Grade (slope)

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Template:Short description Template:About Template:Use dmy dates

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

The grade (US) or gradient (UK) (also called slope, incline, mainfall, pitch or rise) of a physical feature, landform or constructed line is either the elevation angle of that surface to the horizontal or its tangent. 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.

Slopes of existing physical features such as canyons and hillsides, stream and river banks, and beds are often described as grades, but typically the word "grade" is 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
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A tram and gradient sign in Gdańsk, Poland, indicating a 50‰ slope (50Template:Nbspm/km, 5%).

There are several ways to express slope:

  1. as an angle of inclination to the horizontal. (This is the angle Template:Mvar 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 <math> 100 \times \frac{\text{rise}}{\text{run}}</math>, which is equivalent to the tangent of the angle of inclination times 100. At a slope angle of 45 degrees, the run is equal to the rise. Expressed as a percentage, the slope of this angle is 100 percent. As the slope approaches 90 degrees (vertical), the percentage slope approaches infinity.<ref>Template:Cite web</ref> In Europe and the U.S. percentage grade is the most commonly used figure for describing slopes.
  3. as a ‰ figure (denoted with the symbol Template:Char), the formula for which is <math>1000 \times \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. It is sometimes written using mm/m or m/km instead of the ‰ symbol.<ref>Template:Cite web </ref><ref>Template:Cite web</ref>
  4. as a ratio of one part rise to so many parts run. For example, a slope that has a rise of 5 feet for every 1000 feet of run would have a slope ratio of 1 in 200. (The word in is normally used rather than the mathematical ratio notation of e.g. 1:200.) This is generally the method used to describe railway grades in Australia and the UK. It is used for roads in Hong Kong, and was used for roads in the UK until the 1970s.
  5. as a ratio of many parts run to one part rise, which is the inverse of the previous expression (depending on the country and the industry standards). For example, a slope expressed as 4:1 in this system means for a given vertical distance the horizontal distance travelled is four times as long.<ref>Template:Cite book</ref>

Any of these may be used. When the term grade is used, the slope is usually expressed as a percentage. If one looks at red numbers on the chart specifying grade, one can see the quirkiness of using the grade to specify slope; the numbers go from 0 for flat, to 100% at 45 degrees, to infinity at vertical.

Slope may still be expressed when the horizontal run is not known: the rise can be divided by the hypotenuse (the slope length). This is not the usual way to specify slope; this nonstandard expression follows the sine function rather than the tangent function, so it calls a 45 degree slope a 71 percent grade instead of a 100 percent. But in practice the usual way to calculate slope is to measure the distance along the slope and the vertical rise, and calculate the horizontal run from that, in order to calculate the grade (100% × rise/run) or standard slope (rise/run). When the angle of inclination is small, using the slope length rather than the horizontal displacement (i.e., using the sine of the angle rather than the tangent) makes only an insignificant difference and can then be used as an approximation. Railway gradients are often expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In either case, the following identity holds for all inclinations up to 90 degrees: <math>\tan{\alpha} = \frac{\sin{\alpha}}{\sqrt{1-\sin^2{\alpha}}}</math>. Or more simply, one can calculate the horizontal run by using the Pythagorean theorem, after which it is trivial to calculate the (standard math) slope or the grade (percentage).

In Europe, road gradients are expressed in signage as percentage.<ref>Template:Cite web</ref>

Equations

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

Tangent as a ratio

<math>\tan{\alpha} = \frac{\Delta h}{d}</math>

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

<math>\%\,\text{slope} = 100 \tan{\alpha}</math>

Angle from a tangent gradient

<math>\alpha = \arctan{\frac{\Delta h}{d}}</math>

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

<math>\alpha = \arctan{\frac{\%\,\text{slope}}{100}}</math>

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

<math>\alpha = \arctan{\frac{1}{n}}</math>

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
35° 70% 700‰ 1 in 1.428
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
30% warning (France)
30% warning (France)
14.0° 25% 250‰ 1 in 4 Very steep road. Mount Washington Cog Railway (average grade)
25% warning (Wales)
25% warning (Wales)
11.3° 20% 200‰ 1 in 5 Steep road
20% warning (North Yorkshire)
20% warning (North Yorkshire)
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
10% warning (Netherlands)
10% warning (Netherlands)
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.<ref name="NASAsci">Template:Cite web</ref><ref name="facts-transp">Template:Cite web</ref>
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 Template:Ill
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
20‰ Czech Republic
20‰ Czech Republic
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 Template:Ill, 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 TableTemplate:DashDidcot to Swindon
0.0434° 0.07575% 0.7575‰ 1 in 1320 Brunel's Billiard TableTemplate:DashPaddington 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, the maximum grade for federally funded highways is specified in a design table based on terrain and design speeds,<ref name="AASHTO Geometry">Template:Cite book</ref> 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 Template:Convert.

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<ref>Template:Cite web</ref> and Canton Avenue in Pittsburgh, Pennsylvania.<ref>Template:Cite news</ref> 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<ref>Template:Cite web</ref> 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.<ref>Template:Cite news</ref> 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.<ref name=Muni/>

Likewise, the Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.<ref>Template:Cite news</ref> The street has formed part of a bicycle race since 1983.<ref>Template:Cite magazine</ref>

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

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File:Track Grade indicator 150-88.JPG
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. Pulling a heavily loaded train at 20 km/h may require ten times the force climbing a 1% slope than on level track.

Early railways in the United Kingdom were laid out with very gentle gradients, such as 0.07575% (1 in 1320) and 0.1515% (1 in 660) on the Great Western main line, nicknamed Brunel's Billiard Table, because the early locomotives (and their brakes) were feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ assistant engines or cable haulage, such as the Template:Convert section from Euston to Camden Town.

Extremely steep gradients require mechanical assistance. Cable systems are used in cases like the Scenic Railway at Katoomba Scenic World in Australia, which reaches a maximum grade of 122% (52°) and is claimed to be the world's steepest passenger-carrying funicular.<ref>Template:Cite news</ref> For somewhat gentler inclines, rack railways are employed, such as the Pilatus Railway in Switzerland, which has a maximum grade of 48% (26°) and is considered the steepest rack railway.<ref>Template:Cite news</ref>

Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in Template:Mvar, Template:Mvar% or Template:Mvar per mille. Since designers like round figures, the method of expression can affect the gradients selected.Template:Citation needed

File:Skloník-klesání.jpg
A 1371-metre-long stretch of railroad with a 20‰ (2%) slope, Czech Republic

The steepest railway lines that do not use rack systems 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

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References

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