International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Impact of growth characteristics on the fracture perpendicular to the grain of timber Jockwer, Robert; Serrano, Erik; Gustafsson, Per-Johan; Steiger, René Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Impact of Growth Characteristics on the Fracture Perpendicular tothe Grain of TimberRobert JockwerPostdoctoral Fellow, ETH Zurich, Institute of Structural Engineering, Zurich, SwitzerlandErik SerranoProfessor, Lund University, Division of Structural Mechanics, Lund, SwedenPer-Johan GustafssonProfessor, Lund University, Division of Structural Mechanics, Lund, SwedenRené SteigerSenior Researcher, Empa, Swiss Federal Laboratories for Materials Science andTechnology, Dübendorf, SwitzerlandABSTRACT: The natural material wood is commonly graded with regard to the parallel to the grainstrength and stiffness properties and taking into account different growth characteristics such as knotsand grain deviations. In this paper the impact of knots and grain deviations on the fracture perpendicularto the grain of timber is analysed by means of numerical models. The results are used for the calibrationof an analytical model. With this model it is possible to evaluate the impact of growth characteristicson the perpendicular to the grain fracture and compare the results with test data from literature. Theevaluation shows that certain growth characteristics increase the strength perpendicular to the grain. Thisis in contrast with current grading procedures, where such growth characteristics are considered as beingstrength decreasing. The results are compared with a model for the description of the effects of growthcharacteristics on the distribution characteristics of the strength perpendicular to the grain. This strongestlink model can be used to describe phenomena with a parallel system of failure events.1. INTRODUCTIONAlthough wood is a natural material exhibiting vari-ous inhomogeneities, fracture of wood is most oftenstudied on small clear specimens and modeled ne-glecting these inhomogeneities. However, for theprediction of the structural behavior of full scaleglulam members the influence of growth character-istics, like e.g. knots, grain deviations and crackshas to be accounted for.The failure of boards loaded in tension paral-lel to grain is often initiated by local growth char-acteristics. For a safe and reliable use of timberin load-carrying structures it is necessary to gradetimber according to certain criteria. These criteriaare commonly chosen with respect to the bending(or tensile) strength of timber. For other strengthproperties like tension perpendicular to the grain orshear no adequate grading criteria have been spec-ified. However, it is well known that for tensionperpendicular to grain the stressed volume has animportant impact on the effective strength and thatwood checks and other characteristics initiate fail-ure. This effect is often described by means of theWeibull weakest link theory (Weibull, 1939).Cracks in wood normally propagate parallel tothe grain due to the low strength and fracture en-ergy of wood perpendicular to the grain. In thevicinity of knots the wood fibers deviate from theglobal grain direction. This leads to an increase ineffective resistance against crack growth. Thereforeknots and grain deviations along the crack path canlead to an increase of the load-carrying capacity of112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 1: Shear stresses at failure in the reduced cross-section of beams with variable notch height in 3-pointbending, separated into samples with and without knots(Larsen and Riberholt, 1972)allα τu,mean (CoV ) τu,0.05 τu,0.10[−] [N/mm2] (%) [N/mm2] [N/mm2]1 4.38 (3.1) 4.02 4.120.75 2.93 (29.6) 1.70 1.900.5 2.19 (26.5) 1.36 1.510.25 2.00 (38.0) 1.02 1.17with knots1 - - -0.75 3.63 (30.6) 1.78 2.040.5 2.41 (20.9) 1.55 1.710.25 2.77 (31.6) 1.47 1.70without knots1 - - -0.75 2.72 (24.1) 1.75 1.920.5 2.12 (27.9) 1.32 1.450.25 1.78 (31.8) 1.02 1.15e.g. end-notched beams (Jockwer, 2014).Early experimental research on the impact ofknots on the fracture behaviour of timber is reportedin (Larsen and Riberholt, 1972). 200 end-notchedsolid timber beams of the quality grade "ungraded"with varying height of the reduced cross-sectionα · h were tested in 3-point bending. Fracture ofthe notch corner was the failure mode in the testsand the results are summarized in Tab. 1. Thefailure load of the notched beams with knots wasfound to be higher both on mean and 5th– and 10th–percentile levels.Riberholt et al. (1991) tested a large series ofend-notched beams in order to study the influ-ence of various geometrical parameters on the load-carrying capacity. A crack retarding effect of knotswas observed and specimens with knots showedhigher load-carrying capacities. In addition to full-size specimens also tests on small clear specimenswere carried out. One specimen contained a knotalong the crack surface and showed a considerablehigher fracture energy.Similar impacts of knots on the load-carrying be-havior of end-notched beams were detected in thetests reported in (Möhler and Mistler, 1978). A re-duction of load-carrying capacity was observed forbeams with checks along the crack path.Jockwer (2014) analyzed impacts on the vari-ation of load-carrying capacity of end-notchedbeams. In this study it was shown that the largevariation of load-carrying capacities can not be ex-plained only by the variation of the elastic materialproperties and fracture energy in mode 1 (openingmode). The additional variation in the test resultscan be represented by including a model uncer-tainty with a considerably high coefficient of vari-ation CoV ≈ 23%. The variation of this model un-certainty can be justified by the presence of knotsalong the crack path. In (Jockwer, 2014) it is de-scribed how the crack retarding effect of knots wasstudied in tests by means of optical measurementsystems.This paper aims at investigating and quantifyingthe impact of different growth characteristics on thefracture behaviour of timber. In numerical modelsthe impact of the shape of the crack path is anal-ysed. Analytical models are used to study the im-pact of varying fracture energy along the crack path.The results of these studies are used to evaluate thedistribution characteristics of the load-carrying ca-pacity of specimens loaded (locally) in tension per-pendicular to the grain.2. GRADING OF TIMBER AND MATERIAL PROP-ERTIES2.1. Growth characteristicsThe mechanical properties of solid timber mainlydepend on the physical and structural characteris-tics of wood (Forest Products Laboratory, 2010).Key physical parameters are the wood density andthe moisture content (MC). In sound, non-decayedwood, the structural characteristics with the highestimpact on the fracture behavior and the mechanicalproperties are:• Amount and size of knots and knot clusters• Cross grain• Orientation and width of the annual rings (thelatter as a visible indicator for the density)• Presence / absence / distance from pith (as aresult of the cutting pattern).212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 2: Distribution parameters of material properties being of relevance for the fracture of glulam equivalent tostrength class GL24h at MC = 12% according to EN 338 (2009), JCSS (2001) and Jockwer (2014)Parameter Unit Symbol Mean 5th perc. CoV PDFMOE ‖ to the grain[N/mm2]E0,mean 11’500 9’600 13 % lognormalMOE ⊥ to the grain[N/mm2]E90,mean 300 250 13 % lognormalShear modulus[N/mm2]Gv,mean 650 540 13 % lognormalFracture energy Mode 1 (clear wood) [N/mm] Gf,I,mean 0.3 0.218 20 % lognormalFracture energy Mode 2 (clear wood) [N/mm] Gf,II,mean 1.15 0.695 30 % lognormalIn order to reach the intended mechanical prop-erties of glued-laminated products the raw mate-rial has to be strength graded. Selected mechani-cal properties of the resulting glulam of the strengthclass GL24h common in Europe are summarized inTab. 2 (EN 14080 (2013); JCSS (2001)).2.2. Knot clusters and clear wood sectionsThe natural structure of the spruce wood makes adistinction between clear wood and knot sectionspossible. Fink et al. (2013) used a constant lengthof the knot sections of 150 mm for the descriptionof the structure of the boards although the length ofthe knot sections in reality is varying. It is sug-gested by (Fink et al., 2013) to model the clearwood section as being Gamma-distributed with anexpected length of dmean = 530 mm and a standarddeviation of σ = 250 mm.2.3. Cross grainIn a previous investigation reported in (Oscarssonet al., 2014) 450 glulam laminations were scannedfor surface fibre directions. Here, the same data setwas used to quantify the cross grain. The median ofthe nominal grain direction on the edges of the lam-inations (for all laminations) was used as a measurein order not to have the data corrupted by the localgrain deviation close to knots. A median deviationfrom perfectly aligned grain in the range of up to2 degrees was found to be quite common and theextreme 10% fractile values of the median includeddeviations of approximately 4-6 degrees.2.4. Fracture related material parametersThe fracture energy of mode 1 is commonly deter-mined by testing single edge notched beam (SENB)specimens as specified in a Draft Standard of CIB-W18 by Larsen and Gustafsson (1990), also known0 1 2 3 4 5 6 7 8050100150200250300350Deflection [mm]Load[N]  Specimens without knotSpecimen with knotFigure 1: Force-displacment behaviour of a SENBspecimen without and with knots (Jockwer, 2014).as the Nordtest method (Nordtest, 1993). Param-eters for PDFs fitted to the fracture energy valuesof individual data are summarised in Tab. 2. Thecorrelation between density and Gf,I is low for theobserved range of densities being of relevance forstructural applications. No general trend of thefracture energy with regard to the spatial distribu-tion in grain direction can be found. However, thereis a considerable impact of the presence of knots inthe fracture plane as can be seen in Fig. 1. This im-pact can be explained by the resulting larger frac-ture surface due to grain deviations and the dow-eling effect of the knot interfering the separationof the specimen and allowing further load transferalong the fracture plane. Such extended studies donot exist for the mode 2 fracture energy Gf,II. How-ever, a similar impact of growth characteristics canbe assumed.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015rcrack0 5 10 15 20 25 3000.20.40.60.811.21.41.61.82G f,I(r)/Gf,I(0)[-]r [mm]Figure 2: Grain deviation in the region of a knot andimpact of the amplitude of grain deviation on the frac-ture energy Gf,I.3. IMPACT OF GROWTH CHARACTERISTICS ONTHE FRACTURE OF WOODThe impact of knots and grain deviations and ofcross grain on the fracture energy of a SENB andthe crack propagation load of a notched beam, re-spectively, is analyzed by means of numerical mod-els.3.1. Impact of grain deviationsThere are various impacts of knots on the fracturebehaviour of timber. One of them is the deviation ofgrain direction in the vicinity of knots which leadsto an increase of the fracture surface. In Fig. 2 anexample of the deviation of the crack is illustratedas it might occur in the vicinity of a knot. Theimpact of the deviation on the load-deflection be-haviour can be analysed using SENB specimens.The numerical study was performed by means ofthe software ABAQUS applying enriched element(xFEM) as described in (Qiu et al., 2014). The re-sults of the study are summarized in Fig. 2: Theincrease in fracture energy depends on the size andslope of the grain deviation.90° 0° α PPγ  PPhαhßh x0 50 100 150 200 250 300 350 400 450 500050100150200250300350400450500550600650700750800Crack length along grain [mm]Criticalload[N]  γ = -10◦γ = -8◦γ = -6◦γ = -4◦γ = -2◦γ = 0◦γ = 2◦γ = 4◦γ = 6◦γ = 8◦γ = 10◦Figure 3: Model of a beam with cross grain and devel-opment of the critical load causing crack propagation.An additional impact of knots on the fracture en-ergy is caused by the their reinforcing effect on thecrack. The reinforcing effect can be assumed to de-pend on the knot size within one knot cluster. Intests on SENB which included knots in the crackplane a relative increase of up to factor 10 can beobserved (Fig. 1).3.2. Impact of cross grainThe global grain direction of glulam beams is com-monly oriented parallel to the beam axis. Hence, acrack will develop along the beam axis, leading toa separation of the notched beam in an upper and alower part.In case of an inclination of the grain direction,not only the directions of the orthotropic mate-rial properties are different but also the remain-ing cross-section of the lower and the upper beamchange during crack growth.In Fig. 3 an example of a global inclination ofthe grain direction of a notched beam by γ = 10◦is illustrated. The numerical study was performedby means of a MATLAB based FE model using thecompliance method for the calculation of energy re-412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015lease rate and crack propagation load. The ultimateload and also the failure behavior changes consid-erably already for small inclinations in the order of±10◦ as shown in Fig. 3.4. LOAD-CARRYING CAPACITY OF A NOTCHEDBEAM4.1. Notched beam modelGustafsson (1988) proposed an analytical model forthe estimation of the load-carrying capacity of end-notched beams. The strength equation is set upby balancing energies during crack growth initiatedin an end-notched beam. The energy release rateis calculated by derivation of beam deflection as afunction of crack length. The analysis delivers theload-carrying capacity at a given notch length βh.Growth characteristics along the crack path canbe accounted for in the model by assigning vari-ations of the values of fracture energy to the re-spective position along the crack path βh+ x. Thenotched beam shows a brittle failure behavior dueto the strong decrease of the crack propagation loadwith increasing crack length. This leads to a dimin-ishing impact of the variations of fracture energyon the load-carrying capacity with increasing cracklength.Additional lamellas can be inserted in the modelin order to simulate the interaction between theselamellas and in order to model weakest link effectsduring failure of a glulam beam.4.2. Monte Carlo simulationsMonte Carlo simulations were performed using thematerial properties as listed in Tab. 2. Withinthe clear wood sections the fracture energy wasmodeled as lognormally distributed with Gf,mean =0.3 N/mm and CoV = 20%. The variations in stiff-ness and fracture energy in the clear wood only ledto a minor variation of the load-carrying capaci-ties (dash-dotted line "Reference Model" in Fig. 4).Hence, the model has to account for additional vari-ations caused by knots and growth characteristicsalong the crack path.For that reason the crack path was divided intodifferent segments representing the clear wood andknot sections. The length and distribution of thesesections was chosen in a first estimate as explained0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91(3.75GvGf )0.5 [ N/mm3/2 ]Cumulativedensity[-]  Test resultsReference ModellFitted ModellFigure 4: Comparison of the cumulative density dis-tribution of test results with the models accounting fordifferent growth characteristics.in Section 2.2. Within the knot sections the dow-eling effect of the knot or the curving of the crackpath due to grain deviation was modelled by vary-ing the fracture energy.The impact of the knot sections on the load-carrying capacity, expressed by the fracture me-chanics strength parameter(3.75GvG f)0.5, is illus-trated in Fig. 4 and can be described as follows:• Distance between knot sections: The dis-tance between the knot sections has a strongimpact on the distribution of load-carrying ca-pacities. For smaller distances the mean valueof the load-carrying capacity increases. Thevariation of the distances between knot sec-tions has only a minor impact on the distribu-tion of load-carrying capacities.• Disturbance of the crack path: Within knotsections the effective fracture energy is highercompared to clear wood sections due to graindeviations and the reinforcing effect of theknots. A knot factor Fknot is introduced to de-scribe the relative increase of fracture energyin the knot section. This factor has a strong im-pact on the load-carrying capacity. The best fitbetween the analytical model and test resultsfrom literature (Jockwer, 2014) is achieved forFknot,mean = 2.0 with CoV = 40%.The fitted analytical model as illustrated in Fig.4 gives best agreement with the test results when512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015(a)(b)Figure 5: Illustration of the weakest link model (a) andof the strongest link model (b).reducing the load-carrying capacity to 80% of thereference values. This reduction is in line with thestudies by e.g. Franke (2008) and Jockwer (2014).The general trend is, that the presence of knot sec-tions leads to an increase of the load-carrying ca-pacity of notched beams. This effect is in contrastto the current procedures where only the weakeningeffect of the knot sections on the bending strengthis accounted for.5. STRONGEST LINK MODEL5.1. BackgroundAs discussed in Section 2 it is difficult to specifygrading criteria to guarantee for strength propertiesother than bending and tension parallel to the grain,like e.g. tensile strength perpendicular to grain andshear strength. For tension perpendicular to grain itis, however, well known that the size of the stressedvolume is important and also that failure often isinitiated at checks and growth characteristics. Themore or less random distribution and the numberand size of these weak spots causes the volume ef-fect. The probability of failure in tension perpen-dicular to grain of wood is often described using the2-parameter Weibull distribution (Weibull, 1939):p f (σ) = P(R < σ) = 1− e−(σfc)m(1)where σ is the stress in a unit volume of the materialand fc and m are material parameters which definemagnitude and scatter in strength. This distributiontogether with the Weibull weakest link theory givesthe strength distribution as a serial system of fail-ure events like e.g. a linear chain containing n unitvolumes (Fig. 5a) asp f (σ) = 1− e−n(σfc)m(2)A very convenient feature of this extreme value dis-tribution is that its CoV equals the one of the unitvolume strength distribution. A simple generaliza-tion of Eq. 2 makes it applicable to arbitrary vol-umes of material in which the stress may be non-homogenous, but still required to be finite.The Weibull weakest link theory is, however,not applicable to structural elements with notches,cracks or other shaping that reveals a stress singu-larity since the theory for such situations in gen-eral predicts either zero strength or no crack prop-agation, no matter the magnitude of load (Gustafs-son and Enquist, 1988). For end-notched beams theWeibull weakest link theory is contradicted by testresults (Larsen and Riberholt, 1972) as discussedin Section 1. The higher strength observed may in-stead be described by a strongest link concept.5.2. Strongest link modelA strongest link model with a sequential systemof failure events is proposed in (Gustafsson, 2014)and can be illustrated e.g. by the resistance thata zipper gives towards being opened as illustratedin Fig. 5(b): the zipper link providing the high-est resistance is decisive. Such a model can beapplied in cases where global failure is governedby crack propagation along a crack path of certainlength, and more generally where failure of two ormore structural elements, or points, precedes globalstructural failure. If, e.g., the strength distributionof a single link can be described by Eq. 1, then thestrongest link strength distribution of a zipper withn links isp f (σ) =(1− e−(σfc)m)n(3)The strongest link model results for more links in ahigher strength and in a smaller CoV . Moreover, forheterogeneous materials with a given mean link ormaterial strength, an increased structural strength ispredicted.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20150 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91(3.75GvG)0.5 [N/mm3/2]p f[-]  Test resultsn =20, ∆x = 50 mm, m =3.5n =8, ∆x = 125 mm, m =3.5n =1, ∆x = 1000 mm, m =3.5(a)0 5 10 15 20 25 30 35 40 45 5000.10.20.30.40.50.60.70.80.91(3.75GvG)0.5 [N/mm3/2]p f[-]  Test resultsn =8, ∆x = 125 mm, m =2n =8, ∆x = 125 mm, m =3.5n =8, ∆x = 125 mm, m =5(b)Figure 6: Comparison of test results on notched beams (Jockwer, 2014) and numerical predictions applying thestrongest link model in Eq. 5 with (a) variation of n number of links, each link representing the crack propagationlength ∆x, and with (b) variation of the model parameter m.In crack propagation analysis the link strength fccan be interpreted as the fracture toughness Kc ofthe material determined experimentally for constantstress intensity, K, for a crack propagation of length∆x. The parameter m is then a measure of the scat-ter in Kc. σ in the ratio σ/ fc represents the stressintensity K. From conventional fracture mechan-ics analysis the function K = K (P,x) can be deter-mined, P being the external load and x the lengthof the crack. If K is constant along the crack paththen:p f (P) =(1− e−(PK1Kc)m)n(4)K1 represents the value of K when P= 1. The appli-cation of Eq. 4 to strength analysis of end-notchedbeams can be done by representing the fracturetoughness and the stress intensity in terms of thecritical energy release rate Gc and the energy re-lease rate G, respectively, i.e. Kc = (GvGc)0.5 andPK1 = (GvG)0.5. If G varies along the crack propa-gation length L, thenp f (P) =n∏i=11− e−(√G(P,xi)Gc)m (5)where n= L/∆x and xi =(i−0.5)∆x, with ∆x beingthe reference length for the material parameters Gcand m.5.3. DiscussionThe application of the strongest link model (Eq. 5)and the comparison with test data from literature(Jockwer, 2014) is shown in Fig. 6. The materialproperties in Tab. 2 were used with Gc = G f ,I anda total crack length L = 1000 mm. A good fit ofthe model with the test data is achieved for m = 3.5and n = 8 links. This corresponds to a crack prop-agation length of each link of ∆x = 125 mm. Foran increasing number of links n at a constant to-tal crack length L both 5th– and 50th–percentile val-ues increase (Fig. 6 (a)). The 95th–percentile valuesare affected only marginally by n and ∆x. In con-trast, the model parameter m has a major impact onthe upper tail. With an increase of m the variabil-ity of the results and the 95th–percentile values de-crease. The validity of ∆x as a reference length forother fracture mechanic problems has to be evalu-ated more extensively.6. SUMMARYGrowth characteristics have an important impact onthe strength and stiffness properties of timber. Inthis paper their impact on the fracture perpendicularto grain of timber was studied by means of differ-ent models. Changes in fracture energy and crackpropagation load due to growth characteristics wereevaluated by means of numerical models. The re-sults serve as reference for an analytical model in712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015which the crack path is separated into knot and clearwood sections. In this model the estimated strengthof the notched beam increases with increased oc-currence of knot sections. This statistical effect isdescribed by a strongest link model representing aparallel system of failure events. The behavior ofthe strongest link model is discussed in a compari-son with test data of notched beams and a good fitis found.Further applications of the strongest link modelto different situations of tension perpendicular tograin or shear fracture seem possible. Such ap-plications could be e.g. the tension perpendicularto grain strength of large curved or tapered glu-lam beams or the load-carrying capacity of con-nections loaded perpendicular to grain. In additionthe model could help to explain the better fit of aheight instead of a volume based size effect modelwith test data of tensile strength perpendicular tothe grain discussed in Mistler (1998). However,for these further applications of the strongest linkmodel additional calibration is necessary.7. REFERENCESFink, G., Frangi, A., and Kohler, J. (2013). “Modelling thebending strength of glued laminated timber - consideringthe natural growth characteristics of timber.” Proc. of CIB-W18 Meeting 46, Vancouver, BC, Canada. Paper No. CIB-W18/46-12-6.Forest Products Laboratory (2010). Wood handbook: Wood asan engineering material. General Technical Report FPL-GTR-190. U.S. Department of Agriculture, Forest Service,Forest Products Laboratory, Madison, WI, USA.Franke, B. (2008). “Zur Bewertung der Tragfähigkeitvon Trägerausklinkungen in Nadelholz.” Ph.D. thesis,Bauhaus-Universität, Weimar, Germany.Gustafsson, P. J. (1988). “A study of strength of notchedbeams.” Proc. of CIB-W18 Meeting 21, Parksville, Canada.Paper No. CIB-W18/21-10-1.Gustafsson, P. J. (2014). Lecture notes on someprobabilistic strength calculation models. Di-vison of Structural Mechanics, Lund Univer-sity, Lund, Sweden, Report TVSM-7161. URL:http://www.byggmek.lth.se/fileadmin/byggnadsmekanik/publications/tvsm7000/web7161.pdf.Gustafsson, P. J. and Enquist, B. (1988). “Träbalks hållfas-thet vid rätvinklig urtagning.” Report no., Divison of Struc-tural Mechanics, Lund University, Lund, Sweden. ReportTVSM-7042.JCSS (2001). Probabilistic Model Code. Joint Committee onStructural Safety. URL: http://www.jcss.byg.dtu.dk/.Jockwer, R. (2014). “Structural behaviour of glued laminatedtimber beams with unreinforced and reinforced notches.”Ph.D. thesis, Institute of Structural Engineering, ETHZurich, Switzerland.Larsen, H. and Gustafsson, P. J. (1990). “The fracture energyof wood in tension perpendicular to the grain.” Proc. ofCIB-W18 Meeting 23, Lisbon, Portugal. Paper No. CIB-W18/23-19-2.Larsen, H. J. and Riberholt, H. (1972). Forsøg med uklassifi-ceret konstruktionstræ. Number R 31. Danmarks TekniskeHøjskole, Afdelingen for Bærende Konstruktioner.Mistler, H. L. (1998). “Design of glulam beams according toEC 5 with regard to perpendicular-to-grain tensile strength- A comparison with research results.” Holz als Roh - undWerkstoff, 56(1), 51–60.Möhler, K. and Mistler, H.-L. (1978). Untersuchungenüber den Einfluss von Ausklinkungen im Auflagerbereichvon Holzbiegeträgern auf die Tragfestigkeit, Vol. F 1504.Fraunhofer IRB Verlag, Stuttgart, Germany.Nordtest (1993). Wood: Fracture energy in tension perpen-dicular to the grain, Vol. NT Build 422. Nordtest.Oscarsson, J., Serrano, E., Olsson, A., and Enquist, B. (2014).“Identification of weak sections in gulam beams using cal-culated stiffness profiles based on lamination surface scan-ning.” Proceedings of the WCTE 2014, Quebec, Canada.Qiu, L., Zhu, E., and Van De Kuilen, J. (2014). “Model-ing crack propagation in wood by extended finite elementmethod.” European Journal of Wood and Wood Products,72(2), 273–283.Riberholt, H., Enquist, B., Gustafsson, P. J., and Jensen, R. B.(1991). Timber beams notched at the support. Afdelingenfor Baerende Konstruktioner, Lyngby, Denmark.Weibull, W. (1939). “A statistical theory of the strength ofmaterials.” Number 141, Royal Swedish Institute for Engi-neering Research, 45.8

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