Assuming the same figures already considered in § 129, we will consider that the grade against the heaviest traffic and for fully loaded trains is 1%. The grade and tractive resistance for a rating ton on this grade is 22.6 pounds per ton. If the locomotive has a tractive power of 28,200 pounds, it can handle 1248 gross rating tons. The actual weight of the locomotive and tender is 130 tons; multiplying this by 128% for the 1% grade, we have 166 rating tons for the locomotive which leaves 1082 rating tons behind the tender. If the fully loaded trains have a live load equal to twice the weight of the cars, their actual tonnage will be 3/(2+1.28) x l082 = 1081 tons. This means that the cars weigh 360 tons and the freight 721 tons. Assume that the total live load carried in the opposite direction is but 1/3 of the above, we would then have 360 tons of cars and 240 tons of freight, which will aggregate 600 tons. To determine the rating tons corresponding to 600 actual tons of loading, we must make first a trial estimate of the grade in order to determine the value of a rating ton on that grade. We will assume as a trial that the grade is 1.6%. At this grade the ratio of a tare ton to a rating ton is 119%. Since the live load is 1/3 of a nominal full load, which means that it is 2/3 the weight of the cars, to reduce 600 actual tons to rating tons at 1.6% we must divide 600 by (2/3+1)/(2/3+1.19') which equals 668 rating tons.

But on a 1.6% grade the 130-ton locomotive will have the equivalent weight of 155 rating tons. Adding this to 668 rating tons for the load behind the tender, we have 823 rating tons as the total weight of the train. If we divide the total tractive power 28,200 by 34.8, the tractive resistance of a rating ton on a 1.6% grade, we would have 811 as the total capacity in rating tons of the locomotive on the 1.6% grade. This agrees fairly well with 823, but it proves that the trial rate of grade, 1.6%, is a little too high. If we were to carry through another trial calculation on the basis of 1.5%, we would find a far greater discrepancy in the other direction. Considering that our assumption of the probable weight of the loading in the direction of light traffic is at its best a gross approximation, any over-refinement in these calculations is a mere waste of time. We may therefore say that, under the above conditions, a grade of 1% against the heaviest traffic will give as much resistance and require as much work of the engines as a grade of 1.6% against the assumed lighter traffic.

213. Reliability Of Calculations Of This Nature

As before intimated, this is not a question which will ordinarily concern the engineer of a light-traffic, cross-country road. The practical difficulty of predicting the relative amount of traffic on a road before it is constructed, and the probability that such figures, no matter how correct they might be in the early history of the road, will be permanently altered in the course of 20 or 50 years, means that very little reliability can be placed on such computations except in a general way. The great east and west trunk lines, although they find that there are fluctuations in the relative amounts of traffic, have also discovered that, in a rough way, the east-bound traffic is permanently far greater than the west-bound traffic. The Pennsylvania Railroad, in the course of the carefully developed reconstruction of their line and the building of a low-grade line between Philadelphia and Pittsburg, have kept this principle in mind, and have uniformly designed the ruling grade against the east-bound traffic at a considerably lower figure than that against the west-bound traffic. The Canadian Pacific has already begun extensive reconstruction of their line in order to follow this same principle. In an extreme case pusher grades may be used to accomplish the same object, but this does not alter the principle involved. A pusher grade should always be considered as a special case of a ruling grade of a little over one-half the rate which is operated in a special way, and the above principle is one which applies to the relative rate of the ruling grade. Although the engineer of a light-traffic road may not find it justifiable to spend any added amount of money to follow this principle, he should keep it in mind and endeavor to so design his ruling grades to conform to this principle, if it may be done with little, if any, added expenditure.