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| {{about|the grade of a topographic feature or constructed element|other uses|Slope (disambiguation)}}
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| [[File:Grade dimension.svg|thumb|250px|d = run <br>Δh = rise <br>l = slope length <br>α = angle of inclination]] | |
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| The '''grade''' (also called '''slope''', '''incline''', '''gradient''', '''pitch''' or '''rise''') of a physical feature, landform or constructed line refers to the inclination of that surface to the [[horizontal]]. It is a special case of the [[Gradient (calculus)|gradient in calculus]] where zero indicates ''[[Gravitation|gravitational level]]''. 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 and ''rise'' is the vertical distance.
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| One might measure grade or slope of existing physical features (such as [[canyon]] and [[hill]]sides, [[River bank|stream and river banks]] and [[stream bed|bed]]s), or one might specify grades for new construction (such as [[road]]s, [[landscape]] and [[garden]] grading, [[roof pitch]]es, [[Rail tracks|railroad]]s, [[Aqueduct (watercourse)|aqueducts]], and [[pedestrian]]-[[Disability|handicapped]]-[[bicycle]] circulation routes).
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| ==Nomenclature==
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| [[File:Grades degrees.svg|right|thumb|300px|Illustration of grades (percentages) and angles in degrees.]]
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| There are several ways to express slope:
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| # 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.)
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| # as a ''[[Percent sign|percentage]]'', the formula for which is <math> 100 \frac{\text{rise}}{\text{run}}</math> which could also be expressed as the tangent of the angle of inclination times 100. In the U.S., this percentage "grade" is the most commonly used unit for communicating slopes in transportation ([[street]]s, [[road]]s, [[highway]]s and [[Track (rail transport)|rail tracks]]), surveying, construction, and civil engineering. | |
| # as a ''[[per mille]]'' figure, the formula for which is <math> 1000 \frac{\text{rise}}{\text{run}}</math> 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.
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| # 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 100 feet of run would have a slope ratio of 1 in 20. (The word "in" is normally used rather than the mathematical ratio notation of "1:20"). This is generally the method used to describe railway grades in Australia and the UK.
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| Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle α from horizontal or the other expressions.
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| 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; it 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.
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| In Europe, road gradients are signed as a percentage, the exception being Britain, where for [[Traffic sign|road signs]], maps and construction work, the gradient was traditionally expressed as a ratio such as 1 in 12, but signs showing gradient as a percentage are becoming more common.<ref>[http://www.direct.gov.uk/en/TravelAndTransport/Highwaycode/Signsandmarkings/index.htm Highway code: Warning signs]</ref>
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| ===Equations===
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| Grades are related using the following equations with symbols from the figure at top.
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| ;Tangent as a ratio: <math>\tan{\alpha} = \frac{\Delta h}{d}</math><br />This ratio can also be expressed as a percentage by multiplying by 100.
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| ;Angle from a tangent gradient: <math>\alpha = \arctan{\frac{\Delta h}{d}}</math><br />If the tangent is expressed as a percentage, the angle can be determined as:<br /><math>\alpha = \arctan{\frac{\%\,\text{slope}}{100}}</math><br />If the angle is expressed as a ratio ''(1 in n)'' then:<br /><math>\alpha = \arctan{\frac{1}{n}}</math>
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| == Roads ==
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| In [[vehicle|vehicular]] [[engineering]], various [[landform|land]]-based designs ([[Automobile|cars]], [[SUV]]s, [[truck]]s, [[train]]s, etc.) are rated for their ability to ascend [[terrain]]. (Trains typically rate much lower than cars.) 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 [[Embankment (transportation)|fills]] or [[Cut (earthmoving)|cuts]] where these techniques have been used to create them.
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| <gallery>
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| Image:Ten percent slope.svg|10% slope warning sign, [[Netherlands]]
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| Image:Devil's Staircase Wales.jpg|25% slope sign, [[Wales]]
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| Image:Skloník-klesání.jpg|A 1371-metre long stretch of railroad with a 20[[Permille|‰]] (2%) slope, [[Czech Republic]]
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| Image:Pente30%.jpg|Slope warning sign, 30% over 1500 m. La [[Route des Crêtes]], [[Cassis]], [[France]]
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| Image:Jyrkkä alamäki 115.svg|7% descent warning sign, [[Finland]]
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| File:Seattle AM General trolleybus climbing James St near 5th Ave in 1983.jpg|A [[trolleybus]] climbing an 18% grade in [[Seattle]]
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| </gallery>
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| ==Environmental design==
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| 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 aspects of slope considerations in [[environmental design]].
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| == Railways ==
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| [[File:Track Grade indicator 150-88.JPG|thumb|right|200px|'''Grade indicator''' near [[Bellville, Western Cape]], [[South Africa]], showing 1:150 and 1:88 grades.]]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 0.05% (1 in 2000), because the early locomotives (and their brakes) were so feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ [[bank engine|assistant engines]] or [[cable railway|cable haulage]], such as the {{convert|1.2|km|mi|abbr=off}} section from [[Euston railway station|Euston]] to [[Camden Town]]. Extremely steep gradients require the use of cables, or some kind of [[rack railway]], to help the train ascend or descend.
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| Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in n, x% or y per mille. Since surveyors like round figures, the method of expression can affect the gradients selected.
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| The steepest [[Rail adhesion|railway lines]] that do not utilize a rack system include:
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| * 13.5% (1 in 7.40) - [[Trams in Lisbon|Lisbon tram]], [[Portugal]]
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| * 11.6% (1 in 8.62) - [[Pöstlingbergbahn]], [[Linz]], [[Austria]]<ref name=2009brochure>{{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|format=PDF|accessdate=2011-01-06}}</ref>
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| * 11.0% (1 in 9.09) [[Cass Scenic Railway]] USA (former logging line)
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| * 9.0% (1 in 11.11) - [[Ligne de Saint Gervais - Vallorcine]], [[France]]
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| * 7.1% (1 in 14.08) - [[Schneeberg Railway (cog railway)|Erzberg Railway]], [[Austria]]
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| * 7.0% (1 in 14.28) - [[Bernina Railway]], [[Switzerland]]
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| * 6.0% (1 in 16.7) - [[Arica]], [[Chile]] to [[La Paz]], [[Bolivia]]
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| * 6.0% (1 in 16.6) - [[Docklands Light Railway]], [[London]], [[UK]]
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| * 5.6% (1 in 18) - [[Flåm]], [[Norway]]
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| * 5.3% (1 in 19) - [[Foxfield Railway]], [[Staffordshire]], [[UK]]
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| * 5.1% (1 in 19.6) - [[Saluda Grade]], [[North Carolina]], [[United States]]
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| * 5.0% (1 in 20) - [[Khyber Pass Railway]], [[Pakistan]]
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| * 4.0% (1 in 25) - [[Cologne-Frankfurt high-speed rail line]]
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| * 4.0% (1 in 25) - [[Bolan Pass|Bolan Pass Railway]], [[Pakistan]]
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| * 4.0% (1 in 25) - ({{convert|211.2|ft|m|0|}} per {{convert|1|mi|m|}} ) - [[Oberon railway line|Tarana - Oberon branch]], [[New South Wales]], [[Australia]].
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| * 4.0% (1 in 25) - [[Matheran Light Railway]], India<ref>[http://whc.unesco.org/en/tentativelists/2082/ The Matheran Light Railway (extension to the Mountain Railways of India) - UNESCO World Heritage Centre<!-- Bot generated title -->]</ref>
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| * 3.7% (1 in 27) - [[Ecclesbourne Valley Railway]], Heritage Line, [[Wirksworth]], [[Derbyshire]], [[UK]]
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| * 3.0% (1 in 33) - several sections of [[Valley Heights, New South Wales|Valley Heights]] - [[Katoomba, New South Wales|Katoomba]] line, [[Blue Mountains, New South Wales|Blue Mountains]], New South Wales, Australia.<ref>[[Valley Heights railway station]]</ref>
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| * 1.51% (1 in 66) ({{convert|1|ft|m|1}} per {{convert|1|chain|m}}) [[New South Wales Railways]], part of South line.
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| * 1.25% (1 in 80) [[Rudgwick railway station|Rudgwick]] ([[West Sussex]]) platform before regrading - too steep if a train is not provided with [[Railway_brake#Continuous_brakes|continuous brakes]].
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| * 0.77% (1 in 130) [[Rudgwick railway station|Rudgwick]] platform after regrading - not too steep if a train is not provided with continuous brakes.
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| === Compensation for curvature ===
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| Gradients on sharp curves are effectively a bit steeper than the same gradient on straight track, so to "compensate" for this and make the [[ruling grade]] uniform throughout, the gradient on those sharp curves should be reduced slightly.
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| === Continuous brakes ===
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| In the era before trains were provided with [[Railway_brake#Continuous_brakes|continuous brakes]], whether [[Railway air brake|air brake]]s or [[vacuum brake]]s, steep gradients were a serious problem, and it was difficult to stop safely if the line was on a steep grade. In an extreme example, the [[Her Majesty's Railway Inspectorate|Inspector]] insisted that [[Rudgwick railway station]] in [[West Sussex]] be regraded before he would allow it to open. This required the gradient through the platform to be eased from 1 in 80 to 1 in 130.
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| == See also ==
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| {{Div col|cols=3}}
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| * [[Aspect (geography)]]
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| * [[Civil engineering]]
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| * [[Construction surveying]]
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| * [[Tunnel#Cut-and-cover|Cut-and-cover]]
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| * [[Cut and fill]]
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| * [[Cut (earthmoving)]]
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| * [[Embankment (transportation)]]
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| * [[Grade separation]]
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| * [[Inclined plane]]
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| * [[List of steepest gradients on adhesion railways]].
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| * [[Per mil]]le
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| * [[Roof pitch]]
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| * [[Ruling gradient]]
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| * [[Slope]]
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| * [[Surveying]]
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| * [[Trench]]
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| * [[Tunnel]]
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| {{Div col end}}
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| == References ==
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| {{reflist}}
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| == External links ==
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| {{DEFAULTSORT:Grade (Slope)}}
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| [[Category:Physical geography]]
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| [[Category:Construction]]
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| [[Category:Transport engineering]]
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| [[Category:Landscape architecture]]
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| [[Category:Horticulture and gardening]]
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| [[Category:Environmental design]]
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| [[Category:Units of angle]]
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The kid of your , is aware of perseverance and determination are important elements when it comes to an excellent job- . His 1st recording, Keep Me, created the best strikes “All My Friends “Country and Say” Guy,” while his effort, Doin’ Factor, identified the artist-about three direct No. 7 singles: In luke bryan meet and greet tickets - just click the up coming article - addition Getting in touch with Is usually a Very good Factor.”
Within the fall of 2013, Concerts: Bryan & that have an impressive set of , such as Urban. “It’s almost like you are obtaining a approval to go to the next level, says all those artists that were an element of the Tourover right into a greater level of artists.” It twisted among the best trips in their 15-season background.
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