*3.4. Statistical Static Strength Assessment*

To qualify for use on specific aircraft, spars were required to support 115% of the DLL with no permanent deformation, while being capable of supporting 150% of the DLL with no failure. For this test, failure was defined as a 30% drop from the maximum load. Assessing a strength criteria beyond this unidimensional pass/fail condition may provide additional feedback that can be used to evaluate spar behavior across a sample set. For example, the traditional pass/fail test only indicates that a group of spars meets or does not meet a threshold. In this case, the pass/fail classification may not allow for sufficient discretization to determine that one of those spars may have exceeded the criteria significantly. Hypothetically, that spar may have displayed certain favorable progressive damage characteristics that would predict its higher strength. Knowing that extra information could allow for the part production parameters to be studied, and lessons learned to be applied to the next production lot, thus continuously improving part quality.

In order to use the proposed statistical tests, the AE data distribution was examined and verified to follow a normal distribution using the Lilliefors test, a method employed when the population mean and standard deviation is unknown [61]. The Lilliefors analysis was executed using MATLAB R2017a (The MathWorks, Natick, MA, USA), and the results of the normality plot are shown in Figure 6. Both the first damage signal load and spar ultimate load followed a normal distribution (failed to reject the null hypothesis at a 1% significance level).

**Figure 6.** Data normality test for: (**a**) the load at first damage signal, and (**b**) the ultimate load datasets.

Probabilistic assessment (e.g., hypothesis testing) can give additional part performance insight, and the multi-dimensional AE datasets used in this research enable the use of statistical analysis methods. As an example of how AE data could be used to evaluate composite test article performance, a one-sided hypothesis test and lower confidence limit (LCL) was applied to assess both spar strength requirements. Summary statistics for both datasets are listed in Table 4, and are representatively displayed in Figure 7 for comparison.

**Figure 7.** A schematic representation of the probabilistic assessment. (**a**) First damage AE signal distribution, (**b**) ultimate load distribution.


**Table 4.** Summary statistics for the response variables of interest.

The lower confidence limits serve as average predictions for the lowest probable load where damage may initiate and the lowest probable ultimate load, respectively. They could potentially be used as a bound in statistical process control, or as quick indications of part lot performance from a "weakest link" perspective. Alternatively, upper confidence limits (not shown) could also be used to identify spars that perform exceptionally well during testing.

To evaluate spar strength requirements explicitly, the hypothesized population mean was first determined based on the requirement descriptions. Estimated crane-loads at the first AE damage signal were used as an interpretation of the "support 115% design limit load (DLL) with no permanent deformation" requirement, and ultimate loads were used to evaluate "support 150% DLL with no failure". Therefore, the null hypothesis for the first strength requirement was that the first damage signal population mean is equal to 1.15 DLL, and the alternative was that the first damage signal population mean is less than 1.15 DLL. In other words, the analysis was run to examine whether if on average, the first damage signals occurred at or below 1.15 DLL for the total population of spars, given the tested sample population. Figure 7a shows the requirement distance from the sample mean and the sample LCL, schematically. The analysis was run in MATLAB, and the null hypothesis was rejected at a 5% significance level (*p* = 0.0174). According to the results, the probability of the average first damage signal occurring below a crane-load of 115% DLL (the third test load increment and the first strength requirement) was 98%. This suggests that most spars will exhibit observable damage signals at or below the second loading level (100% DLL), a conclusion confirmed by the analysis of multiple AE parameters discussed earlier in this work.

Similarly, the ultimate load distribution can be examined (Figure 7b). The null hypothesis for the second strength requirement was that ultimate crane load population mean is equal to 1.28 DLL (a 30% reduction from the sample population average ultimate load) and the alternative was that the ultimate crane load population mean is less than 1.28 DLL. The analysis was run to examine whether on average, the average ultimate crane load could occur below a "30% drop from the maximum load" in accordance with the requirement. The MATLAB analysis failed to reject the null hypothesis at a 5% significance level (*p* = 1); there was not enough data to support the alternative hypothesis. Based on the results presented, the probability that the average spar ultimate crane-load was less than 128% DLL (30% less than the average ultimate load) was less than 1%. Therefore, almost all spars can withstand the maximum load without failure. Again, this was confirmed by the summary of results presented in

Table 3, to which the probabilistic assessment presented herein adds confidence; the lowest recorded individual spar ultimate load was 151% DLL.
