Monte Carlo Study of Holding Forces for Tank Cars on Grades

This paper describes a numerical procedure to examine the holding forces needed to secure a cut of railroad tank cars staged on a grade during loading and unloading operations. Holding forces are created by applying emergency brake systems and blocking (or chocking) wheels. Moreover, the holding force to secure the cut of cars must be greater than or equal to the gravitational component of force acting on the cars that is parallel to the grade. Engineering statics are applied to examine the forces acting on the individual cars resting on an inclined plane. An equation to calculate holding force is developed that includes two types of factors: constants (i.e. nonrandom or deterministic factors) and probabilistic variables (i.e. factors with inherent uncertainty or randomness). The numerical procedure applies Monte Carlo simulation techniques to study the uncertainties in the engineering analysis. The Monte Carlo approach is well suited to study the uncertainties and inherent variability associated with some of these factors. The factors assumed to be deterministic in this procedure are: steepness of the grade, total number of cars on the grade, number of cars with hand brakes applied, number of chocked wheels, and weight of the tank cars. The factors treated as random variables are: tension in the hand brake chain, mechanical efficiency in the linkages of the brake system, coefficient of friction between the brake pad and the wheel, and the coefficient of friction between the chocks and the rail. Probability distributions are assumed for each of the random variables. In addition, a probabilistic sensitivity analysis is conducted to examine the relative effect of the random variables on the reliability of the braking system to secure the cut of tank cars on a grade.