let x' be the horizontal distance m'q' from m' to the center line, and let d-y' m'k', the elevation of m' above the subgrade.

If the natural surface mcm' is a level line, so that g, c, and g' are at the same elevation, then y=o, y' = o, and b

x=x' = ca=ca' =

(3)

Formulas 1 and 2 are called slope-stake equations and formula 3 is called the level-section equation. The last formula is available when the ground is nearly level. When the ground is sloping or irregular, formula 1 is employed, but not directly, as the value of y is not known until after the stake has been located. The distance x or x' is determined by successive trials. Suppose, for example, that, in Fig. 4, d=6.3, and let the rod reading on the point c be 5.9. Suppose, also, that s=1.5:1 and b=20. Then, if the ground is level, by formula 3,

To find the location of m, the rodman will hold the rod at some point more than 19.5 ft. from cr. Suppose that he holds it at n, 20 ft from cr, and that the

reading on the rod in this position is 2.8. Then, the height of this point above c equals the reading on c minus the reading on n, or 5.9 2.8 3.1 ft.

The computed distance from the rod to cr is, by formula 1, 20+ 1.5 X 6.3 +1.5 X 3.1 =24.2 ft. As the measured distance (20 ft.) is much smaller than this, the rod must be moved much farther out.

Suppose that the rod is car

ried out 7 ft. so that the measured distance to cr is 27 ft., and suppose that the reading on the rod in this position is .8 ft. The elevation of this trial point above c will be 5.9-.8-5.1 ft., and by formula 1, the computed distance x +1.5X6.3+1.5×5.1=27.2 ft. This agrees so closely with the measured distance that the slope stake may be driven at this point.

is

The lower slope stake at m' is set in the same manner as the upper, except that the distance of each trial point below c is measured, and formula 2 is used in computing the corresponding value of x'. The distance of the trial point from cr will in this case be taken less than the distance ca' computed by formula 3. As in the preceding case, if the measured distance from cr to the trial point is less than the computed distance, the point should be moved out; if greater, it should be moved in.

Form of Notes in Cross-Section Work.-When each slope stake has been set as just explained, its distance from the center line and the elevation of the stake above or below subgrade are entered in the field book in the form of a fraction. The numerator of this fraction is the distance of the stake above or below subgrade, and the denominator is the distance of the stake from the center line. Thus, if the slope stakes in the preceding example are set at Sta. 131, the complete entry in the notebook will be as follows:

The fraction

C11.4 indicates that the left slope stake at m, Fig. 4, is 27.2 ft. 27.2 from the center line of the roadbed and 11.4 ft. above subgrade. Similarly, the fraction indicates that the right slope stake m' is 13.5 ft. to the right of the

C2.3

13.5

center line and 2.3 ft. above subgrade. These expressions are called slopestake fractions.

When the ground between the slope stakes and the center stake is irregular, the elevations and distances from the center of the intermediate points where the ground changes abruptly are determined and also entered in the notebook in the form of fractions.

RAILROAD LOCATION

The preliminary survey is made by the methods given, a random line being run along the proposed route and a map of the region made covering several hundred feet on each side of the traverse. The map should show both banks of any streams and enough levels should be taken to show the contours within the width covered by the map. This width will depend on the nature of the ground, being less in hilly than in flat country. In general, the map should cover the ground from the toe of one side hill to the toe of the other and should extend a distance up each hill to an elevation beyond which it would not be practicable to make a cut. After this preliminary line has been mapped, a preliminary estimate of the cost of the proposed work may be made.

Preliminary Estimate.-When making a preliminary estimate, great accuracy is not necessary, and no time should be wasted in useless refinements of calculation. The estimate should be high enough to cover all probable cost, and a liberal allowance should be made to cover unforeseen contingencies that may develop during construction. Most experienced engineers make it a rule to add 10% to a preliminary estimate in order to provide for contingencies. When estimating for earthwork, the cross-sections may be considered as level cuttings; that is, the cross-section surface may be considered as level, unless its slope angle exceeds 10°, in which case a suitable allowance must be made for the slope. The preliminary estimate, which also includes approximate figures for material and labor required for culverts, bridges, trestles, piers, and abutments is then classified and summarized. A sample of a good form of a preliminary estimate of the cost of a proposed railroad follows:

Bear River.

Location. The location is the operation of fitting the line to the ground in such a manner as to secure the best adjustment of the alinement and grade,

D

$31

A30

PT.

26+09

Tangent 275.15
Length of
Curve 525'

20+84'

PC & Curve

consistent with an economical cost of construction. It is then best projected on the map, and it is called a paper location.

An example of such location is illustrated in Fig. 1. Here, the line follows the valley of Bear River, and the gradient is determined by the slope of the stream. The gradient adopted is .5%, or .5 ft. per station. The preliminary line is shown dotted, and the located line is drawn full.

Let the grade elevation for Sta. 16 be 155 ft.; the grade elevation for Sta. 17 will, there fore, be 155 ft.+.5 ft. 155.5 ft. The grade elevation for Sta. 18 will be 155.5+.5=156 ft. By the same process, the grade elevation is found for each station shown in the plat; and by means of interpolation between two contour curves, points having the required elevation are located opposite the corresponding stations of the preliminary survey. Each point is marked by a small dot enclosed in a circle. A line joining the points thus designated will be the grade contour, or the line where the required gradient meets the surface of the ground. The tangents AB and CD are then projected so as to conform as closely as practicable to the grade contour, and a suitable curve is inserted for the intersection angle EFD. This is most conveniently done by means of a curved protractor, an illustration of which is shown in Fig. 2. This instrument, which is made of transparent material, is shifted until there is found a curve that will fit the topography and will close the angle between the tangents, as required.

-150

FIG. 1

Curvature. There is no fixed rule for limiting curvature, but for a permanent track it is desirable to have the curvature as easy as possible. For all ordinary

traffic conditions, it is good practice to use such curves as will best conform to existing topographical conditions. Any curve up to 10° will be no obstacle to a speed of 35 mi. per hr., the average speed of passenger trains. This practice will af

ford a range in

8

curvature that will meet the requirements of almost any locality.

Compensation for Curvature.-The effect of curvature on a railroad line is to cause a resistance to the movement of trains. When a curve occurs on a

gradient, the effect of the curve resistance on ascending trains is practically the same as increasing the gradient. It is customary, when fixing the final grades, to lighten the grade on a curve by an amount sufficient to offset the resistance due to the curvature. This operation is called compensating for curvature. The usual rate of compensation for curvature is .03 to .05 ft. per 100 ft. per degree of curvature. For example, where the maximum gradient on tangents is 1%, the maximum gradient on a 6° curve, allowing a compensation of .03 ft. per degree, would be 1-(.03X6)=.82%. If a compensation of .05 ft. per degree were made, the grade on a 6° curve would be 1-(.05X6) = .70%.

Final Grade Lines.-The establishing of final grade lines is illustrated in Fig. 3, where the uncompensated grade is 1.3%, 30 and the compensation for curvature, as shown on the final grade line, is .03 ft. per degree. The location notes for this line are as given on page 122.

The elevation of the grade at Sta. 27 is fixed at 120 ft., and at Sta. 52, at 152.5 ft., giving between these stations an actual rise of 32.5 ft. and an uncompensated grade of 1.3%. These grade points are marked on the profile with small circles. The total curvature between Sta. 27 and Sta. 52 is 1081°. The resistance due to each degree of curvature being taken as equivalent to an increase of .03 ft. in grade, the total resistance due to 108.5° is equivalent to .03 X 108.5-3.255 ft. additional rise between Sta. 27 and Sta. 52. Hence, the total theoretical grade between these stations is the sum of 32.5 ft., the actual rise, and 3.255 ft. due to curvature, or 40 35.755 ft. Dividing 35.755 by 25, the number of stations in the given distance, there results 35.755÷25-+1.4302 ft., as the grade for tangents on this line. The starting point of this grade is at Sta. 27. The P. C. of the first curve is at Sta. 29, giving a tangent of 200 ft. which is equal to two stations. As the grade for tangents is +1.4302 ft. per station, the rise in grade between Sta. 27 and Sta. 29 is 1.4302 X2 =2.8604 ft. The elevation of grade at Sta. 27 is 120 ft., and the elevation of grade at Sta. 29 is 120+2.8604 122.8604 ft. which is recorded on the profile as shown in the diagram, with the rate of grade, namely, +1.4302, written above the grade line. The first curve is 8°, and, as the compensation per degree is .03 ft., then, for 8°, and a full station, the compensation is .03X8 =.24 ft. The grade on the curve will therefore be the tangent grade minus the compensation, or 1.4302-.24+1.1902 ft. per station. The P. C. of this curve is at Sta. 29, the P. T. at Sta. 33, making the total length of the curve 400 ft. or four stations. The grade on this curve is +1.1902 ft. per station and the total rise on the curve is 1.1902X4-4.7608 ft. The elevation of the grade at the P. C. at Sta. 29 is 122.8604; hence, the elevation of grade at the P. T. at Sta. 33 is 122.8604+4.7608-127.6212 ft., which is recorded on the profile together with the grade, namely, +1.1902, written above the grade line. The P. C. of the next curve is at Sta. 37+50, giving an intermediate tangent of 450 ft., or four and one-half stations. The grade for tangents is +1.4302 ft. per station; hence, the total rise on the tangent is 1.4302X4.5-6.4359 ft. Adding 6.4359 ft., to 127.6212 ft., the elevation

Sta. 52 El Gd-152.5
FIG. 3

of grade at Sta. 37+50 is found to be 134.0571 ft., which is recorded on the profile, together with the rate of grade for tangents.

The

The next curve is 6°, and the compensation in grade per station is .03 ft. X6=.18 ft. The grade on this curve will therefore be 1.4302-.18 1.2502 ft. per station. The length of the curve is 450 ft., or four and one-half stations, and the total rise in grade on this curve is +1.2502 ft.X4.5-5.6259 ft. elevation of the grade at Sta. 37+50, the P. C. of the curve, is 134.0571. The elevation of the grade at Sta. 42, the P. T., is therefore 134.0571+5.6259 = 139.683 ft., which is recorded on the profile, together with the rate of grade on the 6° curve, namely, +1.2502. The P. C. of the next curve is at Sta. 44 +25, giving an intermediate tangent of 225 ft., or two and one-fourth stations, The total rise on the tangent is, therefore, 1.4302X2.25 3.21795 ft. elevation of grade at the P. T. at Sta. 42 is 139.683; therefore, the elevation of grade at Sta. 44+25 is 139.683+3.21795-142.90095 ft., which is recorded on the profile, together with the grade +1.4302.

The

The last curve is 9°, and the compensation in grade per station is .03X9 =.27 ft. The grade on this curve is therefore, 1.4302-.27-1.1602 ft. per station. The length of the curve is 550 ft., or five and one-half stations, and the total rise on the curve is 1.1602 X 5.5-6.3811 ft. The elevation of grade at Sta. 44+25, the P. C. of the 9° curve, is 142.90095; hence, the elevation of grade at the P. T., at Sta. 49 +75, is 142.90095+6.3811=149.28205 ft., which is recorded on the profile, together with the grade, +1.1602. The end of the line is at Sta. 52, giving a tangent of 225 ft., or two and one-fourth stations. The rise on this tangent is 1.4302X2.25 3.21795 ft., which is added to 149.28205, the elevation of the P. T. at Sta. 49+75. The sum, 152.5 ft., is the elevation of grade at Sta. 52.

The sum of the partial grades should equal the total rise between the extremities of the grade line. The points where the changes of grade occur are marked on the profile with small circles, which are connected by fine lines representing the grade line. These points of change are projected on a horizontal line at the bottom of the profile. The portions of this line that represent curves are dotted, and the portions that represent tangents are drawn full. The P. C. and P. T. of each curve are marked with small circles on this horizontal line, and are lettered as shown in the diagram.

Where the grades are light and the curves have large radii, there will be no need of compensation for curvature. Where the grades exceed .5% and the curves 5°, compensation should be made.

VERTICAL CURVES

If the grade of the center line of track changes at any point, the two grade lines that intersect at this point form with each other an angle more or less abrupt. If this angle points upwards, it is called a spur; if it points downwards, it is called a sag. The angles CVD in Fig. 1 (a) and (b) are spurs; the angles CVD in Fig. 2 (a) and (b) are sags.

Vertical Curve at a Spur.-If AV and BV, Fig. 3, are two grade lines meeting at V, a vertical curve CMD must be introduced to join these lines.