This section is from the "The Economics Of Railroad Construction" book, by Walter Loring Webb, C.E.. Also see Amazon: The Economics Of Railroad Construction.

When curvature and grade are combined on a track, the effect of the curvature is to increase the total resistance. This increase may be sufficient to have a material effect on the operation of trains. On minor grades the added resistance is of but little importance, since its virtual effect is the same as increasing the rate of grade somewhat; and if the virtual rate of grade which represents the sum of the two forms of resistance is still within the rate of the ruling grade, the net effect is merely to increase a few items of expense as given previously in this chapter. When the actual grade is nearly or quite equal to the ruling grade of the road, then the additional resistance caused by the curve will be the equivalent of a grade which is perhaps higher than the nominal ruling grade of the road. If we assume that the resistance on a 6° curve is 6 pounds per ton, which is the equivalent of the grade resistance on a 0.3% grade, then if a 6° curve were located on a 1% grade the resistance of a train on that grade would be practically the same as the resistance of that train on a 1.3% grade with straight track. If 1% grades were the ruling grades of that line and freight-trains were made up so that their engines would be taxed to the limit of their capacity on the 1% grade, then they would probably be stalled on the 6° curves, since the total resistance on those curves would be 30% higher than on the straight track having a 1% grade; but if the grade over these 6% curves is cut down to 0.7%, then the total resistance at such a point would still be equal to the resistance on a 1% grade with straight track. This effect can be illustrated by a diagram as in Fig. 32.

Assume that a stretch of track consisting of alternate tangents and curves has an actual grade represented by the line AN. The angle between BN and BC represents the grade which is the equivalent of the added resistance caused by the curve BC. Then the tangent CD is drawn parallel to AN. Similarly the line DE makes an angle with BN equal to the equivalent grade resistance of the curve DE, and the angle of DE with the horizontal line represents a grade on which the resistance would be the equivalent of the total resistance on the curve DE, and then we have the line EF parallel to the line BN. The average resistance throughout that stretch of track would evidently be represented by the line AF, and therefore the angle FAN represents the grade which would cause a resistance equal to the average resistance actually caused by the curves. This figure therefore illustrates the fact that if the grade of a stretch of track, consisting of curves and tangents, is kept actually uniform, the virtual grade of that track is somewhat higher than the actual grade. If it becomes necessary for trains to stop on these curves, then the full effect of the resistance is encountered and the virtual grade would be as represented by the lines BC and DE. If it is possible to operate the trains throughout that stretch of track without any stops, then the virtual grade would be reduced approximately to the grade AF, since the trains would regain on the tangents a portion of the energy which was lost on the curves.

Fig. 32. Effect of uncompensated curvature.

If, on the other hand, the rate of grade is reduced on the curves, so that the actual grade is as shown by the line ABCDEF in Fig. 33, the reduction of the grade on the curves being just equal to the difference of grade which will represent the added resistance of the curves, then the virtual grade of the entire stretch of line will be as represented by the line AG.

Fig. 33. Grade virtually uniform, with compensated curves.

In laying out a ruling grade which is to carry a line to a summit, the compensation for curvature must unquestionably be provided, but it adds a complication which is also illustrated in Fig. 33. An engineer is frequently required to "develop" his line in order to have the necessary length for a given elevation to be overcome, in order that the grade shall be kept within some chosen limitation; but if the grades are actually reduced on the curves, the total horizontal distance required to overcome a vertical elevation of HG at the rate of grade shown by the tangent AB equals AH, but the distance actually required when the curves are compensated is something more, and is represented by the line AK in Fig. 33. The problem is further complicated, owing to the fact that the necessary additional distance can only be obtained by additional "development" which of itself usually implies additional curvature, and perhaps a great deal of it. In order to compensate this additional curvature there is required a still further increase in horizontal distance. The locating engineer therefore is confronted by the problem of introducing considerable added length of track and perhaps considerable added curvature, in order to obtain a ruling grade on which the resistance is virtually constant throughout, whether on a tangent or on a curve, and on which the maximum resistance does not exceed that of the chosen ruling grade for the line. Nevertheless, considering the supreme importance of avoiding an increase in the ruling grade (as will be developed later) and the comparative unimportance of an increase in distance or curvature, such a method is literally the only correct method to follow.

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