Orientation to baselines for building site network.
Kala, Vello
1. Introduction
Relatively high accuracy is needed for setting out and as-built
surveys in the construction sites. Regrettably, the internal accuracy of
national geodetic control points may often be insufficient for some
construction works. In addition, as is well known, one of the basic
requirements for building site network is that reductions in a geodetic
reference ellipsoid and due to map projections (for example, effects
could be relatively large and reduction in the line of 200 m on the
central meridian of TM projection equals 80 mm) should be avoided in the
majority of cases. This is due to the fact that design engineers
customarily work with actual ground distances. There is also a need for
linking a building site to national or regional infrastructure objects
such as pipelines, motorways and railways and power transmission lines;
hence, it is requested to calculate building site network with respect
to the national coordinate system ([TEXT NOT REPRODUCIBLE IN ASCII.]
1970; Smith 1997).
Therefore, building site network is connected to the national
geodetic network with reference to the obtained traverse measurements or
GPS. The length of the traverse could reach as much as a few km. The
adjustment of connecting such a traverse is executed in the usual
manner. Let us review a simple typical case where APQL is connection
traverse and PQMK is building site network (see Fig. 1) with an emphasis
on the problems of adjustment and orientation to building site network.
2. The Essence of the Problem
The coordinates of the national network (points A and L in Fig. 1)
are held fixed, whereas all necessary reductions due to map projection
will be considered. Let us assume that connecting traverse passes via
points P and Q that form a baseline (with direction angle
[[alpha].sub.PQ]) for controlling a network on the building site (see
Fig. 1).
Next, after measuring the distances and angles of the closed loop
traverse PQMK, the coordinates of building site network (in addition to
P and Q, also K and M, see Fig. 1) are to be computed. By fixing the
national coordinates of P and Q and applying all necessary reductions,
we obtain the national coordinates of points M and K. However, we also
need the coordinates of PQMK in the grid system of the site.
[FIGURE 1 OMITTED]
The following two grid systems of the site may be considered:
--a parallel system, when the coordinate axes of the site grid are
parallel to the coordinate axes of the national system (see Fig. 1);
--a non-parallel system, when the coordinate axes of the site grid
are twisted with respect to the coordinate axes of the national system.
The origin of the system is usually placed to the immediate
neighbourhood of the building site so that the whole site would be
located in the first quadrant, see Fig. 3.
Let us examine a more simple first case, see Fig. 1. The national
coordinates of point P and the initial direction angle [[alpha].sub.PQ]
in the national system (from connecting traverse adjustment) are taken
as origin. No reductions in the ellipsoid and map projection to PQMK
sides are used and the coordinates of points Q, M, K will be computed in
the system of the site network. Certainly, due to used initial data and
adopted assumptions, the coordinates in the system of the site network
are expected to be close (but not the same!) to those of the national
system.
Next, we focus on the case when baseline PQ is either close to the
north-south direction (see Fig. 1) or the east-west direction. We
suggest that in such a case only one Q coordinate--either [x.sub.Q] or
[y.sub.Q] will be adjusted. If [[alpha].sub.PQ] [approximately equal to]
0[degrees] (or 180[degrees]), then preferably only [x.sub.Q] will be
adjusted and [y.sub.Q] is to be held fixed, because adjusting [y.sub.Q]
changes predominantly [[alpha].sub.PQ] value as well.
Actually, adjusting [x.sub.Q] also changes [[alpha].sub.PQ], though
it is generally to a lesser degree. The more [[alpha].sub.PQ] differs
from 0[degrees] (or 180[degrees]), the more altering [x.sub.Q] changes
[[alpha].sub.PQ] (see Fig. 2). A similar effect takes place if
[[alpha].sub.PQ] is about 90[degrees] or 270[degrees]; then, adjusting
[y.sub.Q] is only needed. However, a precondition that the initial
direction angle for every geodetic system must remain constant is
necessary.
This contribution attempts to find the most optimum solution for
such a case.
In this case, a problem arises. How much [x.sub.Q] can be altered
for changes in [[alpha].sub.PQ] to remain insignificant?
[FIGURE 2 OMITTED]
3. Recommendations for Orientation to Baselines for Building Site
Network
A differential equation of direction angles ([TEXT NOT REPRODUCIBLE
IN ASCII.] 1981) can be applied to find the permitted value of
([[alpha].sub.PQ], i.e. permitted deviation from NS or EW direction
d[alpha] = -[rho]sin[alpha]xdx/s + [rho]cos[alpha]xdy/s, (1)
where d[alpha]--permitted change in the baseline direction angle
due to change in the coordinates of baseline endpoint (Q); s--the length
of the network side; [alpha]--a direction angle of the network baseline;
dx, dy--are the components of misclosure.
As we do not adjust to [y.sub.Q] (i.e. dy = 0), there is no need to
account for the second term of Eq. (1), and thus we have the absolute
value:
[d[alpha].sub.PQ(perm)] = arcsind[alpha]xs/[rho]xdx, (2)
Assuming:
--that the total station to be used has the nominal distance
measurement accuracy of 2 mm + 2 ppm and the accuracy of angle
measurements (by DIN 18723, 2002) is 1";
--distances PK, KM, MQ, QP are about 200 m and the form of
quadrangle PKMQ is near a quadrate; then the program Local X*Positioning
System gives the mean square error (MSE) of [y.sub.Q] as 5.2 mm and
[v.sub.x] = dx = 2.5 x 5.2 = 13 mm.
Further, for adopting a permissible change in baseline direction
angle d[alpha] = 0.5" (in this case, the initial direction angle
practically remains unchanged), we can find the maximum value of the
deviation of baseline direction angle from NS or EW direction for site
network (see Section 2)
[[alpha].sub.PQ(perm)] = arcsin0.5x200000/206265x13 = 2.14[degrees]
(3)
i.e. 0[degrees] [+ or -] 2.14[degrees], (or 90[degrees] [+ or -]
2.14[degrees] or 180[degrees] [+ or -] 2.14[degrees] or 270[degrees] [+
or -] 2.14[degrees]). This sets rather strict requirements for
orientation to baselines for the site network.
If the required accuracy of building site network is lower/higher,
the network described in this section (i.e. the predicted misclosure
[v.sub.x] = dx (or [v.sub.y] = dy) is larger/ smaller than 13 mm and/or
sides are shorter/longer than 200 and [[alpha].sub.PQ(perm)] value
changes approximately proportionally to
d[alpha]"/[dx.sub.mm] and [s.sub.m]/200, (4)
where [s.sub.m]--the average length of the sides (m); [dx.sub.mm]
is predicted [v.sub.xQ] (mm).
[FIGURE 3 OMITTED]
4. Features of Orientation to Baselines for the Non-Parallel
Reference System
In this section, we briefly deal with the second case--a non
parallel reference system, such as a location grid, site grid or
structural grid (ISO 4463-1), the coordinate axes of which are parallel
to the axes of the main constructions (Fig. 3).
The most important benefit of such non-parallel reference system is
that orientation to the baseline for building site network close to the
direction of the site grid axis is in many cases more convenient than
using a baseline with direction close to the coordinate axis of the
national coordinate system.
Second, the parallelism of the coordinate axes of the site grid and
main constructions simplify significantly the works of design engineers
and surveyors.
Third, the coordinates of the numerical values of the arbitrary
grid are not similar to that of the national system i.e. an accidental
mix between the coordinates of the site grid and that of the national
system can be easily avoided. The permitted deviation
[[THETA].sub.PQ(perm)] of site network baseline PQ from the axis N of
the site grid can be found by means of Eq. (5), similar to Eq. (2), by
replacing da [right arrow] d[THETA] and dx [right arrow] dN.
[[THETA].sub.PQ(perm)] = arcsin d[THETA]xs/[rho]xdN, (5)
5. Summary
One of the basic requirements for processing data on building site
network is that reductions in the ellipsoid and map projection should be
avoided, because design engineers customarily work with actual ground
distances. However, a site network should be connected to the national
coordinate system for linking the building site to regional
infrastructure objects. Thus, the points of building site network should
have the coordinates of both--national and building site grid systems.
A pair of points usually belongs to connection traverse and is
taken as a baseline for the site network. The coordinates of the first
point (as origin) and the direction angle of the baseline are obtained
immediately from connecting traverse adjustment. When adjusting the site
network in the site grid system, no reductions in the ellipsoid and map
projection to the sides of the network will be assigned.
Note that the direction angle of the baseline should remain the
same, although we use free (more exactly semi-free) adjustment.
Nevertheless, after adjusting the coordinates of the direction point
(backsight point) of the baseline, the direction angle may change.
Nevertheless, it is a well-known assumption that the initial direction
angle of a right geodetic system must be constant.
To minimize this change (to become even insignificant), the
direction angle of the baseline should be oriented either to NS or EW
direction approximately and use corrections to the second point of
baseline only, i.e. [v.sub.x] or [v.sub.y] (which is about along the
baseline).
If the direction angle is not exactly equal to 0[degrees],
90[degrees], 180[degrees] or 270[degrees], the change of one coordinate
of the second point of the baseline brings along a slight change in the
other coordinate. The permitted range of the change can be found by
means of the differential equation of direction angles. This study has
demonstrated that the maximum deviation of the direction angle from NS
or EW direction should be no more than 2.14[degrees] for a precise
building site network.
If the required accuracy of the network is lower/ higher and the
sides are shorter/longer, the maximum deviation of the direction angle
will change approximately in proportion to d[alpha]"/[dx.sub.mm]
and s/200, where d[alpha]" is per missible change in the direction
angle; [dx.sub.mm]--predicted [v.sub.x] in the units of mm; s--the
average side of the site network in the units of m. The corresponding
permitted values depend on the features of constructions and should be
determined in cooperation with design engineers.
We have also showed that in the case of the nonparallel reference
system, the permitted deviation can be found by using the equations of
the parallel grid. For instance, the permitted deviation
[[THETA].sub.PQ(perm)] of the baseline for the site network from the
axis N of the site grid can be found by means of Eq. (5), which is
similar to Eq. (2).
Acknowledgment
The author is grateful to Prof. A. Ellmann (TTU) for his
constructive comments.
doi: 10.3846/13921541.2011.558995
References
DIN 18723 Specification for Theodolite Accuracy. 2002. Professional
Surveyor Magazine. Available from Internet: <www.profsurv.com>.
ISO 4463-1 International Standard. Measurement methods for
building--Setting out and measurement. Part 1: Planning and
organization, measuring procedures, acceptance criteria. 1989. 4, 8.
Smith, J. (Ed.). 1997. The management of setting out in
construction. Institution of Civil Engineering Surveyors. Thomas Telford
Publications. London. 1 ... 2, 9, 10.
[TEXT NOT REPRODUCIBLE IN ASCII.] [Levcuk, G. Course of engineering
surveying]. [TEXT NOT REPRODUCIBLE IN ASCII.]: [TEXT NOT REPRODUCIBLE IN
ASCII.]. 344 c.
[TEXT NOT REPRODUCIBLE IN ASCII.], B. 1981. [TEXT NOT REPRODUCIBLE
IN ASCII.] [Selihanovic, V. Geodesy]. [TEXT NOT REPRODUCIBLE IN ASCII.]:
[TEXT NOT REPRODUCIBLE IN ASCII.]. 388 c.
Vello Kala
Chair of Geodesy, Dept of Transportation, Faculty of Civil
Engineering, Tallinn University of Technology, Ehitajate tee 5, 19086
Tallinn, Estonia E-mail: vello.kala@ttu.ee
Received 15 Jan. 2010; accepted 20 Dec. 2010
Vello KALA. Lecturer. MSc from Tallinn University of Technology,
the chair of geodesy, Ehitajate tee 5, 19086, Tallinn, Estonia, Ph +372
6202602, Fax +372 620 2601, e-mail: vello.kala@ttu.ee
A graduate in MIIGAiK, an engineer of engineering geodesy, 1974. A
MSc degree from TTU in 1997. 8 presentations at international and SU
conferences and seminars. The author of 19 course books, handbooks,
standards and more than 30 scientific and technical papers.
Research interests: construction surveys, precise levelling.