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The quotient will be the distance, in stations, which must be
added to the curve A B to bring the P. T. at B 1 ', and the
tangent will pass through the required point C'.

EXAMPLE. Let A B be a 6 curve, and the angle CD C' 4 30'.
Swing the tangent B Cso as to pass through the point C'.

SOLUTION. Reducing 4 30' to decimal form and dividing by 6, the
degree of the curve A B, we have -^- = .75 of a full station = 75 ft.,

which we must add to the curve A B, bringing the P. T. at B' ', and the
tangent B' X' will pass through the point C . Ans.

It will be evident that, had B' X' been the given tangent
and C the required point, it would have been necessary to



move the point of tangent B' backwards to B, i. e., to
subtract 75 ft. from the given curve.


FIG. 367.

1429. Problem VII. To find the distance across a
river in a preliminary survey:

Let the line^ B, Fig. 368,
cross a river, too wide for
direct measurement. With
the instrument at A, sight
to a flag held at B, and turn
an angle B A C = 1. Set a
plug at C, opposite B in the-
line A C, and measure the
distance B C.

The required distance
B C X 100

AB =



EXAMPLE 1. If Z?C=10.6 ft., how long is A B1
10.6 X 100



= 607.45 ft. Ans.



EXAMPLE 2. If B C = 8.8 ft., how long is A />' ?

8 R v 1 00


= 504.3 ft. Ans.

1430. Engineers' Field Books. The problems
given cover the cases which are liable to arise under ordi-
nary conditions, and the explanations have been fully given.
The engineer must necessarily carry a field book containing
the usual tables of reference. All standard field books con-
tain demonstrations of problems covering all those special
cases which do not properly come within the scope of this

1431. Relative Position of Preliminary and
Located Line. The relative position of the preliminary
and of the located line,

where the work has
been intelligently per-
formed, is shown in the
following sketch, Fig.
369, in which the pre-
liminary line is shown
in dotted and the loca-

tion in full lines.

The location is
practically fixed
by the preliminary
line, leaving little
to do but to run in
the curves. The
slight changes in the

,. . ,

direction or the tan-
gents and the degrees of the curves
will be determined by an inspection of
the contour map, which is the basis of
every intelligent location.

1432. Field Profiles. The profile is kept plat-
ted as fast as the line is located, in order that the chief
of party may know how nearly the actual profile ap-
proximates to the theoretical one (the one that is made
from the paper location) and what changes may be



1433. Final Location. After the right of way has
been cleared, affording an unobstructed view of the ground,
it will frequently be seen that slight changes in the located
line will greatly reduce the cost of construction ; and not
until such changes are made will the engineer have made
the final location.

None but experienced engineers can understand how a

Cut 1.5ft

h 3'-*) Area, 4.5 Sq.ft.
L -FillSOft


slight change in location, especially on a side hill line, can
so greatly affect cost; and it is first cost which generally. de-
termines the success or failure of the enterprise.

The accompanying sketches will afford some light where it
is oftenest needed. Fig. 370 is an example of poor location
more often met with than that of any other kind, and yet
one where a little conscientious work, together with common

Fill 5.8ft

Area.l9.2Sqft \ Area=20.3Sq.ft

sense, would have produced amazing results, as shown in Fig.
371, which is decidedly good location. Side hills afford op-
portunity for almost the cheapest form of construction. A


grade tine, i. e., where the grade coincides with the surface
of the ground on the center line, as in Fig. 371, can, unless
rock is encountered, be graded with pick and shovel alone,
the men casting the material taken from the cut directly
into and making the fill. The area of the cut in Fig. 370 is
49.8sq. ft., while the area of the fill is but 4.5 sq. ft., leaving
an excess of excavation of 45.3 sq. ft., or ten times the area
of the fill. There is no way by which this excess of mate-
rial can be utilized; it must, therefore, be wasted, as has
been the labor of excavating it. By moving the center line
4 feet to the right, we obtain the cross-section shown in Fig.
371, in which the calculated areas of cut and fill are as fol-
lows: Cut, 19.2 sq. ft. ; fill, 20.3 sq. ft. ; a difference of less
than 1 sq. ft. , and the excess is on the right side? for a ditch
should be made four feet from the top of the upper slope to
prevent the washing down of the slope, and this material
will more than equal the excess of the fill over the cut.

1434. Referencing Transit Points. Having com-
pleted the final location, the points of curve and points of
tangent must be referenced, and also intermediate points
where a change of grade requires it. Such an intermediate
point is shown in Fig. 372.




The line ABC from the P. T. at A to the P. C. at C is
straight, but the transit pole at C can not be seen through
the transit at A on account of the change of grade at B. It
is, therefore, necessary to establish an intermediate point at
B on the line ABC. The transit being set up at B, both
P. T. and P. C. are in full view.

A good example of referencing is shown in Fig. 373. The
reference points consist of plugs driven flush with the
ground and protected by substantial guard stakes, which are


marked with the letters R. P. Where the located line trav-
erses timber or brushwood, the ordinary stakes on the
center line should be replaced by much larger ones. They
are best cut from saplings about 3 feet in length and from
2 to 3^ inches in diameter. A place for the stake is made
with an iron bar, and the stake driven at least one foot in
the ground with a sledge hammer. Special care is taken in

'lug and
Guard Stake

'lug and
Guard Stake


guarding points of curve and tangent. While the right of
way is being cleared a man is detailed to look after the
stakes and hubs on the center line, as many will be disturbed
or torn out of the ground while hauling logs and timber
from the right of way. When the clearing and burning is
completed, the center line should be rerun, restoring all
lost or disturbed stakes. Transit points, if well set, will
rarely be disturbed. When the center line is restored the
transit points are referenced. A little care and judgment
will enable the engineer to select reference points which will
remain undisturbed during the work of construction.
Where the work is heavy these points will be further re-
moved from the center line than at points where the work
is light.

When the grading is completed, the original points of
curve and tangent can be restored and the center line run
in from both ends of the curve. Any small error in aline-
ment due to inaccuracies in the measurement of the original
line will then be thrown on the middle of the curve, where


they will not in any way affect the excellence of the work,
and the tangents will remain unchanged.

1435. Final Levels. While the transit points are
being referenced, the leveler takes the final levels, reading
all turning points with the target and correcting all bench
marks. He need not hurry, as accuracy is all important.
An error in final levels is unpardonable, as the work of con-
struction is based upon them. Most errors in field work are
directly chargeable to carelessness. A bench mark is estab-
lished at each bridge site, and at all points of the line where
permanent structures, such as arch culverts, trestles, water
tanks, stations, etc., are to be built. The final profile is
platted from these levels and the grade line drawn in pencil.
The points of curve and tangent are marked in small cir-
cles on one of the horizontal lines at the bottom of the pro-
file. That portion of the line corresponding to tangents is
drawn in a full line, and the balance, representing the
curves, in broken line. The stations of the points of curve
and tangent are also numbered on the profile.

The compensations for curvature are then calculated, and
the final grade line drawn in ink.

1436. Compensation for Curvature. From .03

to .05 ft. per degree is the compensation or reduction in
grade, made for the added resistance due to curvature, i. e. ,
where the established grade for tangents is 1 per cent., the
grade on a 6 curve, allowing a compensation of .03 ft. per
degree, would be 1.00 (.03 ft. X 6) = . 82 per cent. Where
a compensation of .05 ft. per degree is made, the grade on
a 6 curve would be 1.00 (.05 X 6) = . 70 per cent.

1437. Final Grade Lines. The establishing of final
grade lines is illustrated in Fig. 374, where the uncompen-
sated grade is 1.3 per cent., and the compensation for
curvature as shown in the final grade line is .03 ft. per

The location notes for the line included in the diagram are
as follows:




Intersection Angles.

52 + 00

.End of Grade.

49 + 75 P. T.

44 + 25 P. C. 9 R.

49 30'

42 + 00 P. T.

37 + 50 P. C. 6 L.

27 00'

33 + 00 P. T.

29 + 00 P. C. 8 R.

32 00'

27 + 00

Beginning of Grade.

The profile is made to standard scales, viz., horizontal
400 ft. = 1 in. ; vertical, 20 ft. = 1 in. The elevation of the
grade at Sta. 27 is fixed at 120 ft., and at Sta. 52 at 152.5 ft.,
giving between these stations an actual rise of 32.5 ft. and
an uncompensated grade of 1.3 per cent. These grade points
we mark on the profile in small circles. The total curvature
between Sta. 27 and Sta. 52 is 108. The resistance due to
each degree of curvature being taken as equivalent to an in-
crease of .03 ft. in grade, the total resistance due to 108.5 is
equal to .03 ft. X 108.5 = 3.255 ft., and is equivalent to
adding 3.255 ft. to the actual rise between Sta. 27 and Sta. 52.
Hence, the total theoretical grade between these stations is
the sum of 32.5 ft., the actual rise, and 3.255 ft. due to
curvature, which is 35.755 ft. Dividing 35.755 by 25, the

number of stations in the given distance, we have


+ 1. 4302 ft. , the grade for tangents on this line. The starting
point of this grade is at Sta. 27. The P. C. of the first curve
is at Sta. 29, giving a tangent of 200 ft. = 2 stations. As
the grade for tangents is + 1.4302 ft. per station, the rise in
grade between Stas. 27 and 29 is 1.4302x2 = 2.8004 ft.
The elevation of the grade at Sta. 27 is 120 ft., and the
elevation of grade at Sta. 29 is 120 + 2.8604 = 122.8604 ft.,
which we record on the profile as shown in Fig. 374, with the


rate of grade, viz., + 1.4302 written above the grade line.
The first curve is 8, and as the compensation per degree is
.03 ft. for 8, or a full station, the compensation is .03 ft. X
8 = .24 ft. The grade on the curve will, therefore, be the
tangent grade minus the compensation, or 1.4302 .24 ft. =
+ 1.1902 ft. per station. The P. C. of this curve is at
Sta. 29, the P. T. at Sta. 33, making the total length of the
curve 400 ft. = 4 stations. The grade on this curve is +
1.1902 ft. per station, and the total rise on the curve is
1.1902 X 4 4.7608 ft. The elevation of the grade at the
P. C. at Sta. 29 is 122.8604; hence, the elevation of grade at
the P. T. at Sta. 33 is 122.8604 + 4.7608 = 127.6212 ft.,
which we record on the profile together with the grade, viz.,
+ 1.1902, written above the grade line. The P. C. of the
next curve is at Sta. 37 + 50, giving an intermediate
tangent of 450 ft. =4.5 stations. The grade for tangents
is + 1.4302 ft. per station; hence, the total rise on the
tangent is 1.4302 X 4.5 = 6.4359 ft. Adding 6.4359 ft.
to 127.6212 ft., we have for the elevation of grade at Sta.
37 + 50, 134.0571 ft., which we record on the profile, to-
gether with the rate of grade for tangents.

The next curve is 6 and the compensation in grade per
station is .03 ft. X 6 = .18 ft. The grade on this curve will,
therefore, be 1.4302 .18 = 1.2502 ft. per station. The
length of the curve is 450 ft. = 4.5 stations, and the total
rise in grade on this curve is + 1.2502 ft. X 4.5 = 5.6259 ft.
The elevation of the grade at Sta. 37 + 50, the P. C.
of the curve, is 134.0571. The elevation of the grade at
Sta. 42, the P. T., is, therefore, 134.0571 + 5.6259 = 139.683
ft., which we record on the profile together with the rate of
grade on the 6 curve, viz. , + 1. 2502. The P. C. of the next
curve is at Sta. 44 + 25, giving an intermediate tangent of
225 ft. = 2.25 stations. The total rise on the tangent is,
therefore, 1.4302 X 2.25 = 3.21795ft. The elevation of grade
at the P. T. at Sta. 42 = 139.683; therefore, the elevation of
grade at Sta. 44 + 25 = 139.683 + 3.21795 ft. = 142.90095 ft.,
which we record on the profile together with the grade, viz.,
+ 1.4302. The last curve is 9 and the compensation in


grade per station is
.03 ft. X 9 = .27 ft.
The grade on 9 curve
is, therefore, 1.4302
.27 = 1.1602 ft. per sta-
tion. The length of
the curve is 550 ft. =
5.5 stations, and the
total rise on the curve
is 1.1602X5.5 = 6.3811
ft. The elevation of
grade at Sta. 44 + 25,
the P. C. of the 9
curve, is 142.90095 ;
hence, the elevation of
grade at the P. T.,
at Sta. 49 + 75, is
142.90095 + 6.3811 =
149.28205ft., which we
record on the profile to-
gether with the grade,
+ 1.1602. The end of
the line is at Sta. 53,
giving a tangent of
225 ft. = 2.25 stations.
The rise on this tan-
gent is 1.4302X2. 25 =
3.21795 ft., which we
add to 149.28205, the
elevation of the P.T. at
Sta. 49+75. The sum,
152.5 ft., is the eleva-
tion of grade at Sta.
52. The sum of the
partial grades should
equal the total rise be-
tween the extremities
of the grade line.


Qdf. 34.(k



The points where the changes of grade occur are marked on
the profile in small circles, which are connected by fine lines
which represent the grade line. These points of change are
projected on a horizontal line at the bottom of the profile.
Those portions of this line representing curves are dotted,
and those portions representing tangents are drawn full.
The points of curve P. C. and P. T. are marked in small
circles on this horizontal line, and lettered as shown in the

Where the grades are light and the curves easy, there will
be no need of compensation for curvature. Where the grades
exceed .5 per cent, and the curves 5, compensation should
be made.

1 438. Changing of Grade Lines. Unforeseen diffi-
culties sometimes arise during construction which warrant
the changing of grade lines, but these occasions are rare.
If the final grade line has been properly considered, it would
better remain unchanged. The engineer should learn to
make up his mind and stick to it.

1439. Vertical Curves. Vertical curves are used to
round off the angles formed by the meeting of two grade

lines. Let A C and C B, Fig. 375, be two grade lines
meeting at C.

These grades are given by the rise per station in going in
some particular direction. Thus, starting from A, the
grades A C and C B may be denoted by g and g' ; that
is, the grade for any station on A C is found by adding the


rate of grade g to the grade of the preceding station, and
the grade for any station on C B is found by adding the rate
of grade g' to the grade of the preceding station. But C B
is a descending grade. Therefore, the rate^', to be added
to each station, is a minus quantity and g' is negative.

The parabola furnishes a simple method of putting in a
vertical curve.

144O. Problem. Given the grade g of A C, Fig. 375,
the grade g' of C B, and the number of stations ;/ on each
side of C to the tangent points A and B, to unite these
points by a parabolic vertical curve:

Let A E B be the required parabola. Through B and C
draw the vertical lines F K and C H, and produce A C to
meet F K in F. Through A draw the horizontal line A K
and join A and /?, cutting C H in D. Then, if a represents
the vertical distance of the first station M on the curve
from the corresponding station 7' on the tangent, the ver-
tical distance at the second station will be the square of
2, or 4 #, and at the third station the square of 3. or 9 a,
and at /?, which is 2 n stations from A, the vertical distance
to the curve will be the square of 2 n or 4 n* a ; that is, F B

77 r>

4 n 9 a. and a = - -. To find a. it will first be necessary
4 a

to find F B. This may be done by means of the following
formula, in which g and g' are the grades mentioned in
Art. 1439, and n is the number of stations between A
and C:

Having determined the value of a, the distances of the
several stations in A C and C B from the curve, viz., a, 4 a >
9 a, 16 #, etc., are readily known. Let T and T' be the
first and second stations on the tangent A C, and if from T
and T' perpendicular lines T P and T' P' be drawn to the
horizontal line A K, T P, the height of the first station T
above A, equals g, and T' P' equals 2 g, and for succeeding
stations we shall find the heights 3 g, 4 g, etc. We have


already found T M a, T' M' = 4 , etc. The heights of
the curve above the level of A will, therefore, be as follows:
At M, height = TP TMga\ at M\ height = T' P'

- T' M' 2 g - 4 a, and at E, height = C H C E = 3 g

9 a, and for succeeding points 4 g 16 tf, etc. To find
the grades for the curve at successive stations from A, that
is, the amount which must be added to the grade or height
of one station to equal the grade of the following station, we
must subtract each height from the next following height.
Thus, calling the height of A 0, we have (g a] = g

a, the height of M above A K, called the grade of M;
(2g 4 a) (g a) = g 3 a, which must be added to the
grade of J/ to find the grade of M'; (%g 9 a) (2 g 4 a)
= g 5 <7, which must be added to the grade of M' to find
the grade of E. The succeeding quantities are (4^16 a)

- (3 "-9 *)=- 7 , (5 g-Ma) - (4^-16 a) = g -
9 a, and (6^- 36 a) (5 g 25 a) g 11 a. The suc-
cessive grades or additions for the vertical curve, Fig.
375, are g a, g 3 a, g5 a, g1 a, g$ a , an d

In finding these grades, strict regard must be paid to the
algebraic signs. The results are then general.

1441. EXAMPLE 1. Let the number of stations on each side
of C, Fig. 375, be 3, and let A Cbe an ascending grade of 1.2 feet per
station, and C B a. descending grade of .8 ft. per station. Assume the
elevation of the grade at Sta. A to be 120 feet, and find the grade at
each station from A to B.

SOLUTION. Here n = 3,g= + 1.2 ft., and^-' = .8 ft. Substitu-
ting known values in formula !Ol,tf = . , we have a = -. ' =

4 n 4 X >


^2 = . 1666 ft. , and the grades from A to B will be

Heights of Curve

above A.

g- a =1.2 .167= 1.033 ft. 1.033 ft.
- 8=1.2 .500= .700 ft. 1.733 ft.
g - 5a = 1.2 - .833 = .367 ft. 2.100 ft.
- 7* =1.2-1.166 = .034ft. 2.134ft.
g - Qa = 1.2 - 1.500 = - .300 ft. 1.834 ft.
g-\\a- 1.2 - 1.833 = - .633 ft. 1.201 ft.



Since the elevation of the grade at Sta. A, Fig. 375, is 120.00 feet, the
grades for the following stations of the vertical curve will be:


EXAMPLE 2. Let A C, Fig. 376, be a descending grade of 1.0 ft. per
station, and C an ascending grade of .5 ft. per station. Let the
vertical curve include 2 stations each side of C. Find the grade at
each station from A to C.

SOLUTION. Here g 1.0 ft., g' = + .5 ft., and = 2. Substituting


Elevation of



Elevation of


121 033 ft


121 834 ft.


121 733 ft


121.201 ft.

E ..


FIG. 376.

these values in formula 1O1 , a . , we have :

4 O


= .1875, and the four grades required will be:


g = 1.0 ( .1875) =-1.0+ .1875= -.8125ft.
g _3 = _i.o_(_ .5625) =-1.0+ .5625 =- .4375 ft.
- _5 rt= _1.0-(- .9375) =-1.0+ .9375 =- .0625 ft.
g - la = - 1.0 - (- 1.3125) = 1.0+ 1.3125 = + .3125 ft.

It will be seen that after finding the first grade, the suc-
ceeding grades are found by a continual subtraction of 2 a.
Thus, in the first example, each grade after the first is .333 ft.
less than the preceding grade. In the second example, a is
a negative quantity, and each grade after the first is .375ft;.
greater than the preceding grade.

The grades in the foregoing examples are calculated for
whole stations, and are sufficient for all purposes except for-
track laying or ballasting, when grade stakes on the vertical
curve should be driven at intervals of 25 feet, and the grades
must be calculated for these sub-stations. To do this, let g


and^' represent the grades for a sub-station of 25 feet, and
n the number of such sub-stations on each side of the
intersection of the grade lines.

EXAMPLE. In the last example divide the curve into sub-stations of
25 ft. each. Assume the grade at A to be 120 feet, and find the grade
at each sub-station.

SOLUTION. Here ^-=-.25 ft., g' - + .125 ft., and n = 8. Sub-
stituting these values in formula 1O1, a. , we have a

25 ~ ( - 125) = ~:f 75 - _ .01172. The first grade is, therefore.

O/w O&

g a = .25 ( .01172) = .23828. Each subsequent grade in-
creases 2 a;- that is, .02344, and we have the following: Grade at
Sta. 2= - .21484; Sta. 3, - .19140; Sta. 4, - .16796; Sta. 5, - .14452;
Sta. 6, -.12108; Sta. 7, -.09764; Sta. 8, -.07420; Sta. 9, -.05076;
Sta. 10, - .02732; Sta. 11, - .00388; Sta. 12, 4- .01956; Sta. 13, + .04300;
Sta. 14, + .06644; Sta. 15, + .08988; Sta. 16, + .11332.

The distance A B, Fig. 376, is 400 feet divided into 16 sub-stations of
25 feet each. Since the grade of A is 120.0 feet, the grades of the fol-
lowing stations will be:


The purpose served by vertical curves will be' at once ap-
parent to the student. The sudden and severe stress upon
the rolling stock caused by passing from one grade to
another results in great harm to rolling stock and much
discomfort to passengers. Vertical curves should always be
put in the grade during construction. Where the meeting
grades are very slight, no curve is necessary.

1 442. Preliminary Estimates. Having established
the final grades, the next work of the engineer is the pre-
liminary estimate. This estimate gives in detail the approx-
imate quantities of all material to be handled in the work of

" Stations.


















12 ; ...






















construction, and of all probable cost attending such work.
Work and materials to be furnished, together with the prices
ruling in the locality where the work is to be done, are
classified as follows:

1443. Classification of Preliminary Estimates.

2. Excava-
tion . .


4. Masonry..



25c. to 30c.


1. Clearing per acre

Earth, per cubic yard

Loose rock, per cubic yard. .
Solid rock, per cubic yard. . .

Overhaul exceeding 1,000 feet, per cubic yard

( Piles, per lineal foot

Frame, per 1,000 ft. bd. meas-
ure, Ga. pine

Ist-class rock-face range

work, per cubic yard $10. 00 to $12 00.

2d-class good lime mortar

rubble, per cubic yard. . . . $8.00.

Dry rubble, per cubic yard. . $4. 00 to $4. 50.
Riprap per square yard, in

place $1. 50 to $2. 50.

j Wooden ,


I Iron and steel

Classification not only affects price, but quantity. Cuts
in solid rock, which are the most costly, stand at a slope of
horizontal to 1 vertical, while earth ordinarily requires a
slope of 1 horizontal to 1 vertical, and sometimes as flat a
slope as 1^ horizontal to 1 vertical. All materials excavated,
and all masonry, are estimated by the cubic yard. Trest-
ling is estimated by the 1,000 feet, board measure, and
piling by the lineal foot. Wooden bridges, of moderate

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