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d = run
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.

Slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described as grades, but typically grades are 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.

There are several ways to express slope:

  1. 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.)
  2. as a percentage, the formula for which is
  • 7% descen

    7% descent warning sign, Finland

  • 25% ascent warnin

    25% ascent warning sign, Wales

  • 30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis, France

  • trolleybus climbing an 18% grade in Seattle

  • ascent of German Bundesstraße 10

  • ascent of German Bundesstraße 10

  • Environmental design

    Grade, pitch, and slope are important components in landscape design, garden design, landscape architecture, and <

    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

    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[10]) 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 railwayEarly 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[10]) 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[11]) 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 designers 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[12]
    • 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 Franciscoruling 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

      References

      1. ^ Strom, Steven; Nathan, Kurt; Woland, Jake (2013). "Slopes expressed as ratios and degrees". Site Engineering for Landscape Architects (6th ed.). Wiley Publishing. p. 71. ISBN 978-1118090862.
      2. ^ "Traffic signs". www.gov.uk. The Highway Code - Guidance. Retrieved 26 March 2016.
      3. ^ A Policy on Geometric Design of Highways and Streets (PDF) (4th ed.). Washington, DC: American Association of State Highway and Transportation Officials. 2001. pp. 507 (design speed), 510 (exhibit 8–1: Maximum grades for rural and urban freeways). ISBN 1-56051-156-7. Retrieved 11 April 2014.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