Prof. M. I. Evdokimov-Rokotovsky

Tracing and Setting Out of Tunnels

Tomsk
1929

Tomsk, Okrlit No. 846

Printing house of the publishing house “Red Banner”
Tomsk, Timiryaz. Ave., 2

Print run: 500 copies.

“Izv. Sib. Tekh. Inst.” vol. 51, issue 2.

Contents

	Page
Preface	1
Part I. Tracing the Tunnel Axis on the Earth’s Surface.
§ 1. Tracing the tunnel axis directly over the top of the mountain mass	5
I. The tunnel axis is straight.
A. Tracing without the use of an angle-measuring instrument	5
B. Tracing with the construction of a tower (observatory) on the pass and with the use of direct sighting directions	6
C. Tracing with the construction of a tower (observatory) on the pass, using direct and reverse sighting directions	9
D. Tracing with the construction of towers (observatories) at the ends of the tunnel	11
II. The tunnel axis is curvilinear.
A. Method of the described polygon	14
B. Method of successive chords	16
§ 2. Determination of the direction of the tunnel axis by means of polygonal surveying	17
§ 3. Tunnel triangulation	28
Types of tunnel triangulations	28
Selection of the system for arranging the points of the tunnel triangulation network	33
Base	35
Selection of a location for the base	35
Reduction of the tunnel length to a specified level	39
Deflection of plumb lines by the attraction of mountain masses	42
§ 4. Marks, signals, and observatories used in tunnel geodetic works on the earth’s surface	48
Marks and signals	48
Centers	49
Signals of tunnel triangulation	51
Observatories (towers)	51
§ 5. Execution of work. Instruments used in tunnel geodetic works on the earth’s surface	55
Execution of work for tracing straight-line tunnels	56
Instruments for tracing tunnels	56
Execution of work	57
Execution of tunnel triangulation works	58
Instruments for tunnel triangulations	58
Measurement of the base	61
Vertical surveying when tracing the tunnel axis	63
Survey of the polygon	64

II

§ 6. Preparatory work on the earth’s surface for setting out the axis of tunnels underground . . . 64
    Calculation of angles and of the direction of the tunnel axis . . . 65
    Calculation of the length of the tunnel . . . 69
    Fixing the direction of the tunnel; arrangement of alignments . . . 70
    Calculation of the angles, direction, and length of side adits (windows) . . . 70

Part II. Setting out the tunnel axis underground.

§ 1. Sighting marks . . . 72
    Sighting marks placed outside the tunnel . . . 72
    Sighting marks placed inside the tunnel . . . 77

§ 2. Setting out tunnels with a straight-line axis . . . 80
    Setting-out marks . . . 82
    The guide adit is located on the tunnel axis . . . 83
    Setting out the axis of hydraulic-engineering tunnels . . . 86
    The guide adit is located above, on the tunnel axis . . . 86
    Transferring the points of the final setting-out . . . 88
    The guide adit is located below, but to the side, away from the tunnel axis . . . 89

§ 3. Setting out tunnels with a curved axis . . . 90
    Sighting adits . . . 93

§ 4. Vertical and inclined shafts. Horizontally arranged adits (windows) . . . 95
    Inclined shafts . . . 97
    Horizontally arranged adits (windows) . . . 97
    Setting out vertical shafts . . . 99
    Setting out the tunnel axis through horizontally arranged adits (windows) . . . 102

§ 5. Setting out during excavation of the tunnel to full section . . . 103
    Method of rectangular coordinates . . . 103
    Method of polar coordinates . . . 105
    Checking the setting-out of the profile by templates . . . 107

§ 6. Execution of work for setting out the tunnel axis. Instruments for underground geodetic work . . . 107
    Leveling work . . . 112
    Determination (measurement) of the length of the driven tunnel . . . 114
    Measurement of small tunnels . . . 114
    Measurement of long tunnels . . . 115

Bibliography . . . 116

PREFACE.

It may be stated decisively that no geodetic work in engineering requires such thoroughness, care, and precision as the setting out of the axis of a more or less long tunnel during its construction. A barely noticeable inaccuracy in setting out may entail distortions of the structure that are difficult to correct and may require costly alterations and corrections. The slightest carelessness in the work or error in the measurements can have enormous consequences and ruin the entire undertaking. Quite a number of examples could be cited in which failure to take all precautions into account was reflected in the setting out of a tunnel in the crudest form.

I remember a case in our USSR when, in 1914, in the Urals, while setting out a tunnel 1107.3 meters long, insufficiently careful work resulted in a broken straight line: a tunnel designed on a straight line received a turn at an angle of \(1^\circ 53'\) with a rounding radius of 2133.6 meters, and another turn at an angle of \(4^\circ 4'\) with a radius of 640 meters; in other words, the tunnel turned out to be absurd in its plan.

I also remember another case: in 1911, while setting out a tunnel 1601.8 meters long, located on one of the outlying roads of our Republic, engineer *** became confused in the setting out and, shaken by the erroneous results, shot himself.

Foreign practice provides curious examples from the history of the tracing of mountain tunnels.

Thus, for example, when setting out the Sasako tunnel in Japan, about 8 km long, the junction of the tunnel came out several meters farther than had been expected according to the calculation.

Miscalculations were also made during the construction of a tunnel on the Bilecik–Karaköy section of the Anatolian Railway in Asiatic Turkey.

An unfortunate error in tracing one tunnel in Italy, situated on a curve, caused a delay of the work for 4 months, involved a considerable excess of rock excavation, and cost a rather large amount in overruns.

During the setting out of one tunnel in Indo-China, the rectilinear axis of the tunnel in plan turned out to be broken at an angle of \(3^\circ 17'\). This disfigured the structure and caused much trouble for the French engineers who were building the road.

Many dreadful, agonizing minutes must be lived through by tunnel builders and by those responsible for tracing and setting out, when the driving of the pilot headings is nearing its end and the junction of the two passages, driven from opposite sides, is expected. How much rejoicing is caused by the moment when the rock separating the two headings collapses and thereby confirms the correctness of the direction of the structure. The persons who carried out the tunnel geodetic work then enjoy special and entirely deserved attention. Scientific institutions and learned societies reward them and sometimes even strike medals in their honor, as was the case, for example, with respect to the surveyors who worked on the triangulation of the Lötschberg Tunnel.

The moment when the junction will occur is usually awaited with feverish impatience. There have been cases abroad when errors, or even delays in the arrival of the expected moment of breakthrough, caused nervous disorders and even ended in suicide.

At times, however, there are curiosities as well: during the construction of the Saint-Gotthard Tunnel, the two parties working from opposite sides—from the side of Göschenen and from the side of Airolo—met vertically almost with mathematical precision, but several days earlier than expected, namely when, “according to the very most exact calculation,” they still had 60 meters to go.

All this indicates how difficult and responsible the geodetic work of setting out tunnels is.

By the tracing of a tunnel, in the broad sense of the word, we shall understand both the determination and marking on the earth’s surface of the direction of the axis of a rectilinear or curvilinear tunnel, and, in general, all those geodetic tunnel works that are carried out on the earth’s surface and that, in their final result, must lead to the creation of the possibility of correctly setting out the tunnel axis underground.

The task before us, in the degree of difficulty of its execution, depends to a considerable extent on the position of the tunnel in plan—whether it is rectilinear or curvilinear—but mainly on the general relief of the terrain, on the mountainous and precipitous nature of the district, and on its accessibility for the performance of geodetic work. Tunnel construction, taking place among mountains, is always associated with difficulties in setting out the structure.

All geodetic tunnel works on the daylight surface, according to their character, may be divided into several groups.

The first group includes those tunnel works when, owing to the insignificant length of the tunnel and under favorable conditions of the terrain relief, it is possible to confine oneself to tracing directly along the earth’s surface through the mountain mass. Here the terrain relief plays the decisive role: the mountain mass being pierced by the tunnel is, in its external outline, accessible for direct tracing. A well-known role is also played here by the simplicity of the tunnel’s position in plan. It may happen that, with a complex position of the tunnel in plan, even with favorable terrain relief, tracing along the surface directly through the mountain mass is, in practice, quite possible to carry out; however, the accuracy of the work may prove somewhat less than when another method is applied for determining the direction of the tunnel axis on the earth’s surface. This consideration regarding the accuracy of the setting out becomes all the more important the more complex the plan of the tunnel and the greater its length.

The second group comprises those operations in which, given the relatively short length of the tunnel, when tracing through a mountain mass, owing to the precipitous terrain, which is not readily accessible for work, concern arises as to the correctness of the direction of the alignment. Therefore, in such cases, in order to mark the direction of the tunnel axis, it becomes necessary to resort to auxiliary operations and to carry out a traverse survey, by closing which a check of the alignment direction is achieved.

Finally, the third group of operations consists of those cases in which one has to deal with a tunnel of considerable length, or one of complex outline in plan. In such cases it is necessary to resort to trigonometric operations and, by means of a sometimes complex triangulation network, to determine accurately the position of the initial and terminal points of the tunnel axis, orienting it in connection with the general position of the triangulation survey of the locality.

The three groups indicated do not exhaust the entire content of the question of carrying the tunnel axis over the surface of the earth. The complexity of the question and the variety of circumstances affecting the execution of tunnel geodetic operations and their accuracy sometimes make it necessary to resort to combined methods of carrying out the work, introducing into the adopted method of tracing the tunnel elements of traverse surveying or of triangulation operations.

Persons engaged in setting out the axis of a tunnel must satisfy certain special requirements. In addition to specialized knowledge, they must possess, to an increased degree, special personal qualities: they must be precise, strictly attentive, patient, and pedantically consistent in their work.

The instruments used in tunnel tracing must be accurate, portable, as simple as possible, and stable.

Finally, geodetic work for tracing and setting out the axes of tunnels in mountainous localities presents a number of difficulties specific to tunneling, consisting in the following:

In localities surrounded by high mountains, the plumb line is deflected by the attraction of these mountains, as a result of which, in computations, it is necessary to introduce the corresponding correction. During the construction of the Simplon Tunnel for the first time, when the triangulation network was computed, both the error and the misclosure of the network of triangles proved so great that there was no possibility of operating with the data obtained; only after the appropriate corrections for the deflection of the plumb line due to the attraction of the mountains had been introduced and the earlier data had been properly processed did the misclosure prove negligible and confirm the correctness of the work performed. A classic example of introducing a correction for the deflection of the plumb line due to the attraction of mountain masses is found in the work of Prof. Level during the construction of the Lötschberg Tunnel. A detailed description of this work will be given by me in the appropriate place.

A second peculiarity of the execution of work in setting out tunnels is the limited working front. To begin with, in connection with the mountainous character of the terrain, geodetic points have to be selected with special skill, so that they will be clearly visible from other places. This presents enormous difficulties, since such points must be visible from at least three other points, which becomes very difficult to achieve because of the relief of the terrain. In addition, one has to make use of comparatively short bases, since it is seldom possible in any valley to set out a line so long that the base would be several kilometers in length, as is quite often the case in triangulation work.

in flat terrain. Here, however, among mountains one has to be content with a base of only 300–500 and rarely 700 m. This was the case in the construction of a whole series of mountain tunnels.

But the greatest difficulty and responsibility in determining the direction of the tunnel axis on the earth’s surface lies in the correct choice of the method of work: whether to trace without towers (observatories) or to use towers? Where should they be placed? How many towers should there be—one or two? Is triangulation necessary? Perhaps it is possible and expedient to use a polygonal traverse? How should it be arranged? What measures should be taken to obtain the most reliable results with the least expenditure of labor and money? These are the questions that inevitably always arise when determining the direction of the tunnel axis on the earth’s surface, and they require the maximum of knowledge and practical preparation.

Still greater difficulty is encountered in setting out, underground, the axis of a tunnel, the lateral adits (windows), and vertical and inclined shafts. The confined, dark spaces of a tunnel are little favorable for such precise work, which requires calm attention and proper conditions, as does setting out a tunnel underground. A number of questions arise: how is the greatest reliability of the work to be ensured? What are the most rational methods of carrying out the work? What instrument should be used? etc.

Work with geodetic instruments in a tunnel is very often surrounded by such difficult conditions and obstacles that it is very necessary for the success of the setting-out work to foresee them in advance and, if they cannot be eliminated, at least to mitigate their influence somewhat.

Thus one cannot but arrive at the conviction that the difficult and responsible work of assigning the tunnel axis on the earth’s surface by means of triangulation or tracing, and then the actual setting out of the tunnel underground, require detailed and comprehensive study. It is far from always possible to entrust these works to geodesists. Usually engineers, the tunnel builders, have themselves to carry out the setting out of the tunnel both above and below ground. Unfortunately, neither in our Union nor abroad is there a special work devoted to the question of tracing and setting out tunnels. Meanwhile, tunnel construction, developing in our Republic with every year, provides occasion and necessity for the fullest possible familiarization of our engineers with the methods of performing geodetic work in setting out tunnels. Having set myself this aim, in the present work I have processed and systematically set forth data obtained by me from my personal experience in tunnel construction, then collected through personal correspondence with engineers, and, finally, drawn from Russian and foreign literature.

Professor M. I. Evdokimov-Rokotovskii.

Tomsk.

Siberian Technological Institute.

Professor M. I. Evdokimov-Rokotovsky.

Part I. Setting Out the Tunnel Axis on the Surface of the Earth

§ 1. Setting out the tunnel axis directly over the surface of a mountain mass. Here there may be two principal cases, sharply different in the technique of carrying out geodetic work:

I. The tunnel axis is straight,
II. The tunnel axis is curvilinear.

I. The tunnel axis is straight. The setting out of the direction of the axis of a straight tunnel directly across a pass, by its nature, may be divided into the following four types of work:

A. Setting out without the use of an angle-measuring instrument.
B. Setting out with the construction of a tower (observatory) on the pass and with the use of direct sighting directions.
C. Setting out with the construction of a tower (observatory) on the pass, with the use of direct and reverse sighting directions.
D. Setting out with the construction of towers (observatories) at the ends of the tunnel.

A. Setting out without the use of an angle-measuring instrument. The essence of this method consists in the fact that the setting out is carried out as ordinary alignment of a line, all “toward oneself,” in one direction: first in the direction toward one of the tunnel portals (the approach to the tunnel). Then, from the tunnel portal along the surface of the earth along the tunnel axis in the direction toward the pass. Then from the pass to the other tunnel portal and, finally, farther beyond the tunnel on the approaches to the tunnel from the opposite side. This method of work is characterized by the fact that, in such alignment, angle-measuring instruments are not used, and only ranging rods and binoculars are employed.

In this way an enormous number of straight tunnels of insignificant length have been set out. Examples could be numerous and, in essence, are uniform and of little interest.

The question may also be posed somewhat differently: the tunnel portals are given by their position, and it is required to set out the tunnel line between two points. Such a formulation of the question may be encountered both in the practice of railway construction and in the case of the construction of hydraulic-engineering tunnels for utilizing the fall of water and for the installation of a power station. In such cases, the choice of the location for the headworks on one side and the place for installing the turbines on the other side is of decisive importance.

Let it be required to trace the axis of a tunnel from portal to portal (Fig. 1). Setting out a straight line across a mountain presents no difficulty and is carried out by the method known from lower geodesy, by means of three range poles. An experienced technician works with two assistants (foremen). The technician with range pole \(c_1\) takes up a position on the pass so that the signals set up at the tunnel portals are visible. Then the technician sends the foremen to points \(a_1\) and \(b_1\) and sets them in the direction of the signals. After this the foremen, turning to face one another, ask the technician to move to \(c_2\), so that he stands with his range pole on the line \(a_1—b_1\), and in such a way that he does not lose sight of the signals at the portals. All three, gradually moving along the crest of the mountain, obtain the straight axis of the tunnel.

Fig. 1. Tracing the axis of a tunnel from portal to portal by simple ranging across a mountain.

At the points between which it is desired to trace the tunnel, signals are set up in the form of masts up to 7, sometimes up to 10, and even more meters in height. The points should be selected not far from the entrance to the tunnel. Sometimes they are placed directly where the tunnel portal will subsequently be. Usually, however, these points—signals—are placed in front of the portals at a distance of 25 to 50 m, so that sighting marks can later be placed there, indicating exactly the direction of the tunnel axis and serving for setting out the tunnel axis underground. It is very expedient to set up signals at the vertices of the angles of turn of the alignment (Fig. 1).

If, owing to local conditions, three range poles are not sufficient, then in tracing the tunnel between the given points one may use four, five, six, or more range poles.

B. Tracing with the erection of a tower (observatory) on the pass and with the use of direct sighting directions. A tower is erected on the pass and equipped with a powerful theodolite. From such an observatory the direction is traced in both directions: forward, toward one portal, and backward, toward the other portal.

The direction of the tunnel axis corresponding to its true position is preliminarily marked out. For this purpose, as preliminary work, one may resort to:

1) either simple ranging of a line from pole to pole,

2) or ranging of the line by means of a theodolite, based on the azimuth of the direction of the tunnel axis,

3) or a small triangulation. The latter method, with sufficient simplicity, is very convenient and fairly widespread. Its essence...

this method is as follows: on the pass two points A and B (Fig. 2) are selected so that from them points I and II, fixed as the portals of the future tunnel, are visible. Points A and B must not be too far apart and must be located in terrain favorable for measurement. The length AB is measured and taken as a base; all the angles are measured. The triangles are computed. After this, determining the direction of the tunnel axis I—II presents no difficulty and makes it possible to indicate the place where the observatory should be placed.

Fig. 2. Determination of the site for the observatory by means of a small triangulation.

In the work on the Karavanken Tunnel we have an example of laying out the tunnel axis under particularly difficult terrain conditions, by means of an auxiliary triangulation.

The Karavanken Tunnel, 7,976 m long, is straight throughout its entire length. The setting-out of this tunnel was based on an extended network of triangles (Fig. 3). The work was complicated by extremely

Fig. 3. Setting-out of the Karavanken Tunnel.

unfavorable mountainous and wooded terrain conditions. In order to use the idea of a network of triangles it was necessary to know the length of the base. The line AE, laid out along the watershed of the ridge, was chosen as such. However, determining its length by direct measurement, despite the good visibility of one point from the other, proved impossible, and therefore new geodetic work had to be undertaken. The length of the base was determined by means of the polygon ABCDE. The sides of the polygon were measured with great care.

with the aid of a rangefinder. As a result of the work and the computations made, the length of the base was determined as 561.261 m, and thus amounted to barely \(1/11\) of the length of the tunnel. Having the dimension of the base length and the triangulation network that had been surveyed, it was possible to represent graphically the location of the triangulation network and to plot the direction of the tunnel axis. For tracing the axis of the tunnel it remained to choose a site for the observatory on the pass such that the observatory would be in the vertical plane of the tunnel axis and that points I and II would be visible from it.

Point \(B\) proved to satisfy these conditions. Points ON and OS are the observation points of the tunnel. Points \(N_p\) and \(S_p\) are the sites of the tunnel portals.

When, by one means or another, the direction of the tunnel axis has been marked out, then, as stated, a tower (observatory) is erected at the highest point of the mountain on the tunnel axis. From the observatory, and from it alone, a sighting line is taken in both directions, sighting on all the poles set along the tunnel route and even taking in the further course of the line already outside the tunnel.

Examples of setting out by the described method of directing the tunnel axis by means of an observatory installed on a pass are quite numerous.

It was precisely such a case of setting out that occurred during the construction of the Khingan Tunnel, 3078 m long, situated on the Chinese Eastern Railway. Despite the considerable length of the tunnel, the setting-out of the axis on the ground surface was carried out without the aid of triangulation. The preliminary setting-out work was performed in the ordinary way, by means of poles and binoculars. The highest point of the route thus laid out served as the point for erecting the tower (observatory) (Fig. 4). A powerful—

Fig. 4. Setting-out of the Khingan Tunnel.

—transit was installed on the tower. After this they proceeded to the final (new) setting-out of the line, setting themselves the aim of obtaining a straight line over a length of approximately 6–7 km. The setting-out of the Khingan Tunnel line had its own special features. The telescope of the transit dominated the entire ground surface beneath which the tunnel passed. A telephone was installed on the tower, the wires from which were arranged along the route. A foreman with a hand telephone and with poles moved along the line and set the poles according to the instructions transmitted from the tower by telephone. The poles were set on special wooden posts driven into the ground at each picket. This last circumstance is important to note because, as a result, the base of the poles was clearly visible. When the line had been set out by means of the tower, a certain misclosure with the former route was obtained. The angle amounted to about \(1^\circ 48'\), and this was easily absorbed on the nearest curve.

The Lagar-Aul Tunnel at the 143rd verst of the Eastern Amur, now the 1576th verst of the Ussuri Railway, 1250.6 m long, is situated under

the Maly Khingan pass. In laying out the tunnel line, a clearing was made, and age-old firs and pines were cut down. In view of the heavy swampiness, especially in the eastern part of the tunnel, and also in view of the large number of stumps, boards up to 10 m long and about 2.5 cm thick were laid in one row along the axis of the route. The boards were nailed to the stumps, which for this purpose were carefully sawn off. Along this bridge, using a 10-sazhen steel tape that had first been carefully checked, the distance was measured three times. All points of inflection were likewise leveled three times with a Gerlach level. To lay out the direction of the tunnel axis on the summit of the mountain, a special tower was erected on the axis of the line (Photo 5), where, during the work, a theodolite with 20-fold magnification was set up. By means of this instrument, a line about 2.5 km long was accurately sighted and the route was carefully fixed at its characteristic points.

Photo 5. View of the tower (observatory) during the laying out of the Lagar-Aul tunnel.

C. Laying out with the construction of a tower (observatory) on the pass, using direct and reverse sighting directions. As in the preceding case, a tower (observatory) is set up on the pass, and from it, by means of a theodolite, the direction of the tunnel axis is taken forward to one of the tunnel portals and backward; but the entrances to the tunnel (the portals themselves), as well as the entrances to shafts, if such are designated for construction, are not visible from the observatory, and therefore such points, unlike in the preceding case, must be taken by the reverse direction of sighting from distant points laid out instrumentally from the observatory. Thus, the present case may be characterized as laying out from a tower (observatory) forward–backward and backward–forward.

An example may be the setting out of the axis of the Musconetcong Tunnel (Fig. 6). This tunnel, located in the United States of North America, has a length of 1471.9 m and lies entirely on a straight line. Its setting out, owing to the convenient situation of the locality, presented no great difficulties. It was necessary, however, to mark the positions of the portals and the positions of the entrances of two shafts, one vertical and one inclined. For setting out the axis of the tunnel, on the pass, at a point determined by preliminary surveys and situated on the axis of the future tunnel, an observatory was built. From the observatory there was a line of sight for a distance of not less than one and a half kilometers in each direction along the tunnel alignment.

Fig. 6. Setting out of the Musconetcong Tunnel.

Therefore it was fully possible to take a sighting line over a considerable distance. This was done: points were established at a distance of 1400 m in one direction and 1350 m in the other, and from these points the direction of the tunnel was then given and the positions of the entrances of the above-mentioned shafts were determined.

An no less interesting example of setting out is provided by the setting out of the axis of the Totley Tunnel in North America. This tunnel, 5695.8 m long, is situated in such terrain that it was necessary to employ a mixed method (Fig. 7). An observatory was built on the summit of the pass. From this observatory the direction along the tunnel was given for a distance of about 6 km in one direction and about 5 km in the opposite direction.

Fig. 7. Setting out of the Totley Tunnel.

Thus, from this observatory a direction about 11 km long was given. At the ends two more observatories were built. The setting out of the tunnel and the determination of the locations of the portals and of the entrances to the vertical shafts were then carried out from the observatories located at the ends. In the course of the work, on both slopes of the mountain, signals in the form of range poles were set up, four on each side. Signal No. 4 was set up above the portal. Together with signals No. 3 and No. 2 it served as the basis for setting out the entrances both to the tunnel and to the shafts. Signal No. 3 was situated in front of the entrance to the tunnel. From the location of both signal No. 3 and signal No. 4 it was possible...

sight directly into the tunnel itself, which proved to be a convenient and very expedient feature in the work of tracing the Gotthard Tunnel.

D. Tracing with the use of two towers (observatories) at the ends of the tunnel. In very mountainous terrain, and for tunnels of considerable length, it is not possible to carry the direction of the tunnel axis by sighting from only one observatory on the summit of the pass. Under conditions corresponding to the terrain relief and tunnel length described above, there is no possibility of seeing the portal entrances from an observatory. In such cases, two independent observatories are set up at the ends of the tunnel, at some distance from the portals. The construction of an observatory on the mountain summit then becomes unnecessary. The direction of the tunnel axis and the orientation of these observatories relative to one another are determined by means of triangulation, which will be discussed in detail in its proper place. For the present, concentrating attention exclusively on the tracing of the direction of the tunnel axis, let us note that the method described gives the most reliable results and that its application is especially successful for long tunnels in terrain with complex relief.

The examples are few, but characteristic.

When tracing the axis of the Lötschberg Tunnel in Switzerland, 13,735 m long, observatories were set up at the tunnel entrances. In the northern part there was the Kandersteg observatory, and in the southern part the Goppenstein observatory (Fig. 8).

Fig. 8. Tracing of the Lötschberg Tunnel.

Both observatories were oriented relative to one another by means of a triangulation network. At Goppenstein the axial signal was installed somewhat higher than the direction of the sighting axis of the tunnel. The pillar of the observatory is located at a distance of 50 m from the “0” point of the tunnel chainage, and the site for the observatory was selected so that it was possible to sight along the tunnel inside it on a gradient of 3.8‰. Subsequently, because it became necessary to build a road, the Kandersteg axial signal was removed, while the signal on the Goppenstein side still stands to this day. As long as these principal signals of the tunnel geodetic works existed, it was possible to give the direction of the tunnel axis without resorting each time to angular measurements. The distance between the axial signals, computed on the basis of triangulation, was reduced to sea level; however, in the direct measurements the reference was taken not from sea level, but from the horizon of \(\pm 1200\) m. The first computations were then recalculated.

to this horizon, and an elongation of 2.69 m was obtained, as a result of which the distance between the axis signals Kandersteg and Goppenstein became 14318.89 m, at the assumed horizon of 1200 m above sea level.

The distance from the Kandersteg observatory to the Goppenstein observatory is 14331.88 m, at the assumed horizon of 1200 m above sea level.

The length of the tunnel, the distance between the portals, is 13744.22 m, at the horizon of 1200 m above sea level.

The Albula Tunnel has a length of 5886 m and is situated on a straight line. The direction of the tunnel was determined by means of triangulation. When the triangulation works had been completed, observatories were erected on each slope of the ridge (Fig. 9): the “northern observatory” and the “southern observatory.” The positions of the points where the tunnel entrances were to be set out were determined by means of two independent triangulations (Fig. 24), located on both sides of the ridge and based on points of the Swiss state triangulation. By computation, knowing the positions of the northern and southern observatories, the angles between the desired axis of the tunnel and the sides of the triangles adjacent to the observation points were determined.

Fig. 9. Tracing of the Albula Tunnel.

After this they proceeded to set out the tunnel axis. For this purpose, from the northern observatory they sighted a line toward the tunnel and set up a signal at the summit of the pass. Then, from the same observatory, they sighted a line in the direction opposite to the tunnel and directed toward the triangulation point Rugnux, where a signal was likewise set up. Then, from the point Rugnux, by means of a theodolite, they sighted in the direction of the tunnel to the northern observatory and to the point set up on the pass and visible from the northern observatory. Meanwhile, sighting was also being carried out from the southern observatory. On the basis of the triangulation data they took the direction of the tunnel axis and, for the convenience of further work, fixed this direction by placing a signal on the mountain slope on the southern side. Then they sighted in the same direction and set up the signal Giurnel on the mountain summit. From the location of the Giurnel signal, because of the mountainous nature of the terrain, it is impossible to see and sight to the northern observatory; however, the signal Rugnux is very clearly visible. The further work was reduced to checking the correctness of the alignment direction by sighting from Rugnux and from Giurnel, and presented no difficulties. The observatories were used for sighting the direction even for some, albeit very short, extensions inside the tunnel; the further setting-out inside the tunnel was carried out by means of sighting marks established outside the tunnel, verified and periodically checked from time to time from the observatories.

An interesting example of setting out an axis is provided by the Suram Tunnel. Here triangulation was employed. The purpose of the triangulation works was to determine the exact length of the tunnel, the axis of which was marked on the surface by only two points I and II (Fig. 10), with

Fig. 10. Triangulation of the Suram Tunnel.

belonging to two points of the triangulation network, one point on each side of the pass. It turned out, however, that the tunnel direction determined by triangulation was unsuitable, since in the south-eastern part of the ridge the railway line near the tunnel fell into the channel of the Monastery-Tskhali River and, following along it, made its diversion extremely difficult and expensive. It was therefore decided at this place to move the line eastward by 27.74 m. As a consequence, the south-eastern portal of the tunnel was likewise shifted eastward, and thus the entire axis of the tunnel was turned through a certain angle about point I. The distance from point I to point II is 4176.3 m.—Knowing this distance and, as stated, having specified the lateral deviation as 27.74 m, we can easily determine the angle formed by the new direction of the tunnel axis with the direction established by triangulation.

\[ \text{The angle is equal to } \operatorname{tg}\alpha=\frac{27.74}{4176.3},\quad \text{whence } \lg 27.74-\lg 4176.3=\lg\operatorname{tg}\alpha \]

\[ 1.4393327-3.6261361=\overline{7}.8191966 \]

Whence \(\alpha=22'40''\).

Knowing the original distance between the entrance signals and the amount of displacement of the new axis, we can easily compute the length of the tunnel along the new alignment. Therefore there was no need to carry out the triangulation anew. However, it was necessary to give the new direction and mark it on both sides of the pass, by tracing the axis over the mountains across the ridge. This work was performed in the following manner. Over triangulation point I a theodolite is set up, its telescope is directed at the signal erected at triangulation point No. 3, and then the telescope of the theodolite is turned to the right toward the direction of the new tunnel axis.

Fig. 11. Setting out of the Suram Tunnel.

The further setting out of the new tunnel axis was carried out with the theodolite and led to the newly designated portal II. Subsequently, the repeated measurement finally refined the angle of deviation of the alignment, \(\alpha=26'\).

II. The axis of the tunnel is curvilinear. The principal and most widespread method of setting out a curvilinear alignment on the surface of the earth is the method of the described polygon. Because of its simplicity and the accuracy of its results, this method deserves the greatest attention. As is known, it consists in the following: first, office work is carried out and the tunnel alignment is laid out on a contour plan. When the tunnel alignment has been laid out and the radius of curvature determined, as well as the beginning and end of the curve and the tunnel entrances, then the curve is divided into separate parts so that the central angles corresponding to the individual parts of the curve are convenient for set-

of the described polygon. The sides of the described polygon are constructed as tangents corresponding to the individual sections of the curve.

An example may be the tracing of the loop tunnels of the Armavir–Tuapse railway (Fig. 12). There are two directions of the line: from the Armavir side and from the Tuapse side, at different levels. When mutually extended, these two alignments intersect in plan at an imaginary point A, at an angle of \(55^\circ 19'30''\), which was determined by direct measurement with a theodolite. It should not be overlooked that the difference in elevation of the two alignments is about 40 m. Therefore it was necessary to develop the line, for which the device of a circular turn (loop) and loop tunnels was adopted.

Fig. 12. Tracing of three loop tunnels of the Armavir–Tuapse Railway.

Before beginning work at the site of construction of the railway line, the following preliminary office work was carried out. A plan of the terrain was made in contours and, after prolonged investigations, the direction of the line was designed and plotted on the plan. The approximate position of the tunnels, and even their length in general terms, was also indicated. In the course of carrying out the office work, the angles of the turns of the loop and the radii of the curves were determined.

They then proceeded to carry the alignment out on the surface of the ground. They began from the Armavir side, from point A. With a radius of 150 sazhen \(= 320.04\) m, they made a turn to the left through an angle of \(66^\circ 10'\). The curve was set out by the method of the described polygon; for this purpose they ran the line A—1, laid off the tangent, took an angle to the left of \(66^\circ 10'\), again laid off the tangent, and obtained point B.—They laid off a straight line 110 sazhen \(= 236.83\) m long and obtained point C.—Then it was necessary to set out a rather long

a curve of radius 153 sazh. = 326.44 m, with a central angle of 147°6′. As in the preceding case of setting out curve A—1—B, the method of the circumscribed polygon was used. This time, however, the terrain had a more uneven, mountainous character than in setting out curve A—1—B, and therefore the circumscribed polygon had to be broken into small sections. The work was carried out very carefully and accurately. They approached the place where, according to the preliminary design, the portal of the first tunnel was to be constructed. A signal was set at the portal. The setting out of the curve was checked once more and adjusted so that the beginning of the tangent would fall exactly at the point where the signal had been set at the portal of the curved tunnel. For this purpose they marked off a central angle of 35°, while the preceding angle, as a remainder, proved to be 28°13′. Then, along the surface of the ground, using the same method of the circumscribed polygon, they began to set out the tunnel route. They approached the second portal. Again they wished to adjust matters so that the beginning of the tangent would fall at the portal, but they recognized that this was of no importance and, even more, that it might even have a negative significance, since in the course of driving the tunnel the tangent could easily be displaced. Therefore they limited themselves to the fact that only at the first portal did such a coincidence remain—the tangent coincides with the portal; in all other cases they not only did not strive for what on the first occasion had cost much labor, but, on the contrary, they even began to avoid such coincidence.

Meanwhile another party of technicians, headed by the section chief, began work on laying out the route from the Tuapse side. The work began from point K. They set out a curve of radius 320.04 m, central angle 78°19′, then a straight line 309.71 m long, and a second curve of radius 320.04 m, central angle 159°27′30″.

The first party of technicians encountered rather difficult terrain conditions: precipices, cliffs, and a small river forced them to introduce a reverse curve, angle to the left 14°2′. The closing of the two routes proved fairly easy and successful, since the problem was solved quickly and readily thanks to this reverse curve. Here there are two straight inserts, 70.77 m and 59.44 m. They were traced to their mutual intersection at point 7. It remained only to set out here a curve with a central angle of 14°2′, which, of course, presented no difficulties. The introduction of the reverse curve, as has been said, is justified by the situation of the terrain. It is clear that under other circumstances it would have been entirely possible to avoid it. When the closing of both routes had been completed, the setting out of the curve with central angle 159°27′30″ was checked once more. This time a small misclosure of 30″ was obtained, and therefore the angle is equal to 159°27′30″, whereas previously it had been 159°27′.

Fig. 13.

The method of successive chords is the second method of tracing curvilinear tunnels on the surface of the ground. This method, as is known (Fig. 13), consists in the following: on the extension of the straight line the quantity \(ab\) is laid off, and then from point \(b\), at a right angle, the quantity \(bb_1\). Then the straight line \(ab_1\) is the chord \(S\).

\[ S = 2R \sin \frac{\varphi}{2} \]

The method under consideration is especially convenient in difficult-to-traverse, narrow and cramped localities.

The method of successive chords is sometimes applied even for complex setting-out of curvilinear routes. An excellent example characterizing the merits of this method is the setting-out of the “Cuchillo” section on the Pacific Railroad in the United States of North America (Fig. 14). Here, over a short section, the line rises steeply into the mountains by approximately 150 m in elevation. Therefore it was necessary to develop the line over a length of 6 km and to give the route a very complex configuration. The tracing of this entire section and of all the tunnels located there was performed by the method of successive chords 20 m long. Such a small chord length was chosen because the traversed section was extremely mountainous, cut up by outcrops of basalt. This made it impossible to apply any other method of geodetic work here for the tracing. To check the route, the azimuths of the line were periodically counted and determined in the field. In addition, the angles of intersection of the extensions of the straight sections of the line route (the tangents between curves) were determined. The results of all the geodetic work on this difficult section proved entirely satisfactory, which attests to the correctness of the choice of method and the correctness of the execution of the tracing work.

§ 2. Determination of the direction of the tunnel axis by means of polygonal survey. Fears about the correctness of the route direction may arise whenever, in the case of very precipitous, poorly accessible terrain, one has to work while suspended on ropes, in cradles, etc. In such cases it is very easy to deviate from the direction of the route to the side and, as is sometimes said in such cases, to “break” the axis of the tunnel. The matter is no better with the measurement of the tunnel length from portal to portal. In such cases, to determine the direction of the tunnel axis and its length, recourse is had to polygonal survey, concentrating the work in terrain accessible for geodetic operations. The polygonal traverse is laid out so that the ends of the polygon adjoin points located at the entrance to the tunnel, strictly on the tunnel axis. In plan, the entire polygon should represent a broken straight line, and in such a way that the sides of the polygon are of the greatest possible length. When the polygonal traverse has been laid out, it is fixed at the angles. During work at the angles, either a theodolite is set up, strictly centered over the given point, or else a vertically set ranging rod is erected, at which the theodolite is sighted from the neighboring point. The measurement of the sides of the polygon and the measurement of the angles must be carried out with special care, on the grounds that the polygon remains unclosed, and therefore its check becomes impossible until all tunnel works have been completed. Thus, the entire responsibility for the work of determining the direction of the tunnel axis is directly connected with the accuracy of measuring the sides and with the accuracy of measuring the angles. When the work of surveying the polygon has been completed, it is drawn, the directions of the tunnel axis are continued, and the angles formed by the direction of this axis with the initial and final sides of the polygon are computed. This gives us the possibility of setting out instrumentally, by setting up alignments, the direction of the tunnel axis, and also of calculating the length of the tunnel. The fewer the angles and sides in the polygon, the fewer, generally speaking, the measurements, i.e. the fewer the sources of accumulation of measurement errors. The number of angles will be the smaller, the longer each side is and the closer the general direction of the polygonal—

Fig. 14. Tracing of the Cuchillo tunnel section on the Pacific Railroad by the method of successive chords 20 m long.

new traverse to the straight line connecting the initial and terminal points of the traverse, located at the tunnel entrances.

The precision with which the results of measuring the length of a traverse side are to be given, and with which the rectangular, rectilinear coordinates of the traverse points are to be computed, must be expressed in hundredths of a meter (centimeters).

The same procedure is followed when computing coordinates from their increments.

The transverse linear deviation of the terminal point of the traverse may be computed by the formula

\[ m_0 = \pm S \cdot \xi \sqrt{\frac{2n^3 + 3n^2 + n}{6}} \]

where

\(S\) is the mean length of a traverse side (in deriving the formula it is assumed that the traverse consists of \(n\) sides, equal in their length \(S\), so that the length of the traverse is \(\Sigma = n \cdot S\)).

\(n\) is the number of sides of the traverse.

\(\pm \xi\) is the mean error made in measuring each angle of the traverse.

The accuracy of sighting with a theodolite may, according to Stampfer’s investigation, be taken as equal to the quotient obtained by dividing the accuracy of sighting with a diopter by the magnification of the telescope. In finding the transverse error of a traverse, one should not lose sight of the fact that

1) the longer the side, the greater the accidental errors in measuring it:

\[ m_s = \pm \mu \sqrt{S} \]

Therefore the lengths of the traverse sides must not exceed a known limit. Prof. Jordan establishes such a limit for traverses at 300 meters.

2) In an elongated traverse, when the angles are close to \(180^\circ\), centering errors have a stronger effect on them.

In view of the fact that the traverse, as stated, is not closed (“open”), it is necessary to make two traverses (out and back), using, if only possible, the same vertices of the traverse. Usually, in view of the responsible nature of the work, a traverse is run by measuring the sides and angles an even number of times—4, and sometimes even 8 times—and the mean is taken.

In order to be able to plot the surveyed traverse on paper, use is made of rectangular rectilinear coordinates. Each point (vertex) of the traverse is determined by two distances to the west–east line passing through the origin of coordinates perpendicularly to the meridian. Along the meridian the axis of abscissas (the \(X\)-axis) is directed positive \((+)\) to the north; in the perpendicular direction the axis of ordinates (the \(Y\)-axis) is positive \((+)\) to the east. Sometimes, however, one of the sides of the traverse is taken as the \(X\)-axis, for example the side adjoining the tunnel portal.

The computation of the coordinates of the vertices of the traverse by which the tunnel has been bypassed is preceded by:

1) Comparison of the measured sum of the interior angles with the corresponding theoretical sum \(180^\circ(n-2)\), where \(n\) is the number of sides (angles) of the traverse.

2) Distribution of the resulting difference. If it is less than three times the accuracy of the vernier, multiplied by the square root of twice the number

angles, the setting-out is carried out on the angles with the shorter (both) sides.

An example of determining the direction of a tunnel axis by means of a traverse survey may be the derivation tunnel of the Adjaris-Tskhali hydroelectric station. The tunnel, 490.5 m long, was driven in the cliffs of the right bank of the Adjaris-Tskhali River. The height of the almost sheer mountain mass reaches 220 and more meters above the water horizon. Thus the working conditions were very unfavorable. Initially the derivation tunnel was intended to be straight. Subsequently, the tunnel was given a curvilinear outline in its entrance part. However, this did not have a substantial effect on the setting-out of the tunnel, since the tunnel was set out as a straight one: in the entrance part, where the turn was subsequently made, a sighting adit had been driven. This adit, later abandoned and even filled with rock, was used for setting out the tunnel underground. To determine the direction of the tunnel axis, a traverse survey was used. The traverse was laid out along the bank of the Adjaris-Tskhali River, which is flooded by high waters (floods) (Fig. 15). The attached table gives the data relating to the traverse survey.

Table 1.

No.	Angles	Length of traverse side, m	No.	Angles	Length of traverse side, m
No. 0	—	86.65	No. 6	160° 08′	77.02
No. 1	125° 52′	64.86	No. 7	147° 19′	150.03
No. 2	231° 49′	177.73	No. 8	190° 18′	64.01
No. 3	146° 53′	120.00	No. 9	130° 28′	19.03
No. 4	155° 22′	37.50	No. 10	128° 26′	27.74
No. 5	124° 50′	47.65	No. 11	—

Having traversed from point No. 0 to point No. 11, i.e., from one alignment to the other, having measured all the sides of the traverse and determined all its angles, the traverse was plotted and the position of the tunnel axis was calculated analytically. For this purpose the coordinates of point No. 11 were determined, taking as the coordinate axis the direction of the first side from point No. 0 to No. 1 and placing the origin of coordinates at point No. 0. The angles No. 0 and No. 11, formed by the direction of the tunnel axis with the first and last sides of the traverse, were unknown. The whole problem reduced to determining them. The distance AB, i.e., the distance from point No. 0 to point No. 11, was also unknown. The traverse was open. In view of this, the magnitudes of angle No. 0 and angle No. 11, and the length AB, were determined by calculation. The attached table gives all the data for the analytical calculation and the calculation itself for determining the coordinates of point No. 11, located at the entrance portal.

Fig. 15. Determination of the direction of the axes of the tunnels of the Adjaris-Tskali hydroelectric station.

As can be seen from the table, the coordinates of point No. 11 are:

\[ X = 217.09\ \text{m},\quad Y = 450.44\ \text{m}. \]

The angle formed by the direction of the tunnel axis with the side No. 0—No. 1 is equal to:

\[ \tg \alpha_0 = \frac{Y}{X} = \tg 64^\circ 16' 7'' \]

\[ \alpha_0 = 64^\circ 16' 7''. \]

The angle formed by the direction of the tunnel axis with the side No. 10—No. 11 is equal to:

\[ \alpha_{11} = 2 \cdot 90^\circ (11 - 1)\ \text{minus all known angles of the polygon}: \]

\[ \alpha_{11} = 143^\circ 18' 53'' \]

The length \(AB\) is equal to

\[ AB = \sqrt{X^2 + Y^2} = 500.01\ \text{m}. \]

Subsequently the length \(AB\) was measured in the field and was found to be 499.97 m. Thus a difference in length of 0.04 m was obtained.

Let us compute, for the case of the polygonal traverse of the Adzharis-Tskhal tunnel, the transverse linear deviation.

The number of sides of the polygon is \(n = 11\).

The total length of the polygon is \(\Sigma a = 872.22\ \text{m} = n \cdot S\). The mean length of a side of the polygon is \(S = 79.293\ \text{m}\).

Therefore the limit of possible error, according to the formula given above, will be

\[ m_0 = \pm 79.293 \xi \sqrt{\frac{2 \cdot 11^3 + 3 \cdot 11^2 + 11}{6}} \]

A 10″ theodolite was used at a magnification of 20 times. Therefore the accuracy of sighting with the telescope is equal to

\[ \xi = \pm 30'' \]

\[ m_0 = \pm 79.293 \cdot 30'' \sqrt{\frac{3036}{6}} \]

\[ = 79.293 \cdot 30'' \cdot 22.494. \]

\[ \lg m_0 = 1.8992348 + 6.1626961 + 1.3520667 \]

\[ = 1.4139976 \]

Whence

\[ m_0 = \pm 0.259\ \text{m}. \]

On the narrow-gauge Albulskaya railway line, between the stations of St. Moritz and Thusis, there is a spiral tunnel 676 m long (Fig. 16). In plan the tunnel is laid out on curves of radii 120 m, 300 m, and again 120 m. The tracing of the tunnel was carried out as follows: a contour plan of the locality was prepared. Then the designed direction of the axis of the line and of both approaches to it was plotted on the plan. The straight sections located on the approaches to the spiral tunnel and lying at different levels were extended on the plan to their mutual intersection at point \(A\). The angle obtained was \(15^\circ 32'\). All this work was transferred to the site of construction of the tunnel and carried out in full. The angle at point \(A\) was measured, and it was found to be equal to \(\alpha = 15^\circ 32'\). Then, at the site of construction of the tunnel, the above-mentioned straight sections, ly—

Table II

No. of polygon angle	Polygon side “a”, m	Angle β	log “a”	log Cos β	log Sin β	log “a” Cos β	log “a” Sin β	a Cos β	a Sin β	Coordinate Xₙ	Coordinate Yₙ
0	86,65	0°	1,9377686	0	—	1,9377686	0	86,65	0	86,65	0
1	64,86	54°8′	1,8119769	9,7678242	9,9086901	1,5798011	1,7206670	38,02	52,56	124,67	52,56
2	177,73	2°19′	2,2497607	9,9864149	8,6066226	2,2494056	0,8563833	177,59	7,18	302,26	59,74
3	120,00	35°26′	2,0791812	9,9110460	9,7632417	1,9902272	1,8424259	97,78	69,57	400,04	129,31
4	37,50	60°4′	1,5740313	9,7980936	9,938220	1,2721249	1,5118533	18,71	32,50	418,75	161,81
5	47,65	115°11′	1,6780629	9,6227211	9,9564466	1,3077840	1,6345095	−20,31	43,10	398,44	204,91
6	77,02	134°6′	1,8866035	9,8425518	9,8562008	1,7291583	1,7428043	−53,60	55,31	344,84	260,22
7	150,03	116°47′	2,1761781	9,6538084	9,9507138	1,8299865	2,1268919	−67,61	133,94	277,23	394,16
8	64,01	106°29′	1,8062478	9,4529151	9,9817744	1,2591629	1,7880222	−18,16	61,38	259,07	455,54
9	19,03	156°1′	1,2794388	9,9607864	9,6090294	1,2402252	0,8884682	−17,39	7,74	241,68	463,28
10	27,74	207°35′	1,4431065	9,9475995	9,6656168	1,3907060	1,1087283	−24,59	−12,84	217,09	450,44
11

…lying, as stated, at different levels, were connected with one another by a network of triangles, whose sides were determined. Then the office computations were again undertaken, and the position of the designated straight lines was computed, as well as their position relative to one another; in this way the angle at which they intersect one another was computed. It proved to be equal to \(15^\circ 32'\). After this, the points of the beginning and end of that complex curve of different radii on which the spiral tunnel was to lie were finally checked, computed, and fixed on the ground. These points are designated on the drawing as B and C. The computation of the angle was reduced to the following. If one proceeds from point C, then the first curve

Fig. 16. Determination of the direction of the axis of the spiral tunnel on the Albulian railway line.

of radius 120 m has the central angle \(\beta = 109^\circ 38'\). Then the curve of radius 300 m makes a further turn through the angle \(\gamma = 25^\circ 18'\), and, finally, the third curve of radius 120 m makes another turn through the angle \(\delta = 240^\circ 36'\) and leads to point B. Thus the angle formed by the straight lines \(O_1C\) and \(O_3B\) is equal to

\[ 109^\circ 38' + 25^\circ 18' + 240^\circ 36' = 375^\circ 32', \]

i.e.

\[ 375^\circ 32' - 360^\circ = 15^\circ 32'. \]

Such is precisely the angle \(\alpha\) at vertex A, whose sides are perpendicular to \(O_1C\) and \(O_3B\). When points B and C had been determined directly on the ground, the setting out of the tunnel axis underground presented no special difficulties and was carried out from both portals.

The bottom tunnel of the Adjaris-Tskhali hydroelectric station has a length of 185.91 m and is situated on two curves directed in different directions (Fig. 17). The difficulty of setting out was due to the ruggedness of the terrain and the inaccessibility of the surface above the tunnel for direct tracing of the tunnel axis. Therefore the polygonal traverse method was used. In view of the fact that the bottom tunnel is located in the cliffs of the right bank, in the same place as the diversion tun—

nel, whose axis had likewise been determined by means of a polygonal traverse; four sides of the already laid-out polygon were used, and the axis of the bottom tunnel was oriented with respect precisely to this part of the polygon. In Fig. 16, angles Nos. 8, 9, 10, and 11 belong to the existing polygonal traverse. The work consisted in determining the direction of all straight inserts on the axis of the bottom tunnel and in orienting these directions with respect to the polygon, with angular adjustment by the analytical method.

The direction of the entrance, shown on the drawing by an arrow, was specified and, according to measurement, proved to be at an angle of \(65^\circ 57'\) to the side of the polygon No. 10—No. 11, at a distance of \(12.16\ \mathrm{m}\) from point No. 10.

The direction of the outlet channel from the tunnel was not governed by any precisely expressed requirement and was marked only approximately.

Fig. 17. Determination of the direction of the axis of the bottom tunnel of the Adjaris-Tskhali hydroelectric station (compare with Fig. 14).

Geodetic work was begun from the entrance part of the tunnel. Before proceeding to the geodetic work, the following office work was carried out. A polygon was drawn, and the direction of the entrance part of the tunnel was plotted. Then they proceeded to represent on the plan the direction of the tunnel axis. Here it was necessary to meet very substantial requirements, determined by the geological structure of the mountain massif. In connection with the fissuring of the cliff in its outer part, it was necessary to go deeper with the tunnel into the massif and to keep to those places where, according to the assumptions of the geologists, the rock was expected to be sufficiently strong and undisturbed.

On the basis of these considerations the tunnel was drawn in plan. The lines of the straight inserts were extended until they intersected the polygon.

The third straight line (at the entrance part), on being extended, when intersecting the line No. 11-B (i.e., the line of the direction of the axis of the diversion tun-

of the tunnel gave point C. This same third straight line, when intersecting the side of polygon No. 8—No. 7, gave point E.

The second straight line, in its extension, when intersecting the same line No. 11—B, gave point B, and when intersecting the side of polygon No. 7—No. 8 gave point D.

The first straight line, in its extension, when intersecting line BD, gave point A.

Then the positions of points A, B, C, D, and E were determined analytically, computing both distances and angles.

Knowing the position of points C and E and the angle formed by line CE with the side of polygon No. 8—No. 7, it was easy to set out and fix on the ground, by means of a sight line, the direction of the exit portion of the bottom adit. The angle formed by this direction with the side of polygon No. 8—No. 7 proved to be equal to 48°. Finally, they proceeded to set out the tunnel axis underground. The work was carried out from the upper and lower ends of the bottom adit. Knowing the position of points A and B, they determined without difficulty the directions of the first and second straight inserts located inside the tunnel.

Fig. 18. Tracing of the tunnel at km 90 of the Circum-Baikal Railway.

An example no less interesting is provided by the determination of the direction of the tunnel axis at km 90 of the Circum-Baikal Railway, now the Chita–Trans-Baikal Railway. Here, by vertical cliffs near Berezovaya Bay, the shores of Lake Baikal are deeply indented (Fig. 18). In bypassing the bridge and the single-track gallery, when reconstructing the road as double-track, it was neces-

...it was necessary to drive a tunnel 491.68 m long. The tunnel has three sections on curves of radius 320.04 m and two straight inserts. It is self-evident that tracing the tunnel line on the ground surface, over the cliffs, was practically impossible; and technically, from the standpoint of geodetic work, it was not even required.

It was decided to proceed so that at each portal the direction of the tunnel axis would be set exactly. This was the extent of the work on the ground surface. All further geodetic work for the tunnel was to be carried out exclusively underground. To determine the directions of the tunnel axis at both portals, recourse was had to a polygonal survey.

The method of tracing the axis of a tunnel by means of a traverse is used quite often for determining the direction of the axes of tunnels of urban underground railways and metropolitan railways. Tracing the axis of such tunnels by any other method is, in most cases, owing to the built-up condition of the urban area and its limited accessibility for work, usually extremely difficult and even impossible. When the method of tracing by a polygonal traverse is applied, the tunnel axis is not traced continuously along its entire length, but is fixed only at several separate points selected according to local conditions. The traverse is laid along streets, across squares, in parks, etc., going around buildings. As an example of determining the direction of a tunnel axis by the method described, a small section of the London Underground Railway may serve. In Fig. 19 three points are given on the tunnel axis, A, O, and J, traced by means of two mutually independent traverses. The first traverse, from A to O, has the side AB 355.06 m long. From vertex B the ver-

Fig. 19. Tracing of the axis of a tunnel of the London urban underground railway.

vertex O. The angle was measured and proved to be equal to \(114^\circ 9' 45''\). However, it was not possible to hang the polygon side \(BO\) and measure it along its entire length with a steel tape. With hanging and measurement it was possible to reach only point “a,” and then, owing to obstacles in the form of buildings, it was necessary to deviate from the polygonal traverse to the side and to pass along the street by the traverse \(a — b — c — O\). The computation carried out gave the direction of the tunnel axis—the angles \(BAO\) and \(AOB\)—and the length \(AO\). Knowing the angles of the direction of the tunnel axis, they confidently set out the line \(AO\) with a theodolite, and along the extension of this line, at various arbitrarily chosen places on the axis of the future tunnel, boreholes 1, 2, 3, and 4 were laid out. When setting out the tunnel underground through these boreholes, plumb lines were subsequently lowered, and thus they served as orientation points.

When it was necessary to trace the tunnel farther from \(O\) to \(J\), the following circumstance was immediately encountered. Point \(O\), very convenient for the preceding work because of its relatively high position, proved entirely unsuitable and inconvenient for the subsequent work. A new point \(D\) had to be chosen as the vertex of a new polygon \(DEFGHJ\), laid out along the streets of London. The sides of the polygon were measured and the angles determined.

§ 3. Tunnel triangulation. As has already been indicated above, the determination of the direction of the axis of a tunnel of more or less considerable length in mountainous terrain is carried out by means of tunnel triangulation. The use of a trigonometric network in tunnel work has the advantage that, by determining a system of principal control points of the survey, evenly distributed over the whole area being surveyed at a considerable distance from one another, and then carrying out a detailed survey between the control points—thus proceeding by a gradual transition from the general to the particular—we avoid the gross accumulation of errors, the unavoidable inaccuracies that result from polygonal, or generally geometric, surveying; of great importance here, of course, is the fact that angles can be measured with much greater precision than lines. Furthermore, in triangulation work it becomes possible to take into account the curvature of the earth’s surface as a part of the geoid, which has a very substantial effect on the length of the tunnel. Finally, in tunnel triangulations the deflection of plumb lines by the attraction of mountains is taken into account.

Types of tunnel triangulations. According to the general arrangement of the triangles, tunnel triangulations are distinguished as being in the form of a network and in the form of a chain. The former consist of a group of triangles filling the entire space of the tunnel section. The latter constitute a row of triangles extended in one direction. Usually, in the practice of carrying out tunnel triangulations, networks are encountered most often. As for the arrangement of triangles in the form of a chain, this type is encountered when setting out a number of tunnels following one after another at an insignificant distance, and also when it is necessary to determine the position of lateral galleries (adits) of vertical or inclined shafts, etc.

The two indicated typical arrangements of triangles in tunnel triangulation—whether in the form of a simple network or of a chain—are in practice complicated by the overlapping of triangles one upon another, then by the insertion of separate points by means of lateral and forward intersection and, finally, by the solution of the well-known Hansen and Pothenot problems.

An example may be the triangulation of the tunnels on the Davos–Filisur line (Fig. 20), in Switzerland, in the canton of Graubünden.

Here, over a distance of 4.2 km, there are 6 tunnels with a total length of about 3724 m, and a considerable extent of the tunnels is situated on curves of radii 400, 300, 250, 200, and even 150 and 120 m. Therefore, the setting out of the axes of all these tunnels, owing to their number and to the complexity of their layout in plan, presented considerable difficulties. Foreseeing all these difficulties, it was decided to resort to triangulation. Now, when the work has been completed and all the tunnels have been driven through, it must be acknowledged that the use of triangulation was the only possible way to give the correct setting out of the tunnels. A great convenience was that it proved possible to make use of the network of the state triangulation of class II. In all, 13 points of the class-II network were included.

An example of the arrangement of tunnel triangulation in the form of a chain may likewise be furnished by the triangulation of the tunnel of the Alb hydroelectric station (Fig. 21). The tunnel, 1416 m long, has four straight sections and three curved ones, of radius 200 m. In selecting the points of the triangulation network, the valley of the Alb River was used, and the network points were placed on the surrounding elevations and cliffs. The entire network, as is apparent from the appended drawing, consists of fifteen triangles arranged in the form of a chain. Network points No. 1, No. 2, No. 4, and No. 6 at one end of the tunnel (the entrance head of the hydraulic-engineering tunnel), and then network points No. 16 and No. 17 at the other end (the exit part of the tunnel), were chosen so that they completely determine the initial and final direction of the tunnel axis. The lengths of the sides of the network range from 80 to 280 m. In view of the fact that, for the development of the working face in driving the tunnel, it was necessary to construct four horizontally arranged—

Fig. 20. Triangulation of tunnels on the Davos–Filisur line.

...side adits (windows) had been laid out, then in order to determine their directions BCA, ED, FH, KL it was necessary to resort to additional geodetic work: it was necessary to determine the position of points which, at the time the tunnel triangulation was carried out, had not been observed at all. For this, as is known (Pothenot’s problem), observations must be made from such a point and the angles between the old trigonometric marks visible from it determined. There was also a need to determine the position of a new point from which only two points of the tunnel triangulation are visible (Hansen’s problem).

Far more often, as has already been said, tunnel triangulations are in the form of a network. In this respect quite a number of examples could be cited.

To characterize triangulations of this kind it will suffice to indicate the triangulations of the Lötschberg, Suram, St. Gotthard, Simplon, Vosges, Bosruck, and Tauern tunnels.

The triangulation of the Lötschberg tunnel (Fig. 22), carried out by the Swiss surveyor Prof. Bäschlin, is based on a network of state triangulation of classes II and III. Six points of this network were taken: First, Birre, Gellihorn, Steghorn, Altels, Hockenhorn. Therefore the base was not measured. The total length of the tunnel according to the first basic design was 13,744 m. Preliminary signals Kandersteg and Goppenstein were established at the portals, and three additional points were taken specially for the tunnel triangulation: Torrenthorn, Rivengrat, and Stritengrat. Thus a network of eleven points was obtained.

On July 24, 1908, a catastrophe occurred: along the alluvial deposits of the Kander River, water broke through into the tunnel. Therefore part of the driven adit had to be abandoned and the line deflected to the north. Instead of a straight alignment, a design was prepared for a curved tunnel 14,535.86 m long. The radii of curvature were 1,100 m. At breakthrough, the discrepancy in the length of the two headings amounted to only −0.41 m.

The triangulation of the Vosges tunnel, 6.3 km long, is very simple (Fig. 23).

The initial side of the network I—II, which served as a baseline, made it possible to determine the direction of the tunnel axis and to compute its length, and then to determine the position of points 1 and 2, laid out precisely on the tunnel axis on the pass. In all, the triangulation network of the Vosges Tunnel, in its general complexity, had seven points, of which 4 lay on the tunnel axis. Unfortunately, despite its simplicity, the triangulation network of this

Fig. 22. Triangulation network of the Leberg Tunnel.

tunnel was nevertheless not chosen quite successfully: the triangles of the network turned out elongated, with acute angles. It is possible that this played a certain role: the discrepancy in the breakthrough of the tunnel amounted to 0.76 m.

Fig. 23. Triangulation of the Vosges Tunnel.

Sometimes tunnel triangulation may rely to such a considerable extent on points of the state triangulation network that the need even disappears to have a continuous network of triangles of the tunnel triangulation along the entire length of the tunnel, and then the task of determi—

the direction of the tunnel axis is reduced to making full use of the points of the network of the existing state triangulation, as a system of control points, and to determining the position of the terminal points of the tunnel. For this purpose it is necessary to carry out triangulation work only at the terminal points of the tunnel.

An example may be the determination of the direction of the axis of the Albulian tunnel, 5866 m long, of a narrow-gauge railway in Switzerland (Fig. 24). Here it proved very successful to make use of the network of the Swiss state triangulation. When, on the basis of survey data and in accordance with geological conditions, the initial and final points (portals) of the tunnel had been marked out, these points were fixed as the “northern observatory” and the “southern observatory.” In addition to these two observatory points, two more were established—one on each side—points which served for work in setting out the tunnel axis on the surface. All four new points—two points on each side—were determined by two special triangulations, carried out independently from each side of the Albula ridge. Each triangulation was connected with the points of the state triangulation. The bases were not measured. By computation, the angles were determined between the required tunnel axis and the sides of the adjacent triangles converging at the observatory points.

Fig. 24. Determination of the direction of the axis of the Albula tunnel.

An equally interesting example is furnished by the setting out of the axis of the Wasserfluh tunnel on the railway line Toggenburg—Lake Constance. The tunnel, 3557.16 m long, single-track, is situated on a gradient of 10.4‰, and in its eastern part on a curve of radius 400 m (Fig. 25). A sighting adit was not driven *). The tracing of the Wasserfluh tunnel was carried out with the aid of two tunnel triangulations independent of one another. Two points of each network of tunnel triangulations were successfully included in the network of the state triangulation of the III class, and this made it possible for both networks of tunnel triangulations to co—

*) In explanation of this it should be noted: the Wasserfluh tunnel is single-track in the direction from east to west. In order to avoid difficulties in dealing with water, the drainage adits were driven according to the plan of work for about 2.5 km from the western portal. In fact, 2419.55 m were driven from the western portal and 936.61 m from the eastern. Here the two drives of the lower pilot adits met. The western section, about 2.5 km long, as stated, lay wholly on a straight line, and its setting out presented no particular difficulties. The eastern section, however, was of comparatively small length, and therefore the sighting adit was not driven. The results of the breakthrough indicated the correctness of the adopted method and the accuracy of the work performed: the misclosure along the tunnel was +0.28 m, the lateral deviation 0.05 m, and in height 0.01 m.

to coordinate them with respect to one another and thus to determine accurately the direction of the tunnel. The points of the tunnel-triangulation network that were included in the state triangulation network were Lichtensteig and signal No. 1 Gruben—on the western side—and Neu Toggenburg and Köbelisberg—on the eastern side. The distances between these points were taken as known and played the role of the principal sides (bases) of the corresponding tunnel triangulations. When the direction of the axis of the future tunnel had been determined by office computation and definitively established, work began on determining in the field the positions of the points located on the tunnel axis, namely: Toggenburg on the summit of the pass above the tunnel, then Gruben on the western slope and Oberberg on the eastern slope of the neighboring mountains. This was accomplished entirely by means of the corresponding, as stated, mutually independent tunnel triangulations. Then, by means of these same tunnel triangulations, it was possible accurately to set out at the work sites the pre-portal points from which the setting-out of the tunnel axis underground was carried out.

Fig. 25. Determination of the direction of the axis of the Wasserfluh Tunnel.

Choice of the system for arranging the points of the tunnel-triangulation network. For preliminary considerations it is very expedient to use a map of as large a scale as possible. Taking account of the relief conditions of the locality, the network points are preliminarily assigned, as office work, so that:

1. The points located at both entrances to the tunnel on the tunnel axis can be sighted from three different points of the triangulation network.

2. The traverse over the mountain massif under which the tunnel is located can be carried out with the smallest possible number of intermediate points of the tunnel triangulation.

3. The triangles of the tunnel-triangulation network should, as far as possible, be equilateral. The length of a side is recommended to be within the limits of 150 to 600 m.

1. It is desirable that the tunnel-triangulation network be connected, at several points—at least two—to the existing network of the state triangulation. This will make it possible to dispense with a special base for the tunnel triangulation.

2. In the case of a long tunnel, it is desirable that the tunnel-triangulation network consist of two independent triangulations, adjusted with one another and having common points located at the tunnel portals.

In this respect, an excellent example is furnished by the triangulation of the St. Gotthard Tunnel (Fig. 26). The direction of the tunnel axis was determined by means of two independent triangulations, carried out by the two surveyors Gelpke and Koppe, quite independently of one another and using different triangulation points selected by them. Gelpke established a triangulation network having 17 points and one base. The base was selected in the valley and had a length of about 3 km. Koppe had an entirely separate triangulation network, having 13 points. No base was measured. For the computation of his network, Koppe tied in to points of Gelpke’s triangulation. After comparison of the two triangulations, the difference between them proved to be 40 millimeters, with agreement at the middle of the tunnel.

Fig. 26. Triangulation network of the St. Gotthard Tunnel, 14,984 m long (Koppe’s network).

When, in the course of office work, the preliminary positions of the points of the tunnel-triangulation network have been laid out, a reconnaissance of the network must be carried out by direct inspection of the locations of the points in the mountains.

With the participation of a local inhabitant familiar with the mountain paths, the person determining the direction of the tunnel axis on the surface of the earth (the triangulator) must visit all points of the proposed network and definitively mark these points. In doing so he must be guided by the following provisions:

1. The points must be selected so that angular measurements can, as far as possible, be carried out from the ground and so that complicated signals need not be erected.

2. From each point, not fewer than two adjacent points must be visible.

3. The angles in successive triangles must not be less than \(40^\circ\) or greater than \(100^\circ\).

4. Access to the points of the tunnel triangulation must not be too difficult.

5. In accordance with the results of the reconnaissance, all corrections are entered into the initially prepared network design. At the same time it is determined what artificial triangulation signals must be constructed, and where.

Tunnel triangulations, by their character, may be equated with triangulations of class III and IV.

Base. For the successive computation of the sides of the triangles of a triangulation network, it is necessary to know precisely the length of one side, called the initial or principal side. Sometimes, when carrying out tunnel triangulation, it is possible to make use of an existing triangulation network. Then the question is solved quite simply: it is only necessary to take two adjacent points of the existing triangulation, connect them with the points of the tunnel triangulation, and then the computation of the sides of the triangles of the entire tunnel-triangulation network presents no difficulties.

The matter is more complicated when there is no possibility of making use of data relating to an existing triangulation network. In such cases one must determine by direct measurement the length of a side called the base.

Sometimes the direct measurement of a side of the triangulation network, owing to the very mountainous character of the locality, becomes so difficult to carry out and so fraught with inaccuracies of measurement that, in order to determine the length of the principal side, recourse is had to the mediation of a so-called base network with a measured small base. For this purpose, a line is selected on the ground that is convenient for direct measurement and is situated entirely independently of the triangulation network; the length of this base is measured, and by successive gradual enlargement of the sides of the triangles (the base network) one passes from the small base to the principal side of the tunnel triangulation network.

The form of the base network may be of any degree of complexity.

Sometimes the base network is rhombic (proposed by Schwerd). The small base intersects a side of the network in such a way that both of these lines become the diagonals of a rhombus. All 8 angles are measured. It is desirable that the ratio of the length of the principal side of the network to the length of the small base not exceed 4. Sometimes the base network is more complex. To pass from the small base to the principal side, it is necessary to make a double transition: not one rhombus but two are constructed; the small base is the small diagonal of the smaller rhombus, and the large diagonal of the large (second) rhombus is the principal side of the network. In Fig. 27 a variant of determining the direction of the axis of the Rickentunnel is presented, as an example of a complex rhombic base network.

Selection of a place for the base. The selection of a place for the base in tunnel triangulation is an important and difficult matter. It is necessary to choose a level section, accessible for direct measurement and possessing the greatest possible length.

Unfortunately, when determining the axis of tunnels, in work in mountains, favorable local conditions are far from always present: a level

site, even if one is found, is not situated on level ground, but on a mountain slope. Often the severe ruggedness and mountainous character restrict the site to very small dimensions: the base must be taken as 600–400, and sometimes even fewer, meters in length.

As an exception one may encounter tunnel triangulations in which the selected base could not in fact be measured directly, and it was necessary to resort to small supplementary triangulations (a base network), and sometimes to the aid of a polygonal traverse. A fine example of the polygonal method for determining the length of a base is provided by the Karawanken Tunnel.

The Karawanken tunnel, 7,976 m long, built in 1902–1906, is situated between Rosenbach station and Assling station in Austria and crosses the Karawanken mountain range in the Eastern Alps. The bare, pale-red limestone chain, cut by wild gorges and presenting a majestic spectacle, created extraordinary difficulties in setting out the tunnel. In determining the direction of the axis of this tunnel, triangulation was used. The base was

Fig. 27. Variant for determining the direction of the axis of the Rikken Tunnel.

laid out and marked along the watershed ridge (Fig. 3). Unfortunately, under local conditions it proved impossible to determine the length of the base by direct measurement. Its length was determined with the aid of a polygonal survey. However, this method too encountered extraordinary difficulties: when the polygon ABCDE was laid out, it became clear that there was absolutely no possibility of determining the lengths of the sides of the polygon by direct measurement. There was no other way out than to resort to tachymetric work. Meanwhile it is known that, when working with rangefinders, the accuracy of distance determination is usually equal to 0.27%, and the limit of error amounts to about 0.8% of the true length. In order to refine the results, the sides of the polygon were taken within the limits from 70 to 130 m, and all the tachymetric work was carried out with the greatest care, by means of repeated measurements using rangefinders. As a result of the computations, the length of the principal side of the network was determined as 561.261 m, while the greatest limit of error amounted to about 0.34 m, i.e., about 0.06%; in actual fact, the inaccuracy of the base length did not exceed ±0.12 m. The subsequent work on …

to determine the tunnel axis. Two observation points, ON and OS, were selected, where observatories were constructed. As a result of all the geodetic work for setting out the axis of the Karavanken Tunnel, both above ground and underground, when the length of the tunnel was measured in the field (inside the tunnel), the difference from the computations amounted to only 0.45 m.

The base should be located in the middle of the section being surveyed. However, because of the presence of a mountain massif in tunnel triangulations, in most cases this is impossible. Therefore the base has to be placed at the end of the tunnel, at one of the portals. Sometimes, in order to have a check on the correctness of the measurements and computations, two bases or two principal sides of the network are assigned at the opposite ends of the tunnel. The end points of the base and the vertices of the triangles are chosen so that from each of them as many other points as possible are visible; for this reason they should not be taken in thickets of shrubbery, beyond cliffs, or in deep valleys. It will be very advantageous to place the base on a mountain plateau, a terrace, or on a mountain slope. It is desirable that the ground at the location of the base not be stony.

The triangulation of the Tauern Tunnel, 8.6 km long (Fig. 28), was carried out with the aid of two principal sides, which served as bases,

Fig. 28. Triangulation of the Tauern Tunnel.

located on both sides of the pass. The northern base between the points Echo and Böckstein has a length of 3138.009 m. The southern base between the points Seebach and Mallnitz is only 930.652 m long. The determination of the length of the principal sides was made with the aid of base networks by triangulation. The measurement of both small bases was made by A. Tichy by an optical method. For a more perfect execution of the triangulation work and in order to avoid possible errors in determining the direction and length of the tunnel axis, Eng. Imhof carried out work on tracing the tunnel axis and on fixing it. As can be seen from the figure, the distance between the terminal signals located on the tunnel axis is about 10 km. This work of tracing required 3 days, 16 working hours each, and the crossing of a mountain pass rising 1.6 km above the elevations of the terminal signals. As a result of the work, the following points were placed on the tunnel axis: two terminal signals, two signals at the tunnel portals (portal

signals), two pre-portal signals and, finally, 8 signals placed at different locations along the tunnel axis. Thus, on the axis of the Tauern Tunnel there are altogether 14 signals. The measures adopted to refine the work in determining the direction of the tunnel axis, in view of the difficult topographical conditions of the locality, were very expedient. And yet, in the final measurement of the tunnel length in the tunnel itself, its true length proved to be greater than the length computed on the basis of the tunnel triangulation by \(+2.93\) m, which is a remarkably irregular quantity.

A similar example of tunnel triangulation with two bases is found in the construction of the Bosruck Tunnel (Fig. 29). The axis of this tunnel, about 4.8 km long, was determined with the aid of a network of triangles having 11 points, of which 5 points are located on the tunnel axis. In addition, there are two more portal points, likewise located on the tunnel axis. In all, therefore, 7 points are situated along the tunnel.

Fig. 29. Triangulation of the Bosruck Tunnel.

The first (southern) base side I—II is 1399.363 m long, and the second (northern) base side VI—VII is 1565.108 m long. This time the results of the triangulation work proved entirely satisfactory: when the length of the tunnel was measured directly inside the tunnel, the difference obtained was only 0.18 m.

One of the ways of developing a railway line in highly mountainous terrain consists in the formation of loops. In doing this, side valleys are used for developing the route and for the reverse turn. Sometimes, in the absence of side valleys, turning and spiral tunnels are constructed. A spiral tunnel is expedient to build where the narrowness of a gorge or local conditions hinder the open placement of loops. Depending on the situation of the place, by choosing one or another radius of curvature it is possible to gain a certain ascent. Spiral tunnels of this kind are found on the St. Gotthard, Schwarzwald, and many other railways of Western Europe, and in the USSR on the Armavir–Tuapse section of the North Caucasus Railway.

The St. Gotthard Railway runs along the valley of the Reuss, and south of the large summit tunnel along the valley of the Ticino River. The valley of the Reuss in its forward part itself has a gradient of 30–40‰ and even 68‰, and therefore much greater than the railway line. For this reason, the railway route passes many times along the valley almost at the level mark of the highest waters. In the presence of such a steep gra—

the valley, an artificial development of the line was necessary and the only possible solution. For this purpose the Pfaffensprung spiral tunnel was constructed, with radii of 280, 300, and 500 m. Its length is 1450 m. Its lower portal is situated at an elevation of 774 m above sea level, and the upper one at an elevation of 809 m above sea level. Thus, thanks to the tunnel, 35 m of height are gained, and in addition another 16 m on the open gradient outside the tunnel.

The next, most interesting point of the St. Gotthard Railway is near Wassen (Fig. 30). Here, in order to carry the route, it was no longer sufficient to build a spiral tunnel. With a great length it would, moreover, have been difficult to ventilate it. It was therefore necessary to construct the route in the form of a loop. This loop begins at the crossing of the Reuss River, at an elevation of 895 m, with the Wattingen turning tunnel 1090 m long. After emerging from this tunnel the line again crosses the Reuss higher up and, along the left bank, returns toward Wassen station and passes beneath the locality of Leggistein by means of a curved tunnel 1095 m long, rising within the tunnel by 25 m. Farther on the road runs above Wassen station along the mountain slope in the direction of Göschenen.

In setting out the railway line at the place indicated, for the precise setting out of the tunnels, a tunnel triangulation was carried out (Fig. 30). At some of its points the triangulation network adjoined the State triangulation of Class II, and therefore the bases were not measured. However, in view of the fact that the tunnel-triangulation network extended for a considerable distance along the railway line, “control bases” were used.

A characteristic feature was also that the annular tunnels were determined and fixed by means of points determined by triangulation and arranged so that the straight lines drawn through these points formed tangents to the circular curves where the tunnels were located. Such points at the Wattingen tunnel were points A and B at one portal, and points C and D at the other portal. The points of tangency BA and BE were determined by setting out on the ground, and it was neither possible nor necessary to tie them by computation into the trigonometric network. In this case the angles \( \alpha \) and \( \beta \) were found trigonometrically, and their sum \( \alpha + \beta \) is equal to the central angle of rotation of the circular curve on which the tunnel is located. The situation was analogous with respect to the Leggistein tunnel: points B and A are situated on the tangent to the curve where the tunnel is. The point of tangency BA was found analytically and served only for setting out the axis of the tunnel underground. At the other portal the alignment CD, determined trigonometrically at its terminal points C and D, gave the point BE.

For setting out the axis of the tunnel underground on the approaches to the tunnel, the instrument (theodolite) was placed over point A (a trigonometric point) and was oriented by points B and BA, where sighting marks had been set up. At the other portal the instrument was placed at point C (a trigonometric point) and was oriented by points D and BE, where sighting marks had likewise been set up. All the triangulation work indicated here gave excellent results: the railway line, complex in plan, was set out quite accurately, and the misclosure amounted to only a few centimeters.

Reduction of the tunnel length to a specified level. In computing tunnel triangulations, these may be reduced to any level. Usually they are reduced to sea level. Meanwhile, the length of the tunnel must be reduced to the mean elevations of the actual

Plate 30. Triangulation network for the Vlasen tunnel on the Sent-Gotthard Railway. P. D.

Drawing 301. Tunnels at the Saint-Gotthard Base. Fig. 1

position of the tunnel in height. Thus it is necessary to introduce a correction depending on the curvature of the earth’s surface and on the mean height of the tunnel above sea level. It is known that the polar diameter of the terrestrial sphere is 12,713 km, and the equatorial diameter 12,756 km. Thus the mean radius of the terrestrial spheroid may be taken as 6367 km.

Let us denote: \(L_0\)—the length of the tunnel reduced to sea level. \(L_h\)—the length of the tunnel at the mean elevation of the tunnel \(h\) above sea level. \(h\)—the mean elevation of the tunnel above sea level; \(\varphi\)—the central angle of the arc of the earth’s surface corresponding to the tunnel.

\[ \Delta L = L_h - L_0 = \pi (R + h)\frac{\varphi^\circ}{180^\circ} - \pi R \frac{\varphi^\circ}{180^\circ} = h \pi \frac{\varphi^\circ}{180^\circ}, \]

and since

\[ L_0 = \pi R \frac{\varphi^\circ}{180^\circ}, \quad \text{i.e. } \frac{\pi \varphi^\circ}{180^\circ} = \frac{L_0}{R}, \quad \text{then } \Delta L = \frac{h L_0}{R}. \]

From this is determined the correction to the reduced length of the tunnel. If \(L_0\) is known, then \(L_h\) is easily found from the formula

\[ L_h = \left(1 + \frac{h}{R}\right)L_0 \]

Deflection of plumb lines by the attraction of mountain masses.—In carrying out geodetic work in mountainous regions, the geodetic coordinates determined for some point (latitude, longitude, and azimuth) are, in their magnitude, different from the same coordinates determined astronomically. Already in the middle of the last century the explanation of this phenomenon was found in the deflection of the observed plumb lines from their true position owing to the attraction of mountain masses¹). The influence of mountain masses on the course of geodetic work is especially pronounced in triangulation work, in particular in tunnel triangulations for setting out the axes of long, deeply situated tunnels. Therefore it is necessary to dwell on this question, important in tunnel construction, and to set forth the basic principles for computing corrections for the attraction of mountains. When Kortazzi and Kuhlberg carried out their astronomical determinations in the Crimea in 1891–93, then, when comparing the astronomical coordinates with the geodetic ones, the deflections of the plumb line produced by the small Crimean ridge²), the highest summit of which is only 1300–1500 m above sea level, proved to be so considerable that in magnitude they equaled the deflections found in high mountain masses such as the Alps, the Caucasus, and the Himalayan mountains, whose principal summits are 3–6 times higher than those of the Crimea.

The force of attraction of the whole earth \(G\), acting at a given point of the earth’s surface whose geographical latitude is \(\varphi\), is equal to

\[ G = \frac{\kappa M}{b^2} \left[ 1 - \varepsilon - \frac{3}{2}m + \left(\frac{5}{2}m - \varepsilon\right)\sin^2 \varphi \right]. \]

where \(\kappa\) is the coefficient of attraction.

¹) Of particular scientific interest in M. I. Lebedev’s geodetic and astronomical work of 1877–1879 in Bulgaria were the considerable deflections of the plumb line found there at many points, not only from the attractive action of the high mountains of the Balkan Peninsula, but also from the opposite action of the depths of the Black Sea.

²) The Crimean or Taurida Mountains are formed of three parallel folded mountain chains. The first chain—the Main Ridge, the southernmost—has an average height of about 750–800 m. The principal summits are: Ai-Petri (1321 m), Orman-Kosh (1543 m), Chatyr-Dag (1526 m), Demerdzhi (1357 m). The second ridge does not rise above 650 m, and, finally, the third barely reaches 270 m in height.

\(M\) — mass of the earth —

\[ \frac{4}{3}\pi a^2 b \delta \]

\(a\) and \(b\) — the major and minor semiaxes of the terrestrial ellipsoid (spheroid).

\[ \varepsilon \]

— the flattening of the earth \(\dfrac{a-b}{a}\), equal, according to Clarke (1880),

\[ \frac{1}{293465} \]

\(m\) — the ratio of the centrifugal force to the force of gravity at the equator —

\[ = \frac{1}{289} \]

\(\delta\) — the mean density of the earth, usually taken to be twice the mean density of the rock masses of which the outer layers of the earth’s crust consist.

With the above-indicated values of \(\varepsilon\) and \(m\), the expression for \(G\) assumes the form:

\[ G=\frac{8}{6}\pi a Z \kappa \delta \qquad \text{where } Z = 0.99479 + 0.00526 \sin^2 \varphi \]

Let \(\varphi\) be the angle of deflection of the plumb line at the point under consideration, due to the additional horizontally directed force of attraction \(F\) of the surrounding mountain masses. This angle is small and may be computed from the formula

\[ \varphi_{\text{(in sec.)}}= \frac{F}{G\sin 1''} = \frac{F}{\frac{8}{6}\pi a\kappa\delta \sin 1''\, Z} \]

Both the magnitude and the direction of the horizontal attraction \(F\) at any given point will easily be found if its components \(X\) and \(Y\) are determined along any two mutually perpendicular horizontal directions \(A_x\) and \(A_y\)—for example, along the meridian and along the parallel of the given point \(A\), or else along the axis of the tunnel and along the line perpendicular to it. Let us agree to count horizontal angles or azimuths \(\alpha\) from precisely that direction \(A_x\) for which it is desired to compute the component \(X\) of local attraction¹. We divide the terrain surrounding point \(A\), first, by a system of vertical radial planes making various azimuths \(\varphi\) with the direction \(A_x\), and, second, by a system of concentric cylindrical surfaces described from \(A\) with various radii \(r\), and consider first the attractive action on point \(A\) of some one of such subdivisions of the terrain, bounded precisely by two cylindrical surfaces with radii \(r_1\) and \(r_2\) and by two vertical planes with azimuths \(\alpha_1\) and \(\alpha_2\).

Denoting the relative height of some point \(M\) of this subdivision above \(A\) by \(z\), and the mean height of all points of its outer part by \(h\).

\[ \xi \quad \text{southern deflection.} \]

\[ \eta \quad \text{eastern deflection} \]

\[ \xi_{\text{(in sec.)}} = \frac{4\theta_m \kappa R}{3\rho_0 \rho''} \sum \left(\sin \varphi_m-\sin \varphi_i\right)F \]

\[ \eta_{\text{(in sec.)}} = \frac{3\theta_0 \rho''}{4\theta_m \kappa R} \sum \left(\cos \varphi_i-\cos \varphi_m\right)F \]

where

\[ F=h\log_{\mathrm{nat}} \frac{r_m+\sqrt{r_m^2+h^2}}{r_l+\sqrt{r_l^2+h^2}} \]

\[ \frac{3\cdot \theta_0 \rho''}{4\theta_m \pi R}=0.000386 \]

For convenience in using these formulas, they are reduced to another form by transformation:

\[ \xi_{(\text{in sec.})} = 0.000386(\sin\varphi_m-\sin\varphi_l)h \frac{r_m-r_l}{r} \left[1-\frac{1}{2}\frac{h^2}{r^2}\right] \cdot \left[1+\frac{1}{12}\left(\frac{r_m-r_l}{r}\right)^2\right]. \]

\[ \eta_{(\text{in sec.})} = 0.000386(\cos\varphi_l-\cos\varphi_m)h \frac{r_m-r_l}{r} \left[1-\frac{1}{2}\frac{h^2}{r^2}\right] \cdot \left[1+\frac{1}{12}\left(\frac{r_m-r_l}{r}\right)^2\right]. \]

These formulas become especially convenient for computations if the sines (or cosines) of the azimuths of the different subdivisions of the locality increase in an arithmetic progression, while the radii of the cylindrical surfaces bounding them increase in a geometric progression; because the numbers \(\sin\varphi_m-\sin\varphi_l\) (or \(\cos\varphi_l-\cos\varphi_m\)) and \(\dfrac{r_m-r_l}{r}\) then become the same for all subdivisions.

Thus, in order to compute the local attraction at some point \(A\) by latitude, it is necessary to construct the above-mentioned angles \(\varphi\) with respect to the meridian of this point, from \(0\) to \(90^\circ\) on both sides of it, for both the northern and the southern halves of the horizon, and then to find the quantities \(X_{\text{north}}\) and \(Y_{\text{south}}\), and substitute them in place of \(F\). For computing the local attraction by longitude, precisely the same must be done for the western and eastern halves of the horizon, with respect to a line perpendicular to the meridian, and the factor \(\sec\varphi\) must also be introduced. In this way, the complete attractions \(\Delta\varphi\) and \(\Delta h\) in latitude and longitude, when \(H\) is expressed in meters, are finally expressed as follows:

by latitude:

\[ \Delta\varphi=k[\Sigma H_c-\Sigma H_{\text{ю}}] \]

by longitude:

\[ \Delta h=k'[\Sigma H_{\text{з}}-\Sigma H_{\text{в}}] \]

For convenience of computation according to the adopted system of subdivisions, it is recommended to draw on tracing paper a diagram consisting of concentric circles with radii in multiple ratios (e.g. \(\frac{1}{2}; 1; 2; 4\), etc.) and of radial straight lines forming, with the principal diametral line, angles whose sines are respectively equal to

\[ \frac{1}{n};\ \frac{2}{n};\ \frac{3}{n};\ \ldots\ \frac{n-1}{n};\ \frac{n}{n}. \]

Such a diagram is laid upon the map of the locality where the tunnel is situated so that the center of its circles coincides with the investigated point \(A\), and so that its principal line coincides with the direction of some line definite in direction, for example, the meridian line or the line of the tunnel axis.

Then, by counting the contour lines falling in its various subdivisions, the absolute mean elevations are estimated and entered for all points of the subdivisions, the number of which is equal to the number of intersections of the radial lines and circumferences. By subtracting from all these numbers the absolute elevation \(h_o\) of the point \(A\) itself, a table is compiled of the mean relative elevations \(h\) of the various subdivisions; then corrections are made to obtain the corrected elevations \(H\), and, finally, by simple summation of the latter for each belt separately, the quantities \(\Sigma H_{\text{north}}\) and \(\Sigma H_{\text{south}}\) are obtained, entering into the expression for the local attraction in latitude \(\Delta \varphi\).

When setting out the axis of the Lötschberg Tunnel, Th. Mathys, in order to introduce the correction for the deflection of the plumb line due to the attraction of the Alpine mountains, took seven cylindrical surfaces, the radii of which were 100, 300, 900, 2700, 8100, 24300, and 72900 m.¹ For the locality within the radius of 8100 m, a topographic plan at a scale of 1:50000 (Siegfriedatlas) was used, and for the remaining part, a hydro-orographic map of Switzerland at a scale of 1:500000. Each quadrant was divided into 10 angles so that \(\cos \varphi\) were successively \(1 > n\), with \(n\) from 1 to 10. The angles were \(0^\circ 0'\), \(25^\circ 51'\), \(36^\circ 52'\), \(45^\circ 34'\), \(53^\circ 08'\), \(60^\circ 0'\), \(66^\circ 25'\), \(72^\circ 33'\), \(78^\circ 28'\), \(84^\circ 16'\), and \(90^\circ 0'\). The cosines of these angles are respectively equal to 1.0; 0.9; 0.8; 0.7; 0.3; 0.2; 0.1; and 0. From these data a diagram was constructed (Fig. 31), the center of which was made to coincide with the middle of the tunnel, and the principal line with the axis of the tunnel,

\[ \eta_{(\text{in sec.})} = -0{,}00772\,\frac{\theta}{\theta_m} \sum \left[ (\cos \alpha_l-\cos \alpha_r)\, \frac{r_k-r_i}{\frac{1}{2}(r_k+r_i)}\, (h-\Delta h) \right], \]

where

\[ \Delta h = \frac{1}{2}\, \frac{h^3}{\left(\dfrac{r_k+r_i}{2}\right)^2}. \]

If the attraction of the earth is taken as \(\theta_z=5.6\), and the attraction of the mountains as 2.8, i.e.

\[ \frac{\theta_m}{\theta_z}=\frac{1}{2}, \]

then

\[ \eta = 0{,}000386 \sum \left[ (\cos \alpha_l-\cos \alpha_r)\, \frac{r_k-r_i}{\frac{1}{2}(r_k+r_i)}\, (h-\Delta h) \right]. \]

With the adopted ratio of cylinder radii \(r_k=3r_i\), one obtains

\[ \frac{r_k-r_i}{\frac{1}{2}(r_k+r_i)}=1. \]

The difference of the cosines of adjacent angles for

at the adopted sizes of the sectors is equal to 0.1. As a result, the formula for computing \(\eta\) took the very simple form

\[ \eta_{\text{(in sec.)}} = 0.000386 \cdot \Sigma (h - \Delta h) \]

The result of the computation was as follows:

Point Kandersteg  \(\eta = +\,0.1''\)
“ First  \(\eta = -\,3.0''\)
“ Wildelsigengrat \(\eta = -\,7.5''\)
“ Immeneigrat \(\eta = -\,16.4''\)
“ Goppenstein \(\eta = +\,9.4''\)

The \(+\) sign denotes a deflection of the plumb line to the east.

[In the diagram: “Axis of the tunnel”; “Scale,” km.]

Fig. 31. Diagram for computing the local attraction of mountains and the deflection of plumb lines.

The triangulation of the Simplon Tunnel, intended to determine the direction of the axis of this tunnel, consisted of 11 points (Fig. 32). Of these, two points were located on the axis of the tunnel, in the immediate vicinity of the portals. The southern axial point had an elevation of 634 m above sea level. The mountain massif beneath which the tunnel was driven rises to a considerable height. One of the points of the triangulation network, situated near the summit of Monte Leone, has an elevation of 3559 m above sea level (photograph 33). For the computation of the tunnel triangulation, a base was not measured separately; instead, the network of triangles was incorporated into the already existing Swiss national triangulation network. For this purpose it was necessary to use three points of the national triangulation network, indicated in Fig. 32 by a dotted line. The angles were measured by Schreiber’s method and then adjusted. In the angles of the triangulation there was a misclosure reaching as much as 8.5″, whereas according to the computations the expected error in the angular misclosure should not have exceeded ±1.0″. In view of this, the results of the angle measurement proved wholly unacceptable. An investigation into the causes of so considerable a misclosure established that the measurement of the angles was affected by the deflection of the plumb lines due to the attraction of the mountain masses. The visible mountain masses were computed on the basis of the contours shown on the topographic plan, the mean density of these masses being taken as 2.8, with the mean density of the terrestrial globe 5.63. In accordance with this, the necessary corrections were introduced, and the misclosure then became insignificant, namely: the greatest error in the measurement of the angles decreased from 8.5″ to 5″, and for the other points from 3.1″ to 1.7″. On the basis of the mean observations, by applying corrections, the triangulation network was computed and plotted in the horizontal plane, tangent at the point Monte Leone, as the mean point of the triangulation network. There were computed 56 directions of the network with 56 error equations, 18 normal equations with 18 unknowns. The probable error of the computed direction proved to be ±0.614″.

Fig. 32. Triangulation network of the Simplon Tunnel.

On the basis of the corrected coordinates, the length of the axial line from one triangulation preportal point to the other was ...

was computed as 20,088.63 m, and, by adding the correction due to the transfer of the direction from the plane to the geoid, the angles from the deflection of the plumb lines at both pre-portal points of the triangulation network were recomputed.

With the final transfer of the length to the geoid, the distance between the pre-portal points of the triangulation network was expressed as 20,091.33 m at an elevation of 696.74 m above sea level (the mean elevation of the tunnel profile). The probable error of this quantity is ±0.10 m, and if all possible errors in the work are taken into account, then ±0.56 m.

§ 4. Marks, signals, and observatories used in tunnel geodetic works on the earth’s surface. In the course of carrying out tunnel geodetic works on the earth’s surface, depending on the scale of the works and on their particular importance and responsibility, it becomes necessary to mark points on the ground by means of marks and signals, and then to set up instruments on towers specially erected for this purpose, called observatories.

Photo 33. Triangulation point on the summit of Monte Leone.

Marks and signals. Points of geodetic tunnel works are assigned in those places which, in one respect or another, are important and necessary in the process of the work. Such points are:

1) Entrances to the tunnel.
2) Entrances to shafts, side adits, and the like.
3) Entrances to sighting adits.
4) Points of tunnel triangulation.
5) Points (vertices) of the angles of a polygonal survey.
6) Alignments established when determining the direction of the tunnel axis.
7) Points of the vertices of the circumscribed polygon when setting out a tunnel on a curve.

In view of the fact that one and the same type of mark or signal may find application in the various cases just enumerated,

designations of points, it is therefore more convenient to concentrate the further consideration of the question on the description of the individual kinds of marks and signals, noting the applicability of a given kind in one case or another.

Centers. All points of tunnel triangulation must be marked and then preserved for the purpose of checks. For this, special underground trigonometric marks are used, called “centers.”

A center is the artificial designation and fixing of a given point on the surface of the earth, by embedding in the soil some object (a bolt, nail, etc.) in order to preserve the given point for as long a period of time as possible.

Special attention must always be paid to the construction and installation of marks.

The centers of tunnel triangulation must possess the simplicity so necessary in carrying out engineering works in general; they must be sufficiently durable and protected against accidental damage. For this, the centers are laid in the ground at a sufficient depth; the centers must possess definiteness and uniformity; they must be easy to find.

The following kinds of centers are used in tunnel triangulation:

1. A pit measuring in plan \(0.9 \times 0.9\) m, with a depth of \(1.0\) m. On the bottom of the pit, in a recess \(0.3\) m deep and \(0.2 \times 0.2\) m in area, concrete is tamped, into which a special mark is embedded in the form of a conical or cylindrical iron blank, or even a simple bolt. Two cross-shaped saw cuts are made on them with a file. The point of intersection of the cuts is the point of the tunnel triangulation network (the center). To protect the center from accidental damage, a stone or concrete mass is erected above the center, brought up to the surface of the earth, and above it a stone pedestal is laid up to a height of \(1.0\) m above the surface of the earth. Through the entire pedestal, as well as to some depth into the foundation, an iron pipe with an internal diameter of \(7.5\) cm is carefully embedded, strictly vertically and centered over the center (the cross made by the file), and in its upper part a wooden stake is inserted. From above the pedestal is covered with a cone of galvanized iron, fastened to the pedestal by means of four iron bolts. The upper edge of the pipe, the top of the stake, and the top of the cone must be strictly centered (Fig. 34).

Fig. 34.

During the performance of triangulation works the cone is removed, the stake is pulled out, and the theodolite is set up on its legs strictly by plumb line over the geometric center of the pipe opening.

The type described above is applicable in soft soils, such as sandy loams, loams, clays, etc.

Thus, the main part of the whole—the “center” itself—exists solely in case of damage and even complete destruction of the entire pedestal, and serves only for restoring the point of the triangulation network.

2. A pit measuring \(1.0 \times 1.0\) m and 1.5 m deep (Fig. 35). Crushed stone or lean concrete is rammed into the bottom of the pit. A mass of bricks is laid, filled inside with concrete. At a height of about 0.3 m an inverted flowerpot is installed, bottom upward, filled with concrete through the hole in the bottom. When the pot has been filled with concrete, lead is poured on top, into which a nail is immersed. The head of the nail is the center of the triangulation network. The center is covered dry with two bricks, and around them the bricks are laid in cement mortar to a certain height; then the entire center is covered with earth.

Fig. 35.

3. A center of rather simple construction is shown in Fig. 36. The pit is \(1.0 \times 1.0\) m in plan and up to 0.7 m deep. A small mass of rubble stone is laid in cement mortar. A log 0.25 m in diameter is embedded. Under the lower end of the log, a bolt or a piece of round iron is embedded in the masonry.

A nail is driven into the log, sawn off at a height of about 0.75 m from the ground surface.

Fig. 36.

4. A pit up to 2.4 m deep, \(1.10 \times 1.10\) m in plan. Crushed stone and concrete are rammed into the bottom of the pit. Then a brick mass is laid in cement mortar in four courses. In the second course a metal plate with a cross-shaped saw cut is placed (the deep main center); on top a piece of rail is laid (Fig. 37).

Fig. 37.

1. Fig. 38 shows a type of center intended for optical sighting. It consists of a concrete mass, hollow inside. At the bottom there is a geodetic center. During sighting an electric lamp is lowered inside the mass.

Signals of tunnel triangulation, in their construction, do not differ in any way from signals of triangulations of classes III and IV.

The timber for pyramids and signals must be straight, of good quality, preferably pine. The platform for setting up the instrument must be solid and not touch the floor of the observer.

After the signal has been built, underground centers are laid. The lower underground center must be below the depth of freezing of the soil. The upper center is level with the ground or somewhat higher.

Observatories (towers). In carrying out geodetic tunnel works for setting out the axis of a tunnel in mountainous terrain, as has already been indicated, observatories (towers) are erected. Tunnel observatories are intended to serve as a place for installing the sighting tube of a large- or small-transit instrument, or, as a particular case, a theodolite, at such a height that the line of sight, by means of the tube, dominates the terrain along the tunnel route for a considerable distance. The height of observatories (towers) reaches 10 and even 12 m, and rarely falls to 5 m.

There are two principal types of tunnel observatories: stone and wooden.

The stone type of observatory is a stone or concrete, sometimes brick, pillar laid on a rubble foundation. The instrument is set up on top of the pillar. Around the pillar a wooden, sometimes stone or concrete, structure is erected, making it possible for the observer to stand on a special platform at the proper height while working with the instrument. The peculiarity of the construction is that the vibrations of the wooden structure under the influence of the observer’s movement (walking) near the instrument must not affect, even in the slightest degree, the stability of the stone mass on which the instrument is installed.

Fig. 38.

In their details the type of stone observatories may be divided into several groups.

To the first group may be assigned those observatories which are a solid monolith, a pillar of square or circular section, resting on a firm foundation.

An example is the tunnel observatory used in the construction of one English tunnel (Fig. 39).

The central part of the tower, intended for installing the transfer instrument, is a mass of square section, laid in cement mortar. The dimensions of the mass are \(2.0 \times 2.0\) m below and \(1.0 \times 1.0\) m above; height, 9.6 m. A ledge is made at the top. The mass is surrounded by a structure standing separately from it, consisting of four stone walls in lime mortar, raised to a height of 8.5 m, and above, to a height of about 3 m, a wooden superstructure is made. This

the superstructure is fastened by means of 8 anchors. Above the roof, for the transmission of signals, there is a special device in the form of a semaphore. The place where the instrument is set up is enclosed by railings.

The second group of masonry observatories should include observatories in the form of pillars of a somewhat lighter construction with respect to the amount of masonry. Usually this is achieved by making the pillar thinner and providing buttresses for stability.

Drawing 39. Tower.

Drawing 40. Tower.

An example may be the tunnel observatory (Drawing 40) used for setting out the Blechingley and Saltwood tunnels.

The main body of rubble masonry was brought up to a height of 10 m and, not counting the buttresses, has a square section of \(1.0 \times 1.0\) m. Around it a wooden structure with an internal staircase was made. At the top there is a platform for obser-

observer. The entire observer’s cabin, to a height of about 4 m, is sheathed with boards; on the sighting side, narrow slits 0.5 m wide are made along the whole wall, where there is no sheathing. In the sheathed walls two small windows with glazed frames are made. Above the cabin there is a roof, and at the top a spire 3.5 m high, on which colored flags were raised, serving as signals in laying out the line.

Finally, the third group consists of observatories in which the stone pier is made as two separate standing piers, with buttresses, so that between them there is free space used for lowering a plumb line when setting up the instrument and centering it over the geodetic point. The division of the principal mass into two piers is made along a line directed along the tunnel alignment and thus in the direction of sighting.

An example of this kind of tunnel observatory may be the observatory of a tunnel in America (Fig. 41). The concrete mass is brought up to a height of about 5 m. The foundations are laid to a depth of 1.6 m; the wooden structure, which has a platform for the observer, consists of 4 posts sunk into the ground. The observer’s cabin is boarded up only on two sides. In the direction of the alignment, however, the cabin is left completely open.

The wooden type of tunnel observatories, as is usual for structures of this kind, consists of two parts: the main part—the structure intended for setting up the instruments—and the other, completely independent of the first and not in contact with it, structure, carrying the platform for the observer.

Fig. 41. Tower.

In Germany, and also in Switzerland, at one time tunnel observatories of the wooden type were used, resembling in their construction a triangulation pyramid.

Instead of this type, subsequently (Fig. 42) wooden observatories of the following arrangement began to be used. The main part consists of four 30 cm posts, placed obliquely and converging at a vertical log, into which they are also notched. A stone block (geodetic point) is laid in the ground plumb beneath the log. In view of the fact that, when laying out and sighting from other points, it is convenient to see the bottom of the signal set over this point, the entire main structure in the form of a pyramid had to be turned in plan, relative to the direction of the tunnel alignment, through an angle of 30°. At the top, a theodolite is set up on the log. Around this structure, on four-

...three posts dug into the ground there is a platform. Through its center, freely, without touching the platform, passes the above-mentioned log. The observer climbs by a ladder to the platform, placed at a height of 4.5 to 6.0 m. The platform is square, measuring 1.6 × 1.6 m. The height of the projecting post on which the theodolite is placed is 1.1 m.

In photograph 5 and Fig. 43 there is shown a wooden tunnel observatory used in 1911 in the tracing of the Lagar-Aul Tunnel on the East Amur, now Ussuri, Railway. The main part of the structure consists of two 7-vershok (31 cm) posts set vertically. The height of the posts is 9.5 m. The distance between the posts is about 2.8 m. Between these posts, in a plane perpendicular to the line of the posts, passes the direction of sighting. For stability, the posts are braced from the sides by inclined struts, three to each post. At the top the posts are sawn off horizontally to level. On top of the posts there is a beam about 6.5 m long, 45 cm wide, and about 11 cm thick. On one post this beam is fixed immovably, while the other end can be raised or lowered slightly by means of wedges. This was used every time before beginning work, in order to attain complete horizontality of the upper face of the beam. This measure is all the more important because there was fear of the posts being forced out under conditions of permanently frozen soil. Subsequently it was necessary to be convinced of the usefulness and expediency of precisely such a construction. In the middle of the beam a longitudinal slot is made, intended for passing the cord for centering when setting up the theodolite. Below, in the ground, a center is embedded, consisting of a small mass with a bolt sunk into it at the top.

Fig. 42. Observatory for tracing tunnels.

The structure with the observer’s platform, according to the general principle of construction of tunnel observatories, is made independent of the main structure. All details are visible from Fig. 43.

The above-described types, illustrated by six drawings and one photograph, do not exhaust all the variety of tunnel observatories. However, there is no need to set forth and describe the remaining types, on the grounds that all those indicated here are the most characteristic and have proved themselves well in the performance of geodetic tunnel works.

In conclusion, it is necessary to say a few words about the methods of signaling used in tunnel tracing. For a long time

there was a widespread method of signaling by means of colored flags, raised on a flagpole at the top of the observatory and serving as a conventional sign indicating where the plumb-line operator should move—to the right or to the left. Recently, the telephone has been becoming more and more widely used. For this purpose, a portable field telephone is used with success. Attempts to use sound signals, as well as light signals at night in the form of lanterns with colored glass, proved unreliable and troublesome.

§ 5. Execution of the Work. Instruments Used in Tunnel Geodetic Work on the Surface of the Earth.

All geodetic work carried out on the surface of the earth for the purpose of determining the direction of the tunnel axis and all preparatory work for setting out the tunnel axis underground may be divided into the following principal types:

Tracing the tunnel axis without the use of an angle-measuring instrument, or with the aid of an angle-measuring instrument, with the erection of towers (observatories) or without them.

Traverse surveying.

Triangulation work.

Vertical surveying on the surface of the earth.

Although, in some of their parts, tunnel geodetic works do not present any special features and are known from courses in geodesy, nevertheless, to a considerable extent all these works undoubtedly

Drawing 43.

have many distinctive traits and specific features; therefore, the execution of tunnel surveying work requires especially detailed consideration.

There is no doubt that the choice of an instrument suited to tunnel surveying work cannot fail to affect the ease and accuracy of the work of determining the direction of the tunnel axis on the surface of the earth; for this reason this question also requires appropriate explanation.

A characteristic feature of tunnel surveying work will also be that, in order to attain greater accuracy, some of it is quite often carried out not only by day, but also at night. This imposes certain requirements on the angle-measuring instruments used in tunnel construction when working on the surface of the earth.

Execution of work in tracing straight-line tunnels. The tracing of straight lines up to 150 and even up to 200 m long is carried out with the aid of binoculars.

The tracing of long straights, from 200 to 750 m in length, is done through the telescope of a theodolite. Sighting the line begins from the far rod, gradually approaching the theodolite; beforehand, however, the perpendicularity of the collimation plane of the telescope to the plane of the limb should be carefully adjusted. For this work a theodolite of any system and make may be used.

Instruments for tracing tunnels. For tracing long straights of more than a kilometer in length, whose direction is usually defined by an azimuth, an instrument with a powerful sighting telescope, with a magnification of 20 to 40, and sometimes even more, is required. As was already indicated in the description of methods for carrying out work on the tracing of long tunnels, towers (observatories) are used on the surface of the earth. Instruments must be installed on these towers. The practice of tunnel construction indicates that such instruments may be:

Large transit instruments (Figure 44), which pass through the zenith and rotate in a vertical plane.

Fig. 44. Large transit instrument for tracing tunnels.

Aliniometers are sighting telescopes passing through the zenith, with a horizontal axis that is shifted in the bearings of the support. Two cylindrical levels—one parallel to the optical axis of the telescope, the other parallel to the axis of rotation of the telescope—serve for setting up the instrument. There are no limbs with graduations or vertical circle. Sometimes there is also another level, allowing the optical axis of the telescope to be set vertically by directing it with the objective downward toward the surface of the earth. This makes it possible to optically center the instrument over the given point (“center”). The vertical axis of the instrument is made hollow. For convenience in such cases the eyepiece is made elbow-shaped. Along the middle of the tripod

a cylinder with a movable pin for centering the instrument is placed.

Small transit instruments (Fig. 45) are recommended by German tunnel specialists for setting out long straight tunnels. They differ from alinometers in their middle part, which consists of two boards adapted so that the instrument can be set accurately on the axis of the line or over a given point.

Targets (sights). In setting out tunnels with an alinometers or a theodolite, it is recommended to have two targets (sights), Drawing 46, set up on tripods with three elevating screws and hollow sleeves, identical both for the theodolite and for the sights. Owing to such an arrangement, the tripods of the theodolite and of the front sight, once set up, can remain in place, while the rear tripod is brought forward and the axes of the theodolite or of the sight are inserted alternately into its sleeve.

Drawing 45. Small transit instrument for setting out tunnels (Alinometer type).

Execution of the work. The essence of prolonging long straight lines consists in the following: having set a stake or a target (sight) by means of the telescope of a theodolite or alinometers installed at the initial point of the given straight line, the instrument is moved to the place of the stake or target and, after making sure that the instrument has stood exactly on the axis of the line at the very point where the first stake or target had been set, the line is continued forward, setting again a second, then a third, etc., target or stake.

The line to be set out is defined, as already indicated, by two signals in the form of stakes. Sometimes, instead of stakes, a rocket, a Bengal light, or even a bonfire is used.

Drawing 46. Target (sight).

Large, as well as small, transit instruments are used exclusively for setting up on towers (observatories). Large transit instruments have a permanent installation. Small transit instruments, upon completion of the work, are each time completely removed and, when work is resumed, require setting up anew.

When setting out from a tower, sighting is carried out by repeated (successive) pointing of the telescope at the stake or target. Until the stake or target has been finally set, its position is each time marked on the geodetic center by a line directed along the setting-out line. Usually no fewer than 8 observations are made, and each time a new setting of the stake or target is performed. From the practice of tunnel setting-out there are known examples where the work was carried out 20 or more times before the tunnel alignment was finally established. The mean is taken from 2 consecutively made readings at different positions of the telescope (circle right,—circle left), and only then is the total mean derived from these partial means. Observations sharply differing from the mean are rejected. In setting out, a deviation of the extreme readings from the mean is considered acceptable at the rate of 25 mm per 1000 m of sighting with the instrument.

In Fig. 47 an example is given of marking strokes on a geodetic center when setting out the alignment in the Hauenstein base tunnel at 1200 m from the southern portal, Oct. 6, 1912.

Here it will be appropriate to say a few words about the error when setting out the axis of a tunnel from a tower (observatory). The permissible error of observation is from \(30''\) to \(45''\), which amounts to from 145 mm to 218 mm per 1 km. Sometimes, when working with large transit instruments, the error falls even to \(10''\), i.e., to 48 mm per 1 km.

Execution of tunnel triangulation work. The time of day most convenient for measuring the angles of the network is from 5 to 8 p.m. If it is possible to use oil, acetylene, or electric lamps, then one can work with great convenience and success from sunset to sunrise. Sometimes, as, for example, in the triangulation of certain tunnels in Indo-China, Bengal lights were successfully used for night work.

Fig. 47.

Instruments for tunnel triangulations. For measuring angles in determining the direction of the tunnel axis it is recommended to use perfected angle-measuring instruments. For this purpose serve large repeating theodolites and universal instruments. The accuracy of reading (readable) on the verniers should be about \(10''\) with \(60'\) and \(60''\) divisions of the degree. In such cases visual estimation can be made with an accuracy up to \(1''\). The magnification of the telescope is usually sufficient within the range from 18 to 30 times, and only in exceptional cases are more powerful telescopes used in tunnel work. For small tunnel works, angle-measuring instruments (theodolites) of any systems may be used. The instruments employed for measuring the angles of a tunnel trigonometric network are, in general, the same as those used for triangulation work in general, in triangulations of classes III and IV. However, on the basis of tunnel practice, chiefly abroad—in Germany, Switzerland, and Italy, as well as here in the USSR—it is possible to point to certain instruments as the most suitable and most frequently used specially for tunnel work.

Theodolite I of the firm Carl Zeiss (Carl Zeiss, Jena) (Fig. 48). This theodolite is very common in the practice of tunnel triangulation work, although, as far as I know, it is intended chiefly for triangulations of classes III and IV.

Fig. 48.

The characteristics of this theodolite are as follows:

Characteristic	Value
Diameter of the telescope objective	30 mm
Magnification	18×
Length of the telescope	137 mm
Diameter of the horizontal circle	75 mm
Diameter of the vertical circle	50 mm
Graduation of both circles	\(\frac{1^\circ}{3}\) at \(360^\circ\) and \(\frac{1}{5}\) grad at 400 grads
Weight of the instrument	3.8 kg
Weight of the case	3 kg
Weight of the tripod	4 kg

The merits of this excellent instrument are its accuracy, brightness of the image, ease of setup, portability, and relatively low price.

This instrument is very widespread abroad. It has been used with great success in tunnel work in Switzerland, on the Davos–Filisur line, in setting out many hydraulic-engineering tunnels, in triangulation work in southern Germany, etc. In the USSR this instrument was used on the Western Amur Railway and on the Armavir–Tuapse Railway, and the reports on it were very good.

Theodolite No. 2 of the Bamberg firm (Bamberg), Berlin-Friedenau (Fig. 49). The characteristics of this instrument are as follows:

Characteristic	Value
Diameter of the telescope objective	40 mm; 40 mm
Magnification, I eyepiece	20×
Magnification, II eyepiece	30×
Diameter of the horizontal circle	135 mm
Graduation of the circle	\(\frac{1^\circ}{6}\) at \(360^\circ\)
Weight of the instrument	8 kg
Weight of the case	8.8 kg
Weight of the tripod	6.1 kg

Fig. 49.

Accuracy of readings (readable) is \(10''\), with estimation down to \(1''\) for \(60'\) and \(60''\) divisions of the degree. The lesser portability of this instrument in comparison with the Zeiss theodolite and its high cost—up to 750 rubles—somewhat limit the use of this excellent instrument in tunnel triangulations. Nevertheless, many cases are known of the use of the Bamberg theodolite in tunnel work. On the head section of the Amur Railway, during the construction of the Sonsky Tunnel, and on the loop tunnels of the Armavir–Tuapse Railway, a Bamberg theodolite was used and proved itself very highly. Owing to its power and magnification up to 30×, this theodolite is used with great success in tracing tunnels when it is set up on a tower (observatory).

Theodolite No. 57 of the Ed. Sprenger firm (Ed. Sprenger), Berlin (Fig. 50). This theodolite has the following characteristics:

Characteristic	Value
Diameter of the telescope objective	30 mm
Magnification with eyepiece I	35×
Magnification with eyepiece II	45×
Magnification with eyepiece III	60×
Diameter of the horizontal circle	176 mm

Taking into account the high power of the telescope, the accuracy of the work, and other merits, one cannot but acknowledge that its price of 600 rubles is comparatively not high. This theodolite is used quite often in complex tunnel triangulations. Some tunnel specialists have successfully used it for carrying out work at night. The greater brightness of the object obtained in the field of the telescope, and the devices for artificial illumination of the cross-hairs, are merits of this instrument. A certain shortcoming is its low portability (bulkiness, great weight).

Theodolite I of the Carl Zeiss firm with an attachment for night work. The above-described theodolite I of the Carl Zeiss firm can have special attachments in the form of an illuminating device that makes it possible to illuminate the circles well and conveniently take readings on them in the dark (Fig. 51). A small accumulator may serve as the source of power. Two small Osram electric lamps of 2.5 volts and 0.5 ampere are installed in front of the illuminating prisms of the circles and illuminate the corresponding illuminators. To illuminate the reticle, a reflector is placed on the objective mount.

Fig. 50.

Fig. 51. Theodolite with attachment for night triangulation work.

Optical plummet of the Carl Zeiss firm (Fig. 52). It is intended to increase the accuracy of centering. There are two mutually perpendicular cylindrical levels. The reticle is set to the eye by rotating the eyepiece mount.

In order to attain the greatest accuracy in determining the angles of a triangulation network, it is necessary to measure each angle several times.

In the St. Gotthard Tunnel, each angle of the triangulation network laid out by Gelpke was measured 24 times. Koppe, working entirely independently of Gelpke, used a theodolite with 30-power ...

magnification. All angles were read on 4 verniers, and the reading was repeated in the reverse order. Then the graduated circle was shifted by \(5^\circ\), and the readings were made again. Each angle at the Koppe was measured in all up to 40 times.

In the Letchberg tunnel, angles were measured with a Kern repeating theodolite graduated into 400 divisions (instead of \(360^\circ\)) and with centesimal-minute and centesimal-second divisions. Each angle was taken from 16 to 32 times. New triangulation points were determined by three or more directions from the principal points. The three angles between these principal directions were measured at least up to 18 times.

In tunnel triangulations the greatest permissible error in measuring angles is taken as equal to from \(15''\) to \(20''\).

Measurement of the base. The accuracy of the entire tunnel triangulation depends to a considerable extent on the accuracy of measuring the base. When the direction of the base has been selected, the base line is carefully aligned with the aid of a theodolite and is fixed by stakes about 8 cm in diameter, driven into the ground every 30 or 50 m. The measurement of bases in tunnel triangulations is carried out in various ways:

either with a steel tape 25 or 30 m long,

or with rods 5 m long,

or with steel wire 24 or 25 m long.

When measuring the base it is necessary to determine the slope of the terrain by leveling or else by means of a suspended semicircle. The air temperature is also determined with a sling thermometer.

Fig. 52. Optical plummet.

Prof. Schmidt proposed a method for measuring the base by means of a measuring tape (ribbon) freely suspended between points. This method gives sufficiently accurate results and has more than once been used in tunnel triangulations. Prof. Schmidt’s method consists of the following: a steel tape 30 m long, 12 mm wide, and 0.4 mm thick is taken. The direction of the base is set out with a theodolite and fixed by stakes driven into the ground 30 m apart and cut off horizontally. Nails are driven into the stakes, over which one of the rings of the tape is placed. The other end of the tape is brought up to the stake, approached to the screw, and stretched with a dynamometer with some specified force, for example, 10 kg. At this moment a reading is taken from the axis of the screw with the aid of a ruler divided into millimeters (Fig. 53). When the distance between the center of the screw and the nearest division of the tape has been determined with an accuracy of up to 1.0 mm, the tape is placed by its ring on this screw, and the free end is carried forward.

Fig. 53.

As an example, one may cite the measurement of the base for one Austrian tunnel in 1913. The base was measured 6 times: three times with ...

at a tension of 5 kg and three times at a tension of 10 kg. The results are as follows:

\[ L_5=\frac{1}{3}(384{,}715+384{,}717+384{,}713)=384{,}715\ \text{m}. \]

\[ L_{10}=\frac{1}{3}(384{,}752+384{,}753+384{,}754)=384{,}753\ \text{m}. \]

Quite often, when measuring the bases of tunnel triangulations, Invar wire is used, tensioned by a load, as shown in Fig. 54.

To obtain the length of the horizontal projection of the base, the following corrections are introduced:

Correction for inaccuracy of length. The given tape (ribbon) is compared with a verified tape. The verification is carried out at the same tension as was used in measuring the base.*)

Fig. 54.

Correction for the difference in temperatures. The temperature at which the base is measured is determined. The normal temperature is considered to be that at which the verified tape has its true length. The correction is computed by the formula:

\[ M=L\alpha(t-t_0), \]

where \(\alpha\) is the coefficient of expansion of steel, equal to \(0{,}000012\).

Correction for the slope of the terrain. The computation is made by the formula:

\[ \sum 2l\sin^2\frac{i}{2}, \]

where \(i\) is the angle of inclination of each part having a slope different from the adjacent section.

\[ \sin i=\frac{h}{l} \]

The value \(h\) is determined by leveling.

Correction for reducing horizontal distances to sea level, or to some other horizon. The computation is carried out by the formula:

\[ l=l_0-\frac{h}{R}. \]

When determining the length of a base by the optical method, the most favorable season of the year is autumn, from the beginning of leaf fall until the first frosts. The time of day is also important. With a sky covered by clouds, the best results are obtained. Good results are also obtained one hour after sunrise and during the last hour before sunset. With partial cloudiness, one should use the periods when the sun is covered by clouds. A light wind is favorable for opti-

*) In the example given for measuring the base of a tunnel triangulation, the length of the tape at a tension of 5 kg proved to be, over 30,000 m, 2.7 mm greater, and at a tension of 10 kg, 5.6 mm greater.

geodetic measurements of the length of the base. The absence of wind can become appreciable only with continuous cloud cover. The most reliable determination of the base, as may be judged from the practice of tunnel work in Switzerland and Austria, is obtained by repeated observations on different days or at different periods of the day.

In determining the lengths of bases for tunnel triangulations, the permissible error is expressed in fractions of the measured length:

Method	Permissible error
by chain	from 0.0003 to 0.00075
by steel tape	from 0.002 to 0.0005
by rod	from 0.00015 to 0.00033
by invar wire	from 0.000001 to 0.000003

Vertical survey in setting out the axis of a tunnel. In geodetic tunnel work on the surface of the earth, the vertical survey has the following aims:

1. To determine the longitudinal profile of the terrain along the axis of the tunnel.
2. To determine the elevation of the portal in marks relative to sea level.
3. If a triangulation survey is being carried out, or if there is a polygon, it is necessary to know the elevation of all points of the network and of all vertices of the polygon in marks relative to sea level.
4. Finally, the purpose of the vertical survey is to provide bench marks which are to serve as the basis for all kinds of setting-out work inside the tunnel.

Geometric leveling from one tunnel portal to the other is carried out by a double closed run either directly through the mountain mass, along the route of the tunnel axis, or around the mountain mass. It is necessary to decide which route is more expedient and advantageous: the season, climatic features, the presence of easily passable communication routes, etc., are taken into account. In leveling, one must take into account the height of both ascents and descents, and include their sum in the calculation.

From the practice of tunnel leveling, the following interesting conclusions may be drawn: in leveling around a mountain mass, the average progress is usually equal to 1.5 km in an 8-hour working day. Thus, if the length of the tunnel, expressed in km, is divided by 1.5 km, and approximately another 30% is added, the probable duration of the time (brutto) required for leveling from one portal to the other will be obtained.

When working with a level on the ground surface, the greatest permissible error is taken as equal to from 2 to 3 mm per km and proportional to the square root of \(L\), expressed in km.

\[ 0.002\sqrt{2n}\ \text{to}\ 0.003\sqrt{2n} \quad \text{or} \quad 0.0063\sqrt{L}\ \text{to}\ 0.0095\sqrt{L} \]

For leveling outside the tunnel it is recommended to set up the instrument in the middle of the setup, and then the first two errors will have no effect on the final result of the leveling.

If it is assumed that, in work outside the tunnel, the instrument has on average \(n = 5\) setups per km, and if it is allowed that the reading on each staff is made with an accuracy of from 0.002 to 0.003 m, then, according to the theory of errors, the difference in levels of the initial and final points of a complex longitudinal leveling will be determined as from \(0.002\sqrt{2n}\) to \(0.003\sqrt{2n}\), or from \(0.0063\sqrt{L}\) to \(0.0095\sqrt{L}\), where \(L\) is the number of km passed by the leveling.

The limiting greatest error of leveling outside tunnels will be equal to three times the mean error, i.e. from \(0.0189\sqrt{L}\) to \(0.0285\sqrt{L}\).

Traverse surveying. Work on surveying the traverse for determining the tunnel axis does not, generally speaking, present any essential special features as compared with the usual methods of carrying out work of this kind.

In determining the direction of the tunnel axis, the vertices of the traverse are assigned in the smallest possible number. The traverse itself must be chosen so that its sides are as long as possible; however, it is recommended that they not exceed 300 m. For checking, a line must be run every 50 m. Measurement of the sides of the traverse is carried out with a steel tape at least 4 times consecutively in opposite directions. To introduce corrections for the slope of the ground, the horizontal position of the line must be plotted on the plan. The angles of ground slope are determined with an eclimeter. When measuring with a steel tape, ground-slope angles of less than \(1^\circ\) are usually not taken into account. With the aid of a compass, bearings or azimuths are measured. Theodolites are used for surveying the traverse.

In recent years abroad, when determining the directions of the axes of certain hydraulic tunnels by means of a traverse, work has successfully been carried out at night. For this purpose, theodolites fitted with an illuminating device were used, and sighting disks were set up at the traverse vertices (see drawing 67). These sighting disks, illuminated by acetylene lamps, are mounted on stands and on tripods; moreover, the theodolite can be set up on the same tripod and on the same stand without resetting and without displacement. According to available reports, the use of night time for traverse surveys has brought greater precision, ease, and speed to the work.

In order to attain the greatest accuracy in the work, the surveying of the polygonal traverse is carried out by repeated, multiple measurement of both sides and angles. From tunnel-work practice the following limits of this work may be noted. When measuring sides, they are usually measured at least 4 times. Quite often the measurement is carried out 8 or more times. Angles are measured, at a minimum, 8 times, and usually 16 or more times for each round. The number of rounds is not less than two, usually four, and sometimes even reaches eight. Thus, each angle proves to have been measured from 16 to 48 and more times.

In traverse surveying for determining the direction of the tunnel axis, it is recommended, in addition to the angles of the traverse, to measure also the supplementary angles to \(360^\circ\). Two values are obtained for each angle: one measured directly and one found by calculation. From the difference of these angles one can determine the mean error of measurement, which in this case is composed of the error of reading and sighting and of the constant error due to dragging of the limb in the direction of rotation of the alidade.

In traverse surveying the greatest permissible error is taken as from \(1'\) to \(1'30''\).

The mean error of measuring an angle will be from \(20''\) to \(30''\).

When measuring the sides of a traverse for determining the tunnel axis, the following error norms may be adopted, these quantities being counted in fractions of the length of the traverse side:

When measuring with a chain . . . . . . . . from 0.001 to 0.0025
When measuring with a steel tape . . . . from 0.0005 to 0.001

§ 6. Preparatory work on the earth’s surface for setting out the tunnel axis underground. Work on the surface of the earth, as has already been indicated, has as its principal and sole purpose to give the direction of the tunnel axis on the daylight surface. The points located at the ends of the direction of the tunnel axis are the preportal points. They fix this direction. When the direction of the tunnel axis on the daylight—

surface has been given and properly fixed, then, before carrying out the setting out of the tunnel axis underground, it is necessary to perform on the earth’s surface certain additional preparatory work, namely:

I. Compute the angles and give the direction (azimuths) of the tunnel axis, and, if the tunnel is curvilinear, of the approaches to it.

II. Compute the length of the tunnel.

III. Mark and fix on the ground the direction of the tunnel axis by constructing the required number of special working points (shafts) necessary for underground work.

IV. Compute the angles, direction, and length of the lateral adits (windows), sighting adits, and vertical and inclined shafts.

V. Mark and fix on the ground the entrances to the lateral adits (windows), the sighting adits, and the shafts.

Only after carrying out the indicated preparatory work is it possible to proceed to the execution of the actual setting out of the tunnel axis underground*).

I. Computation of the angles and direction of the tunnel axis. It has already been indicated that the direction of the axis of a rectilinear tunnel is determined by an azimuth**). In the case of a curvilinear tunnel, the directions both of the rectilinear part of the tunnel and of the rectilinear sections on the approaches to the tunnel are determined by azimuths. In determining the direction of the tunnel axis it is necessary to compute angles and set out curves. Under such circumstances difficulties may arise in the curvilinear connection of two alignments: the tunnel axis on the one hand and the rectilinear sections on the approaches to the tunnel on the other. The determination of the direction of the tunnel axis and the setting out of the curves fall into a whole series of cases, which it is most convenient to consider each separately.

1. The approach to the tunnel consists of two curves of different radius. Given are two directions of the line and the beginning and end of the curvilinear section: points \(A\) and \(E\) (Fig. 55), the angle \(\varphi^\circ\), and the radius \(R_1\) of one of the curves. It is required to find the magnitude of the second radius \(R_2\) and the point \(W\).

Fig. 55.

From \(\triangle AC_1S\) we determine \(\gamma\):

\[ \tg \gamma = \frac{R_1}{AS} \qquad SC_1 = \sqrt{AS^2 + R_1^2} = \frac{R_1}{\Sin \gamma} \]

From \(\triangle EC_1S\) we determine \(\lambda\):

\[ \lambda = \varphi - \gamma \]

\[ C_1E = \sqrt{SE^2 + SC_1^2 - 2SE \cdot SC_1 \cdot \Cos \lambda}. \]

* In antiquity the direction of the tunnel axis underground was first outlined by the sun, and then, as the excavation advanced into the mountain mass, by sound.

* Here an important qualification must be made: it is possible to use an azimuth only in the case where there are no local disturbances (anomalies) of the magnetic needle. (On this subject see my article in Ural’skii tekhnik* for 1927, Nos. 8–9, “The newest methods of setting out mine workings.”) Therefore, in tracing tunnels, when using azimuths, one must be very careful not to overlook a possible anomaly of the magnetic needle. In the case of an anomaly, it is necessary to work with an astronomical theodolite.

\[ \tg S C_1 E=\frac{S E \Sin \lambda}{S C_1-S E \Cos \lambda}=\tg \delta, \]
whence we determine \(\delta\).

\[ \angle S E C_1=180-(\lambda+\delta) \]

From \(\triangle C_1 E C_2\) we find

\[ \angle \delta'=\angle S E C_1-90^\circ \]

\[ C_1 C_2=R_1-R_2 \qquad C_1 C_2^2=(R_1-R_2)^2=C_1 E^2+R_2^2-2R_2 C_1 E \Cos \delta \]

whence

\[ R_2=C_2 E=\frac{C_1 E^2-R_1^2}{2(C_1 E \Cos \delta-R_1)} \]

\[ \Sin \varepsilon=\frac{C_2 E \Sin \delta}{C_1 C_2} =\frac{R_2 \Sin \delta}{R_1-R_2} \qquad \angle \beta=\angle \delta+\angle \varepsilon \]

When all the computations have been made, the execution of the setting out of the approaches to the tunnel will no longer present any difficulty. It is necessary carefully to fix on the ground, by setting out alignments, the directions \(AL\) and \(ED\). Check the angles \(\angle ALW\) and \(\angle EDW\). It is quite advisable, if only this proves possible, also to measure the angle \(\angle ASE\).

1. A curved section on the approaches to the tunnel is defined by the radii \(R_1\) and \(R_2\). The beginning of the curved section and the angle \(\varphi\) are given.

It is required to determine the end of the curved section, the point \(E_1\), and then the point \(W\).

We make auxiliary constructions: extend \(C_1Q\) parallel to \(C_2E\) and lay off \(C_1Q=R_1\). Draw \(S'Q \parallel ES\) and connect \(S'\) with \(C_1\).

Fig. 56.

\[ \angle S=\angle S'=\varphi \qquad C Q=C M+M N+N Q=R_1 \]

\[ \angle C_1 D C_2=\varphi, \]
since the sides of \(\angle C_1 D C_2\) are perpendicular to the sides of angle \(S\).

\[ \angle D C_2 C_1=\angle C_2 C_1 Q=\angle W C_2 E=\alpha \]

\[ C_1 M=C_1 C_2 \Cos \alpha=(R_1-R_2)\Cos \alpha \]

\[ M N=R_2 \qquad N Q=(A S'-A S)\Sin \varphi \]

\[ R_1=C_1 Q=(R_1-R_2)\Cos \alpha+R_2+(A S'-A S)\Sin \varphi =R_1 \Cotg \frac{\varphi}{2} \]

\[ R_1=(R_1-R_2)\Cos \alpha+R_2+\left(R_1 \Cotg \frac{\varphi}{2}-A S\right)\Sin \varphi \]

\[ \Cos \alpha= \frac{R_1-R_2-\left(R_1 \Cotg \frac{\varphi}{2}-A S\right)\Sin \varphi}{R_1-R_2} = 1-\frac{\left(R_1 \Cotg \frac{\varphi}{2}-A S\right)\Sin \varphi}{R_1-R_2} \]

Whence the angle \(\alpha\) is determined.

The angle \(\beta\) is determined from the condition

\[ \angle \beta=180^\circ-(\varphi+\alpha). \]

3. Curvilinear tunnel (Fig. 57). Given are the initial and terminal points \(A\) and \(E\) of an S-shaped curve. Given are the angle \(\varphi\) and the radius of one of the roundings \(R_1\). It is required to determine the radius of the other curve \(R_2\), compute the angles \(\alpha\) and \(\beta\), and set out the curves, for which the points \(P_1\) and \(P_2\) must be determined.

Fig. 57.

Let us denote

\[ AA_1=c \qquad A_1E=l \]

\[ \angle AC_1H=\varphi \qquad A_1H=(R_1-c)\sin\varphi \qquad C_1H=(R_1-c)\cos\varphi \]

\[ KC_2=l-A_1H=l-(R-c)\sin\varphi=b \]

\[ C_1C_2^2=(R_1+R_2)^2=(C_1H+HK)^2+KC_2^2 \]

\[ KC_2=b \qquad HK=EC_2=R_2 \qquad (R_1+R_2)^2=(C_1H+R_2)^2+b^2 \]

\[ R_2=\frac{C_1H^2+b^2-R_1^2}{2(R_1-C_1H)} \qquad \sin\beta=\frac{b}{R_1+R_2} \qquad \alpha=\beta+\varphi \]

If \(R_1=R_2=R\), then

\[ (2R)^2=(C_1H+R)^2+b^2 \]

Let us denote \(C_1H\) by \(z\); then

\[ R=\sqrt{\frac{z^2+b^2}{3}+\frac{z^2}{9}}+\frac{z}{3}. \]

4. Curvilinear tunnel (Fig. 58). Given are the beginning of the curve, point \(A\), the angle \(\varphi\), and the radii of the roundings \(R_1\) and \(R_2\). It is required to compute the end of the curve, find point \(E\), determine the point of inflection \(S'\) of the S-shaped curve, point \(W\), and the angles \(\alpha\) and \(\beta\), and finally set out the curves, for which the points \(P_1\) and \(P_2\) must be found.

Fig. 58.

Let us denote \(AA_1\) by \(c\); make the auxiliary construction

\[ C_2H\perp OA \qquad C_1\lambda \perp C_2\varepsilon. \]

Then

\[ C_2\lambda=(R_2+R_1)\cos\beta \]

\[ C_2\lambda=C_2E+E\lambda=R_2+(R_1-c)\cos\varphi \]

whence

\[ (R_2+R_1)\cos\beta=R_2+(R_1-c)\cos\varphi \]

\[ \cos\beta=\frac{R_2+(R_1-c)\cos\varphi}{R_2+R_1}, \]

from which we determine \(\beta\).

\[ \angle EC_2H=\angle S=\varphi \qquad \alpha=\beta=\varphi \qquad C_2H \parallel C_1A \]

\[ \angle HC_2W=\angle AC_1W=\alpha. \]

If \(R_1=R_2=R\), then

\[ \cos \beta=\frac{R+(R-c)\cos\varphi}{2\cdot R}. \]

5. Curvilinear section on the approach to the tunnel. Straight insert between reverse curves (Fig. 59). Given: the initial point of the curvilinear section \(A\); the length of the straight insert \(MN\), equal to \(a\); the angle \(\varphi\), and one of the radii of curvature \(R_1\). It is required to compute the angles \(\alpha\) and \(\beta\), determine the other radius \(R_2\), and set out the curves, for which it is necessary to know the positions of the points \(P_1\) and \(P_2\).

Fig. 59.

Let us denote: \(AF=c_1\), \(EE_1=c_2\), and the projection of \(FE\) on \(OA\) by \(l\).

From \(\triangle C_1C_2L\) we have:

\[ C_1C_2^2=\bigl[R_1+(R_2-c)\cos\varphi\bigr]^2+\bigl(l+R_2\sin\varphi\bigr)^2. \]

From \(\triangle C_1KC_2\) we have:

\[ C_1C_2^2=(R_1+R_2)^2+a^2 \]

\[ (R_1+R_2)^2+a^2=\bigl[R_1+(R_2-c)\cos\varphi\bigr]^2+\bigl(l+R_2\sin\varphi\bigr)^2 \]

From this \(R_2\) is determined.

\[ \tg \alpha=\frac{KC_2}{C_1K}=\frac{a}{R_1+R_2} \qquad FH=(R_1-c_1)\sin\varphi \qquad C_1H=(R_1-c_1)\cos\varphi. \]

\[ HE=\frac{l}{\cos\varphi}-FH \qquad \triangle HC_1Q \sim \triangle EC_2Q \qquad \text{whence } HQ=HE\,\frac{C_1H}{C_1H+R_2}, \]

\[ \tg \psi=\frac{HQ}{C_1H}=\frac{QE}{C_2E} \qquad \beta=\psi-\alpha \qquad \angle AC_1M=\varphi+\beta-\alpha \]

6. Approach to the tunnel (Fig. 60). The position of the beginning of the curve, point \(A\), is fixed. The angle \(\varphi\) is given. The radii of curvature \(R_1\) and \(R_2\) and the straight insert \(MN=a\) are given. It is required to find the positions of the points \(M\), \(H\), and \(E\), and the angles \(\alpha\) and \(\beta\).

Let us denote \(AF=c\). Then \(C_1F=R_1C\).

\[ \angle EC_1H=\varphi \qquad C_1H=(R_1-c)\cos\varphi \qquad \triangle C_1WM \sim \triangle C_2WN. \]

\[ MW=\frac{MN}{R_1+R_2}\,R_1=\frac{aR_1}{R_1+R_2} \]

\[ \operatorname{tg}\sigma=\frac{MW}{C_1M}=\frac{MW}{R_1}=\frac{a}{R_1+R_2} \qquad C_1C_2=C_1W+C_2W \]

\[ C_1W=\frac{R_1}{\cos\sigma} \qquad C_2W=\frac{R_2}{\cos\sigma} \qquad C_1C_2=\frac{R_1+R_2}{\cos\sigma} \]

\[ \triangle C_1HO \sim \triangle C_2EO \]

\[ \frac{C_1O}{C_1C_2} = \frac{C_1H}{C_1H+C_2E} = \frac{C_1H}{C_1H+R_2} \]

\[ C_1O=\frac{C_1C_2\cdot C_1H}{C_1H+R} \qquad C_2O=C_1C_2-C_1O \]

\[ \cos\varphi=\frac{C_2E}{C_2O} = \frac{R_2}{C_2O} = \frac{C_1H}{C_1O}, \quad \text{whence we determine } \varphi \]

\[ \angle EC_2N=\psi-\sigma=\beta \qquad \angle HC_1M=\angle EC_2N=\beta \]

\[ \angle AC_1M=\angle HC_1H+\angle HC_1M=\varphi+\beta \qquad \alpha=\varphi+\beta\ \text{etc.} \]

If \(R_1=R_2=R\), then

\[ \operatorname{tg}\sigma=\frac{a}{2R} \qquad C_1C_2=\frac{2R}{\cos\sigma} \]

\[ C_1W=C_2W=\frac{C_1C_2}{2}=\frac{R}{\cos\sigma}. \]

Fig. 60.

II. Computation of the tunnel length.

When the angles have finally been computed and the direction of the line has been determined, both for the tunnel and for the approaches to it, then, irrespective of the method by which the tunnel was traced, the tunnel length must be computed. In the section on the triangulation of tunnels it has already been pointed out that the tunnel length must be reduced to the mean elevations of the tunnel profile. If the direction of the tunnel axis on the surface of the earth was determined by means of a polygon, or by direct alignment, or by tracing from towers (observatories), then in these cases as well the length of the tunnel must be computed accurately. If a polygonal traverse was used, then the tunnel length is de-

is determined as part of the side of the closing polygonal traverse. In simple setting out, the length of the tunnel is computed by measuring the alignment on the surface and introducing corrections for the slope of the terrain.

It is very expedient, when computing the length of a tunnel, to determine this length not from portal to portal, but from some geodetic point to another—for example, from one observatory to another, or from one sighting mark to another, or from one vertex of the polygon to another—and only thereafter to pass to the interportal distance. If the tunnel has a single slope, then in tunnel practice one may sometimes encounter the determination of the length along the inclined straight line connecting both end points; however, such a method of determining the tunnel length is not universal, and therefore it cannot be recommended*). In this respect it will be more expedient to refer the tunnel length to the mean elevations of the longitudinal profile of the tunnel.

In the section on triangulation, formulas were given for calculating the length of a tunnel at a height \(h\) above sea level.

To explain the foregoing, one may cite the Lechberg Tunnel, for which \(L_0 = 14316.2\) m; then for elevations of 1200 m above sea level we find

\[ \Delta L=\frac{1200 \cdot 14.316,2}{6367000}=2,692\ \text{m.}\quad L_h=14318,89\ \text{m.} \]

III. Fixing the direction of the tunnel; construction of alignments. The direction of the tunnel axis at each of its entrances must be accurately and firmly marked on the ground by setting an appropriate mark (“center”). In view of possible accidents and of the disturbance of the preservation of the placed “center,” it is necessary to fix it by arranging, if possible outside the area of construction work, an offset along the direction of the tunnel axis at a strictly determined distance. Such an offset, in the form of an alignment, must also be made in the direction perpendicular to the tunnel alignment. The fixing of points on the offset must be at a strictly determined distance, so that in the event of damage to or loss of the main “center” it would be possible to restore it easily and accurately.

The entrances to the tunnel (portals) must be precisely determined and marked on the ground. For this purpose, in the corresponding places on the tunnel axis either thick stakes are driven in, or special concrete blocks are set. Stakes 12 or 15 cm in diameter are sawn off approximately 10 cm below the surface of the ground, in the event that the excavation of the approach cuttings has not been carried out. If, however, the approach cuttings have been completed, then the described mark is set at a depth of 10 cm below the design line of the tunnel excavation. Above the stake there is placed a log crib in one course (a well curb) of approximately \(0.4 \times 0.4\) m in size, and it is covered on top with a wooden shield to protect it from accidental damage. Instead of a stake, a concrete block is sometimes placed with a wooden bar embedded in the concrete, into which a nail is driven or a screw is screwed. The nail or screw must lie strictly on the line of the tunnel alignment.

IV. Calculation of the angles, directions, and lengths of side adits (windows). The direction of the tunnel with all its curvilinear sections is plotted on the plan of the locality. Then, on the drawing, the direction of the side adit is set.

* Determining the length of a single-slope tunnel along an inclined straight line is sometimes needed in hydraulic tunnels.

If the tunnel is curvilinear and a side adit adjoins the tunnel at any angle (Fig. 61), then it is necessary to determine the angle \(\beta\) formed by the direction of the axis of the side adit with the tangent \(T_1T_2\), drawn at the point \(E\) where the side adit meets the axis of the curvilinear part of the tunnel. For this purpose it is highly expedient to make use of the polygonal traverse \(P A B C D E F G P\).

Fig. 61.

The initial point for setting out must be the point \(E\), belonging to the side adit and located on the axis of the tunnel. This point is obtained, in accordance with the plotted polygon, by setting out the angle and by precise measurement of the adit (window). When point \(E\) has been determined on the ground and fixed, it is necessary to set out the tangent straight line \(T—T\). The setting out of the tangent straight line is possible only on the basis of preliminary work in plotting the polygon. The calculation must give the angle \(\beta\).

Knowing \(\beta\), we can construct the tangent \(T—T\) to our curve. The further construction, and indeed the setting out of the curve itself, presents no difficulties: in preliminary setting out it is carried out by the method of successive chords, and in final setting out by the method of the inscribed polygon. In order to be able to plot the surveyed polygon on paper, one may use rectangular, rectilinear coordinates.

Part II. Setting Out the Tunnel Axis Underground.

§ 1. Sighting marks.

Sighting marks must serve for work on setting out the tunnel entrances, and then the guide adit along its entire length, and finally for periodic checks of the correctness of the position of the tunnel axis as the tunnel driving advances into the rock mass.

Sighting marks must possess the following qualities:

1. The best possible visibility, for which there must be an appropriate selection of colors both for the background and for the sighting lines.

2. The arrangement of the sighting lines must be geometrically correct.

3. There must be ease of setting up the sighting mark and ease of adjustment.

4. The construction of the sighting mark must be simple, not intricate, strong, and immovable once installed.

According to their location, all sighting marks are divided into two categories:

1) Sighting marks placed outside the tunnel. They may be intended:

a) for daytime work.

b) for nighttime work.

2) Sighting marks placed inside the tunnel.

Sighting marks placed outside the tunnel. The purpose of these sighting marks is to make it possible to transfer the direction of the tunnel axis, according to the setting-out on the earth’s surface, into the interior of the rock mass.—This responsible and difficult work encounters the major inconvenience that, when working by day, one has to sight the tunnel line under conditions that differ at the various places of work: bright daylight illumination of the sighting marks located on the daylight surface, air heated at the surface—on the one hand; dark spaces, barely distinguishable luminous sighting marks, and a completely different state of the atmosphere inside the tunnel—on the other hand. Added to this is the further inconvenience that there is usually much activity at the tunnel entrance: workers enter and leave the tunnel, rock is taken out, and construction materials are brought in. Therefore, alongside the use outside the tunnel of sighting marks intended for daytime work, sighting marks exclusively for nighttime work are also used, when the working conditions outside and inside the tunnel become somewhat more equal and, in general, more favorable. Nevertheless, there is no possibility of abandoning entirely the use outside the tunnel of sighting marks adapted exclusively for daytime work. For small tunnels and under simplified conditions of tracing and setting out, daytime sighting marks are still used quite often. For long tunnels, however, it may be recommended to use sighting marks adapted exclusively for nighttime work, and to refrain from setting up marks intended for daytime work.

Sight marks must be set up in places clearly visible from the observatories. The plane of the sight mark must be placed strictly perpendicular to the direction of the sight line.

Sight marks should be set so that they do not obscure any of the fixed points of the alignment.

Sight marks for daytime work. The simplest type of sight mark for daytime work is a square board fixed to two posts, made of planks covered with white paint and bearing a black cross (Fig. 62).

When setting out the Albulа Tunnel, the sight marks consisted of wooden plank boards measuring \(2.0 \times 2.0\) m, nailed to two firmly installed posts. On the white background the cross was black, made of strips 2 cm wide. The cross was finally applied after 10-fold sighting.

Fig. 62.

When setting out the Lechberg Tunnel, the sight marks consisted of wooden plank boards measuring \(1.2 \times 1.2\) m. The boards were set on concrete foundations; their coloring, in red or black paint on a white background, was made according to the drawing shown in Fig. 63.

When setting out the tunnel of 521 versts on the North Donets Railway, exactly the same sight marks were used as for the Lechberg Tunnel.

In Fig. 64 a sight mark is presented in the form of a wooden board set on a concrete foundation. The design is applied in black paint on a white background. The narrow white cross in the middle of the design has a width of 1 cm, while the height of the narrow cross is 25 cm. The cross stands out sharply against the black background. In sighting, the white cross is found very easily.

Fig. 63.

Sight marks for nighttime work. In the matter of designing sight marks for nighttime work, two stages of historical development may be noted. Formerly, sight marks were used in which the sighted line was dark (black, shadowed) on an illuminated background. Illumination of such marks was achieved by means of kerosene lamps. In recent years there has been a transition to lamps in which the sighted line, on the contrary, is brightly illuminated against a black shadow background. Illumination is achieved by means of carbide (acetylene) or electric lamps. It has been proved that a bright white light strip on a black background is much easier to observe than a dark body near a bright source of light. Therefore, from the time when the need for sighting increasingly and more insistently arose, ...

…sighting over considerable distances, as well as with the use of light sources of high brightness, the transition to luminous-stroke sighting marks is quite understandable and natural. From tunneling practice one may derive the width of both the black and the white (luminous) stroke of sighting. From the practice of a whole series of tunnels built in the nineteenth century, a black stroke proves clearly visible when 2 mm wide and at a distance of 128–156 m; this gives an angle of approximately 4″.

From the practice of setting out modern tunnels it is known that the visibility of a bright luminous stroke is about 1″ and even less. Thus, the accuracy of sighting with a luminous stroke may be taken as higher than with a shadow stroke.

Fig. 64.

When setting out the Simplon Tunnel, the sighting marks were made so that they would be visible not only by day but also by night. The sighting mark consisted of an iron sheet measuring 1.0 × 1.0 m, painted black, with a white circle drawn on it and with a hole of 1.0 cm. At night a lantern was placed behind the sheet. In Fig. 65 is shown the device for the precise setting of this sighting mark: \(b\)—a cushion, fastened immovably, \(e\)—a screw for displacement, \(h\)—a screw for fixing.

When setting out the Suram Tunnel, the sighting marks consisted of a stone post (Fig. 66) about 1.3 m high. Its upper part, to a height of 0.5 m, was arranged in the form of a niche, into which an ordinary kerosene lamp was placed. In the window of the niche, facing the theodolite, a wooden frame with frosted glass was set in, on which the axis of the tunnel

Fig. 65.

was marked by a plumb opaque line (the old type of sighting mark). It was convenient to prepare such a line from thick paper cut in the form of a strip, painted black and pasted onto the glass. For this purpose the frosted surface of the glass was turned outward. The recess was locked with a door, and the glass was closed with a shutter.

Fig. 66.

In the construction of the Suram Tunnel another type of sighting mark was also used (Fig. 67). To a wooden stand, by means of a screw, there is fastened a long bar with a longitudinal slot into which a frame with frosted glass is inserted. By means of projections the frame is held on the bar and can move along it. Along one edge of the bar there is a millimeter scale in the form of a silvered copper plate, and on the projection of the frame facing the scale there is made an index in the form of a fine vertical line, the end of which moves along the scale when the frame is shifted. One end of the bar is made movable in a vertical plane, for which purpose the opening in the stand through which the screw of this end of the bar passes has an elongated form. Thanks to this it is possible to set the apparatus so that the vertical line drawn on the glass is strictly plumb.

Fig. 67.

Modern types of sighting marks intended for night work outside the tunnel consist of concrete foundations and posts 1.3 to 1.5 m high, with a recess arranged at the top and covered above by a roof of galvanized iron. The open front part of the recess must face the direction of sighting and, during the time when the work...

are not being carried out, is closed with a lid. A metal box is inserted into the niche. One wall facing outward is made of glass or metal and has a narrow slit for sighting. A powerful light source is placed inside (an electric or acetylene lamp).

Another type of sighting mark is as follows: a foundation, with a post on it 1.0 to 1.5 m high. At the top, a gas pipe is embedded in the post to a depth of up to 0.5 m. During the work, a metal rod fitting tightly into the pipe is inserted into this pipe; at the upper end of the rod a cylinder about 10 cm in diameter and 30 cm high is fixed immovably. Along the cylinder there is a narrow slit about 10 cm long. Inside is a powerful light source. Portability and brightness make work with this sighting mark convenient, but each time it is necessary to set the slit in the direction of sighting, so that there is no distortion (phase) during observation.

Finally, it is necessary to mention portable sighting marks intended for night work outside the tunnel. In their details they may be of various designs, but the basic idea of their construction is the same. There is a tripod. A stand is mounted on it. This stand, regulated by three screws, can be strictly centered over the given point by means of a plumb line—either cord or optical. The sighting mark is mounted on the stand. The stand is arranged so that a theodolite or an optical plummet can be mounted on it. The advantage is that, when necessary, without resetting the tripod and without making any shifts, one may use either the theodolite, or the plummet, or the sighting mark.

Fig. 68.

In Fig. 68 a sighting mark is shown in the form of a luminous signal, consisting of a disk of ground glass with some kind of

Fig. 69.

pattern. The signal is arranged so that it can be placed normal (perpendicular) to the line of sight from the instrument’s telescope. For this purpose the signal, in the form of a disk, is mounted on a gimbal ring. Its center coincides with the point of intersection of the axes of the gimbal ring, and after replacing the instrument with the signal, the center of the signal coincides exactly with the center of the sphere about which the instrument rotated when being brought into its various positions. On the disk, as already stated, made of thick ground glass, a series of concentric circles is drawn, the width of whose outline is selected so that there always exists

the possibility of accurately sighting on the signal with the aid of the double vertical thread of the theodolite. In the center of the disk there is an opening for a bright, unobscured beam of light. Any light source may be used for work with the sighting mark described.

In Fig. 69 another type of sighting mark, no less common in tunneling work, is shown; in its details it is of a somewhat different construction, but the idea of the device remains the same: a stand is placed on the tripod, into which either a theodolite, or an optical plumb, or a sighting mark may be set. In this case the sighting mark is made in the form of a black slit of triangular outline on a milky-white transparent (glass) background. In the black triangle there is an opening for a bright, unobscured beam of light. This time the surface of the sighting mark remains invariably vertical.

Speaking of various sighting marks for night work outside the tunnel, one cannot fail to mention an interesting attempt to create a luminous mark visible from all sides—in the form of an electric-arc device. With thin coals, the jumping sparks gave good, sharply defined visibility. The accuracy of setting on a concrete post is easily achieved, which is also an advantage.

Sighting marks placed inside the tunnel. Among the sighting marks used in setting-out work inside the tunnel, various kinds of plumb bobs must be included*).

Fig. 70.

An ordinary plumb bob consists of a weight (bob) weighing from 1 to 4 kg on a copper or iron wire no more than 0.6 mm in diameter. To suspend the plumb bob, the wire is fastened to a bracket (Fig. 70).

When working in wet or highly water-bearing soils, in order that water running down the wire should not change its thickness and thus affect the accuracy of observation, the wire is provided at the top with a metal cap or umbrella (“umbrella wire”) (Fig. 71). The diameter of the umbrella is 0.1 m.

In order that, during observation, no error should arise from lateral illumination of the wire (formation of “phases”), it is recommended to place the lamp behind the wire, while placing between the wire and the lamp, as a screen, a sheet of white paper. Movement of the plumb bob along the bracket when setting it for sighting is usually done by hand. Since setting by hand is slow and involves a number of inconveniences, a special device (Fig. 72) with a micrometer screw is used, by means of which the plumb bob can be set to coincide with the cross of the theodolite threads more quickly, conveniently, and accurately.

Fig. 71.

Sometimes, in order to observe special thoroughness in the work, special electrical devices are used, which signal by a light signal when the wire hangs completely freely, without touching the supports or projections of the rock, and is therefore strictly vertical.

*) Both in the preliminary and in the final setting-out of the tunnel axis, pegs are not used because of their considerable thickness. Pegs, as we shall see below, are used only for setting out the tunnel in the transverse direction when driving to the full profile.

In order that the plumb line not swing like a pendulum, but calm down as quickly as possible, the weights (plummets) are lowered into buckets with water or oil to a depth of about 10–15 cm below the surface of the liquid.

In the construction of small tunnels, under favorable air conditions (cleanliness, absence of dust and, chiefly, fog), and when sighting over a short distance, it is sometimes sufficient to use the following device as a sighting mark (Fig. 73): a candle holder is taken in the form of a suspension and is carefully adjusted so that the axis of the candle coincides exactly with the vertical axis of the suspension. The suspension is passed through a hole in the lower (bent) part of a plate. The plate has a hole in its upper part and can be placed over a nail. By driving a nail into the cap-piece (the upper timber of the drift support) of the guiding adit, we obtain the direction of the tunnel axis. Several small holes are made in the plate, intended not for the nail but for the suspension. When several sighting marks have been set along the tunnel, it is sometimes necessary to place the candle flame, although exactly in one vertical plane, at different levels. This makes sighting more convenient. The sighting marks of the described construction closest to the observer are hooked by the suspension into the lower hole; the next ones, farther from the observer, have their suspensions somewhat higher. It is also important that the cap-pieces of the adit should not be at one and the same level.

Fig. 72.

Fig. 73.

Fig. 74.

Fig. 75.

Another type of suspended sighting mark, used for underground setting out of the tunnel axis, consists of the following (Fig. 74): a cord is taken. At its end, by means of a hook, a ring is attached, from which two small chains run, supporting a plummet (a lead plumb bob). The plummet is made hollow, is filled with colza oil, and has a wick for burning. The bottom of the plummet, in the form of a cone, and the wick are strictly centered along the axis of the cord.

In Fig. 75 an analogous sighting mark is shown in the form of a lamp of a somewhat different design, having no open flame and therefore intended for work in tunnels containing firedamp.

All sighting marks constructed in the form of a lamp suspended on a cord have the important drawback that it is not always easy to arrange for the flame to lie exactly on the axial line of the cord.

The methods of sighting described are applicable only at comparatively short distances, since insufficient illumination makes it impossible to distinguish the cord at distances as little as 150–200 m. If the air inside the tunnel is saturated with vapors, the visibility of the cord rapidly falls, sometimes even to 20–30 m. In such cases, and in general when it is necessary to sight over longer distances, sighting marks with a bright flame are used, closed by a metal screen with a slit, or else by dark glass with a transparent stroke drawn on it. In Fig. 76 various types of such slits, used for metal screens, are shown.

Fig. 76. Types of screens for luminous sighting marks.

The tripod is intended for the temporary installation of an instrument or a sighting mark (Fig. 77). The tripod consists of two parts: a strong, stable tripod up to 1 m high, with a large supporting circle 0.5 m in diameter, and a special cast-iron base, set horizontally by means of three leveling screws. This base is designed so that in its upper part the so-called “slide” can, by means of a special screw, move smoothly sideways up to 30 cm; moreover, the accuracy of the lateral setting can be refined by means of an additional micrometer screw. The instrument is set so that the direction of the indicated movements is perpendicular to the line of sighting. The middle of the “slide” is centered with the aid of a plumb line. On one side of the frame along which the “slide” moves, millimeter graduations are marked.

Fig. 77.

The acetylene lamp has a semi-cylindrical tin screen, polished on the inside and having a narrow slit for sighting, which, by means of

…of a special shutter may, if desired, be made narrower or wider. Usually work is carried out with the slit width at 5 mm. The lamp burner is connected with the gas reservoir by means of a rubber tube. The burning time of the lamp is about 5 hours.

Plumb bobs for setting out a shaft exist in several systems. The mechanized plumb bob of the system of Prof. Schmidt, manufactured by the firm F. Breithaupt et Sohn in Kassel (Germany), is very convenient (Fig. 78). The plumb bob consists of copper wire of suitable length, with a diameter from 0.6 to 1.0 mm. A load (weight) of 8 to 25 kg is placed at the end. The plumb bob is shifted by means of two mutually perpendicular micrometer screws.

§ 2. Setting out tunnels with a straight-line axis.

All geodetic work on the day surface of the earth pursues exclusively the single aim of making it possible to set out the tunnel axis underground absolutely correctly and accurately. Work on setting out the tunnel axis underground is, generally speaking, difficult, since it is carried out in dark and cramped spaces. Water falls in drops from the ceiling, and the atmosphere is saturated with water vapor. Therefore the maximum of attention, accuracy, and technical skill is required.

In tunnel excavation, special attention is invariably always given to the responsible and important work of driving the pilot heading.

The system of work for setting out the tunnel axis must be such that the least expenditure of time is made without detriment to accuracy.

The setting out of the tunnel axis underground differs somewhat in its details depending on the adopted method of tunnel excavation.

In excavation, and consequently also in setting out a tunnel, there may be three characteristic cases:

1. The pilot heading is located below (lower pilot heading). The heading is located on the tunnel axis. This position of the heading occurs when excavating rock by the Austrian method, by the top-heading-and-bench method, and by the sectional method.

Fig. 78. Mechanized plumb bob for vertical tunnel shafts.

1. The pilot heading is at the top of the tunnel profile, on the tunnel axis. This will be the case when excavating a tunnel by the Belgian method, the Austrian method, and the sketch-shoving method.

2. Finally, the pilot heading is laid below, but to the side, and not on the tunnel axis. This will be the case with the German method of tunnel excavation.

The general success of constructing the tunnel also depends to a considerable degree on the success of driving the pilot heading. Therefore the pilot heading is always driven first, with feverish speed. Thus, in terms of the location of construction operations, it always precedes all construction work, such as: excavating and enlarging the tunnel to the full profile, laying the walls, laying the vaults, arranging niches, laying invert arches, and so forth. The driving of the pilot heading precedes the setting out of the tunnel by a length averaging from 200 to 300 m, reaching at most 600 m, and sometimes falling to 50 m.

In connection with such an arrangement of the work, in order to avoid even the slightest delay in driving the guide adit, it is necessary to set out the tunnel as promptly as possible, but in such a way as at the same time to ensure the correctness of the direction. Therefore, in the very process of carrying out the work of setting out the tunnel axis underground, two stages are distinguished: the preliminary setting-out and the final setting-out.

When the tunnel is driven by means of a lower guide adit located on the axis, no substantial peculiarities are introduced into the preliminary setting-out and then into the final setting-out: the final setting-out then has the purpose of finally verifying and accurately and reliably fixing the preliminary setting-out, and the deviations will depend only on the accuracy of the preliminary setting-out and, in practice, the points have to be shifted within negligibly small limits, to the right or to the left.

The matter will be quite different when the tunnel is being developed by means of a guide adit located above, or even below but not on the tunnel axis. In such cases the preliminary setting-out of the tunnel axis must, of course, be concentrated in the guide adit, while the final setting-out, in order to eliminate its temporary character, must be concentrated on the tunnel axis, at its invert; that is, in other words, in an entirely different place from that where the preliminary setting-out was carried out.

The preliminary setting-out of the tunnel axis serves to give the adit its direction and to make possible its further driving. Thus, the basic idea of the preliminary setting-out of the tunnel axis is reduced solely to serving the purposes of excavation of the rock. The preliminary setting-out may be carried out by a technician or by a senior tunnel foreman. The instruments used are: binoculars, a measuring tape or band, and then a rod with a plumb line.

To indicate the direction of the tunnel axis during the preliminary setting-out, it is quite sufficient to use as setting-out marks:

in hard rock—to drill a hole (blast-hole) in the invert and drive into it a wooden peg with a diameter of up to 0.05 cm,

in soft ground—to drive a wooden pile. The diameter of the pile is up to 0.20 m. The driving depth is up to 0.5 m.

Both into the peg and into the pile a nail is driven, the head of which serves to mark the direction of the tunnel axis.

Above each setting-out mark, in the roof of the adit, strictly on the plumb line above the center of the setting-out mark, a nail is driven into the timbering, or, still better, a clamp is fixed, to which the plumb line is attached.

The final setting-out has the purpose of checking the preliminary setting-out and finally fixing it so that, in subsequent tunnel works, it will be possible to have at one’s disposal entirely correct and reliable, accurately fixed points.

During the execution of the final setting-out of the tunnel axis underground, it is absolutely necessary to create the best possible conditions for the setting-out work. For this purpose, all construction work on the setting-out of the tunnel and on the removal of the excavated rock must be stopped. If the ventilation of the tunnel is artificial, then it is necessary to ventilate the tunnel as well as possible, especially the far heading, where the work of setting out the tunnel axis is concentrated. Approximately three hours before the beginning of the setting-out work, artificial ventilation must—

must be stopped. This is necessary so that, as far as possible, a completely calm state of the atmosphere inside the tunnel may be established. It is very convenient and expedient to time the work of final setting-out to coincide with a number of holiday rest days.

The final setting-out must be carried out by an experienced engineer, working carefully and with an awareness of the responsibility of the work entrusted to him. Therefore, in the final setting-out of long tunnels it is sometimes expedient, from time to time, to invite a specialist and entrust him with checking the setting-out of the tunnel. In general, every effort must be made to ensure that the best and most accurate instruments are used, that the most experienced persons work, and that the direction of the tunnel axis is checked periodically.

For the final setting-out of the tunnel axis, before beginning the geodetic work, preparatory work is carried out for the forthcoming marking and final fixing of the tunnel axis underground.

Setting-out marks. Using the results of the preliminary setting-out of the tunnel axis and the planned chainage, setting-out marks are installed. These must be as immovable as possible and arranged so that they are not subject to damage or loss.

The following types of setting-out marks exist for the final setting-out of the tunnel axis underground:

1. In the floor of the heading a drill hole (shpur) is made, into which there is driven:

either a) a round iron rod, on which a file mark is made indicating the tunnel axis,

or b) a gas pipe, the geometric center of which serves as the indication of the tunnel axis.

1. In the floor of the heading a stone of hard rock is embedded. A groove is cut in the stone and filled with lead. Into the lead, along a plumb line, a nail is driven, the head of which serves as the center.

2. In the floor of the heading a concrete block is placed, approximately \(0.5 \times 0.6\) m in size and up to \(0.4\) m high. In the upper part of the concrete block some metal object is embedded: a bolt, an iron rod, and the like, on which a notch is made indicating the tunnel axis. Sometimes a vertically protruding gas pipe is embedded in the block. The use of stones or concrete blocks of this kind is very expedient in tunnels of considerable length and in weak soils excavated without the use of explosives.

3. Into the floor of the heading, to a depth of up to \(0.8\) m, a thick wooden stake up to \(0.12\) m in diameter is driven, with an iron ring (band) at the top. The top of the stake is sawn off horizontally at a height of about \(0.04\) m above the surface of the floor of the excavation. The tunnel axis is marked by a nail driven into the top of the stake. This method of marking the tunnel axis is convenient in soft soils and in tunnels of insignificant length.

4. Into the sleeper of the temporary track laid inside the tunnel, a screw is screwed in, or a dog-spike or nail is driven, marking the tunnel axis. This method of marking the axis, generally speaking, is not very reliable and, thanks to its simplicity, may still be allowed only in tunnels of insignificant length.

1. The tunnel axis is marked on the roof of the heading gallery. If the rock is hard, it is possible to drill a hole, drive in a wooden peg, and then drive a nail into the peg or screw in a hook. If, because of geological conditions, supports have been installed, and if a support happens to be located at the place where the mark must be set, then it is possible to drive nails or screw a hook directly into the frame support (into the cap). The described method of marking the axis is expedient in stable rocky formations and provided that the supports are not deformed under the pressure of the rocks. It is impermissible to drive a nail into a longitudinal support member.

All setting-out marks must be protected against possible damage. For this purpose it is recommended to surround the marks with a wooden curb (a log frame of one course) and to cover them with boards, or to protect them with cast-iron covers, in the same way as is done for hydrants.

Finally, all setting-out marks require some identifying feature—an inscription or indication that there is a mark here determining the direction of the tunnel axis. For this it is often sufficient to make a rough mark on the surface of the rock with oil paint, or else to drill a hole (a shothole) and drive a wooden peg into it, and then nail to the wood a small board with an inscription. One may even limit oneself to inscriptions on the supports. All inscriptions must be uniform and easy to find, and for convenience of orientation they should be placed on one side of the adit.

Setting-out marks are laid at intervals of every 100 m.

The heading gallery is located on the tunnel axis. For the work, a quiet moonless night is recommended, approximately 2–3 hours after sunset. The work begins from the excavation approaching the tunnel. The newly adjusted instrument (theodolite) is set up over the observation point. It is necessary to center the instrument over the point with the greatest care. Then the telescope is sighted on a target mark located outside the tunnel. The center of the crosshairs of the telescope is set exactly on the cross or stroke of the target mark. Then the direction of the tunnel is transferred into the tunnel itself. Here the observer at the theodolite waits for the target mark (a plumb line or light signal). This target mark is moved until it coincides completely with the theodolite crosshair. When the target mark inside the tunnel has been precisely set on the theodolite crosshairs, the direction of the plumb line is marked on the rock mass by applying a geodetic center (a stroke with an inclinometer).

After this the theodolite is transited through the zenith, and the entire operation is repeated, beginning with sighting on the target mark outside the tunnel and ending with the application of a new stroke on the center of the rock mass.

The work is repeated several times, making several sightings with the theodolite in face right and face left, and finally by turning the telescope in the horizontal plane through 180°. At least four observations must be made; however, it is recommended to make 8 observations with different positions of the telescope and take the mean.

If, for the point (center) being marked, the difference in the extreme readings (strokes) is less than 10 mm, then the mean of all eight readings is accepted as the true direction of the tunnel axis. If the deviations are more significant, the number of observations is increased.

When the direction of the tunnel has been determined from the first theodolite station, the latter is moved to another place.

For this purpose, in the floor of the excavation approaching the tunnel, strictly on the tunnel axis, stone posts with a concrete triang—

by a level. If it is impossible to install such marks because of cramped working conditions, then, for the same purpose, low concrete or stone blocks are set up in the floor of the approach excavation, with pieces of iron tubing not more than 20 mm in diameter embedded in them. The center of the tube indicates the place for setting up the instrument.

Having set up the theodolite here, its telescope is directed at the sighting mark and, in the manner described, the direction of the tunnel axis is again taken, sighting into the tunnel. Thus, by moving the instrument to a new position, it is possible to check the direction of the tunnel axis being set into the tunnel, proceeding from two different positions of the instrument.

Sometimes, if there is an observatory at the tunnel entrance and if the observatory is favorably situated in height, it is possible to use it as well and to sight the tunnel axis into the tunnel directly from the observatory.

When the geometric center has been finally marked on the block placed inside the tunnel, the theodolite is transferred and accurately set up over the center just marked. At the place where the theodolite has just stood, a sighting mark is set up. On the next picket, where a block has been prepared, a sighting mark is likewise set up. After this the work is begun. The telescope of the theodolite is directed at the first sighting mark, which served as the base during the first theodolite station, and also at all the sighting marks now visible from the tunnel. If a discrepancy in sighting should occur, it is recommended to interrupt the work and return with the theodolite to the first station and determine the cause of the discrepancy, remembering that any inaccuracy, however seemingly insignificant, must not be passed over in silence, since it may subsequently lead to very large discrepancies.

Thus, when the direction of the theodolite telescope has been finally given and fixed, the telescope is turned through the zenith and directed at the sighting mark located at the next picket inside the tunnel. In a manner analogous to the preceding one, repeating the sighting at different positions of the telescope, strokes are marked on the “center,” and in this way the direction of the tunnel axis is finally established.

It is recommended that the first theodolite station inside the tunnel be used to the greatest possible extent: usually it is possible to sight the tunnel axis not to one, but to several pickets into the tunnel, each time turning the telescope through the zenith and, with the circle right and circle left, taking readings in turn on all sighting marks visible from the given point in the tunnel and located outside the tunnel. The sighting distance inside the tunnel depends on a whole series of causes: on the optical power of the telescope, on the state of the atmosphere inside the tunnel (cleanliness, degree of humidity, air current, and so forth), on the brightness of the illumination of the signals, on the experience of the person carrying out the work, and on certain other causes. Usually the sighting distance inside a tunnel averages about 300 m. Under the most favorable conditions the sighting range reaches even 1.0 and 1.3 kilometers; under unfavorable conditions it falls to 75 and even 50 m.

Direct sighting along the guide points reached, in the St. Gotthard adits (at night), up to 1300 m, and in the Lötschberg it reached up to 2550 m. However, one should not be carried away by the power of the instrument. It is recommended to work with more limited distances, since the air inside the tunnel is not always sufficiently clear to see lamps at a great distance. In the Lötschberg Tunnel they usually worked at distances of 600 m, sometimes even at 400 and even at 200 m, in connection

with fog. In exceptional cases it was scarcely possible to see even 30 m. In driving tunnels on the Western and Eastern Amur railways it was established that the visibility of sighting depends on the density of the fog. There were cases when, at a distance of 25–20 m, absolutely nothing could be seen, while at 10 m the blurred outline of a torch or acetylene lamp was visible. In order to distinguish the signal itself, it was necessary to approach to a distance of 5 m. It is clear that, with the atmosphere in such a condition, the work had to be interrupted and the tunnel ventilated.

When a series of points (pickets) has been established inside the tunnel and their “centers” have been precisely marked along the tunnel axis, some specialists recommend that, before proceeding farther into the tunnel with the theodolite, one should return once more to the point situated outside the tunnel in front of its portal, set up the theodolite there again, and again check all the points marked inside the tunnel. It must be noted that the state of the atmosphere at the entrance to tunnels is usually unsteady and therefore not very favorable for sighting. It is therefore recommended in this case to be especially precise and careful. Thus all points inside the tunnel will be sighted from two different points.

As the tunnel is advanced into the mountain mass, the principle of the work remains the same. Only, owing to the opacity of the air, the distance between instrument stations is usually taken on the average at about 200–300 m, and no more.

The further setting-out of the tunnel axis must include at least three existing principal points, so that by means of them the invariability and reliability of the position of the newly established “centers” may be ensured. Each new “center” is checked from at least two different theodolite stations determined during the final setting-out of the tunnel.

Execution of work in the final setting-out.
In the final setting-out of the tunnel axis underground, a newly adjusted instrument with a good telescope is set up inside the tunnel and centered over a point located not far from the portal. The instrument must be set up so that pointing the telescope is reduced to movements of the horizontal and vertical threads, and not to rotations of the telescope itself. The cross of threads in the telescope is first directed to the sighting mark and then transferred to the direction of the tunnel. A lamp is placed over the point of the tunnel line that is to be fixed. If the length of the sight is long, then for transmitting orders it is often convenient to use a portable telephone. The distance over which sighting takes place depends on a whole series of causes:

On the power of the telescope.
On the state of the atmosphere inside the tunnel.
On the brightness of the illumination of the signal.
On the experience of the person carrying out the work.

Usually in tunnel work this distance reaches about 300 m, although sometimes, under unfavorable conditions, it falls even to 50 m. Under the most favorable working conditions, the sighting distance has reached even several kilometers.

The point obtained by sighting must be marked. The working procedure must be repeated several times, and each time the position of the telescope must be changed.

In the work on the Simplon, the mean of 8 observations was taken with different positions of the telescope. If, at the point being marked, the difference in the read...

was less than 10 mm, the mean of all eight readings was taken as the true direction of the tunnel. If, however, the deviations were more significant, the number of observations was increased.

Setting out the axis of hydraulic tunnels. In essence, setting out the axis of hydraulic tunnels, both pressure and non-pressure, sometimes called adits, is only a special case of the works considered, when the guiding adit is located at the bottom of the tunnel. A distinctive feature will be that, in most cases, the working space of hydraulic tunnels is very small. Thus, for example, in the Kloster Küblis tunnels the diameter of the excavation was barely 2.5 m; in the Löntsch tunnel it did not exceed 2.7–2.9 m. In the Amsteg tunnel the diameter was 2.3 m. The same was true in the Spiez–Thunersee and Engelberg–Obermatt tunnels. In the Ritom tunnel the diameter of the adits over certain lengths was about 2.0 m. In the Lücer a/d. Plessuz tunnel the diameter of the excavation did not exceed 1.75 m. Many other tunnels could be cited as examples, the excavation of which was even smaller.

With such a small cross-section, a hydraulic tunnel has the form of an adit, and the excavation of the rock in such cases is usually carried out immediately to full profile. This cannot fail to affect the performance of geodetic work as well. The influence of certain circumstances accompanying tunnel excavation—water-saturated ground, and even whole streams of water, the cramped and narrow spaces of the working—sometimes makes it necessary to set up the theodolite and sighting marks not on tripods, but on horizontal struts specially installed each time (Fig. 79). A second feature is the transfer of geodetic centers upward. Since lining work in hydraulic tunnels in most cases begins from below and proceeds gradually upward, it is therefore advisable to recommend transferring upward the geodetic points for setting out the tunnel axis.

Setting out the axis of hydraulic tunnels, as usual, begins from the entrance (portals). The theodolite is placed inside the tunnel and is set up by sighting marks placed outside the tunnel. Subsequently, work on setting out the tunnel axis presents no particular difficulties or distinctive features compared with setting out the axis of the adit of a large tunnel.

Hydraulic tunnels are built with a single slope and with a rather considerable longitudinal gradient. When setting out the axis, simultaneously with work by theodolite, leveling work also has to be carried out, which is likewise characteristic of these tunnels.

As the tunnel is advanced, the principle of the work remains the same; only, owing to the opacity of the air, the distance between instrument stations is usually taken on average at about 80–120 m, but not more.

During the period of tunnel construction, its setting out must consist in periodically checking the tunnel axis and leveling its invert within the heading being driven. Such setting out must be carried out as necessary, but not less than once a month.

The guiding adit is located above, on the tunnel axis. The special feature of the work of setting out the tunnel will consist in the fact that, during the preliminary setting out of the tunnel axis,

for directing the upper pilot adit, it is usually necessary to set, below at the sill (the lower timber) of the adit, setting-out marks to indicate the direction of the axis; but then it is no longer possible to rely on them for the final setting-out.

The final setting-out must be carried out when excavating the rock to the full area and must be fixed on the tunnel axis below the outline of the tunnel profile. It is therefore clear that, in the course of construction work on the excavation of the rock, all setting-out marks that have been placed in the upper pilot adit to fix the direction of the tunnel will inevitably be destroyed.

Therefore the preliminary setting-out, concentrated in the upper pilot adit, must be carried out in such a way that, while based wholly on the final setting-out of the tunnel at the preceding

Fig. 79.

lengths (pickets) of the tunnel, it would at the same time not serve as preparatory work for the final setting-out. In other words, the final setting-out must be performed independently of the preliminary setting-out.

For this, the preliminary setting-out concentrated in the upper pilot adit is carried out on the basis of the data of the final setting-out; for this purpose it is periodically necessary to transfer the final setting-out from the lower adit to the upper one and, in this way, to check the preliminary setting-out.

An important circumstance is that the setting-out of the guiding axis of the tunnel, carried out in the upper pilot adit and,

as has been said, while preliminary, must at the same time serve as the basis for setting out the transverse profile in the upper part, the so-called calotte.* Thus we see that setting out the upper guide adit acquires the significance of final setting-out, since any corrections and displacements of the vault, should it be incorrect, become impossible. Therefore the procedure is as follows: the upper guide adit is driven by the usual method of preliminary setting-out, with periodic checking of the direction according to the data of the final setting-out.

The final setting-out is concentrated in the lower guide adit. When it becomes necessary to provide setting-out for excavating the calotte and laying the vault, or when it is necessary to check the direction of the preliminary setting-out in the upper guide adit, the final setting-out is transferred upward into the upper adit. For this purpose vertical workings are used, connecting the upper and lower adits and serving for dumping rock; they are called chutes, and sometimes also serve for the movement of people, in which case they are called traveling shafts. The usual size of traveling shafts is \(2.0 \times 1.5\) m, and of chutes \(0.85 \times 0.85\). Using bremsbergs for tunnel setting-out work is inconvenient and is therefore not recommended.

Transferring the points of the final setting-out from the lower adit to the upper is done as follows: the final setting-out of the tunnel axis in the lower adit is continued as far as possible into the mountain, to the end of the adit in the direction of the face. Then three successive traveling shafts or chutes are selected, the farthest from the portal. Above, in the upper adit, over the opening of the shaft, approximately on the tunnel axis, guided by the preliminary setting-out of the axis, a bracket is placed in the cap-piece of the timbering; if there is no timbering, then it is placed in the rock with the aid of rings (plugs) driven into drill holes, in a direction perpendicular to the tunnel axis. The middle of the bracket should be approximately on the tunnel axis. A plumb line on a thin wire is fastened to the bracket. The upper part of the wire is twisted onto the bracket. The plumb line is passed through the shaft and brought down to the bottom of the adit at a distance not reaching 12–15 cm from the floor of the lower adit. So that the long plumb line will sway as little as possible and will settle more quickly, it is recommended to place tubs with water or with oil below and to lower the plumb weight into them. When the plumb lines are set, the theodolite is set over one of the points (pickets), with the instrument centered strictly over the geodetic “center” of the mark. Plumb lines are set up on two adjacent points, fixing them on brackets and again centering them with proper accuracy. It is recommended not to limit oneself to sighting the telescope on the plumb lines located at the nearest points, but also to set the instrument by the plumb lines of distant points, since the thickness of the wire and its improper illumination (phase) may play some part in the inaccuracy of setting up the instrument. When the instrument has been finally set up, one of the workers goes to the upper guide adit and positions himself at the plumb line passed through the shaft. The plumb line must be visible in the telescope exactly at the intersection of the threads. The telescope remains fixed; the plumb line is moved. For this purpose the worker at the instrument gives orders to his assistant in the upper adit, who is standing at the top of the plumb line. According to these orders (whist—

* This is an all the more responsible matter because, with the Belgian method of carrying out the excavation, the tunnel lining, as is known, is first executed above in the form of a vault resting at its springings directly on the rock.

... (or sometimes by telephone) the plumb line is shifted to the right or to the left. Care must be taken that the plumb line, throughout its entire length, does not touch the lining of the funnel walls or the rock, and that the plumb line has come to rest before a command is given for a new displacement of it. When the plumb line has been brought into its proper position on the tunnel axis, it will be very expedient to fix its position by setting a block beneath it, placing it in the invert of the lower pilot drift, and, in addition, to make a mark (inscription) of the tunnel axis on the bracket.

Then, in succession, the work is carried out in the manner described for the two other plumb lines lowered through the other funnels. By these three plumb lines, which determine the direction of the tunnel axis, setting out is performed in the upper drift, and its direction is given for the excavation of the tunnel.

The pilot drift is located below, but to the side, away from the tunnel axis. As has already been said, such a position is obtained when the German method is used for excavating the rock (the core-support method). As can be seen from Fig. 80, the supporting core is located precisely where the final setting out of the tunnel axis ought to be carried out. It remains for a fairly considerable period of time: the pilot drifts are driven, they meet in two opposite headings, the excavation is extended upward, the rock is excavated along the contour of the tunnel, the foundations of the walls are laid, and even the laying of the tunnel vault is begun, while the supporting core continues to stand. And only when the vault has been brought out and closed at the crown, the need for the supporting core ceases, and it is to be excavated and removed. Thus we see that to wait until the place for the final setting out of the tunnel on the tunnel axis becomes free would take too long and would lose all significance. Therefore both the preliminary and the final setting out have to be carried out in the pilot drift to the side, away from the tunnel axis. In its methods, the execution of the work of setting out the direction of the tunnel differs in no essential way from the execution of setting out in the case of a lower pilot drift on the tunnel axis.

Fig. 80. Setting out a tunnel when the pilot drift is located below, but to the side of the tunnel axis.

The particular features of the execution of the work for setting out the drift are as follows: the cross-section of the tunnel is drawn according to the design, the tunnel lining is plotted, and the position of the drift is assigned. The distance of the drift axis from the tunnel axis is determined.

The axis of the drift is laid off to the side from the tunnel axis and is marked either by setting concrete blocks or by driving stakes. When

the direction of the adit axis has been precisely marked, the actual setting out of the adit axis underground is then carried out. In setting out curves, a correction is introduced for the deviation of the direction of the axis. In all other respects, the setting out of the adit axis underground does not differ in any way from the setting out of the tunnel axis.

Examples of setting-out work toward driven adits are not numerous, since the German method of driving is used rather rarely.

The most remarkable case occurred in the construction of the Triebitz tunnel in Germany. The length of the tunnel is 3860 m. Fig. 80 shows the cross section of this tunnel and the position of the pilot core and the side adit. All the work, both for the preliminary and for the final setting out, was concentrated in the lower side adit (in the drawing the supports are shown as not dismantled). Plumb lines “A” were used. To set out the tunnel in the upper adit, it was necessary to transfer the points of the final setting out. For this purpose, as is clear from the drawing, plumb line “B” was first lowered until it fully coincided with plumb line “A,” which was checked with a theodolite. Then, knowing the distance of the adit-axis offset from the tunnel axis, point “C” was found on the tunnel axis and fixed by placing a plumb line on bracket “C.” The vertical transfer of the tunnel axis and the finding of point “D” on the upper bracket no longer presented difficulties. For the sake of refinement, the entire operation was carried out 8 times, and each time a mark was made on the brackets with a file. Then the mean was taken.

§ 4. Setting out tunnels with a curvilinear axis. The execution of work for setting out the tunnel axis underground, in the case of a curvilinear layout of the structure in plan, in essence does not differ in any way from similar work on the ground surface, provided, of course, that the appropriate correction is introduced for all the special features of tunnel setting-out underground: darkness, poor visibility due to dampness, narrow spaces, cramped conditions for carrying out geodetic work, etc.

Analogously to the setting out of a straight tunnel axis, when setting out a curvilinear axis one should distinguish:

a) preliminary setting out, and
b) final setting out.

Preliminary setting out, as usual, also in the case of a curvilinear axis, has the purpose of giving direction solely for driving the heading adit. The work is performed by a technician and requires a measuring rod or tape.

The preliminary setting out of the tunnel axis is most conveniently carried out by the “method of successive chords.” This method, as is known, consists in the following: from tangent A (Fig. 81), on the extension of the straight line AT, some length \(S\) is laid off, which we take as the chord. Then we have

\[ y = \frac{S^2}{2R}, \]

whence

\[ y_1 = 2y = \frac{S^2}{R}. \]

And therefore, in practice, the work process is reduced to the following operations: for a given radius of the curve \(R\), \(S\) is selected and \(y\) and \(y_1\) are computed. Then \(S\) is laid off on the extension of the straight line and a stake is driven. From the driven stake a tape with measure \(y\) is stretched, and from A a tape with measure \(S\) is stretched. At the place where the stretched tapes meet, point I of the curve is obtained and a stake is driven. Then \(S\) is laid off on the extension of the straight line AL. A tape with measure \(y_1\) and another with measure \(S\) are stretched, obtaining point II of the curve, and so on.

In view of the fact that the preliminary setting out is carried out in the heading adit, in cramped and narrow spaces, the magnitude \(S\) is limited in its size.

If the width of the guide adit “\(a\)” and the radius of the curve on which the tunnel is located, \(R\), are taken, then, according to Fig. 81, we have the following relation:

\[ \frac{S}{2}=R\sin\frac{\alpha}{2} \]

with a sufficient degree of accuracy one may assume that \(y_1=2y=\frac{S^2}{R}\)

and this must be less than \(\frac{a}{2}\),

whence

\[ S<\sqrt{\frac{R\cdot a}{2}} \]

In practice, when setting out adits 1.6 to 2.2 m wide, the angle \(\alpha=3^\circ\), and then the chord \(S\) will be, for \(R=200\) m, \(S=10.48\) m; for \(R=300\) m, \(S=15.72\) m; for \(R=400\) m, \(S=20.96\) m, etc.

From consideration of the above-described method of setting out the curvilinear axis of a tunnel, it is evident that here the errors of measurements and constructions will accumulate. Generally speaking, the accuracy of setting out will, apart from other errors, be the greater the larger the radius of the curve \(R\) and the smaller \(S\).

For marking the points in this preliminary setting-out, the instrument most often used is a plumb line, suspended at the top of the adit from a nail or clamp, and whose cord is well illuminated, especially if a sheet of white paper is placed behind it.

Fig. 81.

Final setting-out. For the final setting-out of the axis of a tunnel situated on a curve, the method of the inscribed polygon is used.

This method of setting out a curve is the best possible for the difficult conditions of tunnel work, connected with the cramped working space.

The setting-out consists in the following (Fig. 82): let point \(A\) be the beginning of the curve. Points I, II, III, etc. must be marked inside the tun—

Let us suppose that \(\overline{A-I}=\overline{I-II}=\overline{II-III}\), etc. Having computed the angle \(\alpha\) for the radius of the curve \(R\) and found from the angle the chord \(S\), we set up the theodolite at point \(A\) and direct the telescope along the tangent \(AT\). Having taken a reading on the limb, we turn the alidade through the angle

\[ \frac{\alpha}{2} \]

and sight along the telescope to the line \(\overline{A I}\). On the line \(A-I\) we measure off the length \(S\) and obtain point \(I\), where we place a mark. In this way we finally fix the first point \(I\) of the curve. After this we transfer the theodolite to point \(I\) and direct the telescope to point \(A\), take a reading on the limb, give it the angle \(\alpha\), transit the telescope through the zenith, and sight along the telescope the line \(I-II\); on it we lay off \(S\), obtain the second point \(II\) of the curve, and finally fix it. The work is carried on in the same manner thereafter.

Fig. 82.

An instructive and interesting example of setting out the axis of a curvilinear tunnel is found at km 5.0–5.3 of the Chur–Arosa road in the canton of Graubünden in Switzerland. Initially there was no tunnel here. Rockfalls in the mountains made it necessary to construct the first tunnel, 154 m long. However, in 1913 such considerable landslides and displacements of the mountain mass through which the tunnel had been driven occurred that the tunnel lining became severely deformed. Concerned with traffic safety, it was decided to reconstruct the tunnel. According to the new design, its total length is 283 m. This second tunnel lies deep within the mountain mass, in stable rock. 216 m of the tunnel was built anew. 32 m remained from the old tunnel without alteration, and 35 m was rebuilt with the lining strengthened by reinforced concrete.—The tracing and setting out of the new tunnel were carried out as follows: the route of the line along the highway was restored and surveyed. This route served as the basis for further constructions (Fig. 83). Having the route of the highway and the plan of the mountain mass surveyed in horizontal contours, they plotted on the plan the projected axis of the new tunnel, marked the directions of the side adits, determined the angles formed by the directions of these adits with the route of the line along the highway. Then they proceeded to set out the side adits by means of alignments. Having determined on the route of the old highway the direction of an adit, they then marked on the ground points \(A, B, C\) and points \(D, E\), and \(F\). Then work was begun from the Chur side. Having accurately restored the axis of the old highway and marked tangents 2 and the vertex of the turning angle 3, corresponding to a central angle of \(28^\circ 40'\), they gave the direction of the axis of the entrance part of the tunnel. From the Arosa side they likewise accurately restored the axis of the old highway and, by the method of the described polygon, set out a curve of radius 60 m with a central angle of \(49^\circ 40'\). On

over a short stretch this curve enters the old tunnel. On this extension the curve was set out by the method of the described polygon; then a straight insert was marked out from the Khur side, and farther on a curve of radius 60 m was set out by chords. Meanwhile work began in the side adits. When the side adits reached the tunnel axis, the adit was enlarged and, guided by the data of the office work, the direction of the tangent to the curved axis of the tunnel was marked. These were angles of 96°32′4″ and 83°27′56″. Having these points, they began in the tunnel to set out the curve by chords. The breakthroughs took place on May 29 and May 25, 1914. After this the final setting-out of the tunnel was checked. The discrepancy proved not to exceed 2 cm.

Drawing 83. Setting-out of a curvilinear tunnel on the Khur–Arosa railway.

A remarkable case of setting out a curvilinear tunnel occurred during the construction of the Weberkogel tunnel, located on a rather sharp curve of radius 189.62 m. The tunnel was driven by the German method, using a supporting core and a side guiding adit. The latter circumstance, in connection with the curvature of the tunnel, involved major complications in setting out the work. The axis of the side adit was located at a distance of 3.26 m to the side, outside the curve. The side adit had to be set out as curvilinear, with radius \(189.62 + 3.26 = 192.88\) m. The setting of plumb lines “A” (the notation is given with reference to Drawing 80) presented no difficulties. There were also no difficulties in carrying out plumb line “B”; but when it was necessary to determine the tunnel axis and to find points “C,” each time corrections had to be computed and introduced for reducing the curvature of the axial longitudinal line of the tunnel. This proved troublesome and, in the course of the setting-out work, was associated with errors. However, the small length of the tunnel—360 m—considerably reduced the amount of troublesome work and lessened the uncertainty in the setting out.

Sight adits. If a tunnel, even over only an insignificant part of its length, is situated on a curve at its entrance portion, this somewhat complicates the tracing of the tunnel. Therefore, for the purpose

of simpler and more convenient tracing, and then, in order to facilitate setting out the tunnel axis underground, recourse is had to the method of the so-called “sighting adits.” Usually, sighting adits have no other technical justification than to serve for a more convenient and accurate setting out of the tunnel axis.

Examples of sighting adits are few.

In the drawing: “sighting adit”; “tunnel axis”; \(R = 277.4\ \text{m}\).

Fig. 84. Sighting adit of the Suram Tunnel. (Surveyed in 1927.)

During the construction of the Suram Tunnel in the Caucasus, \(3972.71\ \text{m}\) long, in its western part, located on a curve of radius \(277.4\ \text{m}\), a “sighting adit” was driven (Fig. 84), now sealed with masonry and visible from outside the tunnel on the side of Tsipa station.

The Simplon Tunnel, as is known, consists of two single-track tunnels located \(17.0\ \text{m}\), axis to axis, from one another. In both its southern and northern parts, on the side of Brig and Iselle stations, the tunnel is set out on a curve of radius \(300\ \text{m}\). In view of the fact that the tracing

this greatest railway tunnel in the world—its length is 19.87 km—presented considerable difficulties: “sighting adits” were driven (Fig. 85).

The St. Gotthard tunnel, in its southern part, toward Airolo, is set out on a curve of radius 300 m. In order to ensure as far as possible the accuracy of the work in setting out the tunnel axis underground, a sighting adit was specified. Its construction made it possible to trace the tunnel axis on the surface of the earth in a straight direction from the northern observatory to the southern observatory; moreover, in the northern part, on the Göschenen side, it was necessary to drive a sighting adit in a small spur of the ridge. Thus, here too there was formed an adit situated separately from the main tunnel.

Fig. 85. General plan of the arrangement of the sighting adits of the Simplon Tunnel.

Sometimes the construction of a sighting adit serves not only to make it possible to carry out correctly the work of tracing and setting out the tunnel axis, but also to provide the best ventilation. A fine example of this is the circular tunnel in Switzerland, set out by means of a sighting adit.

Fig. 86. Sighting adit of the Simplon Tunnel.

Thus, by constructing sighting adits, simplification and accuracy in the setting out of curvilinear tunnels are achieved. The use of sighting adits is highly expedient for curvilinear alignments of the entrances of long tunnels.

§ 4. Vertical and inclined shafts. Horizontally arranged adits (windows). The method of setting out a tunnel by means of

shafts is very convenient and rational. The use of shafts is possible only when the mountain mass above the tunnel is not very large.

It is very rational, for tunnel setting-out work, to make use of those shafts that are intended to enlarge the front of the construction works and that thus have their own economic justification, independently of their use for setting out the tunnel axis. To sink shafts specially, solely for setting out the tunnel axis, is usually very expensive. Moreover, it should be noted that such special shafts are needed only in exceptional cases, when the tunnel has a complex curvilinear layout in plan.

The choice of the locations of shafts, even irrespective of whether they are intended for developing the work front or exclusively for the purpose of setting out the tunnel axis, is determined by the geological and hydrological features of the locality. In those cases where shafts are planned on the axis of the future tunnel or to the side at a small distance from the axis within the limits of the tunnel, it is necessary that the shaft should not subsequently become a path for the inflow of groundwater into the tunnel.

The Gattico tunnel, situated on the Santhià–Borgomanero–Arona line, has a length of 3308.63 m; throughout its entire length it is located on a gradient of 0.00779 and consists in plan of a curve of radius 600 m and length 303.75 m, a straight of length 1762.03 m, a second curve of radius 1000 m and length 412.17 m, a second straight of length 625.45 m, and, finally, a third curve of radius 600 m and length 145.23 m. In view of such a complex layout of the tunnel in plan, its setting out could not but present known difficulties. At the same time, the Gattico tunnel presented difficulties not only for geodetic work underground, but also construction difficulties: the tunnel is on a single slope. Therefore it did not seem possible to drive it from both sides in the usual manner because of the water-bearing nature of the soils. To develop the work front, 4 vertical shafts were sunk. These shafts were also used for setting out the tunnel underground. It is noteworthy that these shafts were sunk not on the tunnel axis, but somewhat to the side.

The Khingan tunnel, 3078 m long, has 18 vertical shafts, intended for developing the work front, for ventilation, for water drainage, and for removal of the excavated rock. In view of the fact that the tunnel is located entirely on a straight line, it was never necessary, for setting out the tunnel axis underground, to resort to the aid of shafts.

Sometimes shafts for setting out the tunnel axis are arranged in the approach cuttings to the tunnel. Let us consider, for example, the arrangement of shafts in setting out the axis of the Dzhadzhur tunnel at km 195–197 on the Tiflis–Dzhulfa line. The Dzhadzhur tunnel, 1705 m long, is located on a straight line. There was neither need nor possibility, from the construction standpoint, to sink shafts along the length of the tunnel itself, since the mountain mass overlying the tunnel is large in its dimensions. On the approaches to the tunnel, however, 8 shafts were sunk: two on the eastern side and six on the western side. On the eastern side, the track on the approaches to the tunnel is located on a curve. Six meters in front of the portal there was a directing and working shaft about 24 m deep. This shaft served for developing the approach cutting and for setting out the tunnel axis. Although the Dzhadzhur tunnel itself is located on a straight line, the approaches to it are located on curves. In setting out the tunnel from the eastern side, it was deemed necessary, for the accuracy of the tunnel-axis setting-out work, to establish a sighting adit 56 m long. This adit was to serve exclusively for setting out the tunnel axis inside the mountain mass. For setting out this...

it was necessary to sink 2 shafts 17 m deep for the gallery. On the western side the approach to the tunnel is likewise situated on a curve of radius 600 m, as on the eastern side, but here the curve runs in the opposite direction and forms the letter S. Then there is a straight insert 1762.03 m long, and thereafter again a reverse curve of radius 1000 m, which again gives the letter S in plan. In view of these two S’s, and chiefly in view of the second S, for the purpose of more accurate setting-out of the approaches to the tunnel, five shafts were sunk on the track axis; of these, 3 shafts were at the beginnings of the curve (characteristic points), and two shafts along the course of the first curve nearest to the tunnel, at approximately equal distances from one another. These five shafts, with depths successively of 23.5 m, 21.5 m, 19.5 m, 16.0 m, and 12.0 m, made it possible to drive the adit correctly and to excavate by the English method. For the purpose of accurately setting out the tunnel itself, it was necessary to set a sighting adit about 48 m long, for which, in the direction of the tunnel line 38 m from the portal, another shaft had to be sunk, 29.8 m deep.

Inclined shafts. Alongside the method of carrying out tunnel works by means of vertical shafts, one may also encounter the use of inclined shafts. Their setting-out undoubtedly presents greater difficulties than that of vertical shafts, but, on the other hand, in construction terms their installation is usually associated with considerable advantages with respect to the removal of rock from the tunnel.

During the construction of the tunnel at 521 versts of the Northern Donets Railway, 75 m from the Rodakovo portal at elevation 166.00 m, an inclined shaft was sunk and brought to the base of the tunnel (to elevation 136.00 m). The shaft was located on the axis of the tunnel and had a gradient of 1 : 1.5. Its length was about 55.5 m, with dimensions of 2.3 × 2.3 m.

Inclined shafts are also used in hydraulic tunnels during the construction of hydroelectric stations. Thus, for example, during the construction of the Trübklamm tunnel, a shaft 134.54 m long was driven at an angle of 45°. The work was complicated by the fact that the beginning of the shaft and its end lie within the mountain mass.

During the construction of the Musconetcong tunnel in North America, 1471.9 m long, two shafts were used: one vertical, about 35 m deep, and the other inclined, with a gradient of 1 : 1.5 and about 82 m long.

During the construction of the Nesquehoning tunnel, an inclined shaft with a gradient of 1 : 1.6 was sunk. The length of the shaft was 160 m.

Horizontally arranged adits (windows). In those cases where a tunnel is located in the slope of a mountain, for carrying out geodetic work for setting out the tunnel axis it is possible to resort to the aid of auxiliary horizontally arranged adits (windows). The use of horizontal adits will be the more rational, the more complex the outline of the tunnels in plan.

Alongside the convenience of using horizontally arranged adits for carrying out the setting-out of the tunnel axis underground, these adits are quite often assigned exclusively for reasons of those advantages that are achieved in the actual execution of construction works: the working face for tunnel excavation is increased, and the removal of rock is conveniently carried out over a short hauling distance. Thus, the use of horizontal adits when setting tunnels in mountain slopes is rather frequent, even independently of considerations connected with setting out the tunnel axis underground.

After works of a general reconnaissance character have been carried out, the points are marked from which the adit workings will be made. All selected locations must have the following justification:

1. The size of the adit excavation (window) should be as small as possible. In practice, on the basis of a number of tunnel works, it may be considered that the excavation (window) on average seldom reaches a length of 100 m, and usually varies within the limits of 40 to 60 m. The section of the window is about \(2.0 \times 2.0\) m.

2. The geological conditions must be sufficiently favorable. It is known (see my work “Fundamentals of Geology in Tunnel Construction,” publ. 1928, p. 10) that when a tunnel is laid into the slope of a mountain there is always a greater chance of encountering weak, strongly disintegrated rock.

3. All places selected for the laying of adits (windows) must possess accessibility and a real possibility of carrying out the excavation.

4. All places selected for the laying of adits must likewise possess good visibility and the possibility of geometrical (polygonal) or trigonometrical (triangulation) connection.

A fine example of setting out the axis of tunnels by means of side adits is provided by a section of the Davos–Filisur railway line (see Fig. 20).

The Silberberg tunnel, 964 m long, has three side adits with lengths of 13.92 m, 21.69 m, and 66.47 m.

The next two tunnels—Brombenz I and Brombenz II—owing to their insignificant length, 113 m and 232.5 m, were successfully set out without the use of side adits.

The fourth tunnel, Berentrit, 968 m long, owing to its curvilinear alignment—three curves with two straight inserts—had to be set out by means of three side adits of 21.31 m, 45.34 m, and 31.86 m.

The fifth tunnel, Wiesener I, is situated on a curve of radius 200 m, has a straight insert, then again is on a curve of radius 250 m, and again on a straight line. The total length of the tunnel is 450 m. However, according to the actual situation of the locality, it proved possible to lay only one side adit, 24.08 m long. The other side adits would have been long and too expensive.

As regards the sixth tunnel, Wiesener II, because of its short length of 96.5 m, setting it out presented no difficulties.

Hydraulic tunnels intended for the conveyance of water, for the purpose of utilizing it for hydroelectric stations, in most cases lie in mountain slopes and therefore, by their natural conditions, have many favorable circumstances for using side adits in setting out their axes.

In addition to the fact that this provides convenience in setting out the tunnel and opens the possibility of developing the work front, in favor of using precisely side adits rather than vertical shafts there is also the circumstance that hydraulic pressure tunnels experience considerable internal pressure and exert pressure on the rock mass; the plugging of side adits is simpler and more reliable than that of vertical shafts.

The Teigitsch tunnel, 5,256 m long, has three side adits of 143 m, 140 m, and 95 m. Fig. 87 presents the plan of the tunnel and shows all three side adits. The direction of the tunnel axis was determined by means of a polygonal survey. The length of the 1st section of the tunnel from the headworks to side adit 1, Langmann, is 1,857 m. The 2nd section of the tunnel from adit 1 to side adit 2, Raffler, has a length of 1,455 m. The 3rd section

of the tunnel to the 3rd lateral adit, the Leitenweber, 1415 m long, and, finally, the 4th section, 1529 m. The 1st and 3rd sections of the tunnel attract attention: they are not rectilinear. Here the influence of the terrain relief and of the geological features of the district made itself felt. It was desired to remain in gneiss favorable for construction and not to bring the tunnel out into weak alluvial deposits.

The direction of a lateral adit is given either by means of triangulation or else through works based on a traverse survey. In this case it is not important that the direction of the lateral adit necessarily form a right angle with the direction of the tunnel axis, as was recommended and practiced at the end of the last century. At the present time there are many excellent examples in which the lateral adit falls on a turn (bend) of the main tunnel.

As has already been said, the use of lateral adits in hydraulic tunnels is very considerable. In this respect, as excellent examples, besides the Teich tunnel, the following hydraulic tunnels may also be indicated.

The Barberine tunnel, 2350 m long, is situated on a straight line and has one lateral adit about 160 m long.

The Amsteg tunnel has, in all, 6 lateral adits.

The Partenstein tunnel has 2 lateral adits and one shaft.

Setting out vertical shafts. After the places for the shafts have been selected, their setting out is begun. Ahead of the shaft on the axis of the route, additional signals—pickets—are set; for this, the theodolite is set up and centered over one of the intermediate points of the route lying closer to the shaft, a direction is taken by the sighting mark or by the nearest signal, and the direction is transferred as accurately as possible to the signals set up near the shaft.

Having accurately set out the center of the shaft, they proceed to install the first rings of the shaft lining.

To check the direction of the shaft during its sinking, the procedure is as follows: ...

approximately along the axis of the route, two construction staples are driven into the inner side of the upper ring, placing them horizontally. Onto these staples, with the aid of cuts made by a triangular file, the direction of the route axis is transferred; for this purpose a theodolite is used, centered over one of the signals located near the shaft.

Setting out the tunnel axis through vertical shafts

When setting out the tunnel axis through a vertical shaft, the procedure is as follows.

Before beginning the work, after the final and precise setting out of the tunnel axis on the daylight surface by one of the methods described above, the vertical axis of the shaft is laid out; moreover, its axis is determined by no fewer than three points and is fixed by stone or concrete blocks with embedded wooden blocks into which nails are driven, marking the points of the shaft axis. In exactly the same way, a guiding line of the tunnel is determined and fixed on the daylight surface in cases where the shaft axis is offset parallel to the tunnel axis. After a very careful setting out of the shaft axis and of the guiding line, the axis points are marked with the theodolite on the staging built over the vertical shaft and are then transferred, by means of plumb lines made of thin wire, to the floor of the shaft and of the tunnel with the greatest possible care, and there they are fixed.

During the sinking of the shaft, the direction of the tunnel axis is determined by a plumb line. When the shaft has been brought down to the floor of the guiding adit, the transfer of the direction of the tunnel axis from the surface of the earth underground is begun. For this purpose the marks on the staples are checked a second time by the above-described method, after which, instead of a cord, a brass wire 1 mm thick is attached to the staples at the marks; to the lower end of this wire, not high above the floor of the guiding adit, a weight of 5 or 6 kg is attached in place of a plumb bob, depending on the depth of the shaft. After this, they once again make sure that the ends of the wire have been accurately fastened to the staples opposite the tunnel axis. For this purpose the theodolite is again centered and the cross-hairs are set on the most distant of the visible route signals.

After the limb and alidade have been fixed, the telescope is turned and sighted on the end of the wire attached to the staple, the axis of the wire having to coincide with the vertical thread of the theodolite telescope.

If the end of the wire, because of the ring, is not visible in the telescope, then exactly above the end of the wire a plumb line is hung from one of the beams of the over-shaft structure in such a way that the point of the plumb bob is directly above the center of the wire, and the cross-hairs are then aimed at the plumb-line cord. All the indicated operations for centering the upper ends of the wire must be repeated periodically whenever it is intended to check or continue the axis of the guiding adit. During setting-out work, all construction work must cease both in the shaft itself and near it. After this the setting-out engineer, with his assistants and instrument, descends into the shaft, leaving a watchman above whose duty it is, during the setting-out work, not to allow anyone near the shaft. While descending into the shaft, necessarily by ladder, the setting-out engineer must carefully watch the wires so that they nowhere touch the walls of the shaft and hang completely freely. If at some point the wire nevertheless touches the wall, then the timber at the point of contact must be hewn away. In order that the weights suspended from the lower ends of the wire may settle more quickly, they are immersed in tubs filled with water or oil. When the weights have completely settled, a second time

they check whether the wires touch the walls of the shaft anywhere and, having finally satisfied themselves that the work performed is correct, they set up the theodolite in the pilot adit, trying to place it so that the rear (farther) wire is not visible because of the front (nearer) one, i.e., so that both wires are in one alignment. The limb or alidade is clamped. At the upper points, plumb bobs are suspended from the roof of the adit, the axes of whose cords must coincide with the vertical line of the tube. So that, during this work, the wires of the shaft plumb bobs do not interfere, they are drawn slightly to the side. To fasten the plumb-bob cords to the roof of the adit, special nails are used, with holes for passing the plumb-bob cord through. Such a nail is driven into the roof of the pilot adit and, with the aid of the theodolite, is set so that the cross of the threads coincides with the middle of the nail eye. If the nail proves to have been driven in not quite accurately, it is centered by light side blows.

Fig. 88.

In view of the fact that, during the work, the nails and frames in the adit may be displaced from their original position, concrete pedestals with rail spikes are therefore placed in the rock under each plumb bob, below the sleepers of the trolley track, so that the center of the spike head lies under the point of the plumb bob. During construction work the concrete pedestals are covered with earth and covered with boards to protect them from accidental damage. Such setting-out, before it is used for directing the adit, should be repeated several times, and the mean points should be used for further underground work. To protect the clamps from blows during work from the shaft, it is recommended that they be shielded with boards in the form of a box without a bottom, but with a lid locked shut.

The setting-out of inclined shafts is, generally speaking, difficult. The difficulty of the work consists in the fact that it is not only necessary to set out accurately the center of the shaft, as is done in the case of a vertical shaft, but it is necessary

by means of a theodolite, to suspend the line of the shaft at all times, and the work is carried out in an inclined position, which is very inconvenient. The difficulty of the work is further increased by the fact that moving the instrument along the shaft is usually extremely difficult. The air current, owing to exchange with the outside, has a very adverse effect on visibility in the tube. Setting up the instrument in an inclined shaft is almost impossible. One has to work with the theodolite all the time on the ground surface, directing it into the inclined space under different illumination and atmospheric conditions.

The setting-out of inclined shafts is carried out as follows. When the line has been traced and the entrance into the inclined shaft has been designated, the entrance into the shaft is set out and the bottom and top of the future shaft, points A and B, are marked with stakes (Fig. 88). Then monuments are installed. The place for the instruments is likewise marked by a geodetic center. The work begins by giving the first direction of the shaft, and after the excavation of the shaft has been begun and the supports have been set, the direction of the trace is transferred to the upper cap of the first supporting frame. A bracket is driven in and a plumb line is lowered. It is very advisable, not far from the entrance into the tunnel, on both sides of the shaft along the trace, to set sighting marks, so as to have permanent points for sighting the direction. As the shaft is deepened, its direction is given into the depth of the mass. For this purpose the theodolite is set up on the surface of the ground, aimed at the sighting marks, and plumb line C is checked. Then it is aimed at the plumb lines placed in the inclined shaft every 25 or 50 m. The work is repeated several times for different positions of the tube.

In parallel with the angular geodetic work, measuring and leveling must be carried out. With respect to inclined shafts, this requirement is an inseparable part of the basic setting-out of the direction of the inclined shaft.

Setting out the axis of the tunnel through horizontally situated adits (windows). As has already been said, all places selected for the laying of adits must possess accessibility, good visibility, and a geometric (polygonal) or trigonometric (triangulation) connection.

When the direction of the adit has been given, it is necessary to fix this direction accurately and firmly on the site by means of an alignment. For this purpose it is sufficient and convenient, in the case of rock, to use three vertical drill holes (boreholes) into which wooden poles are installed. In the case of weak soil, a metal gas pipe is set into the ground to a depth of up to 1.5 m, projecting to a height of up to 1.0 m. A wooden pole is installed in it. With taller high poles, in order to avoid overturning and pulling it out, a special stand is set up, consisting of a metal ring tightly embracing the pole at a height of 3 m and supported by three struts braced like a tripod.

The distance to the point of intersection of the adit line with the tunnel axis is determined by calculation according to the data of the geodetic work.

When the adit has been driven and, according to measurement, the point of intersection of the adit line with the tunnel axis has been found in the field, the setting-out of the tunnel axis itself is begun.

For this, it is necessary to carry out some preparatory work.

First, it is necessary to set the direction of the tunnel axis roughly and to order the excavation and removal of rock to such an extent that there is room for geodetic work. Secondly, it is necessary to make—

the following office work: to compute the angle formed by the axis of the tunnel with the direction of the side adit.

When both these operations have been completed, first of all the direction of the adit is checked carefully once again, all the sight lines along which the adit was worked are checked, and the distance along the adit to the point of its intersection with the axis of the tunnel is again measured in the field; the position of this point is finally determined with precision and fixed by setting out a survey mark in the form of a stone or concrete block.

Then, over the center of this point, a theodolite is set up on a tripod, carefully centered, and the angle at which the tunnel axis must lie is taken. At a distance of about 20–30 m in each direction along the tunnel axis, plumb lines are installed on brackets driven into the caps of the supports. These plumb lines are set exactly according to the indications of the theodolite. At the places where the plumb lines are to be, blocks of the above-described type are installed. Three blocks—two only just set out and the third at the point of intersection of the direction of the adit with the tunnel axis—give a sight line that fully ensures the subsequent work of the preliminary setting out of the tunnel for the needs of excavating the rock.

In the final setting out, the work should be carried out instrumentally, and not simply along one sight line, by setting up the instrument for this purpose at the beginning of the adit, at the point of intersection with the tunnel axis.

§ 5. Setting out during excavation of the tunnel to full section.

The work of setting out the tunnel in the transverse direction, in essence, presents no difficulties. It is only necessary to point out the monotonous repetition of the same operations, carried out along the tunnel at small distances from one another. This monotony at times becomes so tiring and uninteresting that it leads to a decrease of attention in the work and to a careless attitude. Meanwhile, the setting out of the tunnel in the transverse direction, connected with the excavation of rock, does not tolerate errors, since such errors lead either to overbreak of the rock, which requires filling with masonry and is thus associated with excess expenditures, or to underbreak in the excavation of the rock; and since any underbreaks of rock cannot be left, it becomes necessary, as a result of incorrect setting out, to resume drilling blast holes, blasting operations, etc., which again makes construction of the structure more expensive and may entail overbreak of the rock. Therefore the setting out of the tunnel must be carried out so that, with the expenditure of the smallest possible amount of time and labor, the most accurate results are obtained. The problem before us, unfortunately, cannot have one general solution and varies in its details depending on various circumstances, namely: on the size of the tunnel cross-section, on the general form of this section, on the method of excavating the rock, etc.

Considering all methods of setting out the transverse profile of a tunnel, we can distinguish two principal methods of performing setting-out work: the method of rectangular coordinates and the method of polar coordinates. In their details, as has been said, these methods may have various variants, but the essence of each of them in one case or another remains entirely the same.

Method of rectangular coordinates. The essence of this method consists in taking the vertical axis of the tunnel as the coordinate axis. The transverse profile along the outer contour of the tunnel lining is determined by rectangular coordinates perpendicular to the vertical axis of the tunnel and lying in a plane normal-

perpendicular to the longitudinal axis of the tunnel. The origin of coordinates (Fig. 89) is taken to be the point of intersection of the tunnel axis with the upper outline of the lining. Such a point in the drawing will be point 1. The horizontal ordinates are drawn at equal distances from one another, for example at intervals of 1.0 or 1.5 m.

The execution of work in the tunnel is preceded by a small amount of office work. For the given type of lining, the transverse profile of the tunnel lining is drawn. The outer outline of the lining is the required contour that must be set out at the work site. The most convenient scale for the drawing may be recommended as 1:25. The position of the above-mentioned coordinates is determined graphically at equal distances from one another, and the magnitudes of these ordinates for points 2, 3, 4, etc. are determined to scale. The magnitude of the ordinates may also be determined analytically. When the ordinates of the outer contour of the lining are known, the work in the tunnel itself is begun.

Fig. 89. Setting out the transverse profile of a tunnel by the method of rectilinear coordinates.

The work in the tunnel begins with selecting a section from station to station, 100 m in length, and finding in the field the monuments (centers) of the final setting-out of the tunnel. The transverse profile must be set out every 20 m. For this purpose range poles are used. Having measured the distance from station to station and marked points at intervals of 20 m, range poles are set along the length of the station interval. Then, by means of a pantometer, transverse lines are set out every 20 m and the positions of the transverse lines are fixed with range poles.

At the same time, leveling must be carried out every 20 m of the upper clamps driven in at point C.

The work described above is common to several variants of setting out, regardless of the size of the tunnel cross-section.

Case 1. A tunnel of small cross-section, the tunnel having a smaller width than height, for example, a single-track railway tunnel. When the preliminary and final setting-out of the tunnel axis has been made, then, for driving the tunnel, the following procedure is used (Fig. 89): by the method already described, the direction of the tunnel axis is established. The plumb line C serves as the coordinate axis. Knowing from leveling the elevation of the clamp at point C, the position on the plumb line of point 1, corresponding to the upper outline of the arch, is determined, and this point is taken as the origin of coordinates. This may be done in two ways: either by leveling the upper clamps C, using special suspended leveling rods for this purpose. This method is considered the most accurate.

Or by leveling points driven in under the plumb line, and then, with the aid of rods, transferring the elevation to clamp C, using a tape for this purpose.

When point 1 has been determined, some distance, for example 1.0 or 1.5 m, is laid off along the plumb line C. For this purpose a tape is used, and the points on the plumb-line wire are marked with a special clamp or else

they use a wooden rod, graduated to the required dimensions, and during the work hold this rod in a vertical position, resting it against point 1.

Then, by means of the rod and a spirit level, in the plane perpendicular to the longitudinal axis of the tunnel, they lay out the ordinates 2–2, 3–3, 4–4. These distances are set off on the basis of values previously found analytically or graphically. In this way the position of the outer contour is determined. The method described, as has been said, is convenient in the case of a comparatively small tunnel width, when it is possible to use rods of relatively short length (up to 3.75 m).

Second case. A tunnel of considerable width, for example a double-track railway tunnel or a metropolitan-railway tunnel. In this case the ordinates are of such considerable length that rods up to 6.0 m long would have to be used. This would complicate the work in the dark and cramped spaces of tunnel excavation. Therefore the procedure is as follows (Fig. 90). Alongside the principal coordinate axis passing through the axis of symmetry of the tunnel cross-section, the coordinate system is shifted first to the right and then to the left, so as to be able to reduce the size of the ordinates set out in the horizontal direction to the outer contour of the tunnel. Thus the essence of this method in the case of a wide tunnel remains unchanged.

Fig. 90. Setting out the transverse profile of a tunnel of considerable width by the method of rectangular coordinates.

Third case. The tunnel has a complex outline of the transverse profile (Fig. 91). This type of tunnel cross-section is encountered, as is known, when the tunnel is located on a slope, under strong lateral rock pressure. The setting out of a cross-section of such an outline is likewise carried out by the method of rectangular coordinates. The distinguishing feature here is that, with such a complex profile, the coordinate system must sometimes be changed several times. It is expedient to take the axis of the upper heading as the basic coordinate system. However, in the course of driving the tunnel it becomes necessary to deviate to the side and to excavate the rock above the upper heading. This requires not only a new coordinate system, but it is also necessary, by means of a rod and a spirit level, to determine additionally the height of these marks of the tunnel-lining contour in the upper part of the excavation.

Method of polar coordinates. The essence of this method consists in taking, on the tunnel axis, a point characteristic of the profile. Such a

point, taken as the origin of polar coordinates, may, generally speaking, be any point on the tunnel axis, but it is expedient to choose the following: in the case of a circular tunnel outline, as occurs in many pressure hydraulic tunnels, the origin of coordinates should conveniently be placed at the center of the circle. In the case of a profile outline that is a combination of various circular outlines, it is expedient to take as the origin of polar coordinates the center of at least some of the curves.

Fig. 91. Setting out the transverse profile of a tunnel of complex outline.

In the United States of North America, for setting out the profile of certain hydraulic tunnels, a special instrument—the “polar protractor,” Polar Protractor (Fig. 92)—has been successfully used. It consists of a disk from 0.35 to 0.6 m in diameter, divided into degrees and mounted on a tripod in such a way that this disk can be set, by leveling, exactly at the required height. The instrument is placed on a special rod, can be raised in height and rotated about this—

Fig. 92. Polar protractor for work on setting out the transverse profile of a tunnel.

rod. On the front side of the disk there is a wooden support. This support can be set at any angle and is fixed in that position. One edge of the support must pass exactly through the center of the disk. During work the instrument is set up so (Fig. 93) that the center of the disk lies not only exactly on the vertical axis of the tunnel, but also coincides with the center of the circumference of the tunnel-profile outline. When the instrument has been set up and fixed, a long measuring wooden rod, somewhat tapering toward the end, bound with copper and divided into centimeters, is placed on the support located on the outer face of the disk. During work the rod is held by hand. Work with the instrument presents no difficulties and is clear from the appended drawing. Work in the tunnel is preceded by office work to determine the magnitudes of the polar coordinates for the various angles. In working in the tunnel, after the instrument has been set exactly at the selected center of polar coordinates, the support rotating on the disk is successively set at various angles, and the outline of the tunnel excavation is determined by applying the rod.

Fig. 93. Setting out the transverse profile of a tunnel by the polar-coordinate method.

Checking the setting-out of the profile by templates. This check of the profile setting-out may concern equally both the outer contour of the lining, i.e. the outline of the mine excavation, and also the inner outline of the tunnel lining. In its practical implementation this method of work is especially convenient for small tunnels with a small transverse profile, such as hydraulic-engineering tunnels. Here, by the template method, it is very convenient to obtain outlines both for the excavation of the rock and, chiefly, for the inner contour. As regards the use of templates in setting out the transverse profile of large mining tunnels, in practice there are many inconveniences here. Large templates would be required, and this is unattainable in view of the tightness of the tunnel spaces being excavated; the use of small templates, not of large size, may lead to errors that are sometimes of very considerable magnitude. Still, it must be noted that in the excavation of large tunnels the use of templates may sometimes become very expedient. In the appended Fig. 94 a case is shown of the use of a template for the lining of a large railway tunnel.

Fig. 94.

§ 6. Execution of work for setting out the tunnel axis. Instruments for underground geodetic work.

With regard to carrying out work for setting out the tunnel axis underground, it is necessary to distinguish the setting-out of straight-line tunnels, or straight-line sections of a tunnel, on the one hand, and of curvilinear tunnels, or curvilinear sections of a tunnel, on the other.

In the first case, instruments of greater optical power and of simpler construction are used for carrying out the work of setting out the axis of straight-line tunnels.

In the second case, for setting out the axis of curved sections of a tunnel, there is no need for considerable optical power in the theodolite; instead, increased accuracy is required in reading the measured angle.

In both cases, a theodolite for underground tunnel work must be portable, well designed, and have well-balanced parts.

Fig. 95.

The repeating theodolite of the firm Kern (Kern), in Aarau, Switzerland (Fig. 95), is very widely used for setting out long straight tunnels. This instrument was used for setting out the axes of the Simplon, Lötschberg, Hauenstein (base II), Ricken, and many other tunnels. Its specifications are as follows:

Characteristic	Value
Diameter of the objective of the telescope	40 mm
Length of the telescope	320 mm
Magnification	35 times

The axis of rotation is made hollow for illuminating the crosshairs by means of a small electric lamp powered by two dry batteries, one of which is a spare. There is a horizontal micrometer screw. A solid, stable tripod. There are neither horizontal nor vertical circles. This simplicity of construction in no way detracts from the merits of the theodolite; on the contrary, this simplicity is characteristic of the instrument’s principal purpose: to serve for setting out long straight tunnels.

The theodolite described, by the firm Kern, sometimes has certain distinctive details. In setting out the axis of the Simplon Tunnel, two Kern theodolites were used on each side of the tunnel being driven; one instrument (Fig. 96) was intended exclusively for the internal setting-out of the tunnel axis, for which purpose it was mounted on a tripod carried inside the tunnel. The other theodolite, of somewhat larger size, was mounted on a tower (observatory) and was likewise intended for setting-out work on the axis inside the tunnel. The eyepiece crosshairs were illuminated by an acetylene lamp. In addition, there were several other distinguishing features. As with the first type, the second type of instrument lacked both vertical and horizontal circles.

Fig. 96.

Theodolite of the firm Ed. Sprenger (Sprenger), Berlin (Fig. 97). For underground tunnel setting-out, this instrument is quite widespread. Its specifications are as follows:

Horizontal circle diameter . . . . . . . . . . 17.5 cm.
Readings by microscopes are made with an accuracy of up to . . . 5″
Aperture of the theodolite telescope objective . . . . . . 30 mm.
Length of the telescope . . . . . . . . . . 432 mm.
Magnification . . . . . . . 5, 45, and 60 times.

Owing to the sufficient accuracy in readings on the circles and in view of its portability, this instrument is recommended by many specialists in tunnel practice for setting out curved tunnels. The price of this instrument is about 500 rubles.

A similar theodolite by the same firm, but of somewhat smaller size, is likewise suitable for setting out curvilinear tunnels. The diameter of the horizontal circle of this smaller theodolite is 13.5 mm. Reading by microscopes is 10″. Magnification is 24 and 35 times. The price is about 400 rubles.

Fig. 97.

Theodolite of the firm Max Hildebrand (Max Hildebrand) (Fig. 89). This theodolite was used, among other things, in setting out the Lötschberg Tunnel and had great success there. It has electric illumination of the eyepiece threads. The horizontal circle is 210 mm in diameter. The circle has 5-minute divisions, marked with numerals. Both microscopes have 60-fold magnification. Turning the micrometer screw by one revolution moves both threads by 2″. The drum is divided into 60 parts. This makes it possible to take readings directly with an accuracy of up to 2″. With the aid of the vernier, estimation may be made to 0.2″. Three eyepieces provide the possibility—

Fig. 89.

possibility of attaining magnifications of 28, 34, and 42 times. The objective, consisting of two parts, has an aperture of 41 mm and a focal length of 380 mm. Illumination of the thread cross—unusually fine, incidentally—is effected by means of a small gilded mirror; the mirror rotates, making it possible to vary the brightness at will. For the greatest perfection, and so that the theodolite may turn freely, there is a vertical circle with divisions for the microscope of 6″. Illumination of the field of view in the tube aperture and in the four microscopes, as well as in both drum microscopes, is provided by small electric incandescent lamps.

The theodolite of the Breithaupt firm (Breithaupt) in Cassel (Fig. 99) is also not infrequently used for tunnel setting-out. It has horizontal and vertical circles and is thus universal for the setting-out of curvilinear tunnels. Its special feature, among other things, is that it is adapted for centering when plumb lines have been lowered. In tunnel work, especially when vertical shafts are present, there is a need for this kind of centering. A certain shortcoming is its comparatively low portability. Reports on work with this instrument in setting out the Dzhadzhur tunnel and some Crimean tunnels, as well as many Swiss tunnels, are very favorable. The cost of the instrument is about 400 rubles.

Fig. 99.

Errors of misclosure in setting out the axes of tunnels underground. However carefully the work of setting out a tunnel axis underground may be carried out, the results of the measurements will inevitably be associated with a deviation of the measurement results from the true value, which is the object of our search and work. Measurements and setting-out, even when performed in very great quantity and with the greatest possible care, will inevitably differ from one another. The true value of the measured quantity always remains unknown to us.

All errors of measurement, as is known, may be divided into three groups: 1. Gross mistakes. 2. Constant or systematic errors. 3. Accidental deviations and irregularities.

Constant or systematic errors are those which, under identical circumstances, affect the results of measurements in the same way: either always increasing or always decreasing the result. Errors of this kind may be theoretical, instrumental, and personal. To reduce to a minimum the influence of constant (sy-

(systematic) errors of measurement, and as a check on observations previously made, a series of repeated observations of one and the same quantity is carried out under identical observational conditions. This, as is well known, is the problem of the theory of errors of observation or of the adjustment of errors.

Random deviations or irregularities are the most difficult to take into account. Their causes are very varied: oscillations of the theodolite telescope in a current of air during the measurement of angles; improper illumination of the object (the phase of the object), for example, a wire illuminated from the side; imperfections of our senses, etc.—all these produce random irregularities. By acquiring a number of skills and eliminating the possibility of falling into accidental error, some random errors can, to a certain extent, be neutralized.

In the course of tunnel work, both deformations of the supports and displacements of the exposed rock are observed. Under the action of rock pressure, not only do disturbances occur in the upper part of the adit, but the floor of the tunnel also heaves. Thus those places where the setting-out marks (centers) have been placed do not always remain immovable and therefore are not always reliable. This, of course, cannot fail to affect the accuracy of the work of setting out the axis of the tunnel underground. The question is usually further complicated by the fact that the setting-out of the tunnel advances forward, while behind it deformations of the rock and displacements of the established centers begin. Sometimes these displacements occur in a clearly perceptible form, but in most cases

Fig. 100.

they are gradual and scarcely perceptible to the eye. This latter circumstance increases still further the risk of being misled and of making an incorrect setting-out, since, as the setting-out is continued deeper into the tunnel, one must rely on the preceding marks (“centers”), and these are precisely the ones subject to displacement.

The examples are far from isolated and, in their character, are instructive and interesting.

In the Lötschberg Tunnel, at a distance of 2.2 km from the southern portal, a displacement of up to 0.07 m was observed. In the zone of rock pressure, at a distance of 3.5 km from the northern portal, the position of the centers changed, which was detected each time during the check of the final setting-out.

In the Hauenstein base tunnel, cases of heaving of the tunnel floor and disturbance of the integrity of the centers were repeatedly observed.

In a tunnel near Druzhinino station on the Moscow–Kazan Railway, heaving of the floor occurred slowly, but reached a large magnitude. During the check of the final setting-out, displacements of the centers were found to be 0.08 m in height and up to 0.04 m to the side.

In tunnels a very interesting phenomenon is sometimes observed. The line of sight from the theodolite telescope is deflected to the side. The cause may be the escape of compressed air from the flanges of an air pipe. As is evident from the accompanying drawing (Fig. 100), the source of light, visible in the telescope of the the—

the theodolite in fact turns out not to be on the true axis of the tunnel. It is clear that under such circumstances errors in observations and inaccuracies in setting out arise. An example may be the Tauern Tunnel, where the phenomenon described was observed and was discovered quite by chance. The quantities shown on the drawing relate precisely to the case in the Tauern Tunnel.

Leveling work. During the driving of a tunnel, simultaneously with the setting out of the axis, leveling must be carried out in order to have control points for setting out the tunnel vertically and, further, to have control points for setting out the transverse profile of the tunnel.

For leveling inside a tunnel, levels of any systems and designs may be used, preferably the strongest ones and those with inverting telescopes. A distinctive feature of instruments for work inside a tunnel is the need to fit a device for illuminating the cross-hairs of the eyepiece. For this purpose it is very expedient to use small electric bulbs and a portable battery.

Leveling rods for work inside a tunnel are of various designs. They must be portable and light. Ordinary folding rods (Fig. 101) with a level and with a device for illuminating the graduations in the form of a small acetylene lamp, moved along the height on a wire fastened along the rod, may be used. The lamp has a shade so that its light is not directed into the tube of the level. A very convenient type of rod is shown in Fig. 102. The rod is set strictly vertically on a tripod; when the reading has been taken, the instrument itself can also be set up at the same place.

Fig. 101.

Fig. 102.

In tunnel work one rather often encounters the need, for leveling, to use control points located not below, in the invert of the tunnel, but above. The reasons are quite varied: flooding of the invert with water, heavy contamination of the bottom of the tunnel, weak ground soaked with water, heaving of the tunnel invert due to rock pressure from below, etc. When the points for leveling are located above, suspended leveling rods are used (Fig. 103).

The designation and fastening of points for leveling inside a tunnel is governed by the following requirements. It is most convenient for these points to coincide with the stationing, i.e., to be located every one hundred meters. Sometimes, when the tunnel has considerable gradients, certain practical specialists in tunneling indicate the desirability of setting leveling points at intervals of no less than 50 m. Although requirements of this kind do entail extra work, nevertheless it cannot be denied that in many cases they have a technical justification.

Points for leveling inside a tunnel are designated as follows: it is most convenient not to install special points, but to use a bolt or nail in the concrete mass, serving as the geodetic center when setting out the tunnel axis. If the point for leveling is installed independently, regardless of its purpose in serving the setting-out of the tunnel axis, then it is immaterial where it is placed: on the tunnel axis or to the side. In this respect it is necessary only to choose the place so that the point is not subject to deformations either from the rock mass or to mechanical damage when hauling loads, etc. If the tunnel is in hard rock, it is sufficient to drill a hole and drive a metal rod into it. The top of the rod may serve as the point for leveling. It is highly advisable to cover the top of the point with oil paint, and then any sort of mechanical damage, by breaking the integrity of the paint, will indicate that the possibility of damage to the leveling point itself is not excluded. By this simple measure one can guard against unexpected occurrences.

If the rock is soft, yielding clay or similar ground, it is sometimes possible to ensure reliable work by driving a wooden pile into the ground and screwing a screw into it. Sometimes it may be expected that the rock will heave. Unfortunately, in tunneling practice in such cases no type of leveling point has yet been developed or established. Usually in such cases one has to resort to benchmarks located outside the tunnel as the only reliable control points for leveling. In any case, the only and most reliable thing that can be recommended in such cases is the repetition of leveling operations along the entire length of the tunnel.

The work of leveling is carried out in the same way for both the preliminary and the final setting-out of the tunnel axis. The process of the work itself differs in no essential respect from carrying out similar work on the surface of the earth. The only thing to which attention must be paid is that, in underground tunnel leveling, one should by no means become carried away with considerable distances from the instrument to the rod. As has already been indicated with regard to work on setting out the tunnel axis, the visibility through the tube depends on various causes and, first of all, on the state of the atmosphere. Therefore, in leveling, a distance from the instrument to the rod of 50 meters is recommended, and it should rarely exceed 75 meters.

Fig. 103.

Errors in underground leveling. In view of air saturated with water vapor, poor visibility of the rods, poor illu—

… and for some other reasons, the error of leveling when working in a tunnel during its construction will be rather considerable.

If we assume that, when working inside a tunnel, the instrument has on the average \(n = 5\) stations per kilometer, and if we allow that the reading on each staff is made with an accuracy of up to 0.002 m, then, according to the theory of errors, the difference in levels of the initial and final points of compound longitudinal leveling is determined as \(0.002\sqrt{2n}\), or \(0.0063\sqrt{K}\), where \(K\) is the number of kilometers covered by the leveling.

The limiting error of leveling inside the tunnel will be equal to three times the mean error, i.e. \(0.0189\sqrt{K}\) (in m).

German authors give almost the same figure. They consider that, in underground works, the error of leveling should not exceed 18 mm per kilometer; moreover, in the final result the error should not be more than \(f < 18.\sqrt{K}\) (in mm).

Determination (measurement) of the length of the excavated tunnel. After the two heading drifts have broken through, it is very advisable to determine the actual length of the tunnel by direct measurement in such a tunnel. For this purpose it is necessary to make use of holiday interruptions in the work.

Sometimes, in view of the importance of the work to be done, work is not stopped even before an interruption of tunnel works on working days. The measurement of the tunnel length is carried out in different ways, depending on its extent.

Small tunnels up to 2–3 km long are measured with a steel tape or chain; sometimes rods are used.

Longer tunnels should be measured with the aid of invar wire. However, sometimes special measuring chains are also successfully used for measuring long tunnels.

Measurement of small tunnels. If rods are used, there must be two of them. A rod of cross-section \(4.4 \times 2.5\) cm is made of fir wood, as far as possible free of knots, boiled in oil and painted with white paint. The length of a rod is 5 m; one side of the rod is bound with copper, with the exact marking of divisions into decimeters, centimeters, and millimeters. The ends of the rod are sheathed with copper in the form of tips (shoes). For measurement, the rods are laid against one another so that their ends touch. For accuracy, the measurement is made at least twice in the opposite direction and, if the results are close to one another, the arithmetical mean is taken. The direction of the rods is determined by means of a cord. The entire measurement line inside the tunnel is marked by means of an ordinary hemp twisted cord 2–3 mm in diameter, impregnated with wood oil against dampness. The length of the cord is 100 m. The cord is wound on a drum. It is useful to have two drums with cords. Plumb lines are set out along the tunnel, and when the line has been marked by the final setting-out, special trestles are installed over a section about 100 m long (Fig. 104). Light plumb lines (Fig. 105) can be used with great success in this work. At the top of the trestles there are pins, over which the cord is stretched. It is necessary to make sure of the horizontality of the cord, for which a suspended semicircle is used. The trestles are set at distances of approximately 15–20 m from one another.

The use of steel measuring tapes and tapes for measuring small tunnels is more convenient than rods. Their length should be 20 m. When working with measuring tapes or tapes, a cord is sometimes stretched in the same way as when working with rods, but usually the measurement is made along horizontally laid boards.

The accuracy of measurement with rods is much higher than with a steel tape or band, and is expressed as approximately \(1:50000\) of the measured length, whereas in measuring with a tape or band inside a tunnel it is seldom possible to obtain results with an accuracy greater than \(1:10000\).

Fig. 104.

Measurement of long tunnels. The use of invar measuring wire for measuring long tunnels ensures an accuracy from \(1:500000\) to \(1:1000000\) of the measured length.

This enormous accuracy makes the method described highly desirable, although it is associated with large expenditures of time. Invar wire, i.e. wire made of an alloy of nickel and iron, possessing a small coefficient of expansion, is manufactured with a thickness of 5 mm and is provided at its ends with special measuring scales (see Fig. 53); the length of the wire from “0” on the measuring scale at one end to “0” at the other end is 25.000 m.

The accuracy of measuring the length of a tunnel (inside the tunnel), by different methods of measurement, expressed as fractions of the measured length, is as follows:

Fig. 105.

Method	Accuracy
With a chain	from 0.00045 to 0.0012
With a steel tape	from 0.0003 to 0.00075
With invar wire	from 0.000001 to 0.000002
With rods	from 0.0002 to 0.0005

To judge the results of all geodetic work on the tracing and setting out of tunnels, the misclosures (errors) are determined:

a) in the length of the tunnel,
b) in elevation,
c) and the lateral deviation from the tunnel axis at the place where the two headings driven from the portals meet (from opposite sides).

Only by establishing the misclosures is it possible to judge the correctness of the work performed in tracing and setting out the tunnel axis above ground.

How accurately geodetic work in the setting-out of tunnels is carried out may be seen from the following table:

Table III.

Name of tunnel	Length of tunnel, m	Computed error (misclosure), by length, m	Misclosure in length during measurement in the field, m	Misclosure of opposing headings, by height, m	Misclosure of opposing headings, in horizontal direction (to the side), m	Computed error (misclosure) in horizontal direction (to the side), m
Simplon Tunnel . . .	19728	$\delta = 0{,}56$ $\delta_{\max}=1{,}68$	0,79	0,087	0,202	$f = 0{,}05$
St. Gotthard Tunnel . .	14984	—	—	—	0,30	—
Lötschberg Tunnel . . .	14535	$\delta = 0{,}36$ $\delta_{\max}=1{,}08$	0,41	0,102	0,257	$f = 0{,}182$
Hauenstein Tunnel . .	8134	—	0,50	0,010	0,05	$f = 0{,}546$
Vosges Tunnel . . .	6339	—	0,76	—	0,05	—
Karavanken Tunnel . .	7976	—	0,45	—	0,02	—
Bosruck Tunnel . . .	4766	—	0,18	—	0,153	—
Tauern Tunnel . . .	8526	—	2,93	0,056	0,55	—
Wasserfluh Tunnel . .	3557	—	—	0,010	0,05	—
Suram Tunnel . . . .	3973	—	—	0,40	0,13	—
Lagar-Aul Tunnel (through the Lesser Khingan)	1250	—	0,80	0,080	0,106	—
Goit Tunnel (Caucasus) .	1399	—	1.18	0,090	0,195	—
Son Tunnel (Siberia) .	785	—	0,65	0,110	0,145	—
Khadyzhensk Tunnel (Caucasus) . . . . . . .	978	—	1,20	0,075	0,105	—

In concluding here my work on the question of the tracing and setting-out of tunnels, I consider it necessary to emphasize that, for the success of geodetic tunnel work, in addition to knowledge of the procedures for carrying out the work and of the methods of tracing and setting-out presented here, it is required that the engineer carrying out the work fully possess all the necessary technical skills in the performance of geodetic work, and be pedantically accurate, attentive, and precise. Only an awareness of responsibility for the work being performed, together with the observance of the other conditions, can ensure its correct execution.

Tomsk.
Siberian Technological Institute.

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1. Thus, in the works for the Lötschberg Tunnel (3000 m), the influence of the mountain masses was taken into account out to a distance of 73 km. In the investigations of I. I. Stebnitsky on the attraction of the Caucasus Mountains, different limiting radii were adopted at different points, from 80 to 240 versts; on average, for 14 points, \(r=166\) versts. For the Balkan Mountains, M. N. Lebedev adopted a limiting radius of 64 versts. In the recalculation of the triangulation of the Simplon Tunnel, a limiting radius of 40 km was adopted. To achieve equal accuracy of the computations, it is necessary to choose radii the larger, the larger the mountain massif; moreover, the limiting radii will be directly proportional to the volume of the mountain masses (to the square root of the cube of the mean elevations of the mountains). ↩↩