d = run Δh = rise l = slope length α = angle of inclination
The grade (also called slope, incline, gradient, 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 indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which run is the horizontal distance (not the distance along the slope) and rise is the vertical distance.
Illustration of grades (percentages), angles in degrees and ratio.
There are several ways to express slope:
as an angle of inclination to the horizontal. (This is the angle α opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.)
as a percentage, the formula for which is which is equivalent to the tangent of the angle of inclination times 100. In Europe and the U.S. percentage "grade" is the most commonly used figure for describing slopes.
as a per mille figure (‰), the formula for which is which could also be expressed as the tangent of the angle of inclination times 1000. This is commonly used in Europe to denote the incline of a railway.
as a ratio of one part rise to so many parts run. For example, a slope that has a rise of 5 feet for every 1000 feet of run would have a slope ratio of 1 in 200. (The word "in" is normally used rather than the mathematical ratio notation of "1:200".) This is generally the method used to describe railway grades in Australia and the UK. It is used for roads in Hong Kong, and was used for roads in the UK until the 1970s.
as a ratio of many parts run to one part rise, which is the inverse of the previous expression (depending on the country and the industry standards). For example, "slopes are expressed as ratios such as 4:1. This means that for every 4 units (feet or metres) of horizontal distance there is a 1 unit (foot or metre) vertical change either up or down."
Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle α by taking the inverse tangent of the standard mathematical slope, which is rise / run or the grade / 100. If one looks at red numbers on the chart specifying grade, one can see the quirkiness of using the grade to specify slope; the numbers go from 0 for flat, to 100% at 45 degrees, to infinity as it approaches vertical.
Slope may still be expressed when the horizontal run is not known: the rise can be divided by the hypotenuse (the slope length). This is not the usual way to specify slope; this nonstandard expression follows the sine function rather than the tangent function, so it calls a 45 degree slope a 71 percent grade instead of a 100 percent. But in practice the usual way to calculate slope is to measure the distance along the slope and the vertical rise, and calculate the horizontal run from that, in order to calculate the grade (100% × rise/run) or standard slope (rise/run). When the angle of inclination is small, using the slope length rather than the horizontal displacement (i.e., using the sine of the angle rather than the tangent) makes only an insignificant difference and can then be used as an approximation. Railway gradients are often expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In either case, the following identity holds for all inclinations up to 90 degrees:
. Or more simply, one can calculate the horizontal run by using the Pythagorean theorem, after which it is trivial to calculate the (standard math) slope or the grade (percentage).
In Europe, road gradients are signed as a percentage.
Grades are related using the following equations with symbols from the figure at top.
Tangent as a ratio
The slope expressed as a percentage can similarly be determined from the tangent of the angle:
Angle from a tangent gradient
If the tangent is expressed as a percentage, the angle can be determined as:
If the angle is expressed as a ratio (1 in n) then:
In vehicularengineering, various land-based designs (automobiles, sport utility vehicles, trucks, trains, etc.) are rated for their ability to ascend terrain. Trains typically rate much lower than automobiles. The highest grade a vehicle can ascend while maintaining a particular speed is sometimes termed that vehicle's "gradeability" (or, less often, "grade ability"). The lateral slopes of a highway geometry are sometimes called fills or cuts where these techniques have been used to create them.
In the United States, maximum grade for Federally funded highways is specified in a design table based on terrain and design speeds, with up to 6% generally allowed in mountainous areas and hilly urban areas with exceptions for up to 7% grades on mountainous roads with speed limits below 60 mph (95 km/h).
The steepest roads in the world are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales and Canton Avenue in Pittsburgh, Pennsylvania. The Guinness World Record once again lists Baldwin Street as the steepest street in the world, with a 34.8% grade (1 in 2.87) after a successful appeal against the ruling that handed the title, briefly, to Ffordd Pen Llech. The Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue. The street has formed part of a bicycle race since 1983.
Ruling gradients limit the load that a locomotive can haul, including the weight of the locomotive itself. On a 1% gradient (1 in 100) a locomotive can pull half (or less) of the load that it can pull on level track. (A heavily loaded train rolling at 20 km/h on heavy rail may require ten times the pull on a 1% upgrade that it does on the level at that speed.)
Early railways in the United Kingdom were laid out with very gentle gradients, such as 0.07575% (1 in 1320) and 0.1515% (1 in 660) on the Great Western main line, nicknamed Brunel's Billiard Table, because the early locomotives (and their brakes) were feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ assistant engines or cable haulage, such as the 1.2 kilometres (0.75 miles) section from Euston to Camden Town.
Extremely steep gradients require the use of cables (such as the Scenic Railway at Katoomba Scenic World, Australia, with a maximum grade of 122% (52°), claimed to be the world's steepest passenger-carrying funicular) or some kind of rack railway (such as the Pilatus railway in Switzerland, with a maximum grade of 48% (26°), claimed to be the world's steepest rack railway) to help the train ascend or descend.
Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in n, x% or y per mille. Since designers like round figures, the method of expression can affect the gradients selected.
A 1371-metre long stretch of railroad with a 20‰ (2%) slope, Czech Republic
The steepest railway lines that do not use a rack system include:
2.6% (1 in 38) – A slope near Halden on Østfold Line, Norway – Ok for passenger multiple units, but an obstacle for freight trains which must keep their weight down on this international mainline because of the slope. Freight traffic has mainly shifted to road.
0.77% (1 in 130) – Rudgwick, UK platform after regrading – not too steep if a train is not provided with continuous brakes.
Compensation for curvature
Gradients on sharp curves are effectively a bit steeper than the same gradient on straight track, so to compensate for this and make the ruling grade uniform throughout, the gradient on those sharp curves should be reduced slightly.
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