REAL-TIME QUALITY EVALUATION OF STRUCTURAL TIMBER Markus Deublein1, Raimund Mauritz2, Jochen Köhler1 ABSTRACT: Modern grading machines facilitate the integration of the grading process into the industrialized production scheme with its high demand for production rate. Beside the speed the efficiency of the grading machines depends on the machine’s capability to divide the gross supply of ungraded timber into sub-sets of graded timber that fulfil some predefined requirements. The current European standard for machine strength grading of structural timber EN 14081 provides different methods for the control of grading machine settings. These are either machine or output controlled. However, recent insights of timber manufactures, engineers and researchers indicate that both of these methods bear some shortcomings. These are mainly connected to the machine control strategy which is considered as too static and not capable to take into account the large variations of the origin, sawing pattern and growth condition and therefore the large time-, supplier or sawing pattern dependent variability of the properties of the ungraded material that is present in a real market situation of a large or medium sized timber manufacturer. On the other hand, the output control strategy is regarded as cumbersome and expensive since a large amount of timber material has to be assessed by destructive tests, and not all of the available information from the tests is used for the recalibration of the grading machine. In the present paper the basis for enhancing the efficiency of grading machines is shown. The concept builds on utilizing the entire available information continuously gathered by the grading machine during its operational process. Focus is set on the discussion how deviations in the material properties of the ungraded timber can be qualified as being critical and different approaches for automated detection of critical quality shifts in the input material of the timber manufacturing processes are presented. The influence of the sample sizes that are used for the control of the characteristic values in each strength grade is discussed. The outcomes of this paper may serve as a basic approach on how real-time information of a grading device can be utilized for industrial applications aiming at efficient and reliable machine grading strategies for structural timber. KEYWORDS: structural timber, quality control, machine grading, control limits
1 INTRODUCTION 123 In general, when considering the control of manufacturing processes, the problem is one of maintaining a production process in such a state that the output from the process conforms to given design requirements (e. g. characteristic values for strength, stiffness and density). During the operation phase the process will be subject to changes which cause the quality of the output to deteriorate. And also the input material quality of the process may already be subject to significant quality aberrations. A traditional definition of quality is based on the viewpoint that products must meet the requirements of those who use them [2]. In general the quality of conformance is strongly influenced by various factors. These include a. o. the choice of the manufacturing process, the training and supervision of the personnel, the types of process controls, tests, and rates of inspection, the extent to which these procedures are 1
Swiss Federal Institute of Technology, Institute of Structural Engineering, ETH Zurich, Wolfgang-Pauli-Strasse 15, 8093 Zurich, Switzerland. Email:
[email protected] 2 DOKA Industrie GmbH, Reichsstrasse 23, 3300 Amstetten, Austria. Email:
[email protected]
followed and the motivation of the workforce to achieve quality. Furthermore, the quality of a product is considered to be inversely proportional to its variability. That means that if the variability in the input timber material properties of a product decreases, the quality of the product increases. Therefore, reduction of variability leads directly to increased benefit of the manufacturing process [4]. Statistical methods play a central role in quality control procedures since the variability can only be described in statistical terms. On this way it is typical to classify data on quality characteristics as either attributes or variables data. The first are usually discrete data, taken in form of counting processes, e. g. the proof loading of timber boards and the test if one particular specimen survives the captured load or not. The discrete number of failures is counted and evaluated as attributes. The second are continuous measurements, such as knots sizes or modulus of elasticity. Desired values (DV) can be defined for a specific quality characteristic with upper and lower bounds (upper control limit UCL and lower control limit LCL) derived from the desired value. The control limits together with the centre line define an interval within which the statistic under consideration will lie with a high probability (significance level) when the process is in a state of statistical control. Within the current investigations, the grading machine settings are
considered to be the control variable, which can be varied as a function of the measured and indicated material properties so as to change the value of the output characteristics. The output characteristics are equal to the estimations of the grade determining properties (GDP) which are strength, stiffness and density. Timber is by nature a very inhomogeneous building material. On a large scale strength and stiffness related timber material properties are a product of e.g. the specific wood species, the geographical location where the wood has been grown and the sawing pattern which is applied to cut the logs into boards and beams. In daily production processes many timber manufacturing enterprises have to deal with the occurrence of quality shifts due to above mentioned reasons. The current European standard for machine strength grading of structural timber EN 14081 provides different methods for the control of grading machine settings. These are either machine or output controlled. However, recent insights of timber manufactures, engineers and researchers indicate that both of these methods are bearing different shortcomings [European Project: GRADEWOOD]. These are mainly connected to the machine control strategy which is considered as too static and not capable to take into account the large variations of the origin, sawing pattern and growth condition and therefore the large time-, supplier or sawing pattern dependent variability of the properties of the ungraded material that is present in a real market situation of a large or medium sized timber manufacturer. On the other hand, the output control strategy is regarded as cumbersome and expensive since a large amount of timber material has to be tested by destructive tests and not all of the available information from the tests is used for the recalibration of the grading machine. At European level (European Projects: COST Action E53, GRADEWOOD) extensive discussions are conducted concerning the existing grading standard EN 14081 and as a consequence revisions of this standard are presently being considered by the Technical Committee CEN TC124/WG2. The main target of the investigation described in this paper is to improve the outcome of the grading process with regard to the requirements of the standards for different strength classes or with regard to internal quality insurance policies. Aiming to achieve an optimal reliable quality for further production of engineered timber products like e.g. glued laminated timber. In this context two major question arise: (1.) Is it reasonable to tailor the machine settings continuously to the currently observed quality of the input timber material or (2.) do we want to find out the overall quality which is acceptable for delivery to the construction site? In the paper at hand, building on the probabilistic approach for machine grading quality control described in [5] a possible procedure for identification of systematic changes in the tested material quality directly based on the machine grading measurements, at the same time as these are obtained, is outlined. The first step is a detailed description of the dataset which is used for the underlying investigations.
Predictive characteristic values are assessed for subsamples of the total dataset and based on different statistical approaches in a second step. The characteristics of the sub-samples are compared with each other and the influence of the chosen statistical approach and sample size is discussed. The last step is to describe different approaches for the detection of quality shifts in the timber input material. This includes the application of the CUSUM method according to EN 14081-3 with non-destructive machine data, the definition of upper and lower control limits as an alternative approach and the comparison of different source countries by means of regression analysis between the machine measured indicating properties. All investigations are based on observations of the indicating property only. No corresponding strength, stiffness and density values are available for the moment. The aim is to gather as much information as possible just by observations of non-destructively measureable properties. Investigations on large datasets with non-destructive grading machine data enable to describe and analyze the course of the grading process on a very high resolution and without any additional costs. However, the price for this is that the interrelationship between the indicated properties and the real timber material properties cannot be described. Hence, applicability and evaluation of the concepts would require additional destructive testing data.
2 DATA A dataset of a large sized timber manufacturing enterprise is used containing monitoring data of graded Norway Spruce (picea abies) which has been documented over a time period close to one month. Indicating properties for tension strength (IPmor), tension modulus of elasticity (IPmoe) and density (IPdens) are assessed by the grading machine GoldenEye 706 [3]. The GoldenEye 706 grading machine is a combination of an optical vibration measurement device for quantifying the dynamic modulus of elasticity and an X-ray scanning device for measurement of dimension, density and knot values of the boards. While the dimensions (43x85 mm), the sawing pattern (2 ex log) and the log diameters (13-15cm top end) are considered to remain constant over time, source countries of the timber and the corresponding suppliers change every now and then. Every value of the indicating property can be assigned to a certain producer and country. The mentioned constant factors which normally lead to specific variability in the material properties offer now the inimitable chance to investigate solely the effect of varying source countries and the consequences on the observed material properties. In Figure 1 all consecutive observations of IPmor (n=161’265) are plotted over a time period of one month. Definite shifts in the indicated material properties are observable. The most significant steps can be observed
roughly at board numbers 2’000, 22’000, 40’000, 122’000 and 142’000.
Figure 1:
Time series of grading machine measured tension strength indicating property.
3 PROCESS CHARACTERISATION
are taken from the total dataset and, therefore, the process quality is represented continuously in k=160 steps over a time period of one month. The method of the moving average (or moving data window) with s=1000 in 1 adds (1+1000, 2+1000, 3+1000, ...) for the assessment of distribution parameters and characteristic values is also applied. However, compared to the previously explained sampling method, this requires 1000 times more computation time. Therefore, sub-samples are taken consecutively with no overlapping specimens. In structural reliability applications it is necessary to assess the probability distribution function of the relevant material properties. Assuming that a sufficiently large number of experiments have been performed regarding the relevant material property, it is in principle a straightforward task to select a probability density function and to estimate its parameters based on the Maximum Likelihood Method (MLM). The resulting density function might be considered as a prior density function representing the timber material property when no grading procedures are invoked.
For the characterisation of the course of process and for the identification of input material quality shifts the total dataset is split into consecutive sum-samples and quality of each sub-sample is defined by assessing its mean value and 0.5-fractile value probabilistically. Again, the entire investigations for the process characterisations are based just on observations of the non-destructively measured indicating properties of tension strength, tension modulus of elasticity and timber density. Therefore, the assessed 0.05-fractile value just serves as a quality criterion for the timber input material and should not be mixed up with the required characteristic value for tension strength in the context of grading timber into a specific strength class e. g. according to EN 338. 3.1 SAMPLING & PARAMETER ESTIMATION The total dataset is sub-divided into k=160 samples each of size s=1000. The observations of each sub-sample are used to estimate the parameters of an appropriate probability density functions. Due to the fact that in general the un-graded tension strength and tension modulus of elasticity (MOE) can be considered as being lognormal distributed also the indicating properties for these two material properties are treated as to be lognormal distributed. Timber density and its indicating property are considered to be normal distributed. For more straightforward computations during the assessment of the probability distributions and regression analysis the lognormal distributed values of the indicating tension strength and MOE are transformed logarithmically. Hence, using the MLM, the parameters of the prior probability distribution functions are estimated as normal distributed random variables with mean values, standard deviations and correlations. Figure 2 illustrates the assessed probability density functions for every particular sub-sample of the tension strength indicating property. Consecutive sub-samples
Figure 2:
Sampling of sub-samples of size s=1000 and parameter estimation of sub-sample distributions.
The general principle of the Maximum Likelihood Method is that the parameters of the distribution function are assessed such that the likelihood of the observed random sample is maximized. Let the observations of the normal distributed logarithmic indicating property ln IPmor be the random variable of interest, denoted as X . The probability density function of X is then formulated as f X ( x; ș) where ș (T1 , T 2 )T are the distribution parameters to be estimated. In case of ln IPmor the parameters to be estimated are T1
Pln X and T2 V ln X .
If the sub-sample, from which the distribution parameters ș (T1 , T 2 )T are estimated, is collected in the
( xˆ1 , xˆ2, ,.., xˆn )T the likelihood L(ș xˆ ) of the
vector xˆ
observed random sample is defined as:
L(ș xˆ )
n
f
X
( xˆi ș)
(1)
i 1
The maximum likelihood point estimates of the parameters ș (T1 , T 2 )T may now be obtained by solving the following optimization problem:
min( L (ș xˆ ))
The average value of all mean values as well as the average value of all 0.05-fractiles is taken as the desired value for input timber material properties. That means that the timber material as it is delivered to the manufacturing enterprise should in average exhibit these statistical characteristics. The assessed desired values for the overall input material quality are given in Table 1. Table 1:
(2)
T
Instead of the likelihood function it is more straightforward to consider the log-likelihood l (ș xˆ )
Average values of mean values and 0.05fractiles of s=160 sample distributions each with n=1000 observations. Parameters of the distributions are estimated based on Maximum Likelihood Method. IP_MOR [MPa]
IP_MOE [MPa]
IP_DENS [kg/m3]
average of all subsample means
29.2
12638
459.7
average of all subsample 0.05fractiles
15.8
9138
387.3
i.e.:
l (ș xˆ )
n
¦ log( f
X
( xˆi ș))
(3)
i 1
Based on this the covariance matrix CĬĬ for the parameter estimates may be determined through the information matrix H which contains the second-order partial derivatives of the log-likelihood function. The information matrix may be found to be:
H
§ ¨ n ¨ T 22 ¨ ¨ n ¨ 2¦ xi - T1 ¨ i1 ¨ T 23 ©
· ¸ ¸ 3 T2 ¸ ¸ n 2 3¦ xi - T1 ¸ n ¸ 2 i1 4 ¸ T2 T2 ¹ n
2¦ xi - T1 i 1
(4)
For the IPmor and IPdens the average of the 0.05fractiles is taken as desired characteristic of timber material input quality. Lower Control Limits and Upper Control Limits can be defined based on these desired quality characteristics as it will be shown in chapter 4.2. For MOE the average value of the sub-sample means is used as desired quality level. Figure 3 shows for IPmor, IPmoe and IPdens the mean and 0.05-fractile values which are assessed probabilistically based on the observations of each subsample.
whereby the covariance matrix is evaluated as:
CĬĬ
H 1
(5)
In probabilistic modelling where the timber material properties enter as random variables it is then possible to take into account the statistical uncertainty associated with the estimates of the distribution parameters for the distribution function, simply by including the distribution parameters in the reliability analysis as normal distributed variables with the assessed mean values and co-variances. 3.2 CHARACTERISTIC/DESIRED VALUES Based on the calculated distributions parameters the mean values and 0.05-fractile values for every subsample and for every indicating property are calculated and plotted in Figure 3. The straight line in the middle of the figures illustrates the average value of all sub-sample mean values. The lower straight line indicates the average value of all sub-sample 0.05-fractiles. The jagged lines represent the mean and 0.05-fractiles of the three indicating properties for every particular subsample distribution. The fluctuations in the lines clearly confirm the quality aberrations which have already been observable in Figure 1. The most significant shifts are observable exactly at the same point of time regardless which indicating property is taken into account.
Figure 3:
Time series of characteristic mean and 0.05fractile values of all k=160 sub-samples for IPmor, IPmoe and IPdens.
On a second step the influence of the varying sub-sample size is investigated. For this purpose, sub-samples of size s=20, s=100, s=500 and s=1000 are taken consecutively
from the total dataset. Based on the observations of the indicating properties within these sub-samples the parameters for the distribution functions are estimated by MLM and the process characteristics (mean and 0.05fractile) are assessed. Figure 4 illustrates the effect of the different sample sizes Deviations in the lines of the mean values, the 0.05- and 0.95-fractiles can be observed according to the different sample sizes. Also the desired value for the indicating tension strength being the average 0.05fractile value is shifted in dependency to the sample size. Statistical uncertainties are dependent to the amount of observed data and are responsible for the effect of increasing dispersion of the process characteristics with decreasing sample size. For a more apparent illustration just the sample range between board number 80’000 and 130’000 is shown in figure 4.
distribution model, the larger becomes the upper tail of the distribution.
Figure 5:
Time sequence, representing Influence of different statistical approaches on the characteristic values of the individual subsamples.
For a consistent representation of the timber material properties the uncertainties have to be taken into account. That is why within the subsequent steps of the investigations always the method of maximum likelihood is applied.
4 REAL TIME QUALITY EVALUATION
Figure 4:
Illustration of the effect of sub-sample size on the assessed distribution characteristic and desired values.
Further investigations are made to describe the influence of the chosen statistical approaches on the assessed process characteristics. The first approach just applies ordered sample statistics to calculate the sample mean, 0.05- and 0.95-quantile values. For the second approach the sub-sample data is used to assess point estimates of the distribution parameters. Statistical as well as model uncertainties are not taken into account as long as solely point estimates of the parameters are calculated. Therefore, the third approach based on the parameter estimation of the Maximum Likelihood Method, also represents the different types of uncertainties. Hence, in addition to the point estimates of the parameters, also the covariance of the parameters is included into the subsequent assessment of the process characteristics. Within the same sample range as Figure 4, Figure 5 illustrates the characteristic values of IPmor calculated based on the described three different statistical approaches. The mean values for every sub-sample are the same for all three methods. However, the fractile values may differ from each other. Especially the 0.95fractile value shows differences between the different methods. This can be explained by the applied lognormal distribution which is tailored to the right in case of IPmor. The more uncertainties are incorporated into the
For the consistent modelling of the stress grading procedure the strategy has to be subdivided into two major steps. The first is to identify shifts in the characteristics of the input timber material which is intended to be passed through the grading device. Since the current investigations are focused on timber material with constant dimension, sawing pattern and log sizes, these shifts, if they occur at all, may just be caused by different source countries or suppliers. Quality and quantity of the deviating characteristics have to be described and taken into account precisely. Monitoring strategies and real-time information processing of the observed grading machine measurements may serve as an appropriate way to challenge this step of the model development. The second step – and this is the perspective for future investigations – is to use the information about quality aberrations in the input material properties for the development of more efficient grading strategies. Given any indication for quality deviations the grading process in general and the machine settings in particular may be adjusted in accordance to the extent of the observed quality shift. Based on a consistent grading model which incorporates these both steps, recommendations may be provided for individual grading environments to maximize both, the reliability as well as the benefit of the produced timber products. Three methods for the identification of aberrations in the quality of the input timber material are applied to the provided dataset. The first one adapts the control chart method which is given in the European Standard EN 14081, part 3 as one of two possibilities for machine
strength grading in Europe. In the context of the investigations of this paper this method will further on be denoted as the non-destructive CUSUM. The second method, the control limit method, first defines desired values for the overall input timber material properties and based on these, upper and lower control limits are calculated to control the quality of the input timber material. A third method quantifies differences in the regression coefficients where the relationships of the different indicating properties are compared to each other between the different source countries of the timber. The effect of growth areas is discussed based on this method.
IPmor Control of the values of the non-destructively indicated tension strength is realized by the utilization of attributes charts. Framework and requirements of EN 14081-3 are applied as if to control real tension strength values. Therefore, the following assumptions are made: The desired characteristic value for the indicating tension strength is defined to be equal to the average of the 0.05-fractile values of all samples taken from the total dataset. Sample size is s=1000 and distribution parameters for the sample are estimated based on MLM. The desired value for indicating tension strength is equal to the average value of all sub-sample 0.05-fractiles.
4.1 NON-DESTRUCTIVE CUSUM
qIPmor 15.8 .
Statistical process control (SPC) may serve as an effective tool for the reduction of variability of the graded timber material. Control charts are suitable for detecting patterns of variation in datasets which indicate deviations from the desired material quality [1]. There are various types of control charts all of them aiming to identify situations where the production process runs out of control. A very good overview of the different types of control charts and quality assessment methods is given in the annex of ISO 3534. Application of control charts for timber machine stress grading was first presented by [6]. The method found its way first into the European standard EN 519 and still can be found as an approved grading method in the current version of EN 14081-3. In general samples of defined size are randomly taken directly from production after grading and subsequently proof-loaded to a proof-load value which is assessed based on the desired material characteristic (e.g. characteristic bending strength for a particular strength class). For bending strength simply the attribute is recorded stating if the specimen “survived” the proofload or not. Results are documented in so called attributes charts which are also described in ISO 8258. At the same time bending modulus of elasticity is measured and observed values are entered in so-called variables charts. For both types of charts calculation requirements are defined in the mentioned standard and for both different types of charts have to be used when the process is out or in control. This is also considered in the underlying investigations. However, control charts are used for machine strength grading of structural timber in Europe rather rarely. The reason is that they are considered not to be appropriate for grading of manifold dimensions like they are present at the timber market in Europe. Furthermore, due to the required frequent destructive tests, the method appears to be elaborate and expensive. In the investigations at hand, the basic principle of control charts according to EN 14081-3 is adapted to the dataset of machine data. The specialness of this is that just non-destructively measured observations of the indicating properties of tension strength, tension modulus of elasticity and timber density are used. Hence, modifications and assumptions have to be made as follows.
According to the CUSUM method of EN 14081-3 this value is modified by the k h size factor as given in EN 384:
kh
(6)
(150 / width)0.2 .
(7)
The proof load value in the context of these investigations has to be considered as a threshold value for the indicating property. It assessed based on
load
0.96 kh qIPmor .
(8)
The attributes chart parameters for controlling the values of the tension strength indicating property are adapted from the standard as
K 1, Y 1, and Z
6.
IPmoe The values of the indicated modulus of elasticity are controlled by applying the variables control charts of EN 14081-3. No further transformations are made. The variables chart parameters for controlling the values of the tension modulus of elasticity indicating property are adapted from the standard as
K
0.95 Emean 345
(9)
where Emean is the average value of all sub-sample mean values. Furthermore, the parameters A
7381 / Emean as well
as Ba 2.6 and Br =2.8 are used. Based on these, the remaining parameters for the variables chart can be assessed to
Y
0.0467 Emean Ba
N
0.0467 Emean Br
and Z Y N.
IPdens In general, the CUSUM output control method like provided in EN 14081-3 does not require any control of the timber density. However, since density can also be an important material property and since the applied grading machine is capable to assess this property with a high accuracy, its values are also controlled in the underlying investigations. Attributes charts are used for the control of timber density values. The desired value is the average value of the sub-sample 0.05-fractiles, being equal to
qIPdens
387.3 .
(10)
The attributes control chart parameters K, Y and Z are the same as for the IPmor control charts. Results of the applied CUSUM attributes and variables charts are given in Figure 6. These results show that every time when the CUSUM values are larger than the accepted threshold value of Y, the process clearly is indicated to be out of control. This indication works out for all indicating properties at the same point of time during the course of process and fits quite good to the real observed shifts of the data points in Figure 1. The most significant shifts in the input material as they are already observable in figure 1 lead to obvious reactions at the same point of time or board number of the CUSUM control process (Figure 6) regardless which indicating property is taken into account. However, to avoid false alarms and to guarantee an optimum identification of quality aberrations all three indicating properties should be controlled in parallel. The most obvious reactions can be observed for the attributes of the tension strength indicating property (CUSUM IPmor).
different efficient ways of reaction to quality shifts will be part of upcoming investigations. 4.2 DEFINITION OF CONTROL LIMITS Control limits are defined for the indicating properties of tension strength, modulus of elasticity and density. The so called desired values (DV) are based on the probabilistically assessed characteristics of all 160 subsamples. For IPmor and IPdens the DV is the average value of all 0.05-fractiles of the sub-samples. For IPmoe the desired value is equal to the average of all mean values. The UCL and LCL values are formulated as boundary values to control for every particular sample if its characteristic value lies between these boundaries. For IPmor the limits are set as follows:
q.05 0.2 q.05
UCLIPmor q.05
1 k
k
¦q
(11)
i.05
i 1
where qi.05 is the 0.05-fractile of the i th sample each of which contains s 1000 observations of the indicating properties. Accordingly, the lower control limit can be formulated as:
LCLIPmor
q.05 0.2 q.05 .
(12)
For the IPmoe the control limits are assessed as
UCLIPmoe
m 0.05 m
(13)
m 0.05 m
(14)
and
LCLIPmoe
Where m is the average value of all sub-sample means of the indicated modulus of elasticity. For the indicated timber density the control values are calculated to be
UCLIPdens
q 0.02 q
(15)
q 0.02 q .
(16)
and
LCLIPdens Figure 6:
Results of non-destructively applied CUSUM control charts according to EN 14081-3 for all indicating properties.
According to EN 14081-3, when an out of control situation is indicated, the manufacturer first has to check the technical performance of the grading facilities. If no technical causes are detectable he has then the possibility to adjust the settings of the grading device as a reaction to the indicated quality shift. Yield as well as reliability can be optimized in this way. The development of
The factors with which the average 0.05-fractiles are multiplied in equations 1.11 to 1.16 are assessed on iterative steps to gain an optimal control effect without too many false alarms. Figure 7 shows how control limits for the indicating properties may be defined to control that graded material properties remain within pre set ranges around the desired quality characteristics. Upper and lower control limits are defined as given in equations 1.11 to 1.16. Values above the UCL indicate where the yield and monetary benefit of the process may be enhanced by adjusting the grading machine settings. The values
beneath the LCL line represent situations where the desired quality values are not reached with consequences to the reliability of the end product. Values between upper and lower control limits indicate that the process runs efficiently. Figure 7 shows, that this method also clearly indicates situations in which the grading machine settings should be adjusted for optimizing the yield and the reliability of the graded output material.
Figure 8:
Quality shifts can be observed depending on the source country of the timber.
Regression analysis is conducted for the different IPs of timber originating from different countries. It is shown, how these coefficients vary from country to country. In many previous investigations the regression analysis is only conducted for the relationship between the destructively and non-destructively measured timber material properties. For the underlying investigations the regression analysis is conducted between the nondestructively measured indicating properties and the regressions coefficients of each country and they are compared with each other. Regression analysis Bayesian regression analysis is used very often in engineering decision analysis where empirical relationships based on experimental evidence have to be described. To assess the interrelationship e. g. between the indicating properties of IPmor and IPmoe a probabilistic model for the dependent random variable Y (IPmoe) is to be estimated based on a linear function of the explanatory random variable X (IPmor). In general, if it is assumed that n experiment or test
Figure 7:
Upper and lower control limits (UCL and LCL) are defined based on desired values (DV) for IPmor, IPmoe and IPdens.
4.3 COMPARISON OF REGRESSION COEFFICIENTS The third method contains the comparison of the regression coefficients between the three indicating properties assessed on data of remarkably varying input timber material quality. Figure 8 illustrates the influence of the source country of the timber on the characteristic values of the indicated tension strength. Now, the dataset is ordered first by country and then by supplier. Clear downward shifts are detectable from country to country especially at board numbers around 30’000 and 48’000 (country 3), 70’000 and 83’000 (country 5), 117’000 and 132’000 (country 7). There are only minor shifts within one country although timber from one country may be delivered by different suppliers.
X and Y are available as vectors of ˆx , ˆy ˆ can be expressed the interrelationship between ˆx and y T
results of
as follows:
E0 E1 xˆi H i
yˆi where
E0
and
E1
(17)
are the model parameters (regression
coefficients), i ^1, 2 ,...,n` is the index of test results and
Hi
is the error term which is considered to be
normal distributed with zero mean and known standard deviation V H . Hence, it is seen that Y for given observations of XT y
X is also normal distributed. (18)
XT Xȕ
y contains the test results ˆyi while the
The vector
matrix X contains ones in the first column and the test results ˆxi in the second column. Equation 1.17 can be reformulated as
ȕ
X X T
1
XT y
(19)
The standard deviation of the error term between the model and the measurements is assessed by n
VH
¦H
2 i
i 1
nt
(20)
where n is the number of measurements and t is the number of parameters ȕ . The variance of the error can be written in matrix form as
y Xȕ y Xȕ / n t T
V H2
(21)
The conditional distribution of Y given ȕ and X follows a Normal distribution with the following parameters:
E Y | ȕ X ȕX
(22)
Var Y | ȕ X V H2
(23)
The uncertainty of the parameters ȕ can be represented by a covariance matrix, where the parameter variances are contained in the diagonal:
Var ȕ V H2 VE
(24)
with
VE
X X T
1
.
(25)
In case the random variable Y needs to be modelled using not only one but r components in X , the model can be generalized to: r
¦ xˆ E
yˆi
ij
j
Hi
(26)
j 1
This is the case when different grading machine measurements are used to assess one indicating property. Table 3:
Results of regression analysis for country specific interrelationship between the different indicating properties. IPmor / IPmoe
country 1 (n=27’789) -19.43 E0
IPmor / IPdens
IPmoe / IPdens
-25.25
-4676
0.0038
0.1202
38.1405
3.1384
6.5152
1465
country 2 (n=552) -20.34 E
-31.146
-5222
0.0037
0.1209
37.4847
2.7024
6.5152
1343
country 3 (n=19’621) -18.52 E0
-26.78
-5159
E1
0.0037
0.1212
38.9872
VH
3.0464
7.1145
1381
E1 VH 0
E1 VH
country 4 (n=22’061) -16.11 E
-23.93
-5077
0.0036
0.1192
38.8649
2.7362
6.3539
1275
0
E1 VH
country 5 (n=15’369)
E0 E1 VH
-16.99
-19.33
-3699
0.0037
0.1080
35.7621
2.8922
6.7153
1358
country 6 (n=30’028) -19.76 E
-29.12
-4656
0.0038
0.1230
37.0362
2.9970
6.9135
1365
0
E1 VH
country 7 (n=18’378) -20.37 E0
E1 VH
-30.31
-5534
0.0039
0.1315
40.1777
3.0037
7.3703
1419
country 8 (n=27686) -16.65 E
-25.92
-5620
E1
0.0036
0.1171
38.9713
VH
3.0964
7.0923
1423
0
Bayesian regression analysis is applied for the comparison of the different interrelationships between the indicating properties of different source countries of the timber. Results of the analysis are shown in Table 3. Considering only the values in Table 3 no large differences between the regression coefficients of the different countries can be observed. However, plots of the regression lines in Figure 9 indicate especially for country 7 remarkably steeper regression lines than the lines of the remaining countries. Looking again at Figure 8 it can be observed that country 7 is characterised by a significant lower IPmor level. The information achieved by regression analysis can be used in addition to the results of the non-destructive CUSUM or the method of control limits. The results of regression analysis between the different indicating properties give additional evidence for the previously observed quality shifts between the countries.
5 DISCUSSOIN AND OUTLOOK Summing up the results of the underlying investigations it is shown that there may be serious periodical aberrations in the input quality of the timber material which is used for industrial applications. Current frameworks of the relevant European standards for machine strength grading do not take this phenomenon into account sufficiently. The provided ways of grading are either too static and complex or too extensive and expensive. The core element for the detection of quality shifts is the continuous real-time monitoring of the grading machine measured strength, stiffness and density indicating properties. Different approaches can be used to indicate quality deviations just based on these non-destructive data measurements. On this way both can be avoided: 1) Expensive destructive testing procedures although the material quality and production process is “under control”. 2) Inefficient exploitation of the timber material potential since settings of the grading machine would be too conservative for extraordinarily good timber material quality.
Reaction to the currently observed input material quality is just needed if significant shifts in the predicted characteristics are observed. Thus, both can be optimized: the benefit in the context of higher yields and more accurate classification as well as the reliability of the timber material characteristics as required for further manufacturing or by the customer or by the codes and standards. The results of the investigations show that shifts in the input quality may be detected by applying control chart methods to the observed data. Assumptions have to be made in accordance to the desired material quality which is needed for further processing. CUSUM values are assessed by non-destructive proof loading and recorded in the control charts.
desired values (upper and lower control limit) quality of the input material can easily be controlled in real-time. Regression analysis was conducted to show the influence of different source countries on the regression coefficients between the assessable indicating properties. This method gives additional evidence to the detection of quality shifts. The objective for future investigations is seen in the application of the proposed approaches under practical and industrial environment. Furthermore, ways of reaction with regard to adjustment of grading machine settings will be defined and the consequences for the reliability of the output material properties will be assessed.
ACKNOWLEDGEMENT Contributions by DOKA GmbH (Amstetten, Austria) and MiCROTEC (Brixen, Italy) are acknowledged with gratitude for allowing us to publish the results of machine strength grading.
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Figure 9:
Regression lines for the interrelationships between indicating properties within the different countries. The regression line of country 7 shows steeper slopes.
As an alternative approach the general average characteristic values of the timber input material can be assessed based on probabilistic methods. These may serve as a benchmark for the definition of so-called desired values. Accepting a certain range around these
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