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

Extension of data sets for a more reliable prediction of the fire resistance of finger joint connections Fink, Gerhard; Klippel, Michael; Frangi, Andrea 2015-07

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Extension of Data Sets for a more Reliable Prediction of the FireResistance of Finger Joint ConnectionsGerhard FinkResearch Scientist, Empa, Swiss Federal Laboratories for Materail Science andTechnology, Dübendorf, SwitzerlandMichael KlippelPost-Doctoral Fellow, Institute of Structural Engineering, ETH Zurich, SwitzerlandAndrea FrangiProfessor, Institute of Structural Engineering, ETH Zurich, SwitzerlandABSTRACT: Large-scale fire tests are time intensive and expensive. Thus, usually only a limited numberof test results are produced. In order to produce a more reliable description of the parameters of interest,the data set can be extended. This paper shows, as an example, how fire resistance tests investigatingdifferent parameters can be connected and further how censored test results can be considered in theevaluation of the results. The goal of the whole investigation is to extend the information of a limitednumber of fire tests in order to produce a more reliable description of the test results.1. INTRODUCTIONThe influence of adhesives on the fire behaviourof glued structural timber elements was recentlyinvestigated in a comprehensive research project(Klippel, 2014). Within this research project, firetests on finger joint connections have been per-formed investigating different parameters, such asthe adhesive in the finger joint, the width of thecross-section, the load level, the type of fire expo-sure (one and two-dimensional exposure) and thetimber quality. At least two tests were conductedwith the same specimen composition to investigateone parameter.Reliable conclusions about the load-bearing be-haviour of timber members can only be stated if asufficient number of tests is performed, since thematerial properties of timber are subject to largevariation (e.g. Köhler et al., 2007; Fink and Kohler,2011). Conducting large-scale fire resistance testsis very time intensive and expensive and can onlybe performed in certified fire laboratories. Thus,fire test campaigns usually produce only a limitedamount of test results. Therefore, it is sometimeshelpful to extend the data set investigating one pa-rameter (e.g. the fire resistance of finger joint con-nections having a width w = 140 mm) using the re-sults produced by similar tests investigating a dif-ferent specimen set-up (e.g. w = 200 mm). Further,if the failure type in the fire tests showed not theintended failure type, e.g. early failure in the solidwood section occurred and the area of interest (fin-ger joint) was still intact, the test can still be con-sidered as a censored data. Hence, censoring occurswhen the value of a measurement or observation isonly partially known, e.g. the finger joint sustainedat least the strength of the failed solid wood region.On the basis of the fire tests and finite elementheat transfer analysis, Klippel (2014) estimated the“effective” temperature-dependent reduction of thetensile strength for finger-jointed specimens (seeFigure 1). This reduction is described as a simpli-fied bi-linear approach with a breakpoint at 100◦Cin accordance to the bi-linear approach given inEN 1995-1-2 (CEN, 2004) for the timber strengths.It should be noted that the reduction in strength wasdetermined for finger-jointed specimens dependingon the adhesive in the finger joint based on themean fire resistance of tests with the same adhesive112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015on 140 mm wide specimens. Thus, only a limitednumber of fire tests led to the outcome of this inves-tigation. Further, fire tests were neglected, in whichan early failure occurred because of a knot defect.In addition, the fire tests in which failure occurredin the finger joint due to exceeding the timber ten-sile strength were neglected in this evaluation, sincethe adhesive had still sufficient strength. However,by neglecting test results valuable information wasnot included in the evaluation, which could put theresults on a more reliable foundation.As previously described, Klippel (2014) deter-mined the fire resistance for 80, 140 and 200 mmwide finger-jointed specimens exposed to standardISO 834-1 (ISO, 1999) fire and loaded in tensiondepending on the simplified bi-linear reduction ofstrength with increasing temperature, as shown inFigure 2. With the relation shown in Figure 2 it ispossible to determine the fire resistance for finger-jointed specimens glued with the same adhesivebut having a different width, e.g. an adhesive rep-resented by a reduction factor kΘ = 0.5 at 100◦Cleads to a fire resistance of 25.2, 53.8 and 87.5 min-utes for 80, 140, and 200 mm wide specimens, re-spectively. Thus, results determined with differentwidths of the specimens and the same adhesive inthe finger joint are connected. However, this con-nection is based on mean values for the fire resis-tance and additionally neglecting some fire test re-sults, as described above.This paper shows how fire resistance tests inves-tigating different parameters can be connected andfurther how censored test results can be consideredin the evaluation of the results. The goal of thewhole investigation is to extend the information ofa limited number of fire tests in order to produce amore reliable description of the test results.2. FIRE TESTSFire tests were performed on the model-scale hor-izontal furnace with dimensions of 1.0 m × 0.8 musing fire exposure according to ISO 834-1 (ISO,1999) at the Swiss Federal Laboratories for Mate-rials Science and Technology (Empa) in Dubendorf/ Switzerland. A detailed description and the re-sults of the fire tests as well as tests at normal tem-perature are presented in the testing report (Klip-0 50 100 150 200 250 300 Temperature [°C] factor  k   [-]QP2 (PUR)P3 (PUR)P4 (PUR)M1 (MUF)P6 (PUR)P7 (PUR)PRF (timber failure)EPI M2 (MUF)M3 (MUF)PVAc UF Non-structuraladhesiveStructural adhesive Early timber failuredue to knot Figure 1: Bi-linear approach to describe the residualstrength under tension for finger-jointed timber lamel-las, depending on the adhesive (according to Klippel(2014)).0 10 20 30 40 50 60 70 80 90Time [min] factor k              [-]Q=100°C  80 mm140 mm200 mmw Width wReduction factor k [-]Q 20 100 3001.0kQ=100°CTemp [°C] (53.8/0.5) (87.5/0.5)(25.2/0.5)40mmFigure 2: Fire resistance of finger-jointed specimensloaded in tension depending on the temperature-dependent strength reduction factor at a temperatureof 100◦C (according to Klippel (2014)).pel and Frangi, 2014). The fire behaviour of finger-jointed timber boards was studied in 49 fire tests in-vestigating 12 different adhesives in the finger jointconnection. The main objective of this campaign212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015was to evaluate the influence of adhesives on thefire resistance of finger-jointed timber boards. Inaddition, the influence of different other parame-ters on the fire resistance was studied, such as thewidth of the specimens, the applied load level, thetype of fire exposure (one and two-dimensional ex-posure) and the timber quality. It should be no-ticed that the specimens designed for this investi-gation described the real behaviour of finger jointsin a fire situation relevant for glued laminated tim-ber beams. Figure 3 shows the fire resistance ofthe specimens constantly loaded with 30% of themean tensile strength obtained in normal tempera-ture tests. Most of the tests were performed with across-section width of w = 140 mm. Further testswere performed with 80 and 200 mm wide speci-mens. Four typical types of failure were observedin the fire tests, as illustrated in Figure 4:(a) Failure along the fingers in the finger joint(b) Failure of the fingers in the finger joint(c) Mixed-type failure(d) Failure in solid wood regionThe specimens which showed a failure along thefingers show most likely an influence of the adhe-sive on the fire resistance. In the case of the otherthree failure types, the fire resistance was mainlylimited by the tensile strength of timber and the ad-hesive still sustained enough strength.3. EXTENSION OF THE DATA SETAs already mentioned, large-scale fire tests are verytime intensive and costly. Thus, usually only a lim-ited number of specimens can be tested and the re-sults are often not sufficient for a reliable estimationof the investigated parameters; e.g. the fire resis-tance. In this paper, the fire resistance of one typeof finger joint connection is estimated (referred asto reference conditions):• Cross-section: w× t = 140×40 mm2• Type of Adhesive: One-componentpolyurethane (1C PUR) (in Figure 3 ab-breviated as P2)• Tensile load1: F = 0.3 ·Fu• Fire exposure: ISO 834-1 (ISO, 1999)1Fu is the load that corresponds to the mean tensilestrength determined in tests performed at normal temperature.0 10 20 30 40 50 60 70 80Fire resistance t [min]80140200Cross-section width w [mm]dbcba) Failure along fingers (all tests without further explanation)b) Failure of fingers (in the tip of the finger joint)c) Mixed type failured) Failure in solid wood region (due to defect, knot, etc.)cd d cbPVAcUFMUF (M3)Solid wood (V)PUR (P2) PUR (P7)PUR (P3)PUR (P6)EPIPRFMUF (M1)MUF (M2)PUR (P4)d dLoad level: 0.3 Fu90140mm  80mmNon-structural adhesives w40 mmFigure 3: Fire resistance as a function of the cross-section width (80, 140 and 200 mm) depending on theadhesive in the finger joint (according to Klippel andFrangi (2014)).(a)Failure along fingers(b)Failure of fingers(c)Mixed type failure(d)Failure in solid woodFinger joint (intact)Knot(led to failure)Figure 4: Types of failure observed in the fire tests (ac-cording to Klippel and Frangi (2014)).The estimation of the fire resistance considers testresults of (a) fire tests with reference conditions,and (b) fire tests with different dimensions.In the present investigation, altogether nine firetests are taken into account, summarized in Ta-ble 1. Thereby, six tests were performed with ref-erence conditions (test no. 1-6) and three tests were312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 1: Compilation of the fire test results used for the present investigation.No. Width One-dim. Type of Fire resistance Fire resistancecharring rate failure measured estimateda[mm] [mm/min] [min] [min]1 140 0.68 Along fingers 47.5 -2 140 0.70 Along fingers 56.5 -3 140 0.71 Along fingers 47.0 -4 140 0.71 Along fingers 53.0 -5 140 0.78 Along fingers 52.5 -6 140 0.68 Mixed-type 52.0 -7 80 0.78 Along fingers 24.5 52.98 200 0.66 Along fingers 66.5 41.49 200 0.68 Along fingers 87.0 52.8a The fire resistance was estimated on the basis of Figure 3. In this table, the estimation wasapplied for specimens with a width unequal to 140 mm. Thus, given is the estimated fireresistance for those specimens if the specimen tested would have had a width of 140 mm.performed on specimens with different dimensions(test no. 7-9).Three different cross-section widths were testedin the fire test campaign of (Klippel and Frangi,2014): w = 80, 140 and 200 mm. Most of the testswere performed with 140 mm wide specimens. Infurther tests, the width was varied whereas all otherparameters (load level, adhesive in the finger joint,type of fire exposure) were kept the same.4. PREDICTION OF THE FIRE RESIS-TANCEIn the following section, measured values of the fireresistance and estimated values of the fire resistanceare used to estimate the fire resistance of fingerjoints (denoted t). At first, the general approach isdemonstrated. Afterwards, the corresponding pro-cedure is illustrated and the different types of infor-mation explained and how the information can beconsidered.The presented approach is illustrated on an ex-ample by using the measured and estimated fire re-sistance summarised in Table 1. In the example,the fire resistance is assumed to be Lognormal dis-tributed:f (t) =1tζ φ( ln(t)−λζ)(1)here, ζ and λ are the parameters of the Lognor-mal distribution and φ is the the standard normaldensity distribution function. The Lognormal dis-tribution seems to be most accurate to describe thedistribution of the fire resistance, since the flat tailwithin the area of small and large values of the fireresistance (due to a rather constant charring rate inall tests, the probability of small and large valuesfor the fire resistance is rather small). Furthermore,the fire resistance cannot be negative same as theLognormal distribution.4.1. MethodThe maximum Likelihood method will be used forthe estimation of the parameters (e.g. Benjamin andCornell, 1970; Faber, 2012). The basic principle ofthe maximum Likelihood method is to find the pa-rameters of the chosen distribution function whichmost likely reflect the data sample. The parametersof the distribution function are estimated by solvingthe optimisation problem given in Eq. (2). Here,L(θ |tˆ) is the Likelihood of the observed data, θrepresents the parameters, and tˆ are the measuredvalues of the fire resistance.L(θ |tˆ) =n∏i=0Li(θ |tˆi) minθ(−L(θ |tˆ))(2)412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20154.2. Direct information – measured values4.2.1. GeneralFor the most general case, where the quantity ofinterest is measured directly, the Likelihood of theobserved data tˆi is equal to the realisation of thedensity function fT (tˆi|θ), according to Eq. (3). Inthe present example, all experimental investigationswith reference conditions and in which a failurealong the fingers was observed (test no. 1-5), aredirect observations of quantity of interest.Li(θ |tˆi) = fT (tˆi|θ) (3)Often, the measured value does not describe thequantity of interest (here the fire resistance of thefinger joint connection), but can be used as cen-sored information. Examples of censored test re-sults for tensile tests under fire exposure are inves-tigations where (a) the failure is outside the investi-gated area, or (b) the test has to be stopped beforefailure (e.g. technical problems). In such cases, thequantity of interest is not known, but the test re-sult can be used as bonded information since thelower limit is known. E.g. the investigated speci-men failed after a certain time in a knot cluster out-side the investigated finger joint. Thus, the fire re-sistance of the knot cluster is known, but not the fireresistance of the finger joint. However, it is obviousthat the fire resistance of the finger joint is at leastthe fire resistance of the failed knot cluster.For censored observations (denoted tˆi,c), theLikelihood can be calculated with the realisation ofthe cumulative distribution function FT (tˆi,c|θ) ac-cording to:Li(θ |tˆi,c) = 1−FT (tˆi,c|θ) (4)The principle of the Likelihood estimation forboth types of measured values is illustrated in Fig-ure 5. The crosses illustrate the measured quantitiesof interest, the circle illustrates the censored infor-mation, the vertical lines and the dark grey area il-lustrate the corresponding Likelihoods.4.2.2. ExampleAt first, the fire resistance and the associatedstrength reduction of the adhesive due to high tem-peratures is estimated using the measured values ofsix fire tests (test no. 1-6). In five tests (tests no.1-5), a failure occurred along the fingers. Thus, thetest results are realisations of the quantity of inter-est and the Likelihood can be calculated accordingto Eq. (3).In fire test no. 6, a mixed type failure was de-tected (see Figure 4c). For this particular specimenthe major part of the failure occurred in the area ofa knot cluster located about 12 cm next to the in-vestigated finger joint. It is obvious that the tensilecapacity of the finger joint was not reached at thetime of failure. As a result the fire resistance of thefinger joint is at least the measured fire resistancetˆi,c = 52 min. The test result is treated as a censoredinformation and the Likelihood can be calculatedaccording to Eq. (4).In timber engineering, censored information isoften considered incorrect. Usually, censored testresults are neglected and thus the investigated pa-rameters are underestimated. The underestimationdepends on the quantity of the censored observa-tion; the larger the censored observation the largerthe underestimation.In this particular case, the mean fire resistance oftests no. 1-5 is t1−5 = 51.3 min, thus below the re-sult of fire test no. 6. It is obvious that neglectingthis test result leads to an underestimation of thefire resistance and thus to an overestimation of thestrength reduction of the adhesive under high tem-perature.Taking into account all six test results, the ex-pected value of the fire resistance is E[t1−6] =51.9 min, thus slightly larger. In the present exam-ple, neglecting of the test result obtained in the firetest no. 6 would not have such a big influence, how-ever, in other cases the results could be significantlyunderestimated.In Table 2, the estimated parameters and the as-sociated fire resistance (expected value and COV)are summarised and in Figure 6, the estimated dis-tribution function is illustrated (dashed line). Thevariability of the fire resistance of the finger joint isCOV = 7.1%. It is commonly known that the vari-ability of the fire resistance of timber specimens isin general smaller than the variability of test resultson timber specimens obtained at normal tempera-512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 201530 40 50 60 7000. resistance t [min]Probabilitydensity  tˆitˆi,ctˆi,eFigure 5: Schematic illustration of the Likelihood esti-mation for different types of information. The verticallines and the grey areas illustrate the correspondingLikelihoods.ture. This can be addressed to the rather constantcharring rate of timber being a very important pa-rameter in the fire design of timber members (Ta-ble 1).4.3. Indirect information – estimated values4.3.1. GeneralIn addition to measured values, estimations of thequantity of interest (denoted tˆi,e) can be used forthe Likelihood estimation. tˆi,e is an estimation forthe quantity of interest and thus concerned to un-certainties.In the presented example tˆi,e is the estimated fireresistance of a 140 mm wide finger-jointed cross-section, using the test result of a similar experimen-tal investigation and the model described before;i.e. tˆi,e is the estimated fire resistance of the testedspecimens if it would be wider (or thinner). It isobvious that estimated values of the fire resistancecan only be used if the performed tests are (a) verysimilar and (b) the model is able to reflect the rela-tion between the measured and the ’interested’ testconditions. The smaller the differences between theperformed tests and the better the model the moreefficient will be the estimation.In the present investigation the following as-sumptions are made: (a) the model estimates thefire resistance with a certain accuracy – the uncer-tainty is assumed to be known and (b) the modelreflects that the variability of the fire resistance isdifferent for different cross-section widths.The ’real’ fire resistance is, of course, not knownand the estimation is associated to uncertainties; themore precise the estimation the smaller the uncer-tainties. A realistic assumption to describe the dis-tribution function of the ’real’ fire resistance mightbe a Normal distribution with mean value tˆi,e andwith standard deviation σe; hereafter the associateddensity function is denoted hT (tˆi,e,σe). As a result,the Likelihood can be determined:Li(θ |tˆi,e) =∫fT (t|θ) ·hT (t|tˆi,e,σe) ·dt (5)In Figure 5, the principle of the Likelihood esti-mation for estimated values is illustrated. The di-amond represents the estimated fire resistance tˆi,eand the light grey area the associated Likelihood,for the interval σe = 2 min.4.3.2. ExampleOne disadvantage of experimental investigationsunder fire exposure is the relatively large costs, inparticular for large-scale and full-scale fire tests.As a result, usually only a very small amount ofspecimens are tested. In the present investigation,six specimens were tested with the same type ofadhesive, the same cross-section dimensions andthe same load and fire exposure. In this section,the data sample is extended by using test resultstested under the same conditions, but different di-mensions. Using the information given in Figure 2,the associated fire resistance for the reference cross-section tˆi,e was estimated.As already mentioned, tˆi,e is an estimation of thefire resistance under reference conditions.Assuming a given σe, the Likelihood can be cal-culated according to Eq. (5). In Table 2, the esti-mated parameters and the associated fire resistance(expected value and COV) are summarised for dif-ferent σe = 1,2 and 5 min. It is obvious that moreprecise estimations of the fire resistance (smallerσe) have a larger influence on the final outcomes.In the presented example, the variability of thefire resistance of the measured values (tests no. 1-6)612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Table 2: Compilation of the estimated fire resistanceFire tests 1-6 1-9 1-9 1-9σe - 1 min 2 min 5 minζ 3.947 3.929 3.931 3.938λ 0.071 0.092 0.085 0.070E[t] 51.92 51.07 51.13 51.46COV[t] 0.071 0.092 0.085 0.070was rather small COV = 7.1%; the difference be-tween the smallest and largest test results was about10 min. In the two tests performed on 200 mmwide specimens, the difference of the fire resis-tance was significantly larger (about 20 min). Al-though the specimens width is different, the resultsclearly indicate the larger variability. Comparingthe test results with the information given in Figure2, it seems that one of the specimens (test no. 8)failed very early tˆ8,e = 41.4 min, whereas the fireresistance observed in the other test (test no. 9) isaround the estimated value tˆ9,e = 52.8 min (see alsoFigure 6).Taking into account all nine test results (the fivemeasured values of the fire resistance tˆi, the cen-sored observation tˆi,c, and the three estimated val-ues of the fire resistance based on similar experi-mental investigations tˆi,e) the expected value of thefire resistance E[t] ≈ 51.1 min with a variabilityCOV≈ 8.5% is determined.It has to be mentioned that the estimation of theabsolute value of the fire resistance based on testswith different dimensions is associated to large un-certainties, which were not quantified within thisstudy. However, the large differences between thefire resistance of the tests on 200 mm wide speci-mens is a strong indicator that the variability will beunderestimated if only the measured values are con-sidered. Using the presented approach it seems tobe reasonable to quantify the variability more cor-rect. In Figure 6 the associated distributions are il-lustrated, for σe = 2 min (solid lines).4.4. Advantages & limitsThe presented approach to extend data sets, by us-ing estimated values of the fire resistance based onsimilar tests and physical models, is rather simpleand can be applied easily. However, for the ap-plication the quality of the estimation, expressedthrough the error σe, has to be known. Often sucha quantitative description of the quality of the esti-mation does not exist. In such cases, as illustratedin the present examples, the uncertainty has to beestimated and thus the outcome is related to uncer-tainties.Another and more common approach to consideradditional information, such as the estimated fireresistance based on test results from similar in-vestigations, is Bayes updating (see e.g. Rackwitz(1983) or specific to timber engineering Fink andKohler (2014)). Using Bayes updating the uncer-tainty of the additional information can be consid-ered by weighting the prior information against newinformation.5. CONCLUSIONSThis paper demonstrated how a limited number oftest results obtained on timber specimens can beused to extend the data set in order to produce amore reliable prediction of the load-bearing capac-ity of the specimens tested. The application usingthe Maximum Likelihood Method was shown on asimple example using fire test results (Klippel andFrangi, 2014).Large-scale fire tests are usually performed on alimited number of specimens since they are timeintensive and expensive. Due to the limited num-ber of test results, reliable conclusions are often notpossible. Further, an accurate prediction of the fireresistance (expected value and variability) is of par-ticular importance, especially if the results are usedas input parameters for probabilistic models; e.g.for probabilistic models to estimate the fire resis-tance of glued laminated timber beams.The paper shows how test results can be consid-ered in the final evaluation although the test did notshow the intended type of failure and further howtest results can be connected although the specimenset-up (e.g. width of the specimen) was different.However, the presented approach can also be usedto consider other test results (e.g. different load ex-posure) as long as reliable analytical or numerical712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 201530 40 50 60 7000. resistance t [min]Probabilitydensity  Test 1−5Test 6Test 7−930 40 50 60 7000. resistance t [min]Cumulativedistribution  Test 1−5Test 6Test 7−9Figure 6: Estimated fire resistance: (left) Probability density, (right) Cumulative distribution. The dashed linerepresents the estimated distribution considering only the measured fire resistance (test no. 1-6). The solid linerepresents the estimated distribution considering the measured and the estimated fire resistance (test no. 1-9).models for the estimation of the quantity of interestunder reference conditions exist.6. REFERENCESBenjamin, J. R. and Cornell, C. A. (1970). Probability,statistics and decisions in civil engineering. Mc Graw- Hill Book Company.CEN (2004). EN 1995-1-2: Eurocode 5: Design of tim-ber structures – Part 1-2: General – Structural fire de-sign. European Committee for Standardization, Brus-sels, Belgium.Faber, M. H. (2012). Statistics and Probability Theory:In Pursuit of Engineering Decision Support, Vol. 18.Springer.Fink, G. and Kohler, J. (2011). “Multiscale Variabilityof Stiffness Properties of Timber Boards.” Applica-tions of statistics and probability in civil engineering: proceedings of the 11th International Conferenceon Applications of Statistics and Probability in CivilEngineering, Zürich, Switzerland, 1-4 August 2011,1369–1376.Fink, G. and Kohler, J. (2014). “Risk based investi-gations of partly failed or damaged timber construc-tions.” Materials and Joints in Timber Structures,Springer, 67–75.ISO (1999). ISO 834-1: Fire-Resistance Tests - Ele-ments of Building Construction - Part 1: General Re-quirements. International Organization for Standard-ization, Geneva, Switzerland.Klippel, M. (2014). “Fire safety of bonded structuraltimber elements.” Ph.D. thesis, Institute of Struc-tural Engineering, ETH Zurich, Switzerland, Instituteof Structural Engineering, ETH Zurich, Switzerland.Thesis No. 21843.Klippel, M. and Frangi, A. (2014). “Fire tests on finger-jointed timber boards.” Report No. 354, Institute ofStructural Engineering, ETH Zurich.Köhler, J., Sørensen, J. D., and Faber, M. H. (2007).“Probabilistic modeling of timber structures.” Struc-tural safety, 29(4), 255–267.Rackwitz, R. (1983). “Predictive distribution of strengthunder control.” Matériaux et Construction, 16(4),259–267.8


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