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 perpendicular cross slope.
There are several ways to express slope:
Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle α by taking the inverse tangent of the standard mathematical slope, which is rise/run or the grade/100. If one looks at red numbers on the chart specifying grade, one can see the quirkiness of using the grade to specify slope; the numbers go from 0 for flat, to 100% at 45 degrees, to infinity as it approaches vertical.
In Europe, road gradients are signed as a percentage.
Grades are related using the following equations with symbols from the figure at top.
This ratio can also be expressed as a percentage by multiplying by 100.
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:
In vehicular engineering, various land-based designs (automobiles, sport utility vehicles, trucks, trains, etc.) are rated for their ability to ascend terrain. Trains typically rate much lower than automobiles. The highest grade a vehicle can ascend while maintaining a particular speed is sometimes termed that vehicle's "gradeability" (or, less often, "grade ability"). The lateral slopes of a highway geometry are sometimes called fills or cuts where these techniques have been used to create them.
In the United States, maximum grade for Federally funded highways is specified in a design table based on terrain and design speeds, 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 and Canton Avenue in Pittsburgh, Pennsylvania. The Guinness World Record lists Ffordd Pen Llech as the steepest street in the world, with a 37.45% grade (1 in 2.67 slope UK). 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 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.
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.
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) 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) 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.
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
- 11.0% (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 Metro J Church, San Francisco
- 8.65% (1 in 11.95) – Portland Streetcar, Oregon, USA
- 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
- 5.89% (1 in 16.97) – Madison, Indiana, United States
- 5.6% (1 in 18) – Flåm Line, 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
- 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
- 3.0% (1 in 33) – several sections of the Main Western line between Valley Heights and Katoomba in the Blue Mountains Australia.
- 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.
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.
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.
- Aspect (geography)
- Civil engineering
- Construction surveying
- Grading (engineering)
- Cut and fill
- Cut (earthmoving)
- Embankment (transportation)
- Grade separation
- Inclined plane
- List of steepest gradients on adhesion railways.
- Per mille
- Roof pitch
- Ruling gradient
- Slope stability Slope stability analysis
- Wheelchair ramp