RailroadSurveys_CurveCalcs

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Railroad Surveys:History and Curve Computations

Source: <See References Section>

Written By:

Mike Tyler

Nick Battjes

Nate Plooster

Krag Caverly Nathan Ovans

Surveying Engineering

Ferris State University

Big Rapids, Michigan



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T ABLE OF CONTENTS

I. Abstract ............................................................................................................ 3

II. The History of Railroads and Railroad Surveyors ............................................. 4-10

III. Instrumentation Used to Perform Railroad Surveys ..........................................10-16

IV. Simple and Spiral Curves .................................................................................16-23

V. Layout of Simple and Spiral Curves…………………………………………….23-26

VI. Bibliography ...................................................................................................27-28



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ABSTRACT

Railroads are an integral part of the United States’ infrastructure. From the early days of

our nation, railroads have been playing a part in the development of our economy and our nation

as a whole. Since so many railroads exist throughout the land, surveyors must be familiar with

how they were originally placed. All surveyors will, at one time or another, have to perform a

survey that is affected by a previous railroad survey.

This paper takes an in-depth look at the history of the United States railroad system. The

paper then goes on to discuss the instruments and procedures used to map and lay out railroads.

Finally, the paper discusses the methods of calculating and laying out a simple and spiral curves

are described.



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History

Source: <http://www.old-picture.com/old-west/Surveyors-Railroad.htm>



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From the ancient Greeks to modern times, railroads have played an important part in

society. Railroads furthered the early expanse of the United States to the west, and ushered in a

new chapter in American history. Much went into the building of the railroads, and one of the

key people involved was the railroad surveyor. These individuals were responsible for mapping

and laying out railroads. In essence, railroad surveyors defined the look and design of the

railroads. The history of railroads and railroad surveyors is rich and interesting, yet often goes

unnoticed.

In the sixth century B.C., the Corinthians built the Diolkos, which was an artificial

trackway that was used to move boats across the Isthmus of Corinth. The Diolkos is considered

the first predecessor of railways. The first land-based railways were built in the early

seventeenth century in Great Britain. These railways were made of wood and were used

primarily to transport coal. In the late Eighteenth century, the first iron rails, developed by the

British civil engineer William Jessop, appeared. With the development of the first steam

locomotive in 1804, and improvement on the design of the rails and their base in 1811, the

practicality and efficiency of railroads to move things like coal and other resources became

apparent, and rail transportation grew throughout Great Britain.

The railroad systems in the United States were slightly behind Great Britain in

development, but the U.S. soon succeeded them. In 1815, the first railroad charter was given to

John Stevens, who is considered by some to be the father of American railroads. Stevens

demonstrated the feasibility of steam locomotion on tracks in 1826, three years before George

Stephenson perfected his Rocket steam locomotive in England. After Stevens’s success,

government grants to other railway developers soon followed. In 1830, the first major railroad

system was completed, the Baltimore & Ohio. This railway was soon joined by the Mohawk &



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Hudson, the Columbia Railroad of Pennsylvania, and the Boston & Providence. The

construction of railroads continued to increase throughout the United States, as railroads not only

proved efficient in the transportation of material, but also in the transportation of people across

the great distances of the United States. Railroads went from just 23 miles of track in 1830, to

2,808 miles in 1840, 30,626 miles in 1860, and 201,000 miles in 1900 throughout the United

States.

The next great achievement in America’s railroad history was the completion of the First

Transcontinental Railroad in 1869, connecting the East with the West. The railroad covered over

1,700 miles of the U.S. and stretched from Sacramento, California to Omaha, Nebraska. The

First Transcontinental Railroad was a part of the Pacific Railway Act of 1862, which helped to

stimulate construction of numerous railroads throughout the Pacific region. The act also gave

railroad companies monetary aid and land grants from the U.S. government. With the

completion of the First Transcontinental Railroad, railroads increasingly expanded across the

United States. By 1900, five transcontinental railroads linked the East with the West. Trains in

cities evolved with technology and became increasingly powered by means of electricity.

The U.S. railroad services reached their peak in 1916. By then, 98% of all passenger

travel was done by train (never to be outdone by any other method), 77% of all freight was

shipped by train, 254,000 miles of railroad track existed, and railroad employment reached over

two million by 1920. After the railroads reached their apex around 1920, they began to decline

due to rising costs, the Depression, and automobiles; the railroads did, however, make a

comeback during World War II. After the war, railroads resumed their decline. Many railroad

companies were going bankrupt from high costs, competition from the automobile industry, and

the development of the Interstate Highway System. The once-proud railroads were reduced to a



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fraction of what they had been. By 1980, 190,000 miles of track were left; however, later in the

‘80s the efficiency of rail transportation was discovered once again. With the development of

new technology such as diesel trains and container systems, freight trains are increasingly being

used, and many railroad companies are producing big profits once again, totaling over one billion

dollars at present. Also, major U.S. cities depended heavily upon “light rail” commuter trains to

move people in and out of urban areas. The glory days of the American railroad system may be

over, but the railroad industry is making a comeback and many people can see a future for it.

At its peak, the American railroad system had 254,000 miles of track and used a lot of

land for other purposes. Because the railroads used large radius curves and modest gradients, the

right of way was often large. The land that the railroad companies used was obtained in a

number of ways. In the beginning stages of the United States railroad system, land was acquired

through charters from the states that wanted the railroads to be constructed, or the land was

purchased privately bought. After about 1850, extensive land grants from the federal

government were given to the states and railroad companies to promote the building of railroads.

The Pacific Railway Act of 1862, for instance, allowed the federal government permission to

give extensive land grants to the western United States. Section 3 of the Act granted 10 square

miles of public land on each side of the track, every other section, for every mile of track laid,

excluding cities and river crossings (Pacific Railway Act, July 1, 1862). The “every other

section” created a checkerboard land ownership pattern, which can still be seen today in the

Midwest and West.



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Checkerboard pattern of railroad land ownershipSource: <http://memory.loc.gov/ammem/gmdhtml/rrhtml/rrintro.html>

Congress stopped issuing the extensive grants in 1871; by then, the railroad companies

owned more than 123 million acres of federal land, about twice the size of Colorado. At one

point in the late 1800s, western railroad companies established land departments, some with

offices even in Europe, to sell land to immigrants, promote the Western United States, and use

the profits for expanding the railroads. Another way that the land was obtained was through

condemnation and eminent domain. Public railroads were often granted limited rights of

condemnation.

The first railroads were planned and constructed haphazardly, and went without the

supervision of the chartering states. The surveying and construction of these railroads were



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privately financed. The federal government initially helped the railroads indirectly by supplying

them with route surveys made by Army Engineers. With the rising popularity of railroads, the

federal government began to give financial aid to the railroad surveys through the 1824 General

Survey Bill and some later appropriations. The next big step by the federal government in

progressing railroad surveys came in 1853; Congress gave authorization and $150,000 to the

Secretary of War, Jefferson Davis, for the Corps of Topographical Engineers to explore the West

and find the most practical and economic route for a railroad from the Mississippi River to the

Pacific Ocean (Winter, 2007). The Corps of Topographical Engineers, also know as the

“Topogs,” was a division of the U.S. Army whose purpose was to “open up” the West. The

Topogs previously had been surveying along the Mexican border.

In 1854, four groups, one north, two central, and one south, set out to explore and survey

the West. The Topogs brought with them a large assortment of personnel, ranging from

geologist to artists to cooks. The surveys that would be done were very comprehensive and

contained details of the local geology, flora, temperature, routes, and many other data. The

surveying parties brought with them an assortment of instruments, depending upon the need and

the lead surveyor’s liking. Some of the popular instruments used were levels, solar transits,

railroad compasses, and the Jacob staff. For the making of the maps, railroad curve templates,

slide rules, and drafting sets were used. Even as well-equipped as the Topogs were, the

explorations were still very dangerous, as were all early railroad surveying expeditions. The

surveyors faced extreme weather, fire, sickness, wild animals, rough terrain, and hostile natives.

Some of the parties even lost members. Capt. John Gunnison, in charge of the group exploring

the route along the 38th and 39th parallel, was killed by hostile Indians. Despite the risks they



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faced, the Topogs ultimately paved the way for the First Transcontinental Railroad, and later

many other railroads, with their surveys done under the Pacific Railway Act.

With the Topogs finishing some of the last major government railroad surveying

expeditions, other Army and private railroad surveyors would continue on in surveying future

routes for trains and creating maps of current and future railroads. The use and permanency of

the railroad is now taken for granted, but it was the railroad surveyor who ensured the

practicality and use of the railroad in society.

Instrumentation

With the advent of the railroad in the early nineteenth century, surveyors were called to

help survey the land and mark where to lay the tracks. Surveyors were responsible for making

sure that the gradients were not too large for the engines to climb, that curves would not derail

the train, and that the rails would meet when they were started in two different places. In order to

accomplish these and other feats, they had the aid of several instruments including the surveyor’s

compass, transit, solar compass, Jacob staff, chain, and plane table.

The surveyor’s compass would have been used to help run lines at a specific meridian or

angle from the meridian. Its main features are the magnetic needle, limb, levels, and sights. The

magnetic needle is used to determine the meridian line; the limb, also known as the circle, is

often graduated to half degrees. The surveyors compass was used when the surveyor needed to

turn an angle off of the meridian. Each instrument has two spirit levels placed at ninety degrees

to each other, and these help determine when the plane of the instrument is horizontal. The sights

allowed the surveyor to direct the instrument; they are composed of vertical attachments at the

ends the compass. Each sight has slits with small circles that allowed the surveyor to orient the

compass on a line.



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The surveyor’s compass was modified for different purposes. Two variations important to

railroad surveying are the Vernier Compass and the railroad compass. The Vernier Compass was

created by first placing the limb and sights on separate plates that revolve around a common

center, and then attaching a vernier scale. The chief use of the Vernier Compass is to allow the

surveyor to set the sights to the true meridian because the half degrees found on the limb are not

accurate enough to account for magnetic declination. The railroad compass is essentially a

modification of the Vernier Compass. To create the railroad compass, another circle with a

vernier is added. This allowed the surveyor to read angles independently of the needle

(Hodgman, 19).

Vernier Compass,Source: <www.uzes.net>

The next piece of surveying equipment used for railroads was the transit. This piece of

equipment evolved when the railroad compass was modified so that the sights were replaced by a

telescope and leveling screws were added. The main components of this equipment are the

telescope, circular plates, sockets, and leveling head. The telescope is normally about eleven

inches long, and it is fixed to the upper plate so that it can rotate 360 degrees horizontally or

vertically. Two lenses, one adjustable for the object space, composed the object glass, while four

lenses composed the eye piece. The crosshairs were made of either spider web or thin platinum

wire ( Ibid, 53). On the outside of the telescope is attached a vertical circle and a level. The



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vertical circle allowed the surveyor to read vertical angles to within a single minute, while the

level would tell the surveyor when the telescope’s tube was horizontal.

TransitSource: <www.uzes.net>

There are two circular plates on the transit. The upper plate had the telescope attached,

while the lower plate, or limb, is inscribed with a scale to show half degrees. Additionally, the

upper plate also held the compass. Vernier scales are attached to the side of the limb, and the use

of these can give angle readings as small as one minute. The sockets allow the plates to revolve,

and the leveling head permits the surveyor to level the transit.

In order to obtain meridian information during the day, the solar compass was developed.

This instrument contains sockets and plates similar to the transit, except that the sight vanes are

attached to the limb and the vernier is on the upper plate. In place of the needle there is a solar

apparatus composed of a latitude arc, declination arc, solar lens, equatorial sights, hour arc, an

adjuster, needle box, and levels. The three different arcs are used in determining the sun’s

position at a specific time, while the equatorial sights, levels, and adjuster are used to adjust the

whole apparatus (Ibid, 66). The solar lens prevents the sun from causing eye damage while



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sighting on the sun, and the needle box helps align the sights in roughly the north-south

direction.

Solar CompassSource: <www.uzes.net>

A Later models of the solar compass combined the ideas employed by the transit and the

solar compass, and is referred to as the solar transit. This instrument contains a telescope, solar

apparatus, and compass. The instrument is designed so that the compass is placed on the upper

plate, while the transit and solar apparatus are on the revolving lower plate. The telescope and

solar apparatus share the same axis, and movement in one part affects the direction of the other

part. This means that when the surveyor was finished observing the sun, the telescope was

already aligned along the meridian line (Ibid, 83). The surveyor could then look at the compass

and determine the magnetic declination of the current location.

Solar TransitSource: <www.uzes.net>



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Another instrument that was widely used during the nineteenth century was the Jacob

staff. This instrument was used in place of a tripod. The staff is a single rod with a joint at the top

that can be used to accommodate a compass, while the point on the bottom is used to pierce the

ground deep enough to hold the compass and staff upright without movement. The staff itself

was often graduated, and it could be used to help determine the gradient of a survey line

(Krehbiel).

Jacob staff and tripodSource: <www.surveryhistory.org>

In order to measure distances, the railroad surveyors depended on the chain and marking

pins. The chain was a standard sixty-six feet in length composed of one hundred links. The ends

of each chain had a handle or grip on them that allowed surveyors to pull tension on the chain to

keep it level while measuring distances. Terrain often made it hard to level a full chain so many

surveyors would use the half chain, which is composed of fifty links. Many chains also had brass

tags, called tallies, at every tenth link in order to help the surveyors quickly determine which link

they are measuring to (“The Surveyor’s Chain”). The purpose of the marking pins was to help in

the instrument operator keep the linemen on line, and they mark spots where the chain was

broken.



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ChainSource: <www.uzes.net>

The plane table was used when a map was needed to help plan the best route for the

tracks; it is fairly simple, and its main components are the board and an alidade. The use of a

boss allowed the board to be locked in a horizontal position (“Measuring Angles”). The alidade

is a simple telescope with a horizontal and vertical vernier. It allows the surveyor to measure the

angles to the targets. By mapping the targets from two positions, the surveyor could draw a

topographic map based on the intersection of lines. When used with a compass, these maps could

be drawn with the objects correctly oriented to the North.

Plane TableSource: <www.landsurveyors.us>

The thought of laying out miles of rail using only the equipment listed above may make

many of today’s surveyors cringe, but in the 1800s workers did so with gusto. The railroads



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built in the1800s are still used today; this is tribute to the technique and skills of the surveyors of

the past. These accomplishments were aided with the use of the surveyor’s compass, transit,

solar compass, Jacob staff, chain, and the plane table.

Simple and Spiral Curves

Proper computations are a necessary component for any survey-related project, and

railroads are no exception. Many of the computations done when building a railroad dealt with

curves of one type or another. Due to the considerable amount of weight and force from trains,

horizontal curves were generally made simple so trains could easily maneuver around them. The

majority of curves found in railroad survey and design is the simple curve. In order to prevent

the bending of the train cars, the change from the level tangent line to the curve must be gradual.

This gradual change is called the transition curve. The inclination of the track is proportional to

the centrifugal force at a point. Curvature will increase and decrease directly with the distance

from the starting point.

Simple Curves

Circular curves may be classified generally as simple or spiral. A simple curve is a

circular arc extending from one tangent to another with a constant radius (Allen, 20). A spiral

curve is defined as any curve inserted to provide a gradual transition between a straight and

circular path. The effect of the spiral curve is to introduce centrifugal force gradually, therefore

reducing shock to the railroads’ tracks and equipment and making high speed train travel

attractive to passengers. In addition to reduced shock, the spiral curve also allows a place for

accomplishing the gradual change from zero to full super elevation (Meyer, 64).

Just like in road and highway curve design there are specific standards and guidelines that

need to be followed for railroads. A few include: (1) the inside rail may not be higher than the



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outside rail or have more than six inches of elevation change while the inside rail remains at a

constant grade. (2) Maximum allowable speed around a curve is determined by

(Ea+3/0.0007*D).5 where Ea is the elevation of the outside rail. (3) Elevation runoff and

transition must be at a uniform rate. In normal practice, recommendations dictate that runoff

should be about 1” per 60’.

Before getting into the calculations, it is important to know some of the terminology that

is used to describe curves. The point where the curve leaves the first tangent is called the “P.C.”,

meaning the point of curvature, and the point where the curve meets the second tangent is called

the “P.T.”, meaning the point of tangency. The P.C. and the P.T. are often called the Tangent

Points. Once the tangents have been generated, they will intersect at a point called the “P.I.”,

meaning the point of inflection. The distance from the vertex to the P.C. or P.T. is called the

Tangent Distance, T. The distance from the vertex to the curve (measured toward the center) is

called the External Distance, E. The line joining the middle of the Chord, C, with the middle of

the curve subtended by this chord, is called the Middle Ordinate, M. The radius of the curve is

called the Radius, R. The angle of deflection between the tangents is called the Intersection

Angle, I. The angle at the center subtended by a chord of 100 feet is called the Degree of Curve,

D. A chord of less than 100 feet is called a sub-chord, c; its central angle a sub-angle, d (Allen,

20).

Surveyors use many formulas to determine the necessary information required for staking

curves prior to the actual laying out of the curve in the field. In the United States, for railroad

purposes, a curve is generally designated by its degree, D. Once a value for D is obtained, the

radius of the simple curve can be found using the formula:

2sin

50

D R = . Other formulas that were



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used to find the radius include: )2

cot( I

T R = ,

)2

sec( I

ex

E R = ,

)2

( I

vers

M R = , and

)2

sin(2 I

C R =

where A A tan

1

cot=

, )2tan(tan1secsec

A

A A Aex=-=

, and 2tansincos1

A

A AversA=-=

. If

the sub-chord, c, was required, then it could be obtained using the formula; )2

sin(2 d

Rc = where

d is the sub-angle. If the sub-angle, d, was required, then the simple formula100

cDd = could be

used. If the degree and intersection angle is known and the tangent distance T is needed, then the

formula: )2

tan( I RT = could be used to find T (Allen, 24).

On the other hand, vertical curves are also necessary in railroads to account for the

changes of grade to prevent shock and the wearing out of train parts. Length will depend on total

change of the grade and the length of the longest train. Typically the length is found by dividing

the total change by the allowable change per station. Rates of change fluctuate between 0.05 and

0.1 percent per station. Unlike horizontal curves, vertical curves use the tangent offset method of

layout instead of using the degree of curve or the radii.

Spiral Curves

Spirals have been used on railroad tracks since about 1880. All simple spirals obey the

fundamental law that “the radius of the spiral at any point is inversely proportional to its length”

(Meyer, 89). There are two common types of spiral curves, arc spirals and chord spirals. Arc

spirals are commonly used in highway design, whereas chord spirals are used in railroad design.

The difference between the two types is in how they are defined either by degree of curvature arc

definition or degree of curvature chord definition; both types will be discussed later. The use of

spiral curves is best summed up in the following statement:



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Spiral curves are applied to railroad layout to lessen the sudden change in curvature at the

junction of a tangent and a circular curve. A spiral makes an excellent easement curve because

its radius decreases uniformly from infinity at the tangent to that of the curve it meets. Spirals

are used to connect a tangent with a circular curve, a tangent with a tangent (double spiral), and a

circular curve with a circular curve and compounded or reversed curves (Brinker & Wolf, 436).

A spiral curve starts with a simple horizontal curve that is placed between two tangent

lines. From there, the curve is designed in accordance with factors such as design speed,

minimum length, and others that can be found on a table. Now the horizontal simple curve

becomes a spiral with three distinct parts, the spiral in, the reduced horizontal curve, and the

spiral out.

Diagram of a spiral curveSource: <http://www.usace.army.mil/publications/eng-manuals/em1110-1-1005/c-9.pdf>

Even though spirals sound much more difficult to compute, they are no more involved

than the computations for a simple curve. For a spiral, the radius is infinitely long at the T.S. and

decreases uniformly to R at the S.C. Since the length of the spiral is measured along its curve,



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and R is inversely proportional to D, it follows that the degree of curve of the spiral increases

uniformly from zero at the T.S. to D at the S.C. over the distance L s. Generally, the sta. P.I., I,

and D are known quantities to begin spiral computations. The first step is to select Ls to fit the

imposed conditions. Next, calculate Δ using the formula,200

D L s =D . Then, take X and Y from a

table, and calculate X0 and o from the following formulas, D-= sin0 R X X and

D-= RversY o . The distance o, is the distance through which the circular curve must be moved

inward in order to provide clearance for inserting the spiral. The next step is to calculate Ts

using the one of the following formulas, 0)2tan()( X

I

o RT s ++= or )2tan(0

I

o X T T s ++= .

Then, calculate the stationing of the T.S., which is sta. P.I.-Ts; the stationing of the S.C., which is

sta. T.S.+Ls; the stationing of the C.S., which is sta. S.C.+ D

I )2(100 D- ; and the stationing of the

S.T., which is sta. C.S.+Ls. After the stations have been computed, the deflection angles at

selected points on the approach spiral must be calculated. This is done using the following

formulas, A L

l a

s

2)( = and3

D= A (Meyer, 70).

Simple curve and the same curve as a spiralSource: <http://www.usace.army.mil/publications/eng-manuals/em1110-1-1005/c-9.pdf>



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Spirals need to be superelevated in order to dissipate the centrifugal force that is created

when trains travel along curves. “The effect of centrifugal force on a vehicle passing around a

curve can be balanced by superelevation of the outer rail of a track. Correct superelevation on a

spiral increases uniformly with the distance from the beginning of the spiral, and is in inverse

proportion to the radius at any point. Properly superelevated spirals ensure smooth and safe

riding qualities with less wear on equipment” ( Brinker & Wolf, 436).

Degree of Curve

The degree of curve (D) has been the preferred method of identifying curves using either

the chord or arc definition. The degree of curve is defined by the angle at the center of a circular

arc subtended by a chord of 100 feet; this is the chord definition (see figure 1-1 below).

The arc definition is denoted by the angle at the center of a circular arc subtended by an arc

length of 100 feet (See figure 1-2 below).

Source: <http://www.civil.ubc.ca/courses/civl235/Course%20Notes%20-

%20Student%20Access/13-Civl235-Route_Curves.pdf.>

The radius of the spiral at any point is inversely proportional to its length. In contrast to

the circular arc, the spiral is a curve of variable radius or variable degree of curve. The degree of

curve of the spiral increases at a uniform rate from zero at the T.S. (tangent to spiral) to the

degree D of the circular arc at the S.C. (spiral to curve) (Meyer, 88).



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The constant rate of increase in degree of curve per station along a spiral is represented by K.

The degree of curvature for a spiral increases from 0 at the T.S. to D a at the S.C., the rate of

change of curvature K per station (one full station = 100 feet) is represented as, K =100D/ L s.

Ls (length of spiral) is selected to fit the imposed conditions on American railroads that

have been formulated as a result of many years of experience in operating over spiraled

superelevated curves. The minimum spiral length is usually based upon the rate of rotational

change. For many years, the American Railway Engineering Association (A.R.E.A.) has

recommended that spiral length be based on attaining superelevation across standard-gauge track

at a desirable maximum rate of 1.25 inches per second. The superelevation (e) is made equal to

approximately 0.00067V 2 D (A.R.E.A. , 1995) expressed in inches where V=train speed in mph

and D=degree of curve (chord definition). The amount of superelevation should not exceed

seven or eight inches (Anderson, & Mikhail, 866 ).

Deflection angles from the T.S. can be computed for any point on the spiral by the

equation a=l s2 /6RL s. In a simple spiral curve, where the two spirals on either side of the simple

curve are equal, the deflection angles from above can be used from the S.T. assuming equal

stationing.

Since calculators were not around during this period, surveyors had to use different ways

to solve trigonometric functions. The surveyor would carry books containing tables wherever he

would go. In the books were natural sine, cosine, and tangent tables that would list the decimal

solution to these and more trigonometric functions. The tables would have the degree in the

column and the minute in the rows. If more precise angles were measured, then the surveyor

could interpolate in between the minute values. The decimals were usually carried out to five

places, after which the surveyor could multiply or divide with it in long-hand (Allen, 161).



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Simple and Spiral Curve Layout

Simple Curves

As technology and knowledge has grown through history, the evolution of techniques has

also grown. Railroad layouts throughout history have been completed through methods such as

tangent offset, ordinate offset, and chord offset. Although these methods are very useful, 100

years ago the technology to perform some of these methods made the methods a bit more

challenging. Due to the challenges, the most commonly used method for laying out a curve was

the deflection method. This method can be used to lay out both a simple and spiral curve.

One of the most common ways to layout a curve is by deflection angles.

For simple curves, usually the curve is placed in position from the Point of Curvature

(P.C.). Before the actual curve is laid out, the P.C., Point of Intersection (P.I.), and the Point of

Tangency (P.T.) must be located. The first step is to use the P.I. as the backsight point while

occupying the P.C.; the line from the P.I. to the P.C. is commonly called a tangent line. After all

the deflection angles are computed, angles can be turned from the backsight to the station along

the curve, and the proper distance can be measured from the P.C. to the first even full station.

Often, the P.C. is not at an even station, for instance if the P.C. is station 24+63; the first station

that would be placed along the curve is station 25+00 and the following points along the curve

would be staked at full stations.

Once the first station has been placed, the distance to the next station is determined based

on the distance from the previous station to the next desired station. For example, if the P.C.

held a station of 240+00 and the first point along the curve was at a deflection angle of 1º 30’ at

station 241+00, then the angle is turned and the distance of 100 feet is measured from station

240+00 to station 241+00. Each deflection angle that can be sighted from the P.C. is done so. If



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all the stations along the curve are not visible, then it requires another setup. Known as moving

up on the curve, this method is required when obstacles block visibility, when a curve is very

large, and when terrain causes a lack of visibility. The instrument is moved up on the curve by

placing the instrument at one of the points that have already been set.

Once the instrument is moved up the curve, the previous point that was occupied is used

as a backsight point. When moving in a forward manner along a curve, the backsight direction is

set to zero and then the scope is turned to 180º. The deflection angle that was used to place the

currently occupied station from the P.C. is subtracted from each of the remaining deflection

angles. The resulting angles are used to place the remaining stations along the curve until

another move up the curve is needed or the curve is completed. Once the direction is

determined, a distance is measured from the preceding station to the station being laid out to

determine the exact location of point. This process can be repeated until all the points along a

curve have been located. All this can be done quite simply with a theodolite and tape.

Spiral Curves

The setup procedure of spirals involves locating numerous components of the spiral and

then using deflection angles to place the curve itself. While occupying the P.I., the instrument

should sight the back tangent to set the tangent to the spiral (T.S.) and D, a point 100 feet along

the tangent from the T.S. The next step is to measure a direction for the forward tangent through

a process of double deflection and turning the interior angle (I). Once the direction has been

established, the measurement of the C” and S.T. points can be placed; these are the same points

as the D and T.S., but along the forward tangent line with the C” being the angle subtended by

100 feet, and the S.T. being known as the tangent of the spiral. The next step is then to occupy D

and C” in order to set the S.C. and C.S. This is done by turning a right angle from the P.I.



25

toward the curve from both the forward and back tangent lines. Once the S.C. and C.S. have

been established, the setup will move to the T.S. The process here is to set stations along the

curve from the T.S. to the S.C. Each of these points is set by deflection angles, and often an

equal number of chords are used; because of this, most flat spiral curves have a nominal

difference in stationing that is equal to the chord length. Once the points along the curve up to

the S.C. have been set, the setup is moved up to the S.C. The backsight at this point is the T.S.

point. The horizontal circular reading is 180 ± 2A, with A being the total deflection angle from

the T.S. to the S.C. Once the backsight and circular reading is set, the instrument is turned

through 180º. The stations all the way to the C.S. are to be set by deflection angles and chords,

as previously done from the T.S to the S.C. As a check, the instruments’ circular reading should

read the deflection angle for the station where the C.S. is located.

After completing the measurements at the S.C., the setup is moved to the S.T. The

stations that leave the spiral are to be set from this point. For any curves that are equally

spiraled, the approach deflections can be used to set the leaving stations of the curve. If even

stations are being set, on the other hand, new calculations are required in order to set the

remaining stations. The backsight for this setup is to the S.C. with a circular reading on the

instrument of 180º ± (I - 2Δ) / 2. In some cases, the use of the osculating-circle principle is used

to compute the deflections. The osculating-circle principle helps to utilize an intermediate setup

on spirals. The method calculates deflection angles to the desired points along a spiral curve

from C.S. to S.T.

For many years, it has been the deflection method that ahs continued to excel in the field,

whether the curve being laid out is a spiral or simple, and even whether the curve is high way or

railroad. The continued success of this method will always be useful, whether our technology is



26

digital or if it is simple a tape and compass. There will always be a time and place for the

deflection method to be used.



27

References

Allen, Frank C. Railroad Curves and Earthwork . Boston: May, 1931. Pg 20.

Anderson, James M. & Edward M. Mikhail. Surveying Theory and Practice: 7th edition. The

McGraw-Hill Companies Inc. USA. 1998. p.840-900.

Brinker, Russell C. & Paul R. Wolf. Elementary Surveying: 6th edition. Harper & RowPublishers, NY. 1977. p.436.

Bureau of Land Management Staff. Science and Children magazine: Steel Rails and Iron Horses. Nov/Dec 1995. National Science Teachers Association.[http://www.blm.gov/education/00_resources/articles/steel_rails_and_iron_horses/article.html]

Ferguson, Isabelle. Surveying for the San Diego and Arizona. DISPATCHER. Issue #22. Dec.

20, 1958.

Hodgman, Francis. A Manual of Land Surveying . Columbus, Ohio: Buckner HistoricalSurveying Book Reprints. 1976.

Krehbiel, David G. Jacob’s Staff. The Ontario Land Surveyor. Spring 1990.[http://www.surveyhistory.org/jacob's_staff1.htm]

Measuring Angles. [http://www.surveyhistory.org/measuring_angles1.htm]

Meyer, Carl F. Route Surveying . Scranton, Pennsylvania: International Textbook Company.

1949. Pg 64.

Meyer, Carl F. Route Surveying and Design: 4th edition. International Textbook Company.Scranton, Pennsylvania. 1969. p.88.

Modelski, Andrew M. Railroad Maps of North America: The First Hundred Years. WashingtonLibrary of Congress. 1984. pp. ix-xxi.

Modelski, Andrew M. Introduction to Railroad Maps of the United States. Washington Libraryof Congress. 1975. pp. 1-14.

National Railroad Museum[http://www.nationalrrmuseum.org/collections-exhibits/index.php]

Pacific Railway Act. July 1, 1862. Enrolled Acts and resolutions of Congress. 1789-1996.Record Group 11. General Records of the United States Government. National Archives.

The Pacific Railroad Surveys: Whipple on the 35th Parallel.[http://www.southwestexplorations.com/whipplehist.htm]



The Surveyor’s Chain.[http://www.surveyhistory.org/surveyor's_chain.htm]

Winter, Rebecca C. Eastward to Promontory.[http://cprr.org/Museum/Eastward.html#Pacific%20Railroad%20Surveys].

Online Image. <http://images.google.com/imgres?imgurl=http://www.picture-history.com/old-

west/pictures/Surveyors.jpg&imgrefurl=http://www.picture-history.com/old-west/Surveyors- Railroad.htm&h=721&w=572&sz=107&hl=en&start=3&tbnid=X44me7DKykcAeM:&tbnh=140&t bnw=111&prev=/images%3Fq%3Drailroads%2Band%2Bsurveyors%26ndsp%3D18%26svnum%3D

10%26hl%3Den%26sa%3DN>

Documents

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