The theodolite is most accurate instrument used mainly for measuring horizontal and vertical angles. It can also be used for locating points on a line, prolonging survey lines, finding difference in elevations, setting out grades, ranging curves etc.
Theodolites are primarily classified as:
(1) Transit, and
(2) Non transit.
A theodolite is called to be transit when its telescope can be transited i.e. revolved through a complete revolution about its horizontal axis in the vertical plane, whereas in non-transit type, the telescope cannot be transited. The non-transit theodolites are inferior in utility and have now become obsolete.
Theodolites are also classified as:
(1) Vernier theodolites, and
(2) Micrometer theodolites according as vernier or micrometer is fitted to read the graduated circle.
The first type is more commonly used.
The diameter of the graduated circle on the lower plate determines the size of theodolite. The common sizes are 8 cm to 12 cm while 14 cm to 25 cm instruments are used for triangulation work.
Practical Hints on Measuring Horizontal Angles with a Theodolite:
(i) Always use the lower clamp and its tangent screw for left-hand object or for back sight.
(ii) Always use the upper clamp and its tangent screw for sighting an object on the right.
(iii) If the two objects are situated at different levels, use the vertical clamp and its tangent screw.
(iv)Tighten the clamping screw first before using its corresponding tangent screw.
(v) For accuracy of work, coincide exactly the point of intersection of the cross-hairs with some well-defined point on the object, such as nail driven on the top of a peg etc.
Laying out a Horizontal Angle:
Suppose it is required to lay out the angle AOB (55° – 27′) with the line OA.
Proceed as follows:
(i) Set up the instrument at station-point O and level it accurately.
(ii) Set the vernier A to zero by using the upper clamp and its tangent screw.
(iii) Loosen the lower clamp and turn the telescope to sight A. Then tighten the lower clamp and bisect A exactly by means of the lower tangent screw.
(iv) Check the reading of vernier A, which should still be zero.
(v) Loosen the upper clamp and set the vernier A to the angle to be set out (i.e. 55° – 27′) exactly by using this clamp and its tangent screw. The line of sight is thus set in the required direction of OB.
(vi) Establish the point B in the line of sight by first looking over the top of the telescope and directing the flag man approximately into line, and then looking through the telescope and ranging the rod exactly in line.
Measuring a Vertical Angle:
A vertical angle is an angle between the inclined line of sight and the horizontal. It may be an angle of elevation or depression according as the object is above or below the horizontal plane.
To measure the vertical angle of an object A at a station O,
(i) Set up the theodolite at station-point O and level it accurately with reference to the altitude bubble.
(ii) Set the zero of vertical vernier exactly to the zero of the vertical circle by using the vertical circle clamp and tangent screw.
(iii) Bring the bubble of the altitude level in the central position by using the clip screw. The line of sight is thus made horizontal, while the vernier reads zero.
(iv) Loosen the vertical circle clamp screw and direct the telescope towards the object A and sight it exactly by using the vertical circle tangent screw.
(v) Read both ve rniers on the vertical circle. The mean of the two vernier readings gives the value of the required angle.
(vi) Change the face of the instrument and repeat the process. The mean of two vernier readings gives the second value of the required angle.
(vii) The average of the two values of the angle thus obtained, is the required value of the angle free from instrumental errors.
To measure the vertical angle between two points A and B:
(i) Sight A as before, and take the mean of the two vernier readings at the vertical circle. Let it be α.
(ii) Similarly sight B and take the mean of the two vernier readings at the vertical circle, Let it be β.
(iii) The sum or difference of these readings will give the value of the vertical angle between A and B according as one of the points is above and the other below the horizontal plane (Fig. 9.8) or both points are on the same side of the horizontal plane (Fig. 9.9 a & b).
Reading Magnetic Bearing of a Line: (Fig. 9.10):
To find the bearing of a line AB,
(i) Set up the instrument over A and level it accurately.
(ii) Set the vernier A to the zero of the horizontal circle.
(iii) Release the magnetic needle and loosen the lower clamp.
(iv) Rotate the instrument in the horizontal plane until the magnetic needle takes the normal position, i.e. the zeros of the small scales in the trough compass, or the N and S graduations in the circular box compass, or the index mark in the tubular compass are opposite to the ends of the needle. Tighten the lower clamp and use its tangent screw for the exact coincidence. The line of sight is now parallel to magnetic meridian and the vernier A reads zero.
(v) Loosen the upper clamp. Turn the telescope and sight the object B, bisect B exactly by using the upper tangent screw.
(vi) Read both verniers on the horizontal circle. The mean of the two vernier readings gives the magnetic bearing of the line AB.
If greater accuracy is desired, change the face, take a second reading and record the mean of the two.
Prolonging a Straight Line:
There are two methods of prolonging a given line such as AB:
(1) Fore sight method, and
(2) Back sight method.
1. Fore Sight Method (Fig. 9.11.):
(i) Set up the theodolite at A and level it accurately. Bisect the point B correctly. Establish a point C in line beyond B approximately by looking over the top of the telescope and accurately by sighting through the telescope.
(ii) Shift the instrument to B, take a fore sight on C and establish; point D in line beyond C.
(iii) Repeat the process until the last point (Z) is established.
For this method, the instrument must be in perfect adjustment.
2. Back Sight Method. (Fig. 9.11):
(i) Set up the instrument at B and level it accurately.
(ii) Take a back sight on A.
(iii) Tighten the upper and lower clamps, transit the telescope and establish a point C in the line beyond B.
(iv) Shift the theodolite to C, back sight on B, transit the telescope and establish a point D in line beyond C.
(v) Repeat the process until the last point (Z) is established.
This method also requires the instrument to be in perfect adjustment.
When the line is to be prolonged with high precision or when the instrument is in imperfect adjustment, the process of double sighting or double reversing, is used. Suppose the line AB is to be prolonged to a point Z.
Procedure (Fig. 9.12):
(i) Set up the theodolite at B and level it accurately.
(ii) With the face of instrument left, back sight on A and clamp both the upper and lower motions.
(iii) Transit the telescope and set a point C1 ahead in line.
(iv) Loosen the lower clamp, revolve the telescope in the horizontal plane and back sight on A, Bisect A exactly by using the lower clamp its tangent screw. Now the face of the instrument is right.
(v) Transit the telescope and establish a point C2 in line beside the point C1. The exact position of the true point C must be midway between C1 and C2.
(vi) Measure C1, C2 and establish a point C exactly mid-way, which lies on the true prolongation of AB.
(vii) Shift the instrument to C, double-sight on B, establish the points D1 and D2 and establish the true point D as before.
(viii) Continue the process until the last point Z is established.
Levelling with a Theodolite:
The process is similar to ordinary levelling except that the motion of the telescope in the vertical plane is stopped by clamping it such that the vertical circle verniers, C and D, read zero-zero. If the instrument is in perfect adjustment, the line of sight will be horizontal when the bubble is in its central position.
In this case, the instrument may be set anywhere between two points and their difference in elevation is found as usual. But if the instrument is out of adjustment, it will give an inclined line of sight even though the bubble is brought into centre. In this case, the back and fore sight distances should be equalised to eliminate the error due to inclined line of sight.
Traversing with a Theodolite:
In theodolite traversing, the field-work consisting of:
(ii) Selection of stations,
(iii) Marking and locating stations,
(iv) Running of survey lines,
(v) Locating the details, and
(vi) Booking of field-notes is almost the same as for compass traversing.
The traverse may be open or closed like the compass or plane table traverses.
A theodolite is used for determining the relative directions of the lines of a traverse, and a steel tape is commonly used for the linear measurements.
Checks on Closed Traverse:
1. Check on Angular Measurements.
2. Check on Linear Measurements.
1. Check on Angular, Measurements:
(a) Traverse by Included Angles:
(i) Sum of the measured interior angles should equal (2N – 4) right angles.
(ii) The sum of the measured exterior angles should equal (2N + 4) right angles. where N is number of sides of the traverse.
(b) Traverse by Deflection Angles:
The algebraic sum of the deflection angles should equal 360°.
Consider the right-hand deflection angles as positive, and left-hand ones as negative.
(c) Traverse by Direct Observation of Bearings:
The back bearing of the last line observed at the first station, and the fore bearing of this line observed at the last station should differ exactly by 180°.
2. Checks on Linear Measurement:
In a closed traverse, the sum of the northings (distances measured towards north) should be equal to the sum of the southings (distances measured towards south). Similarly the sum of the eastings (distances measured towards east) should be equal to the sum of the westings (distances measured towards west).
Checks on Open Traverse:
There are no direct checks on the field measurement of an open traverse.
However, these can be approximately checked by the following methods:
1. By Means of Tie Lines:
By means of Tie Lines Fig. 9.20 shows an open traverse for a road. The lines A to G are surveyed with measurements of angles and distances, and a lie line AG is later surveyed with its bearing and distance. It the survey is correctly done, the later will serve as a check on the correctness while plotting.
2. By Means of an Auxiliary Station:
While surveying, bearing of a well-defined object or a point fixed on one side of the traverse such as P (Fig 9.20), may be observed from stations A, E, G, etc. If the survey is correctly done and accurately plotted, all these bearings, when plotted, must meet in the point P.
Permissible Errors in Theodolite Traversing:
The plotting work is started after the field-work is over. The positions of different points are plotted on a plan with reference to two lines YY1(Y – axis) and XX 1(X-axis) which are respectively parallel and perpendicular to the meridian. These reference lines are called the axis of the co-ordinates and the point of their intersection O, the origin.
(Fig. 9.21.). They origin may either be any traverse station or entirely out-side the traverse. The distances of a point from each of the axis are called its co-ordinates.
If the length and bearing of a line are known, its projections on the Y- axis and X-axis may be obtained. These projections are called latitude and departure of the line respectively.
Latitude if measured upward or northward is known northing; and if measured down-ward or south ward is called southing. Nothing is assumed to be positive while southing is assumed to be negative.
Similarly, departure if measured right- ward or eastward is known as easting; and if measured left-ward or westward is called westing. Easting is assumed to be positive while westing is assumed to be negative.
To find the latitude of a line, multiply the length of the line by the cosine of its reduced bearing; and to find its departure, multiply the length of the line by the sine of its reduced bearing (Fig. 9.22).
If I is the length of the line and 0 is its reduced bearing, then:
Latitude of the line = l cos θ
Departure of the line = I sin θ.
Then reduced bearing of a line determines the sign of its latitude and departure, the first letter N or S of the bearing defines sign of the latitude and the last one E or W, the sign of the departure. If the bearing is give as W.C.B., first convert it into R.B. and then determine the sign of latitude and departure.
The latitude and departure of any point with reference to the preceding point are called consecutive co-ordinates of the point; while the coordinates of any point with reference to a common origin are called independent co-ordinates of the point. The independent co-ordinates are also known as total latitude and the total departure of a point.
Gale’s Traverse Table:
The computations for a closed traverse may be made in the following steps and entered in a tabular form known as Gale’s Traverse Table as in Table 9.2.
(i) Add up all the included angles. Their sum should be equal to (2N – 4) right angles or (2N + 4) right angles according as the angles measured are interior or exterior, where N is the number of the sides of a traverse.
(ii) If not, apply the necessary corrections to the angles so that the sum of the corrected angles be exactly equal to (2N ± 4) right angles.
(iii) Calculate the W.C. bearings of the other lines from the observed bearing of the first line and the corrected included angles.
The calculated bearing of the first line must be equal to its observed bearing.
(iv) Convert the W.C. bearings of the lines into reduced bearings (R.B.) and determine the quadrants in which the lines lie.
(v) From the known lengths and calculated reduced bearings of the lines, compute their latitudes and departures (consecutive co-ordinates).
(vi) Add up all northings, and also add up all southings, and find the difference between the two sums. Similarly determine the difference between the sum of all eastings and the sum of all westings.
(vii) Apply the necessary corrections to the latitudes, and departures so that the sum of the northings equals the sum of the southings and the sum of eastings equals the sum of the westings.
(viii) Find the independent co-ordinates of the lines from the consecutive co-ordinates so that they all are positive and thus the whole traverse may lie in the first (N.E.) quadrant.
Problems in Theodolite Traversing:
The following rules will be much useful while solving problems on traverse surveying:
Refer to fig 9.24.
If I is the length of a line, and θ is its reduced bearings; then:
The co-ordinates of two points A and B are as follows:
Find the length and bearing AB.
Let I = the length of AB.
θ = the reduced bearings of AB
Latitude of AB = the difference between the north co-ordinates of A and B = 840.78 – 500.25 = 325.53.
Departure of AB = the difference between the east co-ordinates of A and B = 315.60 – 640.75 = 325.15
∴ θ = 43⁰ 41′
Since the latitude is +v and the departure is –ve, the line AB lies in the fourth (N.W.) quadrant.