.7 >6.5 >9.9 >1.8 D DA-AN 19.0 2 3.0 2 18.1 9.1 1.8 0.8 B XA-AN 19.0 4 3.0 2 27.3 14.8 6.1 1.0 B XB-Pa 9.5 2 1.5 4 9.8 6.1 6.1 1.5 B, D ZB-Pa 9.5 3 1.5 4 >10.1 >6.2 >9.4 >1.5 D DB-Pa 9.5 4 1.0 4 5.8 5.3 1.0 1.0 B ZB-Pa 9.5 4 1.5 4 >10.0 6.2 >8.4 1.8 D ZB-Pa 13.0 2 1.5 3 14.3 11.1 3.6 1.0 B ZB-Pa 13.0 3 1.0 3 >17.9 12.0 >8.1 1.3 B, D ZB-Pa 13.0 4 1.5 3 >17.0 12.0 >6.9 1.3 D ZB-Pa 19.0 2 1.5 2 24.4 16.3 6.1 1.0 B ZB-Pa 19.0 3 1.5 2 31.4 17.4 4.3 1.3 B ZB-Pa 19.0 4 1.0 2 14.9 10.3 1.3 0.8 B ZB-Pa 19.0 4 1.5 2 30.9 17.4 6.4 1.3 B ZC-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 DC-Pa 9.5 4 1.0 4 9.6 >7.1 6.6 >2.5 B, D DC-Pa 9.5 4 1.5 4 >11.0 >7.0 >9.9 >2.5 D DC-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 DC-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°, RC-Pa 19.0 4 1.0 2 30.0 14.7 2.3 0.8 B Z, RC-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, RD-Pa 9.5 3 1.5 4 >10.7 >5.5 >9.9 >1.0 D DD-Pa 9.5 4 1.0 4 8.2 5.5 3.6 1.0 D Z, RD-Pa 9.5 4 1.5 4 >10.7 >6.3 >11.9 >1.8 D DD-Pa 13.0 2 1.5 3 13.8 11.8 4.1 1.0 B Z, RD-Pa 13.0 4 1.5 3 >20.6 12.6 >11.7 1.3 B, D Z, RD-Pa 19.0 2 1.5 2 27.0 17.8 2.0 1.0 B Z, RD-Pa 19.0 3 1.5 2 34.9 20.6 5.6 1.0 B, D Z, RD-Pa 19.0 4 1.0 2 24.7 12.5 2.0 0.8 B Z, RD-Pa 19.0 4 1.5 2 32.6 20.7 7.1 1.0 B Z, RD-Pe 9.5 2 3.0 4 >10.4 >6.0 >9.1 >1.3 D DD-Pe 9.5 4 3.0 4 >11.0 >6.2 >13.7 >1.3 D D

Table 3. Single-dowel connection geometry, experimental results, and predicted results.



stress plot at ultimate load, shown in Fig. 5, shows thatstresses are not uniform throughout the thickness. The modelwas found to be well behaved for predicting the failuremodes in all loading orientations. For panel layup types C,D, and E the model was able to detect the layer in whichfailure initiated. Shear stress concentrations in Fig. 6 showthe distinct differences between fully oriented specimensand eight-layer specimens. Figure 6a has a more or less uni-form distribution throughout its thickness, whereas Fig. 6bhas stress concentrations at the interfaces of the layers.

Multiple-bolt connections

In most practical designs, multiple-bolt connections arenecessary. Rather than developing cumbersome 3-D finite el-ement models of many bolted connections, and given thehighly localized effects of stress concentrations around boltholes, a simplified one-dimensional (1-D) model is proposedfor multiple-bolt connections. This model requires outputfrom the 3-D model.

Isyumov (1967) proposed a 1-D multiple-spring model,shown in Fig. 7, to analyse bolted connections. The springsrepresent the main and side members and the interaction ofeach bolt with wood. This model simulates the load redistri-bution among the bolts as load is applied. The interactionstiffness can be nonlinear to capture the ductile behaviour ofa single-bolt connection until failure. Jorissen (1998) usedthis model to predict brittle failure using fracture mechanicsand assumed stress distributions based on the load level ineach spring. Tan and Smith (1999) used a similar modelwith reasonable success using fitted load–slip curves forsingle-bolt connections. A different approach, which isbased on the relationship between the load level and theWeibull weakest link prediction, will be used here.

The load–displacement curves developed using the finiteelement model for single-dowel connections have distinct

behaviours that depend on, for example, edge distance andslenderness ratio. These curves were found to be eitherroughly linear or nonlinear (depending on the situation), andthis behaviour can be incorporated into the spring modelthrough the wood–dowel interaction stiffness, kb, shown inFig. 7. The load–displacement curves were found to be verysimilar if only end distance was changed; the only differencebetween these curves, as shown in Fig. 8 for LSL, was theend point (ultimate load) as a result of brittle failure. Thus,for this particular configuration, the spring stiffness, kb, isthe same for both, regardless of end distance. The point offailure, on the other hand, must be predicted using the rela-tionship between stress intensity and load level.

At each load level prior to failure, the volume integral onthe left side of eq. [1] was calculated for each of the connec-tions listed in Table 3. In Fig. 9, the relationship betweenvolume integral and load level is shown for one connectiongeometry for each of the three brittle stresses that werefound to typically govern in these specimens, i.e., tensionperpendicular to the main strand axis (X direction), tensionperpendicular to the panel surface (delamination in the Z di-rection), and shear in the plane of the panel (XY). In general,a logarithmic relationship appears to exist, though this rela-tionship could be further refined. These lines were fitted us-ing linear regression, as were a corresponding set of curvesfor 19 mm dowels with end distance 4d.

Finite element software was used to create a spring modelsimilar to that in Fig. 7 for each of the four cases shown inFig. 10. For each case, a single central LSL member and two6.4 mm steel side plates with four dowels were analysed.The stiffness constants km and ks, corresponding to the mainwood member and the steel side plates, respectively, werealso included. The end distance and spacing between dowelswere fixed at 4d.

The single-bolt 3-D finite element model was used to de-termine kb and the volume integral – load relationship for

© 2003 NRC Canada

562 Can. J. Civ. Eng. Vol. 30, 2003

GeometryUltimate load(kN)

Ultimatedisplacement(mm) Failure mode

Typea d (mm) e/d Edge/d l/d Expt. Predicted Expt. Predicted Expt.b Predictedc

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 DE-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 DE-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 DE-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

aA–E, panel layup types; AN, specimens cut at 45° to main strand axis; Pa, specimens cut parallel to mainstrand axis; Pe, specimens cut perpendicular to main strand axis.

bB, brittle fracture; D, no fracture and ductile.cX 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°.dNo failure achieved due to grip limitations.

Table 3 (concluded).



cases (a) and (b), and a new 3-D finite element model of tworows with one dowel per row (to allow for possible stress in-teraction between rows) was used to determine these rela-tionships for cases (c) and (d). These relationships couldthen be used in the 1-D spring model by assuming independ-ence between connector groups. The load was applied to thespring model in increments. At each load step, the load ineach connector group was determined according to the rela-tionships determined from the 3-D models. No interaction instresses between neighbouring dowels in a row was as-sumed; only the redistribution in load (simulated by thespring model) was used to determine the current state ofstress. The volume integral for each critical stress was deter-mined for each connector group. Then, the volume integralsfor each critical stress were summed for all connectors toprovide a measure of the total stress state in the entire LSLmember and to check for failure using eq. [1].

The predicted load–displacement behaviour of these fourconnections is shown in Fig. 11. The curve for case (a) ex-

hibits the same basic shape as that of a single dowel withl/d = 4 prior to brittle failure. However, the single dowelwith this slenderness ratio and end distance was not found tohave brittle failure either experimentally or by 3-D modelpredictions. Shear stresses were the cause of this brittle fail-ure. As a result, the ultimate load per connector for the four-dowel connection is only 94% of the single-dowel connec-tion strength.

Case (b) load–displacement was found to be linear as a re-sult of the linear behaviour observed and predicted for thesingle-dowel connections with l/d = 2 with the same end dis-tance. Brittle tension perpendicular-to-grain failure gov-erned, resulting in the average load per connector of only76% of the load expected for a single-dowel connection.Non-uniform load distribution between dowels resulted inthe end dowel carrying the highest load and led to the lowerefficiency of this connection.

Case (c) has two rows of closely spaced dowels. The 3-Dmodel predicted shear failure with load per connector of the

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Moses and Prion 563

(A) Material properties for uniaxial behaviour of LSL

Uniaxialconstant

Type A fullyoriented panels(MPa)

Type B randomlyoriented panels(MPa)

Ex 655 5516

Ey 11 700 5516

Ez 90 103

ETx 31 57

ETy 345 57

ETz 25 23

σ–x 6.6 16.0

σ–y 24 16

σ–z 5.6 9.0

Tensionvalues

Yield A σ+i

(MPa)Ultimate A σi*(V = 16.4, p = 0.5)

Yield B σ+i

(MPa)Ultimate B σi*(V = 16.4, p = 0.5)

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

Shearconstant

Type A fullyoriented panels(MPa)

Type B randomlyoriented panels(MPa)

Gxy 1379 2068

Gyz 421 345

Gxz 179 345

GTxy 3.4 3.4

GTyz 3.4 3.4

GTxz 3.4 3.4

σ±xy 34 55

σ±yz 55 55

σ±xz 55 55

Note: Poisson’s ratios not shown.

Table 4. Material properties for uniaxial and shear behaviour of LSL with anisotropic plas-ticity model.



two-dowel connection at only 73% of the load expected fora single-dowel connection. Using the spring model for thetwo 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 connectionefficiency (i.e., 100% efficient or independence betweenconnectors). Using the spring model, the efficiency droppedto 68%; however, this is greater than the 53% efficiency forcase (c). This corresponds with experimental findings byMischler and Gehri (1999). In both of these eight-dowelconnections, it was found that the end connectors carriedroughly twice as much load as each of the others.

The reduction in efficiency of multiple-dowel connectionsis pronounced because of a combination of unequal load dis-tribution and the logarithmic relationship between load andvolume integral: at higher load levels, the end connectorswere found to carry a much greater amount of load, whilethe stress intensity increased exponentially. We note that al-though some single-bolt connections were ductile, the sameconnector-units became brittle in a multiple-bolt connection.This corresponds to experimental observations by Massé etal. (1988) and Quenneville and Mohammad (2000).

Conclusions

In Canada, only the design of timber rivet connections isbased on stress and failure analysis using the finite elementmethod (Foschi and Longworth 1975). The analysis providesdesigners of timber rivet connections with a sense of thegoverning mode of behaviour for a connection. Given the ac-ceptance of the timber rivet model, why then has the designcommunity only recently started to address the behaviour ofbolted connections with computer models? It is likely thatincreased computing speed and the development of sophisti-cated material models now make such analyses less prohibi-tive.

Although stress analysis of bolted connections can be per-formed using a variety of analytical models, a 3-D finite ele-ment 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 simu-late the embedment test. It provides load–displacement be-haviour and is well suited to evaluate the uneven stressdistributions that lead to failure in bolted connections.

These models and procedures describe the stress field, ul-timate strength, and mode of failure in single- and multiple-

© 2003 NRC Canada

564 Can. J. Civ. Eng. Vol. 30, 2003

Fig. 4. Load–displacement curves for type A LSL loaded parallelto 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, σx, in type A LSL atpeak load (7.3 kN) for e/d = 4, l/d = 4, edge distance 1.5d, d =9.5 mm, and loading parallel to grain.



bolt connections in solid wood and in wood composites.They provide a rational approach to bolted connection de-sign and, unlike the current European yield model designpractice with code-stipulated group factors, the proposedstress-based models can be used to explain many of the phe-nomena described in earlier experimental studies by others.

In practice, these models can be used for parametric stud-ies to determine the transition between brittle and ductilefailure as a result of changes in geometry. As with anymodel, the material properties and failure criterion can un-doubtedly be fine-tuned; however, the concept and proceduredescribed herein can be used, with engineering judgement,to develop new design standards for bolted connections inwood.

© 2003 NRC Canada

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 = 19 mm:(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 Isyumov1967). kb, stiffness of the bolt–member interaction; km, stiffnessof the main member; ks, stiffness of the side members.

Fig. 8. Load–displacement curves for e/d = 2, 3, and 4 and d =9.5 mm.

Fig. 9. Stress volume integral variation in LSL for e/d = 4, l/d =4, edge distance 1.5d, and d = 9.5 mm.



Acknowledgements

Financial support for this project was provided by theScience Council of British Columbia, Trus Joist – AWeyerhaeuser Business, and Forest Renewal British Colum-bia.

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Fig. 10. Multiple-dowel connections used in 1-D spring modelstudy. Central LSL member shown only.

Fig. 11. Multiple-dowel connection load–displacement curves(see Fig. 10).



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List of symbols

d bolt or dowel diameter (mm, in.)e end distance (mm, in.)

Ei modulus of elasticity (MPa, psi)ET tangent modulus (MPa, psi)Gij shear modulus (MPa, psi)GT target modulus in shear

k shape parameter for Weibull distributionkb stiffness of the bolt–member interactionki spring stiffness

km stiffness of the main memberks stiffness of the side membersl thickness of wood member (mm, in.)p probability of failureV model volumeVi specimen volume

V* reference volume (m3, in.3)X, Y, Z principal axes of orthotropyx, 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)σij shear stress in material coordinate system (MPa, psi)σ* reference stress for Weibull distribution (MPa, psi)



a three-dimensional model for bolted connections in wooda three-dimensional model for bolted...

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