Slope gradients are written as Y:X, where Y is a single unit in rise and X is the run. Both numbers must use the same units. For instance, if you travel 3 inches vertically and 3 feet (36 inches) horizontally, the slope would be 3:36 or 1:12. Degrees: Most of us are familiar with slopes measured in degrees. Measure would be 100% (1.0. 100) and the second measure would be 30% (0.3. 100).
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Mowing a sloped yard can be a challenge at best and is potentially unsafe if done improperly or with the wrong equipment. Determining whether your slope is safe to mow is much easier than actually mowing it. If your slope is mowable, you will have to make sure that you have the right equipment to handle the job safely. Some slopes that can be mowed with a walk-behind mower, for example, may not be safe to mow with a riding mower. If your slope is not mowable, you have other options for effectively maintaining the space.
1
Measure the slope of the yard by planting a stake at the top of the steepest part of the slope and another at the bottom. Tie a string between the two and slide the string up the bottom stake until a level says the string is even. Measure the height from the ground to the string on the bottom of the stake. This is the “rise.” Measure the distance between the two stakes: this is the “run.” Divide the rise by the run to determine the slope. Multiply your answer by 100 to convert your slope into a percentage if necessary.
2
Consult the owner’s manual for your mower to determine whether your slope is safe to mow. In general, you should never mow a slope greater than 20 degrees with a walk-behind mower or more than 15 degrees on a riding mower. Zero-turn-radius riding mowers are safest on slopes of less than 10 degrees.
3
Replace slope turf with a ground cover or low-growing grass varieties that do not require mowing such as buffalograss (Buchloe) and fine-leaf fescue (Festuca), both suitable for growing in U.S. Department of Agriculture plant hardiness zone 8. Remote controlled mowers can also be used to more areas unsafe to mow traditionally.
Things You Will Need
- 2 stakes
- String
- Measuring tape
- Calculator
- Lawnmower owner’s manual
- Ground cover plants (optional)
- Remote control lawn mower (optional)
Tip
- Always mow across slopes when using a walk behind mower and up and down the slope when using a riding mower.
Warning
- Never mow a slope if you feel unsafe on it or feel the need to shift your weight to keep the mower stable.
References (8)
About the Author
Writing professionally since 2008, Michelle Miley specializes in home and garden topics but frequently pens career, style and marketing pieces. Her essays have been used on college entrance exams and she has more than 4,000 publishing credits. She holds an Associate of Applied Science in accounting, having graduated summa cum laude.
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- Jupiterimages/Photos.com/Getty Images
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d = run
Δh = rise
l = slope length
α = angle of inclination
Δh = rise
l = slope length
α = angle of inclination
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.
The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described. Grades are typically specified for new linear constructions (such as roads, landscape grading, roof pitches, railroads, aqueducts, and pedestrian or bicycle circulation routes). The grade may refer to the longitudinal slope or the perpendicularcross slope.
- 1Nomenclature
- 1.1Equations
- 4Railways
Nomenclature[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 α 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 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 (streets, roads, highways and rail tracks), surveying, construction, and civil engineering.
- 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.
- 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. 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 meters) of horizontal distance there is a 1-unit (foot or meter) vertical change either up or down.'[1]
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 math 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% x 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 any 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.[2]
Equations[edit]
Grades are related using the following equations with symbols from the figure at top.
Tangent as a ratio[edit]
This ratio can also be expressed as a percentage by multiplying by 100.
Angle from a tangent gradient[edit]
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:
Roads[edit]
In vehicularengineering, 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,[3] 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 are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales[4] and Canton Avenue in Pittsburgh, Pennsylvania.[5] The Guinness World Record lists Baldwin Street as the steepest street in the world, with a 35% grade (19°,1 in 3 slope UK) overall and disputed 38% grade (21°) at its steepest section. The Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.[6] The street has formed part of a bicycle race since 1983.[7]
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.[8]
- 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 Crêtes, 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, Pennsylvania
Environmental design[edit]
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[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 0.05% (1 in 2000), 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[9]) 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[10]) 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 surveyors like round figures, the method of expression can affect the gradients selected.
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) – Pöstlingbergbahn, Linz, Austria[11]
- 11.0% (1 in 9.09) Cass Scenic Railway USA (former logging line)
- 9.0% (1 in 11.11) – Ligne de Saint Gervais – Vallorcine, France
- 9.0% (1 in 11.11) – Muni MetroJ Church, San Francisco[8]
- 8.65% (1 in 11.95) – Portland Streetcar, Oregon, USA[12]
- 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. 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
- 6.0% (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[13]
- 5.89% (1 in 16.97) – Madison, Indiana, United States[14]
- 5.6% (1 in 18) – Flåm, Norway
- 5.3% (1 in 19) – Foxfield Railway, Staffordshire, UK
- 5.1% (1 in 19.6) – Saluda Grade, North Carolina, United States
- 5.0% (1 in 20) – Khyber Pass Railway, Pakistan
- 4.5% (1 in 22.2) – The Canadian Pacific Railway's Big Hill (prior to the construction of the Spiral Tunnels)
- 4.0% (1 in 25) – Cologne-Frankfurt high-speed rail line
- 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[15]
- 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[16]
- 3.0% (1 in 33) – several sections of the Main Western line between Valley Heights and Katoomba in the Blue Mountains Australia.[17]
- 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
- 2.7% (1 in 37)- Braganza Ghats, Bhor Ghat and Thull ghat sections in Indian Railways.
- 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 Østfold 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 (after the construction of the Spiral Tunnels)
- 2.0% (1 in 50) - Numerous locations on New Zealand's railway network
- 1.51% (1 in 66) - (1 foot (0.3 m) per 1 chain (20 m)) New South Wales Government Railways, part of Main South line.
- 1.25% (1 in 80) - Wellington Bank, Somerset.
- 1.25% (1 in 80) - Rudgwick (West Sussex) platform before regrading – too steep if a train is not provided with continuous brakes.
- 0.77% (1 in 130) - Rudgwick 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 trains were provided with continuous brakes, whether air brakes or vacuum brakes, 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 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.
See also[edit]
- List of steepest gradients on adhesion railways.
- Slope stability
References[edit]
- ^page 71, 'SLOPES EXPRESSED AS RATIOS AND DEGREES' in Site Engineering For Landscape Architects 6th Edition. (c)2013, Steven Strom, Kurt Nathan, & Jake Woland. Wiley Publishing. ISBN978-1118090862
- ^'Traffic signs - The Highway Code - Guidance - GOV.UK'. www.gov.uk. Retrieved 2016-03-26.
- ^Staff (2001). A Policy on Geometric Design of Highways and Streets(PDF) (4th ed.). Washington, DC: American Association of State Highway and Transportation Officials. pp. 507 (design speed), 510 (Exhibit 8–1: Maximum Grades for Rural and Urban Freeways). ISBN1-56051-156-7. Retrieved April 11, 2014.
- ^'Bricks don't usually roll': the Welsh town vying for world's steepest street | The Guardian | 10 January 2019
- ^Kiwi climb: Hoofing up the world's steepest street – CNN.com
- ^Here: In Beechview
- ^The Steepest Road On Earth Takes No Prisoners | Autopia | WIRED
- ^ ab'General Information'. San Francisco Metropolitan Transportation Agency. Retrieved September 20, 2016.
- ^'Top five funicular railways'. Sydney Morning Herald.
- ^'A WONDERFUL RAILWAY'. The Register. Adelaide: National Library of Australia. 2 March 1920. p. 5. Retrieved 13 February 2013.
- ^'The New Pöstlingberg Railway'(PDF). Linz Linien GmbH. 2009. Archived from the original(PDF) on 2011-07-22. Retrieved 2011-01-06.
- ^'Return of the (modern) streetcar - Portland leads the way' (October 2001). Light Rail Transit Association. Tramways & Urban Transit. Retrieved 15 December 2018.
- ^'Il Piano Tecnologico di RFI'(PDF). Collegio Ingegneri Ferroviari Italiani. 15 October 2018. Retrieved 23 May 2019.
- ^'Madisonview'. www.oldmadison.com. Retrieved 2017-04-07.
- ^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
External links[edit]
- British railway gradients and their signsRailsigns
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