04 Angles Directions And Angle Measuring Equipment s2n1d

Dr. Mazen Abualtayef

Surveying

Chapter 5 Angles directions & angle measuring equipment

Content 5.1 Introduction 5.2 Horizontal, vertical and zenith angles 5.3 Reference direction 5.4 Reduced bearing of a line 5.5 Azimuth or whole circle bearing 5.6 Back reduced bearing and back azimuth 5.10 Main applications of the Theodolite

5.1 Introduction Surveying aims to determine the relative location of points on

the surface of the Earth. To locate a point, both distance and angular measurements are usually required. Angular measurements are either horizontal or vertical and they are

accomplished with a Theodolites.

5.2 Horizontal, vertical & Zenith angles Horizontal Angles: In the horizontal plane, direction of all lines of a survey are referenced to the meridian (true or magnetic).

5.2 Horizontal, vertical & Zenith angles

Vertical Angles:

A vertical angles is measured in a vertical plane in two ways depending on the reference from which the angle is measured: 1- Elevation or depression angle: using the horizontal plane as a reference (the value of the angle is between -90º to 90º) a- when the point being sighted on is above the horizontal plane, the angle is called an angle of elevation with positive value. b- when the point being sighted on is below the horizontal plane, the angle is called an angle of depression with negative value.

2- Zenith angle: using the overhead extension of the plumb line as a reference line, its value ranges from 0º to 180º.

5.3 Reference Direction It is convenient to choose a reference line to which directions of all the lines of surveying are referenced. Three types of direction lines are used in surveying. These are: •

true or geographic north,

•

the magnetic north and

•

the assumed north.

5.3 Reference Direction 5.3.1 True meridian ‫( خط الطول‬at any point): the great circle that es through that point and the geographic north and south poles of the earth. There are two methods to determine the north: The watch method

The shadow method

B 10AM D C 2PM

5.3 Reference Direction 5.3.2 Magnetic meridian: a direction that the magnetic needle takes when allowed to come to rest in the earth’s magnetic needle. Declination: The angle between the magnetic and true meridian is called the magnetic declination and expressed as the angular distance east or west of the true meridian, and its value is about 3º. 5.3.3 Assumed meridian: if neither the geographic north, nor the magnetic north is known in a certain local area for which the surveyor is performing measurements, a line can be arbitrarily chosen and assumed to be in the direction the north .

5.4 Reduced Bearing of a Line Bearing is the acute angle that the line makes with the meridian, it expressed as north or south and how many degrees to the east or west and its value ranges from 0º to 90º and called true bearing if the true meridian is used and magnetic bearing if the magnetic meridian is used.

5.5 Azimuth or Whole Circle Bearing Azimuth is the clockwise horizontal angle that the line makes with the north end of the reference meridian, and its value ranges from 0º to 360º. And called true azimuth if the true meridian is used as the reference and called magnetic azimuth if the magnetic meridian is used.

5.6 Back Reduced Bearing & Back Azimuth Back Reduced Bearing and Back Azimuth: Back bearing or Back azimuth of a line going from A to B is the azimuth or Bearing of the same line going from B to A. N Reference meridian N

A 60º 60º O

5.10 Main applications of the theodolite The main use of the theodolite is to measure horizontal

and vertical angles. These angles are used for the calculation of object heights, distances and point coordinates.

5.10.1 Measurement of Object Heights

Two cases can be distinguished: •

The determination of the elevation of a point whose horizontal distance to Theodolite can be measured

•

The determination of the elevation of a point whose horizontal distance to Theodolite can’t be measured

5.10.1 Measurement of Object Heights Case (1): Horizontal distance to Theodolite can be measured

D and  are measured and the elevation of building can be calculated by

HC  H A  i  D  tan 

H is vertical distance b/w C & C”

 1 1    H  Dtan α - tan β   D  tan z1 tan z 2 

5.10.1 Measurement of Object Heights Case (2): Horizontal distance to Theodolite can’t be measured A’B’ distance, a, b,  and  angles are measured then elevation of HC can be calculated by

A' C' B' C' A' B'   sin b sin a sin c

A' B' A' C'  sin b sin c

HC  H A  iA  A' C ' tan  HC  H B  iB  B' C ' tan 

A' B' B' C'  sin a sin c

5.10.2 Tacheometry Tacheometry is a system of rapid surveying, by which the positions, both horizontal and vertical, of points on the earth surface relatively to one another are determined without using a chain or tape or a separate leveling instrument. (http://en.wikipedia.org/wiki/Tacheometry) Two methods will be discussed here: • Tangential method and • Stadia method

5.10.2 Tacheometry Tangential method Measure vertical angles θ and Φ on a staff held vertically at B at two points M and N.

OM  D tan θ ON  D tan 

OM - ON b  D tan tan  - tan   b b D  tan  - tan    1 1     tan z1 tan z 2  ΔH  i  V  BN  i  D tan  t

5.10.2 Tacheometry Stadia method For this method, a Theodolite is equipped with a rectile that has one vertical and three horizontal cross-wires should be available. Vertical Wire

Upper Wire Middle Wire Lower Wire

5.10.2 Tacheometry Stadia method

5.10.2 Tacheometry Stadia Geometry for Horizontal Sight

d F  r i

 F D   r  F  C i D  kr

F is the focal length of lens. F and i are constant. (F+C) = 0.25 to 0.3m & can be neglected. k is called a stadia coefficient and = 100 for most instruments

5.10.2 Tacheometry Stadia Geometry for Inclined Sight

m' n'  mn cos , or r'  r cos S  kr'F  C

5.10.2 Tacheometry Stadia Geometry for Inclined Sight

V  S sin θ  kr cos sin   F  C sin D  S cos θ  kr cos   F  C cos 2

1 V  kr sin 2 2 1 2 D  kr cos  2

1 V  kr sin 2 z 2 1 D  kr sin 2 z 2

Example 5.2

Example 5.2

Example 5.2

Example 5.2