Handbook of Railroad Construction; For the use of American engineers. / Containing the necessary rules, tables, and formulæ for the location, construction, equipment, and management of railroads, as built in the United States.

by George L. Vose

EQUATING FOR GRADES.

53. In comparing the relative advantages of several lines having different systems of grades, it is customary to reduce them all to the level line involving an equal expenditure of power.

The question is to find the vertical rise, consuming an amount of power equal to that expended upon the horizontal unit of length. This has been estimated by engineers all the way from twenty to seventy feet. For simple comparison it does not matter much what number is used if it is the same in all cases; but to find the equivalent horizontal length to any location, regard must be had to the nature of the expected traffic.

The elements of the problem are, the length, the inclination or the total rise and fall, and the resistance to the motion of the train upon a level, which latter depends upon the speed and the state of the rails and machinery.

From chapter XIV. we have the following resistances to the motion of trains upon a level:—

Velocity, in miles, per hour.	Resistance, in lbs. per ton.
10	8.6
15	9.3
20	10.3
25	11.6
30	13.3
40	17.3
50	22.6
60	27.1
100	66.5

The power expended upon any road is of course the product of the resistance per unit of length, by the number of units. Calling R the resistance per unit upon a level, and R′ the resistance per unit on any grade, and designating the lengths by L and L′, that there shall be in both cases an equal expenditure of power, we must have

RL = R′L′,

whence the level length must be

L = L′R′
R.

Thus assuming the resistance on a level as twenty lbs. per ton, that on a fifty feet grade is

20 + 50
5280 of 2240, or 20 + 21.2 or 41.2,

and if the length of the inclined line is ten miles, the equivalent level length is

L = 41.2 × 52800
20 = 108768 feet, or 20.6 miles.

Also 10 miles × 41.2 lbs. = 412,

and 20.6 miles × 20 lbs. = 412.

54. The above may be somewhat abridged as follows: Let R be the resistance on a level. The resistance due to any grade is expressed by

W × 1
a,

where 1
a is the fraction showing the grade, and W the weight of the load.

The vertical height in feet, to overcome which we must expend an amount of power sufficient to move the train one mile on a level, must be such that

W × 1
a = R,

or

1
a = R
W;

and to find the number by which to equate, we have only to place the values of R and W in the formula. For example, let the speed be twenty miles per hour, the corresponding resistance is 10.3 lbs. per ton. W being one ton, or 2240 lbs., we have

1
a = R
W = 10.3
2240 = 1
218 of 5280, (the number of feet in one mile,)

1
218 of 5280 = 24 feet.

In the same manner we have

Speed, in miles, per hour.	Equating number.
15	22
20	24
30	32
40	41
50	53
60	67
100	155

Thus when we take the speed as thirty miles per hour, for each thirty-two feet rise we shall consume an amount of power sufficient to move the train one mile on a level. In descending, the grade instead of being an obstacle, becomes an aid; indeed the incline may be such as to move the trains independently of the steam power. Thus if on account of ascending grades we increase the equated length, so also in descending we must reduce the length. The amount of reduction is not, however, the reverse of the increase in ascending, as after thirty or forty feet any additional fall per mile instead of being an advantage is an evil; as too much gravity obliges us to run down grades with brakes on. Twenty-five feet per mile is sufficient to allow the train to roll down, and any more than this is of very little use. Therefore for every mile of grade descending at the rate of twenty-five feet per mile we may deduct one mile in equating, and for every mile of grade descending twelve and one half feet per mile deduct a proportional amount; but for any more fall per mile than twenty-five feet, no allowance should be made; i. e., if we descend at the rate of forty feet per mile, we may deduct one mile in equating for the twenty-five feet of fall, and throw aside the remaining fifteen feet.

55. This is a common method of equating for grades, and represents a length which is proportional to the power expended, but not proportional to the cost of working, as the ratios of power expended and cost of working under different conditions are very different, double power requiring only twenty per cent. more working capital. The above rules, therefore, require a correction.

The cost of working a power represented by unity being expressed by	100;
That of working a power 2 is expressed by	125;
That of working a power 3 is expressed by	150;
That of working a power 4 is expressed by	175;
That of working a power 5 is expressed by	200.
(See Appendix F.)

Now the resistance on a level being at a velocity of twenty miles per hour, 10.3 lbs. per ton by the formula

1
a = R
W,

the vertical height in feet causing a double expenditure of power is twenty-four; but as above, the whole expense of a double power is increased by only twenty-five per cent.; we should not add one mile for twenty-four feet rise, but one fourth of a mile only, or one mile for each ninety-six feet; and by correcting our former table in this manner, we have the following table:—

Speed, in miles, per hour.	Equating number.
15	88
20	96
25	110
30	128
40	164
50	212
60	268
100	620

So much for equating for the ascents. In descending, we have allowed one mile reduction for each mile of twenty-five feet of descending grade; but as in ascending we correct the first made table, so in descending we must also correct as follows. If we needed no steam power either while descending or afterwards, we should only save wood and water; as a general thing the fire must be kept up while descending, and the only gain is a small part of the expense of fuel; so small, in fine, that with the exception of roads which incline for the whole or a great part of their length, no reduction should be made.