Reliability Analysis of Corrosion at Multiple Locations  
For a large and complicated structure such as a railroad tank car, multiple failure modes (e.g. buckling, fatigue, fracture, corrosion, etc.) may exist simultaneously at certain stages and service conditions. One or more of the failure modes may occur at more than one location in a tank car at a given time. System reliability theory and solutionmethods are required to deal with such situations. This section focuses on the development and application of a reliability analysis methodology for tank cars with corrosion damage at multiple locations. When a tank car has corrosion at multiple locations, the failure probability of the tank car depends on failure probabilities at individual locations and can be strongly influenced by the degree of correlation between individual failure events. The lowerleft figure shows failure probability for a system with five different corrosion sites. Four different values of correlation coefficient in Cr between different corrosion sites are considered. The key cccr=0.50 in the figure stands for that the correlation coefficient between corrosion rate Crj and Crk is equal to 0.50, similarly for cccr=0.80, etc. The component of the highest failure probability of the five is also included in the figure as dashed line. It can be seen that (1) the system failure probability can be significantly higher than that of the highest individual failure probability; and (2) the higher the correlation between the individual failure modes (locations), the lower the system failure probability. It is helpful to know the relative importance of the components in determining the system failure probability, so that a proper priority can be decided in inspection, repair and maintenance of certain locations in tank car structures. The sensitivity of the system reliability index with respect to the component reliability index is a measure of importance for the components, and is useful for identifying the relative importance of the components to the system reliability. The equation in the lowerright figure defines such a measure. The pie chart shows the relative importance determined using the equation. It can be seen that in this case none of the components is dominant, with site 1 more important than any of the other sites, and sites 2 through 5 being about equally important. This information indicates that an improvement in the reliability at site 1 has more beneficial effect on the system reliability than a similar degree of improvement at one of the other sites. 
