d = run
Δh = rise
l = slope length
α = angle of inclination
The grade (or gradient or pitch or slope) of any physical feature such as a hill, stream, roof, railroad, or road, refers to the amount of inclination of that surface where zero indicates level (with respect to gravity) and larger numbers indicate higher degrees of "tilt". Often slope is calculated as a ratio of "rise over run" in which run is the horizontal distance and rise is the vertical distance.
Expression
Angles for various tangent gradients
There are three ways in which slope may be defined:
- 1. as an angle of inclination from the horizontal of a right triangle. (This is the angle α opposite the "rise" side of the triangle.)
- 2. as a percentage (also known as the grade), the formula for which is (rise ÷ run) * 100 which could also be expressed as the (tangent of the angle of inclination) * 100. The grade is the most commonly used unit for stating slope in transportation, surveying, construction and civil engineering.
- 3. as a ratio of 1 part rise per 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.
Any one of these expressions may be used interchangeably to express the characteristics of a slope. Grade is usually expressed as a percentage, but this may easily be converted to the angle α from horizontal since that carries the same information.
There is a fourth method in which slope may be expressed: the rise is divided by the hypotenuse (the slope length). This is not a usual way to measure slope but it is useful when one only knows the slope length and not the horizontal run. This follows the sine function rather than the tangent function and this method diverges from the "rise over run" method as angles start getting larger (see small-angle formula).
Many of the mathematical principles of slope, that follow from the definition, are applicable in topographic practice. 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 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.
Mathematical equations
Grades can be related using the following equations with symbols from the figure at the top of article.
- Tangent as a ratio
This ratio can also be expressed as a percentage by multiplying both sides by 100.
- Angle from a tangent gradient
If the tangent is expressed as a percentage, the angle can be determined as:
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.
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A 1371 meters distance of a raiload with a 20‰ slope (or 2%). Czech Republic
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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:
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. [2]
See also
References
- ^ Highway code: Warning signs
- ^ C.Michael Hogan, Analysis of highway noise, Journal of Water, Air, & Soil Pollution, Volume 2, Number 3, Biomedical and Life Sciences and Earth and Environmental Science Issue, Pages 387-392, September, 1973, Springer Verlag, Netherlands ISSN 0049-6979
External links
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