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 in calculus where zero indicates 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.
One might measure grade or slope of existing physical features (such as canyon and hillsides, stream and river banks and beds), or one might specify grades for new construction (such as roads, landscape and garden grading, roof pitches, railroads, aqueducts, and pedestrian-handicapped-bicycle circulation routes).
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.
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.
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.
In Europe, road gradients are signed as a percentage, the exception being Britain, where for 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.
Grades are related using the following equations with symbols from the figure at top.
- Tangent as a ratio
This ratio can also be expressed as a percentage by multiplying by 100.
- 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:
In vehicular engineering, various land-based designs (cars, SUVs, trucks, trains, 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 fills or cuts where these techniques have been used to create them.
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.
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/hr 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 assistant engines or cable haulage, such as from Euston to Camden Town, about 1.2 km. Extremely steep gradients need the help of cables, or some kind of rack railway.
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 utilize 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
- 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
- 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
- 4.0% (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 per mile) - Tarana - Oberon branch, New South Wales, Australia.
- 4.0% (1 in 25) - Matheran Light Railway, India
- 3.7% (1 in 27) - Ecclesbourne Valley Railway, Heritage Line, Wirksworth, Derbyshire, UK
- 3.0% (1 in 33) - several sections of line Valley Heights - Katoomba, Blue Mountains, New South Wales, Australia.
- 1.66% (1 in 60) Rudgwick platform with non-continuous brakes - too steep.
- 1.51% (1 in 66) (one foot per chain) New South Wales Railways, part of South line.
- 0.77% (1 in 130) Rudgwick platform with non-continuous brakes - not too steep.
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.
In the era before continuous brakes, whether air brake or vacuum brakes steep gradients were a serious problem making it difficult to stop on the grade safely. In an extreme example, the Inspector insisted that a station 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.
- Highway code: Warning signs
- "The New Pöstlingberg Railway" (PDF). Linz Linien GmbH. 2009. Retrieved 2011-01-06.
- The Matheran Light Railway (extension to the Mountain Railways of India) - UNESCO World Heritage Centre
- Valley Heights railway station