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.

TOPOGRAPHICAL SKETCHING.

39. Topographical drawing includes every thing relating to an accurate representation upon paper, of any piece of ground. The state of cultivation, roads, town, county, and state boundaries, and all else that occurs in nature. The sketching necessary in railroad surveying, however, does not embrace all of this, but only the delineation of streams and the undulations of ground within that limit which affects the road, perhaps 500 feet on each side of the line. The making of such sketches consists in tracing the irregular lines formed by the intersection of the natural surface, by a system of horizontal planes, at a vertical distance of five, ten, fifteen, or twenty feet, according to the accuracy required.

40. Suppose that we wish to represent upon a horizontal surface a right cone. The base m m, fig. 13, is shown by the circle of which the diameter is m, m. If the elevation is cut by the horizontal planes a a, b b, c c, the intersection of these planes with the conical surface is shown by the circles a, b, c, in plan. The less we make the horizontal distances, on plan, between the circles, the less also will be the vertical distance between the planes.

Wishing to find the elevation of any line which exists on plan, as 1, 2, 3, 3, 2, 1, we have only to find the intersection of the verticals drawn through the points 1, 2, 3, 3, 2, 1, and the elevation lines a a, b b, c c; this gives us the curve 4, 5, 6, 7, 6, 5, 4.

41. Again, in fig. 14, the cone is oblique, which causes the circles on plan to become eccentric and elliptic. Having given the line 1, 2, 3, as before, we find it upon the elevation in the same manner.

42. In the section of regular and full lined figures, the horizontal and vertical projections are also regular and full lined; but in a broken surface like the ground, the lines become quite irregular.

Suppose we wish to show on plan the hill of which we have the plan, fig. 15, and the sections figs. 16, 17, and 18. Let AD be the profile (made with the level) of the line AD on plan, fig. 15. B E that of B E, and C F that of CF.

To form the plan from the profiles proceed as follows:—

Intersect each of the profiles by the horizontal planes a a, b b, c c, d d, equidistant vertically. In the profile A D, fig. 18, drop a vertical on to the base line from each of the intersections a, b, c, d, d, c, b, a. Make now A 1,1 2, 2 3, 3 4, etc., on the plan equal to the same on the profile. Next draw, on the plan, the line B E, at the right place and at the proper angle with A D; and having found the distances B 1, 1 2, 2 3, etc., as before, transfer them to the line B E on plan. Proceed in the same manner with the line C F.

The points a a a, b b b, c c c, are evidently at the same height above the base upon the profiles, whence the intersections of these lines with the surface line or 1 1 1, 2 2 2, 3 3 3, etc., on the plan, are also at the same height above the base; and an irregular line traced through the points 1 1 1, or 2 2 2, will show the intersection of a horizontal plane, with the natural surface.

When as at A we observe the contour lines near to each other, we conclude that the ground is steep. And when the distances are large, as at 6, 7, 8, we know that the ground falls gently. This is plainly seen both on plan and profile.

Having now the topographical sketch, fig. 15, we may easily deduce therefrom at any point a profile. If we would have a profile of G E, on plan, upon an indefinite line G E, fig. 19, we set off G 1, 1 2, 2 3, 3 4, etc., equal to the same distances on the plan. From these points draw verticals intersecting the horizontals a a, b b, c c; and lastly, through the intersections draw the broken line (surface line or profile) a, b, c, d, d, c, b, a. Thus we see how complete a knowledge of the ground a correct topographical sketch gives.

43. Field sketches for railroad work are generally made by the eye. The field book being ruled in squares representing one hundred feet each. When we need a more accurate sketch than this method gives, we may cross section the ground either by rods or with the level.

By making a very detailed map of a survey, and filling in with sketches of this kind, the location may be made upon paper and afterwards transferred to the ground.

So far we have dealt with but one summit; but the mode of proceeding is precisely the same when applied to a group or range of hills, or indeed to any piece of ground.

44. As a general thing, the intersection of the horizontal planes with the natural surface (contour lines) are concave to the lower land in depressions, and convex to the lower land on spurs and elevations. Thus at B B B b b, fig. 20, upon the spurs, we have the lines convex to the stream; and in the hollows c c c, the lines are concave to the bottom.

45. Having by reconnoissance found approximately the place for the road, we proceed to run a trial line by compass. In doing this we choose the apparent best place, stake out the centre line, make a profile of it, and sketch in the topography right and left.

Suppose that by doing so we have obtained the plan and profile shown in figs. 21 and 22, where A a a B is the profile of A C D B, on the plan. The lowest line of the valley though quite moderately inclined at first, from A to C, rises quite fast from C to the summit; and as the inclination becomes greater, the contour lines become nearer to each other.

Now that the line may ascend uniformly from A to the summit, the horizontal distances between the contour lines must be equal; this equality is effected by causing the surveyed line to cut the contours square at 1, 2, 3, 4, and obliquely at 5, 8, 10. Thus we obtain the profile A 5 5 B.

46. Having given the plan and profile, figs. 23 and 24, where A C D B represents the bed of the stream, in profile, if it were required to put the uniformly inclined line A m m B, upon the plan, we should proceed as follows. Take the horizontal distance A m from the profile, and with A (on plan) as a centre, describe the arc 1, 3. The point m on the profile is evidently three fourths of a division above the bed of the stream. So on the plan we must trace the arc 1, 3, until we come to a, which is three fourths of b c, from b. Again, m′ is nine and one half divisions above m. From a, with a radius m n on profile, describe the arc 4, 5, 6. Now, as on the profile, in going from m to m′, we cross nine contour lines, and come upon the tenth at m′, so on the plan we must cross nine contour lines and come upon the tenth, and at the same time upon the arc 4, 5, 6.

Proceeding in this way, we find A, a, b, B, on the plan, as corresponding to A m m′ B on the profile.

To establish in this manner any particular grade, we have first to place it upon the profile, and next to transfer it to the plan.

47. It may be remembered as a general thing, that the steepest line is that which cuts the contour line at right angles; the contour line itself is level, and as we vary between these limits we vary the incline.

GENERAL ESTABLISHMENT OF GRADES.

48. Considerable has been written upon the relation which ought to exist between the maximum grade, and the direction of the traffic. Some have given formulæ for obtaining the rate and direction of inclines as depending upon the capacity of power. This seems going quite too far, as the nature of the ground and of the traffic generally fix these in advance.

49. Between two places which are at the same absolute elevation, there should be as little rise and fall as possible.

50. Between points at different elevations, we should if possible have no rise while descending, and consequently no fall while on the ascent.

51. Some engineers express themselves very much in favor of long levels and short but steep inclines. There are cases where the momentum acquired upon one grade, or upon a level, assists the train up the next incline. The distance on the rise during which momentum lasts, is not very great. A train in descending a plane does not receive a constant increase of available momentum, but arrives at a certain speed, where by increased resistance and by added effect of gravity, the motion becomes nearly regular. Up to this point the momentum acquired is useful, but not beyond.

Any road being divided into locomotive sections, the section given to any one engine should be such as to require a constant expenditure of power as nearly as possible; i. e., one section, or the run of one engine, should not embrace long levels and steep grades. If an engine can carry a load over a sixty feet grade, it will be too heavy to work the same load upon a level economically. It is best to group all of the necessarily steep grades in one place, and also the easy portions of the road; then by properly adapting the locomotives the cost of power may be reduced to a minimum.

As to long levels and short inclines the same power is required to overcome a given rise, but quite a difference may be made in the means used to surmount that ascent.

52. Suppose we have the profiles A E D and A B D, fig. 25. The resistance from A to D by the continuous twenty feet grade is the same as the whole resistance from A to B and from B to D. The reason for preferring A E D is, that an engine to take a given load from B to D would be unnecessarily heavy for the section A B; while the same power must be exerted at each point, of A E D. Also the return by A E D is made by a small and constant expenditure of power, being all of the way aided by gravity; while in descending by B, we have more aid from gravity than we require from D to B, after which we have none.

When the distances A B, B C, are sixty and twenty miles in place of six and two, we may consider the grades grouped at B D, and use a heavier engine at that point, as we should hardly find eighty miles admitting of a continuous and uniform grade.

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:—

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

whence the level length must be

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

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

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

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

or

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

In the same manner we have

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.

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

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:—

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.

COMPARISON OF SURVEYED LINES.

56. The requisite data for an approximate comparison of lines are, the measured length, total rise, total fall.

Assume the number by which to equate, as ninety-six, and we shall have

The cost of construction being assumed as the actual length, and that of working as the equated length, we have the final approximate comparison thus:—

Assume the construction cost as $25,000 per mile, and the cost of maintenance $4,000 per mile, and we have

The line A to the line B as