Advanced Civil Engineering Handbook › Chapter 12

Chapter 12 of 14 · 40 formulas · 79 one-liners

Surveying — formulas & one-liners

Chain, compass, traverse, levelling, theodolite, tacheometry, contouring, plane table, curves, areas and volumes, and basics of total station, GNSS and photogrammetry.

12.1 Linear Measurement (Chain & Tape) and Errors

Correction for incorrect chain length

L_{t r u e} = \frac{l ^{'}}{l} L_{m e a s}, A_{t r u e} = (\frac{l ^{'}}{l})^{2} A_{m e a s}

l' = actual length of chain; l = nominal length; L, A = measured length and area

For volume use (l'/l)³

Temperature correction

C_{t} = α (T_{m} - T_{0}) L

α = coefficient of thermal expansion; T_m = field temperature; T₀ = standardisation temperature; L = measured length

Positive when T_m > T₀

Pull (tension) correction

C_{p} = \frac{( P - P _{0} ) L}{A E}

P = applied pull; P₀ = standard pull; A = tape cross-section; E = modulus of elasticity

Sag correction

C_{s} = \frac{W ^{2} L}{24 P ^{2}} = \frac{w ^{2} L ^{3}}{24 P ^{2}}

W = total weight of tape between supports; w = weight per unit length; P = pull

Always negative

Slope correction

C_{s l} = \frac{h ^{2}}{2 L} or L (1 - cos θ)

h = difference in level of ends; θ = slope angle

Always negative

Reduction to mean sea level

C_{m s l} = \frac{L h}{R}

h = mean elevation of line above MSL; R = radius of earth ≈ 6370 km

Negative for lines above MSL

Probable error of single observation and of the mean

E_{s} = 0.6745 \frac{\sum v ^{2}}{n - 1}, E_{m} = \frac{E _{s}}{n}

v = residual (observation − mean); n = number of observations

Metric chains (IS 1492) are 20 m long with 100 links or 30 m long with 150 links, each link being 0.2 m.
Gunter's chain is 66 ft long and the engineer's chain 100 ft long, each with 100 links.
Brass tallies are fixed at every 5 m on a metric chain, with small rings at every metre in between.
If a chain is too long, measured distances are too short and the correction is positive.
Normal tension is the pull at which the elongation due to pull exactly cancels the shortening due to sag.
A well-conditioned triangle in chain surveying has no angle smaller than 30° or larger than 120°.
Reciprocal (indirect) ranging is used when the two ends of a line are not intervisible because of rising ground.
The optical square sets out right angles using two mirrors inclined at 45° to each other.
Systematic (cumulative) errors grow in proportion to the length or number of observations, while accidental (compensating) errors grow as its square root.
The weight of an observation is inversely proportional to the square of its probable error, and the mean of n equal-weight observations has weight n.

12.2 Compass Surveying

Back bearing (whole circle)

B B = F B \pm 18 0^{\circ}

FB = fore bearing

Add 180° if FB < 180°, subtract if FB > 180°

True bearing from magnetic bearing

θ_{t r u e} = θ_{ma g} \pm δ

δ = magnetic declination

+ for east declination, − for west declination

Whole circle bearing is measured clockwise from north from 0° to 360°; reduced (quadrantal) bearing is measured from north or south towards east or west from 0° to 90°.
A WCB of 210° is the reduced bearing S 30° W.
In the prismatic compass the graduated ring moves with the needle, has 0° at the south end and is read through a prism, so figures are written inverted.
In the surveyor's compass the graduated card is fixed to the box, reads quadrantal bearings with 0° at N and S, and has E and W interchanged.
Local attraction is suspected when the fore and back bearings of a line do not differ by exactly 180°.
Bearings affected by local attraction are corrected starting from a line whose fore and back bearings differ by exactly 180°.
Isogonic lines join places of equal declination, and the agonic line joins places of zero declination.
Dip of the magnetic needle is zero at the magnetic equator and 90° at the magnetic poles; a rider is used to balance it.
Secular variation is the slow long-period swing of magnetic declination, the largest of its variations.

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12.3 Traversing & Theodolite

Latitude and departure

L = l cos θ, D = l sin θ

l = length of line; θ = reduced bearing

Northing and easting are positive

Closing error of a traverse

e = (\sum L)^{2} + (\sum D)^{2}, tan θ = \frac{\sum D}{\sum L}

ΣL, ΣD = algebraic sums of latitudes and departures

Relative precision = e ÷ perimeter

Bowditch (compass) rule

C_{L} = \sum L \cdot \frac{l}{\sum l}, C_{D} = \sum D \cdot \frac{l}{\sum l}

C_L, C_D = corrections to latitude and departure of a line; l = length of that line; Σl = perimeter

Transit rule

C_{L} = \sum L \cdot \frac{∣ L ∣}{\sum ∣ L ∣}, C_{D} = \sum D \cdot \frac{∣ D ∣}{\sum ∣ D ∣}

|L|, |D| = arithmetic latitude and departure of the line

Sum of angles of a closed traverse

\sum interior = (2 n - 4) \times 9 0^{\circ}, \sum exterior = (2 n + 4) \times 9 0^{\circ}

n = number of sides

Algebraic sum of deflection angles = 360°

Least count of a vernier

L C = \frac{s}{n}

s = value of one main scale division; n = number of vernier divisions

Bowditch's rule is used when angles and distances are measured with equal precision (compass traverse); the transit rule when angles are more precise than distances (theodolite traverse).
A closed traverse can be checked for angular and linear closure, but an open traverse such as a route survey cannot.
Independent (total) coordinates are computed in Gale's traverse table by adding consecutive coordinates from a station of known coordinates.
A vernier transit theodolite commonly has a least count of 20″.
Fundamental relations of a theodolite: plate level axis ⊥ vertical axis, line of collimation ⊥ horizontal axis, horizontal axis ⊥ vertical axis, and altitude bubble axis ∥ line of collimation.
Face left is the position with the vertical circle on the observer's left when sighting.
Averaging face left and face right readings eliminates collimation error, trunnion axis error and vertical circle index error, but not the error of a non-vertical vertical axis.
The repetition method measures a single angle with greater precision than the least count; the reiteration method measures several angles at a station.
Temporary adjustments of a theodolite are setting up (centring), levelling up and elimination of parallax.
Transiting (plunging) means turning the telescope over 180° about its horizontal axis.

12.4 Levelling

Height of instrument method

H I = R L_{B M} + B S, R L = H I - (I S or F S)

BS = back sight; IS = intermediate sight; FS = fore sight

Arithmetic check

\sum B S - \sum F S = \sum Rise - \sum Fall = R L_{l a s t} - R L_{f i r s t}

Rise and fall method checks every reduced level

Curvature and refraction corrections

C_{c} = 0.0785 d^{2}, C_{r} = 0.0112 d^{2}, C = 0.0673 d^{2}

d = distance in km; corrections in m

C_c negative, C_r positive (≈ 1/7 of C_c); combined is negative

Distance to the visible horizon

d = 3.855 h

d in km; h = height of eye above ground in m

Reciprocal levelling: true difference of level

H = \frac{( a _{1} - b _{1} ) + ( a _{2} - b _{2} )}{2}

a₁, b₁ = staff readings on A and B with level near A; a₂, b₂ = readings with level near B

Sensitivity of a bubble tube

α = \frac{s}{n D} \times 206265 seconds

s = difference in staff readings; n = divisions the bubble moved; D = distance to staff

The first reading after setting up the level is a back sight and the last before shifting it is a fore sight.
A change point is a staff station at which both a fore sight and a back sight are taken.
The HI method is quicker for profile work with many intermediate sights but has no arithmetic check on intermediate reduced levels.
Keeping back sight and fore sight distances equal eliminates collimation error and the effect of curvature and refraction.
Reciprocal levelling across a river eliminates collimation, curvature and average refraction errors.
The two-peg test is used for the permanent adjustment of the line of collimation of a dumpy level.
A staff held inverted against a ceiling or bridge soffit is booked as a negative reading.
Sensitivity of a bubble tube increases with the radius of curvature of the tube.
GTS benchmarks of known reduced level above mean sea level are established by the Survey of India.
An automatic level uses a compensator to keep the line of sight horizontal once the instrument is roughly levelled.

12.5 Tacheometry, Contouring & Plane Table

Stadia distance, horizontal sight

D = \frac{f}{i} s + (f + d) = K s + C

s = staff intercept; f = focal length; i = stadia hair interval; d = objective to instrument axis; K = multiplying constant; C = additive constant

Horizontal distance, inclined sight, staff vertical

D = K s cos^{2} θ + C cos θ

θ = vertical angle of line of sight

Vertical component, inclined sight, staff vertical

V = \frac{K s sin 2 θ}{2} + C sin θ

RL of staff station = RL of instrument axis ± V − central hair reading

Subtense bar method

D = \frac{s}{2} cot \frac{θ}{2}

s = length between targets (usually 2 m); θ = angle subtended at instrument

For a 2 m bar, D = cot(θ/2)

Tacheometers are usually made with multiplying constant K = 100 and additive constant C = 0.
An anallactic lens is fitted in an external-focusing telescope to make the additive constant zero.
Contour interval is the vertical distance between consecutive contours, and horizontal equivalent is the horizontal distance between them.
Closely spaced contours indicate a steep slope, widely spaced ones a gentle slope and equally spaced ones a uniform slope.
Contours never cross each other except at an overhanging cliff, and they merge at a vertical cliff.
Contours cross ridge and valley lines at right angles; V-shapes point downhill for a ridge and uphill for a valley.
Closed contours with higher values inside represent a hill, and with lower values inside a depression.
Plane table methods are radiation, intersection, traversing and resection.
Orientation of a plane table by back sighting is more accurate than orientation by trough compass.
The three-point problem is solved by the tracing paper method, Bessel's graphical method or Lehmann's trial-and-error method.
The three-point problem becomes indeterminate when the table station lies on the circle through the three known points (great circle).
The two-point problem needs an auxiliary station and is less accurate than the three-point problem.

12.6 Curves

Radius from degree of curve

R = \frac{1718.9}{D}

D = degree of curve for a 30 m arc; R in m

For a 20 m arc, R = 1146/D

Tangent length

T = R tan \frac{Δ}{2}

Δ = deflection angle of the curve

Length of curve

l = \frac{π R Δ}{18 0 ^{\circ}}

Δ in degrees

Length of long chord

L = 2 R sin \frac{Δ}{2}

R = radius; Δ = deflection angle

External distance and mid-ordinate

E = R (sec \frac{Δ}{2} - 1), M = R (1 - cos \frac{Δ}{2})

E = apex distance; M = versed sine of the curve

Rankine's tangential (deflection) angle

δ = \frac{1718.9 c}{R} minutes

c = chord length; R = radius

Total deflection to any point = sum of individual δ

Length of transition curve

L = \frac{v ^{3}}{C R}

v = speed in m/s; C = rate of change of radial acceleration in m/s³; R = radius

Shift of circular curve

S = \frac{L ^{2}}{24 R}

L = length of transition curve

The deflection angle of a curve equals 180° minus the angle of intersection of the two straights.
Chainage of the point of tangency is the chainage of the point of curvature plus the length of the curve, not plus the tangent length.
A compound curve has two arcs of different radii bending in the same direction, while a reverse curve bends in opposite directions.
Rankine's method of tangential angles uses a theodolite and tape and is the standard method for setting out long curves.
The ideal transition curve is the clothoid (spiral), in which radius × length is constant; Indian Railways use the cubic parabola.
The spiral angle of a transition curve is L/(2R) radians.
Railway cant is e = GV²/(127R), with e and gauge G in mm, V in km/h and R in m.
Vertical curves are parabolic because a parabola gives a constant rate of change of grade.
Length of a vertical curve is the algebraic difference of grades divided by the permissible rate of change of grade.

12.7 Areas & Volumes

Trapezoidal rule

A = d [\frac{O _{1} + O _{n}}{2} + O_{2} + O_{3} + \dots + O_{n - 1}]

d = common interval; O = offsets

Simpson's one-third rule

A = \frac{d}{3} [(O_{1} + O_{n}) + 4 \sum O_{e v e n} + 2 \sum O_{o dd}]

Σ O_even = O₂ + O₄ + …; Σ O_odd = O₃ + O₅ + … (intermediate)

Needs an odd number of ordinates

Area from coordinates

A = \frac{1}{2} \sum (x_{i} y_{i + 1} - x_{i + 1} y_{i})

x_i, y_i = coordinates of traverse stations taken in order

Prismoidal formula for volume

V = \frac{d}{3} [(A_{1} + A_{n}) + 4 \sum A_{e v e n} + 2 \sum A_{o dd}]

A = cross-sectional areas at interval d

Needs an odd number of sections; trapezoidal (end-area) form is V = d[(A₁ + A_n)/2 + A₂ + … + A_n−1]

Amsler planimeter

A = M (F - I \pm 10 N + C)

M = multiplying constant; F, I = final and initial readings; N = complete turns of the dial; C = constant added when the anchor is inside the figure

The trapezoidal rule assumes straight boundaries between offsets, while Simpson's rule assumes parabolic arcs and is more accurate.
The area of a level cross-section is A = (b + sh)h, where b is formation width, h depth and s:1 the side slope.
For level sections the prismoidal correction is d·s(h₁ − h₂)²/6, so the end-area method gives a larger volume than the prismoidal formula.
On a mass haul diagram a rising curve indicates cutting and a falling curve indicates embankment.
A maximum on the mass haul curve marks the change from cut to fill, and a minimum the change from fill to cut.
The area between the mass haul curve and a balancing line gives the haul in m³·m.
The zero circle of a planimeter is the circle along which the tracing point moves without rotating the wheel.
Reservoir capacity is computed from the areas enclosed by successive contours using the trapezoidal or prismoidal formula.

12.8 Total Station, GNSS & Photogrammetry

Scale of a vertical photograph

S = \frac{f}{H - h}

f = focal length of camera; H = flying height above datum; h = ground elevation

Relief displacement

d = \frac{r h}{H}

r = radial distance of image from principal point; h = height of object above datum; H = flying height above datum

Displacement is radially outward for points above datum

A total station combines an electronic theodolite, an EDM and a microprocessor to measure angles and slope distance and compute coordinates and elevations.
EDM instruments are electro-optical (infrared or laser) or microwave types and measure distance by comparing the phase of a modulated carrier wave.
Distances measured by a total station need prism constant and atmospheric (temperature and pressure) corrections.
A GNSS receiver needs signals from at least four satellites for a 3D position, the fourth resolving the receiver clock error.
The GPS constellation has 24 nominal satellites in 6 orbital planes at about 20,200 km altitude with a period of about 12 hours.
GPS positions are referred to the WGS 84 datum.
Differential GPS uses corrections from a reference station of known position, and RTK uses carrier-phase corrections to reach centimetre-level accuracy.
A lower PDOP value indicates better satellite geometry and better position accuracy.
India's regional navigation system NavIC (IRNSS) is designed with a constellation of 7 satellites (3 geostationary and 4 geosynchronous).
Aerial photographs are taken with about 60 % forward (end) overlap and about 25–30 % side overlap for stereoscopic coverage.
A photograph with tilt less than about 3° is treated as a vertical photograph.

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Values follow the current Indian Standards and IRC / Indian Railways documents named in each item. Always check the latest amendment of a code before using a value in design.