d = run
Δh = rise
l = slope length
α = angle of inclination
Δh = rise
l = slope length
α = angle of inclination
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.
Is the 30 d angle is the angle between the line whose slope we are talking about and the positive x-axis, then Slope = tan30d = 1/SQRT(3) 30% slope means = 0.3 slope. These two values are not equal. That is what can be said from your question that seems somewhat ambiguous to me.
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 perpendicularcross slope.
- 1Nomenclature
- 1.1Equations
- 4Railways
Nomenclature[edit]
Illustration of grades (percentages), angles in degrees and ratio.
There are several ways to express slope:
- 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.)
- 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.
- 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.
- 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.
- 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 by taking the inverse tangent of the standard math slope, which is rise/run or the grade/100. If one looks at red numbers on the chart specifying grade, one can see the quirkiness of using the grade to specify slope; the numbers go from 0 for flat, to 100% at 45 degrees, to infinity as it approaches vertical.
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; this nonstandard expression 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, in order to calculate the grade (100% x rise/run) or standard slope (rise/run). 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 and can then be used as an approximation. Railway gradients are often 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:. Or more simply, one can calculate the horizontal run by using the Pythagorean theorem, after which it is trivial to calculate the (standard math) slope or the grade (percentage).
In Europe, road gradients are signed as a percentage.[2]
Equations[edit]
Grades are related using the following equations with symbols from the figure at top.
Tangent as a ratio[edit]
This ratio can also be expressed as a percentage by multiplying by 100.
Angle from a tangent gradient[edit]
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[edit]
In vehicularengineering, 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]
- 10% slope warning sign, Netherlands
- 7% descent warning sign, Finland
- 25% ascent warning sign, Wales
- 30% descent warning sign, over 1500 m. La Route des Crêtes, Cassis, France
- A trolleybus climbing an 18% grade in Seattle
- ascent of German Bundesstraße 10
- A car parked on Baldwin Street, Dunedin, New Zealand
- Looking down Canton Avenue, Pittsburgh, Pennsylvania
Environmental design[edit]
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[edit]
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.
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:
- 13.5% (1 in 7.40) – Lisbon tram, Portugal
- 11.6% (1 in 8.62) – Pöstlingbergbahn, Linz, Austria[11]
- 11.0% (1 in 9.09) Cass Scenic Railway USA (former logging line)
- 9.0% (1 in 11.11) – Ligne de Saint Gervais – Vallorcine, France
- 9.0% (1 in 11.11) – Muni MetroJ Church, San Francisco[8]
- 8.65% (1 in 11.95) – Portland Streetcar, Oregon, USA[12]
- 8.33%(1 in 12) – Nilgiri Mountain Railway Tamil Nadu, India
- 8.0% (1 in 12.5) - Just outside the Tobstone Jct. Station in the Tombstone Junction Theme Park. The railroad line there had a ruling grade of 6% (1 in 16.7).
- 7.1% (1 in 14.08) – Erzberg Railway, Austria
- 7.0% (1 in 14.28) – Bernina Railway, Switzerland
- 6.0% (1 in 16.7) – Arica, Chile to La Paz, Bolivia
- 6.0% (1 in 16.6) – Docklands Light Railway, London, UK
- 6.0% (1 in 16.6) - Ferrovia Centrale Umbra, Italy[13]
- 5.89% (1 in 16.97) – Madison, Indiana, United States[14]
- 5.6% (1 in 18) – Flåm, Norway
- 5.3% (1 in 19) – Foxfield Railway, Staffordshire, UK
- 5.1% (1 in 19.6) – Saluda Grade, North Carolina, United States
- 5.0% (1 in 20) – Khyber Pass Railway, Pakistan
- 4.5% (1 in 22.2) – The Canadian Pacific Railway's Big Hill (prior to the construction of the Spiral Tunnels)
- 4.0% (1 in 25) – Cologne-Frankfurt high-speed rail line
- 4.0% (1 in 25) – Bolan Pass Railway, Pakistan
- 4.0% (1 in 25) – (211.2 feet (64 m) per 1 mile (1,600 m) ) – Tarana – Oberon branch, New South Wales, Australia.
- 4.0% (1 in 25) – Matheran Light Railway, India[15]
- 4.0% (1 in 26) – Rewanui Incline, New Zealand. Fitted with Fell center rail but was not used for motive power, but only braking
- 3.6% (1 in 27) – Ecclesbourne Valley Railway, Heritage Line, Wirksworth, Derbyshire, UK
- 3.6% (1 in 28) - The Westmere Bank, New Zealand has a ruling gradient of 1 in 35, however peaks at 1 in 28
- 3.33% (1 in 30) – Umgeni Steam Railway, South Africa[16]
- 3.0% (1 in 33) – several sections of the Main Western line between Valley Heights and Katoomba in the Blue Mountains Australia.[17]
- 3.0% (1 in 33) - The entire Newmarket Line in central Auckland, New Zealand
- 3.0% (1 in 33) - Otira Tunnel, New Zealand, which is equipped with extraction fans to reduce chance of overheating and low visibility
- 2.7% (1 in 37)- Braganza Ghats, Bhor Ghat and Thull ghat sections in Indian Railways.
- 2.7% (1 in 37) – Exeter Central to Exeter St Davids, UK (see Exeter Central railway station#Description)
- 2.7% (1 in 37) - Picton- Elevation, New Zealand
- 2.65% (1 in 37.7) – Lickey Incline, UK
- 2.6% (1 in 38) - A slope near Halden on Østfold Line, Norway – Ok for passenger multiple units, but an obstacle for freight trains which must keep their weight down on this international mainline because of the slope. Freight traffic has mainly shifted to road.
- 2.3% ( 1 in 43.5) – Schiefe Ebene, Germany
- 2.2% (1 in 45.5) – The Canadian Pacific Railway's Big Hill (after the construction of the Spiral Tunnels)
- 2.0% (1 in 50) - Numerous locations on New Zealand's railway network
- 1.51% (1 in 66) - (1 foot (0.3 m) per 1 chain (20 m)) New South Wales Government Railways, part of Main South line.
- 1.25% (1 in 80) - Wellington Bank, Somerset.
- 1.25% (1 in 80) - Rudgwick (West Sussex) platform before regrading – too steep if a train is not provided with continuous brakes.
- 0.77% (1 in 130) - Rudgwick platform after regrading – not too steep if a train is not provided with continuous brakes.
Compensation for curvature[edit]
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[edit]
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[edit]
- List of steepest gradients on adhesion railways.
- Slope stability
References[edit]
- ^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. ISBN978-1118090862
- ^'Traffic signs - The Highway Code - Guidance - GOV.UK'. www.gov.uk. Retrieved 2016-03-26.
- ^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). ISBN1-56051-156-7. Retrieved April 11, 2014.
- ^'Bricks don't usually roll': the Welsh town vying for world's steepest street | The Guardian | 10 January 2019
- ^Kiwi climb: Hoofing up the world's steepest street – CNN.com
- ^Here: In Beechview
- ^The Steepest Road On Earth Takes No Prisoners | Autopia | WIRED
- ^ ab'General Information'. San Francisco Metropolitan Transportation Agency. Retrieved September 20, 2016.
- ^'Top five funicular railways'. Sydney Morning Herald.
- ^'A WONDERFUL RAILWAY'. The Register. Adelaide: National Library of Australia. 2 March 1920. p. 5. Retrieved 13 February 2013.
- ^'The New Pöstlingberg Railway'(PDF). Linz Linien GmbH. 2009. Archived from the original(PDF) on 2011-07-22. Retrieved 2011-01-06.
- ^'Return of the (modern) streetcar - Portland leads the way' (October 2001). Light Rail Transit Association. Tramways & Urban Transit. Retrieved 15 December 2018.
- ^'Il Piano Tecnologico di RFI'(PDF). Collegio Ingegneri Ferroviari Italiani. 15 October 2018. Retrieved 23 May 2019.
- ^'Madisonview'. www.oldmadison.com. Retrieved 2017-04-07.
- ^The Matheran Light Railway (extension to the Mountain Railways of India) – UNESCO World Heritage Centre
- ^Martin, Bruno (September 2005). 'Durban - Pietermaritzburg main line map and profile'(PDF). Transport in South and Southern Africa. Retrieved 7 April 2017.
- ^Valley Heights railway station
External links[edit]
- British railway gradients and their signsRailsigns
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Grade_(slope)&oldid=902575200'
Slope
The steepness of any incline can be measured as the ratio of the vertical change to the horizontal change. For example, a 5% incline can be written as 5/100, which means that for every 100 feet forward, the height increases 5 feet.
In mathematics, we call the incline of a line the slopeThe incline of a line measured as the ratio of the vertical change to the horizontal change, often referred to as rise over run. and use the letter m to denote it. The vertical change is called the riseThe vertical change between any two points on a line. and the horizontal change is called the runThe horizontal change between any two points on a line..
The rise and the run can be positive or negative. A positive rise corresponds to a vertical change up and a negative rise corresponds to a vertical change down. A positive run denotes a horizontal change to the right and a negative run corresponds to a horizontal change to the left. Given the graph, we can calculate the slope by determining the vertical and horizontal changes between any two points.
Example 1: Find the slope of the given line:
Solution: From the given points on the graph, count 3 units down and 4 units right.
Answer:
Here we have a negative slope, which means that for every 4 units of movement to the right, the vertical change is 3 units downward. There are four geometric cases for the value of the slope.
Reading the graph from left to right, we see that lines with an upward incline have positive slopes and lines with a downward incline have negative slopes.
If the line is horizontal, then the rise is 0:
The slope of a horizontal line is 0. If the line is vertical, then the run is 0:
The slope of a vertical line is undefined.
Try this! Find the slope of the given line:
Answer:
Video Solution
' href='http://www.youtube.com/v/xLYz6p3JIDM'>(click to see video)Calculating the slope can be difficult if the graph does not have points with integer coordinates. Therefore, we next develop a formula that allows us to calculate the slope algebraically. Given any two points and , we can obtain the rise and run by subtracting the corresponding coordinates.
This leads us to the slope formulaGiven two points and , then the slope of the line is given by the formula .. Given any two points and , the slope is given by
Example 2: Find the slope of the line passing through (−3, −5) and (2, 1).
Solution: Given (−3, −5) and (2, 1), calculate the difference of the y-values divided by the difference of the x-values. Since subtraction is not commutative, take care to be consistent when subtracting the coordinates.
Answer:
We can graph the line described in the previous example and verify that the slope is 6/5.
Certainly the graph is optional; the beauty of the slope formula is that we can obtain the slope, given two points, using only algebra.
Example 3: Find the slope of the line passing through (−4, 3) and (−1, −7).
Solution:
Answer:
When using the slope formula, take care to be consistent since order does matter. You must subtract the coordinates of the first point from the coordinates of the second point for both the numerator and the denominator in the same order.
Example 4: Find the slope of the line passing through (7, −2) and (−5, −2).
Solution:
Answer: . As an exercise, plot the given two points and verify that they lie on a horizontal line.
Example 5: Find the slope of the line passing through (−4, −3) and (−4, 5).
Solution:
Answer: The slope m is undefined. As an exercise, plot the given two points and verify that they lie on a vertical line.
Try this! Calculate the slope of the line passing through (−2, 3) and (5, −5).
Answer:
Video Solution
' href='http://www.youtube.com/v/UT0QPj1QckY'>(click to see video)When considering the slope as a rate of change it is important to include the correct units.
Example 6: A Corvette Coupe was purchased new in 1970 for about $5,200 and depreciated in value over time until it was sold in 1985 for $1,300. At this point, the car was beginning to be considered a classic and started to increase in value. In the year 2000, when the car was 30 years old, it sold at auction for $10,450. The following line graph depicts the value of the car over time.
a. Determine the rate at which the car depreciated in value from 1970 to 1985.
b. Determine the rate at which the car appreciated in value from 1985 to 2000.
Solution: Notice that the value depends on the age of the car and that the slope measures the rate in dollars per year.
a. The slope of the line segment depicting the value for the first 15 years is
Answer: The value of the car depreciated $260 per year from 1970 to 1985.
b. The slope of the line segment depicting the value for the next 15 years is
Answer: The value of the car appreciated $610 per year from 1985 to 2000.