A MULTIPLE ORTHOGONAL POLYNOMIAL BIRNBAUM-SAUNDERS

A MULTIPLE ORTHOGONAL POLYNOMIAL BIRNBAUM-SAUNDERS MODEL FOR FATIGUE DATA

Mariano MARTÍNEZ-ESPINOSA1 Francisco LOUZADA-NETO2 Carlito CALIL JUNIOR3 ABSTRACT: The log-linear Birnbaum-Saunders model with p independent variables is used for the study of fatigue in materials. A special variant of this model is obtained when the values of the independent variables are coded using a factorial design. In this case, the independent variables are orthogonal, allowing for more accurate parameter estimates. Another advantage of this model is that it is derived by considering the basic characteristics of the fatigue process. Therefore, it is more accurate for the study of fatigue in wood. In this paper, we propose a multiple orthogonal polynomial Birnbaum-Saunders model to estimate fatigue in wood, with its respective maximum likelihood estimates end statistical inferences based on the asymptotic normal approximation of maximum likelihood estimates. Experimental data are presented on fatigue in wood under tension, obtained in the Laboratory of Wood and Timber Structures LaMEM-EESC-USP-Brazil. The estimation procedure based on the multiple orthogonal polynomial Birnbaum-Saunders model is compared with that obtained by considering the normal multiple orthogonal polynomial model. The results demonstrate that the parameter estimates obtained through the multiple orthogonal polynomial Birnbaum-Saunders model are at least 9% more accurate. Therefore, this model can be used for the study of fatigue data in wood and its derivatives. KEYWORDS: Birnbaum-Saunders distribution; maximum likelihood estimation; statistical inferential procedure; log-linear mode; orthogonal polynomia; fatigue in wood.

1 Introduction A great industrial problem is related to failures of materials which are related to different causes. Fatigue is the process of progressive localized permanent structural change occurring in a material subjected to conditions that produce fluctuating stress and strains at some point or points and may culminate in cracks or complete fracture after a sufficient number of fluctuations (ASTM E1150/87, 1987). The properties of fatigue in

1 Departamento de Estatística, Instituto de Ciências Exatas e da Terra, Universidade Federal do Mato Grosso – UFMT, CEP: 78060-900, Cuiabá, MT, Brasil. E-mail: [email protected] 2 Departamento de Estatística, Universidade Federal de São Carlos – UFSCar, Caixa Postal 676, CEP: 13565905, São Carlos, SP, Brasil. E-mail: [email protected] 3 Departamento de Engenharia de Estruturas, Escola de Engenharia de São Carlos, Universidade de São Paulos _USP, SCar, Caixa Postal 359, CEP: 13566-590, São Carlos, SP, Brasil. E-mail: [email protected]

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53

wood and wood products (e.g., laminated wood) are influenced by the following factores (Hansen, 1991): 1) The wood species, place of origin, density, etc. 2) The size and shape of the test specimen. 3) The moisture content. 4) The type of the applied load. Experiments Normally consist of tension, compression, bending or shear tests or a combination of these. Most fatigue experiments are made with harmonic loads and there are three important factors in this load type: • The R-ratio (the ratio between the minimum and maximum tension). • The value of stress. • The frequency (loading period) which must be taken in account. 5) Other factors affecting the properties of fatigue in wood are temperature, chemical treatments, adhesives, gluing, etc. Although some studies have focused on wood fatigue strength, statistical estimates are still quite limited due to the lack of statistical experimental design and an appropriate statistical model. The rationale underpinning the use of statistical experimental design for the study of the fatigue strength of wood is based on several factors, i.e., the need for fewer tests, shorter testing times and hence, lower costs, and less variable, more reliable results (Martínez, 2001). In this paper, we propose a multiple orthogonal polynomial Birnbaum-Saunders model to estimate fatigue data in wood and derivatives. Such a model may motivated by applications in the characterization of this material. Such characterizations are important for predicting the performance of the wood under different conditions. The Birnbaum-Saunders distribution with two parameters was derived by Birnbaum and Saunders (1969a) and was developed by considering the basic characteristics of the fatigue processes (Birnbaum and Saunders, 1969a). The Birnbaum-Saunders probability density function for a random variable T > 0 denoting fatigue time is given by (Birnbaum and Saunders, 1969a):

f (t ; α , β ) =

(t + β ) exp{

−1

2α 2

[ βt + 1

3

2 2π αβ 2 t 2

β t

− 2]}

,

(1)

where α and β are the shape and scale parameters, respectively, so that α, β > 0 . The joint maximum likelihood estimators for α and β were derived by Birnbaum and Saunders (1969b). Figure 1 shows the density graphs given by equation (1) for several values of α ( α = 0,10; 0,25; 0,50; 1,0; 1,5; 2,1), with β = 1 . The distribution shape is asymmetric when the α value increases. This is an advantage of equation (1) over the normal distribution for fitting fatigue data (Martínez and Calil, 2003).

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Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

FIGURE 1 - Density plot of equation (1) for several values of α , with β = 1 .

2 The log-linear Birnbaum-Saunders model 2.1 The model with one independent variable A common way to present fatigue results is the stress (S)-number per cycles (N) curve (ASTM: E739/80, 1981). This curve is composed of the plot of ln(N ) versus a oneto-one function of S . In practice, a logarithmic function ( ln(S ) ) is normally used, leading to:

ln( N ) = a + bx ,

(2)

where x equals S or ln(S ) . To estimate parameters a and b in equation (2), the following assumptions are considered (Rieck and Nedelman, 1991): Assumption 1: The number of cycles to failure (N) has a Birnbaum-Saunders distribution.

Assumption 2: The shape parameter α is independent of the stress per cycle ( S ). Birnbaum and Saunders (1969b) showed that, if N has a Birnbaum-Saunders distribution and if c > 0 is a constant, then cN has a Birnbaum-Saunders distribution, and if assumptions 1 and 2 are valid, then the equation (2) can be expressed as: N = exp[a + bx]δ

(3)

where δ has a Birnbaum-Saunders distribution. Applying the natural logarithm to equation (3), one has:

ln( N ) = a + bx + ln(δ)

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(4)

55

Thus, equation (4) is a log-linear model in an additive form, where ln(δ) has a sinhnormal distribution (sinus-hyperbolic-normal distribution). In other words, if the random variable N has a Birnbaum-Saunders distribution with parameters α and β , then the random variable Y = ln( N ) is a special case of the two-parameter sinh-normal distribution with the probability density function given by (Martínez, 1993):

f ( y; α , γ ) = {

2 2 2πα

}cosh{

y−γ −2 y−γ }exp{ 2 senh 2 ( )} α 2 2

(5)

where α is a shape parameter and γ = ln(β) is a location parameter. Note, however, that the general form of the sinh-normal probability density function is given by (Martínez and Achcar, 1991):

f ( y; α , γ , υ ) = {

2 2 2π α

} cosh{

y −γ

υ

} exp{

−2

α

2

senh 2 (

y −γ

υ

)}

(6)

where α is a shape parameter, υ is a scale parameter and γ is a location parameter. Shown below are the principal properties of the sinh-normal distribution (Rieck and Nedelman, 1991): • The distribution is symmetric around location parameter γ ;

• The distribution is strongly unimodal for α ≤ 2 and bimodal for α > 2 ; • The mean of the distribution is given by E ( y ) = γ . Figure 2 shows the density graphs given by equation (5) for several special cases of α ( α = 0,10; 0,25; 0,50; 1,0), with β = 1 , i.e., γ = 0 . Note that the variability increases from α towards 1.

FIGURE 2 - Density plot of equation (7) for several values of α < 2 , with γ = 0 .

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2.2 The model with p independent variables The Log-Linear model (equation 4) can be rewritten for p independent variables if one considers ~ X * = ( X 1* , X 2* , , X *p ) (7) where X 1* , X 2* ,

, X *p are the independent variables, so that the assumptions of equations (1) and (2) can be generalized as follows (Martínez and Achcar, 1991): ~ ~ ~ Assumption 3: N = exp[ X *' θ ] , where θ ' = (θ1 , , θ p ) is a vector of unknown parameters that must be estimated and the number of cycles to failure (N) has a Birnbaum-Saunders distribution. ~ Assumption 4: The shape parameter α is independent of the explanatory vector X * . Moreover, if c > 0 , then cN has a Birnbaum-Saunders distribution with shape parameter α and scale parameter cβ . Based on this fact and the above-mentioned assumptions 3 and 4, N can be expressed as: ~ ~ (8) N = exp[ X * ' θ ]δ where δ has a Birnbaum-Saunders distribution with shape parameter α and scale parameter 1. Therefore, when one applies the natural logarithm in equation (8) one has: ~ ~ (9) Y = ln( N ) = X * 'θ + φ which is the log-linear model (equation 4) in its generalized form, where φ = ln(δ) is the error term of the model, which has a sinh-normal distribution with parameters α , γ = 0 and υ = 2 (Rick and Nedelman, 1991).

3 The multiple orthogonal polynomial Birnbaum-Saunders model Although Rieck and Nedelman (1991) consider the log-linear model in its generalized form (equation 9), they do not consider an orthogonal polynomial approach, which has the advantage of not inducing any correlation between the independent variables, thus allowing for more accurate parameter estimates. The values of the independent variables are coded by a factorial design (Martínez and Calil, 2000). To illustrate this statement, consider a 3k design with -1, 0 and 1 denoting the lower, medium and higher factor levels, respectively. The coded variables can be obtained by: x ic =

x i* − x i* s i*

(10)

x−*1 + x0* + x1* and si* = x0* − x−*1 or s i* = x1* − x 0* , where x−*1 , x0* e x1* are, 3 respectively, the lower, medium and higher factor levels. with xi* =

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57

~ Based on equation (10), the X * matrix in equation (9) can be rewritten as:

X =

1 x11 x12

x1 p

1 x21 x22

x2 p

1 xn1 xn 2

xnp

,

(11)

where vectors xip and xiq are orthogonal if xip xiq = 0 , for p ≠ q (Khuri and Cornell, 1996). Considering the matrix given by equation (11), the model given by equation (9) can be written as: ~ (12) Y = ln( N ) = X θ + φ , which is the multiple orthogonal polynomial model of the Birnbaum-Saunders distribution, where φ = ln(δ) is the random error of the model with a sinh-normal distribution, with parameters α , γ = 0 e υ = 2 . An important and special case of the model given by equation (12) is obtained for k = 2 independent variables, in which quadratic terms are considered, leading to the model given by: Y = θ0 + θ1 x1 + θ2 x2 + θ3 x12 + θ4 x22 + θ5 x1 x2 + φ ,

(13)

where θ0 , θ1 , , θ5 are the unknown parameters to be estimated. This model can be used in studies of fatigue in wood to estimate the effects of stress and frequency in the total number of cycles to failure and is considered here, in Section 6, to fit the data obtained at LaMEM-EESC-USP-Brazil. For fittings using the model given by equation (13), it is important to note that a minimum of 8 observed values are required, considering that there are seven parameters to be estimated. Moreover, since the model counts quadratic terms in both variables, at least three levels of each variable must be used, corresponding to at least 9 observed values for the parameters of the model (equation 13) to be estimated. Thus, we will consider a special orthogonal form of equation (11), which, without replications, is given by (Box and Draper, 1987): 1 −1 −1 1 1

1

1 0 −1− 2 1 0 1 1 −1 1 1 −1 1 −1 0 1− 2 0 X = 1 0 0−2−2 0 1 1 0 1−2 0 1 − 1 1 1 1 −1

(14)

1 0 1− 2 1 0 1 1 1 1 1 1

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However, experimental studies in engineering, physics, chemistry and other fields of research should always consider replicates ( r ) (Montgomery, 1991). In this case, each row of the orthogonalized matrix given by equation (14) is repeated r times.

3.1 Maximum likelihood estimators for the parameters of the quadratic model (equation 12) Let us assume that y1 ,

, yn are n independent variables of the model (equation 12),

with φi = yi − (θ0 + θ1 x1i + θ2 x2 i + θ3 x12i + θ4 x22i + θ5 x1i x2 i ) , where φi has a sinh-Normal distribution. From equation (5), the probability density function of

f (φi ) = (

φ i is given as follows:

2

1 ) × Wi × exp{− Z i2 } 2 2 2π

(15)

where

Wi =

2 y − (θ0 + θ1 x1i + θ2 x2 i + θ3 x12i + θ4 x22i + θ5 x1i x2 i ) cosh i , α 2

Zi =

2 y − (θ0 + θ1 x1i + θ2 x2 i + θ3 x12i + θ 4 x22i + θ5 x1i x2i ) sinh i , α 2

for −∞ < θ0 , θ1 , , θ5 < ∞ ; α > 0 and −∞ < yi < ∞ . The likelihood function for θ0 , θ1 , , θ5 and α is given by:

L(φ θ 0 ,θ 1 ,

,θ 5 , α ) = (

1

n

2 2π

)∏ Wi × exp{− i =1

1 n 2 Zi } 2 i =1

The natural logarithmic likelihood function (of the informative part) for θ0 , θ1 , and α is:

l (φ θ 0 ,θ 1 ,

,θ 5 , α ) =

n i =1

ln(Wi ) −

1 n 2 Zi 2 i =1

Differentiating the equation (17) with respect to θ0 , θ1 , zero, one obtains the following likelihood equations: ∂l (φ θ 0 ,θ 1 ,

,θ 5 ,α )

,θ 5 , α )

∂θ 1 ∂l (φ θ 0 ,θ 1 , ∂θ 2

,θ 5 , α )

, θ5

(17)

, θ5 and α , and equaling to

Z 1 n {Z iWi − i } = 0 2 i =1 Wi

(18)

=

Z 1 n x1i {Z iWi − i } = 0 2 i =1 Wi

(19)

=

Z 1 n x 2i {Z iWi − i } = 0 2 i =1 Wi

(20)

∂θ 0 ∂l (φ θ 0 ,θ 1 ,

(16)

=

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59

∂l (φ θ 0 ,θ 1 ,

,θ 5 , α )

∂θ 3 ∂l (φ θ 0 ,θ 1 ,

,θ 5 , α )

∂θ 4 ∂l (φ θ 0 ,θ 1 ,

,θ 5 ,α )

∂θ 5 ∂ l (φ θ 0 , θ 1 ,

=

=

Z 1 n 2 x1i {Z iWi − i } = 0 2 i =1 Wi

(21)

=

Z 1 n 2 x 2i {Z iWi − i } = 0 2 i =1 Wi

(22)

Z 1 n x1i x 2 i {Z iWi − i } = 0 2 i =1 Wi

(23)

,θ 5 , α )

∂α

n

= −

+

α

1

n

α

i =1

(24)

Z i2 = 0

Solving equation (24) leads to the maximum likelihood estimator of α 2 in terms of θˆ 0 , θˆ 1 , , θˆ 5 , which is given by: αˆ 2 =

4 n

n i =1

sinh

2

y i − (θˆ 0 + θˆ1 x 1 i + θˆ 2 x 2 i + θˆ 3 x 12i + θˆ 4 x 22i + θˆ 5 x 1 i x 2 i ) 2

where θˆ 0 , θˆ 1 , , θˆ 5 are the maximum likelihood estimators of θ0 , θ1 , obtained by numerical methods.

,

(25)

, θ5 , which are

3.2 Inference for the parameters For large values of n it is common practice to consider inferences of parameters θ0 , θ1 , , θ5 and α based on the normal asymptotic approximation of the maximum likelihood estimators θˆ 0 , θˆ 1 ,

, θˆ 5 and αˆ (Mood, et al., 1974).

Considering that: a

(θˆ 0 , θˆ 1 ,

, θˆ 5 , αˆ ) ~ N {(θ0 , θ1 ,

(26)

, θ5 , α ); I 0−1 )} ,

where I 0 is the observed information matrix, given by: −

n 1 4

v02 uˆi

−

n 1 4

i =1

−

n 1 4

v0 v1uˆi

−

n

i =1

v1v0 uˆi −

n 1 4

i =1

i =1

n

n

v0 v5uˆi

1 4

−

n 1 αˆ

i =1

v12 uˆi

−

n

v1v5uˆi

1 4

v0 Zˆ iWˆi

i =1

−

n 1 αˆ

i =1

v1Zˆ iWˆi

i =1

I0 = −

1 4

v5 v0 uˆi −

1 4

i =1

−

n 1 αˆ i =1

60

v5 v1uˆi

−

n 1 4

i =1

v0 Zˆ iWˆi −

n 1 αˆ i =1

v uˆi 2 5

−

n 1 αˆ

i =1

v1 Zˆ iWˆi

−

n 1 αˆ i =1

,

(27)

v5 Zˆ iWˆi

i =1

v5 Zˆ iWˆi

−

1 αˆ 2

(3

n

Zˆ i2 − n )

i =1

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with

2 y − (θˆ 0 + θˆ 1 x1i + θˆ 2 x2 i + θˆ 3 x12i + θˆ 4 x22i + θˆ 5 x1i x2 i ) Wˆi = cosh i , αˆ 2 2 y − (θˆ 0 + θˆ 1 x1i + θˆ 2 x2 i + θˆ 3 x12i + θˆ 4 x22i + θˆ 5 x1i x2i ) Zˆ i = sinh i , αˆ 2 Zˆ 2 ~ uˆi = 1 − Wˆi 2 − Zˆ i2 − i 2 , v0 = x 0i = 1 , v1 = x1i , v 2 = x 2i , v3 = x12i , v4 = x22i e v5 = x1i x 2i . Wˆ i

The elements of the matrix (equation 27) are the minus of the partial second derivatives of l (θ 0 ,θ 1 , ,θ 5 , α ) with respect to θ 0 ,θ 1 , ,θ 5 , α , which are given in detail in Appendix A. ~ One can also use the Fisher information matrix I (θ * ) instead of I 0 (equation 27), ~ where θ * = (θ 0 ,θ 1 , ,θ 5 , α ) , in which case the Fisher information matrix is given by: ~ ~ ∂l (θ * ) ∂l (θ * ) − } E { } ∂θ 0 ∂θ 1 ∂θ 02 ~ ~ ∂l (θ * ) ∂l (θ * ) } E {− } E {− ∂θ 1 ∂ θ 0 ∂θ 12

E {−

~ I (θ * ) =

~*

~*

∂l (θ ) ∂l (θ ) } E {− } ∂θ 5 ∂ θ 0 ∂θ 5 ∂θ 1 ~ ~ ∂l (θ * ) ∂l (θ * ) E {− } E {− } ∂α ∂ θ 0 ∂α ∂θ 1

E {−

~ ∂l (θ * ) } ∂θ 0 ∂θ 5 ~ ∂l (θ * ) } E {− ∂θ 1 ∂θ 5 E {−

~ ∂l (θ * ) } ∂θ 0 ∂α ~ ∂l (θ * ) } E{− ∂θ 1 ∂α E {−

~*

~*

,

(28)

∂l (θ ) ∂l (θ ) } E {− } ∂θ 5 ∂α ∂θ 52 ~ ~ ∂l (θ * ) ∂l (θ * ) E {− } E {− } ∂α ∂θ 5 ∂α 2

E {−

which is obtained based on the assumptions of equation (27). The calculations relating to equation (28) are given in Appendix B. Therefore, considering the matrix given by equation (28), the final simplified form of the Fisher information matrix for θ0 , θ1 , , θ5 and α is given by: n

v 02

i =1 n

~ 1 4 I ( θ * ) = (1 + 2 ) 4 α

n

v 0 v1

i =1

v1 v 0

i =1

n

n

v0 v5

0

v1 v 5

0

i =1

v12

n

i =1

i =1

n

n

(29) n

v5 v0

i =1

0

v 5 v1

i =1

i =1

0

0

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v

0

2 5

4(1 +

4 −1 2n ) α2 α2

61

Considering the estimated variances of θˆ0 , θˆ1 ,

, θˆ5 and αˆ , which are directly obtained by inverting the matrix given by equation (29), the 100(1 − ψ )% confidence intervals for θ 0 ,θ 1 ,

,θ 5 and α are given by: IC (α ) = [αˆ ± Zψ / 2 vâr (αˆ ) ] ,

(30)

IC (θ j ) = [θˆ j ± Zψ / 2 vâr (θˆ j ) ] ,

(31)

and

for j = 1, ,5 , where Zψ / 2 is the upper ψ / 2 percentile of the standard normal distribution (Kalbfleisch, 1985).

4 Materials and methods The aspects relating to the materials and methods used here to estimate the values of fatigue strength in solid wood under tension are described below. • Wood Species: Caribbean Pine (Pinus caribaea var. hondurenses) and Eucaliptus (Eucalyptus grandis). • Sampling Method: 54 specimens were selected randomly from the two species (Martínez et al., 2000). • Program and preparation of the specimens: the specimens used to determine tensile fatigue strength in wood were prepared according to the dimensions established by Macêdo (1996) and (2000) and adopted by the Brazilian standard, NBR 7190/97 (1997). • Testing: static and cyclic tests were carried out using a DARTEC M1000/RC universal testing machine at the Laboratory of Wood and Timber Structures (LaMEM), São Carlos School of Engineering EESC, USP, Brazil. The static tests for the solid wood specimens followed the recommendations of the NBR 7190/97 standard (1997) and Macêdo (1996). The data obtained from the static tests on the two specimens were used to establish the levels of maximum and minimum tensile strength for the fatigue tests. Thus, the maximum stress levels for the cyclic loads were 90%, 75% and 60% of the material’s tensile strength ( f t 0 ), estimated from the static test of the twins specimens, and the minimum level stress ( σ min ) found was 10% of the maximum material strength ( σ max ). Reversed sinusoidal stress cycles were used in the cyclic tests, considering frequencies of 1, 5 and 9 HZ for the specimens. • Statistical design: the values of the factors (independent variables) were coded using a 3k factorial design. They also can be coded by Central Orthogonal Composite Design (Martínez, 2001) and the total number of cycles to failure N (response variable) was studied considering six replicates ( N ir for r = 1, ,6 ) and using combinations of two factors (stress and frequency), each tested on three levels. The levels for each factor

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and the coding for each level are given in Table 1, where -1, 0 and 1 are the codes x * − 75 x* − 5 obtained by equation (10): x1c = 1 , x 2c = 2 , where, 15 4 60 + 75 + 90 1+ 5 + 9 x1* = = 75 , x 2* = = 5 , s1* = 75 − 60 = 15 and s2* = 5 − 1 = 4 . 3 3 Table 1 - Number of cycles under three levels of stress and three types of frequencies

Coded levels Stress (S) in % Frequency (f) in %

-1 60 1

xic x1* x 2*

0 75 5

1 90 9

5 Results Tables 2 and 3 present the experimental results considering the notation given in Table 1. The data in Tables 2 and 3 represent the number of cycles corresponding to the tensile fatigue strength of solid wood of the two wood species, Pinus Caribaea var. Hondurensis and Eucaliptus Grandis. The tests were carried out at the Laboratory of Wood and Timber Structures - LaMEM of EESC-USP, Brazil (Macêdo, 2000). Table 2 - Solid wood fatigue data for the species Pinus Caribaea variety Hondurensis

Run (i) 1 2 3 4 5 6 7 8 9

N i1 211480 93432 91 900005 360056 187 793417 150375 103

Ni 2 430899 118610 57 703389 124050 417 1106942 241314 325

Response ( N ) Ni3 Ni 4

Ni5

Ni6

204017 178388 21 1794615 309413 348 1814582 454026 649

252510 72917 113 1382568 253120 584 1330109 297954 669

400197 87134 87 791034 219832 381 1016584 373929 925

229270 66818 139 1149067 163539 617 1264336 264474 619

Factors f S 60 75 90 60 75 90 60 75 90

1 1 1 5 5 5 9 9 9

Table 3 - Solid wood fatigue data for the species Eucaliptus Grandis

Run (i) 1 2 3 4 5 6 7 8 9

N i1 300060 153472 13 1601572 339606 181 963383 175454 355

Response ( N ) Ni 4

Ni 2

Ni3

447727 124896 45 902906 255226 120 1367100 241199 157

256633 81872 204 1310678 356979 236 1410236 310731 430

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

483367 94734 104 867411 249317 639 1145759 473653 284

Ni5

Ni6

428444 77639 34 1413229 182182 594 1145760 332578 745

180865 114332 134 1887691 290385 897 1535709 329510 1077

Factors f S 60 75 90 60 75 90 60 75 90

1 1 1 5 5 5 9 9 9 63

6 Data analysis The methodology described in the previous sections was applied to the two data sets. The results are summarized in Tables 4 and 5, which present the maximum likelihood estimator (MLE) obtained numerically, using equations (18) to (25) with their standard errors (SE) and their respective 90% confidence interval (CI), considering equations (30) and (31). A comparison was made of our results with the fitting of a normal orthogonal polynomial model, based on the results obtained by Martínez and Calil (2003). We found that, although the point estimates for the parameters were very close, the use of the Birnbaum-Saunders model led to smaller confidence intervals for the parameters, even though we used one parameter more than the normal model, since the estimated variances obtained by the Birnbaum-Saunders model are at least 9% smaller than those obtained with a normal model. Also, Figures 3-6 show the quantile-quantile plots for the residual fits, which confirm the slight better fit of the Birnbaum-Saunders model to the data. As pointed out by a referee, it is important to note that the diference between the plot should be relativized. Table 4 - Estimator, SE and CI of the parameters of the Birnbaum-Saunders multiple

orthogonal polynomial model and normal distribution for the data in Table 2

Parameter

θ0 θ1

Birnbaum-Saunders Model Estim. SE CI (90%)

Normal Model SE CI (90%)

Estim.

10.3295

0.0629

(10.2263 ; 10.4327)

10.3445

0.0660

(10.2362 ; 10.4527)

-4.0249

0.0771

(-4.1513 ; -3.8985)

-4.0014

0.0808

(-4.1339 ; -3.8689)

θ2

0.7205

0.0771

(0.5941 ; 0.8470)

0.7234

0.0808

(0.5909 ; 0.8560)

θ3

-0.8927

0.0445

(-0.9657 ; -0.8197)

-0.8854

0.0466

(-0.9619 ; -0.8090)

θ4

-0.1956

0.0445

(-0.2686 ; -0.1226)

-0.1893

0.0466

(-0.2657 ; -0.1129)

θ5 α

0.0922

0.0944

(-0.0627 ; 0.2470)

0.0908

0.0990

(-0.0716 ; 0.2532)

0.4754

0.0457

(0.4004 ; 0.5504)

Table 5 - Estimator, SE and CI of the parameters of the Birnbaum-Saunders multiple

orthogonal polynomial model and normal distribution for the data in Table 3

Parameter

64

Birnbaum-Saunders Model

Normal Model

Estim.

SE

CI (90%)

Estim.

SE

CI (90%)

θ0

10.3790

0.0707

(10.2631 ; 10.4950)

10.3882

0.0743

(10.2664 ; 10.5101)

θ1

-4.1416

0.0866

(-4.2836 ; -3.9995)

-4.1315

0.0910

(-4.2808 ; -3.9823)

θ2

0.7310

0.0866

(0.5889 ; 0.8730)

0.7139

0.0910

(0.5646 ; 0.8631)

θ3

-0.9227

0.0500

(-1.0048 ; -0.8407)

-0.9183

0.0525

(-1.0044 ; -0.8322)

θ4

-0.2176

0.0500

(-0.2996 ; -0.1356)

-0.2102

0.0525

(-0.2963 ; -0.1241)

θ5 α

0.1708

0.1061

(-0.0032 ; 0.3447)

0.1461

0.1114

(-0.0366 ; 0.3288)

0.5381

0.0518

(0.4532 ; 0.6230)

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

FIGURE 3 - Quantile-Quantile residual plot for the data in Table 2, considering the BirnbaumSaunders model.

FIGURE 4 - Quantile-Quantile residual plot for the data in Table 2, considering the normal model.

FIGURE 5 - Quantile-Quantile residual plot for the data in Table 3, considering the BirnbaumSaunders model. Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

65

FIGURE 6 - Quantile-Quantile residual plot for the data in Table 3, considering the normal model.

Conclusions It has been shown that the use of the proposed multiple orthogonal polynomial Birnbaum-Saunders model can be effectively used to estimate fatigue in wood and is of great practical interest. The advantage of this model is that it derives from the consideration of the basic characteristics of the fatigue process. Moreover, the parameters of the multiple orthogonal polynomial Birnbaum-Saunders model are not inducing any correlation between the independent variables, thus allowing for more accurate parameter estimates. The fatigue data analyzed revealed that the number of cycles to failure is significantly reduced when stress levels are increased and increases when frequency increases (Tables 4 and 5). From a practical standpoint, although only stress is usually considered to estimate the total number of cycles to failure, our experiments showed that the frequency is also statistically significant and must be taken into account when estimating fatigue in wood. Because the effects of S and f are significant, the proposed multiple orthogonal polynomial Birnbaum-Saunders model can be effectively used to estimate the total number of cycles to failure in wood as a function of frequency and stress rather than only as a function of stress, as it is the common practice. To conclude, the use of the model proposed here is of great practical interest for tensile fatigue studies in wood and wood derivatives. From the practical point of view, the maximum likelihood estimation procedure is implemented using an executable program. The software used for calculations of the Birnbaum-Saunders model is properly documented for use in the field of fatigue in wood and by-products. Interested readers can obtain an executable version of the program by writing to the first author. MARTÍNEZ-ESPINOSA, M. ; LOUZADA-NETO, F.; CALIL JÚNIOR, C. O modelo polinomial ortogonal múltiplo da distribuição de Birnbaum-Saunders para dados de fadiga. Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004. RESUMO: O modelo Log-Linear da distribuição de Birnbaum-Saunders para p variáveis independentes é utilizado para o estudo da fadiga dos materiais. Um caso especial deste modelo 66

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

é quando os valores das variáveis independentes são codificados utilizando um planejamento fatorial. Neste caso, as variáveis independentes podem ser ortogonalizadas possibilitando estimativas dos parâmetros mais precisas. Uma outra vantagem deste modelo é que o mesmo é derivado a partir das considerações das características básicas dos processos de fadiga, por conseguinte é mais preciso para o estudo da fadiga em madeira. Portanto, o objetivo principal deste trabalho é apresentar este modelo para estimar a fadiga em madeira, com seus respectivos estimadores de máxima verossimilhança e inferências estatísticas baseadas na aproximação normal assintótica dos estimadores de máxima verossimilhança. Também são apresentados dados experimentais de fadiga em madeira, obtidos no Laboratório de madeira e de Estruturas de Madeiras - LaMEM-EESC-USP-Brasil, nos quais os estimadores dos polinômios ortogonais múltiplos da distribuição de Birnbaum-Saunders são comparados com os polinômios ortogonais múltiplos da distribuição Normal. Os resultados desta comparação mostram que as estimativas obtidas por meio do modelo polinomial ortogonal múltiplo da distribuição de BirnbaumSaunders são 9% mais exatas, que as obtidas pelo modelo da distribuição normal, sendo assim o mesmo pode ser utilizado para o estudo de dados da fadiga em madeira e derivados. PALAVRAS-CLAVE: Distribuição de Birnbaum-Saunders, estimadores de máxima verossimilhança; inferência estatística; modelo log-linear; polinômio ortogonal; fadiga em madeira.

References AMERICAN SOCIETY FOR TESTING AND MATERIALS. ASTM E739/80: Statistical analysis of linear or linearized stress-life and strain-life fatigue data. Pittsburgh, 1981. p.29-137. AMERICAN SOCIETY FOR TESTING AND MATERIALS. E1150/87: Standard Definitions of Terms Relating to Fatigue. Philadelphia, 1987. p.1-10. ASSOCIAÇÃO BRASILEIRA DE NORMAS TÉCNICAS. NBR 7190/97: Projeto de Estruturas de Madeira. Rio de Janeiro, 1997. 107p. BIRNBAUM, S. W.; SAUNDERS, S. C. A new family of life distribution. J. Appl. Prob., Sheffield, v.6, p.319-27, 1969a. BIRNBAUM, S. W.; SAUNDERS, S. C. Estimation for a Family of life distribution with Application to Fatigue. J. Appl. Prob., Sheffield, v.6, p. 328-47, 1969b. BOX, G. E. P.; DRAPER, N. R. Empirical model – Building and response surfaces. New York: John Wiley, 1987. 669p. HANSEN, L. P. Experimental investigation of fatigue properties of Laminated Wood Beams, Timb. Eng. Conf., London, v.5, p.3742, 1991. KALBFLEISCH, J. G. Probability and statistical inference, 2 ed. New York: SpringVerlag, 1985. v.2, 360p. KHURI, A. I.; CORNELL, J. A. Response surfaces: designs and analyses. New York: Marcel Dekker, 1996. 668p. LEITHOLD, L. The Calculus with Analytic Geometry. New York: Harpers and Row, 1972, 1014p.

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MACEDO, A. N. Estudo de emendas dentadas em madeira laminada colada (MLC): Avaliação de métodos de ensaios. 1996. 132f. Dissertação (Mestrado em Estruturas) Escola de Engenharia de São Carlos, Universidade de São Paulo, São Carlos, 1996. MACEDO, A. N. Fadiga em emendas dentadas em madeira laminada colada (MLC). 2000. 216f. Tese (Doutorado em Estruturas) - Escola de Engenharia de São Carlos, Universidade de São Paulo, São Carlos, 2000. MARTÍNEZ, M. E. Desenvolvimento de um Modelo Estatístico para Aplicação no Estudo da Fadiga em Emendas Dentadas de Madeira. 2001. 191f. Tese (Doutorado em Ciência e Engenharia de Materiais) – Interunidades- EESC/IFSC/IQSC, Universidade de São Paulo, São Carlos, 2001. MARTÍNEZ, M. E. Uso de métodos bayesianos nas inferências para a distribuição de Birnbaum-Saunders. 1993. 106f. Dissertação (Mestrado em Estatística) - Instituto de Ciências Matemáticas de São Carlos, Universidade de São Paulo, São Carlos, 1993. MARTÍNEZ, M. E.; ACHCAR, J. A. Bayesian methods in accelerated life test considering a log-linear model for the Birnbaum-Saunders distribution. Rev. Bras. Estat., Rio de Janeiro, v.52, n.197/198, p.47-68, 1991. MARTÍNEZ, M. E.; CALIL, C. J. Statistical design and orthogonal polynomial model to estimate the tensile fatigue strength of wooden finger joints. Int. J. Fatigue, Guildford, v.25, p.237-43, 2003. MARTÍNEZ, M. E.; CALIL, C. J. Statistical fatigue experiment design in medium density fiberboard. J. Mater. Res., Pittsburgh, v.3, n.3, p.84-91, 2000. MARTÍNEZ, M. E.; CALIL, C. J.; SALES, A. Un método de muestreo para la determinación de las propiedades físicas y mecánicas de la madera. Rev. Madera Cien. y Tecnol., Concepción, v.2, n.1, p.5-20, 2000. MONTGOMERY, J. S. Diseño y análisis de experimentos. México: Panamericana, 1991. 589p. MOOD, A. M.; GRAYBILL, F. A.; BOES, D. C. Introduction to the theory of statistics. 3 ed. New York: McGraw-Hill, 1974. 564p. RIECK, J. R.; NEDELMAN, J. R. A. Log-linear model for the Birnbaum-Saunders distribution. Technometrics. Washington, v. 33, p.51-60, 1991.

Appendix A ~ ~ Partial derivatives according to l ( θ * ) of equation (17) with respect to θ * are given by: ~ ∂ 2l ( θ * ) 1 = ∂θ20 4 ~ ∂ 2l ( θ * ) 1 = ∂θ0∂θ1 4 68

n

i =1

v0 v1ui =

~ ∂ 2l ( θ * ) , ∂θ1∂θ0

,

n

v02ui ,

(A.1)

i =1

~ ∂ 2l ( θ* ) 1 = ∂θ 0 ∂ 5 4

n

i =1

v0 v5ui =

~ ∂ 2l ( θ * ) , ∂θ5∂θ0

(A.2)

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

~ ∂ 2l ( θ * ) 1 }= 2 ∂θ1 4 ~ ∂ 2l ( θ * ) 1 = ∂θ1∂θ2 4

~ ∂ 2l ( θ * ) , v1v2ui = ∂θ2 ∂θ1 i =1 n

n

v2 v3ui =

i =1

~ ∂ 2l ( θ * ) , ∂θ3∂θ2

,

n

n

v3v4ui =

i =1

~ ∂ 2l ( θ * ) 1 =− ∂θ0∂α α

n

i =1

n

i =1

v0ui* =

n

v2 v5ui =

i =1

,

~ ∂ 2l ( θ * ) , ∂θ 5 ∂θ 2

v32ui ,

(A.6)

(A.7)

i =1

n

n

v3v5ui =

i =1

~ ∂ 2l ( θ * ) , ∂θ5∂θ3

v42ui ,

(A.8)

(A.9)

i =1

~ ∂ 2l ( θ * ) 1 =− ∂θ5∂α α

~ ∂ 2l ( θ ) n 3 = 2 − 2 ∂α 2 α α

~ where θ * = (θ0 , θ1 ,

n

~ ~ ∂ 2l ( θ * ) ∂ 2l ( θ * ) 1 ,e }= 2 ∂θ5 ∂θ5∂θ4 4

~ ∂ 2l ( θ * ) , ∂α∂θ0

(A.4)

(A.5)

~ ~ ∂ 2l ( θ * ) ∂ 2l ( θ * ) 1 ,e = ∂θ 4 ∂θ 3 4 ∂θ3∂ 5

v4v5ui =

~ ∂ 2l ( θ * ) , v1v5ui = ∂θ5∂θ1 i =1 n

v22ui ,

~ ∂ 2l ( θ * ) 1 = ∂θ 2 ∂ 5 4

~ ∂ 2l ( θ * ) 1 = ∂θ24 4 ~ ∂ 2l ( θ * ) 1 = ∂θ 4∂θ5 4

(A.3)

i =1

~ ∂ 2l ( θ * ) 1 = 4 ∂θ32 ~ ∂ 2l ( θ* ) 1 = ∂θ3∂θ 4 4

v12ui ,

i =1

~ ∂ 2l ( θ * ) 1 , }= ∂θ1∂θ5 4

~ ∂ 2l ( θ * ) 1 = ∂θ22 4 ~ ∂ 2l ( θ * ) 1 = ∂θ2 ∂θ3 4

n

n

Z i2 ,

n

i =1

n

v52ui ,

(A.10)

i =1

v5ui* =

~ ∂ 2l ( θ * ) , ∂α∂θ5

(A.11)

(A.12)

i =1

, θ5 , α) , ui = 1 − Wi 2 − Z i2 −

Z i2 ~ , ui* = Z iWi , v0 = x0i = 1 , v1 = x1i , 2 Wi

v2 = x2 i , v3 = x12i , v4 = x22i and v5 = x1i x2 i .

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

69

Appendix B ~ The expected partial derivatives according to l ( θ * ) of equation (17) with respect to

~ ( θ * ) are given by: E −

E −

E −

~ ∂ 2l ( θ * ) 1 = ∂θ0 ∂θ1 4

~ 1 ∂ 2l ( θ * ) = 4 ∂θ0 ∂θ5

E −

E −

E −

E −

70

n

~ 1 ∂ 2l ( θ * ) = 4 ∂θ1∂θ5

~ ∂ 2l ( θ * ) 1 = ∂θ 2∂θ5 4

i =1

v0 v5 1 +

Z2 4 +E 2 i 2 2 Zi + 4 / α α

=E −

~ ∂ 2l ( θ * ) , ∂θ5∂θ0

(B.3)

,

(B.4)

n

i =1

4 Z i2 + E α2 Z i2 + 4 / α 2

v1v2 1 +

4 Z2 +E 2 i 2 2 α Zi + 4 / α

=E −

~ ∂ 2l ( θ * ) , ∂θ2 ∂θ1

(B.5)

v1v5 1 +

Z2 4 +E 2 i 2 2 Zi + 4 / α α

=E −

~ ∂ 2l ( θ * ) , ∂θ5∂θ1

(B.6)

,

(B.7)

n

v22 1 +

i =1

4 Z i2 + E α2 Z i2 + 4 / α 2

v2v3 1 +

4 Z2 +E 2 i 2 2 α Zi + 4 / α

=E −

~ ∂ 2l ( θ * ) , ∂θ3∂θ2

(B.8)

v2v5 1 +

4 Z2 +E 2 i 2 2 α Zi + 4 / α

=E −

~ ∂ 2l ( θ * ) , ∂θ5∂θ 2

(B.9)

i =1

n

v12 1 +

i =1

i =1

n

(B.1)

(B.2)

i =1

n

,

~ ∂ 2l ( θ * ) , ∂θ1∂θ0

i =1

n

4 Z i2 + E α2 Z i2 + 4 / α 2 =E −

i =1

n

v02 1 +

4 Z2 +E 2 i 2 2 α Zi + 4 / α

~ ∂ 2l ( θ * ) 1 = 2 ∂θ2 4

~ ∂ 2l ( θ * ) 1 = ∂θ 2∂θ3 4

n

v0 v1 1 +

~ ∂ 2l ( θ * ) 1 = 2 ∂θ1 4

~ ∂ 2l ( θ * ) 1 = ∂θ1∂θ2 4

E −

E −

~ ∂ 2l ( θ * ) 1 = 2 ∂θ0 4

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

E −

~ ∂ 2l ( θ * ) 1 = ∂θ32 4

~ ∂ 2l ( θ * ) 1 E − = ∂θ3∂θ 4 4 E −

~ 1 ∂ 2l ( θ * ) = 4 ∂θ3∂θ5

E −

E −

E −

E −

v32 1 +

i =1

4 Z i2 + E α2 Z i2 + 4 / α 2

4 Z2 v3v4 1 + 2 + E 2 i 2 α Zi + 4 / α i =1 n

n

4 Z i2 E + Z i2 + 4 / α 2 α2

v3v5 1 +

i =1

~ ∂ 2l ( θ * ) 1 = ∂θ24 4

~ ∂ 2l ( θ * ) 1 = ∂θ 4∂θ5 4

n

n

n

i =1

i =1

n

i =1

v52 1 +

E −

(B.11)

~ ∂ 2l ( θ * ) , ∂θ5∂θ3

(B.12)

,

(B.13)

~ ∂ 2l ( θ * ) , ∂θ5∂θ 4

(B.14)

,

(B.15)

=E −

=E −

4 Z2 +E 2 i 2 2 α Zi + 4 / α

~ ~ ∂ 2l ( θ * ) ∂ 2l ( θ ) =E − = ∂θ0 ∂α ∂θ1∂α

=E −

(B.10)

~ ∂ 2l ( θ * ) =E − , ∂θ 4∂θ3

4 Z2 +E 2 i 2 2 α Zi + 4 / α

4 Z i2 + E α2 Z i2 + 4 / α 2

v4v5 1 +

~ ∂ 2l ( θ * ) 1 = ∂θ52 4

v42 1 +

,

~ ∂ 2l ( θ * ) =0, ∂θ5∂α

~ ∂ 2l ( θ * ) 2n = 2 , ∂α 2 α

(B.16)

(B.17)

For calculations of the elements of the Fisher information matrix, Rieck and Nedelman (1991) show that, for small values of α ( 0 < α < 1 ), Z i given by equation (15) is normally distributed with mean zero and variance 1, i.e., Zi =

2 y − (θ0 + θ1 x1i + θ2 x2 i + θ3 x12i + θ4 x22i + θ5 x1i x2 i ) a senh i ~ N (0,1) , α 2

(B.18)

Considering equation (B.18), random variable Z i2 has a Chi-Square distribution with 1 degree of freedom ( χ12 ), which has mean 1 and variance 2 (Mood et al., 1974). Thus, for the hyperbolic trigonometric functions, one has the following result (Leithold, 1972): cosh 2 x − senh 2 x = 1 ,

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004

(B.19)

71

The use of equations (B.18) and (B.19) gives Wi 2 − Z i2 =

Wi 2 = Z i2 +

4 (1) , i.e., α2

4 . α2

(B.20)

Thus, substituting equations (B.18) and (B.20) in the expected partial second ~ ~ derivative of l ( θ * ) with respect to ( θ * ) , and defining:

c (α ) = 1 +

4 Z2 +E 2 i 2 , 2 α Zi + 4 / α

(B.21)

one obtains the Fisher information matrix, given by: 1 4

c (α )

n

v02

1 4

i =1 1 4

c (α )

n

v0v1

c (α)

1 4

i =1

v1v0

1 4

n

c(α)

i =1

~ I ( θ* ) =

n

c (α )

n

v0 v5

0

v1v5

0

i =1

v12

1 4

c(α )

i =1

n

i =1

(B.22) 1 4

c (α )

n

v5v0

1 4

i =1

c(α )

n

v5 v1

1 4

i =1

0

c (α )

n

v52

0

i =1

0

0

2n α2

Rieck and Nedelman (1991) and Martínez (1993) show that, for small values of α ( 0 < α < 1 ), E

Z i2 ≈0, Z i2 + 4 / α 2

(B.23)

hence,

c (α ) ≈ 1 +

4 . α2

(B.24)

Note that the expected partial derivatives according to equation (b.16) are equal to zero; therefore, parameter α is orthogonal to parameters θ0 , θ1 , , θ5 . Recebido em 01.09.2003. Aprovado após revisão em 24.09.2004.

72

Rev. Mat. Estat., São Paulo, v.22, n.2, p.53-72, 2004