Determination of Direction

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Triangulation procedures

Intermediate Engineering

Adjusting a chain of quadrilaterals

Determination of Direction

As you learned earlier in this chapter, most astronomical observations are made to determine the true meridian from which all azimuths are referred In first-order triangulation systems, these observations are used to determine latitude and longitude. Once the true meridian is established, the azimuths of all other sides are computed from the true meridian.

To compute the coordinates of triangulation stations, you must determine the latitudes and departures of the lines between stations; to do this, you must determine the directions of these lines. The latitude of a traverse line means the length of the line as projected on the north-south meridian running through the point of origin. The departure of a traverse line means the length of the line as projected on the east-west parallel running through the point of origin Latitudes and departures are discussed in detail in chapter 7 of this TRAMAN.

Figure 15-26.Field notes for base line measurement.

Base Line Measurement The accuracy of all directions and distances in a system depends directly upon the accuracy with which the length of the base line is measured. Therefore, base line measurement is vitally important. A transit must be used to give precise alignment while measuring a base line. For third-order triangulation measurement with a steel tape, you are required to incorporate all the tape corrections described in the EA3 TRAMAN. For measurement over rough terrain, end supports for the tape must be provided by posts driven in the ground or by portable tripods. These supports are usually called chaining bucks. The slope between bucks is determined by measuring the difference in elevation between the tops of the bucks with a level and rod.

On the top of each buck, a sheet of copper or zinc is tacked down, which provides a surface on which tape lengths can be marked Bucks are setup along the base line at intervals of one-tape length. The tape, with thermometers fastened at each end, is stretched between the supports and brought to standard tension by a tensiometer (spring balance). When the proper tension is indicated, the position of the forward end is marked on the metal strip with a marking awl or some other needle-pointed marker. At the same time, the thermometer readings are taken.

If stakes, driven at tape-length intervals, are used as set back or set forward to bring the end again on the marking strip. The set back or set forward is entered in the field notes and deducted from or added to the tape length for that particular interval.

Figure 15-26 shows field notes for a base line measurement. In this case the tape was supported on stakes, driven at full-tape, 100-foot intervals. With the exception of the interval between stakes 5 and 6, the amounts to the standard tape length (with the tape supported at both ends, and with standard tension applied), as corrected for temperature and for slope. For the interval between stakes 5 and 6 (where there is, as you can see, a forward set), the horizontal distance amounts to the standard tape length plus 0.104 foot, as corrected for temperature and for slope. The length of the base line will, of course, amount to the sum of the horizontal distances.

Note that in this case the line is being measured forward. After the forward measurement, the line is again measured in the backward direction. If the backward measurement varies slightly from the forward measurement, the average is taken as the length of the base line. A large discrepancy would, of course, indicate a mistake in one measurement or the other.

Rather than using chaining operations to perform base line measurements, an electronic distance meter (EDM) can be used. The use of EDM equipment greatly simplifies the measurement of base lines in triangulation. Chapter 12 of this TRAMAN gives a general discussion of EDMs and EDM principles.

Computations

In triangulation of ordinary precision or higher, the observed angles are adjusted before the lengths of the least squares method that involves the computation of the most probable values of the adjusted quantities. In many advanced surveying textbooks, the least squares method

Figure 15-27.-Chain of triangles.

is preferred; however, calculation of the probable values of the unknowns involves a level of mathematics (calculus) that is beyond that required of the Engineering Aid. Therefore, in this text we will discuss more elementary adjustment procedures that, while less accurate than the method of least squares, yield satisfactory results.

There are two steps in angle adjustment, called station adjustment and figure adjustment. Station (n 2) x 180, with n representing the number of sides of the polygon.

ADJUSTING A CHAIN OF TRIANGLES. In station adjustment you compute the sum of the

Table 15-6.Station Adjustment for Chain of Triangles, Figure 15-27

Table 15-7.Figure Adjustment for Chain of Triangles, Figure 15-27

measured angles around each station, determine the extent to which it differs from 360, and distribute this difference over the angles around the station according to the number of angles.

Figure 15-27 shows a chain of triangles. Station adjustment for this chain of triangles is given in table 15-6.

At station A, as you can see, the sum of the observed interior angles 3, 5, and 8 plus the observed exterior closing angle 12 comes to 3600025. This differs from 360 by 25 seconds. The number of angles around the station is four; therefore, the correction for each angle is one fourth of 25, or 6 seconds, with 1 second left over. The sum of the observed angles is in excess of 360; therefore, 6 seconds was subtracted from the observed value of each interior angle and 7 seconds from the observed value of the exterior angle. The angles around the other stations were similarly adjusted, as shown.

The next step is the figure adjustment for each of the triangles in the chain. For a triangle, the sum of the interior angles is 180. The figure adjustment for each of the three triangles illustrated in figure 15-27 is shown in table 15-7.

As you can see, the sum of the three adjusted observed interior angles in triangle ABC (angles 1, 2, and 3) comes to 1795940. This is 20 seconds less than 180, or 20/3, or 6 seconds for each angle, with 2 seconds left over. Therefore, 6 seconds was added to the station adjusted value of angle 1, and 7 seconds each was added to the measured values of angles 2 and 3. The angles in the other two triangles were similarly adjusted.