At all changes in the rate of inclination of the grade line a heavier vertical line should be drawn.
Gradient. The grade of a line is its longitudinal slope, and is designated by the proportion between its length and the difference of height of its two extremes. The ratio of these two qualities gives it its name; if the road ascends or fall one foot in every twenty feet of its length, it is said to have a grade of 1: 20 or a 5 per cent grade. Grades are of. two kinds, maximum and minimum. The maximum is the steepest which is to be permitted and which on no account is to be exceeded. The minimum is the least allowable for good drainage. (For method of designating grades see Table 9).
Determination of Gradients. The maximum grade is fixed by two considerations, one relating to the power expended in ascend ing, the other to the acceleration in descending the incline.
There is a certain inclination, depending upon the degree of perfection given to the surface of the road, which cannot be exceeded without a direct loss of tractive power. This inclination is that in descending which, at a uniform speed, the traces slacken, or which causes the vehicles to press on the horses; the limiting inclination within which this effect does not take place is the angle of repose.
The angle of repose for any given road surface can be easily ascertained from the tractive force required upon a level with the same character of surface. Thus if the force necessary on a level to overcome the resistance of . the load is of its weight, then the same fraction expresses the angle of repose for that surface.
On all inclines less steep than the angle of repose a certain amount of tractive force is necessary in the descent as well as in the ascent, and the mean of the two drawing forces, ascending and descending, is equal to the force along the level of the road. Thus on such inclines, as much mechanical force is gained in the descent as is lost in the ascent. From this it might.be inferred that when a vehicle passes alternately each way along the road, no real loss is occasioned by the inclination of the road; such is not, however, practically the fact with animal power, for while it is necessary in the ascending journey to have either a less or a greater number of horses than would be requisite if the road were entirely level, no corresponding reduction can be made in the descending journey. On inclines which are more steep than the angle of repose, the load presses on the horses during their descent, so as to impede their action, and their power is expended in checking the descent of the load; or if this effect be prevented by the use of any form of drag or brake, then the power expended on such a drag or brake corresponds to an equal quantity of mechanical power expended in the ascent, for which no equivalent is obtained in the descent.
The maximum grade for a given road will depend (1) upon the class of traffic that will use it, whether fast and light, slow and heavy, or mixed, consisting of both light and heavy; (2) upon the character of the pavement adopted; and (3) upon the question of cost of con struction. Economy of motive power and low cost of construction are antagonistic to each other, and the engineer will have to weigh the two in the balance.
For fast and light traffic the grades should not exceed 2 per cent; for mixed traffic 3 per cent may be adopted; while for slow traffic combined with economy 5 per cent should not be exceeded. This grade is practicable but not convenient.
Minimum Grade. From the previous considerations it would appear that an absolutely level road was the one to be sought for, but this is not so; there is a minimum or least allowable grade which the road must not fall short of, as well as a maximum one which it must not exceed. If the road was perfectly level in its longitudinal direc tion, its surface could not be kept free from water without giving it so great a rise in its middle as would expose vehicles to the danger of overturning. The minimum grade commonly used is 1 per cent.
Undulating Grades. From the fact that the power required to move a load at a given velocity on a level road is decreased on a descending grade to the same extent it is increased in ascending the same grade, it must not be inferred that the animal force expended in passing alternately each way over a rising and falling road will gain as much in descending the several inclines as it will lose in ascend ing them. Such is not the case. Theaanimal force must be sufficient, either in power or number, to draw the load over the level portions and up the steepest inclines of the road, and in practice no reduction in the number of horses can be made to correspond with the decreased power required in descending the inclines.