Grade (slope)

The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which run is the horizontal distance (not the distance along the slope) and rise is the vertical distance.

The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described. Grades are typically specified for new linear constructions (such as roads, landscape grading, roof pitches, railroads, aqueducts, and pedestrian or bicycle circulation routes). The grade may refer to the longitudinal slope or the perpendicular cross slope.

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

Nomenclature

Slope quadrant
Illustration of grades (percentages), angles in degrees and ratio.

There are several ways to express slope:

  1. as an angle of inclination to the horizontal. (This is the angle α opposite the "rise" side of a triangle with a right angle between vertical rise and horizontal run.)
  2. as a percentage, the formula for which is which could also be expressed as the tangent of the angle of inclination times 100. In the U.S., this percentage "grade" is the most commonly used unit for communicating slopes in transportation (streets, roads, highways and rail tracks), surveying, construction, and civil engineering.
  3. as a per mille figure, the formula for which is 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 to 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. (The word "in" is normally used rather than the mathematical ratio notation of "1:20"). This is generally the method used to describe railway grades in Australia and the UK. It is used for roads in Hong Kong, and was used for roads in the UK until the 1970s.
  5. as a ratio of many parts run to one part rise, which is the inverse of the previous expression (depending on the country and the industry standards). For example, "slopes are expressed as ratios such as 4:1. This means that for every 4 units (feet or meters) of horizontal distance there is a 1-unit (foot or meter) vertical change either up or down."[1]

Any of these may be used. Grade is usually expressed as a percentage, but this is easily converted to the angle α from horizontal or the other expressions.

Slope may still be expressed when the horizontal run is not known: the rise can be divided by the hypotenuse (the slope length). This is not the usual way to specify slope; it follows the sine function rather than the tangent function, so it calls a 45-degree slope a 71-percent grade instead of a 100-percent. But in practice the usual way to calculate slope is to measure the distance along the slope and the vertical rise, and calculate the horizontal run from that. When the angle of inclination is small, using the slope length rather than the horizontal displacement (i.e., using the sine of the angle rather than the tangent) makes only an insignificant difference. Railway gradients are usually expressed in terms of the rise in relation to the distance along the track as a practical measure. In cases where the difference between sin and tan is significant, the tangent is used. In any case, the following identity holds for all inclinations up to 90 degrees: .

In Europe, road gradients are signed as a percentage.[2]

Equations

Grades are related using the following equations with symbols from the figure at top.

Tangent as a ratio

This ratio can also be expressed as a percentage by multiplying by 100.

Angle from a tangent gradient

If the tangent is expressed as a percentage, the angle can be determined as:

If the angle is expressed as a ratio (1 in n) then:

Roads

In vehicular engineering, various land-based designs (automobiles, sport utility vehicles, trucks, trains, etc.) are rated for their ability to ascend terrain. Trains typically rate much lower than automobiles. 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 fills or cuts where these techniques have been used to create them.

In the United States, maximum grade for Federally funded highways is specified in a design table based on terrain and design speeds,[3] with up to 6% generally allowed in mountainous areas and hilly urban areas with exceptions for up to 7% grades on mountainous roads with speed limits below 60 mph (95 km/h).

The steepest roads in the world are Baldwin Street in Dunedin, New Zealand, Ffordd Pen Llech in Harlech, Wales[4] and Canton Avenue in Pittsburgh, Pennsylvania.[5] The Guinness World Record lists Baldwin Street as the steepest street in the world, with a 35% grade (19°,1 in 3 slope UK) overall and disputed 38% grade (21°) at its steepest section. The Pittsburgh Department of Engineering and Construction recorded a grade of 37% (20°) for Canton Avenue.[6] The street has formed part of a bicycle race since 1983.[7]

The San Francisco Municipal Railway operates bus service among the city's hills. The steepest grade for bus operations is 23.1% by the 67-Bernal Heights on Alabama Street between Ripley and Esmeralda Streets.[8]

Nederlands verkeersbord J6

10% slope warning sign, Netherlands

Finland road sign 115

7% descent warning sign, Finland

Devil's Staircase Wales

25% ascent warning sign, Wales

Pente30%

30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis, France

Seattle AM General trolleybus climbing James St near 5th Ave in 1983

A trolleybus climbing an 18% grade in Seattle

B10 Enzweihinger Steige 20060528

ascent of German Bundesstraße 10

Environmental design

Grade, pitch, and slope are important components in landscape design, garden design, landscape architecture, and architecture; for engineering and aesthetic design factors. Drainage, slope stability, circulation of people and vehicles, complying with building codes, and design integration are all aspects of slope considerations in environmental design.

Railways

Track Grade indicator 150-88
Grade indicator near Bellville, Western Cape, South Africa, showing 1:150 and 1:88 grades.

Ruling gradients limit the load that a locomotive can haul, including the weight of the locomotive itself. On a 1% gradient (1 in 100) a locomotive can pull half (or less) of the load that it can pull on level track. (A heavily loaded train rolling at 20 km/h on heavy rail may require ten times the pull on a 1% upgrade that it does on the level at that speed.) 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 feeble. Steep gradients were concentrated in short sections of lines where it was convenient to employ assistant engines or cable haulage, such as the 1.2 kilometres (0.75 miles) section from Euston to Camden Town. Extremely steep gradients require the use of cables (such as the Scenic Railway at Katoomba Scenic World, Australia, with a maximum grade of 122% (52°), claimed to be the world's steepest passenger-carrying funicular[9]) or some kind of rack railway (such as the Pilatus railway in Switzerland, with a maximum grade of 48% (26°), claimed to be the world's steepest rack railway[10]) to help the train ascend or descend.

Gradients can be expressed as an angle, as feet per mile, feet per chain, 1 in n, x% or y per mille. Since surveyors like round figures, the method of expression can affect the gradients selected.

Skloník-klesání
A 1371-metre long stretch of railroad with a 20 (2%) slope, Czech Republic

The steepest railway lines that do not use a rack system include:

Compensation for curvature

Gradients on sharp curves are effectively a bit steeper than the same gradient on straight track, so to compensate for this and make the ruling grade uniform throughout, the gradient on those sharp curves should be reduced slightly.

Continuous brakes

In the era before trains were provided with continuous brakes, whether air brakes or vacuum brakes, steep gradients were a serious problem, and it was difficult to stop safely if the line was on a steep grade. In an extreme example, the Inspector insisted that Rudgwick railway station in West Sussex be regraded before he would allow it to open. This required the gradient through the platform to be eased from 1 in 80 to 1 in 130.

See also

References

  1. ^ page 71, "SLOPES EXPRESSED AS RATIOS AND DEGREES" in Site Engineering For Landscape Architects 6th Edition. (c)2013, Steven Strom, Kurt Nathan, & Jake Woland. Wiley Publishing. ISBN 978-1118090862
  2. ^ "Traffic signs - The Highway Code - Guidance - GOV.UK". www.gov.uk. Retrieved 2016-03-26.
  3. ^ Staff (2001). A Policy on Geometric Design of Highways and Streets (PDF) (4th ed.). Washington, DC: American Association of State Highway and Transportation Officials. pp. 507 (design speed), 510 (Exhibit 8–1: Maximum Grades for Rural and Urban Freeways). ISBN 1-56051-156-7. Retrieved April 11, 2014.
  4. ^ 'Bricks don't usually roll': the Welsh town vying for world's steepest street | The Guardian | 10 January 2019
  5. ^ Kiwi climb: Hoofing up the world's steepest street – CNN.com
  6. ^ Here: In Beechview
  7. ^ The Steepest Road On Earth Takes No Prisoners | Autopia | WIRED
  8. ^ a b "General Information". San Francisco Metropolitan Transportation Agency. Retrieved September 20, 2016.
  9. ^ "Top five funicular railways". Sydney Morning Herald.
  10. ^ "A WONDERFUL RAILWAY". The Register. Adelaide: National Library of Australia. 2 March 1920. p. 5. Retrieved 13 February 2013.
  11. ^ "The New Pöstlingberg Railway" (PDF). Linz Linien GmbH. 2009. Archived from the original (PDF) on 2011-07-22. Retrieved 2011-01-06.
  12. ^ "Return of the (modern) streetcar - Portland leads the way" (October 2001). Light Rail Transit Association. Tramways & Urban Transit. Retrieved 15 December 2018.
  13. ^ "Madisonview". www.oldmadison.com. Retrieved 2017-04-07.
  14. ^ The Matheran Light Railway (extension to the Mountain Railways of India) – UNESCO World Heritage Centre
  15. ^ Martin, Bruno (September 2005). "Durban - Pietermaritzburg main line map and profile" (PDF). Transport in South and Southern Africa. Retrieved 7 April 2017.
  16. ^ Valley Heights railway station

External links

Cable railway

A cable railway is a railway that uses a cable, rope or chain to haul trains. It is a specific type of cable transportation.

The most common use for a cable railway is to move vehicles on a steeply graded line that is too steep for conventional locomotives to operate on - this form of cable railway is often called an incline or inclined plane. One common form of incline is the funicular - an isolated passenger railway where the cars are permanently attached to the cable. In other forms, the cars attach and detach to the cable at the ends of the cable railway. Some cable railways are not steeply graded - these are often used in quarries to move large numbers of wagons between the quarry to the processing plant.

Cut and fill

In earthmoving, cut and fill is the process of constructing a railway, road or canal whereby the amount of material from cuts roughly matches the amount of fill needed to make nearby embankments, so minimizing the amount of construction labor.

Downhill creep

Downhill creep, also known as soil creep or commonly just creep, is the slow downward progression of rock and soil down a low grade slope; it can also refer to slow deformation of such materials as a result of prolonged pressure and stress. Creep may appear to an observer to be continuous, but it really is the sum of numerous minute, discrete movements of slope material caused by the force of gravity. Friction, being the primary force to resist gravity, is produced when one body of material slides past another offering a mechanical resistance between the two which acts to hold objects (or slopes) in place. As slope on a hill increases, the gravitational force that is perpendicular to the slope decreases and results in less friction between the material that could cause the slope to slide.

Embankment (transportation)

A road, railway line or canal is normally raised onto an embankment made of compacted soil (typically clay or rock-based) to avoid a change in level required by the terrain, the alternatives being either to have an unacceptable change in level or detour to follow a contour. A cutting is used for the same purpose where the land is originally higher than required.

Gradient (disambiguation)

Gradient in vector calculus is a vector field representing the maximum rate of increase of a scalar field or a multivariate function and the direction of this maximal rate.

Gradient may also refer to:

Gradient sro, a Czech aircraft manufacturer

Image gradient, a gradual change or blending of color

Color gradient, a range of position-dependent colors, usually used to fill a region

Texture gradient, the distortion in size which closer objects have compared to objects farther away

Spatial gradient, a gradient whose components are spatial derivatives

Grade (slope), the inclination of a road or other geographic feature

Grading (engineering)

Grading in civil engineering and landscape architectural construction is the work of ensuring a level base, or one with a specified slope, for a construction work such as a foundation, the base course for a road or a railway, or landscape and garden improvements, or surface drainage. The earthworks created for such a purpose are often called the sub-grade or finished contouring (see diagram).

Helderberg Escarpment

The Helderberg Escarpment also known as Helderberg Mountains is an escarpment and mountain range in eastern New York, roughly 11 miles (18 km) west of the city of Albany. The escarpment rises steeply from the Hudson Valley below, with an elevation difference of approximately 700 feet (from 400 to 1,100 feet) over a horizontal distance of approximately 2,000 feet. Much of the escarpment is within John Boyd Thacher State Park, and has views of the Hudson Valley and the Albany area.

Hillclimbing (railway)

Hillclimbing is a problem faced by railway systems when a load must be carried up an incline. While railways have a great ability to haul very heavy loads, this advantage is only significant when the tracks are fairly level. As soon as the gradients increase, the tonnage that can be hauled is greatly diminished.

Incline

Incline, inclined, inclining, or inclination may refer to:

Grade (slope), the tilt, steepness, or angle from horizontal of a topographic feature (hillside, meadow, etc.) or constructed element (road, railway, field, etc.)

Slope, the tilt, steepness, or angle from horizontal of a line (in math and geometry)It can also refer to:

Cable railway, a steeply graded railway that uses a cable or rope to haul trains

Funicular (or funicular railway, a type of cable railway), a cable railway in which a cable attached moves cars up and down a steep slope

Incline, California

Inclined loop, a feature found on some roller coasters

Inclined orbit, an orbit that does not lie on the equatorial plane

Inclined plane, a flat surface whose endpoints are at different heights

Inclined rig, a method of rigging a sail to direct the force of the sails in such a way as to reduce heeling

Inclined tower, a tower that was intentionally built at an incline

Inclining test, a test that determines a ship's stability and the coordinates of its center of gravity

Orbital inclination, the tilt of an object's orbit around a celestial body

Inclinometer

An inclinometer or clinometer is an instrument used for measuring angles of slope (or tilt), elevation, or depression of an object with respect to gravity's direction. It is also known as a tilt indicator, tilt sensor, tilt meter, slope alert, slope gauge, gradient meter, gradiometer, level gauge, level meter, declinometer, and pitch & roll indicator. Clinometers measure both inclines (positive slopes, as seen by an observer looking upwards) and declines (negative slopes, as seen by an observer looking downward) using three different units of measure: degrees, percent, and topo (see Grade (slope) for details). Astrolabes are inclinometers that were used for navigation and locating astronomical objects from ancient times to the Renaissance.

A tilt sensor can measure the tilting in often two axes of a reference plane in two axes.

In contrast, a full motion would use at least three axes and often additional sensors. One way to measure tilt angle with reference to the earth's ground plane, is to use an accelerometer. Typical applications can be found in the industry and in game controllers. In aircraft, the "ball" in turn coordinators or turn and bank indicators is sometimes referred to as an inclinometer.

Matlock Cable Tramway

Matlock Cable Tramway was a cable tramway that served the town of Matlock between 28 March 1893 and 30 September 1927.

Percentage

In mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", or the abbreviations "pct.", "pct"; sometimes the abbreviation "pc" is also used. A percentage is a dimensionless number (pure number).

Regrading

Regrading is the process of grading for raising and/or lowering the levels of land. Such a project can also be referred to as a regrade.

Regrading may be done on a small scale (as in preparation of a house site) or on quite a large scale (as in major reconfiguration of the terrain of a city, such as the Denny Regrade in Seattle).Regrading is typically performed to make land more level (flatter), in which case it is sometimes called levelling.) Levelling can have the consequence of making other nearby slopes steeper, and potentially unstable or prone to erosion.

Slope

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line. Slope is often denoted by the letter m; there is no clear answer to the question why the letter m is used for slope, but it might be from the "m for multiple" in the equation of a straight line "y = mx + b" or "y = mx + c".

Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical - as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan.

The steepness, incline, or grade of a line is measured by the absolute value of the slope. A slope with a greater absolute value indicates a steeper line. The direction of a line is either increasing, decreasing, horizontal or vertical.

The rise of a road between two points is the difference between the altitude of the road at those two points, say y1 and y2, or in other words, the rise is (y2y1) = Δy. For relatively short distances - where the earth's curvature may be neglected, the run is the difference in distance from a fixed point measured along a level, horizontal line, or in other words, the run is (x2x1) = Δx. Here the slope of the road between the two points is simply described as the ratio of the altitude change to the horizontal distance between any two points on the line.

In mathematical language, the slope m of the line is

The concept of slope applies directly to grades or gradients in geography and civil engineering. Through trigonometry, the slope m of a line is related to its angle of incline θ by the tangent function

Thus, a 45° rising line has a slope of +1 and a 45° falling line has a slope of −1.

As a generalization of this practical description, the mathematics of differential calculus defines the slope of a curve at a point as the slope of the tangent line at that point. When the curve is given by a series of points in a diagram or in a list of the coordinates of points, the slope may be calculated not at a point but between any two given points. When the curve is given as a continuous function, perhaps as an algebraic formula, then the differential calculus provides rules giving a formula for the slope of the curve at any point in the middle of the curve.

This generalization of the concept of slope allows very complex constructions to be planned and built that go well beyond static structures that are either horizontals or verticals, but can change in time, move in curves, and change depending on the rate of change of other factors. Thereby, the simple idea of slope becomes one of the main basis of the modern world in terms of both technology and the built environment.

Slope (disambiguation)

Slope or gradient of a line describes its steepness, incline, or grade, in mathematics.

Slope may also refer to:

Grade (slope) of a topographic feature or constructed element

Piste, a marked track for alpine skiing.

Roof pitch, steepness of a roof

Slope (album) by Steve Jansen

A racial slur against Asians

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