ESPOO 2003 VTT RESEARCH NOTES Goran Turk & Alpo Ranta-Maunus. Analysis of strength grading of sawn timber based on numerical simulation

Transcription

1 ESPOO 003 VTT RESEARCH NOTES 4 Goran Turk & Alpo Ranta-Maunus Analysis o strength grading o sawn timber based on numerical simulation

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3 VTT TIEDOTTEITA RESEARCH NOTES 4 Analysis o strength grading o sawn timber based on numerical simulation Goran Turk, Alpo Ranta-Maunus VTT Building and Transport

4 ISBN X (sot back ed) ISSN (sot back ed) ISBN (URL: ISSN (URL: Copyright VTT 003 JULKAISIJA UTGIVARE PUBLISHER VTT, Vuorimiehentie 5, PL 000, 0044 VTT puh vaihde (09) 4561, aksi (09) VTT, Bergsmansvägen 5, PB 000, 0044 VTT tel växel (09) 4561, ax (09) VTT Technical Research Centre o Finland, Vuorimiehentie 5, POBox 000, FIN 0044 VTT, Finland phone internat , ax VTT Rakennus- ja yhdyskuntatekniikka, Puumiehenkuja A, PL 06, 0044 VTT puh vaihde (09) 4561, aksi (09) VTT Bygg och transport, Träkarlsgränden A, PB 06, 0044 VTT tel växel (09) 4561, ax (09) VTT Building and Transport, Puumiehenkuja A, POBox 06, FIN 0044 VTT, Finland phone internat , ax Technical editing Marja Kettunen Otamedia Oy, Espoo 003

5 Turk, Goran & Ranta-Maunus, Alpo Analysis o strength grading o sawn timber based on numerical simulation Espoo 003 VTT Tiedotteita Research Notes 4 38 p + app 8 p Keywords timber, construction timber, sawn timber, spruce, materials properties, strength, strength grading, numerical simulation, accurancy Abstract Numerical simulation is used to analyse machine strength grading o sawn timber Starting rom correlations between grade determining properties and measured grading properties, sets o values o properties are generated and machine grading simulated including the determination o settings and application o the settings to another generated sample Results are obtained concerning the statistical eect o sample size to the accuracy o grading and yield to various grades Also the lower tail o strength distribution o various grades was analysed with the conclusion that the highest grade has relatively better strength distribution than the lower grades 3

6 Preace This report is based on numerical calculations made by Dr Goran Turk during his visit at VTT The intention o the publication is to document results o various numerical exercises, also in cases when the applicability o results to practical grading situations is not obvious Due to the limited duration o the stay in Finland, most o the practical applications ollow aterwards and will be published separately The work belongs to the area o COST Action E4, VTT s work is supported by TEKES, Wood Focus Finland Ltd and VTT 4

7 Contents Abstract3 Preace 4 1 Introduction7 Random generation o sample and population8 1 Generating a sample o normally distributed random vector 9 11 Linear transormation9 1 Generating a random vector with a given covariance matrix 9 13 Generating procedure 10 3 Optimum grading11 31 Characteristic value determination1 4 Machine grading 14 5 Numerical examples16 51 Old sample generation16 5 Sample generation based on descriptive statistics0 53 Combined grading 5 54 The inluence o dependent sampling31 6 Applications Eect o sampling size to quality o grading 34 7 Conclusions37 Reerences 38 Appendix Appendix A: Generated grading results 5

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9 1 Introduction Basic problem in understanding and development o strength grading o wood is related to the statistical nature o the strength o timber: there is no method which could predict the strength o a certain piece o wood However, there are known correlations between strength and measurable physical properties The consequence is that thousands o specimens need to be tested to establish settings or grading machines, and ater having done it, there is still an uncertainty about the strength distribution o produced strength graded timber as well as on the yield o timber to dierent grades This report concerns grading o timber according to the orthcoming European standard [1] Based on earlier experiments, correlations between grade determining properties, strength, modulus o elasticity and density, will be assumed, and numerical simulation o the standard grading procedure will be applied to larger populations than can be aorded in testing Numerical procedure is needed because requirements concerning all three grade determining properties has to be ulilled This method is expected to be useul in analysis o issues as ollows: sensitivity o grading result to initial distributions o properties and to the eectiveness o the grading method sensitivity o grading result to sample size and to statistical methods used when settings o machine are determined yield to dierent grades when graded simultaneously to one or more grades orm o lower tail o strength distribution o dierent grades 7

10 8 Random generation o sample and population It is assumed that grade determining parameters: strength, modulus o elasticity E and density ρ, and observed grading parameter model are all normally distributed random variables The characteristics o this random vector = = model E ρ (1) are presented by its mean value vector m and its covariance matrix Σ [ ] ( )( ) [ ] = = = =,,,,,,,,,,,, T E m m m m E m y m y Σ m () The mean value vector as well as covariance matrix are estimates rom a sample o real experimental results n k i y ik, 1,,4, 1,, = = (n is the sample size) by the ollowing equations ( ) ( )( ) 1 ˆ 1 ˆ ˆ = = = = = = n m y m y n m y n y m n k jk ik n k ik n k ik i j i j i i i (3) Two types o sample generation will be used later on: (1) generation o independent random samples, () generation o random samples rom a given inite size population The techniques to generate a inite size population and independent random sample are equal

11 1 Generating a sample o normally distributed random vector The basic idea or generating a sample o dependent normally distributed random variables is to generate a sample o independent normally distributed random variables and then use a linear transormation to obtain a sample o dependent random variables [] 11 Linear transormation The linear transormation o random vector X, which is taken to be a set o independent random variables with zero mean and unit variances, is deined as = H X (4) where H is transormation matrix, which deorms the coordinate system A very important linear transormations are rotations which correspond to orthonormal transormation matrix H The variance-covariance matrix o random vector X is an identity matrix An important property o normally distributed random variables is that the linear transormation o normally distributed random variable is also normally distributed 1 Generating a random vector with a given covariance matrix Lets assume that the random vector has zero mean values In this case the variancecovariance matrix is deined as ollows T [ ] Σ = E, (5) where E [] denotes expected values I we use Eqn (4) in Eqn (5), we obtain a useul relationship T T T T T [ ] = E[ H X X H ] = H E[ X X ] H Σ = E, (6) T Since the variance-covariance matrix o X is identity matrix [ X X ] = I reduces to E, the Eqn (6) Σ T = H H, (7) 9

12 We have to ind such matrix H, or which the Eqn (7) holds Since the variancecovariance matrix is nonsingular symmetric matrix, the problem can easily be solved by Cholesky decomposition (see eg [3]) 13 Generating procedure The generation o normally distributed random vector with known variance-covariance matrix is carried out by the ollowing procedure: 1 Perorm a Cholesky decomposition o variance-covariance matrix Σ o random vector X (determination o H) Generate a set o independent random variables with standardized normal distribution (generation o random vector X) 3 Perorm a transormation which transorm independent variables X into randomvector with known variance-covariance matrix (use o equation = HX) 4 Perorm an additional transormation to obtain a random vector Z with non-zero mean values (use o equation Z = + E[Z]) 10

13 3 Optimum grading When settings o a grading machine are determined according to EN 181, both nondestructive testing by the use o grading machine, and destructive testing or determination o the real grade determining properties are made or a representative sample The irst step in the analysis o results is the determination o the real grade o each specimen based on inormation o destructive testing: bending strength, modulus o elasticity and density This grading is made in such a way that each specimen would be place in as high grade as possible, and is called thereore optimum grading In our case a simultaneous grading to three grades was perormed: C, C and C The modiied requirements or these grades according to pren 181 [1] are shown in Table 1 Table 1 Modiied requirements or grades C, C and C Grade r [N/mm ] E r [N/mm ] ρ r [kg/m 3 ] C C C During the optimum grading determination we would like to grade as many pieces as possible to higher grades, with the ollowing requirements (constraints): 1 the sample (grade) 5% percentile o the strength 0 05 is higher than r, the sample (grade) mean (average) o E is higher than E r, 3 the sample (grade) 5% percentile o the density ρ 0 05 is higher than ρ r, a 4 the sample 5% percentile o the strength 0 05 is higher than r or the sub-sample obtained just by ranking according to the strength, a 5 the sample mean o E is higher than E r or the sub-sample obtained just by grading according to the modulus o elasticity, and a 6 the sample 5% percentile o the density ρ 005 is higher than ρ r or the sub-sample obtained just by ranking according to the density A computer code (written in MATHEMATICA [4]) was prepared or automatic determination o optimum grading In addition to these requirements (constraints) the objective unction S was deined as 11

14 S = 005 r r + E E E r r + ρ 005 ρ ρ r r + a 005 r r + E a E E r r + ρ a 005 ρ ρ r r (8) It is our goal to ind such settings o limits used in optimum grading determination that the unction S reaches its minimum These set o limits gives the optimum grading o the sample or population The optimization procedure itsel utilizes a relatively simple step-by-step bisection-like method, in which the limits or any o grading determining parameters are decreased by a certain step-size i the dierence (eg 0 05 r ) is positive and increased i it is negative The increasing is our times aster than decreasing When all three limits (or, E and ρ) are as low as possible with the conditions ulilled, the step-size is halved and the procedure is repeated This simple method, which emulates the manual determination o optimal grading, gives very accurate limits or the grade determining parameters Thus, the obtained grading is almost certainly the optimal one 31 Characteristic value determination One o the tasks that has to be repeated requently is the determination o characteristic value (eg 5% percentile) rom the sample There are several options o how to deal with this problem One o them is to presume the distribution o the random variable which is represented by the sample We can use either method o moments or maximum likelihood method to determine the characteristic value In case the sample is very small a special consideration should be put into the act the moments themselves are not determined exactly, only the estimates based on the sample are known Alternatively, the characteristic value can be determined by the use o order statistics Eg, let's determine the 5% percentile y k rom the sample y i = 1,, n where n is sample size The sample is sorted so that y i y i+ 1 The sample element y j which satisies the condition = 0, (9) j 05n is the highest among the elements which are below the characteristic value The notation j = x denotes the loor o the number x, ie the greatest integer less than or equal to x The characteristic value is inally determined by the equation y ( 05n j)( y y ) k = y j + j +1 0, (10) j 1

15 which represents the linear interpolation between the two values which embrace the characteristic value I the sample size is lower than 1/005 = 0 the equation (10) can't be used The reliable determination o characteristic value is in that case diicult An approximate ormula can be used y k = 005n y 1 In grading procedure a second method based on order statistics, has been used 13

16 4 Machine grading Basis o machine grading lies on settings which will be given based on comparison o machine readings and real grade determining parameters Here is a procedure introduced or automatic determination o settings according to pren 181- The three limits or grades C, C and C will be set so that the ollowing conditions are ulilled: 1 the sample (grade) 5% percentile o the strength 0 05 is higher than r, the sample (grade) mean (average) o modulus o elasticity E is higher than E r, 3 the sample (grade) 5% percentile o the density ρ 0 05 is higher than ρ r, 4 none o the cells in the global cost matrix which indicate wrongly upgraded is grater than 0, 5 the number o rejected pieces is grater or equal to 05% o the total number o pieces in the sample Similarly as or optimum grading determination, an automatic optimization procedure was developed in machine settings determination too There are several options or the objective unction: one can choose to use a sum o all terms in global cost matrix (GCM) to be the objective unction S = m m i= 1 j= 1 GCM ( i, j), (11) where m is the number o grades used in this particular grading The minimum value o S is sought In this case we obtained relatively low values or yield in higher classes The other two options that were tried have greater yields in higher classes: Objective unction is the sum o all terms in global cost matrix which indicate wrongly downgraded pieces (upper triangular part only) S = m m i= 1 j= i+ 1 GCM ( i, j), (1) The minimum o S is sought 14

17 Objective unction is the weighted number o pieces assigned to a particular grade In our case the ollowing objective unction is used: S = + (13) 9nc + 3nc nc The latter objective unction is used or inal evaluations Since the variance o the assigned values o machine grading parameter model is relatively high and its correlation to the actual grade determining parameter and speciically to the density ρ low, the relationship between machine settings limits and objective unction is not clear Thereore we start rom random choices o settings, and among the cases that ulil the requirements the one with the highest objective unction is chosen From that point the machine settings limits were lowered as much as possible so that the requirements are ulilled and the objective unction is maximized This procedure usually gives relatively high yields in higher grades 15

18 5 Numerical examples 51 Old sample generation Old sample title reers to an earlier paper [5] Here, the generation o samples has been repeated by using techniques described in this publication, whereas the earlier version was made with Excel and using smaller sample sizes and lesser repetitions o generation In both cases the grade determining properties are based on an earlier research on spruce (45 x 150) grown in Finland [6] A random vector o our dependent random variables (, E, ρ, model ) is considered here We assume that random variables E, ρ and model are linear unctions o, which is a normally distributed random variable with mean m and standard deviation E = a + b + X, ρ = c + d +, = + model Z, (14) where a, b, c and d are known deterministic constants, and X, and Z are normally distributed random variables with zero mean ( m X = m = mz = 0 ) and standard deviations X, and Z In this case we can derive the mean values, variances and covariances in terms o nine parameters m,, a, b, c, d, X, and Z mean values: variances: covariances: m m = a + b m, E ρ = c + d m, E b X = +, ρ d + =, model E b = +, =, ρ = d, E ρ = bd, =, model model E = b, ρ = d model Z From these relations is easy to see, that correlation coeicients are 16

19 r r r E ρ Eρ r r r E = model modele modelρ E E ρ = ρ Eρ = = ρ = = = = = model model modele model modelρ model b d b E ρ b d = = = X b X d,, bd d, + Z b + X d + +, + Z + Z, The ollowing values were chosen (or estimated rom experimental data) to be: m = 45, = 114, a = 480, b = 165, X = 10, c = 36, d = 03, = 35, = 15, 85 or 55 Standard deviation o random variable Z ( Z ) depends o the quality o grading The lowest value o 55 corresponds to excellent grading, whereas the highest value o 15 corresponds to poor grading The mean values vector and variance-covariance matrix in the case o poor quality grading ( Z = 15) are Z m = Σ = In the cases o good and excellent grading, only the variance o the ourth random variable changes poor grading var[ model ] = good grading var[ model ] = 0 1 excellent grading var[ model ] = 160 1,, 17

20 The corresponding correlation matrix or all three qualities o grading are: poor grading good grading excellent grading R =, R =, R = Dierent sample size or machine settings were chosen: 50, 500, 750, 1000, 150 and 500 Machine settings determination was repeated 100 times with independent samples The results or mean machine settings parameters are shown in Table Because all generations o values did not allow grading to C, the number o non-zero generations or which a non-zero yield to C was resulted, is given, as also mean values o settings i all zero-yield-to C-cases are omitted

21 Sample size Table Mean machine settings parameters or dierent sample sizes Number o nonzero Poor grading Z = 15 Include all Include only nonzero yield in C yield in C Grades 50 15/ / / / / /100 Mean St dev Mean St dev Mean St dev Mean St dev Mean St dev Mean St dev Sample size Number o nonzero Good grading Z = 8 5 Include all Include only nonzero yield in C yield in C Grades 50 61/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev Several things could be concluded rom the results presented in Table : The number o samples which give non-zero yield in the highest class C is greater i the sample size is higher However, in the case o poor grading the sample size 19

22 1000, 150 and 500 have approx same number o non-zero yield in C samples, and in the case o good grading, the sample size o 750 have almost the same number o non-zero yield in C samples I we take only results or non-zero yield in C, we obtain somehow surprising result: the average limit or C is lower or larger samples, which will result in lower yield in the highest class C This result is due to the act, that in the case o smaller samples there is more chance to have a zero yield in C, but the remaining samples which have a yield in C have better chance to be "good" samples, which will result in a larger (and too optimistic) yield in C Obviously, it is not right to take only those samples with non-zero yield in consideration, since the result is clearly biased The limit or class C is lower i the sample size is larger As a result we may expect higher combined yield in C and C However, the sample sizes larger than 1000 do not contribute a lot to the result The values o grading settings obtained by 100 repetitions were applied on the same population o 000 and pieces The results are shown in Appendix Table 1 With this calculation the yields in individual classes are determined The ratio between 5% and 05% percentile o strength is determined or all three classes In the case there were not enough pieces in a particular grade, 05% percentile has not been determined 5 Sample generation based on descriptive statistics Based on the available data o 589 pieces o spruce with the depth o 150 mm, 1508 pieces o spruce with various depth and 1995 pieces o spruce and pine with variable depth [6], the descriptive statistics were determined It has to be noted, that in descriptive statistics E mach related to bending type o grading machine is used Descriptive statistics may be summarized in the mean vector m and covariance matrix Σ spruce (150 mm depth) m = Σ =

23 spruce (all depths) m = Σ = spruce and pines m = Σ = Correlations between these variables are more evident in correlation matrices R: spruce (150 mm depth) R = spruce (all depths) R = spruce and pine R = I we compare these correlation matrices with correlation matrices used in "Old sample generation" we may observe that correlation between density and modulus o elasticity is higher than one assumed beore, whereas the correlation between density and strength depends on which sample we are observing It is higher i only spruce with 150 mm depth is considered and it is lower i all spruce specimens or all spruce and pine specimens are considered The correlation between strength and modulus o elasticity is similar to the one used in "old sample generation" The correlations between strength, modulus o elasticity, density and model can't be directly compared to the correlation between strength, modulus o elasticity, density and 1

24 E mach However we may observe that the correlation between model and modulus o elasticity as well as model and density were probably underestimated This is particularly true in the case o "poor grading" The latter conclusions should be veriied by computing model rom complete data set and recalculating descriptive statistics In urther analysis only the data or spruce with 150 mm depth and all spruce data are used The covariance matrix in the case o combined spruce and pine data is very similar to the covariance matrix o all spruce data Thereore similar results could be expected Dierent sample size or machine settings were chosen: 15, 50, 500, 750, 1000, 150 and 500 Machine settings determination was repeated 100 times with independent samples The results or mean machine settings parameters are shown in Table 3 Table 3 Mean machine settings or dierent sample sizes Spruce with depth o 150 mm Samle size Number o nonzero Include all Include only nonzero yield in C yield in C Grades 15 61/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev

25 All spruce Samle size Number o nonzero Include all Include only nonzero yield in C yield in C Grades 15 4/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev From Table 3 we can see the importance o separate grading o pieces o 150 mm In that case the nonzero-yield in the highest grade C was 100% or all sample sizes but the smallest one (15 pieces only) The values o machine settings do not dier much or dierent sample size rom 50 to 500 pieces On the other hand i the data rom dierent depth o the pieces were combined lower correlations between grading parameters and machine value results in lower yield in the highest class This is speciically true or smaller sample size Thereore, in this case it is beneiciary to have a larger sample size The values o grading settings obtained by 100 repetitions were applied on the same population o pieces The results are shown in Appendix Table With this calculation the yields in individual classes are determined The ratio between 5% and 05% percentile o strength is determined or all three classes In the case there were not enough pieces in a particular grade, 05% percentile has not been determined The results clearly show the dierence between the eectiveness o machine grading the spruce with the depth o 150 mm and all spruce data 3

26 In the case o 150 mm deep spruce specimens the machine grading was quite successul The yield in C was approx 36% and in C between 50 and 55% (compared to optimum grade yield o 6% and 9% in C and C, respectively) The results slightly improve i the sample size is increased rom 50 to 500 (see Fig 1) However there is an evident improvement in COV o yield which decrease orm above 05 in the case o n = 50 to below 01 in the case o n = 150 (see Fig ) This reduction in COV results in much narrower conidence interval or the cases when the machine grading were determined rom only ew samples Other parameters don't change or dierent sample sizes, eg the ratio is virtually constant or all sample sizes In the case o all spruce data the improvement can be seen in a number o samples that have non-zero yield in C (see Fig 3) Due to low correlation between grade determining properties and grading parameter it may easily happen that the machine settings can't be set so that a yield in C is obtained I the case o smaller samples the possibility or this is even higher as it was clearly shown by this example All other parameters were not eected by the sample size Figure 1 Average yield in C and C or the generated samples o spruce with 150 mm depth 4

27 Figure Coeicient o variation or yield in C and C or the generated samples o spruce with 150 mm depth Figure 3 Number o samples with non-zero yield in C and average yield in C and C or the generated samples o all spruce 53 Combined grading Our inal objective is to improve the machine grading One o the procedures which may improve the machine grading is to include additional indicating property in machine grading Thereore, we assumed that in addition to modulus o elasticity the density o the beams are estimated by the machine Since we don't have any data about the covariances 5

28 between the density evaluated by machine ρ m and others parameters we assumed, that ρ m depends on actual density ρ and can be generated by the ollowing equation ρ m = ρ +W, (15) where W is normally distributed random variable with zero mean and standard deviation ρ W and represent the error in machine evaluation o the density As a result we obtain a new mean vector and covariance matrix in which the moments o a new parameter is added In the case o spruce specimens o various depth, we have spruce (150 mm depth) m = Σ = W Correlations between these variables are more evident in correlation matrices R: R = spruce (all depths) m =

29 Σ = W Correlations between these variables are more evident in correlation matrices R: R = In last equations the ollowing relationships between the moments are used (they can easily be derived) m ρm = m, [ ρ] + var[ W ], cov[ ρm, F ] = cov[ ρ, F ], cov[ ρm, E] = cov[ ρ, E], cov[ ρm, ρ] = var[ ρ], cov[ ρ, ] = cov[ ρ, ] m ρ E mach E mach In this analysis the data or spruce o the depth o 150 mm and all spruce data are used Dierent sample size or machine settings were chosen: 15, 50, 500, 750, 1000, 150 and 500 Machine settings determination was repeated 100 times with independent samples The results or mean machine settings parameters are shown in Table 4 and 5 Optimum grading was obtained by the same procedure as in the previous cases In machine settings determination, besides the limits or E mach a set o limits or ρ m was sought Thus, or each grade two limits are given, a piece is selected i both modulus o elasticity and density equal or exceed the appropriate limits The search procedure is similar as in the previous cases - a combination o random search and inal lowering the limits so that the required conditions are ulilled From the data we may conclude that the additional indicating property in machine grading is beneiciary or the case o all spruce data In this case the number o non-zero yield in the highest class C increases considerably This is speciically true i small sample size is chosen (15 to 500 pieces) (see Fig 4) From the Appendix A Table 4 we can see that yield in C increases rom about 6% in ordinary grading to above 8% in 7

30 combined grading Both values are quite low with respect to the value o optimum grading (544%) Also we can see that the overall yield in grades C and C does not change i the combined grading is used: it is approx 8% This value is relatively close to the optimum grading value (867%) (Appendix Table 3) In the case o spruce specimens with the depth o 150 mm the dierences between ordinary grading and combined grading are negligible Table 4 Mean machine settings parameters or dierent sample sizes (spruce - depth o 150 mm) Combined grading (modulus o elasticity and density) Spruce (150 mm)) Sample Number o Include all size nonzero Machine modulus o yield in C elasticity Machine density Grades 15 67/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev

31 Grading to modulus o elasticity only Spruce (150 mm) Sample Number o Include all size nonzero Machine modulus o yield in C elasticity Machine density Grades 15 60/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev Figure 4 Number o samples with non-zero yield in C or two types o grading 9

32 Table 5 Mean machine settings parameters or dierent sample sizes Combined grading (modulus o elasticity and density) All spruce Sample Number o Include all Size nonzero Machine modulus o yield in C elasticity Machine density Grades 15 13/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev Grading to modulus o elasticity only All spruce Sample Number o Include all Size nonzero Machine modulus o yield in C elasticity Machine density Grades 15 6/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev

33 54 The inluence o dependent sampling In real machine settings determination we will not be able to have a large number o independent samples In act we may have only a relatively large population (eg N = 100) o all available measurements rom which we may draw several samples These samples are obviously not independent since the population rom which we take samples is not ininite In this numerical example we illustrate the eect o dependent sampling, ie sampling rom a inite population First we generate the population o 1000 pieces (N = 100) Then we randomly select samples rom this population In this case it does not make any sense to analyse larger samples, thus only samples o size 15, 50, 500 and 750 have been analysed Table 6 Mean machine settings parameters or dierent sample sizes Spruce with depth o 150 mm Samle size Number o nonzero Independent sampling yield in C Grades 15 61/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev Number o Sampling rom inite population Samle size nonzero yield in C Grades 15 67/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev

34 All spruce Sample size Number o nonzero Independent sampling yield in C Grades 15 4/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev Sample size Number o nonzero Sampling rom inite population yield in C Grades 15 8/100 Mean St dev /100 Mean St dev /100 Mean St dev /100 Mean St dev In the case o spruce with the depth o 150 mm we can see that the eect rom sampling rom the population o 1000 is not very evident We can see a slight reduction in variance which is clearly illustrated by the Fig 5 In the case o all spruce data the eect o sampling rom inite population o 1000 depends rom the population o 1000 pieces itsel For all spruce data the variation is so high and correlation is so low that the variation o the results obtained rom the sample o 1000 is still to high to consider this a representative sample which may act as a population rom which the samples may be drawn In the example shown in Table 6 we can see that the number o non-zero yield in C increased considerably However i we repeated the generation o population o 1000 and than repeat sampling rom that population we might observe decreasing o the same parameter We should conclude that in such cases where the variation is high and correlations low the sampling rom a population o 1000 is not very reliable 3

35 Figure 5 Coeicients o variation o machine settings or the cases o independent sampling and sampling rom the inite population o 1000 pieces 33

36 6 Applications 61 Eect o sampling size to quality o grading The sample size used or determination o settings o grading machine may have eect on the grading result when the machine is used First we analyse how the yield to dierent grades is inluenced by the random actors, and how they can be counteracted by increasing the sample size Based on the analysis o the most homogenous sample, spruce 150 mm, described in Appendix Table, we observe that when sample size is increased, yield to higher grades increases and COV o yield decreases (see Fig 6) Results are summarised in Table 7 indicating that when settings are determined by the use o standard sampling, the machine gives on average 35% yield to C, but it may give on a yield rom 6 to 44% within conidence limits o 95% When our times larger sample is used or determination o settings, we obtain mean yield 37% and conidence limits are rom 3 to 4% Table 7 Conidence intervals or yield to dierent strength classes [%] It is assumed that independent sampling is repeated 4 times sample size yield C yield C yield C ±93 196/sqrt(4) 486± /sqrt(4) 136±79 196/sqrt(4) ±61 196/sqrt(4) 508±97 196/sqrt(4) 110±64 196/sqrt(4) ±54 196/sqrt(4) 544±71 196/sqrt(4) 81±31 196/sqrt(4) ±50 196/sqrt(4) 53±73 196/sqrt(4) 84±48 196/sqrt(4) ±36 196/sqrt(4) 559±45 196/sqrt(4) 70±4 196/sqrt(4) ± 196/sqrt(4) 554±39 196/sqrt(4) 64±16 196/sqrt(4)

37 Figure 6 in C and C - 95 % conidence intervals (4 repetitions o sampling) The ratios between 05% and 5% percentile o the strength are 088, 0753, and 0605 in classes C, C, and C, respectively The average values o this ratio don t dependent on the sample size However, the variance o the ratio is in general lower i larger samples were taken into account (see Fig 7) Figure 7 Ratio between 05% and 5% percentile o the strength Also the orm o strength distribution o dierent grades has been studied Based on data rom 150 mm spruce as above, we have analysed the strength distributions o specimens when graded simultaneously to C, C and C with settings based on mean values o 100 simulations o independent samples We learned that result is quite independent o the sample size, and is illustrated in Figure 8, where the 5% percentile strength o each grade is denoted by 1 and cumulative distribution o the 35

38 relative strength is shown in lower tail area The curves are lognormal distributions which are itted to simulated results in two points: 0005 and 005 COV-parameter o the itted lognormal distribution is indicated in Figure 8, 0 or C and 06 or C We observe that C is qualitatively much better material than lower grades Cumulative probability 0,05 0,045 0,04 0,035 0,03 0,05 0,0 0,015 0,01 0,005 0 C~LN 00 C~LN 031 C~LN 06 0,5 0,6 0,7 0,8 0,9 1 Relative strength Figure 8 Relative strength o timber graded simultaneously to C, C and C 36

39 7 Conclusions It was important to use actual data to determine random vector characteristics (mean and variance-covariance matrix) used in sample generation The data or spruce pieces with the depth o 150 mm showed quite high correlation between grade determining parameters and grading parameters As a result the machine grading proved to be quite successul The correlations between the parameters or all spruce data and combined data or spruce and pine were lower Thereore, the machine grading is less eective Probable reasons may be: (1) the procedure to adjust values to the values valid or 150 mm deep spruce pieces was not adequate or at least not accurate enough, () the values or pieces o dierent size may have been obtained rom dierent laboratories, (3) the pieces may have been taken rom the timber rom dierent region, etc Sampling rom inite size population results in reducing the variances in machine settings, but this reduction is not due to better procedure but due to dependent sampling, which may lead to biased results However, at least or the case o spruce o 150 mm the population o 1000 gives satisactory results Adding additional grading parameter slightly improved the machine grading in the case o spruce o 150 mm (the yield in C increased or approx 1%) It seems that a grading procedure or the data with as low correlations as in the case o all spruce specimens has to be changed, revised, ie improved One o the possibilities is the use o artiicial neural networks 37

40 Reerences 1 pren 181-, Timber structures - Strength graded structural timber with rectangular cross section - Part : Machine Grading - Additional requirements or initial type testing, European Committee or Standardization, Brussels, September 00 Devroy, L, Non-Uniorm Random Variate Generation, Springer-Verlag: New ork, Press, W H, Teukolsky, S A, Vetterling, W T, Flannery, B P, Numerical Recipes in Fortran, The Art o Scientiic Computing, Second Edition, Cambridge University Press, Cambridge (UK), Wolram, S, Mathematica, A System or Doing Mathematics by Computer, Second Edition, Addison-Wesley Publishing Company, Inc, Redwood City (Ca), Ranta-Maunus, A, The eect o machine strength grading on the strength distribution o sawn timber based on numerical simulation COST Action E4 & JCSS Workshop in Zürich, October 10 11, 00 6 Ranta-Maunus A, Fonselius M, Kurkela J, Toratti T, 001, Reliability analysis o timber structures VTT Research Notes 109 Espoo, Finland 10 p + app 3 p 38

41 Appendix A: Generated grading results Table 1:Grading results o populationo 000,settings arebased on100 samples(n sample =50) Good grading quality r =064 All samplesareincluded Optimum grading /36/ /3475/ /3875/ /375/ /55/ Mean 6751/377/ mean COV Good grading quality r =064 Onlynon-zeroyield in Cisincluded Optimumgrading /3475/ /4575/ /38/ /3875/ /55/ Mean 5665/3956/ mean COV Poor grading quality r =036 All samplesareincluded Optimum grading /53/ /435/ /3975/ /4175/ /65/ Mean 9313/381/ mean COV Poor grading quality r =036 Onlynon-zero yield in Cisincluded Optimum grading /3975/ /195/ /445/ /54/ /475/ Mean 6675/47/ mean COV A/1

42 Table 1(cont): Grading results opopulation o000, settings arebased on 100samples(n sample =500) Good gradingquality r =064 Allsamplesareincluded Optimum grading /75/ /3975/ /365/ /48/ /35/ Mean 6607/3659/ mean COV Good gradingquality r =064 Onlynon-zeroyieldinCisincluded Optimum grading /75/ /3975/ /365/ /48/ /35/ Mean 5901/3745/ mean COV Poor grading quality r =036 All samplesareincluded Optimumgrading /335/ /365/ /475/ /45/ /475/ Mean 931/3866/ mean COV Poor grading quality r =036 Only non-zeroyieldincisincluded Optimumgrading /45/ /475/ /35/ /365/ /475/ Mean 6970/4147/ mean COV A/

43 Table 1(cont): Grading results opopulation o000, settings arebased on 100samples(n sample =750) Good gradingquality r =064 All samplesareincluded Optimum grading /39/ /315/ /3675/ /37/ // Mean 64708/36135/ mean COV Good gradingquality r =064 Onlynon-zeroyieldinCisincluded Optimum grading /39/ /315/ /3675/ /4/ /37/ Mean 60649/3667/ mean COV Poor grading quality r =036 All samplesareincluded Optimum grading /485/ /45/ /365/ // /415/ Mean 89555/38045/ mean COV Poor grading quality r =036 Only non-zeroyield in Cisincluded Optimum grading /485/ /365/ /55/ /415/ /415/ Mean 717/4109/ mean COV A/3

44 Table 1(cont): Grading resultsopopulation o000, settings arebased on 100samples (n sample =1000) Good grading quality r =064 All samples are included Optimum grading /355/ /3/ /575/ /345/ /345/ Mean 6437/3515/ mean COV Good grading quality r =064 Onlynon-zero yield in C is included Optimum grading /355/ /3/ /675/ /3175/ /345/ Mean 6147/3547/ mean COV Poor grading quality r =036 All samples are included Optimum grading /5/ /36/ /36/ /335/ /395/ Mean 8707/3756/ mean COV Poor grading quality r =036 Only non-zero yield in C is included Optimum grading /5/ /36/ /36/ /435/ /395/ Mean 7/04/ mean COV A/4