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RISE AND FALL.

By the amount of rise and fall is meant the total vertical height through which a load must be lifted in passing in each direction over the road. It is distinct from and independent of the rate of gradient.

The minimum amount of rise and fall is found where the rise is all in one direction and the fall in the other, each being equal to the difference of eleva tion of the terminal points. Any increase in the rise and fall beyond this amount is represented by the rise encountered in passing from the higher to the lower terminus. This may be considered as avoidable rise and fall. If the cost of developing the work necessary to overcome rise and fall be the same as that of develop ing an equal amount of work to overcome distance, the rise and fall may be evaluated in terms of distance, and any change in rise and fall may be considered as though it were a difference in distance and treated as in Art. 17.

The value of rise and fall in terms of distance will depend upon the nature of the road surface, as the work necessary to lift a given load to a given height is a constant, while the work done in hauling a load over a given distance will vary with the resistance offered to traction by the surface. Thus, taking the surface as above, the work of lifting one ton through a rise of 1 foot is 200o foot-pounds, while with a tractive force of Ioo pounds per ton 200o - Ioo = 20 feet, the distance a ton may be moved on the level surface in developing 2000 foot-pounds of work. Therefore 1 foot of rise or fall may be considered as equivalent to 20 feet of level distance, and the value of reducing the amount of rise and fall may be found from that for reducing distance. If the road considered were a first-class macadam road, with resistance of 40 pounds per ton, i foot of rise or fall would equal 2000 _ 40 = feet of distance.

Where the rate of grade is less than the angle of repose of the wheels upon the road surface (see Art. 2) no additional work is imposed, by avoidable rise and fall, upon teams hauling loads over the road. The

amount of work done in lifting the loads up the rise is equal to that done by the grade in diminishing trac tion in descending the fall, and the total work required is equal to that necessary to haul the loads from one terminus to the other upon a uniform gradient. Upon an undulating road, therefore, where the grades are light, there is no economic advantage in reducing the rise and fall of the road.

When the rate of grade is greater than the angle of repose, the amount of work imposed by avoidable rise and fall is equal to twice that caused by the excess of fall above that at the angle of repose. In this case an additional amount of work must be done in applying a resistance to prevent the too rapid descent of the vehicle in going down the grade. The amount of this work in any case equals the work done in lifting the load to a height equal to the difference between the actual rise of the grade in question and the rise of a grade of the same length and a rate equal to the angle of repose. Thus on an ordinary earth road whose resistance to traction where level is I00 pounds per ton, suppose a grade to occur of 8 feet per Ioo, Iwo feet in length. For the road surface we have Ex) 2000 = .05, and the angle of repose is a 5 per cent grade. Then 8 per cent — 5 per cent = 3 per cent, or the brake-power necessary to secure uniform motion is the same as would be necessary to haul the load up a 3 per cent grade, and a grade of 3 in no for i000 feet gives 30 feet. The work to be done in holding back the load for the 1000-ft. grade is therefore the same as would lift the load through a vertical height of 30 feet, or the fall of 8 feet per zoo for moo feet has the same effect as 3o feet of rise in the same direction, pro vided brake-power costs the same as animal power. The work saved to the traffic passing down this grade, by eliminating it as avoidable rise and fall (without changing the ruling gradients), would be twice the above amounts or equal to lifting the loads through 6o feet of rise.