69 th Annual International Conference on Mass Properties Engineering

1. The Use of Inferential Statistics in Ships’ Stability Analysis 69th Annual International Conference on Mass Properties Engineering May 26, 2010 Michael Diggs, Melissa Cooley & David Hansch Naval Architects

2. Purpose of Damage Stability Analysis Produce a family of allowable KG curves to provide an overall assessment of stability Demonstrate that the ship meets vulnerability or recoverability requirements Provide technical guidance to ship designers for areas where improvements can be made Ensure that damage stability performance is not degraded during ship design and construction

3. Traditional Approach to Damage Stability Navy Ships Deterministic approach Uses engineering judgment to analyze the worst cases Commercial Ships Safety of Life at Sea (SOLAS) and International Maritime Organization (IMO) probability studies Provides an attained subdivision index Useful and important but does not answer What if…?

4. What if…? • Captain asks, “How likely am I to withstand a certain type of damage? • Requirement written in percentage • Ship able to survive grounding X% of time. • Ship able to perform it’s mission X% of time after damage. Traditional approach does not address these questions • Would have to analyze all cases to determine a passing percentage • Does not allow you to determine the actual KG value to survive damage a certain percentage of time. Must use statistics 4

5. Statistical Options Two statistical options exist • Bernoulli Trial • Results from this approach are binomial, all damage cases are either pass or fail • Case KG is compared to Ship’s KG limit • Calculate Percentage at Ship’s KG limit • Requires a large sample size • Inferential Statistics • a mathematical method that employs probability theory for inferring the properties of a population from the analysis of a sample taken from that population 5

6. Advantages of Inferential Statistics • Uses the required KG of each case to predict the population of required KGs • Allows a smaller sample size • Allows greater flexibility in determining the percentage of cases which will pass for a given KG 6

7. Assumptions / Sources of Uncertainty Assumptions • A Gaussian or normal distribution of the population of required KGs • The analyzed sample is randomly selected from the population Sources of Uncertainty • Population may not be truly Gaussian • The sample mean and standard deviation are not exactly the population mean and standard deviation 7

8. Estimating the Population from a Sample The sample mean and standard deviation provide estimates for the population mean and standard deviation Confidence Intervals are attached to these estimates to yield the range of values within which the population mean and standard deviation are likely to reside 8

9. Validating the Assumption of a Normal Distribution • The validity of the inference to the population can be assessed using a Chi-Square Goodness of Fit Test 9

10. Introduction of Trim • Resultant trim is insensitive to initial KG • If there are trim criteria for passing, failure of these criteria must be separately analyzed and then combined with the failure due to lack of KG cases • This is accomplished by raising the KG required in order that the overall passing rate is acceptable 10

11. Differences between Inferential Statistics Analysis and the SOLAS Probabilistic Analysis • This inferential statistics analysis considers passing the criteria to provide 100% likelihood of survival while SOLAS determines survival probability for individual cases • The SOLAS method requires analysis of every location on the ship while the inferential statistics method allows for a sample of damage locations to be used • SOLAS method provides an Attained Subdivision Index which is comparable to other ships while the inferential statistics method presented provides a likelihood of survival for a given KG or a KG required for a given likelihood of survival 11

12. Conclusions • Inferential Statistics is another tool for analyzing damage stability • For ships of complex subdivision this tool reduces the amount of calculations required to determine the likelihood of survival 12