Grade (slope)

The gradient (derived from Gradient " slope ", in Austria also Nivellette ) describes in road and railway construction in the altitude of a planned or existing route in relation to the route ( axis). It is composed of inclined straight line ( pitch ) as well as peaks and Wannenausrundungen and is shown in the profile view.

Unit of measurement and calculation

Gradients are typically expressed as a mathematical fraction, percentage or per mille or as an angle in degrees.

When breaking term in the numerator is a difference in height H, in the denominator, the horizontal distance to be traveled D. If, for example, per 100 m in the horizontal direction distance traveled D the height H 10 m, so the slope is 10:100 = 1:10 = 0.1 = 10% = 100 ‰.

To obtain the pitch angle α as one uses the arc tangent function:

Which in the example results in ≈ 5.7 °. The distance to be traveled on the inclined plane E is according to the Pythagorean Theorem

In the example, ≈ 100.5 m.

In practice, the difference between, for example, read from a map, the horizontal distance D and the distance to be covered E is usually negligible. In contrast, some occurring in the example vehicle inclination of ≈ 5.7 ° is quite noticeable and can be made dynamic driving reasons not be neglected.


The road must comply with certain parameters to comply with the dynamic driving parameters and visibility. The predetermined by the terrain heights must be overcome by sufficiently shallow ramps. In addition to ensuring the dehydration, the ratio between the height and the curvature evolution and cross- slope development must be considered in the site plan. This results in a spatial line routing.

Bow change (S- curves) should not come to lie in the tub, as the drainage of the Verwindungsstrecke via bank becomes difficult then, for example. The required rounding of the crests or troughs as well as the allowable gradients arise from the road category according to RAS -L ( Guidelines for the construction of roads - lines ).

As the slope of the lost to by a subsequent gradient piece again overcome height difference is called.

The sign for the increase or the slope of a road used the percent notation.


The gradient refers at railway lines within the German rail network generally on the top of the rail of the rail is not excessive. Deviations occur within shear ramps (so-called track scissors ), since this results in special dynamic performance of a particular geometric arrangement ( reverse curve ). The elevation of the outer rail arc needs to move from one to the other rail in this case.

Gradients were as a ratio ( fraction ) indicated in railway construction in Germany until the early 20th century, such as 1:80. For flat routes was 1: indicated ∞. Today ( = 0.0125 = 1/80 ) the indication is common in parts per thousand, for example, 12.5 ‰.

After experience of Deutsche Bahn (as of 1994) about 15 % of the costs could be saved in new buildings with 40 ‰ instead of 12.5 ‰ maximum inclination to 20 %.