| angle | tangent | sine |
|---|---|---|
| 0° | 0% | 0% |
| 5° | 9% | 9% |
| 10° | 18% | 17% |
| 30° | 58% | 50% |
| 45° | 100% | 71% |
| 60° | 173% | 87% |
| 90° | ∞ | 100% |
A grade (or gradient) is the pitch of a slope, and is often expressed as a percent tangent, or "rise over run". It is used to express the steepness of slope on a hill, stream, roof, railroad, or road, where zero indicates level (with respect to gravity) and increasing numbers correlate to more vertical inclinations. There are three common numbering systems:
- the angle from horizontal in degrees,
- as a percentage: the tangent of the angle of inclination: the ratio of the altitude change to the horizontal distance (this is the more common percentage type), or
- an alternative definition as a percentage: the sine of the angle: the ratio of the altitude change to the surface length 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 unrecognized, 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 [1]. 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.
 slope warning sign, Netherlands |
 slope warning sign, Poland |
 A 1371 meters distance of a raiload with a 20‰ slope (or 2%). Czech Republic |
Contents |
Railways
Steep gradients limit the size of load that a locomotive can haul, including the weight of the locomotive itself. A 1 in 100 gradient (1%) halves the load. Early railways in the United Kingdom were laid out with very gentle gradients, such as 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:
- 1 in 18 - Flåm, Norway.
- 1 in 25 - Tarana - Oberon, New South Wales.
- 1 in 33 - Valley Heights - Katoomba, New South Wales
- 1 in 40 - Goulburn - Cootamundra, New South Wales (southbound)
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. Since steeper gradients require greater energy to climb, rail lines are normally built along a route that permits the use of the gentlest gradients possible. Civil engineering works such as cuttings, embankments and tunnels are employed to achieve this.
Effects of grade
Vehicles proceeding upgrade demand more fuel consumption with typically increased air pollution generation. Sound level increases are also produced by motor vehicles traveling upgrade..[1]
