METHODS OF DETERMINING THE CENTER LINE AND FORMS AND DIMENSIONS OF
CROSS-SECTION.
DETERMINING THE CENTER LINE.
Tunnels may be either curvilinear or rectilinear, but the latter form is
the more common. In either case the first task of the engineer, after
the ends of the tunnel have been definitely fixed, is to locate the
center line exactly. This is done on the surface of the ground; and its
purpose is to find the exact length of the tunnel, and to furnish a
reference line by which the excavation is directed.
=Rectilinear Tunnels.=--In short tunnels the center line may be
accurately enough located for all practical purposes by means of a
common theodolite. The work is performed on a calm, clear day, so as to
have the instrument and observations subjected to as little atmospheric
disturbance as possible. Wooden stakes are employed to mark the various
located points of the center line temporarily. The observations are
usually repeated once at least to check the errors, and the stakes are
altered as the corrections dictate; and after the line is finally
decided to be correctly fixed, they are replaced by permanent monuments
of stone accurately marked. The method of checking the observations is
described by Mr. W. D. Haskoll[2] as follows:
“Let the theodolite be carefully set up over one of the stakes, with
the nail driven into it, selecting one that will command the best
position so as to range backwards and forwards over the whole length
of line, and also obtain a view of the two distant points that range
with the center line; this being done, let the centers of every stake
... be carefully verified. If this be carefully done, and the centers
be found correct, and thoroughly in one visual line as seen through
the telescope, there will be no fear but that a perfectly straight
line has been obtained.”
[2] “Practical Tunneling,” by F. W. Simms.
[Illustration: FIG. 1.--Diagram Showing Manner of Lining in Rectilinear
Tunnels.]
The center line which has thus been located on the ground surface has to
be transposed to the inside of the tunnel to direct the excavation. To
do this let _A_ and _B_ be the entrances and _a_ and _b_ be the two
distinct fixed points which have been ranged in with the center line
located on the ground surface over the hill _A f B_, Fig. 1. The
instrument is set up at _V_, any point on the line _A a_ produced, and a
bearing secured by observation on the center line marked on the surface.
This bearing is then carried into the tunnel by plunging the telescope,
and setting pegs in the roof of the heading. Lamps hung from these pegs
furnish the necessary sighting points. This same operation is repeated
on the opposite side of the hill to direct the excavation from that end
of the tunnel. These operations serve to locate only the first few
points inside the tunnel. As the excavation penetrates farther into the
hill, it becomes impossible to continue to locate the line from the
outside point, and the line has to be run from the points marked on the
roof of the heading. Great accuracy is required in all these
observations, since a very small error at the beginning becomes greater
and greater as the excavation advances. To facilitate the accurate
location of points on the roof of the tunnel, a simple device was
designed by Mr. Beverley R. Value, shown in Fig. 2. Two iron spikes,
each having a small hole in the flat end, are driven into the rock about
9 ins. apart. A brass bar, 1 in. high, ¹⁄₄ in. thick and 10 ins. long,
having a hole near one end and a 1 in. slot at the other, is screwed
tightly into the head of the spikes. The middle part of the bar is
divided into inches and tenths of an inch. A separate brass hanger is
fitted to the bar, having a vernier with its zero at the middle of the
hanger and corresponding to a plumb line attached below. The hanger is
moved back and forth until it coincides with the line of sight of the
transit, and then the readings of the vernier are recorded. Any time
that the hanger is placed on the bar and the vernier marks the same
reading, the plumb line will indicate the center line of the tunnel.
When, instead of one bar, two are inserted at a distance of 20 or 30 ft.
apart, the plumb lines suspended from the hangers will represent the
vertical plane passing through the axis of the tunnel in coincidence
with the one staked out on the surface ground.
[Illustration: FIG. 2.--B. R. Value’s Device for Locating the Center
Line Inside of a Tunnel.]
The location of the center line of a long tunnel, which is to be
excavated under high mountains, is a very difficult operation, and the
engineers usually leave this part of the work to astronomers, who fix
the stations from which the engineers direct the work of construction.
The center lines of all the great Alpine tunnels were located by
astronomers who used instruments of large size. Thus, in ranging the
center line of the St. Gothard tunnel, the theodolite used had an object
glass eight inches in diameter.[3] Instead of the ordinary mounting a
masonry pedestal with a perfectly level top is employed to support the
instrument during the observations. The location is made by means of
triangulation. The various operations must be performed with the
greatest accuracy, and repeated several times in such a way as to reduce
the errors to a minimum, since the final meeting of the headings
depends upon their elimination.
[3] See also the Simplon Tunnel, Chapter X.
[Illustration: FIG. 3.--Triangulation System for Establishing the Center
Line of the St. Gothard Tunnel.]
The St. Gothard tunnel furnishes perhaps the best illustration of
careful work in locating the center line of long rectilinear tunnels of
any tunnel ever built. The length of this tunnel is 9.25 miles, and the
height of the mountain above it is very great. The center line was
located by triangulation by two different astronomers using different
sets of triangles, and working at different times. The set or system of
triangles used by Dr. Koppe, one of the observers, is shown by Fig. 3;
it consists of very large and quite small triangles combined, the latter
being required because the entrances both at Airolo and Goeschenen were
so low as to permit only of a short sight being taken. The apices of the
triangles were located by means of the contour maps of the Swiss Alpine
Club. Each angle was read ten times, the instrument was collimated four
times for each reading, and was afterwards turned off 5° or 10° to avoid
errors of graduation. The average of the errors in reading was about one
second of arc. The triangulation was compensated according to the method
of least squares. The probable error in the fixed direction was
calculated to be 0.8″ of arc at Goeschenen and 0.7″ of arc at Airolo.
From this it was assumed that the probable deviation from the true
center would be about two inches at the middle of the tunnel, but when
the headings finally met this deviation was found to reach eleven
inches.
Comparatively few tunnels are driven by working from the entrances
alone, the excavation being usually prosecuted at several points at once
by means of shafts. In these cases, in order to direct the excavation
correctly, it is necessary to fix the center line on the bottom of the
shaft. This is accomplished in two ways,--one being employed when the
shaft is located directly over the center line, and the other when the
shaft is located to one side of the center line.
When the shaft is located on the center line two small pillars are
placed on opposite edges of the shaft and collimating with the center
line as shown by Fig. 4. On these two pillars the points corresponding
to the center line are correctly marked, and connected by a wire
stretched between them. To this wire two plumb bobs are fastened as far
apart as possible. These plumb bobs mark two points on the center line
at the bottom of the shaft, and from them the line is extended into the
headings as the work advances. In these operations, heavy plumb bobs are
used. In the New York subway plumb bobs of steel, weighing 25 lbs. each,
were used, and to prevent rotation they were made with cross-sections,
in the shape of a Greek cross, and were sunk in buckets filled with
water. Owing to the difference between the temperature at the top and
that at the bottom of the shaft, strong currents of air are produced,
which keep in constant oscillation the wires to which the bobs are
suspended.
[Illustration: FIG. 4.--Method of Transferring the Center Line down
Center Shafts.]
To determine the center line at the bottom of the shaft, the headings
are first driven from both sides of the shaft, after which a transit is
set up on the same alignment with the two wires, and this will indicate
the vertical plane passing through the axis of construction. Two points
are then fixed on the roof of the tunnel in continuation of this
vertical plane. When the plumb bobs are removed from the shaft and two
small plumb bobs are suspended to the two points mentioned, they will
always give the same vertical plane passing through the axis of
construction transferred from the surface.
Because of the continuous moving of the wires, the fixing of the points
on the roof of the tunnel is very troublesome, and the operation should
be repeated by different men at different times before the points are
permanently fixed.
[Illustration: FIG. 5.--Method of Transferring the Center Line down Side
Shafts.]
When the shaft is placed at one side of the tunnel the pillars or bench
marks are placed normal to the center line on the edges of the shaft as
shown by Fig. 5. Between the points _A_ and _B_ a wire is stretched, and
from it two plumb bobs are suspended, as described in the preceding
case; these plumb bobs establish a vertical plane normal to the axis of
the tunnel. The excavation of the side tunnel is carried along the line
_BW_ until it intersects the line of the main tunnel, whose center line
is determined by measuring off underground a distance equal to the
distance _BO_ on the surface. By setting the instrument over the
underground point _O_, and turning off a right angle from the line _BO_,
the center line of the tunnel is extended into the headings.
=Curvilinear Tunnels.=--There are various methods of locating the center
lines of curvilinear tunnels, but the method of tangent offsets is the
one most commonly employed.
At the beginning the excavation is conducted as closely as may be to
the line of the curve, and as soon as it has progressed far enough the
tangent _AT_, Fig. 6, is ranged out. At _B_ a point is located over
which to set the instrument, and the distance _AB_ is measured for the
purpose of finding the ordinate of the right angle triangle _OAB_. Now
_OA_ = _r_, _AB_ = _d_, and φ = angle _ABO_. Then:
_r_
Tang. φ = ---.
_d_
[Illustration: FIG. 6.--Method of Laying Out the Center Line of
Curvilinear Tunnels.]
Doubling the value of φ and making the angle _ABC_ = 2 φ, the line _BC_
will be fixed and the point _C_ located by taking _AB_ = _BC_. On _BC_
the ordinates are laid off to locate the curve. Prolong _CB_ so that
_CD_ = _CB_. Then the portion of the curve _CF_ is symmetrical with
_CE_, and the ordinates used to locate _EC_ may be employed to locate
_CF_, by laying them off in the reverse order.
In curvilinear tunnels several cases may be considered.
(1) When the tunnel for almost its entire length is driven on a tangent
with a curve at each end.
(2) When the tunnel begins with a curve and ends with a straight line.
(3) When the whole tunnel is in curve from portal to portal.
(4) The helicoidal or corkscrew tunnel.
* * * * *
(1) The axis of every one of the great Alpine tunnels is a straight
line, with a curve at each end. To range out the center line of one of
these long tunnels from a curve, no matter how accurately laid out, will
certainly cause an error, which, magnified with the distance, may
produce serious results. To avoid these inconveniences, the
determination of the axis of the tunnel should be made from a straight
line. This means that the tunnel is at first excavated on a straight
line for its entire length and after the headings driven from both
portals have met, the two portions of the tunnel or curve are excavated
and constructed. The portions of the tunnel excavated on straight lines
for conveniences of construction may then be abandoned or used in cases
of accidents or repairs.
When the axis of a short tunnel has a curve at each end and a straight
line in the middle, it is driven directly from the entrances; first,
however, excavating the curvilinear portions of the tunnel. In such a
case it would be advisable to proceed in the following manner. Drive the
headings on the curvilinear portions of the tunnel, staking out the
center line by means of the offsets from the tangents. At the ends of
the curves lay out from both fronts the rectilineal portion of the
tunnel. Only very narrow headings should be excavated at first while the
whole section could be enlarged near the entrances. The excavation of
the headings at the front should advance very rapidly, in order that the
headings may meet in the shortest possible time. When communication is
established, it is comparatively easy to correct an error resulting from
driving the tunnel from the curves.
(2) When a tunnel begins with a curve and ends with a straight line, the
work of excavation should proceed from both ends. From the straight end
of the tunnel only the heading should be driven, while from the
curvilinear end the whole section could be opened at once. By this
arrangement the excavation progresses slowly from the curvilinear end
and rapidly from the straight end of the tunnel. Once communication has
been established and any error corrected, the work of enlarging the
profile of the tunnel may be pushed with the same activity from both
ends.
(3) When the center line of the entire tunnel is a curve, there is more
probability of slight deviations from the true axis of the proposed
work. In such a case it would be advisable to first excavate a narrow
heading and to concentrate all the efforts in driving the headings as
rapidly as possible in order that they may meet in the shortest time.
The center line of these headings is staked out by the usual method of
the offsets from the tangent. The enlarging of the section of the tunnel
could be commenced at both portals and be driven slowly until the
headings have met and any errors corrected, when the work could be
pushed with the greatest activity all along the line.
(4) In corkscrew or helicoidal tunnels the entire center line is on a
curve. In these tunnels, as a rule, there is a great difference of level
between the two portals, one being much higher than the other, so
careful attention should be paid to the tunnel grade. Working in the
limited spaces afforded by narrow headings it is very probable that
errors may be made in fixing both the alignment and the grade of the
tunnel. To prevent these almost unavoidable errors, it would be well to
excavate at first only the headings, to stake the center line in the
roof of these headings and then to lay the grade of the tunnel as
accurately as possible. The work on the headings should be pushed as
rapidly as possible in order that they may meet quickly, so that the
center line, as temporarily laid out, may be corrected and permanently
fixed for the direction of successive operations. In these tunnels the
headings should be excavated near the center of the tunnel cross-section
so that the sides and roof of the heading would be at some distance from
the sides and roof of the proposed tunnel. This arrangement will easily
permit corrections to be made in case any slight difference from the
true line was erroneously made during the excavation of the headings.
FORM AND DIMENSIONS OF CROSS-SECTION.
In deciding upon the sectional profile of a tunnel two factors have to
be taken into consideration: (1) The form of section best suited to the
conditions, and (2) the interior dimensions of this section.
=Form of Section.=--The form of the sectional profile of a tunnel should
be such that the lining is of the best form to resist the pressures
exerted by the unsupported walls of the tunnel excavation, and these
vary with the character of the material penetrated. These pressures are
both vertical and lateral in direction; the roof, deprived of support by
the excavation, tends to fall, and the opposite sides for the same
reason tend to slide inward along a plane more or less inclined,
depending upon the friction and cohesion of the material. In some rocks
the cohesion is so great that they will stand vertically, while it may
be very small in loose earth which slides along a plane whose
inclination is directly proportional to the cohesion.
From the theory of resistance of profiles we know that the resistance of
a line to exterior normal forces is directly proportional to its degree
of curvature, and consequently inversely proportional to the radius of
the curve. Hence the sectional profile of a tunnel excavated through
hard rock, where there are no lateral pressures owing to the great
cohesion of the material, and having to resist only the vertical
pressure, should be designed to offer the greatest resistance at its
highest point, and the curve must, therefore, be sharper there, and may
decrease toward the base. In quicksand, mud, or other material
practically without cohesion, the pressures will all be normal to the
line of the profile, and a circular section is the one best suited to
resist them. These theoretical considerations have been proved correct
by actual experience, and they may be employed to determine in a general
way the form of section to be adopted. Applying them to very hard rock,
they give us a section with an arched roof and vertical side walls. In
softer materials they give us an elliptical section with its major axis
vertical, and in very soft quicksands and mud they give us the circular
section. These three forms of cross-section and their modifications are
the ones commonly employed for tunnels. An important exception to this
general practice, however, is met with in some of the city underground
rapid-transit railways built of late years, where a rectangular or box
section is employed. These tunnels are usually of small depth, so that
the vertical pressures are comparatively light, and the bending strains,
which they exert upon the flat roof, are provided for by employing steel
girders to form the roof lining.
[Illustration: FIG. 7.--Diagram of Polycentric Sectional Profile.]
From what has been said it will be seen that it is impossible to
establish a standard sectional profile to suit all conditions. The best
one for the majority of conditions, and the one most commonly employed,
is a polycentric figure in which the number of centers and the length of
the radii are fixed by the engineer to meet the particular conditions
which exist. In a general way this form of center may be considered as
composed of two parts symmetrical in respect to the vertical axis. Fig.
7 shows such a profile, in which _DH_ is the vertical axis. The section
is unsymmetrical in respect to the horizontal axis _GE_. The upper part
forming the roof arch is usually a semi-circle or semi-oval, while the
lower part, comprising the side walls and invert of floor, varies
greatly in outline. Sometimes the side walls are vertical and the invert
is omitted, as shown by Fig. 8; and sometimes the side walls are
inclined, with their bottoms braced apart by the invert, as shown by
Fig. 9. In more treacherous soils the side walls are curved, and are
connected by small curved sections to the invert, as shown by Fig. 10.
In the last example the side walls are commonly called skewbacks, and
the lower part of the section is a polycentric figure like the upper
part, but dissimilar in form.
In a tunnel section whose profile is composed entirely of arcs the
following conditions are essential: The centers of the springer arcs
_Ga_ and _Ea′_, Fig. 7, must be located on the line _GE_; the center of
the roof arc _bDb′_ must be located on the axis _HD_; the total number
of centers must be an odd number; the radii of the succeeding arcs from
_G_ toward _D_ and _E_ toward _D_ must decrease in length, and finally
the sum of the angles subtended by the several arcs must equal 180°.
[Illustration:
~Fig. 8~
~Fig. 9~
~Fig. 10~
FIGS. 8 to 10.--Typical Sectional Profiles for Tunnel.]
=Dimensions of Section.=--The dimensions to be given to the
cross-section of a tunnel depend upon the purpose for which it is to be
used. Whatever the purpose of the tunnel, the following three points
have to be considered in determining the size of its cross-section: (1)
The size of clear opening required; (2) the thickness of lining masonry
necessary; and (3) the decrease in the clear opening from the
deformation of the lining.
Railway tunnels may be built either to accommodate one or two, three and
four tracks. In single-track tunnels a clear space of at least 2¹⁄₂ ft.
on each side should be allowed for between the tunnel wall and the side
of the largest standard locomotive or car, and a clear space of at least
3 ft. should be allowed for between the roof and the top of the same
locomotive or car. Since the roof of the tunnel is arch-shaped, to
secure a clearance of 3 ft. at every point will necessitate making the
clearance at the center greater than this amount. In double-track
tunnels the same amounts of side and roof clearances have to be provided
for, and, in addition, there has to be a clearance of at least 2 ft.
between trains. On the three- and four-track tunnels only the width
varies while the height remains almost equal to the two track. Referring
to Fig. 7, and assuming the line _AB_ to represent the level of the
tracks, then the ordinary dimensions in feet required for both single-
and double-track tunnels are as follows:--
+--------------+--------+----------+----------+---------------+
| | HEIGHT, | WIDTH, | HEIGHT, | HEIGHT, |
| | D. F. | G. E. | C. F. | C. H. |
| | FEET. | FEET. | FEET. | FEET. |
+------------+----------+----------+----------+---------------+
|Single track|17.6 to 18|16.5 to 18|6 to 7.4|¹⁄₄ to ¹⁄₈ _AB_|
|Double track|26.6 to 28|26.6 to 28|6.3 to 6.9|¹⁄₄ to ¹⁄₈ _AB_|
+------------+----------+----------+----------+---------------+
The dimensions of tunnels built for aqueduct purposes are determined so
as to have an area of cross-section equal to the required waterway. In
the Croton Aqueduct two different types of cross-sections were used, the
circular one for tunnels through rock and the horseshoe section for
tunnels through loose materials. In the Catskill aqueduct three
different cross-sections have been selected, the circular one for
tunnels under pressure and the horseshoe for tunnels at the hydraulic
gradient. These, however, through rock have a cross-section formed of a
semi-circular arch and vertical side walls, while through earth the
semi-circular arch is supported by skewback walls.
In tunnels built for railroad aqueduct sewer and any other purpose the
thickness of the masonry lining to be allowed for varies with the
material penetrated, as will be explained in a succeeding chapter where
the dimensions for various ordinary conditions are given in tabular
form. The lining masonry is subject to deformation in three ways: by the
sinking of the whole masonry structure, by the squeezing together of the
side walls by the lateral pressures, and by the settling of the
roof-arch. The whole masonry structure never sinks more than three or
four inches, and merits little attention. The movement of the side walls
towards each other, which may amount to three or four inches for each
wall without endangering their stability, has, however, to be allowed
for; and similar allowance must be made for the settling of the
roof-arch, which may amount to from nine inches to two feet, when the
arch is built first as in the Belgian system and rests for some time
upon the loose soil.