11
LINDBERG AND KULACHENKO
FIGURE 13
Total normal strain distribution sampled from the midplane. In (A) Board A MD strain, (B) Board B MD strain, (C) Board A CD
strain and (D) Board B CD strain
FIGURE 14
Failure evaluation with Maximum Strain theory. In (A) Board A and in (B) Board B
3.4
Failure evaluation
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numbers of the tensile and the compressive stresses are picked from their respective distribution. 6. Analyse the results to see the probability for failure, that is, proba- bility to pass σ TW = 1, in the critical location. So far, point 1 to 4 has been performed. The tensile tests from point 2 are shown in Figure 4 and consist of 10 curves in each direction (MD, CD and 45). The number of points is too low to make a reliable estimate of how the tensile strengths spread. However, it has been seen in previous studies 37 that the strength parameters for paper can be well described with the Weibull distribution. Hence, the Weibull probability density function, PDF, is fitted to the tensile strengths from the tensile tests performed for this study. The fit is performed utilizing the Matlab wblfit- function. The two-parameter Weibull PDF reads
The Maximum Strain Theory was shown to be too conservative for this study. The Tsai – Wustress σ TW for Board A implies that the risk of fail- ure in the lower corner is very high, as seen in Figure 15A, where σ TW reaches 1.7 [-] which is far beyond the failure limit. Board B, however, has an area where σ TW reaches 0.8 [-] in the lower corner. In the fol- lowing, these values are further analysed to show the probability of failure in these locations. The analysis is performed as follows: 1. Run FE analysis. 2. Identify the distributions for the tensile and compressive strengths given by tensile tests. 3. Use the modes, that is, the most frequent value , of the tensile and compressive strength distributions to post-process the FE-model for the Tsai – Wu stress σ TW . 4. Identify critical locations in the model. 5. With the stress state ( σ 11 , σ 22 , σ 12 ) in the critical location, run Monte Carlo simulations of the σ TW (Equation 7) where random
β 1
x η
β η
β
x η ð Þ
f x ð Þ¼
e
ð 12 Þ
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