Simulation of correlated modulus of elasticity and compressive strength of lumber with gain factor

ProbabilisticEngineeringMechanics10 (1995) 63-71

ELS EVIER

0266-S920(94)00009-3

© 1995 Elsevier Science Limited Printed in Great Britain. All fights reserved 0266-8920/95/$09.50

Simulation of correlated modulus of elasticity and compressive strength of lumber with gain factor Yin-Tang Wang*, Frank Lam & J.D. Barrett Department of Wood Science, Faculty of Forestry, University of British Columbia, 2357 Main Mall, Vancouver, B.C., Canada V 6 T 1Z4

(Received 1 August 1994; revised 21 September 1994; accepted 21 September 1994) This paper discusses a gain factor model for the simulation of correlated stationary material properties of lumber (modulus of elasticity and compressive strength) in standard normalized space. Coupled with trend and multivariate standard normal simulation models, the gain factor model has been used for simulation of nonstationary properties of lumber. The mathematical formulae of ensemble autocorrelation functions and power spectra have been obtained with test data based on nonlinear least square regression with constraints minimization. The method can estimate the spectrum of data profiles with few points which can be used to simulate the strength of lumber by the spectral representation method.

Lam and Varoglu 11'12 experimentally evaluated the spatial variation of tensile strengths along the length of lumber and developed a random process model using the moving average technique to represent the spatial variation of the tensile strength of lumber. Xiong 13 combined the spectral approach, which generates MOE profiles, with the bivariate normal distribution transformation method, which simulates correlated MOE and strength properties in a study of the performance of glulam beams. The model considered both MOE and strength as stationary random processes. Also, the within member variability of strength was considered indirectly through the spectral approach of the MOE process. Based on random process theory, Wang and Foschi 14 developed a stochastic finite element procedure to study the reliability of glutam beams. The model required MOE profiles that were represented by a random field as input. Utilizing the spectral representation method, the random field that was considered as a stationary process was quantified by the variation of MOE along each lamination. The model examined the serviceability limit state of glulam beams under maximum mid-span deflection. Lain and Barrett 15 studied a method using trend removal and kriging techniques to model nonstationary, within member, tensile and compressive strengths of lumber, without consideration of the correlation between MOE and strength. Lam, Wang and Barrett 16 developed a stochastic model to simulate spatial variation of MOE and compressive strength. The model considered MOE

INTRODUCTION Many simulation models have been developed to predict the structural performance and to estimate the reliability of engineered wood components or systems. Finite element method was used in some of these models to analyze engineered wood structures. Examples of applications of finite element techniques include sheathed floor systems model, 1-3 diaphragm model,4 truss model,5 glued-laminated beam model6 and floor's model7 which is a combination of finite elements and Fourier series. To further study the reliability of these structures, more information on material properties, including strength and stiffness properties of lumber and their correlation, is needed. In the last ten years, a number of researchers have made great contributions to this area. Kline et al. 8 developed a model to represent the lengthwise variability of modulus of elasticity (MOE) of lumber. Showalter et al. 9 extended the MOE generation model to consider the variation of tensile strength in lumber within a member, Taylor and Bender (1988) 1° presented an approach for simulating long-span correlated lumber properties using a simple transformation of the multivariate normal distribution model. The method was shown to exactly preserve the marginal distribution of each variable and also satisfactorily preserved the correlation between the long-span lumber properties. *On leave from Department of Engineering Mechanics, Hohai University, Nanjing, China. 63

Yin-Tang Wang, Frank Lam, J.D. Barrett

64

and compressive strength as correlated nonstationary random processes that can be converted to a stationary random process and a random trend process. The spectral representation and bivariate normal distribution transformation methods were used to simulate the trend removal profiles of M O E and of compressive strength, which are considered stationary processes. Trends of M O E and strength were added to the trend removal profiles to preserve the nonstationary nature of the two processes. The moving average model was used to refine the spatial correlation of simulated compressive strength profiles. In this paper, a gain factor model was developed to simulate the correlated stationary compressive strength and stiffness properties of lumber in standard normalized space. The gain factor model was then combined with trend and multivariate, standard, normal simulation models to simulate nonstationary correlation within member strength and stiffness properties of lumber.

DATABASE

Two grades of 38 × 89 m l T l 2 Spruce-Pine-Fir machine stress rated lumber, 2400f-2"0E and 1650f-l.5E, were tested to obtain the flatwise M O E and internal compressive strength profiles parallel to the grain. Each piece was 4"88 m in length. The specimen was air dried for a period of four months to achieve uniform moisture content of 10-12%. The sample sizes were 51 for 2400f2.0E and 56 for 1650f-l-5E. The flatwise M O E of each piece was determined nondestructively using a Cook Bolinders AG-SF stress grading machine. Simple beam bending theory was used to calculate the M O E profile from the measured load profile which has approximately 1800 points one piece. After the M O E test, each specimen was cut into 32 segments for parallel-to-grain compression testing. The strength profile accuracy may be influenced by cutting a board into segments in two different ways: (a) since only

the minimum strength is measured within a segment, higher strength profile resolution can be achieved with an increased number of segments; (b) with an increasing number of cuts, there is an increased chance of cutting through defects which may change the stress distribution and hence the measured strength. In this study, we try to balance the two effects by choosing a reasonable specimen width of 152-4 mm. The specimen dimensions and failure load were recorded to estimate compressive strength. The 32 compression strength data of each specimen provide the database for model development and verification. 16 Parameters for the Normal and 2-parameter Weibull distributions fitted to the test data within member mean, minimum and standard deviation of MOE and compressive strength are shown in Tables 1 and 2, respectively. Figure 1 shows M O E and compressive strength test data profiles. For both grades, the cumulative probability distributions of within member mean, minimum and standard deviation of MOE are shown in Figs 2-4. Similarly, the cumulative probability distributions of within member mean, minimum and standard deviation of strength are shown in Figs 5-7.

MODEL DEVELOPMENT The MOE and compressive strength profiles, en(X) and sn(x), can be considered as samples of random processes E(x) and S(x), respectively. As shown in Fig. 1, trends exist in both M O E and strength profiles. Therefore, E(x) and S(x) may be considered nonstationary. Since the internal mean and the standard deviation of MOE and compressive strength varies from specimen to specimen, E(x) and S(x) are also non-ergodic processes. E(x) and S(x) can be represented by the sum of two random processes as

E(x) = E°(x) + TE(x) S(x) = S°(x) + Ts(x)

(la) (lb)

Table 1. Comparison of simulated results to test data for within member means, minimum and standard deviation (S.D.) of MOE

Distribution Types

Parameter

Normal

Test Data Mean S.D. Simulation Mean S.D.

2-P Weibull

Test Data m k Simulation m k

2400f-2.0E Mean

1650f-1.5E

Minimum

S.D.

Mean

Minimum

14 105 1022

12643 1007

14017 995

S.D.

634 190

11 621 725.6

10085 738-5

570 170

12 550 1061

643 173

11 601 738"5

10 188 822.5

595 154

14487 18.572

13022 16.688

685 4.23

11934 19.738

10410 16.196

617 4.31

14430 18.101

12988 15.058

696 4-80

11907 20.316

10530 15.727

637 5.42

Simulation o f correlated modulus o f elasticity

65

Table 2. Comparison of simulated results to test data for within member means, minimum and standard deviation (S.D.) of compression strength Distribution Types

2400f-2.0E

Parameter Test Data Mean S.D. Simulation Mean S.D. Test Data m k Simulation m k

Normal

2-P Weibull

Mean

Minimum

S.D.

Mean

Minimum

S.D.

55-23 4-974

45.50 6.457

4.405 1.171

45.63 2-990

36.61 3.661

4.361 0.818

54.72 5.634

45.77 6.407

4.517 1.186

45.62 3.922

36.60 4.478

4-515 0.909

57.12 14-44

47.84 9.092

4.766 4.758

46.87 19.20

38.13 12.11

4.650 6.703

57.04 12.22

48 "37 8.743

4-960 4.320

47-22 14.90

38.46 9-995

4.851 6.121

The processes E(x) and S(x) can be converted to nonergodic stationary random processes, E°(x) and S°(x), and random trend processes Te.(x) and Ts(x), using trend removal techniques. 14 The ergodic stationary processes E*(x) and S*(x) can be obtained from the nonergodic stationary processes E ° and E°(x) by applying standard normalized procedures to the trend-removed test data. For individual records, they can be expressed as:

en(X) = (e°(x) - em)/es

(2a)

S ; ( X ) = ( S ° ( X ) -- S m ) I S s

(2b)

Here, e°(x) and s°(x) are trend removed data profiles of M O E and strength. The within-member mean and standard deviation of M O E and compression strength

Odglnal Data

Trend Removed

!.

are em and e~, Sm and s s, respectively. Therefore, the M O E and strength functions can be expressed as follows:

e~(x) = e,, + es" e*(x) + et" x

(3a)

s,(x)

(3b)

= s , , + Ss

+ s, . x

where e t and st are the first-order trends of MOE and strength for each record, respectively. On the basis of time series theory, the single-input/ single-output model can be applied to simulate the two correlated stochastic processes. ]7A8 Here, the profile e~,(x) is assumed to be an input to a system and may be considered as a sample from an ergodic stationary process E* (x). The profile Sn (x) is the output (response) of the system and belongs to another ergodic stationary process S* (x). The output of the system can be defined by the convolution integral as:

s*(x) =

i0o h('r), e*(x -

"r)dr

(4)

where h(r) is a weighting function. For a physical system, the effective lower limit o f integration in eqn (4) is zero, rather than - o o , since the system responds only

!

2Looatbn(m) $

4

17

1.0

:

.~, 0.8

1860f-1.SE i /

~'°1 •

..°. •o• o //"

° 2400f-2.0E

Trend Removed

le

0.6 ~~ ; 0.4 E ~ 0.2 •s" 0,0 8,000 10,000

Data

15

12 11 lo

1650f-1.5E

i

Odgln$1Data o'.5

'

1.5

'

='.5

'

~

'

415

Location (rn)

Fig. 1. MOE and compressive strength profiles.

i/

.:

L =.~d,.

L 12,000

14,000

16,000

18,000

Mtmlmun MOE (MPa)

Fig. 2. Cumulative probability distribution of internal mean MOE.

Yin-Tang Wang, Frank Lain, J.D. Barrett

66 ,.0

.

,.0

., ." ..." ...- •

oI

/

O.e

. . . / . . . /,.. :

~

.']

Iol o

//

O.e

o.,

o.,

0

,t ..t'," 400

200

, eoo

, 800

°

."

.V.."

2400f-2.OE

0.0

/:/

11mOl-l.6E

, 1,000

0.0

1,200

10

20

80

Standard Deviationof MOE (MPa)

"~"/" " ~'~'~ 40 50

eo

70

Mean Strength (MPa)

Fig. 3. Cumulative probability distribution of internal minimum MOE.

Fig. 5. Cumulative probability distribution of internal mean compressive strength.

to past inputs, that is, h(r) = 0 for r < 0. In this study, there is no meaning for e(x) or e*(x) when x < 0; e(x) is the real MOE sample and e*(x) is its standard normalized form. Therefore, eqn (4) implies that the higher limit of integration is x rather than + ~ . A transfer function H(co) can be defined as the Fourier transform of h(r):

where Re(r) and Rs(7.) are autocorrelation functions of MOE and compressive strength processes, respectively. Direct Fourier transform of eqn (8) yields the input/ output autospectrum relation as:

H(w) = I : h(r) exp(-arr)d'r

(5)

H(co) is generally complex and can be expressed in complex polar notation as: H(co) = IH(co) l • exp[-hb(co)]

(6)

where [H(co) l and 4~(co)are commonly referred to as gain and phase factors, respectively. The product s* (x)s* (x + r) is given by:

s* (x)s* (x + 7") =

i0oj0o h(~)h(rl)e* (x

-

where SPs(co) and SPe(co) are the one-sided power spectral density functions of the MOE and compressive strength processes, respectively. Equation (9) permits the determination of SPs(co) from knowledge of SPe(co) and In(co)l. SPs(CO) and SPe(~v) may be obtained from the Fourier transform of autocorrelation functions Rs (7.) and Re (7.), respectively. Since autocorrelation functions are always even functions of 7., it follows that the autospectra are given by the real part of the Fourier transform only; therefore,

SPs(co) = -~ SPe(co)

(7)

Taking the expected values of both sides yields the input/output autocorrelation relation as:

h(~)h(77)Re(r + ~ - o)d~dT1

1.0

~ E ~

~<.<.<.=-""'" ..-.

(8)

Fig.

j0

(10b)

~ O,a~

i

1660f-1.6E / "

~

0.4

: : " "~ '/

1e~°1.~

o., 400 600 800 1,000 Standard Deviationof MOE (MPa)

"

Tmt 0,6

0.2

4.

(10a)

2f~Re(co)cos(w7.)dT. (co _> O) "it

1.0 0.8

200

(co _> 0)

-

0.4

0.0

Rs(r) cos(azr)dr

Figures 8 and 9 show the ensemble autocorrelation functions of the trend removed MOE and strength data in standard normalized space. Relative large fluctuations occur within the first few lags and tend to zero when the lag increases. Here, a lag is defined by a

~. o.s 0.8

(9)

~)

x e*(x + 7"- rl)d~dr1

Rs(7") =

SPs(co) = IH(co)I ~- SPE(CO)

1,200

Cumulative probability distribution of internal standard deviation of MOE.

0.0

10

0..~

Z...j ! 20

80 40 50 Mlmlmun Strength (MPa)

80

70

Fig. 6. Cumulative probability distribution of internal minimum compressive strength.

Simulation of correlated modulus of elasticity

l

1.0

0.10

0.8

0.08

0.6

0

0.4

0.04 ~

67 RI~ Cunnl lllmuldan ...........

1e60t-1.SE

i

!

O Tat

0.2

0.0 0

///

1.0

/

•~

0.8

=

0.6

t°¢',

0.02 0.~

2 4 Standard Deviationof Strength (MPa)

.

10

0

10

0.10

° •

I

0.08

0.06

8t==1~0.

0.2

0 Tea

0.04 ~

0,02 O.(Xl

'6

4

8

lo

i0

Fig. 7. Cumulative probability distribution of internal standard deviation of compressive strength. separation distance of 152.4mm. In this study, the autocorrelation function for both MOE and strength was assumed to take the following form: N

I~ I)" Z cos(Bi~.)

exp(_/32~.2)

i=1

-~
(11)

2o

~

~

Fig. 9. Ensemble power spectral density functions of MOE.

This function satisfies the following conditions: R(0) = 1 and

R(c~) = 0

(12)

R(T) and SP(w) constitute a Wiener-Khinchine transform pair, therefore the corresponding power spectral density function SP(w) is given by a N[ 1 1 sP(~) = ~ ..~ ~2 + (Bi + ~)2 + ~2 + (~i -

where N is the number of cosine terms. 1.0 tU

2~v/-~ exp - ~ - g

]

(13)

"6

i

~)2

( - e e < ~o < ee)

I0501-1.5E

0.5

5o

Frequenc~(11m)

Standard Deviationof Strensth (MPa)

R('r) -- ~2e x p ( - a

60

40

°°

0.4

2

.............

30

Frequenoy (l/m)

." " /

0.0 0

20

The function SP(w) should also satisfy the following conditions

0.0

SP(O) = 0 and SP(oo) = 0

-0.~ ,

-I.0 0.0

i

0.5

1.0 Lag (m)

From Eqns (13) and (14), it follows that

i

1.5

2.0

/3= 4a /

1.0

Rllll~l~n¢o

T~ ~

0.5

-0.~

-1.0 0.0

(14)

/=1

(a 2 + B~/)-]

(15)

mmuldon 1[ J

MOE process modeling

,,,

0.5

2400f-2.0E ' 1.0 (m)

' 1.5

2,0

Fig. 8. Ensemble autocorrelation functions of MOE.

For a profile en(x) sampled over a finite length L, an ergodic stationary record en(x) can be obtained using trend removal and standardization procedures [eqn (3a)]. The power spectral density of the record e~(x) is approximated by

spe,,(w) -- ~ -1£ IF.(w) 12

(16)

Yin-Tang Wang, Frank Lam, J.D. Barrett

68

where F.(03) is the Fourier spectrum, corresponding to a frequency w and can be obtained by performing Fast Fourier Transform procedure) 7 For each record, ~ and Bi(i = 1, N) in eqn (13) can be estimated by nonlinear least square fitting procedure with constraints minimization as:

~.o~

I Mr~c=" =.~...'~. ~

15

M m

q~E

=

~[Spen(03, ) -- SPe(03i)] 2

(17)

/=1

-1.0

subject to eqn (15), where SPe(03,) takes the form of eqn (13). In this study, N = 8 and 12 are selected to best fix the data for the two grades of lumber, 2400f-2.0E and 1650f-l.5E respectively. With the coefficients ~ and B, ( i = 1,N), the power spectral density of each record can be obtained from eqn (13). The power spectral density for the ergodic stationary process E* (x), SPe(03) can be obtained by averaging the power spectral densities of individual records over an ensemble of all samples: 1

M

s e E(60) = -~ ~

Wen (03)

(18)

0.0

'

'

0.5

Lag(m)

J

1.0

1.5

2.0

1.0

i

-0.5

°°1 0 f~

0.0

\'/

240062,0E

~ 0.6

, 1.0

.........

.

........

1.5

2.0

Lag (m)

Here, the subscript E refers to the MOE process.

Fig. 10. Ensemble autocorrelation functions of compressive strength.

Strength process modeling The autocorrelation function of each strength record can be calculated as

r;n(7-) = E[sn(t)" sn(t + 7")1

(19)

where E[.] denotes expected value. The function r~n(7-) varies from specimen to specimen with an ensemble average

1 N R*(7-) = ~ _ _ rs*n(7-)

(20)

n=l

The c~ and Bi(i = 1,N) in eqn (11) for the strength process can also be estimated by nonlinear least square fitting procedure with constraints minimization as: N

• s =

- Rs(7-, )]2

(21)

i=l subject to eqn (15), where Rs(7-i) takes the form of eqn (l l) where the subscript s refers to the strength process. The error in the approximate expression for the autocorrelation function in eqn (11) can be estimated by the formula as follows: M

=

[Rs(7-,) - Rs(7-,)] 2

in which M is the number of sample points within a record. In this paper, the errors in the strength process are 0.004488 for 2400f-2.0E and 0-004947 for 1650f-l.5E.

Here, N = 5 is selected to closely fit the data from both grades. Given a and Bi(i = I,N) for the strength process, the ensemble autocorrelation function can be obtained with eqn (11) and the ensemble power spectral density functions SPs(w) can be integrated directly from the ensemble autocorrelation functions. Gain Factor can then be obtained from the knowledge of the ensemble power spectra of MOE and strength. Figures 9 and 10 show the fitting curves of autocorrelation functions of MOE and strength data, respectively. Figure 11 shows the Gain Factor of both grades.

Trend process modeling Consider wi(i = 1, NL) as a series of data sampled at a constant interval Ax, and N L as the number of datum values. A trend ¢vi(i = 1,NL) for this data set can be determined as a polynomial of degree k k

~vi = y ~ bj(i. Ax) j

(i = 1, NL)

j=0 The coefficients bj can be estimated by solving following equations

NL k y ~ w i ( i . Ax) m= Z b j . i=l j=O

NL Z ( i . Ax)J+m i=l

( m = 0 , k)

In this study, the trends in each MOE and strength profile are assumed to be fifth order (k = 5) and third order (k = 3).

Simulation of correlated modulus of elasticity 10

randomly generate the parameters b = {a, B I , . . . BN} r is as follows

8

(a) generate zi, i = 1, k(k = N + 1) as independent standard random normal variables;

1650f-1.SE

6 4

k

(b) bi

2 0

69

=

~bi at- Z

Cijzj

(i = 1, k),

j=l i

0

20

40

~

The individual power spectral density Spe,(W) can then be obtained with eqn (13). The spectrum amplitudes can be calculated as:

i

6O

100

Fmquenoy (l/m) 10

ae, (o.;i) = 2 VSPen(Odi )" AOd Generation of records e~,(x) can then be accomplished by representation method '9 as

2400f-2.0E

6

M

aen(O')i ) COS(OdiX + Oi )

e;(x) = E i=1

where Oi are the random phase angles that vary uniformly between 0 and 27r. The trend removed function of M O E can, therefore, be estimated as

0 0

20

40

60

80

1O0

Frequenoy (l/rn)

Fig. 11. Gain Factor.

M

e ° (x) =em + es" Z The ensemble means of the first-order, within-member trend of M O E and strength, defined as the slope of the first order trend, #t and their covariance matrices ~t are obtained as well. For 1650f-l.5E, /~t --- {193"56,0"002} t and 1"0000

0"8184]

Gt = [0"8184

l'0000J

For 2400f-2.0E,/~t = {244.21,-0.073} t, and

= I ,.0000 -0.=199] Zt

L-0.2199

aen(°3i ) cos(0JiX + 0 i )

(23)

i=1

The internal mean and standard deviation, e m and e~, follow the 2-parameter Weibull distributions of the test data given in Table 1. From MOE test data, the ensemble mean /% and covariance matrix ~b of trend coefficients h can be obtained from trend removal process. The joint density function of the trend coefficient vector b can be obtained with eqn (22). The same procedure can be used to generate the trend coefficient bi, i = 1, k, then the record trend t(i • Ax) is given as follows k

1-0000

t(iAx) = Z b j ( i A x ~

(i : 1,NL)

j=0

The trends were added to each stationary profile e°(x) to preserve the nonstationary characteristics of within member data. Finally, the simulated function can be expressed as follows:

DATA REGENERATION MOE simulation In M O E processes, individual power spectral density Spen(Od) c a n be generated by multivariate standard normal simulation method with the ensemble means of parameters b = { o G B t , . . . B N } T and their covariance matrices Eb. The joint density function of the coefficient vector b can be got as:

f(b)

= (27r) "/21~ b [-1/2 • e x p { - 0 . 5 • (b - # b ) r E b ] ( b - #b)}

(22)

Cholesky decomposition is applied to Gb to obtain the lower triangle matrix C as CC r = ~'b. The algorithm to

M

k

i=1

j=0

(24) Parameters for the Normal and 2-parameter Weibull distributions fitted to the simulated internal mean, minimum and standard deviation of M O E are compared with the test data in Table 1. The cumulative probability distributions of internal mean, minimum and standard deviation o f M O E are also shown in Figs 2 to 4 to compare with the distributions of test data. Figure 8 shows the ensemble autocorrelation functions

Yin-Tang Wang, Frank Lam, J.D. Barrett

70

of simulated MOE profiles. Good agreement is found in all cases between prediction and test data.

Strength simulation Using the gain factor, the individual power spectral density of simulated strength spsn(w) can be estimated as:

Spsn(W) = IH(w)12nSpen(W)

(25)

Here, the sample gain factor [H(w) 12 was assumed to be the same as the ensemble one IH(w) I2. The spectrum amplitude of strength profile can be obtained

asn(wi ) = 2 V/SPsn(wi)Aw

(26)

Generation of records s*,(x) can also be accomplished by representation method 19 as M

s*(x) = Z

asi(~i) cos(wix + Oi)

i=1

The bivariate standard normal distribution model is used in the strength simulation process to represent the correlation between the ensemble means, standard deviations and the first-order trends of MOE and strength. From the test data, the ensemble means of average and the standard deviation of MOE and strength t~m and t~s, and their covariance matrices Zm and Es can be obtained. For 1650f-l-5E, t ~ = {11621,45.63} t, /~s = {568.05,4.36} t and

E"

= [1"0000 [0-4782

For 2400f-2-0E, 4"40}t and [1'0000 Era= 0"7484

0"4782] 1.0000J

Zs

= [1"0000 [0"3840

0"3840] l'0000J

/.Lm={14090,55"23} t, /~s={634"12, 0"7484] l'0000J

=[1"0000 Es [0"2416

0"2416] l'0000J

The compressive strength record can be generated according to M

sn(x) =sm + ss E a~i(wi) cos(a;ix + Oi) + s,x

(27)

i=1

Parameters for the Normal and 2-Parameter Weibull distributions fitted to the simulated internal mean, minimum and standard deviation of compression strength are compared with the test data in Table 2. The cumulative probability distributions of internal mean, minimum and standard deviation of the strength are also shown in Figs 5-7. Figure 9 shows the ensemble autocorrelation functions of strength for both the fitted curve from test data and the simulated results. Good agreement between the prediction and test results can be observed.

CONCLUSION This paper developed a single-input/single-output Gain Factor Model to simulate the correlated MOE and strength profiles of lumber in standard normalized space. It also provided a method to obtain the mathematical formulae of ensemble autocorrelation functions and power spectra from the test data. The method can estimate the spectrum of data profile with few points, with which the strength of lumber can be estimated by the spectral representative method. The Gain Factor Model coupled with the trend removal model and multivariate standard normalization model can be used to simulate the real records of correlated non-stationary and non-ergodic MOE and strength processes. Good agreement between model prediction and test data is obtained. This study only considered the ideal situation with no extraneous noises at input and output points. Furthermore, only single-input/single-output cases are considered. In the future, multiple-input/multiple-output cases with noises should be considered where additional inputs correspond to other non-destructively measured parameters and outputs may correspond to other strength properties.

ACKNOWLEDGEMENTS This research is financially supported by the Council of Forest Industry of British Columbia in Canada, which is gratefully acknowledged by the authors.

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