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Railroad track spirit level in place indicating 5 in (130 mm) of superelevation between the inside and outside rails of a curve along the Keystone Corridor near Narberth, Pennsylvania In curved track, it is usually designed to raise the outer rail, providing a banked turn , thus allowing trains to maneuver through the curve at higher speeds ...
To compensate for this, the gradient should be a little less steep the sharper the curve is; the necessary grade reduction is assumed to be given by a simple formula such as 0.04 per cent per "degree of curve", the latter being a measure of curve sharpness used in the United States. On a 10-degree curve (radius 573.7 feet) the grade would thus ...
The relationship between speed and tilt can be calculated mathematically. We start with the formula for a balancing centripetal force: θ is the angle by which the train is tilted due to the cant, r is the curve radius in meters, v is the speed in meters per second, and g is the standard gravity, approximately equal to 9.81 m/s²:
For the United States, with a standard maximum unbalanced superelevation of 75 mm (3 in), the formula is this: v m a x = E a + 3 0.00066 d {\displaystyle v_{max}={\sqrt {\frac {E_{a}+3}{0.00066d}}}} where E a {\displaystyle E_{a}} is the superelevation in inches, d {\displaystyle d} is the curvature of the track in degrees per 100 feet, and v m ...
l = slope length α = angle of inclination. The grade (US) or gradient (UK) (also called stepth, slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero indicates horizontality. A larger number ...
A narrow-gauge railway (narrow-gauge railroad in the US) is a railway with a track gauge narrower than 1,435 mm (4 ft 8 + 1 ⁄ 2 in) standard gauge. Most narrow-gauge railways are between 600 mm ( 1 ft 11 + 5 ⁄ 8 in ) and 1,067 mm ( 3 ft 6 in ).
Sine, cosine, and versine of θ in terms of a unit circle, centered at O. The Hallade method, devised by Frenchman Emile Hallade, is a method used in track geometry for surveying, designing and setting out curves in railway track.
The actual equation given in Rankine is that of a cubic curve, which is a polynomial curve of degree 3, at the time also known as a cubic parabola. In the UK, only from 1845, when legislation and land costs began to constrain the laying out of rail routes and tighter curves were necessary, were the principles beginning to be applied in practice.