For those who might be interested (or bored).
Here is a real-world example of a Railroad Right-of-Way retracement that includes Searles Spirals. In this particular case, the Railroad constructed ROW Fences concurrently with the railroad construction (1902-1903). These ROW Fences are easily identified (steel-reinforced cast concrete posts of two types, typically ~6x6 inches on 'corners' & ~3x4 inches on 'line posts'), most of which are still extant (in some areas, ALL of them appear to be extant).
I have attached a snap-shot of the original “as built” alignment document submitted to the Secretary of Interior in 1903 (see Act of March 3, 1875).
In the case of this particular curve, the Railroad was passing through Private Lands, and the ROW was acquired by Deed. All of the deeds contained verbiage (paraphrased) as follows: “xx feet either side of the centerline as constructed.”
The ROW width varies from 100, to 150 to 250 and back to 100 feet within the subject area.
My well used copy of FIELD ENGINEERING, A handbook or the THEORY AND PRACTICE OF RAILWAY SURVEYING, LOCATION AND CONSTRUCTION. By William H. Searles (Eighteenth Edition w/Ives), published in 1918 (I think), contains the following statements:
“The spiral curve is constructed upon a series of equal chords, and the angle subtended by the first chord is made the common difference for the angles subtended by the succeeding chords. It is a multi-compounded curve in which the degree of curve progresses in arithmetical ratio from chord to chord.”
Those two statements contain everything you need to know to solve the spirals.
Spoiler alert: All of the extant “original” ROW Fence Posts (concrete) were tied in (spacing varies), as well as the existing tracks (every 50 feet or so). Solve the Centerline, and I'll post some ROW width and fence data.
Those who believe that a circular curve can (or should) be computed for the ROW are encouraged to whip one up (good luck with that).
Enjoy...
Oh Boy!!!
I am going to go and put the popcorn on now.
I may compute this, as I think I know how, except I'm not sure how to express the answers for comparison to what you may post later.
N & E with the PS assumed to be 10,000 10,000? Something else?
I may compute this, as I think I know how, except I'm not sure how to express the answers for comparison to what you may post later.
N & E with the PS assumed to be 10,000 10,000? Something else?
That would work fine, although the total measured delta came out to 67?ø00'10" (best fit of the rail shots on the tangents North and South East of the curves), and the Tangent bearings (County LDP) were S 8?ø42'35" W and S 58?ø17'35" E. There was no SCALE variation. LDP Coordinate at the PI was N=120,617.92 & E=46,515.85 USSF.
Loyal,?ÿI would play with?ÿC&GS FORM 128?ÿbefore I would play with this stuff.
I do have quite amount of information on Spirals?ÿbut?ÿGeodesy is more fun.
?ÿ
JOHN NOLTON :d ?ÿ
Hint: The two "magic numbers" in this case are 120 and 16...(but you probably already know that).
😎
I think I have it (within hundredths) in this spreadsheet
Download (Loyal_spiral.xls)I would hate to have to do this without the spreadsheet, like the original designer did.
I never did figure out what the 51?ø 50' means.
Hint: The two "magic numbers" in this case are 120 and 16...(but you probably already know that).
16 or 15?
tan ...?ÿ sin ...
"cos" that's more fun?
Looks like you nailed it!
“The spiral curve is constructed upon a series of equal chords, and the angle subtended by the first chord is made the common difference for the angles subtended by the succeeding chords."
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15 = 120 (magic number 1). Spiral Delta (9°) / Magic Number (120) = 0°04'30" which is the Delta of the first 30' chord (circular curve), from there you are off to the races (120 in you sum column).
It is a multi-compounded curve in which the degree of curve progresses in arithmetical ratio from chord to chord.”
The Degree of Curve (chord definition) of the Central Curve is 4°, there are 15 chords in the "spiral," so the NEXT Degree of Curve must equal 4°, therefore 15+1 = 16 (magic number 2) and 4°/16 = 0°15'00" for the first curve of the spiral.
Nothing to it, but like you said above, a computer "program" makes short work of what can be easily done with a pencil and paper, but that would take a LOT MORE time.
Loyal
I encountered my first (and to date only) Searles Spiral in 1998 for a project in Grand County Colorado shortly after moving to Colorado and prior to becoming registered. Luckily for me, my boss at the time had run into a Searles Spiral before so after he saw me scratching my head and read the deed & realized the geometry described in the deed was a Searles Spiral, he pushed me (a little) in the right direction and let me make a run at figuring it out before he stepped in to impart some wisdom on me. Once he fully explained the Searles Spiral and pointed me to some applicable reference material (William Searles' "A Handbook Of The Theory and Practice of Railway Surveying, Location and Construction" if my notes are correct)?ÿ it all came together nicely on that project. Was glad I took notes the first time I encountered one and looking back was glad I was able to have that enigma to solve on a project. Cool stuff for sure.
The Searles Spiral is one of many . When Highways were being built I had read an article about Superelevation and spirals back in?ÿ
around 1958. I thought that would be interesting to see the?ÿmath so I found a book called?ÿHighway Curves by Howard Ives
and Prof. Philip Kissam (who also has a surveying book plus other books). The 4th edition , 3rd printing was March 1958.
Chapter 8 is , Spirals, Superelevation, and Widening (page 110).?ÿOn page 111 starting at the bottom ,
The Searles Spiral; #131. Theory?ÿ (my book still looks like new).
JOHN NOLTON
I think that my first encounter with a Searles Spiral as around 1980 or so. Since then I have bumped into Searles Sprirals on dozens of projects around the Great Basin, including a few State Highway Projects (early 1900s) when chord definition curves and (from time to time) spirals were used instead of ARC definition curves. Due to "open range" conditions (cattle) in much of the Great Basin, Railroads constructed ROW Fences concurrently with the tracks on a regular basis. Also of note, when acquiring ROW(s) through Private Lands, the DEEDs contained the "either side of the track centerline AS CONSTRUCTED" verbiage more often than not.
Loyal
I've run in to railroad spiral curves a dozen times; did not know they're called Searles Spirals; we called 'em track transition curves.?ÿ An interesting math exercise but not too difficult to resolve.?ÿ I was puzzled by their purpose, 'till I talked with a railroad engineer and he explained since trains can't steer, they would get slammed around by an abrupt tangent section transitioning to a curve at the PC & similarly at the EC.?ÿ So the railroad guys engineered an alignment that is "almost" a spiral, but still stakeable without calculus.?ÿ I asked if each subcurve would result in a "jerk" at the transitions and he said yes maybe a little on newly laid track but the train dynamics bash it out in to a true spiral over years of use and maintenance activities.
An interesting sidenote is?ÿ US highway design does not use spirals instead controlling transitions through preemptive superelevation.?ÿ But the German Autobahn does have spiral curves apparently because they designed for essentially unlimited vehicle speeds.
I'll point out superelevation both for trains and cars softens direction transitions and side loads in curves but is not a cure-all. For example, when DOT rebuilt/constructed what is presently I84 in the '70s, some lanes featured the tightest curves allowed on the Interstate Highway system, with near racetrack superelevation.?ÿ One particularly tight steep downhill curve (55mph advisory) west of Baker City, Or. worked fine in good weather, but when iced up in winter and traffic slowed to a crawl, even chained up cars inexorably slid into the lowside guardrail below a certain speed. ODOT band-aided it with intensive sanding maintenance, grooving, signage, etc., for decades but I've heard they spent millions finally fixing the alignment in the 2000s.
Not all Railroad "spirals" are Searles Spirals (although the Searles is the most common in my area). I have also encountered the Talbot Spiral, 10 chord Spiral, and a Union Pacific spiral unique (I believe) to that Railroad and used in the late 20th Century. On one project I worked on, all of the centerline geometry used Seales Spirals, but in some cases (where practical) the ROW used simple circular curves, but others used the centerline spirals. In fact (as I recall), there was one curve that used a "bestest fit" circular curve (ROW) on one end of the curve, and a centerline spiral at the other end (the section line running across the alignment separated USA from Private ownership (and YES, the ROW jogged at the section line).
Interestingly, Interstate 80 through Evanston Wyoming, used clothoid (as I recall) spirals for both the centerline and ROW Lines (much to every ones chagrin).
As always, it depends, and the need for in depth research is paramount.
The truth is out there...
😉
Speaking of Highways. Here's some alignment data for US-40 (1980s) which runs through one of my projects:
Thankfully, UDOT did NOT spiral the ROW lines, which are actually "straight" vectors between various offsets from centerline stations NOT within any of the spirals.
Loyal
How did the railroad guys lay out the spiral??ÿ I can't imagine they did a setup every 30 ft, or held accuracy over the whole spiral if they did that many 30-ft sights.?ÿ
The method of laying out a circular curve from one setup at the PC has simple deflection angles, but the angles for a spiral are messy.?ÿ Did they have to compute all those coordinates and arctangents, or have I overlooked a simple method?
Searles (& Ives) provide both formulas and tables for laying out the Searles Spiral. I have never used them, but I'm pretty sure that's how the "old Boys" rolled.
Loyal