Diploma Civil and Mechanical Engineering for Engineering Assistant · Chapter 7

Surveying (Chain, Compass, Theodolite, Levelling)

What to remember

Surveying finds the relative positions of points on or near the earth. Plane surveying treats the earth as flat and is used for small areas. Geodetic surveying allows for the curvature of the earth.
The basic principle is working from the whole to the part. A framework of main control points is fixed first with high accuracy, and the details are then added inside it.
Levelling uses Height of Instrument (HI) and Reduced Level (RL). HI = RL of the benchmark + backsight. RL of a point = HI − foresight.

Chain surveying

Chain surveying is used for small, open areas where only linear measurements are taken. The area is divided into well-conditioned triangles, whose angles should be between 30° and 120° (ideally close to equilateral).

Instruments and terms:

Metric chains are 20 m and 30 m long. A 20 m chain has 100 links, each 0.2 m. Gunter's chain is 66 ft with 100 links. The engineer's chain is 100 ft with 100 links.
Tags (tallies) mark metre points along the chain. Brass tags help read the length quickly.
Arrows (chaining pins) are about 40 cm long, and a set has 10. They mark each chain length. Ranging rods are 2 m or 3 m long, painted in alternate bands of black and white (or red and white), and used to mark stations.
A base line is the longest main line. Check lines (proof lines) test the accuracy of the work. Tie lines join points on the main lines for details.
Ranging is fixing intermediate points on a line between two end stations, so that they lie on a straight line. It can be direct or indirect (reciprocal).
Offsets are lateral measurements from the chain line to details. A perpendicular offset is at 90°. An oblique offset is not at 90°. Offsets are usually limited to about 15 m for ordinary work.
Instruments for setting out right angles are the cross staff, optical square and prism square.

Corrections and errors in chain surveying:

If the chain is too long, the measured length is too short. A correction is added. True length = (L′ ÷ L) × measured length, where L′ is the actual length of the chain and L is the nominal length.
Area correction = (L′ ÷ L)² × measured area.
Slope correction = h² ÷ (2 l), where h is the height difference and l is the slope length. Always subtract it. The horizontal distance is l cos θ.
Sag correction = w² l³ ÷ (24 P²) per span, where w is the weight per unit length and P is the pull. Always subtract it.
Temperature correction = α (T − T0) L, where α is the coefficient of expansion. It can be positive or negative.
Tension (pull) correction = (P − P0) L ÷ (A E).

Worked example: A line measured with a 20 m chain which is 0.1 m too long is recorded as 400 m. True length = (20.1 ÷ 20) × 400 = 402 m.

Compass surveying

A prismatic compass reads bearings through a prism, the card being graduated in the whole circle bearing (WCB) system from 0° to 360°. A surveyor's compass has a graduated card in the quadrant system and is read through a slit.

Bearings:

Magnetic bearing is measured from the magnetic meridian. True bearing is from the true (geographic) meridian. Arbitrary bearing is from any assumed meridian.
Whole circle bearing (WCB) runs from 0° to 360° clockwise from the north. Quadrantal (reduced) bearing (QB) is measured from the north or south, towards the east or west, with a value from 0° to 90°, such as N 40° E.
Fore bearing (FB) is in the direction of survey. Back bearing (BB) is in the opposite direction. BB = FB ± 180°. Use + when FB is less than 180° and − when it is more.
Magnetic declination is the angle between the true and magnetic meridians. Dip is the angle by which the needle tilts from the horizontal. A rider (a small weight) balances the dip.
Local attraction is the disturbance of the needle by magnetic materials such as iron or electric cables nearby. In a line affected by it, the FB and BB do not differ by exactly 180°.

Conversion of WCB to QB:

WCB range	Quadrant	QB
0° to 90°	NE	N θ E, where θ = WCB
90° to 180°	SE	S θ E, where θ = 180° − WCB
180° to 270°	SW	S θ W, where θ = WCB − 180°
270° to 360°	NW	N θ W, where θ = 360° − WCB

Worked example: FB of a line is 125°30′, so BB = 125°30′ + 180° = 305°30′. A WCB of 230° gives S 50° W.

Sum of the interior angles of a closed traverse with n sides = (2n − 4) × 90°. Included angle = FB of the next line − BB of the previous line (adjusted by 360° if needed).

Theodolite

A theodolite measures horizontal and vertical angles precisely. A transit theodolite can rotate its telescope fully in a vertical plane about the horizontal axis. Vernier theodolites commonly have a least count of 20″ or 10″. The least count of a vernier = the value of one main scale division ÷ the number of vernier divisions.

Parts: levelling head with foot screws, lower plate (graduated horizontal circle), upper plate (with vernier), telescope, vertical circle, altitude bubble, plate bubble and plumb bob.

Axes: the vertical axis, the horizontal (trunnion) axis, the line of collimation and the bubble line axis. The fundamental relationship is that the vertical axis is perpendicular to the horizontal axis, and the line of collimation is perpendicular to the horizontal axis.

Temporary adjustments done at every station: setting up (over the station), centering (using the plumb bob), levelling (with foot screws using the plate bubble) and elimination of parallax (by focusing the eyepiece and the objective).

Terms: face left (vertical circle on the left of the observer) and face right. Changing face means transiting the telescope and swinging the instrument. Measurements are taken on both faces and averaged to remove instrumental errors. Swing is the rotation of the telescope in the horizontal plane. Transiting (plunging) is the rotation in the vertical plane. Methods of measuring horizontal angles: ordinary (single) method, repetition method (reduces least count errors) and reiteration method.

Tacheometry measures distances using the stadia hairs. For a horizontal line of sight, D = k s + c, where s is the staff intercept, k is the multiplying constant (usually 100) and c is the additive constant (nearly zero for modern internal-focusing instruments). For an inclined sight, horizontal distance D = k s cos² θ.

Worked example: For a staff intercept of 1.50 m with k = 100 and c = 0 and a horizontal sight, D = 100 × 1.50 = 150 m.

Levelling

Levelling finds the difference in elevation between points. Instruments are the dumpy level, tilting level, automatic level and levelling staff (usually 4 m or 5 m, telescopic or folding).

Terms:

Benchmark (BM): a fixed point of known RL. GTS benchmarks are those of the Great Trigonometrical Survey.
Backsight (BS): the first staff reading after setting up the instrument. Foresight (FS): the last reading before shifting. Intermediate sight (IS): any reading in between.
Change point (turning point): the point where both FS and BS are taken.
Line of collimation is the line of sight. Height of instrument (HI) is the RL of the line of collimation.

Methods:

Height of instrument (HI) method: HI = RL of BM + BS. RL of a point = HI − reading (FS or IS). On changing the instrument, the new HI = RL of the change point + BS.
Rise and fall method: the difference between consecutive readings. A rise means the second point is higher, so RL = previous RL + rise. A fall means the point is lower.

Arithmetic checks: ΣBS − ΣFS = Σrise − Σfall = last RL − first RL.

Corrections:

Curvature correction = 0.0785 d² metres (d in km), and is always subtracted from the staff reading.
Refraction correction = one-seventh of the curvature correction, and is added.
Combined correction = 0.0673 d² metres (d in km), subtracted from the staff reading.
Reciprocal levelling eliminates the errors of collimation, curvature and refraction.

Worked example: The BM has an RL of 100.000 m. BS on it is 1.500 m, so HI = 101.500 m. The FS on a point is 2.300 m, so its RL = 101.500 − 2.300 = 99.200 m.

Contouring: A contour is a line joining points of equal elevation. The vertical gap between two consecutive contours is the contour interval. Properties: contour lines do not cross each other except in an overhanging cliff or a cave. Closely spaced contours show a steep slope. Widely spaced contours show a gentle slope. Equally spaced contours show a uniform slope. Contours run perpendicular to the direction of the steepest slope. Closed contours with higher values inside show a hill, and lower values inside show a depression. Contours cross a ridge line or valley line at right angles. A V-shape pointing towards a lower value shows a ridge, and a V-shape pointing towards a higher value shows a valley.

Areas and volumes: area by the trapezoidal rule = d × [(O1 + On) ÷ 2 + O2 + ... + On−1]. Simpson's rule needs an odd number of ordinates: Area = (d ÷ 3) × [first + last + 4 × (sum of even-numbered ordinates O2, O4, ...) + 2 × (sum of the remaining odd-numbered inner ordinates O3, O5, ...)]. Volume by the prismoidal formula = (h ÷ 6) × (A1 + 4 Am + A2).

Exam traps

BB = FB ± 180°. Do not add 90°.
A chain that is too long gives a measured length that is too short, so the correction is added.
Slope and sag corrections are always subtractive. Temperature can be either.
Curvature correction is subtractive, refraction is additive, and the combined value is 0.0673 d².
HI is the RL of the line of sight, and not the height of the instrument above the ground.
IS readings are not used to find the HI at a change point.
WCB uses 0° to 360°, and QB uses 0° to 90°.
Closely spaced contours mean a steep slope. Widely spaced contours mean a gentle slope.

One-liners

1. A 20 m chain has 100 links of 0.2 m each.
2. Ranging rods are painted black and white in alternate bands.
3. Prismatic compass uses the WCB system.
4. BB = FB ± 180°.
5. Magnetic declination is the angle between the true and magnetic meridians.
6. Theodolite measures horizontal and vertical angles.
7. HI = RL of benchmark + BS.
8. RL = HI − FS.
9. ΣBS − ΣFS = last RL − first RL.
10. Combined curvature and refraction correction = 0.0673 d² (d in km).
11. A contour joins points of equal elevation.
12. Tacheometric distance for a horizontal sight = 100 × staff intercept.

Practice questions

The fore bearing of a line is 125°30′. Its back bearing is
1. 215°30′
2. 54°30′
3. 35°30′
4. 305°30′
Answer
D. 305°30′
BB = FB + 180° when FB < 180°: 125°30′ + 180° = 305°30′.
A whole circle bearing of 230° is equal to the quadrantal bearing
1. S 40° W
2. S 50° W
3. S 50° E
4. N 50° W
Answer
B. S 50° W
For 180° to 270°, QB = S (WCB − 180°) W = S 50° W.
The quadrantal bearing S 30° E is equal to the whole circle bearing
1. 30°
2. 210°
3. 330°
4. 150°
Answer
D. 150°
For the SE quadrant, WCB = 180° − 30° = 150°.
A line measured with a 20 m chain which is 0.1 m too long is recorded as 400 m. The true length is
1. 404 m
2. 402 m
3. 400.1 m
4. 398 m
Answer
B. 402 m
True length = (20.1 ÷ 20) × 400 = 402 m.
For a slope length of 50 m with a height difference of 2 m between the ends, the slope correction h² ÷ 2l is
1. 0.02 m
2. 0.4 m
3. 0.08 m
4. 0.04 m
Answer
D. 0.04 m
4 ÷ (2 × 50) = 0.04 m, to be subtracted.
The combined curvature and refraction correction (0.0673 d², d in km) for a sight of 2 km is about
1. 0.135 m
2. 0.538 m
3. 0.0673 m
4. 0.269 m
Answer
D. 0.269 m
0.0673 × 2² = 0.269 m.
A benchmark of RL 100.000 m is read with a backsight of 1.500 m. The height of the instrument is
1. 151.500 m
2. 101.500 m
3. 98.500 m
4. 100.000 m
Answer
B. 101.500 m
HI = RL + BS = 100.000 + 1.500 = 101.500 m.
For a horizontal line of sight with k = 100, c = 0 and a staff intercept of 1.50 m, the tacheometric distance is
1. 100 m
2. 150 m
3. 15 m
4. 1.5 m
Answer
B. 150 m
D = k s + c = 100 × 1.50 = 150 m.
The sum of the interior angles of a closed traverse with 6 sides is
1. 720°
2. 1080°
3. 900°
4. 540°
Answer
A. 720°
(2n − 4) × 90° = 8 × 90° = 720°.
In a level book ΣBS = 8.2 m and ΣFS = 5.7 m. If the first RL is 100.00 m, the last RL is
1. 100.00 m
2. 105.70 m
3. 97.50 m
4. 102.50 m
Answer
D. 102.50 m
Last RL − first RL = ΣBS − ΣFS = 2.5 m.
For an inclined line of sight, D = k s cos² θ. With k = 100, s = 2 m and θ = 30°, the horizontal distance is
1. 200 m
2. 100 m
3. 150 m
4. 173.2 m
Answer
C. 150 m
cos² 30° = 0.75, so D = 100 × 2 × 0.75 = 150 m.
A 30 m steel tape (α = 1.2 × 10⁻⁵ per °C) is used at 20 °C above its standard temperature. The temperature correction is
1. +7.2 mm
2. +72 mm
3. +0.72 mm
4. −7.2 mm
Answer
A. +7.2 mm
α (T − T0) L = 1.2 × 10⁻⁵ × 20 × 30 = 0.0072 m = 7.2 mm; it is added since the tape is longer.
A 20 m chain is actually 20.2 m long. An area measured on the plan as 1000 m² has a true area of about
1. 1010 m²
2. 980.2 m²
3. 1000 m²
4. 1020.1 m²
Answer
D. 1020.1 m²
True area = (20.2 ÷ 20)² × 1000 = 1.0201 × 1000 = 1020.1 m².
In the rise and fall method, a point has a backsight of 1.200 m and the next foresight is 1.800 m. If the first RL is 50.00 m, the RL of the next point is
1. 50.60 m
2. 49.40 m
3. 48.20 m
4. 51.80 m
Answer
B. 49.40 m
BS − FS = −0.600 m, so it is a fall: 50.00 − 0.60 = 49.40 m.
A metric chain 20 m long has
1. 200 links of 0.1 m each
2. 66 links of 0.3 m each
3. 50 links of 0.4 m each
4. 100 links of 0.2 m each
Answer
D. 100 links of 0.2 m each
Metric chains of 20 m have 100 links each 0.2 m long.
For good chain survey work, the angles of the triangles should lie between
1. 30° and 120°
2. 90° and 150°
3. 10° and 60°
4. 60° and 180°
Answer
A. 30° and 120°
Such triangles are well-conditioned.
Ranging rods are usually painted in alternate bands of
1. green and yellow
2. orange and brown
3. black and white
4. blue and green
Answer
C. black and white
The contrast makes the rods visible from far away.
A prismatic compass measures bearings in the
1. quadrantal system only
2. decimal degree system
3. whole circle bearing system
4. azimuth from south only
Answer
C. whole circle bearing system
The prismatic compass card is graduated from 0° to 360°.
The angle between the true meridian and the magnetic meridian is called
1. dip
2. magnetic declination
3. convergence
4. local attraction
Answer
B. magnetic declination
Magnetic declination is the angle between the two meridians.
Local attraction is caused by
1. the rotation of the earth
2. magnetic materials near the compass
3. tilt of the telescope
4. an error in the chain
Answer
B. magnetic materials near the compass
Iron objects and electric cables disturb the needle.
The rider on the needle of a prismatic compass is used to counter
1. dip
2. refraction
3. parallax
4. declination
Answer
A. dip
A small weight balances the tilt of the needle caused by the earth's dip.
A theodolite is mainly used to measure
1. only slopes of land
2. only the heights of benchmarks
3. horizontal and vertical angles
4. only the length of lines
Answer
C. horizontal and vertical angles
Its horizontal and vertical circles read angles.
Which is the correct order of the temporary adjustments of a theodolite?
1. Centering, focusing, setting up, levelling
2. Levelling, focusing, setting up, centering
3. Focusing, setting up, levelling, centering
4. Setting up, centering, levelling, focusing
Answer
D. Setting up, centering, levelling, focusing
The instrument is first set up over the station, then centered, levelled and focused.
Observing on both the face left and the face right of a theodolite helps to
1. eliminate instrumental errors
2. increase the range of the telescope
3. change the least count
4. measure only vertical angles
Answer
A. eliminate instrumental errors
Averaging the two faces cancels many instrumental errors.
The repetition method of measuring a horizontal angle mainly reduces
1. errors in the staff
2. errors due to curvature
3. errors of declination
4. errors due to the least count
Answer
D. errors due to the least count
Repeated readings accumulate the angle, so the least-count error is divided.
A backsight in levelling is
1. any reading in between
2. the first staff reading after setting up the instrument
3. the last reading before shifting
4. a reading on the instrument
Answer
B. the first staff reading after setting up the instrument
A backsight is taken on a point of known RL.
A change point (turning point) is the point at which
1. only a backsight is taken
2. only an intermediate sight is taken
3. the instrument is placed
4. both a foresight and a backsight are taken
Answer
D. both a foresight and a backsight are taken
It links two instrument stations.
The curvature correction in levelling is applied to the staff reading by
1. subtraction
2. multiplying by 0.0673
3. dividing by 7
4. addition
Answer
A. subtraction
Curvature makes the staff reading appear too high, so it is subtracted.
Reciprocal levelling eliminates the errors of
1. only parallax
2. collimation, curvature and refraction
3. only staff reading
4. only temperature
Answer
B. collimation, curvature and refraction
Readings from both sides cancel these errors.
Closely spaced contour lines indicate
1. a gentle slope
2. a lake
3. a steep slope
4. a flat ground
Answer
C. a steep slope
A small horizontal gap for the same interval means a steep slope.
Contour lines can cross each other only in the case of
1. a ridge
2. an overhanging cliff
3. a flat plain
4. a valley
Answer
B. an overhanging cliff
At an overhanging cliff, contours may overlap or cross.
Simpson's rule for area requires
1. an odd number of ordinates
2. an even number of ordinates
3. only two ordinates
4. equal heights
Answer
A. an odd number of ordinates
The rule takes the ordinates in pairs of strips, so the number of strips must be even.
Which of the following statements is/are correct? 1. If a chain is too long, the measured length is less than the true length. 2. Slope correction is always added to the measured length.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Slope correction is always subtracted.
Which of the following statements is/are correct? 1. The back bearing of a line is its fore bearing ± 180°. 2. Magnetic bearing is measured from the true meridian.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Magnetic bearing is measured from the magnetic meridian.
Which of the following statements is/are correct? 1. Sag correction is additive. 2. Temperature correction can be positive or negative.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Sag correction is always subtractive.
Which of the following statements is/are correct? 1. Swing means rotation of a theodolite telescope in the vertical plane. 2. Transiting means rotating the telescope in the vertical plane about the horizontal axis.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Swing is rotation in the horizontal plane.
Which of the following statements is/are correct? 1. HI = RL of benchmark + backsight. 2. The RL of a point = HI + foresight.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
RL = HI − foresight (or intermediate sight).
Which of the following statements is/are correct? 1. The curvature correction is subtracted from the staff reading. 2. The refraction correction is added to the staff reading.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Refraction is one-seventh of the curvature correction and acts in the opposite sense.
Which of the following statements is/are correct? 1. A prismatic compass reads whole circle bearings. 2. A surveyor's compass reads quadrantal bearings.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Prismatic compass uses WCB; surveyor's compass uses the quadrant system.
Which of the following statements is/are correct? 1. A contour joins points of equal elevation. 2. Widely spaced contours indicate a steep slope.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Widely spaced contours indicate a gentle slope.
Which of the following statements is/are correct? 1. The sum of the interior angles of a closed traverse is (2n − 4) × 90°. 2. For a closed four-sided traverse this sum is 360°.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
For n = 4, (8 − 4) × 90° = 360°.
Which of the following statements is/are correct? 1. In tacheometry D = k s + c, where k is usually 100. 2. For modern internal-focusing instruments the additive constant c is nearly zero.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
C. Both 1 and 2
Both are standard facts.
Which of the following statements is/are correct? 1. Arrows mark each chain length during chaining. 2. Ranging rods are about 40 cm long.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Ranging rods are 2 m or 3 m long; arrows are about 40 cm.
Match the sight with its meaning: (a) Backsight (b) Foresight (c) Intermediate sight. Meanings: (1) last reading before shifting the instrument (2) first reading after setting up (3) any reading between the first and the last
1. a-3, b-1, c-2
2. a-2, b-3, c-1
3. a-1, b-2, c-3
4. a-2, b-1, c-3
Answer
D. a-2, b-1, c-3
Backsight is first, foresight is last, intermediate is in between.
Match the whole circle bearing with its quadrantal bearing: (a) 60° (b) 150° (c) 300°. Values: (1) N 60° W (2) S 30° E (3) N 60° E
1. a-1, b-2, c-3
2. a-2, b-3, c-1
3. a-3, b-1, c-2
4. a-3, b-2, c-1
Answer
D. a-3, b-2, c-1
60° is N 60° E; 150° is S 30° E; 300° is N 60° W.

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