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.
Slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described as grades, but typically grades are used for human-made surfaces such as roads, landscape grading, roof pitches, railroads, aqueducts, and pedestrian or bicycle routes. The grade may refer to the longitudinal slope or the perpendicular cross slope.
There are several ways to express slope:
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.^{[3]}
Grades are related using the following equations with symbols from the figure at top.
The slope expressed as a percentage can similarly be determined from the tangent of the angle:
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:
For degrees, percentage (%) and per-mille (‰) notations, larger numbers are steeper slopes. For ratios, larger numbers n of 1 in n are shallower, easier slopes.
The examples show round numbers in one or more of the notations and some documented and reasonably well known instances.
Degrees | Percentage (%) | Permillage (‰) | Ratio | Remarks |
---|---|---|---|---|
60° | 173% | 1732‰ | 1 in 0.58 | |
47.7° | 110% | 1100‰ | 1 in 0.91 | Stoosbahn (funicular railway) |
45° | 100% | 1000‰ | 1 in 1 | |
30.1° | 58% | 580‰ | 1 in 1.724 | Lynton and Lynmouth Cliff Railway (funicular railway) |
30° | 58% | 577‰ | 1 in 1.73 | |
25.5° | 47% | 476‰ | 1 in 2.1 | Pilatus Railway (steepest rack railway) |
20.3° | 37% | 370‰ | 1 in 2.70 | Mount Washington Cog Railway (maximum grade) |
20° | 36% | 363‰ | 1 in 2.75 | |
18.4° | 33% | 333‰ | 1 in 3 | |
16.9° | 30% | 300‰ | 1 in 3.3 | Extremely steep road |
14.0° | 25% | 250‰ | 1 in 4 | Very steep road. Mount Washington Cog Railway (average grade) |
11.3° | 20% | 200‰ | 1 in 5 | Steep road |
8.13° | 14.2% | 142‰ | 1 in 7 | |
7.12° | 12.5% | 125‰ | 1 in 8 | Cable incline on the Cromford and High Peak Railway |
5.71° | 10% | 100‰ | 1 in 10 | Steep road |
4.0° | 7% | 70‰ | 1 in 14.3 | |
3.37° | 5.9% | 59‰ | 1 in 17 | Swannington incline on the Leicester and Swannington Railway |
2.86° | 5% | 50‰ | 1 in 20 | Matheran Hill Railway. The incline from the Crawlerway at the Kennedy Space Center to the launch pads.^{[4]}^{[5]} |
2.29° | 4% | 40‰ | 1 in 25 | Cologne–Frankfurt high-speed rail line |
2.0° | 3.5% | 35‰ | 1 in 28.57 | LGV Sud-Est, LGV Est, LGV Méditerranée |
1.97° | 3.4% | 34‰ | 1 in 29 | Bagworth incline on the Leicester and Swannington Railway |
1.89° | 3.3% | 33‰ | 1 in 30.3 | Rampe de Capvern on the Toulouse–Bayonne railway |
1.52° | 2.65% | 26.5‰ | 1 in 37.7 | Lickey Incline |
1.43° | 2.5% | 25‰ | 1 in 40 | LGV Atlantique, LGV Nord. The Schiefe Ebene. |
1.146° | 2% | 20‰ | 1 in 50 | Railway near Jílové u Prahy. Devonshire Tunnel |
0.819° | 1.43% | 14.3‰ | 1 in 70 | Waverley Route |
0.716° | 1.25% | 12.5‰ | 1 in 80 | Ruling grade of a secondary main line. Wellington Bank, Somerset |
0.637° | 1.11% | 11.11‰ | 1 in 90 | Dove Holes Tunnel |
0.573° | 1% | 10‰ | 1 in 100 | The long drag on the Settle & Carlisle line |
0.458° | 0.8% | 8‰ | 1 in 125 | Rampe de Guillerval |
0.2865° | 0.5% | 5‰ | 1 in 200 | Paris–Bordeaux railway , except for the rampe de Guillerval |
0.1719° | 0.3% | 3‰ | 1 in 333 | |
0.1146° | 0.2% | 2‰ | 1 in 500 | |
0.0868° | 0.1515% | 1.515‰ | 1 in 660 | Brunel's Billiard Table - Didcot to Swindon |
0.0434° | 0.07575% | 0.7575‰ | 1 in 1320 | Brunel's Billiard Table - Paddington to Didcot |
0° | 0% | 0‰ | 1 in ∞ (infinity) | Flat |
In vehicular engineering, 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,^{[6]} 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 according to the Guinness Book of World Records are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales^{[7]} and Canton Avenue in Pittsburgh, Pennsylvania.^{[8]} 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^{[9]} against the ruling that handed the title, briefly, to Ffordd Pen Llech.
A number of streets elsewhere have steeper grades than those listed in the Guinness Book. Drawing on the U.S. National Elevation Dataset, 7x7 (magazine) identified ten blocks of public streets in San Francisco open to vehicular traffic in the city with grades over 30 percent. The steepest at 41 percent is the block of Bradford Street above Tompkins Avenue in the Bernal Heights neighborhood.^{[10]} The San Francisco Municipal Railway operates bus service among the city's hills. The steepest grade for bus operations is 23.1% by the 67-Bernal Heights on Alabama Street between Ripley and Esmeralda Streets.^{[11]}
Likewise, the Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.^{[12]} The street has formed part of a bicycle race since 1983.^{[13]}
Grade, pitch, and slope are important components in landscape design, garden design, landscape architecture, and architecture; for engineering and aesthetic design factors. Drainage, slope stability, circulation of people and vehicles, complying with building codes, and design integration are all aspects of slope considerations in environmental design.
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^{[14]}) 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^{[15]}) 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.^{[citation needed]}
The steepest railway lines that do not use a rack system include:
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.
In the era before they were provided with continuous brakes, whether air brakes or vacuum brakes, steep gradients made it extremely difficult for trains to stop safely. In those days, for example, an inspector insisted that Rudgwick railway station in West Sussex be regraded. He would not allow it to open until the gradient through the platform was eased from 1 in 80 to 1 in 130.