12
LINDBERG AND KULACHENKO
FIGURE 15
Failure evaluation with Tsai – Wu theory. In (A) Board A; and in (B) Board B
TABLE 4 Weibull parameters for the tensile and compressive strengths from Figure4
Paperboard
Strength
Scale η [MPa]
Shape β [MPa]
Mode [MPa]
BoardA
MDtension
60.0
39.6
60.0
( ) 30.0 a
MD compression
30.0
39.6
CDtension
29.4
45.3
29.4
( ) 14.7 a
CD compression
14.7
45.3
BoardB
MDtension
70.6
57.0
70.6
( ) 35.3 a
MD compression
35.3
57.0
CDtension
32.3
60.3
32.3
( ) 16.1 a
CD compression
16.2
60.3
a The negative sign must be added after the fitting procedure.
1 η
where the scale η and shape β must be determined. The results of the fit of Equation 12 to the tensile tests in Figure 4 are shown in Table 4. Since the Weibull distribution returns 0 for a negative variable, the distributions for the compressive stresses were created by divid- ing the tensile test data by two (since the failure in compression is 50% of that in tension) for the MD and the CD respectively, which were then used for fitting of the Weibull function. The analysis continues with Monte Carlo simulations of the σ TW in the studied locations. As seen in Equation 7, the current stress state ( σ 11 , σ 22 , σ 12 ) in the studied location is a part of the σ TW expression. In Figure 16, a close-up of the corners of the trays is shown. For Board A, the σ TW is about 1.7 [-] in the most critical point, and for Board B the σ TW is about 0.8 [-]. Note that the locations are not the same. The stress state in the studied location is σ 11 ¼ 75MPa, σ 22 ¼ 50 MPa and σ 12 ¼ 1 : 5 MPa for Board A, and for Board B the stress state is σ 11 ¼ 9 : 5MPa, σ 22 ¼ 34MPa and σ 12 ¼ 2 : 7MPa. Now Monte Carlo simulations are performed of Equation 7 where the F i and F ij coefficients are re-calculated each time with values ran- domly picked from the Weibull tensile strength distributions. The values picked from the distributions for the compressive stresses are multiply by 1 to make the stresses negative. The distribution of the σ TW Monte Carlo simulations is expected to have a skewness, partly due to the Weibull distributed input strength parameters, and partly due to the non-linearity of the Tsai – Wu stress equation itself and its parameters, see Equations 7 and 8. The distribution with the best fit to the Monte Carlo simulations was the Generalized Extreme Values (GEV) distribution. The GEV PDF reads
β þ 1 e t x ð Þ
f x ð Þ¼
t x ð Þ
ð 13 Þ
1 = β
x μ η
t x ð Þ¼ 1 þ β
if β ≠ 0
Equation 13 is fitted to the Monte Carlo simulations for the two boards using the Matlab gevfit -function, and the results are shown in Figure 17. The fitted GEV distribution parameters and the modes are shown in Table 5. With the cumulative distribution function, CDF, for the fitted GEV functions, the probability of staying under the failure value σ TW = 1 can now be derived. The CDF for the GEV function reads F x ð Þ¼ e t x ð Þ ð 14 Þ and derives the probability of staying below x . With Equation 14 the probability of passing x ¼ 1, that is, σ TW ¼ 1, becomes for Board A in practice 100%. For Board B, this probability is about 1%, or 1 out of 100 trays can be expected to fail in this location. The parameters in Table 5 only apply for the specific stress state ( σ 11 , σ 22 , σ 12 ) since these are included in the expression for the Tsai – Wu stress, and hence affect the results of the Monte Carlo simu- lations. Hence, both point 5 and 6 must be performed each time a new location in the model is evaluated. That is, new Monte Carlo sim- ulations must be performed for the studied stress state and then the risk of getting failure must be evaluated again. Other examples of how
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