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Grade (slope)

From Wikipedia, the free encyclopedia
Angle to the horizontal plane
This article is about the grade of a topographic feature or constructed element. For other uses, seeSlope (disambiguation).

d = run
Δh = rise
l = slope length
α = angle of inclination

Thegrade (US) orgradient (UK) (also calledslope,incline,mainfall,pitch orrise) of a physical feature, landform or constructed line is either theelevation angle of that surface to thehorizontal or its tangent. It is a special case of theslope, where zero indicateshorizontality. 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 whichrun is the horizontal distance (not the distance along the slope) andrise is the vertical distance.

Slopes of existing physical features such ascanyons and hillsides,stream and river banks, andbeds are often described as grades, but typically the word "grade" is used for human-made surfaces such as roads,landscape grading,roof pitches,railroads,aqueducts, and pedestrian or bicycle routes. The grade may refer to thelongitudinal slope or theperpendicularcross slope.

Nomenclature

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Illustration of grades (percentages), angles in degrees, and ratio
A railroad gradient sign in Gdańsk, Poland, indicating a 50 per mille (5%) slope.
A tram and gradient sign inGdańsk, Poland, indicating a 50‰ slope (50 m/km, 5%).

There are several ways to express slope:

  1. as anangle of inclination to the horizontal. (This is the angleα opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.)
  2. as apercentage, the formula for which is100×riserun{\displaystyle 100\times {\frac {\text{rise}}{\text{run}}}}, which is equivalent to the tangent of the angle of inclination times 100. At a slope angle of 45 degrees, the run is equal to the rise. Expressed as a percentage, the slope of this angle is 100 percent. As the slope approaches 90 degrees (vertical), the percentage slope approaches infinity.[1] In Europe and the U.S. percentagegrade is the most commonly used figure for describing slopes.
  3. as a‰ figure (denoted with the symbol‰), the formula for which is1000×riserun{\displaystyle 1000\times {\frac {\text{rise}}{\text{run}}}}, which could also be expressed asthe tangent of the angle of inclination times 1000. This is commonly used in Europe to denote the incline of a railway. It is sometimes written usingmm/m orm/km instead of the ‰ symbol.[2][3]
  4. as aratio 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 wordin is normally used rather than the mathematical ratio notation of e.g.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.
  5. as aratio 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, a slope expressed as4:1 in this system means for a given vertical distance the horizontal distance travelled is four times as long.[4]

Any of these may be used. When the termgrade is used, the slope is usually expressed as a percentage. 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 at vertical.

Slope may still be expressed when the horizontal run is not known: the rise can be divided by thehypotenuse (the slope length). This is not the usual way to specify slope; this nonstandard expression follows thesine 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:tan⁡α=sin⁡α1−sin2⁡α{\displaystyle \tan {\alpha }={\frac {\sin {\alpha }}{\sqrt {1-\sin ^{2}{\alpha }}}}}. 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 expressed in signage as percentage.[5]

Equations

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Grades are related using the following equations with symbols from the figure at top.

Tangent as a ratio

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tan⁡α=Δhd{\displaystyle \tan {\alpha }={\frac {\Delta h}{d}}}

The slope expressed as a percentage can similarly be determined from the tangent of the angle:

%slope=100tan⁡α{\displaystyle \%\,{\text{slope}}=100\tan {\alpha }}

Angle from a tangent gradient

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α=arctan⁡Δhd{\displaystyle \alpha =\arctan {\frac {\Delta h}{d}}}

If the tangent is expressed as a percentage, the angle can be determined as:

α=arctan⁡%slope100{\displaystyle \alpha =\arctan {\frac {\%\,{\text{slope}}}{100}}}

If the angle is expressed as a ratio(1 in n) then:

α=arctan⁡1n{\displaystyle \alpha =\arctan {\frac {1}{n}}}

Example slopes comparing the notations

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For degrees, percentage (%) and per-mille (‰) notations, larger numbers are steeper slopes. For ratios, larger numbersn of 1 inn are shallower, easier slopes.

The examples show round numbers in one or more of the notations and some documented and reasonably well known instances.

Examples of slopes in the various notations
DegreesPercentage (%)Permillage (‰)RatioRemarks
60°173%1732‰1 in 0.58
47.7°110%1100‰1 in 0.91Stoosbahn (funicular railway)
45°100%1000‰1 in 1
35°70%700‰1 in 1.428
30.1°58%580‰1 in 1.724Lynton and Lynmouth Cliff Railway (funicular railway)
30°58%577‰1 in 1.73
25.5°47%476‰1 in 2.1Pilatus Railway (steepestrack railway)
20.3°37%370‰1 in 2.70Mount 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 3.3Extremely steep road
30% warning (France)
30% warning (France)
14.0°25%250‰1 in 4Very steep road.Mount Washington Cog Railway (average grade)
25% warning (Wales)
25% warning (Wales)
11.3°20%200‰1 in 5Steep road
20% warning (North Yorkshire)
20% warning (North Yorkshire)
8.13°14.2%142‰1 in 7
7.12°12.5%125‰1 in 8Cable incline on theCromford and High Peak Railway
5.71°10%100‰1 in 10Steep road
10% warning (Netherlands)
10% warning (Netherlands)
4.0°7%70‰1 in 14.3
3.37°5.9%59‰1 in 17Swanningtonincline on theLeicester and Swannington Railway
2.86°5%50‰1 in 20Matheran Hill Railway. The incline from theCrawlerway at theKennedy Space Center to the launch pads.[6][7]
2.29°4%40‰1 in 25Cologne–Frankfurt high-speed rail line
2.0°3.5%35‰1 in 28.57LGV Sud-Est,LGV Est,LGV Méditerranée
1.97°3.4%34‰1 in 29Bagworthincline on theLeicester and Swannington Railway
1.89°3.3%33‰1 in 30.3Rampe de Capvern on theToulouse–Bayonne railway [fr]
1.52°2.65%26.5‰1 in 37.7Lickey Incline
1.43°2.5%25‰1 in 40LGV Atlantique,LGV Nord. TheSchiefe Ebene.
1.146°2%20‰1 in 50Railway nearJílové u Prahy.Devonshire Tunnel
20‰ Czech Republic
20‰ Czech Republic
0.819°1.43%14.3‰1 in 70Waverley Route
0.716°1.25%12.5‰1 in 80Ruling grade of a secondary main line.Wellington Bank, Somerset
0.637°1.11%11.11‰1 in 90Dove Holes Tunnel
0.573°1%10‰1 in 100The long drag on the Settle & Carlisle line
0.458°0.8%8‰1 in 125Rampe de Guillerval
0.2865°0.5%5‰1 in 200Paris–Bordeaux railway [fr], except for the rampe deGuillerval
0.1719°0.3%3‰1 in 333
0.1146°0.2%2‰1 in 500
0.0868°0.1515%1.515‰1 in 660Brunel's Billiard Table – Didcot to Swindon
0.0434°0.07575%0.7575‰1 in 1320Brunel's Billiard Table – Paddington to Didcot
0°0%0‰1 in ∞ (infinity)Flat

Roads

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Invehicularengineering, variousland-based designs (automobiles,sport utility vehicles,trucks,trains, etc.) are rated for their ability to ascendterrain. 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 calledfills orcuts where these techniques have been used to create them.

In the United States, the maximum grade for federally funded highways is specified in a design table based on terrain and design speeds,[8] 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 areBaldwin Street in Dunedin, New Zealand,Ffordd Pen Llech in Harlech, Wales[9] andCanton Avenue in Pittsburgh, Pennsylvania.[10] The Guinness World Record once again listsBaldwin Street as the steepest street in the world, with a 34.8% grade (1 in 2.87) after a successful appeal[11] against the ruling that handed the title, briefly, toFfordd 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 theBernal Heights neighborhood.[12] TheSan Francisco Municipal Railway operates bus service amongthe city's hills. The steepest grade for bus operations is 23.1% by the67 Bernal Heights on Alabama Street between Ripley and Esmeralda Streets.[13]

Likewise, the Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.[14] The street has formed part of a bicycle race since 1983.[15]

Environmental design

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Grade, pitch, and slope are important components inlandscape design,garden design,landscape architecture, andarchitecture 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 inenvironmental design.

Railways

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Grade indicator nearBellville, Western Cape,South Africa, showing 1:150 and 1:88 grades

Ruling gradients limit the load that alocomotive can haul, including the weight of the locomotive itself. Pulling a heavily loaded train at 20 km/h may require ten times the force climbing a 1% slope than on level track.

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 theGreat 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 employassistant engines orcable haulage, such as the 1.2-kilometre (0.75-mile) section fromEuston toCamden Town.

Extremely steep gradients require mechanical assistance. Cable systems are used in cases like the Scenic Railway atKatoomba Scenic World in Australia, which reaches a maximum grade of 122% (52°) and is claimed to be the world's steepest passenger-carryingfunicular.[16] For somewhat gentler inclines,rack railways are employed, such as thePilatus Railway in Switzerland, which has a maximum grade of 48% (26°) and is considered the steepest rack railway.[17]

Gradients can be expressed as an angle, as feet per mile, feet perchain, 1 inn,x% ory per mille. Since designers like round figures, the method of expression can affect the gradients selected.[citation needed]

A 1371-metre-long stretch of railroad with a 20‰ (2%) slope,Czech Republic

The steepestrailway lines that do not use rack systems include:

Compensation for curvature

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Gradients on sharp curves are effectively a bit steeper than the same gradient on straight track, so to compensate for this and make theruling grade uniform throughout, the gradient on those sharp curves should be reduced slightly.

Continuous brakes

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In the era before they were provided withcontinuous brakes, whetherair brakes orvacuum brakes, steep gradients made it extremely difficult for trains to stop safely. In those days, for example, aninspector insisted thatRudgwick railway station inWest 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.

See also

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References

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  1. ^"Understanding slope—ArcGIS Pro | Documentation".
  2. ^Michael W. Sayers; Thomas D. Gillespie; William D.O. Paterson (January 1986)."Directives pour la mesure de l'uni des routes et l'étalonnage des appareils"(PDF) (in French).World Bank. p. 2.L'indice IRI est une mesure de l'uni des routes standardisée, apparentée aux mesures obtenues à l'aide des appareils de type-réponse. Les unités recommandées sont: les mètres par kilomètres (m/km) = millimètres par mètres (mm/m) = pente x 1000.
  3. ^ZEMBA, Baowendzooda Joël (July 2015)."ETUDE D'AVANT-PROJET DETAILLE DE L'AMENAGEMENT D'UN PERIMETRE IRRIGUE DE 100 HA À BAGRE EN RIVE DROITE DU NAKANBE (BURKINA FASO)" (in French). p. 37. Archived fromthe original on 19 November 2023.Pente longitudinale i (m/km ou ‰): 0.4
  4. ^Strom, Steven; Nathan, Kurt; Woland, Jake (2013). "Slopes expressed as ratios and degrees".Site Engineering for Landscape Architects (6th ed.). Wiley Publishing. p. 71.ISBN 978-1118090862.
  5. ^"Traffic signs".gov.uk. The Highway Code – Guidance. Retrieved26 March 2016.
  6. ^"Crawler-Transporter". NASA. 21 April 2003. Archived fromthe original on 1 June 2020. Retrieved18 June 2020.
  7. ^"Countdown! NASA Launch Vehicles and Facilities"(PDF). NASA. October 1991. pp. 16–17. PMS 018-B, section 3. Archived fromthe original(PDF) on 27 January 2005. Retrieved21 August 2013.
  8. ^A Policy on Geometric Design of Highways and Streets(PDF) (4th ed.). Washington, DC: American Association of State Highway and Transportation Officials. 2001. pp. 507 (design speed), 510 (exhibit 8–1: Maximum grades for rural and urban freeways).ISBN 1-56051-156-7. Retrieved11 April 2014.
  9. ^"Welsh town claims record title for world's steepest street".Guinness World Records. 16 July 2019.
  10. ^"Kiwi climb: Hoofing up the world's steepest street".CNN.com.
  11. ^"Baldwin street in New Zealand reinstated as the world's steepest street".Guinness World Records. 8 April 2020.
  12. ^"The Real Top 10 List of Steepest Streets in San Francisco".7x7.
  13. ^ab"General Information". San Francisco Metropolitan Transportation Agency. Archived fromthe original on 3 December 2016. Retrieved20 September 2016.
  14. ^"Canton Avenue, Beechview, PA".Post Gazette.
  15. ^"The steepest road on Earth takes no prisoners".Wired. Autopia. December 2010.
  16. ^"Top five funicular railways".Sydney Morning Herald.
  17. ^"A wonderful railway".The Register. Adelaide, Australia. 2 March 1920. p. 5. Retrieved13 February 2013 – via National Library of Australia.
  18. ^"The New Pöstlingberg Railway"(PDF). Linz Linien GmbH. 2009. Archived fromthe original(PDF) on 22 July 2011. Retrieved6 January 2011.
  19. ^"Pantele din Iaşi pun probleme ofertanţilor" (in Romanian). 5 March 2019.
  20. ^"Return of the (modern) streetcar – Portland leads the way" (Press release). Tramways & Urban Transit. Light Rail Transit Association. October 2001. Archived fromthe original on 27 September 2013. Retrieved15 December 2018.
  21. ^Massachusetts Bay Transportation Authority (6 March 2017).MBTA Green Line Track Charts. pp. 14–15.
  22. ^"Il Piano Tecnologico di RFI"(PDF). Collegio Ingegneri Ferroviari Italiani. 15 October 2018. Retrieved23 May 2019.
  23. ^"Madisonview".oldmadison.com. Retrieved7 April 2017.
  24. ^"Going under the hills".BBC. 24 September 2014.
  25. ^"The Matheran Light Railway (extension to the Mountain Railways of India)". UNESCO World Heritage Convention.
  26. ^Martin, Bruno (September 2005)."Durban-Pietermaritzburg main line map and profile"(PDF).Transport in South and Southern Africa. Retrieved7 April 2017.
  27. ^Valley Heights railway station
  28. ^"Improving the Southeast Expressway: A Conceptual Plan"(PDF).bostonmpo.org. Retrieved24 February 2021.
  29. ^"Network Statement 2023"(PDF). Irish Rail.

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