For most applications, slope is usefully expressed as a rate of change of the sloped surface, and is most often calculated as the algebraic division of "units of rise over units of run" expressed as a percentage; the run being a base distance, and rise a change in elevation.
In mathematical terms, the percentage of rise over run is the trigonometric tangent of the face of the hill. In civil engineering applications and physical geography, the slope is a special case of the gradient of calculus calculated along a particular direction of interest which is normally the route of a highway or railway road bed.
Expression
| angle | tangent | sine |
|---|---|---|
| 0° | 0% | 0% |
| 5° | 9% | 9% |
| 10° | 18% | 17% |
| 30° | 58% | 50% |
| 45° | 100% | 71% |
| 60° | 173% | 87% |
| 90° | ∞ | 100% |
There are three common numbering systems, each base in the features of an ideal right triangle and on similar triangles in trigonometry:
- the angle from horizontal in degrees (the Angle opposite the "rise" side of the triangle),
- as a percentage: the tangent or the angle of inclination: the ratio of the altitude change to the horizontal distance (this is the most commonly used percentage type in transportation, surveying, construction and civil engineering), or
- an alternative definition as a percentage: the sine of the angle: the ratio of the altitude change to the surface length (i.e hypotenuse between any two points on the grade—also known as rise to run (not to be confused with the "rise over run" taught in grade-school geometry).
The difference between the latter two is small for gentle slopes (see small-angle formula). The ambiguities and the small differences that result may permit these two inconsistent approaches to coexist unrecognised, especially where grades considered are 15% or less.
Many of the mathematical principles of slope, that follow from the definition, are applicable in topographic practice. Grade is usually expressed as a percentage. Expressing it as the angle from horizontal carries the same information, but may lead to confusion for readers who are not proficient in trigonometry: they may confuse degree with percent, and/or not know how to do the conversion. In the UK, for road signs, maps and construction work, the gradient is often expressed as a ratio such as 1 in 12, or as a percentage 
Road
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 fill or cuts.
Railways
Steep gradients limit the size of load that a locomotive can haul, including the weight of the locomotive itself. A 1% gradient (1 in 100) halves the load. 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 8 km. Extremely steep gradients need the help of cables, or some kind of rack railway.
The steepest non-rack railway lines include:
- 13.5 % - Lisbon tram, Portugal
- 11.6 % - Pöstlingbergbahn, Linz, Austria
- 9.0 % - Ligne de Saint Gervais - Vallorcine, France
- 7% - Bernina Railway, Switzerland
- 5.6% (1 in 18) - Flåm, Norway.
- 4.0% - Cologne-Frankfurt high-speed rail line
- 4.0% (1 in 25) - Tarana - Oberon, New South Wales.
It is customary for civil engineers to refer to the steepest grade on a section of rail line as the ruling grade for that section. Civil engineering works such as cuttings, embankments and tunnels are employed to achieve this.
Effects of grade
The greater a grade, the more energy an animal or a machine spends climbing it; therefore routes with lower grades are preferred, so long as they do not have other disadvantages, such as causing significantly increased overall travel distance.Vehicles proceeding upgrade demand more fuel consumption with typically increased air pollution generation. Sound level increases are also produced by motor vehicles travelling upgrade.
See also
Notes, links and references
External links
References
Footnotes
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Last updated on Saturday July 05, 2008 at 11:15:04 PDT (GMT -0700)
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