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Grade (slope)

January 13, 2023

This article is about the grade of a topographic feature or constructed element. For other uses, see Slope (disambiguation).

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.

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

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.

Nomenclature

 
Illustration of grades (percentages), angles in degrees and ratio.

There are several ways to express slope:

  1. 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.)
  2. 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.
  3. 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. It is sometimes written as mm/m instead of the ‰ symbol.[1]
  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 "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, "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."[2]

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]

Equations

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:

 

Example slopes comparing the notations

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.

Examples of slopes in the various notations
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 [fr]
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 [fr], 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

Roads

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]


Environmental design

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.

Railways

 
Grade indicator near Bellville, Western Cape, 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. 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]

 
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:

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.

Continuous brakes

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.

See also

References

  1. ^ Description of the Pau-Canfranc railway line - Operations (English Wikipedia) - Tracé (French Wikipedia)
  2. ^ 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.
  3. ^ "Traffic signs". www.gov.uk. The Highway Code - Guidance. Retrieved 26 March 2016.
  4. ^ . NASA. 21 April 2003. Archived from the original on 1 June 2020. Retrieved 18 June 2020.
  5. ^ (PDF). NASA. October 1991. pp. 16–17. PMS 018-B, section 3. Archived from the original (PDF) on 27 January 2005. Retrieved 21 August 2013.
  6. ^ 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. Retrieved 11 April 2014.
  7. ^ "Welsh town claims record title for world's steepest street". Guinness World Records. 16 July 2019.
  8. ^ "Kiwi climb: Hoofing up the world's steepest street". CNN.com.
  9. ^ "Baldwin street in New Zealand reinstated as the world's steepest street". Guinness World Records. 8 April 2020.
  10. ^ "The Real Top 10 List of Steepest Streets in San Francisco". 7x7.
  11. ^ a b . San Francisco Metropolitan Transportation Agency. Archived from the original on 3 December 2016. Retrieved 20 September 2016.
  12. ^ "Canton Avenue, Beechview, PA". Post Gazette.
  13. ^ "The steepest road on Earth takes no prisoners". Wired. Autopia. December 2010.
  14. ^ "Top five funicular railways". Sydney Morning Herald.
  15. ^ "A wonderful railway". The Register. Adelaide, Australia. 2 March 1920. p. 5. Retrieved 13 February 2013 – via National Library of Australia.
  16. ^ (PDF). Linz Linien GmbH. 2009. Archived from the original (PDF) on 22 July 2011. Retrieved 6 January 2011.
  17. ^ "Pantele din Iaşi pun probleme ofertanţilor" (in Romanian). 5 March 2019.
  18. ^ (Press release). Tramways & Urban Transit. Light Rail Transit Association. October 2001. Archived from the original on 27 September 2013. Retrieved 15 December 2018.
  19. ^ "Boston's Light Rail Transit Prepares for the Next Hundred Years" (PDF). onlinepubs.trb.org. Retrieved 23 February 2021.
  20. ^ Daniel, Mac (11 November 2005). . Boston.com. Archived from the original on 6 March 2007. Retrieved 23 February 2021.
  21. ^ "Il Piano Tecnologico di RFI" (PDF). Collegio Ingegneri Ferroviari Italiani. 15 October 2018. Retrieved 23 May 2019.
  22. ^ "Madisonview". www.oldmadison.com. Retrieved 7 April 2017.
  23. ^ "The Matheran Light Railway (extension to the Mountain Railways of India)". UNESCO World Heritage Centre.
  24. ^ Martin, Bruno (September 2005). "Durban-Pietermaritzburg main line map and profile" (PDF). Transport in South and Southern Africa. Retrieved 7 April 2017.
  25. ^ Valley Heights railway station
  26. ^ "Improving the Southeast Expressway: A Conceptual Plan" (PDF). www.bostonmpo.org. Retrieved 24 February 2021.

External links

  • . Railsigns. Archived from the original on 16 October 2017. Retrieved 16 October 2017.

January 13, 2023
grade, slope, this, article, about, grade, topographic, feature, constructed, element, other, uses, slope, disambiguation, grade, also, called, slope, incline, gradient, mainfall, pitch, rise, physical, feature, landform, constructed, line, refers, tangent, an. This article is about the grade of a topographic feature or constructed element For other uses see Slope disambiguation 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 d run Dh rise l slope length a angle of inclination 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 Contents 1 Nomenclature 1 1 Equations 1 1 1 Tangent as a ratio 1 1 2 Angle from a tangent gradient 1 2 Example slopes comparing the notations 2 Roads 3 Environmental design 4 Railways 4 1 Compensation for curvature 4 2 Continuous brakes 5 See also 6 References 7 External linksNomenclature Edit 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 a 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 100 rise run displaystyle 100 times frac text rise 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 as a per mille figure the formula for which is 1000 rise run displaystyle 1000 times frac text rise 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 as mm m instead of the symbol 1 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 2 Any of these may be used Grade is usually expressed as a percentage but this is easily converted to the angle a 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 tan a sin a 1 sin 2 a 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 signed as a percentage 3 Equations Edit Grades are related using the following equations with symbols from the figure at top Tangent as a ratio Edit tan a D h d displaystyle tan alpha frac Delta h d The slope expressed as a percentage can similarly be determined from the tangent of the angle slope 100 tan a displaystyle text slope 100 tan alpha Angle from a tangent gradient Edit a arctan D h d displaystyle alpha arctan frac Delta h d If the tangent is expressed as a percentage the angle can be determined as a arctan slope 100 displaystyle alpha arctan frac text slope 100 If the angle is expressed as a ratio 1 in n then a arctan 1 n displaystyle alpha arctan frac 1 n Example slopes comparing the notations Edit 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 Examples of slopes in the various notations Degrees Percentage Permillage Ratio Remarks60 173 1732 1 in 0 5847 7 110 1100 1 in 0 91 Stoosbahn funicular railway 45 100 1000 1 in 130 1 58 580 1 in 1 724 Lynton and Lynmouth Cliff Railway funicular railway 30 58 577 1 in 1 7325 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 7518 4 33 333 1 in 316 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 77 12 12 5 125 1 in 8 Cable incline on the Cromford and High Peak Railway5 71 10 100 1 in 10 Steep road 4 0 7 70 1 in 14 33 37 5 9 59 1 in 17 Swannington incline on the Leicester and Swannington Railway2 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 line2 0 3 5 35 1 in 28 57 LGV Sud Est LGV Est LGV Mediterranee1 97 3 4 34 1 in 29 Bagworth incline on the Leicester and Swannington Railway1 89 3 3 33 1 in 30 3 Rampe de Capvern on the Toulouse Bayonne railway fr 1 52 2 65 26 5 1 in 37 7 Lickey Incline1 43 2 5 25 1 in 40 LGV Atlantique LGV Nord The Schiefe Ebene 1 146 2 20 1 in 50 Railway near Jilove u Prahy Devonshire Tunnel 0 819 1 43 14 3 1 in 70 Waverley Route0 716 1 25 12 5 1 in 80 Ruling grade of a secondary main line Wellington Bank Somerset0 637 1 11 11 11 1 in 90 Dove Holes Tunnel0 573 1 10 1 in 100 The long drag on the Settle amp Carlisle line0 458 0 8 8 1 in 125 Rampe de Guillerval0 2865 0 5 5 1 in 200 Paris Bordeaux railway fr except for the rampe de Guillerval0 1719 0 3 3 1 in 3330 1146 0 2 2 1 in 5000 0868 0 1515 1 515 1 in 660 Brunel s Billiard Table Didcot to Swindon0 0434 0 07575 0 7575 1 in 1320 Brunel s Billiard Table Paddington to Didcot0 0 0 1 in infinity FlatRoads EditIn 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 10 slope warning sign Netherlands 7 descent warning sign Finland 25 ascent warning sign Wales 30 descent warning sign over 1500 m La Route des Cretes Cassis France A trolleybus climbing an 18 grade in Seattle Ascent of German Bundesstrasse 10 A car parked on Baldwin Street Dunedin New Zealand Looking down Canton Avenue Pittsburgh PennsylvaniaEnvironmental design EditGrade 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 Railways Edit Grade indicator near Bellville Western Cape 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 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 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 13 5 1 in 7 40 Lisbon tram Portugal 11 6 1 in 8 62 Postlingbergbahn Linz Austria 16 11 0 1 in 9 09 Cass Scenic Railway US former logging line 9 0 1 in 11 11 Ligne de Saint Gervais Vallorcine France 9 0 1 in 11 11 Muni Metro J Church San Francisco US 11 8 8 1 in 11 4 Iași tram Romania 17 8 65 1 in 11 95 Portland Streetcar Oregon US 18 8 33 1 in 12 Nilgiri Mountain Railway Tamil Nadu India 8 0 1 in 12 5 Just outside the Tobstone 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 1 1 in 14 08 Erzberg Railway Austria 7 0 1 in 14 28 Bernina Railway Switzerland 6 5 1 in 15 4 Incline from the Causeway Street Tunnel up to the Lechmere Viaduct on the Green Line MBTA Boston Massachusetts US 19 This incline is the steepest grade of tracks in the T system 20 6 0 1 in 16 7 Arica Chile to La Paz Bolivia 6 0 1 in 16 6 Docklands Light Railway London UK 6 0 1 in 16 6 Ferrovia Centrale Umbra Italy 21 6 0 1 in 16 6 Link Light Rail Seattle US 5 89 1 in 16 97 Madison Indiana US 22 5 6 1 in 18 Flam Line Norway 5 3 1 in 19 Foxfield Railway Staffordshire UK 5 1 1 in 19 6 Saluda Grade North Carolina US 5 0 1 in 20 Khyber Pass Railway Pakistan 4 5 1 in 22 2 The Canadian Pacific Railway s Big Hill British Columbia Canada prior to the construction of the Spiral Tunnels 4 0 1 in 25 Cologne Frankfurt high speed rail line Germany 4 0 1 in 25 Bolan Pass Railway Pakistan 4 0 1 in 25 211 2 feet 64 m per 1 mile 1 600 m Tarana Oberon branch New South Wales Australia 4 0 1 in 25 Matheran Light Railway India 23 4 0 1 in 26 Rewanui Incline New Zealand Fitted with Fell center rail but was not used for motive power but only braking 3 6 1 in 27 Ecclesbourne Valley Railway Heritage Line Wirksworth Derbyshire UK 3 6 1 in 28 The Westmere Bank New Zealand has a ruling gradient of 1 in 35 however peaks at 1 in 28 3 33 1 in 30 Umgeni Steam Railway South Africa 24 3 0 1 in 33 several sections of the Main Western line between Valley Heights and Katoomba in the Blue Mountains Australia 25 3 0 1 in 33 The entire Newmarket Line in central Auckland New Zealand 3 0 1 in 33 Otira Tunnel New Zealand which is equipped with extraction fans to reduce chance of overheating and low visibility 3 0 1 in 33 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 26 2 7 1 in 37 Braganza Ghats Bhor Ghat and Thull ghat sections in Indian Railways India 2 7 1 in 37 Exeter Central to Exeter St Davids UK see Exeter Central railway station Description 2 7 1 in 37 Picton Elevation New Zealand 2 65 1 in 37 7 Lickey Incline UK 2 6 1 in 38 A slope near Halden on Ostfold 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 2 3 1 in 43 5 Schiefe Ebene Germany 2 2 1 in 45 5 The Canadian Pacific Railway s Big Hill British Columbia Canada after the construction of the Spiral Tunnels 2 0 1 in 48 Beasdale Bank West Coast Scotland mainline UK 2 0 1 in 50 Numerous locations on New Zealand s railway network New Zealand 1 51 1 in 66 1 foot 0 3 m per 1 chain 20 m New South Wales Government Railways Australia part of Main South line 1 25 1 in 80 Wellington Bank Somerset UK 1 25 1 in 80 Rudgwick UK West Sussex platform before regrading too steep if a train is not provided with continuous brakes 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 Edit 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 Continuous brakes Edit 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 See also EditAspect geography Civil engineering Construction surveying Grading engineering Cut and cover Cut and fill Cut earthmoving Embankment transportation Grade separation Inclined plane List of steepest gradients on adhesion railways Percentage Per mille Rake Roof pitch Ruling gradient Slope Slope stability Slope stability analysis Stream slope Surface gradient Surveying Trench Tunnel Wheelchair rampReferences Edit Description of the Pau Canfranc railway line Operations English Wikipedia Trace French Wikipedia 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 Traffic signs www gov uk The Highway Code Guidance Retrieved 26 March 2016 Crawler Transporter NASA 21 April 2003 Archived from the original on 1 June 2020 Retrieved 18 June 2020 Countdown NASA Launch Vehicles and Facilities PDF NASA October 1991 pp 16 17 PMS 018 B section 3 Archived from the original PDF on 27 January 2005 Retrieved 21 August 2013 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 Retrieved 11 April 2014 Welsh town claims record title for world s steepest street Guinness World Records 16 July 2019 Kiwi climb Hoofing up the world s steepest street CNN com Baldwin street in New Zealand reinstated as the world s steepest street Guinness World Records 8 April 2020 The Real Top 10 List of Steepest Streets in San Francisco 7x7 a b General Information San Francisco Metropolitan Transportation Agency Archived from the original on 3 December 2016 Retrieved 20 September 2016 Canton Avenue Beechview PA Post Gazette The steepest road on Earth takes no prisoners Wired Autopia December 2010 Top five funicular railways Sydney Morning Herald A wonderful railway The Register Adelaide Australia 2 March 1920 p 5 Retrieved 13 February 2013 via National Library of Australia The New Postlingberg Railway PDF Linz Linien GmbH 2009 Archived from the original PDF on 22 July 2011 Retrieved 6 January 2011 Pantele din Iasi pun probleme ofertanţilor in Romanian 5 March 2019 Return of the modern streetcar Portland leads the way Press release Tramways amp Urban Transit Light Rail Transit Association October 2001 Archived from the original on 27 September 2013 Retrieved 15 December 2018 Boston s Light Rail Transit Prepares for the Next Hundred Years PDF onlinepubs trb org Retrieved 23 February 2021 Daniel Mac 11 November 2005 Lechmere Science Park stations reopen Boston com Archived from the original on 6 March 2007 Retrieved 23 February 2021 Il Piano Tecnologico di RFI PDF Collegio Ingegneri Ferroviari Italiani 15 October 2018 Retrieved 23 May 2019 Madisonview www oldmadison com Retrieved 7 April 2017 The Matheran Light Railway extension to the Mountain Railways of India UNESCO World Heritage Centre Martin Bruno September 2005 Durban Pietermaritzburg main line map and profile PDF Transport in South and Southern Africa Retrieved 7 April 2017 Valley Heights railway station Improving the Southeast Expressway A Conceptual Plan PDF www bostonmpo org Retrieved 24 February 2021 External links Edit British railway gradients and their signs Railsigns Archived from the original on 16 October 2017 Retrieved 16 October 2017 Retrieved from https en wikipedia org w index php title Grade slope amp oldid 1130772494, wikipedia, wiki, book, books, library,

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