Grade (slope)#Expression

File:Slope quadrant.svg

File:50 per mille.png, indicating a 50‰ slope (50{{nbsp}}m/km, 5%).]]

There are several ways to express slope:

1. as an angle of inclination to the horizontal. (This is the angle {{mvar|α}} opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.)
2. as a percentage, the formula for which is $100\; \backslash times\; \backslash frac\{\backslash text\{rise\}\}\{\backslash text\{run\}\}$, 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.
3. as a per mille figure (denoted with the symbol {{char|‰}}), the formula for which is $1000\; \backslash times\; \backslash frac\{\backslash text\{rise\}\}\{\backslash text\{run\}\}$, 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. It is sometimes written using mm/m or m/km instead of the ‰ symbol.{{cite web |title=Directives pour la mesure de l'uni des routes et l'étalonnage des appareils |publisher=World Bank |author1=Michael W. Sayers |author2=Thomas D. Gillespie |author3=William D.O. Paterson |url=https://documents1.worldbank.org/curated/en/864001468334203776/pdf/WTP460PUB0FREN0s0Appareils000French.pdf |page=2 |quote=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. |lang=fr |date=January 1986}} {{cite web |url=http://documentation.2ie-edu.org/cdi2ie/opac_css/doc_num.php?explnum_id=1975 |title=ETUDE D'AVANT-PROJET DETAILLE DE L'AMENAGEMENT D'UN PERIMETRE IRRIGUE DE 100 HA À BAGRE EN RIVE DROITE DU NAKANBE (BURKINA FASO) |last=ZEMBA |first=Baowendzooda Joël |page=37 |quote=Pente longitudinale i (m/km ou ‰): 0.4 |lang=fr |date=July 2015 |archive-url=https://web.archive.org/web/20231119164344/http://documentation.2ie-edu.org/cdi2ie/opac_css/doc_num.php?explnum_id=1975 |archive-date=19 November 2023}}
4. 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 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 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, a slope expressed as 4:1 in this system means for a given vertical distance the horizontal distance travelled is four times as long.{{cite book |page=71 |chapter=Slopes expressed as ratios and degrees |title=Site Engineering for Landscape Architects |edition=6th |year=2013 |first1=Steven |last1=Strom |first2=Kurt |last2=Nathan |first3=Jake |last3=Woland |publisher=Wiley Publishing |isbn=978-1118090862}}

Any of these may be used. When the term grade 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 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:

$\backslash tan\{\backslash alpha\}\; =\; \backslash frac\{\backslash sin\{\backslash alpha\}\}\{\backslash sqrt\{1-\backslash sin^2\{\backslash alpha\}\}\}$. 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.{{cite web |url=https://www.gov.uk/guidance/the-highway-code/traffic-signs |title=Traffic signs |series=The Highway Code – Guidance |website=gov.uk |access-date=2016-03-26}}

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

:$\backslash tan\{\backslash alpha\}\; =\; \backslash frac\{\backslash Delta\; h\}\{d\}$

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

:$\backslash \%\backslash ,\backslash text\{slope\}\; =\; 100\; \backslash tan\{\backslash alpha\}$

:$\backslash alpha\; =\; \backslash arctan\{\backslash frac\{\backslash Delta\; h\}\{d\}\}$

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

:$\backslash alpha\; =\; \backslash arctan\{\backslash frac\{\backslash \%\backslash ,\backslash text\{slope\}\}\{100\}\}$

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

:$\backslash alpha\; =\; \backslash arctan\{\backslash frac\{1\}\{n\}\}$

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.

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, the maximum grade for federally funded highways is specified in a design table based on terrain and design speeds,{{cite book |title=A Policy on Geometric Design of Highways and Streets |edition=4th |year=2001 |location=Washington, DC |publisher=American Association of State Highway and Transportation Officials |isbn=1-56051-156-7 |pages=507 (design speed), 510 (exhibit 8–1: Maximum grades for rural and urban freeways) |url=https://law.resource.org/pub/us/cfr/ibr/001/aashto.green.2001.pdf#page=556 |access-date=11 April 2014}} 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 {{convert|60|mph|abbr=on|round=5}}.

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{{cite web |url=https://www.guinnessworldrecords.com/news/2019/7/welsh-town-claims-title-for-worlds-steepest-street-582452 |title=Welsh town claims record title for world's steepest street |website=Guinness World Records |date=16 July 2019}} and Canton Avenue in Pittsburgh, Pennsylvania.{{cite news |title=Kiwi climb: Hoofing up the world's steepest street |website=CNN.com |url=https://edition.cnn.com/travel/article/worlds-steepest-street-residents/index.html}} 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{{cite web |url=https://www.guinnessworldrecords.com/news/2020/4/baldwin-street-in-new-zealand-reinstated-as-the-worlds-steepest-street-614287 |title=Baldwin street in New Zealand reinstated as the world's steepest street |website=Guinness World Records |date=8 April 2020}} 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.{{cite news |url=https://www.7x7.com/the-real-top-10-list-of-steepest-streets-in-san-francisco-1786501295.html |title=The Real Top 10 List of Steepest Streets in San Francisco |newspaper=7x7}} 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.

Likewise, the Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.{{cite news |title=Canton Avenue, Beechview, PA |newspaper=Post Gazette |url=http://old.post-gazette.com/pg/05030/448976.stm}} The street has formed part of a bicycle race since 1983.{{cite magazine |url=https://www.wired.com/2010/12/the-steepest-road-on-earth-takes-no-prisoners/ |title=The steepest road on Earth takes no prisoners |series=Autopia |magazine=Wired |date=December 2010}}

File:Nederlands verkeersbord J6.svg|10% slope warning sign, Netherlands

File:Finland road sign 115.svg|7% descent warning sign, Finland

File:Devil's Staircase Wales.jpg|25% ascent warning sign, Wales

File:Pente30%.jpg|30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis, France

File:Seattle AM General trolleybus climbing James St near 5th Ave in 1983.jpg|A trolleybus climbing an 18% grade in Seattle

B10 Enzweihinger Steige 20060528.JPG|Ascent of German Bundesstraße 10

File:DunedinBaldwinStreet_Parked_Car.jpg|A car standing on Baldwin Street, Dunedin, New Zealand

File:CantonAve_Top.jpg|Looking down Canton Avenue, Pittsburgh, Pennsylvania

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.

{{anchor|Railroad gradient}}

File:Track Grade indicator 150-88.JPG, South Africa, showing 1:150 and 1:88 grades]]

Ruling gradients limit the load that a locomotive 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 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 {{convert|1.2|km|mi|abbr=off|adj=on}} section from Euston to Camden Town.

Extremely steep gradients require mechanical assistance. Cable systems are used in cases like the Scenic Railway at Katoomba Scenic World in Australia, which reaches a maximum grade of 122% (52°) and is claimed to be the world's steepest passenger-carrying funicular.{{cite news |url=http://www.smh.com.au/news/take-five/top-five-funicular-railways/2005/10/29/1130400400382.html |title=Top five funicular railways |newspaper=Sydney Morning Herald}} For somewhat gentler inclines, rack railways are employed, such as the Pilatus Railway in Switzerland, which has a maximum grade of 48% (26°) and is considered the steepest rack railway.{{cite news |url=http://nla.gov.au/nla.news-article62617786 |title=A wonderful railway |newspaper=The Register |place=Adelaide, Australia |date=2 March 1920 |access-date=13 February 2013 |page=5 |via=National Library of Australia}}

Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in {{mvar|n}}, {{mvar|x}}% or {{mvar|y}} per mille. Since designers like round figures, the method of expression can affect the gradients selected.{{Citation needed|date=May 2021}}

File:Skloník-klesání.jpg (2%) slope, Czech Republic]]

The steepest railway lines that do not use rack systems include:

• 13.5% (1 in 7.40){{dash}}Lisbon tram, Portugal
• 11.6% (1 in 8.62){{dash}}Pöstlingbergbahn, Linz, Austria{{cite web

|title=The New Pöstlingberg Railway

|year=2009

|publisher=Linz Linien GmbH

|url=http://www.linzag.at/cms/media/en/linzagwebsite/dokumente/mobilittverkehr_1/pstlingbergbahn_1/folder_bergbahn.pdf

|access-date=2011-01-06

|url-status=dead

|archive-url=https://web.archive.org/web/20110722040709/http://www.linzag.at/cms/media/en/linzagwebsite/dokumente/mobilittverkehr_1/pstlingbergbahn_1/folder_bergbahn.pdf

|archive-date=2011-07-22}}

• 11.0% (1 in 9.09){{dash}}Cass Scenic Railway, US (former logging line)
• 9.0% (1 in 11.11){{dash}}Ligne de Saint Gervais – Vallorcine, France
• 9.0% (1 in 11.11){{dash}}Muni Metro J Church, San Francisco, US{{cite web

| url=http://archives.sfmta.com/cms/rhomemu/genmuinfo.htm

| title=General Information

| publisher=San Francisco Metropolitan Transportation Agency

| access-date=September 20, 2016

| archive-url=https://web.archive.org/web/20161203183047/http://archives.sfmta.com/cms/rhomemu/genmuinfo.htm

| archive-date=December 3, 2016

| url-status=dead

}}

• 8.8% (1 in 11.4){{dash}}Iași tram, Romania{{cite web |language=Romanian |url=https://www.ziaruldeiasi.ro/stiri/tramvaiele-noi-vor-ajunge-peste-doi-ani-pantele-din-iasi-pun-probleme-ofertantilor--213251.html |title=Pantele din Iaşi pun probleme ofertanţilor |date=2019-03-05}}
• 8.65% (1 in 11.95){{dash}}Portland Streetcar, Oregon, US{{cite press release

|title=Return of the (modern) streetcar – Portland leads the way

|url=http://www.lrta.org/mag/articles/art0110.html

|url-status=dead

|access-date=15 December 2018

|series=Tramways & Urban Transit

|date=October 2001

|publisher=Light Rail Transit Association

|archive-url=https://web.archive.org/web/20130927202015/http://www.lrta.org/mag/articles/art0110.html

|archive-date=27 September 2013}}

• 8.33%(1 in 12){{dash}}Nilgiri Mountain Railway Tamil Nadu, India
• 8.0% (1 in 12.5){{dash}}Just outside the Tombstone Jct. Station in the Tombstone Junction Theme Park, Kentucky, US. The railroad line there had a ruling grade of 6% (1 in 16.7).
• 7.85% (1 in 12.7){{dash}}Near Washington Street station on the Green Line B branch, Boston, Massachusetts, US.{{cite book |last=Massachusetts Bay Transportation Authority |title=MBTA Green Line Track Charts |date=2017-03-06 |pages=14–15 |url=http://archive.org/details/mbta-green-line-track-charts}}
• 7.1% (1 in 14.08){{dash}}Erzberg Railway, Austria
• 7.0% (1 in 14.28){{dash}}Bernina Railway, Switzerland
• 6.0% (1 in 16.7){{dash}}Arica, Chile to La Paz, Bolivia
• 6.0% (1 in 16.6){{dash}}Docklands Light Railway, London, UK
• 6.0% (1 in 16.6){{dash}}Ferrovia Centrale Umbra, Italy{{cite web

|title=Il Piano Tecnologico di RFI

|date=15 October 2018

|publisher=Collegio Ingegneri Ferroviari Italiani

|url=http://www.cifi.it/UplDocumenti/Firenze15102018/7%20Ing.Rosini.pdf

|access-date=23 May 2019}}

• 6.0% (1 in 16.6){{dash}}Link light rail, Seattle, U.S.
• 5.89% (1 in 16.97){{dash}}Madison, Indiana, US{{cite web |title=Madisonview |website=oldmadison.com |url=http://www.oldmadison.com/madview2.html |access-date=2017-04-07}}
• 5.6% (1 in 18){{dash}}Flåm Line, Norway
• 5.3% (1 in 19){{dash}}Foxfield Railway, Staffordshire, UK
• 5.1% (1 in 19.6){{dash}}Saluda Grade, North Carolina, US
• 5.0% (1 in 20){{dash}}Khyber Pass Railway, Pakistan
• 4.5% (1 in 22.2){{dash}}The Canadian Pacific Railway's Big Hill, British Columbia, Canada (prior to the construction of the Spiral Tunnels)
• 4.3% (1 in 23){{dash}}Colwall Old Tunnel, Malvern Hills, UK{{cite web |url = https://www.bbc.co.uk/herefordandworcester/features/malverns/new_malverns_railways.shtml |title = Going under the hills |publisher = BBC |date = 24 September 2014}}
• 4.0% (1 in 25){{dash}}Cologne-Frankfurt high-speed rail line, Germany
• 4.0% (1 in 25){{dash}}Bolan Pass Railway, Pakistan
• 4.0% (1 in 25){{dash}}({{convert|211.2|ft|m|0|}} per {{convert|1|mi|m|}} ){{dash}}Tarana – Oberon branch, New South Wales, Australia.
• 4.0% (1 in 25){{dash}}Matheran Light Railway, India{{cite web |title=The Matheran Light Railway (extension to the Mountain Railways of India) |publisher=UNESCO World Heritage Convention |url=https://whc.unesco.org/en/tentativelists/2082/}}
• 4.0% (1 in 26){{dash}}Rewanui Incline, New Zealand. Fitted with Fell center rail but was not used for motive power, but only braking
• 3.6% (1 in 27){{dash}}Ecclesbourne Valley Railway, Heritage Line, Wirksworth, Derbyshire, UK
• 3.6% (1 in 28){{dash}}The Westmere Bank, New Zealand has a ruling gradient of 1 in 35, however peaks at 1 in 28
• 3.33% (1 in 30){{dash}}Umgeni Steam Railway, South Africa{{cite web |title=Durban-Pietermaritzburg main line map and profile |last=Martin |first=Bruno |date=September 2005 |website=Transport in South and Southern Africa |url=http://www.sa-transport.co.za/trains/maps/NML-Dbn-Pmb.pdf |access-date=7 April 2017}}
• 3.0% (1 in 33){{dash}}several sections of the Main Western line between Valley Heights and Katoomba in the Blue Mountains Australia.Valley Heights railway station
• 3.0% (1 in 33){{dash}}The entire Newmarket Line in central Auckland, New Zealand
• 3.0% (1 in 33){{dash}}Otira Tunnel, New Zealand, which is equipped with extraction fans to reduce chance of overheating and low visibility
• 3.0% (1 in 33){{dash}}The approaches to the George L. Anderson Memorial Bridge crossing the Neponset River, Boston, Massachusetts, US. The Ruling Gradient of the Braintree Branch of the Red Line (MBTA).{{cite web |title=Improving the Southeast Expressway: A Conceptual Plan |website=bostonmpo.org |url=https://www.ctps.org/data/pdf/studies/highway/se_expr_concept.pdf |access-date=2021-02-24}}
• 2.7% (1 in 37){{dash}}Braganza Ghats, Bhor Ghat and Thull ghat sections in Indian Railways, India
• 2.7% (1 in 37){{dash}}Exeter Central to Exeter St Davids, UK (see Exeter Central railway station#Description)
• 2.7% (1 in 37){{dash}}Picton- Elevation, New Zealand
• 2.65% (1 in 37.7){{dash}}Lickey Incline, UK
• 2.6% (1 in 38){{dash}}A slope near Halden on Østfold Line, Norway{{dash}}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.
• 2.3% (1 in 43.5){{dash}}Schiefe Ebene, Germany
• 2.2% (1 in 45.5){{dash}}The Canadian Pacific Railway's Big Hill, British Columbia, Canada (after the construction of the Spiral Tunnels)
• 2.0% (1 in 48){{dash}}Beasdale Bank (West Coast Scotland mainline), UK
• 2.0% (1 in 50){{dash}}Numerous locations on New Zealand's railway network, New Zealand
• 1.66% (1 in 60){{dash}}Section of track on the Dublin-Cork line between mile posts 164 and 162.{{cite web |title=Network Statement 2023 |publisher=Irish Rail |url=https://www.irishrail.ie/getmedia/3ec0febe-9be1-43a7-92ae-04e965146165/IE-2023-Network-Statement-updated-29-04-2024.pdf}}
• 1.51% (1 in 66){{dash}}({{convert|1|ft|m|1}} per {{convert|1|chain|m}}) New South Wales Government Railways, Australia, part of Main South line.
• 1.25% (1 in 80){{dash}}Wellington Bank, Somerset, UK
• 1.25% (1 in 80){{dash}}Rudgwick, UK (West Sussex) platform before regrading{{dash}}too steep if a train is not provided with continuous brakes.
• 0.77% (1 in 130){{dash}}Rudgwick, UK platform after regrading{{dash}}not too steep if a train is not provided with continuous brakes.

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.

{{Div col|colwidth=25em}}

• Angle of repose
• Aspect (geography)
• Civil engineering
• Construction surveying
• Grading (engineering)
• Cut-and-cover
• Cut (earthmoving)
• Cut and fill
• Embankment (transportation)
• Grade separation
• Inclined plane
• List of steepest gradients on adhesion railways
• Per mille
• Percentage
• Rake
• Roof pitch
• Ruling gradient
• Slope stability
• Slope
• Slope stability analysis
• Stream slope
• Strike and dip
• Surface gradient
• Surveying
• Trench
• Tunnel
• Wheelchair ramp

{{Div col end}}

{{Reflist|25em}}

• {{cite web |title=British railway gradients and their signs |website=Railsigns |url=http://www.railsigns.uk/sect24page1/sect24page1.html |access-date=16 October 2017 |url-status=usurped |archive-url=https://web.archive.org/web/20171016174532/http://www.railsigns.uk/sect24page1/sect24page1.html |archive-date=16 October 2017}}

{{Geotechnical engineering}}

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