Modeling lumber strength spatial variation using trend removal and kriging analyses

Wood Sci. Technol. 26:369-381 (1992) W o o d S c i e n c e a n d T e c h n o l o g y

�9 Springer-Verlag 1992

Modeling lumber strength spatial variation using trend removal and kriging analyses

E Lain and J. D. Barrett, Vancouver B.C., Canada

Summary. The techniques of kriging and trend removal analysis for modeling the strength properties of lumber have been discussed. Using these techniques, the within member compressive strengths of 38 mmx 89 mm 2100f-l.8E and the within member tensile strengths of 38 mm x 89 mm No. 2 spruce-pine-fir lumber have been modeled. The suitability of these techniques to model the strength properties of lumber has been evaluated by comparing the kriged compressive and tensile strength data to baseline experimentally collected compressive strength data and simulated tensile strength data, respectively. Comparisons of the results of statistical analyses of the baseline data and the kriged data show good agreement. The use of kriging and trend removal techniques to generate strength values in simulation studies for finite element analyses of wood structures is judged to be effective.

Introduction

Engineered wood products such as metal plated wood trusses, wood I-beams, and glulam beams are commonly use in construction. There is strong interest to evaluate the structural performance of these products because the information is needed by the wood products industry to promote their products and by code development agencies to ensure public safety. Since the variability o f wood products is typically large, many tests are needed to experimentally evaluate and quantify performancel Alternately, reliable structural analysis models can be developed to evaluate and quantify the performance of these products using Monte Carlo simulation studies or reliability studies if factors contributing to product variability are well understood.

Advances in finite element techniques have led to the development of sophisticated structural analysis models for these engineered wood products. Examples are models to analyze glulam beams (Foschi, Barrett 1980), diaphragms (Foschi 1977a), trusses (Foschi 1977b), and floors (Foschi 1982). In these models, structural members are typically divided into sub-members (elements) and key points (nodes) on the elements are identified where conditions o f equilibrium and compatibility are enforced. Dis- placement functions within each element are assumed so that the displacement at any point in the element is specified by the displacements at the nodal points. Satisfying strain-displacement and stress-strain relationships within each element, the stiffness of an element is established using energy principles from continuum mechanics. Equilibrium equations for the entire structure are obtained by considering the contri-

370 F. Lain and J. D. Barrett

bution of the stiffness of individual element, nodal compatibility, nodal forces, and boundary conditions. The equilibrium equations are then solved for the nodal displacements and the stress states in the elements. In order to fully utilize the finite element techniques in structural analysis of engineered wood products, detailed information on the strength and stiffness properties of the lumber is needed. Specifically, information on the variation of strength and stiffness properties between the members and within each member is needed.

In the past, a great deal of research focused on the variation of strength and stiffness properties between members. Recently researchers have begun to focus atten- tion on within members variation of strength and stiffness properties of lumber. Kline, Woeste and Bendtsen (1986) developed a model to generate the lengthwise variability of the modulus of elasticity (MOE) of lumber. A second order Markov model was used to generate serially correlated MOE values along 762 mm segments for each piece of lumber. Model parameters were developed from bending tests of lumber. In each piece of lumber, four equally spaced apparent MOE values along the member length were obtained from third point bending at a 762 mm span. Showalter, Woeste and Bendtsen (1987) extended the MOE generation model to consider within member variation of tensile strengths in lumber. They used the specimens from the lengthwise variability of MOE study (Kline, Woeste, Bendtsen 1986) and divided each piece of lumber, 4572 mm long, into two specimens. Each 2286 mm long specimen was tested in tension where a distance of 762 mm at each end was allowed for the grips. Therefore, test results yielded the tensile strength of the central 762 mm por- tions of the specimens. Considering the two tensile strengths (in log-space) in each piece of lumber, separated by two 762 mm segments, it was found that their coefficients of correlation, r, were 0.397, 0.560, 0.321 and 0.029, respectively, for the following four size/grade combinations of Southern Pine lumber: 38 mm x 89 mm 2250f-l.9E, 3 8 m m x 8 9 m m No. 2 KD15, 38mmx235 mm 2250f-l.9E, and 38 mm x 235 mm No. 2 KD15. Assuming that the tensile strengths in adjacent lumber segments were also correlated, coefficients of correlation of tensile strengths in adjacent lumber segments were obtained. Using second order Markov generation of MOE values followed by first order Markov generation of tensile strength values, tensile strength values along a piece of lumber composed of 762 mm long segments were generated. It can be noted that the resolution of the models is fairly coarse. Finally, the basic assumption of correlation of between adjacent segments requires verification.

Lam and Varo~lu (1991 a) experimentally evaluated the spatial variation of tensile strengths along the length of lumber. By using a gauge length of 610 mm, within member tensile strengths of more than 240 pieces of 6.096 m long lumber were obtained. The spatial correlation of tensile strength properties were characterized by semivariogram and autocorrelation analyses. Results indicated that tensile strength values within a piece of lumber separated by a distance of 1.83 m can be considered statistically independent. Lam and Varo~lu (1991 b) also developed a random process and moving average model to represent the spatial variation of tensile strength of lumber. Good agreement between model predictions and experimental results were obtained. However, the within member variation of tensile strength data were incomplete because of the 610 mm gauge length limitation and the spacing requirements of the gripping devices in the tension testing machine. Therefore, the experimentally

Modeling spatial variation of lumber strength 371

obtained tensile strength profiles were not continuous and the spatial variation of tensile strength in lumber, with element length less than 610 ram, is unavailable. This information is important for the structural analysis models of engineered wood products which may require fine finite element mesh size.

This paper describes techniques to provide refined estimation of spatial variation o f strength properties in lumber. One of the techniques, known as kriging, is pio- neered by the mining industry for mineral resources evaluation (Krige 1951). By applying the kriging technique to the spatial variation of strength data, a best linear unbiased estimator of the unknown strength at any point along the length of lumber can be obtained from known strengths values along the length of lumber. The second technique, known as trend removal analysis, was used to preprocess the data since kriging analysis assumes a stationary underlying process. Therefore, incomplete strength profiles can be refined. Case studies of applying the kriging technique to the spatial variation of compressive and tensile strength data are presented.

M e t h o d

Kriging analysis

Let S (x) denote the strength of a piece of lumber at any location along its length. It is difficult to measure S (x) experimentally. However, suppose it is possible to test for the strength of the lumber at certain locations along its length yielding a series of n strength values S (xl) (i = 1 . . . . . n) where xl is the i 'h location along its length. The unknown strength at any location x along the length of a piece of lumber, S* (x), can be approximated by:

n

S*(x) = • aiS(xi) (1) i = l

where a~ is a set of weights to be evaluated such that S* (x) is the best linear unbiased estimator of S (x).

Here, the term best linear unbiased estimator of S (x) means that the evaluation of the set of weights, ai, is subject to the following conditions: 1) the mean of the squared difference between S (x) and S* (x) is minimized and 2) E IS* (x)] = E [S (xi)], where E [ �9 ] denotes the expected value.

The mean of the squared difference between S (x) and S* (x), ar z , can be expressed a s :

a 2 = E[{S (x) - S* (x)} z]

= E [S (x) 2 - 2 S (x) S* (x) + S* (x) 2] (2)

z _ 2 5Z a i Oxx t 4- a i a j O-xi ~j i = l i = l j - 1

where

G~ = E[S(x)2],

~rxx ~ = E [S (x) S (xl)],

~rxi xj = E [S (xi) S (xj)] .

372 F. Lam and J. D. Barrett

It can be sh~ that c~176 2 requires that E I ~i=1 ai S (xl)l = E[S (xl)]; theref~

~ - ' ~ a i = ] i s a constraint condition. i = 1

Using the method of Lagrange multiplier, the following function, IF, should be minimized with respect to the unknowns ai and the Lagrange multiplier ~t:

2 2 ~ a iaxxi+ ~ ~ a i a j a x ~ x j + 2 # ( ~ a i _ ] ) (3, F = fix-- i = l i = l j = l i = l

The derivatives of F are computed with respect to the unknowns and set to zero yielding:

D E m n ~al 20xx i Jr- 2 j=IZ aj Crx~ xj Jr- 2 p = 0 , (4)

The following system of linear equations can, therefore, be obtained:

n

Z aj ax~ xj +/z = ax~ ~ , a i = 1 (i = 1 . . . . . n). (6) j = l i = l

Equat ion (6) can be written in matrix form as:

[C] {A} = {R} (7)

where

[C] =

~ X 1 X 1 O'X1 X2 " �9 �9 O'X1Xn

~ ' X2 Xl O-X2X2 �9 �9 �9 O'X2Xn

O-XnXI O'XnX2 �9 �9 . G 'XnXn

t 1 . . . 1

1-

1

1

0

{A} t = {al, a2 . . . . . an ,P} ,

{R}' = {a . . . . 0"XX2, . . . . 0"XXn, 1} .

For any location x, the vector of unknowns is given by:

{A} = [C1-1 {R}.

Note that the covariance matrix, [C], depends solely on axixj = E [S(xi) S (xj)] (i and j = 1 . . . . . n). axlxj can be directly evaluated from the experimental strength data S (xi) ( i= 1 . . . . . n); therefore, the covariance matrix needs to be inverted only once in the kriging analysis process�9 Also note that the vector {R} depends on axx i = E [S (x) S (xi) ] (i = 1 . . . . . n) which is a function of S (x) and S (xl). Here S (x) is an unknown which can be expressed as S (x i + zi) where z i is the separation distance between the location of interest x and the ph location where strength values have been experimentally


measured, x i . Therefore, axx t (i = 1 , . . . , n) can be expressed as axx, = E [S (xi) S (x i + zi)]- From the experimental strength data S (x~) (i = 1 . . . . . n), an autocorrelation function, Auto (z), can be obtained where Auto (r) = E [S (x~) S (x i + z)]. Therefore, for any location x or separation distance z~, a ~ can be interpolated from Auto (~) to estimate the vector {R} so that Eq. (8) can be solved to obtain the unknowns, a~ (i = 1 . . . . . n) and p. Knowing the weights a~ (i = 1 . . . . . n), Eq. (1) can be evaluated for S* (x).

Trend removal analysis

Trends exist when the data contain frequency components whose period are longer than the record length. As mentioned earlier, [C] depends on the estimation of axixj (i and j = 1 . . . . . n) of the experimental strength data S (xi) (i = 1 . . . . . n). Since trends can influence the estimation of axix~ (i and j - -1 . . . . . n), trend in the data must be removed to avoid error in the estimation of [C]. In general, trends in data cannot be removed by filtering techniques; therefore, special removal techniques must be used. Here, the least squares procedure, which is suitable for removal of linear or higher order trends, is introduced. The idea is to estimate the trend in the data first. Trend is then removed from the data prior to estimating [C]. Kriging analysis can then be applied to the trendless data. Finally, the trend can be reintroduced to the kriged results.

Let {u~} (i = 1 . . . . . n) be the n data values sampled at a constant interval of h. Suppose the data can be fitted with a polynomial of degree k defined by:

k Ui = ~ bs(i h) j, i = 1 . . . . . n . (9)

j=0

Using least square fitting techniques, the coefficients b0 , . . . , bk are chosen so that the following expression, Q, is minimized:

Eu Q = ~ [ u i - f i i ] 2= ~ i - ~ bs(ih) �9 (10) i=1 i=1 j=0

The coefficients b o . . . . . bk are found from 8 Q = 0, oQ ~Q k + 1 equations as: Obo ~ = 0 . . . . . Ob, = 0, yielding

~b moQ = 2 i=1 ~- I ul . . . . j=O~ bs(ih)51[-(ih)m] 0 ' m 0 '1 . . . . . k (11)

Equation (11) can be rewritten as:

k ui(ih) m = Y~ b 5 ~ (ih) j+m, m = 0 , 1 . . . . . k

i = l j=0 i=1

which can be solved to obtain the k + 1 coefficients bj ( j= 0, 1, . . . , k) required to describe the trend fil (i = 1 . . . . . n).

The zeroth order trend represents the mean of the data. The linear, quadratic, and cubic trends in the data are represented by k = 1, 2, and 3, respectively. The choice of k for trend removal depends on the nature of the problem, the desired accuracy and the sampling interval h. As k increases, it is expected that the trend would converge


60

oo 40

e-

30 D Experimental Data

Trend 20

Trend Removed Data

u) 10 U)

- 10

_ _ _ 1 i i r i i i i i I -20 0 1 2 3 4 5 Distance (m)

Fig. 1. Typical baseline compressive strength profiles in 38 mm • 89 mm 2100f-1.8E spruce-pine- fir lumber

60

50 [] Experimental Data

~, Trend rt 40 ~*" 2030 ~ ~ Trend Removed D a t t ~

�9 ~ 10 r

I-- 0 . ~ j f * ~ ~/ -10

i I i i i i -20 0 2 4 6 Distance (m)

Fig, 2. Typical baseline tensile strength profiles in 38 mm • 89 mm No. 2 spruce-pine-fir lumber

to the shape of the input function. However, it is not recommended to use a large k because numerical difficulties may result. Therefore, judgement must be exercised to properly choose the order of trend removal for the problem of interest.

Case studies

Two sets of data were available in the case studies: 1) experimental data on the within member compressive strengths of 54 pieces of machine stress rated 2100f-l.8E 38 mm • 89 mm 4.876 m long spruce-pine-fir lumber (Xiong 1991) and 2) simulated within member tensile strength data, based on a verified two stage random process


and moving average model, of No. 2 38 mm x 89 ram, 6.096 m long spruce-pine-fir lumber (Lam, Varo~lu 1991 b).

In the compression data set, each piece of lumber was cut into 32 specimens (cells). Each 152.4 mm long specimen was tested in compression parallel to grain to obtain the within member compressive strength profile. It was assumed that the compressive strength in each cell represented the strength at the middle of each cell. Therefore the compressive strengths in each profile were evenly spaced at 152.4 mm. First, a partial database was obtained by considering only the compressive strengths of alternate cells. Thus, each strength profile in the partial database contained the strengths of 16 cells spaced at 304.8 mm. Trend removal and kriging analyses were applied to the partial database to estimate the strength values of the cells omitted from the original database; therefore, kriged compressive strength profiles with strength values spaced at 152.4 mm were obtained. In this study, the statistics of the kriged and the experimental compressive strength data were compared to verify the trend removal and kriging analyses.

For the within member tensile strengths, 200 pieces of 6.096 m long lumber were considered. Each piece was divided into 10 cells, spaced at 0.610 m on center. The two-stage model was used to generate the within member tensile strengths in each cell. It was also assumed that the tensile strength in each cell represented the strength at the middle of the cell. However, the two-stage model was limited to generate within member tensile strengths for cells with a minimum length of 0.610 m because the model parameters were developed from experimental data of specimen tested at a 0.610 m gauge length (Lam, Varo~lu 1991 a). In this study, trend removal and kriging analyses were applied to the database to extend the simulated baseline tensile strength data to a smaller cell size of 0.305 m.

Results and discussions

Figures 1 and 2 show two representative examples of compressive and tensile strength data sets, the estimated trends, and the trend removed strength data, respectively. Here, it was judged that k = 2 would adequately represent the trend in the strength data. The results show that the second order trend in the data was successfully removed.

In order to check the results of trend removal and kriging analyses, comparisons of statistical analyses of the original strength profiles and the kriged strength profiles were performed. The mean, minimum, and standard deviation of the compressive and tensile strengths within each piece of lumber obtained from the baseline data sets were calculated (using the 32 strength values in each compressive strength profile and the 10 strength values in each tensile strength profile). Similarly, the mean, minimum, and standard deviation of the kriged compressive and tensile strengths in each profile were obtained. In each kriged compressive strength data set, the 16 kfiged values and the 16 input compressive strengths were considered. Similarly, the 10 kriged values and the 10 input tensile strengths were considered in each kriged tensile strength data set. Regression analyses were performed on the mean, minimum, and standard deviation of the within member compressive strengths between the original data and the kriged data. The coefficients of determination, r 2, for the mean, minimum, and standard


J : m

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I1. Q .>

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Mean Compressive Strength (MPa) Fig. 3. Cumulative probability distributions of within member mean compressive strengths of 38 mm • 89 mm 2100f-1.8E spruce-pine-fir lumber from experimental data and kriging analyses

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L i I I I I 1 3 5 7

Compressive Strength Standard Deviation (MPa) Fig. 4. Cumulative probability distributions of within member standard deviation of compressive strengths of 38 mm • 89 mm 2100t"-1.8E spruce-pine-fir lumber from experimental data and kriging analyses

deviat ion of the within member compressive strengths between the baseline and the kriged da ta were 0.95, 0.82, and 0.77, respectively. Figures 3 - 5 show comparisons of the cumulat ive probabi l i ty distr ibutions of the within member mean, s tandard deviation, and min imum compressive strengths for the experimental da ta and the kriged data. The coefficients of determination, r 2, for the mean, minimum, and s tandard deviat ion of the within member tensile strengths between the baseline and the kriged da ta were 0.99, 0.99, and 0.95, respectively. Figures 6 - 8 show comparisons of the cumulat ive probabi l i ty distr ibutions of the within member mean, s tandard deviation,


0.9

0,8

:= 0.7 . o (o

J:l 0.6 o

0.5

~ 0,4 3 ~ 0.3

0.2

0,1

. Krlglng Analyses

+ Experimental Data

+ = a m

+L . . . . 'o . . . . . ,;' ' '6 0 22 26 3 34 38 2 4

Minimum Compressive Strength (MPa)

~+ m m

Fig. 5. Cumulative probability distributions of within member minimum compressive strengths of 38 mm x 89 mm 2100f-1.8E spruce-pine-fir lumber from experimental data and kriging analyses

0.9 ~ 0 ~

.~" 0 8

,~ 0.7

0.6

>0 0.5

~ 0.4 _ _ _ _

~ 0.3 Krlglng Analyses

0.2 j ~ . + Simulated Baseline Data

t / . 0,1

0 i f ~ i i i i i 10 30 5O 7O

Mean Tensile Strength (MPa) Fig. 6. Cumulative probability distributions of within member mean tensile strengths of 38 mm x 89 mm No. 2 spruce-pine-fir lumber from baseline data and kriging analyses

and min imum tensile strengths for the simulated baseline da ta and the kriged data. It is clear that the kriging technique was able to preserve the characterist ics o f the within member mean, minimum, and s tandard deviat ion of compressive and tensile strength of the baseline data.

Semivar iogram and autocorre la t ion analyses were performed on the baseline and the kriged data. These analyses can be used to define the spatial correlat ion of within member compressive and tensile strengths. Semivar iogram and autocorrelat ion, 7 (Q and Auto (r), are respectively defined by:

378 F. Lam and J, D. Barrett

o,9 ~ ~ = + + a

/ 0.8

, , , K r lg lng Analyses

Simulated Baseline Data

0.1

0 i i i i J i i i i 0 4 8 12 16 20 24 28

Tensile Strength Standard Deviation (MPa)

Fig. 7. Cumulative probability distribution of within member standard deviation of tensile strengths of 38 m m x 89 mm No. 2 spruce-pine-fir lumber from baseline data and kriging analyses

0.7

0.6 2 II.

O.5

O.4 -i

H 0.3 (3

0.2

1 IP ~ . t,

0,9

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~- o.7 .Q t~

.o 0.6 2

IX 0,5

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[] Kdglng Analyses 0,3

C.} ~ + Simulated Baseline at 0.2 / 0.1

~ = r i r i i 0 0 20 40 610

Minimum Tensile Strength (MPa)

Fig. 8. Cumulative probability distributions of within member minimum tensile strengths of 38 m m x 89 mm No. 2 spruce-pine-fir lumber from baseline data and kriging analyses

3' (r) = �89 E [{S (xi) - S (xj)}21, i, j = 1 . . . . . n , (13)

A u t o (r) = E [S (xi)" S (x~)], i, j = 1 . . . . . n (14)

where r is the separa t ion dis tance between x i and xj. Figures 9 and 10 show compar i sons of the semivar iogram and the au tocor re la t ion

of compressive strengths between the exper imental da ta and the kriged data. Similar- ly, Fig. 11 and 12 show compar i sons of the semivar iogram and the au tocor re la t ion of tensile s t rengths between the s imula ted basel ine da ta and the kriged data. It is clear

Modeling spatial variation of lumber strength

3 5

379

3O

r 25

E 20

O �9 = 15 ca .> E 10 00

R Krlglng Analyses

+ Experimental Data

0 , ~ 0 1 2 3 4 5

Separation distance (m)

Fig. 9. Comparisons of semivariogram of within member compressive strengths of 38 mm x 89 mm 2100 f-l.8E spruce-pine-fir lumber from experimental data and kriging analyses

2 , 2 2

r 2.21 rt : [ o

2 . 2

X

C 0 . m

2.18 0 o

2 2 . 1 7

2.16

Krlglng Analyses

+ Experimental Data

- 5

i

-3 -1 I 3 5

Separation distance (m) Fig. 10. Comparisons of autocorrelation of within member compressive strengths of 38 mm x 89 mm 2100f-l.8E spruce-pine-fir lumber from experimental data and kriging analyses

that the semivar iogram and autocorre la t ion of the kriged da ta matched the semivar iogram and the autocorre la t ion of the baseline da ta reasonably well.

Conclusions

Trend removal and kriging techniques are shown to provide a basis for generating within member strength proper ty distr ibut ions for structural wood products where propert ies are correlated along the length of the member. These techniques allow the


100

9O

80

E 6o

50

"~ 40

"~ 30 O0 / / <> Simulated Baseline Data

Y 2O i

10

0 - - ~ 2 4

Separation distance (m) Fig. 11. Comparisons of semivariogram of within member tensile strengths of 38 mm• 89 mm No. 2 spruce-pine-fir lumber from baseline data and kriging analyses

1.2

1.15 o~ el

1.1

O 1.05 re- X

C 1 O .m

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O o 0,9

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[] Krlglng Analyses

+ Simulated Baseline Data

-6 -4 -2 0 2

Separation distance (m)

i t i i

4 6

Fig. 12. Comparisons of autocorrelation of within member tensile strengths of 38 mm • 89 mm No. 2 spruce-pine-fir lumber from baseline data and kriging analyses

generation of properties values at locations where physical test results are not available while retaining the correlation structure of the within member property data derived from experimental data.

These techniques were used to evaluate the within member compressive strength variation in 2100f-l.8E 38 mm • 89 mm machine stress rated lumber and the within member tensile strength variation of No. 2 grade 38 mm • 89 mm spruce-pine-fir lumber. Regression analyses of within member mean, minimum and standard deviation of kriged and baseline data show that r 2 ranged from 0.77 to 0.99. Cumulative probabili ty distributions of compression and tensile strengths of baseline and kriged


results also compared well. The semivariogram and autocorrelation analyses of the baseline and kriged data showed the within member correlation structures of compressive and tensile strength were retained.

Analytical and numerical studies of the strength and structural response of wood structures have largely been based on strength data derived from studies on between member behavior. Further the influence of within member correlations of tension, compression and bending strength properties on structural behavior of members and systems have not been adequately explored. This paper presents techniques which provide a basis for describing the correlation structure and simulating the within member variation of a single strength property such as tension, compression, or bending strength. In future work, these concepts will be extended to evaluate the cross correlation between the various within member strength properties.

References

Foschi, R. O. 1977a: Analysis of wood diaphragms and trusses. Part I: Diaphragms. Canad. J. Civil Eng. 4:345-352

Foschi, R. O. 1977 b: Analysis of wood diaphragms and trusses. Part I: Truss-plate connections. Canad. J. Civil Eng. 4:353-362

Foschi, R. O. 1982: Structural analysis of wood floor systems. J. Struct. Div. Amer. Soc. Civil Eng. 108:1557-1574

Foschi, R. O.; Barrett, J. D. 1980: Glued-laminated beam strength: a model. J. Struct. Div. Amer. Soc. Civil Eng. 106:1735-1754

Lain, E; Varofglu, E. 1991 a: Variation of tensile strength along the length of lumber - Part I: Experimental. Wood Sci. Technol. 25:351-359

Lam, E; Varoglu, E. 1991 b: Variation of tensile strength along the length of lumber - Part II: Model development and verification. Wood Sci. Technol. 25:449-458

Kline, D. E.; Woeste, F. E.; Bendtsen, B. A. 1986: Stochastic model for modulus of elasticity of lumber. Wood Fiber Sci. 18:228-238

Krige, D. G. 1951: A statistical approach to some basic mine valuation problems on the Witwatersrand. J. Chem. Metallurg. Mining Soc. of South Africa 52:119-139

Showalter, K. L.; Woeste, F. E.; Bendtsen, B. A. 1987: Effect of length on tensile strength of structural lumber. U.S. Dept. of Agric. For. Serv. For. Prod. Lab. Res. Paper FPL-RP-482. Madison, WI., U.S.

Xiong, P. 1991: Modeling strength and stiffness of glued-laminated timber using machine stress sated lumber. Thesis of M.Sc., Dept. of Forestry, Univ. of British Columbia.

(Received January 2, 1991)

Frank Lain Research Engineer Department of Harvesting and Wood Science Faculty of Forestry The University of British Columbia 2357 Main Mall Vancouver, British Columbia Canada V6T IZ4

J. D. Barrett Professor, Department of Harvesting and Wood Science Faculty of Forestry The University of British Columbia 2357 Main Mall Vancouver, British Columbia Canada V6T IZ4

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Modeling lumber strength spatial variation using trend removal and kriging analyses