LECTURE 4 - Angularrrr Measurements.pptx

ANGULAR MEASUREMENT OF SURFACES USING SURVEYING PRINCIPLES

1. Lecture 4 Angular Measurements Isham C. Pelpuo [email protected] July 2023 1

2. Understand the purpose of angular measurements in surveying; Understand the difference between horizontal and vertical angles; Describe all the components of a theodolite and explain how these are used when measuring and setting out angles; Describe the field procedures that are used to set up and measure angles with a theodolite and total station; Book and calculate horizontal and vertical angles from theodolite readings; and Understand meridians and bearings used in giving directions of survey lines. The objective of the Lecture is to help students to: 2

3. Purpose of Angular Measurements Angular measurements are carried out in surveying for various purposes: Angles are directly measured in the field using: Theodolite; Total Station; An angle can be measured indirectly by the tape method (or linear measurements) and its value computed from the relationships of known quantities in a triangle or other geometric figure. 3 Compass; Sextant;

4. Definition of Horizontal and Vertical Angles Angles measured in surveying are classified as horizontal or vertical angles depending on the plane in which they are formed and measured. A horizontal angle is formed by two intersecting lines in a horizontal plane, and is measured in a horizontal plane. A vertical angle is formed by two intersecting lines in a vertical plane, and is measured in a vertical plane. A horizontal angle is considered clockwise if it is measured to the right or anticlockwise if it is measured to the left. A vertical angle is considered clockwise if it is measured downward or anticlockwise if it is measured upward. 4

5. Q W P K F W Q F Graphics of Horizontal and Vertical Angles 5 5 Horizontal angle formed by lines TA and TF at T = θ Vertical Angle formed at T by lines TA and TQ = α A = Elevation Angle Zenith Angle of line TA = Z A = 90 ᵒ - α A Vertical Angle formed at T by lines TW and TF = α B = Depression Angle Zenith Angle of line TF = Z B = 90 ᵒ + α B A was downward-projected to Q and F was upward-projected to W, and the horizontal angle was formed in a horizontal plane PKQW

6. Definition of Zenith Angle Zenith and Nadir are terms describing points on a Celestial sphere, i.e. a sphere of infinitely large radius with its centre as the earth. The zenith is directly above the observer and the nadir is directly below the observer. A zenith angle is an angle measured in a vertical plane downward from an upward directed vertical line through the instrument. Its value is between 0 º and 180 º . Zenith angle (Z) of a line may be defined as the clockwise vertical angle measured from the vertical line through the starting point of the line to the line. When zenith angle of a line is zero, it means the line is an upward directed vertical line. When zenith angle of a line is 180 ᵒ , it means the line is a downward directed vertical line. Z + α = 90 ᵒ , where α = Angle of Elevation (positive alpha) or Depression (negative alpha). Vertical (or zenith) angles are used in trigonometric levelling, stadia tacheometry, and for reducing measured slope distances to horizontal. 6

7. Definition of Inclination and Gradient of a Line Inclination of a line is the vertical angle ( α ) the line makes with the positive direction of x-axis. Value of this angle is between 0 ᵒ and 180 ᵒ . It is measured in an anti-clockwise direction from the positive direction of x-axis. Inclination of line AB may be defined as the anticlockwise vertical angle measured from an eastward directed horizontal line passing through A to line AB. The slope (or gradient) of a line is the tangent of the angle of inclination ( α A ) of that line. Slope of a line can be positive, negative, zero or undefined. Positive slope : tan α > 0, when α is acute angle; Negative slope : tan α < 0, when α is obtuse angle; Zero slope is obtained when α = 0, and the line is horizontal; Undefined slope is obtained when α = 90 ᵒ , and the line is vertical. Elevation Angle is an upward directed acute vertical angle measured with reference to a horizontal plane. A downward directed acute vertical angle measured with reference to a horizontal plane is called Depression Angle . 7

8. Relevance of Horizontal and Vertical Angles Horizontal angles are used for determining bearings and azimuths of lines, locating details when mapping, and for setting out structures during construction. Vertical angles are used for height determination and slope distance corrections. 8 Horizontal distance, D = L x cos α = L x sin Z Vertical Component, V = L x sin α = L x cos Z α = Vertical Angle, Z = Zenith Angle α + Z = 90 ᵒ

9. Theodolite Theodolites are precision instruments used extensively in construction work for measuring angles in the horizontal and vertical planes. Many different theodolites are available for measuring angles and they are classified according to the smallest reading (least count) that can be taken with the instrument. For example, a 1'' theodolite is one which can be read to 1'' directly without estimation. 9

10. Setting up: Theodolite The sequence of operations required to get the theodolite ready for angle measurements is as follows: Set up the tripod over the ground point or peg such that the top or tripod head is approximately horizontal; Mount the theodolite on the tripod and firmly secure it to the tripod head; Centre the theodolite over the ground mark using the optical plummet or the plumb line; Level the theodolite by centralizing the horizontal plate bubble. The purpose of levelling a theodolite or total station is to make its vertical axis coincide with the vertical through the instrument, i.e. the local direction of gravity. Remove parallax. 10

11. Double Sighting with the theodolite When a theodolite is set up over a survey mark and properly levelled, the position of the vertical circle with respect to the observer when looking through the eyepiece of the telescope is used to define the two positions or faces of the theodolite: ( i ) Face Left (FL): vertical circle is to the left of observer; (ii) Face Right (FR): vertical circle is to the right of observer. A theodolite can be swung horizontally in two directions: Anticlockwise (i.e. swinging to the left) and Clockwise (i.e. swinging to the right). Swinging a theodolite is rotating the theodolite horizontally about its vertical axis. 11

12. Double Sighting with the theodolite When a theodolite is in face left position when sighting one of the points, it is usually swung to the left to sight the other point. Similarly, when it is in face right position, it is swung to the right to sight the other point. Combining the face and swing we have: LL meaning face left, swing left; and RR meaning face right, swing right. Double sighting consists of making a measurement of a horizontal or a vertical angle once with the telescope in the direct or face left position and again with the telescope in the reversed, inverted, plunged or face right position. The act of turning the telescope upside down, that is, rotating it about the transverse or tilting axis, is called “ plunging ” or “ transiting ” the telescope. 12

13. Total Station 13

14. Total Station 14

15. Booking and Reducing Horizontal Circle Readings 15

16. Booking and Reducing Horizontal Circle Readings 16 Question 1: A horizontal angle is measured at E from D to F by the method of double sighting. A backsight on D gives a circle reading of 30º 50' 40'' and a foresight on F gives a circle reading of 146º 40' 20''. After plunging the telescope, face-right readings made to F and D were 326º 40' 30'' and 210º 50' 40'' respectively. Prepare a horizontal angle booking sheet, book the horizontal circle readings, and deduce the mean included angle at E?  

17. END OF LECTURE 4: Angular Measurements 17

18. Assignment 18 Question 2: A backsight was made at K to J, and the horizontal circle read 312º 14' 30''. A foresight was made to L, and the circle read 14º 42' 00''. The telescope was reversed, and face-right readings made to L and J were 194º 42' 10'' and 132º 14' 30'' respectively. What is the value of the mean included angle at K?