Curves, and how they are measured

[Ian J.]

Hi,

 

A question: when a curve on a railway line is described as '50 chains', what does this mean for how transitions relate to it?

 

To expand on that question, what I'm looking to understand from the figure is how and where the transitions from the line before and after come into it.

 

Additionally, I'm interested to know how the line builders calculated their curves with the transitions in place.

 

Ultimately, this information will help me create my fictional map for the Sayersbridge & Penmouth (see my blog).

 

TIA

[bike2steam]

80 chains equal one mile, or ten chains equal one furlong.

In 1896 G.H.A. Krohnke of the Prussian Railways set out his ideas in his 'Handbook For Laying Out Curves On Railways, and Tramways' translated, and used by many railways across the globe, regarded by many as a bible for permanent way. All a bit involved - just depends how complicated you need to be drawing out various radii, and transitions between for a model, or just do it by eye. 

[asmay2002]

Hi,

 

A question: when a curve on a railway line is described as '50 chains', what does this mean?

 

It means the radius of the curve is 50 chains (i.e 50*22*3 =) 3300 feet.

[rockershovel]

The curve will be designed to suit the required entry and exit tangents (alignments). This defines the deflection (total curvature) and arc length of the curve, and may define the radius, if the required tangents are fixed in the X,Y plane. Alternatively, a predefined radius may be applied and in this case, the entry and exit tangent points can be calculated along the respective tangents.

 

The transition curves are, strictly speaking, Euler spirals or clothoids in which the rate of change of radius defines the curve (starting with infinite radius and ending on the radius of the circular curve). Early railways often used a simple type known as the "cubic spiral" but modern designs are much more complex, bearing in mind that the train will have design parameters for maximum speed and rate of change of velocity (velocity does not equal speed).

 

I have seen simple transitions created by distributing the change of radius along a predetermined length, but these are not usual and really, only used for lowspeed industrial lines constructed ad hoc

[Ian J.]

It means the radius of the curve is 50 chains (i.e 50*22*3 =) 3300 feet.

 

Apologies, the original question may have come across as unclear, as I know what a chain is (66 feet), but I was wondering how transition curves fit in with the radius quoted.

 

 

The curve will be designed to suit the required entry and exit tangents (alignments). This defines the deflection (total curvature) and arc length of the curve, and may define the radius, if the required tangents are fixed in the X,Y plane. Alternatively, a predefined radius may be applied and in this case, the entry and exit tangent points can be calculated along the respective tangents.

 

The transition curves are, strictly speaking, Euler spirals or clothoids in which the rate of change of radius defines the curve (starting with infinite radius and ending on the radius of the circular curve). Early railways often used a simple type known as the "cubic spiral" but modern designs are much more complex, bearing in mind that the train will have design parameters for maximum speed and rate of change of velocity (velocity does not equal speed).

 

I have seen simple transitions created by distributing the change of radius along a predetermined length, but these are not usual and really, only used for lowspeed industrial lines constructed ad hoc

 

This is the somewhat more complex answer that is along the lines of what I was looking for, but can't for the life of me understand. You don't happen to know of any illustrations to show it do you?

[LNER4479]

Rockershovel has made reference to 'curvature' and it can be useful to talk of this rather than radius, particular where very gentle curves are involved. The maths can be as complex as you want it to be but in essence curvature is the reciprocal of radius, ie 1 / radius. ('one divided by radius').

 

When I dabbled in track alignment in the early years of my railway career (mainly to understand its relationship to vehicle ride), I was shown curvature diagrams of some of the principal UK mainlines and this is a useful way of representing track in a diagrammatic form. The beauty of this is that a straight line has a curvature of 0 (zero), entering a curve the curvature increases (the transition) until a constant curvature is reached, represented by a straight line parallel to the x-axis.

 

The track alignment machines of the day (ie 30 years ago) measured curvature by stretching out a 20m steel wire chord line between their outer bogies and then a central wheelset registered the offset from that chord (they may still do it this way, for all I know?). The resulting measurement was known as a versine. I have adopted an equivalent technique when setting out curves on model railways and find it a good way of mapping out curves and transitions. Afterall, if you're after a very gentle curve, the length of string to draw an arc of large radius becomes exponentially impractical!

[Derekstuart]

Ian,

 

Try this link. I think it explains it all fairly well

 

https://www.thepwi.org/member/network_rail_track_delivery_update__steve_featherstone/nr_track_delivery_update_28_may_2014/pway_design_guide_2011

 

If you want to know more about track geometry, look at Templot .com website. Even if not using the programme, the amount of information regarding prototype track is staggering- but you will need to look around for it as there's a lot there.

 

Unless you are build real track for a living (unlikely as you are asking this) or have years to study it, using Templot to plan this (even if you use it for off the shelf track) will help you get it as per prototype.

[Joseph_Pestell]

Rockershovel has made reference to 'curvature' and it can be useful to talk of this rather than radius, particular where very gentle curves are involved. The maths can be as complex as you want it to be but in essence curvature is the reciprocal of radius, ie 1 / radius. ('one divided by radius').

 

When I dabbled in track alignment in the early years of my railway career (mainly to understand its relationship to vehicle ride), I was shown curvature diagrams of some of the principal UK mainlines and this is a useful way of representing track in a diagrammatic form. The beauty of this is that a straight line has a curvature of 0 (zero), entering a curve the curvature increases (the transition) until a constant curvature is reached, represented by a straight line parallel to the x-axis.

 

The track alignment machines of the day (ie 30 years ago) measured curvature by stretching out a 20m steel wire chord line between their outer bogies and then a central wheelset registered the offset from that chord (they may still do it this way, for all I know?). The resulting measurement was known as a versine. I have adopted an equivalent technique when setting out curves on model railways and find it a good way of mapping out curves and transitions. Afterall, if you're after a very gentle curve, the length of string to draw an arc of large radius becomes exponentially impractical!

 

Fortunately, these days we don't need a long piece of string to draw such a curve. Any of the track planning programs will allow you to print out a curve template with whatever radius you need. Interesting though the idea of creating a chord between the bogies of a vehicle to do it.

 

It's an interesting subject. Some time ago, Jamie 92208 published on here a document that he has for the building of the West Riding Line at Dewsbury. The survey drawing goes direct from a straight track to a 66 chain curve but it is annotated that this is a centre line from which the track will deviate between certain boundaries. Is there a point at which a transition is no longer necessary or would the final arrangement see a ?65 chain curve linked to the straight track by a transition length of ?a couple of chains length. If of this sort of order, it would hardly show up on a model.

[martin_wynne]

A question: when a curve on a railway line is described as '50 chains', what does this mean for how transitions relate to it?

 

To expand on that question, what I'm looking to understand from the figure is how and where the transitions from the line before and after come into it.

 

Hi Ian,

 

Straight track has no cant (superelevation), the two rails are level with each other. Curved track needs cant (outer rail higher than inner rail) to allow vehicles to negotiate it at speed comfortably.

 

So if you simply join a straight track to a curved track, you would have a vertical step in the rails, which is obviously daft.

 

A transition curve (easement) is therefore used between the straight and the curve, along which the radius gradually reduces, and the corresponding cant gradually increases.

 

There is a lot of maths in doing a detailed design. The amount of cant needed is determined by the radius of the curve and the vehicle speed. The latter obviously varies, so a design is found where the cant is sufficient for fast vehicles, if a bit less than ideal, and not so much as to upset slow or stationary vehicles. There are design limits on the total amount of cant, and the amount by which it can vary from the ideal (called the "cant deficiency"), and the rate at which it can rise or fall along a transition curve (called the "cant gradient").

 

I believe your question boils down to "What is the length of this transition curve?".

 

Fortunately, in a paper read to the Institution of Civil Engineers in January 1909, W. H. Shortt provided some rules of thumb about this. These wouldn't satisfy modern permanent way engineers, but they are very likely to have been used in laying out bullhead track in the steam era. So applicable to your fictional map.

 

The rules are:

 

Rule 1. Minimum length of transition in feet = line speed in feet per second, cubed, divided by radius in feet.

 

So for your 50 chain curve, the radius is 50 x 66 = 3300 ft.

 

If designed for say a steam era speed of 60mph, this is 88 ft/sec.

 

So the minimum transition length would be 88 cubed, divided by 3300 = 206 ft.

 

Then there is a general rule to provide 50% more than the minimum length if space permits, taking the transition length to 309 ft.

 

Using that length we can calculate the required Shift. That is the amount by which the straight track must be moved sideways from the curved track, to allow space for a transition curve to be created between them.

 

Rule 2. Shift in feet = length of transition in feet, squared, divided by the radius in feet, and divided again by 24.

 

So for your 309 ft transition, the shift would be  309 squared, divided by 3300, divided by 24 =  1.2 ft.

 

regards,

 

Martin.

[Compound2632]

 

So for your 309 ft transition, the shift would be  309 squared, divided by 50, divided by 24 = 79 ft.

 

If I've correctly understood the shift to be the sideways displacement of the end of the curve from the position it would be in without transition, 79 ft doesn't seem quite right? It's about a quarter of the length of the transition curve.

 

[martin_wynne]

So for your 309 ft transition, the shift would be  309 squared, divided by 50, divided by 24 = 79 ft.

 

If I've correctly understood the shift to be the sideways displacement of the end of the curve from the position it would be in without transition, 79 ft doesn't seem quite right? It's about a quarter of the length of the transition curve.

 

Hi Stephen,

 

I agree, the 79ft figure is clearly wrong. I did however quote the rule exactly as given in BRT3. Clearly there is an error in it -- the radius should be in feet, not chains. I have amended the calculations, which now gives a shift of 1.2ft (which matches the result in Templot).

 

Thanks.

 

regards,

 

Martin.

[Ian J.]

OK, that's the thing I was after! Many thanks, Martin. Now all I have to do is work out how to convert that into bezier curves for drawing on an HTML5 Canvas!

[Derekstuart]

Hello Martin

 

If it is sufficiently related to Ian's OP, can I please ask how Templot calculates the transition length without knowing the target speed of the road in question? ie, does it calculate that if you are going from straight to 100 chains you are likely to be going very fast and if it is from straight to 10 chains you are likely to be going very slow.

The reason I ask is that I used your calculation above and compared it to the transition length in a recent Templot print and the line speed would have been only 1.4MPH difference between them, which is very close and yet Templot could not have have 'known' what target speed I had assumed. Clever stuff indeed.

 

Thanks.

[martin_wynne]

If it is sufficiently related to Ian's OP, can I please ask how Templot calculates the transition length without knowing the target speed of the road in question?

 

Hi Derek,

 

I'm sorry to disappoint you, but Templot knows nothing about line speed or making any clever guesses. In most cases the transition curve is based solely on the existing geometry of the tracks. Given two existing curves, or a curve and a straight, there is only one transition curve which will fit between them. It is for the user to set the initial radii and position of the curves to achieve the desired transition length. The transition maths used is the clothoid curve (Euler Spiral) which is used for most prototype transitions nowadays, and allows long transitions swinging through significant angles if needed.

 

If a transition curve is set from scratch, Templot uses the existing template radius, or defaults to 10 chains radius if the existing template is straight. The transition zone length is set initially to 60% of the existing template length, centralized within the existing template length. It is then very easy to adjust the zone markers or the radii as required by mouse action. There are 2 modes for this, normal mode and roll-out mode.

 

Alternatively the user can enter the dimensions directly of course.

 

In most cases for a model, the prototype considerations are irrelevant, because our radii are so much smaller than the prototype. We run fast trains round curve radii which would normally be found only in goods yards, where no p.w. engineer would waste time designing a transition curve.

 

The primary purpose of transition curves in a model is to disguise the severity of the curves, and to allow tracks to join smoothly in cramped locations. They can also be a big help in avoiding buffer-locking, and tangles with some types of coupling such as Alex Jackson.

 

There is a lot more about all this on the Templot Club forum.

 

regards,

 

Martin.

[Bomag]

In the dim and distant past curvature of existing track was measured by versines - this was the offset in the middle of a cord. When realigning curves to get the transitions back in the right location the maths/geometry conveniently resulted in the versines increasing linearly along a transition. There was an art in realigning a curve so the end tied into next bit of track.

[Derekstuart]

Martin, there is never a need to put "Templot" and "disappoint" in the same sentence (unless adding "...does not" in between).

 

Thanks for that explanation- it must have just been good fortune that the default figure just happened to be right for what I was trying to do.

 

But you wrote above that there is only one transition that will work for 2 given radii; that does not seem to me to match your previous calculation that shows line speed affecting transition length. I'm sure I've misunderstood somewhere, but I don't want to hijack the thread.

 

I will make one last comment if I may- if you have space, do model the transition zone(s); you CAN see the loco/stock running on gradually tightening curves and it looks far more realistic, IMO.

[Ian J.]

The thread is titled for a general conversation on the topic, and Martin has provided me with what I believe I need for the purpose of my original post, so it can happily wander into other railway curve related areas, real, model, or virtual

[Dungrange]

But you wrote above that there is only one transition that will work for 2 given radii; that does not seem to me to match your previous calculation that shows line speed affecting transition length. I'm sure I've misunderstood somewhere, but I don't want to hijack the thread.

 

Mathematically, there is only one transition curve that will fit between two given radii.  That will have a fixed length related to the positions and radii of the two curves.  You could then work backwards to obtain the appropriate line speed for this transition.  If this is not as intended (ie you have designed a 5 mph line speed) then you would have to alter one or both of the adjacent curves to allow the transition length to be increased.  All Martin was highlighting is that Templot 'designs' the transition based on the mathematical solution.  It would then be the Tempot users responsibility to ensure that the transition length is appropriate for the intended design speed (bearing in mind the fact that curves are normally tighter than the prototype we are often trying to represent).

[Ian J.]

Mathematically, there is only one transition curve that will fit between two given radii.  That will have a fixed length related to the positions and radii of the two curves.  You could then work backwards to obtain the appropriate line speed for this transition.  If this is not as intended (ie you have designed a 5 mph line speed) then you would have to alter one or both of the adjacent curves to allow the transition length to be increased.  All Martin was highlighting is that Templot 'designs' the transition based on the mathematical solution.  It would then be the Tempot users responsibility to ensure that the transition length is appropriate for the intended design speed (bearing in mind the fact that curves are normally tighter than the prototype we are often trying to represent).

 

If it can be mathematically retro-calculated then, and Martin might wince at this suggestion, could Templot be programmed to show the user the advised line speed of a given curve+transition+curve arrangement?

[Derekstuart]

Thanks. That is the **APPARENT** contradiction I was referring to.

 

"...there is only one transition curve that will fit..." and "...to allow the transition length to be increased.."

 

Surely they are contradictory, aren't they?

Mathematically, there is only one transition curve that will fit between two given radii.  That will have a fixed length related to the positions and radii of the two curves.  You could then work backwards to obtain the appropriate line speed for this transition.  If this is not as intended (ie you have designed a 5 mph line speed) then you would have to alter one or both of the adjacent curves to allow the transition length to be increased.  All Martin was highlighting is that Templot 'designs' the transition based on the mathematical solution.  It would then be the Tempot users responsibility to ensure that the transition length is appropriate for the intended design speed (bearing in mind the fact that curves are normally tighter than the prototype we are often trying to represent).

[Derekstuart]

Probably only if it was also able to calculate cant and percentage of stopping, slow and line speed trains.

(just a thought- I am out of my depth programming computers beyond "hello World" and as for working out track geometry each time I think I understand, some soandso comes along and says "...now you need to factor in this XYZ for the next level of understanding)

 

If it can be mathematically retro-calculated then, and Martin might wince at this suggestion, could Templot be programmed to show the user the advised line speed of a given curve+transition+curve arrangement?

 

EDIT: Now understood by Keith's follow up explanation.

[Grovenor]

 

Mathematically, there is only one transition curve that will fit between two given radii.

That statement is incomplete and is only true if a number of other conditions are specified, in particular the positions of the curves are also fixed, which Martin did mention elsewhere in his explanation. In practice, when designing neither the radii or the locations start off fixed so you have a fair bit of flexibility.

Regards

[martin_wynne]

I said that there was only one clothoid transition curve which can fit between two existing curves.

 

That was of course a lie. There are always 2 such curves, both the same length, but only one of them is of any practical use in a given situation:

 

 

In this Templot screenshot, a track is being laid out from A to B. In the process it runs round two circular curves at 1 and 2. Templot is then asked to find a transition curve between them. It comes up with the 2 transition S-curves shown here in yellow and violet, which it refers to as left-hand and right-hand. These are shown to user, who is asked to say which one is required.

 

Clearly the violet curve links the two circular arcs in a transition, but it is no help at all in getting from A to B if there is a mountain at M and a lake at L.

 

The yellow transition T is the one needed.

 

This is the make transition function in Templot, which is distinct from the geometry > transition curve > functions to set up or adjust a specific transition.

 

Martin.

[Trog]

At one time some railways put a small amount of cant on the straight to help reduce hunting. You can also sometimes find old transitions where the cant runs up from 0 to 1/4" then stays at a 1/4" for about a rail length before building up to the maximum cant in the normal regular even manner.

[28XX]

Given that the OP has freed this topic to roam in the wild, are preserved railways obliged to perform these calculations and check alignments accordingly, or are they exempt under their LRO's by keeping speed under 25mph?