![Grade dimension](https://upload.wikimedia.org/wikipedia/commons/d/dc/Grade_dimension.svg)
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
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
, 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
bed
A bed is an item of furniture that is used as a place to sleep, rest, and relax.
Most modern beds consist of a soft, cushioned mattress on a bed frame. The mattress rests either on a solid base, often wood slats, or a sprung base. Many beds ...
s are often described as grades, but typically grades are used for human-made surfaces such as roads,
landscape grading,
roof pitch
Roof pitch is the steepness of a roof expressed as a ratio of inch(es) rise per horizontal foot (or their metric equivalent), or as the angle in degrees its surface deviates from the horizontal. A flat roof has a pitch of zero in either insta ...
es,
railroad
Rail transport (also known as train transport) is a means of transport that transfers passengers and goods on wheeled vehicles running on rails, which are incorporated in tracks. In contrast to road transport, where the vehicles run on a pre ...
s,
aqueducts
Aqueduct may refer to:
Structures
*Aqueduct (bridge), a bridge to convey water over an obstacle, such as a ravine or valley
*Navigable aqueduct, or water bridge, a structure to carry navigable waterway canals over other rivers, valleys, railw ...
, and pedestrian or bicycle routes. The grade may refer to the
longitudinal
Longitudinal is a geometric term of location which may refer to:
* Longitude
** Line of longitude, also called a meridian
* Longitudinal engine, an internal combustion engine in which the crankshaft is oriented along the long axis of the vehicl ...
slope or the
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It ca ...
cross slope
Cross slope, cross fall or camber is a geometric feature of pavement surfaces: the transverse slope with respect to the horizon. It is a very important safety factor. Cross slope is provided to provide a drainage gradient so that water will run ...
.
Nomenclature
![Slope quadrant](https://upload.wikimedia.org/wikipedia/commons/b/b0/Slope_quadrant.svg)
There are several ways to express slope:
# as an ''angle'' of inclination to the horizontal. (This is the angle 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
Per mille (from Latin , "in each thousand") is an expression that means parts per thousand. Other recognised spellings include per mil, per mill, permil, permill, or permille.
The associated sign is written , which looks like a percent sig ...
'' 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.
# 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 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
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse e ...
(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.
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.
Roads
In
vehicular
A vehicle (from la, vehiculum) is a machine that transports people or cargo. Vehicles include wagons, bicycles, motor vehicles (motorcycles, cars, trucks, buses, mobility scooters for disabled people), railed vehicles (trains, trams), wa ...
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, various
land
Land, also known as dry land, ground, or earth, is the solid terrestrial surface of the planet Earth that is not submerged by the ocean or other bodies of water. It makes up 29% of Earth's surface and includes the continents and various isla ...
-based designs (
automobiles,
sport utility vehicle
A sport utility vehicle (SUV) is a car classification that combines elements of road-going passenger cars with features from off-road vehicles, such as raised ground clearance and four-wheel drive.
There is no commonly agreed-upon defini ...
s,
truck
A truck or lorry is a motor vehicle designed to transport cargo, carry specialized payloads, or perform other utilitarian work. Trucks vary greatly in size, power, and configuration, but the vast majority feature body-on-frame constructi ...
s,
train
In rail transport, a train (from Old French , from Latin , "to pull, to draw") is a series of connected vehicles that run along a railway track and transport people or freight. Trains are typically pulled or pushed by locomotives (often ...
s, etc.) are rated for their ability to ascend
terrain
Terrain or relief (also topographical relief) involves the vertical and horizontal dimensions of land surface. The term bathymetry is used to describe underwater relief, while hypsometry studies terrain relative to sea level. The Latin wo ...
. 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 .
The steepest roads in the world are
Baldwin Street
Baldwin Street, in Dunedin, New Zealand is located in the residential suburb of North East Valley, northeast of Dunedin's central business district. ''Guinness World Records'' calls it the steepest street in the world, meaning no street gai ...
in Dunedin, New Zealand,
Ffordd Pen Llech
Ffordd Pen Llech () is a public road in the town of Harlech which lies within Snowdonia National Park, North Wales. It was once considered the steepest street in the world, although that title reverted to the previous holder Baldwin Street on 8 ...
in Harlech, Wales and
Canton Avenue
Canton Avenue is a street in Pittsburgh's Beechview neighborhood which is the steepest officially recorded public street in the United States.
Canton Avenue is long (the hill is about 213 feet long) and is claimed to include a 37% grade long. ...
in Pittsburgh, Pennsylvania. The Guinness World Record once again lists
Baldwin Street
Baldwin Street, in Dunedin, New Zealand is located in the residential suburb of North East Valley, northeast of Dunedin's central business district. ''Guinness World Records'' calls it the steepest street in the world, meaning no street gai ...
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
Ffordd Pen Llech () is a public road in the town of Harlech which lies within Snowdonia National Park, North Wales. It was once considered the steepest street in the world, although that title reverted to the previous holder Baldwin Street on 8 ...
. 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.
The
San Francisco Municipal Railway
The San Francisco Municipal Railway (SF Muni or Muni), is the public transit system for the City and County of San Francisco. It operates a system of bus routes (including trolleybuses), the Muni Metro light rail system, three historic cabl ...
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.
File:Nederlands verkeersbord J6.svg, 10% slope warning sign, Netherlands
)
, anthem = ( en, "William of Nassau")
, image_map =
, map_caption =
, subdivision_type = Sovereign state
, subdivision_name = Kingdom of the Netherlands
, established_title = Before independence
, established_date = Spanish Netherl ...
File:Finland road sign 115.svg, 7% descent warning sign, Finland
Finland ( fi, Suomi ; sv, Finland ), officially the Republic of Finland (; ), is a Nordic country in Northern Europe. It shares land borders with Sweden to the northwest, Norway to the north, and Russia to the east, with the Gulf of B ...
File:Devil's Staircase Wales.jpg, 25% ascent warning sign, Wales
Wales ( cy, Cymru ) is a country that is part of the United Kingdom. It is bordered by England to the east, the Irish Sea to the north and west, the Celtic Sea to the south west and the Bristol Channel to the south. It had a population in ...
File:Pente30%.jpg, 30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis
Cassis (; Occitan: ''Cassís'') is a commune situated east of Marseille in the department of Bouches-du-Rhône in the Provence-Alpes-Côte d'Azur region, whose coastline is known in English as the French Riviera, in Southern France. In 201 ...
, France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans. Its metropolitan area ...
File:Seattle AM General trolleybus climbing James St near 5th Ave in 1983.jpg, A trolleybus
A trolleybus (also known as trolley bus, trolley coach, trackless trolley, trackless tramin the 1910s and 1920sJoyce, J.; King, J. S.; and Newman, A. G. (1986). ''British Trolleybus Systems'', pp. 9, 12. London: Ian Allan Publishing. .or trol ...
climbing an 18% grade in Seattle
Seattle ( ) is a seaport city on the West Coast of the United States. It is the seat of King County, Washington. With a 2020 population of 737,015, it is the largest city in both the state of Washington and the Pacific Northwest regio ...
B10 Enzweihinger Steige 20060528.JPG, Ascent of German Bundesstraße 10
The Bundesstraße 10 (abbr. B10) is a German federal highway. It leads from Eppelborn, near the city of Lebach in Saarland, eastward to Neusäß near Augsburg in Bavaria. The Bundesautobahn 8 mostly runs in parallel to the Bundesstraße 10.
Aft ...
File:DunedinBaldwinStreet_Parked_Car.jpg, A car parked on Baldwin Street
Baldwin Street, in Dunedin, New Zealand is located in the residential suburb of North East Valley, northeast of Dunedin's central business district. ''Guinness World Records'' calls it the steepest street in the world, meaning no street gai ...
, Dunedin, New Zealand
Dunedin ( ; mi, Ōtepoti) is the second-largest city in the South Island of New Zealand (after Christchurch), and the principal city of the Otago region. Its name comes from , the Scottish Gaelic name for Edinburgh, the capital of Scotland. Th ...
File:CantonAve_Top.jpg, Looking down Canton Avenue
Canton Avenue is a street in Pittsburgh's Beechview neighborhood which is the steepest officially recorded public street in the United States.
Canton Avenue is long (the hill is about 213 feet long) and is claimed to include a 37% grade long. ...
, Pittsburgh, Pennsylvania
Pittsburgh ( ) is a city in the Commonwealth of Pennsylvania, United States, and the county seat of Allegheny County. It is the most populous city in both Allegheny County and Western Pennsylvania, the second-most populous city in Pennsylva ...
Environmental design
Grade, pitch, and slope are important components in
landscape design
Landscape design is an independent profession and a design and art tradition, practiced by landscape designers, combining nature and culture. In contemporary practice, landscape design bridges the space between landscape architecture and gard ...
,
garden design
Garden design is the art and process of designing and creating plans for layout and planting of gardens and landscapes. Garden design may be done by the garden owner themselves, or by professionals of varying levels of experience and expertise. ...
,
landscape architecture, and
architecture
Architecture is the art and technique of designing and building, as distinguished from the skills associated with construction. It is both the process and the product of sketching, conceiving, planning, designing, and constructing building ...
; 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
Environmental design is the process of addressing surrounding environmental parameters when devising plans, programs, policies, buildings, or products. It seeks to create spaces that will enhance the natural, social, cultural and physical environm ...
.
Railways
Ruling gradient
The term ruling grade is usually used as a synonym for "steepest climb" between two points on a railroad. More simply, the steepest grade to be climbed dictates how powerful the motive power (or how light the train) must be in order for the run to ...
s 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 section from
Euston to
Camden Town.
Extremely steep gradients require the use of cables (such as the Scenic Railway at
Katoomba Scenic World
Scenic World is a private, family-owned tourist attraction located in Katoomba in the Blue Mountains, New South Wales, Australia, about 100 kilometres west of Sydney. Scenic World is home to four attractions, the Scenic Railway, the Sce ...
, Australia, with a maximum grade of 122% (52°), claimed to be the world's steepest passenger-carrying funicular) or some kind of
rack railway
A rack railway (also rack-and-pinion railway, cog railway, or cogwheel railway) is a steep grade railway with a toothed rack rail, usually between the running rails. The trains are fitted with one or more cog wheels or pinions that mesh with th ...
(such as the
Pilatus railway
The Pilatus Railway (german: Pilatusbahn, links=no, PB) is a mountain railway in Switzerland and the steepest rack railway in the world, with a maximum gradient of 48% and an average gradient of 35%. The line runs from Alpnachstad, on Lake Al ...
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 , % or per mille. Since designers like round figures, the method of expression can affect the gradients selected.
![Skloník-klesání](https://upload.wikimedia.org/wikipedia/commons/5/5a/Sklon%C3%ADk-kles%C3%A1n%C3%AD.jpg)
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) –
Pöstlingbergbahn
The Pöstlingbergbahn () is a narrow-gauge electric railway, or "mountain tramway", in Linz, Austria. It connects the main square in the centre of Linz with the district of Pöstlingberg, located at the top of a hill (or small mountain) at the n ...
,
Linz
Linz ( , ; cs, Linec) is the capital of Upper Austria and third-largest city in Austria. In the north of the country, it is on the Danube south of the Czech border. In 2018, the population was 204,846.
In 2009, it was a European Capital ...
, Austria
* 11.0% (1 in 9.09) –
Cass Scenic Railway, US (former logging line)
* 9.0% (1 in 11.11) –
Ligne de Saint Gervais – Vallorcine
The ''ligne'' ( ), or line or Paris line, is a historic unit of length used in France and elsewhere prior to the adoption of the metric system in the late 18th century, and used in various sciences after that time. The ''loi du 19 frimaire an V ...
, France
* 9.0% (1 in 11.11) –
Muni Metro
Muni Metro is a light rail system serving San Francisco, California, United States. Operated by the San Francisco Municipal Railway (Muni), a part of the San Francisco Municipal Transportation Agency (SFMTA), Muni Metro served an average of 15 ...
J Church,
San Francisco
San Francisco (; Spanish for " Saint Francis"), officially the City and County of San Francisco, is the commercial, financial, and cultural center of Northern California. The city proper is the fourth most populous in California and 17th ...
, US
* 8.8% (1 in 11.4) –
Iași tram, Romania
* 8.65% (1 in 11.95) –
Portland Streetcar
The Portland Streetcar is a streetcar system in Portland, Oregon, that opened in 2001 and serves areas surrounding downtown Portland. The NS Line runs from Northwest Portland to the South Waterfront via Downtown and the Pearl District. The Lo ...
,
Oregon
Oregon () is a state in the Pacific Northwest region of the Western United States. The Columbia River delineates much of Oregon's northern boundary with Washington, while the Snake River delineates much of its eastern boundary with Idaho. T ...
, US
* 8.33%(1 in 12) –
Nilgiri Mountain Railway
The Nilgiri Mountain Railway (NMR) is a railway in Nilgiris district, Tamil Nadu, India, built by the United Kingdom, British in 1908. The railway is operated by the Southern Railway zone, Southern Railway and is the only rack railway in India. ...
Tamil Nadu, India
* 8.0% (1 in 12.5) – Just outside the Tobstone Jct. Station in the
Tombstone Junction
Tombstone Junction was a small, Western-town-themed park located on Kentucky Route 90 in McCreary County, Kentucky near the Cumberland Falls State Resort Park. It began operating in the 1960s and continued uninterrupted until it was heavily ...
Theme Park,
Kentucky
Kentucky ( , ), officially the Commonwealth of Kentucky, is a state in the Southeastern region of the United States and one of the states of the Upper South. It borders Illinois, Indiana, and Ohio to the north; West Virginia and Virginia ...
, US. The railroad line there had a
ruling grade
The term ruling grade is usually used as a synonym for "steepest climb" between two points on a railroad. More simply, the steepest grade to be climbed dictates how powerful the motive power (or how light the train) must be in order for the run ...
of 6% (1 in 16.7).
* 7.1% (1 in 14.08) –
Erzberg Railway
, photo=HochtorOedsteinGroup FromS HoherZinken.jpg
, photo_caption=Hochtor - Ödstein Group (Gesäuse) (from right to left) as seen from Hoher Zinken in the south
, country= Austria
, subdivision1_type=States
, subdivision1=
, parent= Northe ...
, Austria
* 7.0% (1 in 14.28) –
Bernina Railway
The Bernina Railway (german: Berninalinie; it, Linea del Bernina; rm, Lingia dal Bernina) is a single-track railway line forming part of the Rhaetian Railway (RhB). It links the spa resort of St. Moritz, in the canton of Graubünden, Switze ...
, Switzerland
* 6.5% (1 in 15.4) – Incline from the Causeway Street Tunnel up to the
Lechmere Viaduct
The Lechmere Viaduct is a concrete arch bridge connecting the West End neighborhood of Boston to East Cambridge, Massachusetts. Opened in 1912, the viaduct carries the MBTA's Green Line over the Charles River. It is adjacent to the Charles Riv ...
on the
Green Line (MBTA)
The Green Line is a light rail system run by the Massachusetts Bay Transportation Authority (MBTA) in the Boston, Massachusetts, metropolitan area. It is the oldest MBTA subway line, and with tunnel sections dating from 1897, the oldest subway ...
,
Boston
Boston (), officially the City of Boston, is the state capital and most populous city of the Commonwealth of Massachusetts, as well as the cultural and financial center of the New England region of the United States. It is the 24th- mo ...
,
Massachusetts
Massachusetts (Massachusett: ''Muhsachuweesut Massachusett_writing_systems.html" ;"title="nowiki/> məhswatʃəwiːsət.html" ;"title="Massachusett writing systems">məhswatʃəwiːsət">Massachusett writing systems">məhswatʃəwiːsət'' En ...
, US. This incline is the "steepest grade of tracks in the T system."
* 6.0% (1 in 16.7) –
Arica
Arica ( ; ) is a commune and a port city with a population of 222,619 in the Arica Province of northern Chile's Arica y Parinacota Region. It is Chile's northernmost city, being located only south of the border with Peru. The city is the capita ...
, Chile to
La Paz, Bolivia
* 6.0% (1 in 16.6) –
Docklands Light Railway
The Docklands Light Railway (DLR) is an automated light metro system serving the redeveloped Docklands area of London, England and provides a direct connection between London's two major financial districts, Canary Wharf and the City of Lo ...
, London, UK
* 6.0% (1 in 16.6) –
Ferrovia Centrale Umbra, Italy
*6.0% (1 in 16.6) -
Link Light Rail
Link light rail is a light rail rapid transit system serving the Seattle metropolitan area in the U.S. state of Washington. It is managed by Sound Transit in partnership with local transit providers, and consists of two non-connected lines: ...
,
Seattle
Seattle ( ) is a seaport city on the West Coast of the United States. It is the seat of King County, Washington. With a 2020 population of 737,015, it is the largest city in both the state of Washington and the Pacific Northwest regio ...
, US
* 5.89% (1 in 16.97) –
Madison, Indiana
Madison is a city in and the county seat of Jefferson County, Indiana, United States, along the Ohio River. As of the 2010 United States Census its population was 11,967. Over 55,000 people live within of downtown Madison. Madison is the larges ...
, US
* 5.6% (1 in 18) –
Flåm Line
The Flåm Line ( no, Flåmsbana) is a long railway line between Myrdal and Flåm in Aurland Municipality, in Vestland county, Norway. A branch line of the Bergen Line, it runs through the valley of Flåmsdalen and connects the mainline with So ...
, Norway
* 5.3% (1 in 19) –
Foxfield Railway
The Foxfield Railway is a preserved standard gauge line located south east of Stoke-on-Trent. The line was built in 1893 to serve the colliery at Dilhorne on the Cheadle Coalfield. It joined the North Staffordshire Railway line near Blythe Bri ...
,
Staffordshire, UK
* 5.1% (1 in 19.6) –
Saluda Grade
Saluda Grade is the steepest standard-gauge mainline railway grade in the United States.. Owned by the Norfolk Southern Railway as part of its W Line, Saluda Grade in Polk County, North Carolina, gains in elevation in less than between Melros ...
,
North Carolina
North Carolina () is a state in the Southeastern region of the United States. The state is the 28th largest and 9th-most populous of the United States. It is bordered by Virginia to the north, the Atlantic Ocean to the east, Georgia and ...
, US
* 5.0% (1 in 20) –
Khyber Pass Railway
The Khyber Pass Railway ( ur, ) was one of several railway lines in Pakistan, operated and maintained by Pakistan Railways. The line begins at Peshawar City and ended at Landi Khana. The total length of this railway line is with 13 railway st ...
, Pakistan
* 4.5% (1 in 22.2) – The
Canadian Pacific Railway's
Big Hill
The Big Hill on the Canadian Pacific Railway main line in British Columbia, Canada, was the most difficult piece of railway track on the Canadian Pacific Railway's route. It was situated in the rugged Canadian Rockies west of the Continental Div ...
,
British Columbia
British Columbia (commonly abbreviated as BC) is the westernmost province of Canada, situated between the Pacific Ocean and the Rocky Mountains. It has a diverse geography, with rugged landscapes that include rocky coastlines, sandy beaches, ...
, Canada (prior to the construction of the
Spiral Tunnels
Lists of spiral (helicoidal) tunnels and tunnels on a curved alignment on roads and railway lines worldwide.
Road tunnels
* Churchischleif, road to Isenfluh (Switzerland)
* Drammen Spiral, Norway
* on SS 659 near Formazza, Italy (full 360° turn ...
)
* 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) – ( per ) –
Tarana – Oberon branch,
New South Wales
)
, nickname =
, image_map = New South Wales in Australia.svg
, map_caption = Location of New South Wales in AustraliaCoordinates:
, subdivision_type = Country
, subdivision_name = Australia
, established_title = Before federation
, es ...
, Australia.
* 4.0% (1 in 25) –
Matheran Light Railway
The Matheran Hill Railway (MHR) is a narrow-gauge heritage railway in Maharashtra, India, which is administered by the Central Railway. It covers a distance of , cutting a swathe through forest and connecting Neral to Matheran in the Wester ...
, India
* 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
Wirksworth is a market town in the Derbyshire Dales district of Derbyshire, England. Its population of 5,038 in the 2011 census was estimated at 5,180 in 2019. Wirksworth contains the source of the River Ecclesbourne. The town was granted a mar ...
,
Derbyshire
Derbyshire ( ) is a ceremonial county in the East Midlands, England. It includes much of the Peak District National Park, the southern end of the Pennine range of hills and part of the National Forest. It borders Greater Manchester to the nor ...
, 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
The Umgeni Steam Railway is a gauge heritage railway at Inchanga, near Durban. The Durban to Pietermaritzburg line was built in the 1880s; it runs through a long tunnel at Drummond built in 1878, which is probably the oldest tunnel in use ...
,
South Africa
South Africa, officially the Republic of South Africa (RSA), is the Southern Africa, southernmost country in Africa. It is bounded to the south by of coastline that stretch along the Atlantic Ocean, South Atlantic and Indian Oceans; to the ...
* 3.0% (1 in 33) – several sections of the
Main Western line between
Valley Heights
Valley Heights is a small township of the City of Blue Mountains in New South Wales, Australia. It is about from Sydney and is located east of the township of Springwood. At the , Valley Heights had a population of 1,337 people.
Valley Heig ...
and
Katoomba in the
Blue Mountains Australia.
* 3.0% (1 in 33) – The entire
Newmarket Line
The Newmarket Line is a railway line in Auckland, New Zealand, that runs between Quay Park Junction, near Britomart Transport Centre, and Newmarket Train Station. It is 2.64 km long. It connects the North Island Main Trunk (NIMT), whic ...
in central Auckland, New Zealand
* 3.0% (1 in 33) –
Otira Tunnel
The Otira Tunnel is a railway tunnel on the Midland Line in the South Island of New Zealand, between Otira and Arthur's Pass. It runs under the Southern Alps from Arthur's Pass to Otira – a length of over . The gradient is mainly 1 in 33, a ...
, 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
The Neponset River is a river in eastern Massachusetts in the United States. Its headwaters are at the Neponset Reservoir in Foxborough, near Gillette Stadium. From there, the Neponset meanders generally northeast for about to its mouth at ...
, Boston, Massachusetts, US. The
Ruling Gradient
The term ruling grade is usually used as a synonym for "steepest climb" between two points on a railroad. More simply, the steepest grade to be climbed dictates how powerful the motive power (or how light the train) must be in order for the run to ...
of the Braintree Branch of the
Red Line (MBTA)
The Red Line is a rapid transit line operated by the Massachusetts Bay Transportation Authority (MBTA) as part of the MBTA subway system. The line runs south and east underground from Alewife station in North Cambridge through Somerville and Ca ...
.
* 2.7% (1 in 37) –
Braganza Ghats,
Bhor Ghat
Bhor Ghat is a mountain pass located between and for railway and between Khopoli and Khandala on the road route in Maharashtra, India, on the crest of the Western Ghats.
History
In February 1781, Bhorghat was the site of a battle between t ...
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
The Lickey Incline, south of Birmingham, is the steepest sustained main-line railway incline in Great Britain. The climb is a gradient of 1 in 37.7 (2.65% or 26.5‰ or 1.52°) for a continuous distance of two miles (3.2 km). Constructed ...
, UK
* 2.6% (1 in 38) – A slope near
Halden
Halden (), between 1665 and 1928 known as Fredrikshald, is both a town and a municipality in Viken county, Norway. The municipality borders Sarpsborg to the northwest, Rakkestad to the north and Aremark to the east, as well as the Swedish mun ...
on
Østfold Line
The Østfold Line ( no, Østfoldbanen) is a railway line which runs from Oslo through the western parts of Follo and Østfold to Kornsjø in Norway. It continues through Sweden as the Norway/Vänern Line. The northern half is double track and ...
, Norway – Ok for passenger
multiple units
A multiple-unit train or simply multiple unit (MU) is a self-propelled train composed of one or more carriages joined together, which when coupled to another multiple unit can be controlled by a single driver, with multiple-unit train contro ...
, 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
The Schiefe Ebene ( literally: 'inclined plane') is a steep incline on Bamberg–Hof section of the Ludwig South-North Railway in the region of Upper Franconia, in Bavaria, Germany.
Location and construction
The Schiefe Ebene is locat ...
, Germany
* 2.2% (1 in 45.5) – The
Canadian Pacific Railway's
Big Hill
The Big Hill on the Canadian Pacific Railway main line in British Columbia, Canada, was the most difficult piece of railway track on the Canadian Pacific Railway's route. It was situated in the rugged Canadian Rockies west of the Continental Div ...
,
British Columbia
British Columbia (commonly abbreviated as BC) is the westernmost province of Canada, situated between the Pacific Ocean and the Rocky Mountains. It has a diverse geography, with rugged landscapes that include rocky coastlines, sandy beaches, ...
, Canada (after the construction of the
Spiral Tunnels
Lists of spiral (helicoidal) tunnels and tunnels on a curved alignment on roads and railway lines worldwide.
Road tunnels
* Churchischleif, road to Isenfluh (Switzerland)
* Drammen Spiral, Norway
* on SS 659 near Formazza, Italy (full 360° turn ...
)
* 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) – ( per )
New South Wales Government Railways
The New South Wales Government Railways (NSWGR) was the agency of the Government of New South Wales that administered rail transport in New South Wales, Australia, between 1855 and 1932.
Management
The agency was managed by a range of differen ...
, Australia, part of
Main South line
The Main South Line, sometimes referred to as part of the South Island Main Trunk Railway, is a railway line that runs north and south from Lyttelton in New Zealand through Christchurch and along the east coast of the South Island to Inverca ...
.
* 1.25% (1 in 80) –
Wellington Bank, Somerset, UK
* 1.25% (1 in 80) –
Rudgwick
Rudgwick is a village and Civil parishes in England, civil parish in the Horsham (district), Horsham District of West Sussex, England. The village is west from Horsham on the north side of the A281 road. The parish's northern boundary forms par ...
, UK (
West Sussex
West Sussex is a county in South East England on the English Channel coast. The ceremonial county comprises the shire districts of Adur, Arun, Chichester, Horsham, and Mid Sussex, and the boroughs of Crawley and Worthing. Covering an ...
) platform before regrading – too steep if a train is not provided with
continuous brakes
A railway brake is a type of brake used on the cars of railway trains to enable deceleration, control acceleration (downhill) or to keep them immobile when parked. While the basic principle is similar to that on road vehicle usage, operational f ...
.
* 0.77% (1 in 130) –
Rudgwick
Rudgwick is a village and Civil parishes in England, civil parish in the Horsham (district), Horsham District of West Sussex, England. The village is west from Horsham on the north side of the A281 road. The parish's northern boundary forms par ...
, 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
The term ruling grade is usually used as a synonym for "steepest climb" between two points on a railroad. More simply, the steepest grade to be climbed dictates how powerful the motive power (or how light the train) must be in order for the run ...
uniform throughout, the gradient on those sharp curves should be reduced slightly.
Continuous brakes
In the era before they were provided with
continuous brakes
A railway brake is a type of brake used on the cars of railway trains to enable deceleration, control acceleration (downhill) or to keep them immobile when parked. While the basic principle is similar to that on road vehicle usage, operational f ...
, whether
air brakes or
vacuum brake
The vacuum brake is a braking system employed on trains and introduced in the mid-1860s. A variant, the automatic vacuum brake system, became almost universal in British train equipment and in countries influenced by British practice. Vacuum br ...
s, 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
West Sussex is a county in South East England on the English Channel coast. The ceremonial county comprises the shire districts of Adur, Arun, Chichester, Horsham, and Mid Sussex, and the boroughs of Crawley and Worthing. Covering an ...
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
*
Aspect (geography)
In physical geography and physical geology, aspect (also known as exposure) is the compass direction or azimuth that a terrain surface faces.
For example, a slope landform on the eastern edge of the Rockies toward the Great Plains is descr ...
*
Civil engineering
Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...
*
Construction surveying
Construction surveying or building surveying (otherwise known as "staking", "stake-out", "lay-out", "setting-out" or "BS") is to stake out reference points and markers that will guide the construction of new structures such as roads or buildings. T ...
*
Grading (engineering)
*
Cut-and-cover
A tunnel is an underground passageway, dug through surrounding soil, earth or rock, and enclosed except for the entrance and exit, commonly at each end. A pipeline is not a tunnel, though some recent tunnels have used immersed tube cons ...
*
Cut and fill
In earthmoving, cut and fill is the process of constructing a railway, road or canal whereby the amount of material from cuts roughly matches the amount of fill needed to make nearby embankments to minimize the amount of construction labor.
...
*
Cut (earthmoving)
In civil engineering, a cut or cutting is where soil or rock from a relative rise along a route is removed. The term is also used in river management to speed a waterway's flow by short-cutting a meander.
Cuts are typically used in road, rail ...
*
Embankment (transportation)
A road, railway line, or canal is normally raised onto an embankment made of compacted soil (typically clay or rock-based) to avoid a change in level required by the terrain, the alternatives being either to have an unacceptable change in lev ...
*
Grade separation
*
Inclined plane
An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle from the vertical direction, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six clas ...
*
List of steepest gradients on adhesion railways.
*
Percentage
*
Per mille
*
Rake
*
Roof pitch
Roof pitch is the steepness of a roof expressed as a ratio of inch(es) rise per horizontal foot (or their metric equivalent), or as the angle in degrees its surface deviates from the horizontal. A flat roof has a pitch of zero in either insta ...
*
Ruling gradient
The term ruling grade is usually used as a synonym for "steepest climb" between two points on a railroad. More simply, the steepest grade to be climbed dictates how powerful the motive power (or how light the train) must be in order for the run to ...
*
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
*
Slope stability
Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of earth and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. Slope stability refers to the condition of i ...
**
Slope stability analysis
Slope stability analysis is a static or dynamic, analytical or empirical method to evaluate the stability of earth and rock-fill dams, embankments, excavated slopes, and natural slopes in soil and rock. Slope stability refers to the condition of i ...
*
Stream slope
*
Surface gradient
*
Surveying
*
Trench
A trench is a type of excavation or in the ground that is generally deeper than it is wide (as opposed to a wider gully, or ditch), and narrow compared with its length (as opposed to a simple hole or pit).
In geology, trenches result from ero ...
*
Tunnel
A tunnel is an underground passageway, dug through surrounding soil, earth or rock, and enclosed except for the entrance and exit, commonly at each end. A pipeline is not a tunnel, though some recent tunnels have used immersed tube cons ...
*
Wheelchair ramp
A wheelchair ramp is an inclined plane installed in addition to or instead of stairs. Ramps permit wheelchair users, as well as people pushing strollers, carts, or other wheeled objects, to more easily access a building, or navigate between are ...
References
External links
*
{{DEFAULTSORT:Grade (Slope)
Physical geography
Construction
Transportation engineering
Landscape architecture
Horticulture
Environmental design
Angle