I see the grade of certain sections of track indicated in percentage. Is this percentage related to degrees? What is it a percentage of?
It’s a true percentage – it compares one number as a percent of another number, in this case the rise (in any unit of measurement) to the run. For example, if a given track has 1’ rise in 100’ of run, it would be a 1.0% grade (1 = 1% of 100), a 2’ rise in 100’ = a 2% grade, and so on. A 100% grade – a track that rises 100 feet in 100 feet of run, would be at a 45 degree from the horizontal.
Optional reading from here on.
In U.S. practice, a grade of 1.8% or greater is a sort of a break-even point between a “moderate grade” and a “heavy grade.” (But that’s no hard and fast rule; it very much depends on custom.) Western transcontinental railways with land grants or bonding based on land grant railways used a maximum grade of 2.2%, a grade determined by the Pacific Railway Act to be the maximum economically practical grade (it was the grade on the B&O West End, which became the example). Grades of 3.0% were not uncommon on western main lines that were not land-grant lines, and some main lines which were intended as temporary facilities to get a line open (but often ran for many years) had 4.0% to 4.5% grades. Some main lines had very short grades in the 5.5-6%% range. Some special-purpose railways intended to haul minerals or logs on a permanent basis, both narrow and standard-gauge, used grades of 5 to 8%, and some temporary logging railways used grades in the 10-12% range. There were some adhesion logging railways with 14-16% grades too, for very short distances. Beyond that amount of grade, adhesion locomotives become impractical and cog railways are used instead.
Generally, the grade that matters is called the “ruling grade” – it is the grade on a given subdivision that determined the maximum trailing tonnage for a single locomotive (in steam engine days) regularly assigned to that district. For example, if the district was regularly assigned a heavy 2-8-2, and the maximum grade that locomotive could consistently ascend a
The Virginian engineered their railroad very tidily between Princeton yard (where road trains were to be assembled) and tidewater: the “ruling grade” eastward was 0.2%. Say they planned on hauling 100 cars up 0.2% with one 2-8-2-- then they could reasonably hope to haul the same train up 0.6% with two of the same 2-8-2s, so that became their helper grade eastward. Westward they would still like to haul 100-car trains, empty this time; they figured the 2-8-2 was good for 0.6% with 100 MTs and two engines could handle 1.5%. So those became the westward grades.
Sometimes it’s impossible to say what the ruling grade on a line is, if it’s short enough for momentum to be a big factor. An extreme example of that appears on the SP main line west from Sacramento, where the climb to the Benicia bridge on the former eastward (descending) track is something like 1.9% for 0.7 mile or so, preceded and followed by level.
To help you visualize this, a 1.0% grade is very close to a 1/8 inch rise over the length of a 12" long ruler (there’s 96 of the 1/8" segments in that 12" length).
In degrees, a 1.0% grade is about* 0.55 degrees (in decimal degrees) or 0 degrees 33 minutes in Deg. - min. - sec. format. For small angles such as this, other percentage to degree conversions are close to proportional (linear relationship).
[* - I don’t have my trig calculator with me at the moment; Angle = arc-tangent of grade (as a decimal - 1.0% = 0.010).]
- Paul North.
Sorry, I think I may be confused. By “100 feet of run” I take you to mean 100 feet of track. However, if that is indeed what is intended, then the angle is not 45 degrees. Is “run” the distance that the track would cover if it were level ground?
A 1% grade rises 1 foot in 100 feet of run, which is the projected horizontal line, not the hypotenuse of the triangle (which would be the track).
RWM
Of course, we can’t forget the finagle factor, which is curvature, which creates the compensated grade.
Now with calculator, I can confirm and tell you that a 1.00 % grade = 0.5729 degrees (decimal) = 0 Degrees 34 Minutes 22.6 Seconds of arc. [Those familiar with the definition of a radian - there being 2pi radians in a 360 degree circle, or 57.2958 degrees per radian - or the highway engineer’s [}:)] “100-ft. arc” definition of radius and degree of curvature, R = 5729.58 / Degree of Curvature, or vice-versa - will recognize the significance of the decimal degrees version of this.]
A 1.0 % grade is imperceptible to most laypersons without a nearby reference line, such as a level line or surface, or a really good sense of equilibrium, balance, and level. To stand there and look at it, you wouldn’t be able to tell if a track on that grade is level, ascending at 1 %, or decending at 1 %, etc. Those of us with “calibrated eyeballs” and “feel” from some years of field experience can usually call it to within 1/2 % or so - some are better than others at that acquired skill.
Railroad cars and locomotives can roll away on surprisingly low grades. The figures vary a little bit, but I usually use around 0.2 % (4 lbs. of resistance per 2,000 lb. ton of weight), depending on how soft the track roadbed is and how good the bearings are on the rolling stock - that figure is for roller-bearing equipment, with a little bit of a margin of safety.
For context, the highway people think of railroad grades as essentially flat, as most road grades can can get into the 5 to 10 % range. I believe Interstate Highways are usually limited to 8 % max. grades. Local roads are usually allowed to go up to 10 %, depending on the hilliness of the local terrain, economics, and the locality; likewise, driveways can be as steep as 12 to 15 %. Most parking lots and other small drainage channels (swales and low-flow channels in detention basins) as well as the cross-slopes for sidewa
It can be observed by the excellent engineering responses to this posting that grades are not of much concern in relatively flat terrain. Just arrive at the destination, but be sure to have adequate head room where high water in creeks and rivers may be encountered.
Where grades must be engineered to overcome significant changes in elevation nineteenth and early twentieth century engineers were constrained by two things; tools and equipment which were available and the financial limitations placed by their employers. Therefore the grades were undulating to fit, as nearly as practicable, upon the terrain without significant cut and fills. This created lots of slack action in trains as they traversed the summits and sags in the profile, with substantial operating expense from equipment wear and failure a result.
By the mid 20th century new railroad lines or reallignments were engineered to eliminate undulation and verticle curves were created minimalizing the rate of change between opposing grades. A classic example would be on the Santa Fe’s Williams - Crookton line change completed in 1960. From MP 374.5 to MP 406.3 a 1% decending grade was consistant. Then a 0.88% ascending grade was encountered which was engineered with a 10,00 foot verticle curve. The result was no noticeable slack action.
[#oops]a 10,000 foot verticle curve.
An excellent example ! If I recall correctly, the AREA/ AREMA recommended practice for the length of vertical curves in main line tracks is that in “sags”, a change in grade should preferably not exceed 0.05 % per 100-ft. surveying “station” - to minimize the slack run-in as you point out - and 0.1 % per 100-ft. “station” at summits. In this instance, with a total change of grade at a sag of from -1.00 % to + 0.88 % = 1.88 % total, the recommended length would be 1.88 / 0.05 x 100 ft. = 3,760 ft. So, the Santa Fe’s vertical curve here - at almost 2 miles long !- is about 2.66 times as long = only 37.6 % as sharp as would otherwise be considered acceptable. Chico’s engineers did well ! I hope to see it in person myself in about 3 months.
- Paul North.
RWM: Thanks for the history lesson. Makes a lot of sense and gives pause for the proposed HSR. I do recall that the French built much of their TGV at much steeper grades to cut track milesbut am not sure . You know anything?
Railroad grades and track locating via engineers, etc…has always been an interesting subject to me in the hobby and the recent posts were really intelligent and interesting.
blue streak 1 -
Pending RWM’s response, here’s what I know:
Yes, the TGV does have much steeper grades - up to 3% as I recall. But I believe that was done more to reduce the earthwork costs from what a line with lesser grades would require. Either that, or as you indicate, a line with a lower grade standard would have to meander a lot more to create extra miles to achieve the required rise and fall (technically known as “development”), and that extra mileage would likewise incur more costs.
The TGV may be “the exception that proves the rule” of what RWM said. Staying under 0.4 % is basically correct - though with the undertanding (unstated above) that such a grade is the desired maximum where self-contained motive power units that have to carry their prime mover and source of power around with them, and the output of which are limited by same - such as steam and diesel locomotives. - are likely to operate. But as you know, the TGV is powered by electricity from the overhead catenary, and so is not subject to those inherent limitaitons - in fact, if properly designed the catenary can supply to the train virtually all the power it needs. Plus, the electric motors can accept short-term overloads to “goose it” up those steep but hence short grades. Perhaps one of our more electrical engineer-oriented members here can enlighten us on the typical magnitude and durations of such overloads - I believe that the 5-minute rating can be as high as 150% of the continuous rating of the motors. [:-^]
For example, using the 5-minute overload rating as a guide, at 150 MPH the train will cover 12.5 miles during that time (1/12 hour). A 3 % grade will rise a little over 150 feet per mile, so in that 12.5 mile distance the train can climb 12.5 x 150 = 1,875 feet. There are a few grades that have
Paul,
Minor nitpick - Donner Pass is a crossing of the Sierra’s, not the Rockies…
One reason that the grades for the TGV are of less concern than for slower trains is that all but a few per cent of the drag at top speed is from aerodynamic drag, not rolling resistance. A 3% grade would increase rolling resistance by a factor of 15 over the case for running on flat ground, at which point the combined grade and rolling resistance is maybe a bit higher than aero drag.
- Erik
The newest LGV, LGV Est, has a 3% ruling grade, older LGVs over flatter ground had lesser gradients. The German Rhine-Main NBS has ruling grades of 4%, but is limited to ICE3 or ICE3M units which have distributed traction which allows them to cope with the stiffer gradients. All TGV sets and earlier German ICE sets used powercars (specialized locomotives) at each end. German ICE3 versions, and ICE-Ts have at least one powered bogie under each vehicle except for the restaraunt car. ICE-Ts (tilting) are banned from the Rhein-Main NBS due to their 250kph. maximum speed.
Erik -
Thanks for adding that. Do I correctly understand that the aero drag at top speed is approaching the 3 % range ? That would indicate a rolling resistance on level track in the range of 3.25 % = 65 lbs. per ton of weight. If so, then 1 HP (= 550 ft.-lbs. / sec.) per ton would result in a speed of 550 / 65 = 8.46 ft. / sec. / 1.47 = 5.75 MPH. So, if 1 HP per ton results in 5.75 MPH, then 150 MPH would need 150 / 5.75 = about 26 HP per ton.
The Wikipedia article on the TGV - http://en.wikipedia.org/wiki/TGV - indicates that their power-to-weight ratio is in the range of from 16 to 24 W(atts) / KiloGram. If I’m doing the units conversion properly (1 HP per ton = 0.8 W / KG), then that translates to roughly from 20 to 30 HP per ton, and the figure derived above (26 HP/ton) is right in the middle of that range, so probably consistent. This is more like sports-car P/W ratios than trains !
What this leads to is that the TGV has plenty of power, but it can’t both go at full speed (150 MPH +/-) and climb those 3 % grades at the same time, at least not in compliance with the continuous motor rating - that would take in the range of 50 to 60 HP per ton. Instead, it is either running on a short-term motor rating overload, and/ or the speed is dropping back a little bit to reduce the aero drag and use that power instead to climb the grade. Sparing you all most of the math here, since aero drag is proportional to the speed squared, the cahnged equilibirum on the grade (“balancing speed”) would be at about (square root of 2 = 71 %) of max. speed of 150 MPH = 105 MPH or so. I can believe and accept either or both of those scenarios. So we’ve dissected this pretty well, eh ?
On the nam
Paul,
I don’t remember the exact percentage for aero drag as far as the total train resistance of the TGV, but do remember it was huge ( >90%). I would strongly suspect that the TGV would have to slow down a bit when hitting a 3% grade, but bear in mind that anything under a 1,000’ change in elevation is pretty much a momentum grade. I would also guess that even with the slowdown, the loss in time going up a 3% grade is less than the slowdown due to curves and circuitry with a lesser grade.
- Erik
P.S. Someone actually thinks the Appalachians are mountains??? [swg]
Erik -
Excellent point, which goes back to some of what RWM said in an earlier post here. We can explore more of that later - including what’s a mountain, anyway ? [:-^] - when I have more time in a day or two, than right at the moment.
- Paul.
This is too good an opportunity to pass up to use this one - from NPR’s “Car Talk” show, sometime during the middle of 2000. [EDIT: Looks like it was asked on July 1st, so it would have been answered on July 8th.] You’ll see why when you get to the end.
- Paul North.
Locomotive Puzzler
RAY: Hi, we’re back. You’re listening to Car Talk with us, Click and Clack, the Tappett Brothers, and we’re here to talk about cars, car repair, and, duh, the answer to last week’s Puzzler. And this one came from someone named Tim Sullivan, and I don’t know if the facts are right, but the flavor was just so good, I had to use it.
TOM: I’m beginning to remember it. It was good.
RAY: Anyway, Tim writes, years ago, when the railroads used steam locomotives, and that isn’t even relevant.
TOM: No.
RAY: The Baltimore & Ohio Railroad had a busy freight line running south from Rochester, New York, and they used a locomotive of the 2-8-2 type, also irrelevant, meaning there are two wheels in the front which don’t do anything, eight wheels behind those, which are the drivers. Those are the ones that are connected to the steam engine. And then, two trailing wheels. And a 2-8-2 could handle a train of 80 cars.
TOM: Mmm-hmm.
RAY: But on this particular run, it couldn’t handle a train of 60 cars. It had to have 80.
TOM: That doesn’t make sense.
RAY: Doesn’t make sense. And the hint is, there’s something unusual between Rochester and wherever the train is headed. And what’s unusual –
TOM: Bandits. No. Butch Cassidy and the Sundance Kid.
RAY: Bandits would be good.
TOM: Bandits would be good, but that’s not the answer.
RAY: No, this route consists of a bunch of hills which are pretty closely spaced. Imagine the following scenario.
TOM: OK.