This section is from the "The Economics Of Railroad Construction" book, by Walter Loring Webb, C.E.. Also see Amazon: The Economics Of Railroad Construction.

The basis of the computation of this least objectionable form of grade is as follows: The resistance to the movement of a train on a straight level track is variable, depending on the velocity, the number and character of the cars, and on the character of the road-bed and track. No one figure that can be stated may be considered accurate for all cases, but for average conditions and for average velocities we may consider that the round number of 10 pounds per ton is a reasonable figure. This value agrees fairly well with the results of some dynamometer tests made by Mr. P. H. Dudley, using a passenger-train of 313 tons running at about 50 miles per hour. It also agrees with Searles's formula (based on experiments) for the resistance of a freight-train with 40 cars running 25 miles per hour. Using the very approximate resistance formula published by the "Engineering News," which makes the resistance in pounds per ton equal to [2+(V/4)], in which is the velocity in miles per hour, this value would be true for a train moving at a speed of 32 miles per hour. A comparison of the three cases mentioned above shows at once the wide variations in the values given by different formulae. Therefore this value of 10 pounds per ton may be considered to be as nearly correct for an average value as any other one value that can be chosen. Ten pounds per ton is the grade resistance of a 0.5% grade, or a grade of 26.4 feet per mile. On this basis a 0.5% grade will just double the tractive resistance on a straight level track. We may compute, as in the previous chapter, the cost of doubling the tractive resistance for one mile, but, since the extra resistance is due to lifting the train through 26.4 feet of elevation, we may divide the extra cost of a mile of 0.5% grade by 26.4 and we will have the cost of one foot of difference of elevation. If the rate of grade is not so great that it has an effect in limiting the length of trains, we may then say that the cost of this one foot of difference of elevation is independent of the rate of grade. On account of the compensating character of the effect of grade in the operation of trains down the grade or in the operation of a train down the other side of an elevation which has just been climbed, we must consider the total effect of one foot of rise and fall. Although we may say in a general way that the cost of one foot of rise and fall is independent of the rate of grade, it is true, as will be seen, that the cost of a foot of rise and fall of a very light grade is very much less than the cost of a foot of a much heavier grade.

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