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ENGINEERING SURVEYING (221 BE)
Distance & Angle
Sr Tan Liat Choon Email: [email protected]
Mobile: 016-4975551
INTRODUCTION
Types of Measurements in Surveying: Surveying is the art of making suitable
measurements in horizontal or vertical planes. This is one of the important subjects of civil engineering. Without taking a survey of the plot where the construction is to be carried out, the work cannot begin
2
INTRODUCTION
From the above definition, we conclude on two types of measurements in surveying. They are as follows: Linear measurements Angular measurements
3
INTRODUCTION
Linear measurements are further classified as follows: Horizontal Distance Vertical Distance
4
INTRODUCTION
Horizontal Distance A horizontal distance is measured
in horizontal plane if a distance is measured along a slope, it is reduced to its horizontal equivalent
5
INTRODUCTION
Vertical Distance A vertical distance is measured
along the direction of gravity at that point. The vertical distance is measured to determine difference in elevations in various points
6
INTRODUCTION
Angular Measurements Two sides meeting at an angle are
measured. The angle between them is measured and represented in degrees or radians
7
DISTANCE MEASUREMENT
Linear measurement is the basis of all surveying and even through angles may be read precisely, the length of at least one line in tract must be measured to supplement the angles in locating points
8
DISTANCE MEASUREMENT
Is generally regarded as the most fundamental of all surveying observations
Many angles may be read, the length of at least one line must be observed to supplement the angles in locating points
In plane survey, the distance between two points means the horizontal distance
9
DISTANCE MEASUREMENT
Methods of measuring a horizontal distance: Tacheometry (Stadia), Taping, EDM and GPS Distance from stadia: (High wire – Low wire) *
100 = Distance EDM & GPS are most common in today’s survey Direct measurement Indirect measurement
10
DISTANCE MEASUREMENT
As simple as ABC:
11
New building site – how big is it?
60.159 m
60.159 m
32
.57
9 m
32
.57
9 m
DISTANCE MEASUREMENT
Measurement must be straight:
12
DISTANCE MEASUREMENT
Measurement around obstacle:
13
starting point closing point Obstacle to sight line
DISTANCE MEASUREMENT
Measurement around obstacle: Horizontal angles α and β are used to transform the resulting horizontal lengths to an equivalent horizontal length along the measurement line H 2-4 = h 2-3 cos α + h 3-4 cos β
14
1
β α
5 4
3
2
DISTANCE MEASUREMENT
Measurement must be straight:
15
DISTANCE MEASUREMENT
Measurement must be straight:
16
DISTANCE MEASUREMENT
Measurement must be straight:
17
DISTANCE MEASUREMENT
Generally, measurements are made horizontally, but on even, often man-made slopes the distance can be measured directly on the slope, but the vertical or zenith angle must be obtained V = Vertical Distance S = Slope Distance H = Horizontal Distance
18 A
S V
H
B
C
DISTANCE MEASUREMENT
c2 = a2 + b2 C = 90° = A + B sin A = a/c cos A = b/c tan A = a/b
19
A
c a
b
B
C
DISTANCE MEASUREMENT
A + B + C = 180° Sine Rule: a/sin A = b/sin b = c/sin C Cosine Rule: a2 =b2 + c2 – 2bc cos A cos A = (b2 = c2 - a2)/2bc Area = 1/2 bc sin A = √ s(s - a) (s - b) (s – c) where s = (a + b + c)/2
20
A
c
b
a B
C
INTRODUCTION ON ANGLE MEASUREMENT
Measuring distances alone in surveying does not establish the location of an object. We need to locate the object in 3 dimensions. To accomplish that we need: Horizontal length (distance) Difference in height (elevation) Angular direction
21
INTRODUCTION ON ANGLE MEASUREMENT
Angles measured in surveying are classified as either horizontal or vertical, depending on the plane in which they are observed
Horizontal angles are the basic observations needed for determining bearing and azimuths
Vertical angles are used in trigonometric levelling stadia and for reducing slope distances to horizontal
22
INTRODUCTION ON ANGLE MEASUREMENT
Determining the locations of points and orientations of lines frequently depends on measurements of angles and directions
In surveying, directions are given by azimuths and bearings
23
INTRODUCTION ON ANGLE MEASUREMENT
An angle is defined as the difference in direction between two convergent lines
A horizontal angle is formed by the directions to two objects in a horizontal plane
A vertical angle is formed by two intersecting lines in a vertical plane, one of these lines horizontal
A zenith angle is the complementary angle to the vertical angle and is formed by two intersecting lines in a vertical plane, one of these lines directed toward the zenith
24
INTRODUCTION ON ANGLE MEASUREMENT
25
TYPE OF ANGLE MEASUREMENT
Interior angles are measured clockwise or counter-clockwise between two adjacent lines on the inside of a closed polygon figure
Exterior angles are measured clockwise or counter-clockwise between two adjacent lines on the outside of a closed polygon figure
Deflection angles, right or left, are measured from an extension of the preceding course and the ahead line. It must be noted when the deflection is right (R) or left (L)
26
TYPE OF ANGLE MEASUREMENT
Angles to the right are turned from the back line in a clockwise or right hand direction to the ahead line
Angles to the left are turned from the back line in a counter-clock wise or left hand direction to the ahead line
Angles are normally measured with a transit or a theodolite, but a compass may be used for reconnaissance work
27
TYPE OF ANGLE MEASUREMENT
28
ANGLE MEASUREMENT
Angle is a difference in direction of 2 lines
To turn an angle we need a reference line, direction of turning and angular distance
Angular units: Degree, minutes, second Circle divided into 360 degrees Each degree divided by 60 minutes Each minute divided into 60 seconds
A check can be made because the sum of all angles in
any polygon must equal. (n-2) * 180, where n is the number of angles
29
MEASURING HORIZONTAL ANGLE
Set a bearing on the horizontal plate, and lock the upper motion
Release the lower motion, sight the backsight, lock the lower motion, and perfect the sighting with the lower tangent screw
Release the upper motion, turn to the foresight, lock the upper motion, and perfect the sighting
Record the horizontal bearing
Release the lower motion, invert the scope and point to the backsight in the reverse position, lock the lower motion, and perfect the sighting
Release the upper motion, turn to the foresight, lock the upper motion, and perfect the sighting
Record the second bearing 30
MEASURING ZENITH ANGLE
Point the instrument to the target object in a direct position
Lock the vertical motion, perfect the sighting and record the zenith angle
Loosen both the horizontal and vertical motions, plunge the scope, rotate the alidade 180° and re-point to the target in the reverse position
Lock the vertical motion, perfect the pointing and record the zenith angle
31
MEASURING ZENITH ANGLE
Direct 83° 28’ 16”
Reserve 276° 31’ 38”
Sum 359° 59’ 54”
360° Minus Sum 00° 00’ 06”
Half Value (error) 00° 00’ 03”
Plus Original Angle 83° 28’ 16”
Final Angle 83° 28’ 19” 32
USEFUL CONCEPTS Degrees: full circle = 360°
0°
45°
90°
135°
180°
225°
270°
315°
33
USEFUL CONCEPTS
34
BEARING AND AZIMUTH
35
COMPARISON OF AZIMUTH AND BEARING
Because bearings and azimuths are encountered in so many surveying operations, there is important to know the conversion of these two. Example 1 The azimuth of a boundary line is 128° 13’ 46”. Convert this into bearing. The azimuth places the line in the southeast quadrant. Thus the bearing angle is: 180° 13’ 46” - 128° 13’ 46” = 51° 46’ 14”, and the equivalent bearing is S 51° 46’ 14” E Example 2 The first course of a boundary survey is written as N 37° 13’ W. What is its equivalent azimuth? Since the bearing is in the northwest quadrant, the azimuth is: 360° - 37° 13’ = 322° 47’
36
COMPARISON OF AZIMUTH AND BEARING
37
Azimuths Bearings
Vary from 0° to 360° Vary from 0° to 90°
Require only a numerical value Require two letters and a numerical value
May be geodetic, astronomic, magnetic, grid, assumed, forward or back
May be geodetic, astronomic, magnetic, grid, assumed, forward or back
Are measured clockwise only Are measured clockwise and counterclockwise
Are measured either from north only, or from south only on a particular survey
Are measured from north and south
COMPARISON OF AZIMUTH AND BEARING
38
Quadrant Formulas for computing bearing angles from
azimuths
I (NE) Bearing = Azimuth
II (SE) Bearing = 180° - Azimuth
III (SW) Bearing = Azimuth - 180°
IV (NW) Bearing = 360° - Azimuth
COMPARISON OF AZIMUTH AND BEARING
39
Example directions for lines in the four quadrants (Azimuths from north)
Azimuths Bearings
54° N 54° E
112° S 68° E
231° S 51° W
345° N 15° W
AZIMUTH
40
Line Azimuth
O – A 54°
O – B 133°
O – C 211°
O – D 334°
COMPUTING AZIMUTH
41
N
B
A
C N
COMPUTING AZIMUTH
41° 35’ = AB + 180° 00’ 221° 35’ = BA + 129° 11’ 350° 46’ = BC - 180° 00’ 170° 46’ = CB + 88° 35’ 259° 21’ = CD - 180° 00’ 79° 21’ = DC + 132° 30’ 211° 51’ = DE
42
211° 51’ = DE - 180° 00’ 31° 51’ = ED + 135° 42’ 167° 33’ = EF + 180° 00’ 347° 33’ = FE + 118° 52’ 466° 25’ - 360° = 106° 25’ = FA + 180° 00’ 286° 25’ = AF + 115° 10’ 401° 35’ - 360° = 41° 35’ = AB
When a computed azimuth exceeds 360°, the correct azimuth is obtained by merely subtracting 360°
BEARING
43
Line Bearing
O – A N 54° E
O – B S 47° E
O – C S 31° W
O – D N 26° W
COMPUTING BEARING
44
Line Bearing
AB N 25° W
BC N 68° E
CD S 17° W
DA S 62° W
BEARING Bearing of a line is the direction of the line Azimuth of a line is the horizontal angle between 2 lines
Designation of Bearings Whole circle bearing Reduced Bearing (RB) or quadrantal bearing (QB) Fore Bearing (FB) or forward bearing (FB) Back bearing or Backward bearing (BB) Calculated bearing
Whole Circle bearing Bearings measured from north in a clockwise direction is termed as whole
circle bearing The value varies from 0 degrees to 360 degrees
Reduced bearing/Quadrantal bearing The bearings measured either from the north or from the south towards east
or west whichever is nearer is known as reduced bearing The values vary from 0 degrees to 90 degrees for a particular quadrant It is also known as quadrantal bearing (QB)
45
BEARING
Fore Bearing (FB) The bearings measured in the progress of surveying i.e. in the forward
direction of survey lines is known as fore bearing or forward bearing Back Bearing (BB) The bearings measured in opposite to the progress of surveying i.e. in
backward direction of survey line is known as Backward Bearing
Observed Bearing The bearings taken in a field with an instrument is known as Observed
Bearing
Calculated Bearing The bearings calculated from the field observation is known as
calculated bearing
46
RECTANGULAR COORDINATE
47
POLAR COORDINATE
48
+P (r, θ)
y
x
r
u
P (x,y)
y
x
Cartesian coordinates
Perpendicular axes Origin at (0,0) Coordinates increase right & up of origin Coordinates decrease down & left of origin
(0,0) x +
+
-
-
COORDINATE IN A PLANE
49
Cartesian coordinates
Coordinates of point given by bracketed pairs of numbers: (right,up)
(0,0) x
(3,4) x
(-3,-2) x
(x,y) (Easting,Northing) -depending on coordinate system used
COORDINATE IN A PLANE
50
Often easier to avoid negative values by increasing origin coordinates
+
+
(1000,1000) x
(1004,1006) x
(1001,1002) x
(998,999) x
NOTE: Some countries (incl. Sweden) use on maps: y=East x=North Others use opposite (e.g. (England, USA)
We’ll use (Easting,Northing)
COORDINATE IN A PLANE
51
p0 x
p1 x
Find coordinates of p1 in relation to p0
Easting
No
rth
ing
FINDING COORDINATE
52
p0 x
p1 x
Reference bearing (N)
Instrument
p0 (instrument) has known coordinates (0,0) for the moment reference bearing is known (N)
p1 has a unknown coordinates
FINDING COORDINATE
53
ө Nort
hin
g =
d c
os(ө
)
Easting = d sin(ө)
Use instrument to measure: d = (horizontal) distance p0-p1 ө = angle between North & bearing of p1 from p0
p0
p1 x
Reference bearing (N)
ө = bearing from reference d = distance from p0 to p1
d
with trigonometry...
Polar Coordinates
FINDING COORDINATE
54
ө
Nort
hin
g =
d c
os(ө
)
Easting = d sin(ө) p0
x
p1 x
Reference bearing (N)
d 10m
36.87°
Easting = d sin(ө)
= 10 sin(36.87)
= 10*0.6
= 6m
Northing = d cos(ө)
= 10 cos(36.87)
= 10*0.8
= 8m
ө also called the azimuth
FINDING COORDINATE
55
ө
No
rth
ing
= d
co
s(ө
)
Easting = d sin(ө) p0
x
p1 x
Reference bearing (N)
d 10m
36.87°
(6,8)
p1(Easting) = p0(Easting) + (d sin(ө)) p1(Northing) = p0(Northing) + (d cos(ө)) So if p0=(1000,1000) then p1(Easting,Northing) = (1006,1008)
FINDING COORDINATE
56
SURVEYING COORDINATE
57
For surveying we use a slightly different form of notation, instead of x, y,
+P ( E, N)
N
E
D
u
We use E, N (Easting, Northing)
SURVEYING COORDINATE
Note:
Easting is always quoted first and then Northing
Θ is always measured in a clockwise direction from North
Θ is known as the whole circle bearing
We must be able to convert from Rectangular to Polar and from Polar to Rectangular very quickly
58
SURVEYING COORDINATE
59
Any line has two bearings: N
N
u PQ QP
Q
P
u
We consider that the line PQ is a different line to line QP
SURVEYING COORDINATE
60
Given the coordinates of two points, calculate the distance and bearing
between of the line joining them
Q N
u PQ P(E,N)
Q(E, N)
Q(1127.37) – P(937.77) =189.69
Q(850.04) – P(1341.50) = 491.46
b=491.46
a=18
9.6
9
sin A = a/c c2 = a2 + b2
c2 = 189.602 + 491.462
c=526.77
c (PQ) = 526.77
sin A = 189.60/526.77
A = 21° 05’ 45”
Azimuth PQ = 270° + 21° 05’ 45” = 291° 05’ 45” Bearing PQ = N 68° 54’ 15” W
A
Point Easting (E) Northing (N)
P 1341.50 937.77
Q 850.04 1127.37
LOCAL ATTRACTION
The deflection of a magnetic needle from its true position due to the presence of magnetic influencing material such as iron rod, magnetic rock, underground pipeline, electric cables, iron pipes and electric pole in its vicinity is called local attraction
61
METHOD OF CORRECTING THE BEARING
There are two methods of correcting the bearing affected by local attraction: Included angle method Error computation method
62
METHOD OF CORRECTING THE BEARING
Included angle method: The included angles of the traverse
are calculated first, then starting from the line which is unaffected by local attraction and using the included angles, the corrected bearings of the traverse are computed
63
METHOD OF CORRECTING THE BEARING
Error computation method: The direction and the amount of local attraction
at each survey station is determined
Start from the line which is unaffected by local attraction, the corrected bearing of the traverse are computed
More accurate than the included angle method
It is adopted by most of the surveyors
64
Question 1. Give the comparison between Azimuth and Bearing 2. From the figure below, please give the bearings of lines AB and BC.
65
N
B
A
C N
66
T h a n k Yo u & Q u e s t i o n A n d A n s w e r