Grade (slope)

d = run
Δh = rise
l = slope length
α = angle of inclination

The grade (incline 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 several systems for expressing slope:

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 $100 \frac{rise}{run}$ which could also be expressed as the tangent of the angle of inclination times 100. In the U.S., the grade is the most commonly used unit for communicating slopes in transportation, surveying, construction, and civil engineering.
3. as a per mille figure, the formula for which is $1000 \frac{rise}{run}$ 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.
4. as a ratio of one 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 method in which slope may be expressed when the horizontal run is not known: rise divided by the hypotenuse (the slope length). This is not a usual way to measure slope. 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.

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 top.

Tangent as a ratio

$\tan{\alpha} = \frac{\Delta h}{d}$
This ratio can also be expressed as a percentage by multiplying both sides by 100.

Angle from a tangent gradient

$\alpha = \arctan{\frac{\Delta h}{d}}$
If the tangent is expressed as a percentage, the angle can be determined as:
$\alpha = \arctan{\frac{\%\,\mbox{slope}}{100}}$

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.

• 

  slope warning sign, Netherlands

• 

  25% slope sign, Wales

• -

  A 1371 meters distance of a raiload with a 20‰ slope (or 2%). Czech Republic

• 

  Slope warning sign, 30 % over 1500 m. La route des Crêtes, Cassis, France

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.
• 5.1% - Saluda Grade, North Carolina, United States
• 4.0% - Cologne-Frankfurt high-speed rail line
• 4.0% (1 in 25) - Tarana - Oberon branch, New South Wales, Australia.
• 3.7% (1 in 27) - Ecclesbourne Valley Railway, Heritage Line, Wirksworth, Derbyshire, UK

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 power an animal or a machine requires to climb 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

Civil engineering Construction surveying Cut-and-cover Cut and fill

Cut (earthmoving) Cutting (transportation) Embankment (transportation) Grade separation

Slope Surveying Trench Tunnel

External links

• A Simple Explanation and Example of Grade Calculation