ALIGNMENT.
57. The broken line furnished by the survey is of course unfit for the centre line of a railroad. The angles require to be rounded off to render the passage from one straight portion to the other easy.
Fig. 26.
58. Let A X B, fig. 26, represent the angle formed by any two tangents which it is required to connect by a circular curve. It is plain that knowing the angle of deflection of the lines A X, B X, we obtain also the angles A C X, X C B. The manner of laying these curves upon the ground is by placing an angular instrument at any point of the curve, as at A, and laying off the partial angles E A a, E A M, E A G, etc., which combined with the corresponding distances A a, a M, M G, fix points in the curve.
These small chords are generally assumed at one hundred feet, except in curves of small radius (five hundred feet) when they are taken less.
The only calculation necessary in laying out curves, is, knowing the partial deflection to find the corresponding chord, or knowing the chord, to get the partial angle.
As the radius of that curve of which the angle of deflection is 1° is 5730 feet, the degree of curvature for any other radius is easily found. Thus the radius 2865 has a degree of curvature per one hundred feet of
5730
2865 = 2°;
again,
5730
2000 = 2°.86 or 2° 51.6.
The radius corresponding to any angle is found by reversing the operation. If the angle is 3° 30′, or 210′, we have
5730 × 60
210 = 1637 feet radius.
The following figures show the angle of deflection for chords one hundred feet long, corresponding to different radii:—
| Angle of deflection. | Radius, in feet. | |
|---|---|---|
| ¼° | or 15′ | 22920.0 |
| ½° | or 30′ | 11460.0 |
| ¾° | or 45′ | 7640.0 |
| 1° | or 60′ | 5730.0 |
| 1¼° | 4585.0 | |
| 1½° | 3820.0 | |
| 1¾° | 3274.0 | |
| 2° | 2865.0 | |
| 2½° | 2292.0 | |
| 3° | 1910.0 | |
| 3½° | 1637.0 | |
| 4° | 1433.0 | |
| 4½° | 1274.0 | |
| 5° | 1146.0 | |
| 5½° | 1042.0 | |
| 6° | 955.4 | |
| 6½° | 822.0 | |
| 7° | 819.0 | |
| 7½° | 764.5 | |
| 8° | 716.8 | |
| 10° | 573.7 | |
Points in any curve may also be fixed by ordinates, as a b, M D′, G F, or by E a, K M, etc.
For the details of locating, of running simple and compound curves, and of the calculations therefor, the reader is referred to the works of Trautwine, and of Henck.
Fig. 27.
59. Suppose now that we have the surveyed lines m m, and n n, fig. 27, one of which is to be finally adjusted to the ground. The shortest line is the straight one, which is generally impracticable. The most level line is the contour line, which is also impracticable. Between these two lies the right line, which is to be found by an instrumental location. The line A n n n n B, on the plan, gives the profile A n n n n B. The line A m m m m B gives the profile A m m m m B, while the finally adjusted line A 1 2 3 4 5 6 gives the profile A 1 2 3 4 5 6 B.
Fig. 28.
60. Again, in fig. 28, the straight line A n n n B gives the profile A n n n B, requiring either steep grades or a great deal of work. By fitting the line to the ground, as by the line A a b c d ... m n o B, we obtain the profile A a b c ... m n o B.