Similarly, for an earth road having a cost of 15 cents per ton-mile and a tractive power of 100 pounds per ton, 1 foot of rise costs 0.028,4 cents, and the foot of ascent Essurned above will cost 0.028,4 cents for each ton going over the road. If this road has a traffic of 5 tons one way for 300 days of the year, the annual cost of the foot of rise and fall is 0.028,4X 5X 300= $44.60, which is the sum that can be spent annually to eliminate the foot of rise and fall.
From the point of view of the last solution, it appears that the cost of Class A rise and fall increases with the steepness of the grade, that is, increases as the rate of the grade approaches the angle of repose. In all probability this is correct, but all the data involved are too uncertain to warrant any further discussion of the subject here. However, the engineer should bear such relations in mind in solving a particular problem.
Distance vs. Rise and Fall. In locating a road the question may arise between the relative desirability of introducing rise and fall and of increasing the length of the line. The problem then is to determine the relative value of distance and of rise and fall.
If the conclusion in § 70 is correct, that the cost of Class A rise and fall is not appreciable, then the distance should not be increased at all to eliminate Class A rise and fall.
For Class B rise and fall an approximate solution can be obtained by assuming that it costs the same to develop a certain amount of energy in overcoming Class B rise and fall as to develop a like amount of energy in moving a load on a level road. This assumption is probably reasonably correct.
For example, the tractive resistance of the best broken-stone road is 33 pounds per ton, and the work necessary to raise 1 ton through 1 foot of rise is 2,000 foot-pounds; therefore to develop 2,000 foot-pounds of work on a level broken-stone road, a ton must be moved 2,000 4- 33 =60 feet. Hence the cost of operating 60 feet of distance on this road may be considered as equivalent to 1 foot of rise and fall. Therefore to eliminate a foot of rise and fall of Class B. the length of the road may be increased 60 feet. Table 12, page 59, gives the corresponding distance for other road surfaces.* Apparently writers on roads have not made a distinction between the several classes of rise and fall. Herschel says:* "To determine whether it is more advisable to go over than around a hill, all other considerations being equal, we have this rule: Call the difference between the distance around on a level and that over the hill d (the distance around being taken as the greater), and call h the height of the hill. Then in case of a first-class road, we go round
when d is less than 16h; and in case of a second-class road, we go round when d is lgss than 10h." Although not specially so stated, the above rule was plainly intended for broken-stone roads.
The above rule (which has been frequently quoted) recognizes no distinction between the several classes of rise and fall. It makes the avoidance of a foot of rise in going over a small culvert or of a foot of fall in crossing an open ditch, equally as important as the elimination of a foot of rise and fall on the maximum grade It is not possible to draw sharp lines between the several classes of rise and fall, but it is certain that there is a great difference in cost tween a foot of rise and fall on a flat grade and the same quantity on the maximum or limiting grade. Notice that the above rule makes the horizontal distance equivalent to a foot of rise much less than that stated in Table 12 above.
As an Ascent. The load which a team can draw over any road is determined by the length and steepness of the maximum grade; or, in other words, the length and rate of the permissible maximum grade depends upon the endurance of the team. The method of computing the load that a team can draw up any grade was explained in § 65, page 51. That investigation shows that the maximum grade varies greatly with the conditions of the surface; and that the better the surface the less should be the ruling grade. In other words, unless the maximum grade is light, the amount that can be hauled on a broken-stone road does not differ greatly from that on an earth road.