FUNDAMENTALS
Survey → An art of determining the relative position of points above or beneath the earth's surface by Direct or Indirect Measurements
Earth shape → Oblate spheroid of revolution
The motion of earth related to the sun is in a plane inclined at an angle of 23° 27’
180°= 1°=60'=3600"=/180 radians
Geodetic Survey of India was done using triangulation
Plumb lines are radial inwards in geodetic survey
For the control establishment in surveying → Triangulation, Traversing, Trilateration
Digital elevation model(DEM) → Generated using digitization
Conformal projection → The shape of any small geographical area is preserved
Isopach lines → indicates equal thickness of a bed in a map
Based on Nature of field survey
Land survey → Cadastral, City survey
City syr → Locating Premises, streets, water supply, sanitary system.
Cadastral or Public Land survey → To fix property line, calculation of land area, to fix boundary of municipality and of state jurisdiction,Plans of property boundaries for legal purposes, Revenue chain used
Hydrographic or Marine syr → Large water bodies (lake,river,harbour), navigation, harbour work, Depth, Q, fluctuation of ocean tide
Astronomical syr → Heavenly bodies (sun, stars), azimuths, latitude, longitude, absolute location of a point on earth
Based on Object of Survey
Engineering syr → Design & construction of new routes, roads & railway
Geological syr → Diff strata of earth surface
Archeological syr → Old & Nelic str info
Military survey →
Mine survey →
Reconnaissance survey → Determining feasibility and estimation of scheme
Base on Type of instrument used
Chain, tacheometer, theodolite, Plane table, Triangulation, traverse
Other Survey
Topographical syr → Natural and artificial features Mountains, valley, lake, river, buildings, monuments
Longitudinal land syr → Linear bars used
Cross-sectⁿ/Profile → Sewage disposal & water supply work
Traffic (Topographic) → Reconnaissance → Preliminary → Detail/location/final survey
New Highway → Map Study (Topographic) → Reconnaissance → Preliminary → Location of final alignment→ Detail location Survey.
Town planning → 1st topographic survey
Principle Of Surveying
a) Work from whole to part
Localise error & prevent their accumulation → Error are minimised
major control points are measured with lower degree of precision
minor control points are measured with Higher degree of precision
b) Locate a point by at least two measurement
At least two, already fixed points of reference
Two side, One side one angle, Two angle & One side one right angle
SCALE
Scale=Lmap/Loriginal = Amap/Aoriginal = 3Vmap /Voriginal
Representative fraction (R.F.) = map distance/ground distance
Accuracy = least count / RF
Comparative Scale → Pair of scales having a common R.F
Engineer scale: 1cm = 30m → RF = 1/3000
Scale 1/100 is larger than 1/1000
Building = 1 : 1000
Town planning = 1 : 5000
Route (Rd & rail) = 1 : 10000
Topographical or forest = 1 : 25000
Toposheet = 1 : 50000
SOI → Toposheet 1:50000 (1:50k)
Types of scale
i) Plain scale
Two dimensions → Units, Tenth
ii) Diagonal
Three dimensions → units, tenth & hundreds :m dm cm
Based on the principle of similarity of triangles
iii) Shrunk scale
SS = Original scale x shrinkage factor (SF)
SF or SR = shrunk/original length = Shrunk RF/original = Shrunk scale/original.
Corrected Area = meas Area / SF²
Correct L = Lm / S.R
A = Am / (SR)²
Correct = ( Std ± δ ) x L,A,V → (+) longer than std & (-) shorter than std
Correct L x correct RF = wrong L x wrong RF
Graphical scale → Not affected due to Shrinkage of map → Better than Numerical scales
iv) Vernier Scale
DRE 10 = 9 11 19
Least count of vernier scale = s/n=value of one div of main scale/No of div of vernier scale
Least count of combination = s - v =value of one div of main scale-value of one div of the Vernier scale
a) Direct vernier
Shorter than div of main scale
Reading /graduatⁿ in directⁿ of main scale
n div of DV = ( n - 1 ) div of main scale
b) Retrograde Vernier.
Longer than div of main scale
opp direction of main scale
n div of RV = (n + 1) div of main scale
c) Extended Vernier.
Calibrated in both direction
n div of EV = ( 2n - 1 ) div of main scale
Maps and plan
Locality map cum site plan → Combination of key map to scale of 1:250,000 and index map to a scale of 1:50,000
Accuracy → Micro-optic theodolite > theodolite > compass > chain
Planimetr formula
Area = M(FR - IR 10N + C) → take N = -1 if not given
Additive constant (C) is used only when the anchor point is placed inside the plan (area measured)
THEORY OF ERRORS
Error = Measured - True value
Correction = True - Measured
E = -C
True error = Observed value -True value
Residual error = Observed value -Most probable value
Most probable value Close to true value than any other
Apparent error = 2 x Actual Error
e1:e2:e3=(1/w1):(1/w2):(1/w3)
Negative error → Measured < True value
Positive error → Measured > True value
Discrepancy → Difference between two measured values of same quantity
Probable error
Em=Es/n
Em → Probable error of mean, Es → Probable error of single observation, n → Number of observation
Probable error = ± Standard error → if n = 1 or observation of unit weight
Permissible error
max allowable limit up to measured value can vary from True value
Permissible error hilly/rough region = 1 in 250
Mistake or Blunder or Gross error
Due to inexperience, carelessness, fatigue, miscommunication, poor judgement
ex. Improper levelling of instrument, setting instrument over the wrong point
Personal error → Mistake in rod handling
Cumulative or Systematic errors
∝ L, (+ve or -ve )
Same shape, Size, Sign under same condition
May increase or decrease with increase in measurement
Occur in same direction & Tend to Accumulate
Due to → Faulty instruments,
+ Cumulative error → Bad ranging, Bad sighting, wrong alignment
Accidental or Random or Compensating error
∝ √L , ∝ 1/√n → L= length, n = no of observation
associates with Surveyors Skills & vigilance
can't be eliminate & are beyond the control of surveyor
obey's Law of chance
occur in both directions & tends to compensate
Law of accidental error/ Theory of probability
Applied to accidental error
Minimise the effects of personal and accidental errors
Small error tend to be more frequent than the large ones this means they are more probable
Large errors occur infrequently and are impossible
Positive and negative errors of same size happen with equal frequency, that is they are equally probable
Follows normal probability distⁿ curve Gaussian distⁿ
Smaller the value of standard deviation smaller error and great precision
Standard deviation is also known as Root mean square error
Theory of least square
finding the best fitting curve or line of best fit for a set of data
Σ (error)² = minimum
Methods of least square adjustments.
i) method of correlates (conditⁿ eqⁿ method)
ii) Normal eqⁿ method
iii) method of diff
Weight theory
Higher weight → Lower error
Lower weight → Higher error
Sum or Difference (AB) 1/(1/w1+1/w2)
KW → W/K2
W/K K2W
KWW
The weight of the weighted Arithmetic mean of observation = sum of the weight of observation
e=/2
Rectangle of side (l ±el)(d ± ed) → Error in area =± ld (el/l)2+(ed/d)2 =(eld)2+(edl)2
LINEAR MEASUREMENT
Plans required on a large scale (1/10 , 1/100) & Ground Fairly level
Accuracy in linear measurement = L/S
Area is divided into triangles
Chain Survey
Field work is limited to Linear measurement only
Reconnaissance → marking & fixing survey stations → Running survey lines
Principle of chain Surveying is Triangulation
A triangle is the only simple figure that can be plotted from the length of its sides measured in the field
Length of chain → Outside of one handle to outside of the Other handle or Centre to centre distance b/w the last end of links
When very high accuracy is not required
Small surveys in fairly level & open ground with small details
Plans are required on a large scale
Base Line
Longest survey line , Measured By invar tape
Baseline L = 10 - 20 km → 3rd order triangulation system
Apparatus for measuring BL → Colby apparatus and Wheeler base apparatus → using rigid bar
SOI → Colby apparatus
It is difficult and expensive to measure long base lines but It is greater than sides of triangle
Main survey line
Join two main survey station
Tie or Subsidiary station
Join fixed points on the main station
Line joining tie stations for taking offsets from it
Helpful for locating interior details & details of objects in an Area
Collect the details of nearby objects in an area.
Proof or Check line
Check accuracy of field work
Offset
Lateral measurement w.r.t. main survey line
Offset may be Perpendicular or Oblique
Limiting length of offset = S/40sinθ
where S = Scale = 100 if RF = 1:100., θ = error in sec
Long offset > 15 m
Short offset < 15 m
Measuring offset → Butt rod, Steel tape
Position of a point can be fixed more accurately → By perpendicular offset
Field book
Chain or tape measurement recorded.
Size = 20 cm x 12 cm
A. Equipment used for measuring line
i) Chain
Types of Chain
Revenue → 33 ft = 16 links
Gunter's/Surveyor's → 66ft (20.12m) = 100 links
Engineer's → 100 ft = 100 links
Metric → 30m (150 links), 20m(100 links), 10m(50 links)
1link of metric chain = 20 cm
Metric chains are used for indirect measurements
Accuracy of 30, 20, 10, 5m Chain → ±8mm, ±5mm, ±3mm, ±2 mm respectively
1 mile = 80 Gunter's Chain
1 furlong = 10 Gunter's Chain
Adjustment of Chain
When the chain is too short
Straightening the bent links
Opening the joints of the rings
Replacing the old rings by some larger rings
When the chain is too long
Closing up the joints of the rings
Hammering the elongated rings
Replacing some old things with new rings.
ii) Tapes
Accuracy → Invar > Steel > Metallic > Linen
invar tape → Alloy of nickel (36%) + Steel (64%), Baseline measurement, More accurate, Low thermal expansion coefficient.
Cloth/linen tape
Metallic → linen tape with bronze or brass or copper wire → Cloth + wire
Steel tape
iii) Pegs
To mark temporary points on ground or to mark survey stations.
ht = 15 cm
c/s A = 2.5 x 2.5 cm²
to recognise main station
iv) Ranging rods
Dia = 30mm & L = 2 - 3m
Locating no of intermediate points on a long survey line
White & red
c/s = circular & octagonal
v) Arrows
Size = 40cm
intermediate station
vi) Offset rods
L = 2m
Plotting offsets
Plumb bomb
Center of the instrument.. transfer points to ground
Made up of bronze & brass
Pacing
Measure Distance by counting paces
Avg length of pace = 80 cm
B.Equipment use for Right Angle
By prismatic compass & theodolite
a) Cross staff
Open cross staff → Angle & altitudes, More accurate, 90°
French Cross staff → Octagonal form of cs, Used to set angles of 45°, 90°, 135°, Less accurate
Adjustable cross staff → An offset at any angle
b) Optical square
Best → more convenient & accurate
Pocket instruments
Laying of 90° (right angle) or establish two point at right angle
Principal → Double reflection → Angle b/w two mirror = 45°
Angle b/w 1st & last incident ray = 90°
Taking offset with an optical square on the right hand side of the chain line it is held by Left Hand Upright
c) Prism square
two reflecting surfaces at 45° no adjustments required.
laying of 90° (right angle) or establish two point at right angle
C. Equipment for establishing intermediate points
Ranging
To locate intermediate point on survey line b/w two fixed end point
Error due to bad ranging → Cumulative (+ve)
i). Direct Ranging
Possible when stations are intervisible
Done by eye or line Ranger
n > 3
Minimum persons required = 02
ii). Indirect Ranging
Stations are invisible due to elevated RE or long sight distance.
No of rods require = 04
Minimum persons required = 02
Carried out either by reciprocal method or by random line method
Reciprocal Ranging → Two station for which ranging is to be done are not intervisible
D. Chaining
i) on smooth level: with chain, peg, arrows
ii) on sloping ground:
Direct/Stepping method → Steps banao
it is easier to work downhill while Stepping than to work uphill
Indirect/ Hypotenuse allowance method → By measurement of inclination,diff in level & Hypotenuse allowance Along slope by Abney level
In order to achieve higher accuracy → The distance between successive steps for measuring along a hill decreases with increase of slope
Permissible limits of error in chaining (or Accuracy)
Rough & hilly ground = 1 : 250
Ordinary chain survey = 1 : 1000
Steel chain or band used = 1 : 2000
Std. Steel or invar tape used = 1 : 5000
Obstacles
Ranging → Forest, Hill, hillock
Chaining Obstacles → Small Pond, Small bend in River, Reciprocal ranging is used
Both → A big building
a) Chain measurement Correction
Length of chain Longer than std length → Error = -ve cumulative, correction = +ve
Length is shorter → Error = +ve cumulative , correction = -ve
Correction due to sag and pull → Equalised by normal tension
Standardisation correction
± ve → Cumulative in nature
Correction = True - measured length
Slope correction
(-ve) cumulative
Along Slope (θ) = L(1- cosθ)
Along Perpendicular AB = -h²/2L
Along slope or Hypotenusal Allowance = L(secθ -1) = -h²/2L
Along the Horizontal line(Base) = L(cosθ - 1)
Correction per chain length
for 100 links along a slope of α radian = 100 α
slope having rise of 1 unit in n horizontal unit = 100/n²
correction per chain length of 100 links along slope of α°= 1.5α²/100
b) Tape correction
Standardization & Slope → Same as chain
Pull correction
Cp = ± ve
Cp = (Pm - Ps) x L / AE
Ps = standard pull
Steel tape → E = 2.1 x 10⁵ N/mm²
invar tape → E = 1.54 x 10⁵ N/mm²
Temperature correction
Ct → ± ve
Ct = α x (Tm - To) x L
Tm = mean temperature
Mean sea level correction
Ch → -ve
Ch = -Lh/R
Sag Correction
Cs = -ve
Parabolic shapes is assume to be followed
Cs = -W²L/24Pm² = -w²L³/24Pm² W = wl
Misalignment or Wrong alignment
-ve
Note
Standardised tapes → Short base in plan ground
Hunter’s short base → Used for meas 80 m long base
Tacheometric base → Undulating ground for small base
EDM → Fairly long distance
COMPASS SURVEY
A magnetic compass needle is generally supported on Jewel bearing
CS → error ≤ 5 min → Accuracy ≤ 5 min.
Principle → Traversing → angular & linear measurement to est. control point.
CS is more useful than chain Survey when a large area needs to be covered
Suitability of CS →When the area to be survey is large having undulating Grounds and higher accuracy is not required
The line of force of Earth's magnetic field are parallel to the Earth surface → Near the equator
Local attraction → Due to magnetic fields produce by some magnetic material/object around the location where surveying is taking place, Gives error in observation of either fore bearing or back bearing or both
Errors in CS → Irregular variations due to magnetic storms, local attraction due to proximity of local attraction forces, magnetic changes in atmosphere due to clouds and storms
Bearing
Clockwise or anticlockwise
Magnetic bearing → The horizontal angle that a line makes with the magnetic meridian
Azimuth or True bearing
Horizontal angle between the true meridian and survey line
Always clockwise from true north
The true bearing of a line (Azimuth) doesn't change with time & can be reestablished even after hundreds of years
Azimuth = True bearing
BB = FB ± 180° → (+ve if FB < 180 & -ve if FB > 180°)
Open traverse → no of FB = BB = no. of station - 1
Close traversing → no. of FB = BB = no. of station
FB - BB = Either external or internal angle
FB → Depends on the direction of progress of survey
Meridian
it's a reference line
Arbitrary Meridian → Taken in any convenient arbitrary directⁿ
Standard meridian of india = 82.5°(82°30') west
TB = MB ± declination
East (+ve) declination → Magnetic north is east of true north
West (-ve) declination → Magnetic north is west of true north
True meridian
Line in which earth's surface is intersected by a plane through North and South Geographic poles
True meridian → Converge at pole
Magnetic meridian
Directⁿ indicated by a freely suspended & properly influenced by local attraction.
Changes gradually with time
Determine either by Surveyor or Prismatic compass
Magnetic declination
δ = horizontal angle b/w TM & MM
Declination at noon = 180° or 360° - Bearing of sun at noon
22 Dec → Sun declination = 23°27’S, Right ascension = 270°
Agonic lines → Zero declination
isogonic → Same declination, Isogonic lines doesn't form complete great circle, it radiates from North and South magnetic regions and follow irregular paths
Variation of Magnetic declination
Varies from place to place
Secular variatⁿ → Gradual shift in earth's magnetic field.
Annual or yearly variation → Revolution of earth around sun
Diurnal variation → Rotation of earth about its own axis , more near the pole in the day & summer time.
irregular variation → Magnetic storms
Dip
Vertical angle made by lines of magnetic force with earth's surface or Inclination of compass needle to the horizontal towards the pole
Dip → Equator = 0° & Poles = 90°
isoclinic → Equal dip, Aclinic → Zero dip
Dip of the horizon → Angle between the line of sight and the tangent to the level surface
Strike is always perpendicular to true dip
THEODOLITE
Most accurate for both Horizontal & Vertical angle in surveying → Arbitrary bearing
Use → Horizontal and vertical angles, Prolonging survey lines, measuring horizontal distance
least count = 20sec (Vernier theodolite) & = 1sec (Electronic theodolite)
Lower clamp screw → Used while taking backsight reading in Vernier Theodolite
Tangent screw → Fine adjustment in a theodolite
Common size = 8 - 12 cm → but for Triangulation = 14 - 25cm
A simple circular curve can be set by two theodolite methods. In this method only angular measurements are taken with the help of two theodolites
Aplanatic combination → A compound lens free from spherical aberration
Types
Transit theodolite → Can rotate about its horizontal axis in the vertical plane
Non - Transit theodolite → Can't rotate 180° in vertical plane
Vernier theodolite → Levelling-head + Horizontal-Circle + Alidade assembly
Direction theodolite → Has only one vertical axis
Inverted → Vertical circle is to the right of the observer and the Bubble of telescope is down
Horizontal Circle/Lower plate/Main Scale plate
WCS → 0°- 360° each graduation at 20'
Size of theodolite is defined by lower Graduate circle → Lower plate or Scale plate dia
Dia = 100mm - 130mm
Vertical Circle
0°- 90°
The two zeros of VC are on the Horizontal Dia of Circle.
Important Terms
Centering → with help of Plumb bob
Face left → Vertical circle is on left hand side of observer
Face right → Vertical circle is on right hand side of observer
Axis of telescope → Optical centre of objective to the centre of eyepiece
Line of sight → intersection of cross-hair of diaphragm & optical centre of objective lens and its continuation
LOS reverse → Revolving 180° in a vertical plane.
Line of collimation → When LOS is perfectly horizontal, Centre of diaphragm & optical centre of objective lens
Vertical axis / Azimuth axis
Horizontal axis / trunnion axis
Telescope Normal → Vertical circle on left side & Bubble is Up
Telescope inverted → VC on right & Bubble down
Changing face → Bringing face left to right & vice-versa.
Swinging → Revolving in Horizontal plane & about Vertical axis
Transiting / Plunging / Reversing → Revolving in vertical plane & about horizontal axis
Shifting centre → By means of plumb bob → Helps in easy and Rapid performance of the centreing
Lining in → Est intermediate points on straight line whose points are intervisible
Balancing in → Est intermediate points on line whose end are not intervisible
Temporary Adjustments of a theodolite
Done at every station the instrument is set up
Setting/fixing(setup) → Centering → Levelling → Focussing Eyepiece → Focusing Objective → elimination of parallax
Elimination of parallax → Operation of forming the clear image of the object in the plane of cross hair, By focusing both obj. & Eye piece
Permanent Adjustment of a Theodolite
Deals → with maintaining the relationships between fundamental lines
Plate level test → Axis of level tube ⟂ Vertical axis
Cross-hair ring test → Vertical hair ⟂ horizontal axis
Collimation in Azimuth test → LOS ⟂ Horizontal axis
Spire test → Adjustment of Horizontal axis ⟂ vertical axis
Bubble tube adjustments →
Vertical circle test → indicate zero when LOS is ⟂ Vertical axis
Vertical Arc test → LOC ∥ Bubble tube axis
Vertical collimation error → Line of Altitude bubble is not parallel to the line of collimation
Horizontality of the trunnion axis(HA) of theodolite is checked by the Striding level
Methods
Reiteration method of series/direct method
The angle is measured and the instrument turn to close the Horizon
Preferred in triangulation, where no. of angles are taken at one station
Repetition method
Preferred for Horizontal single angle measurement
Eliminate → line of collimation error, Error due to eccentricity of verniers, due to wrong adjustment of line and trunnion axis, due to inaccurate graduation
Rotate the instrument without changing the readings → lower clamp screw is loosened and upper clamp is tightened
To change reading on circle while meas angle → lower clamp screw is tightened and upper clamp is loosened
Ordinary method or Direction method
Eccentricity of Vernier & centre → Eliminated by reading both vernier
inaccurate graduation → Take reading on diff part of circle
LOS & HA → Taking both face reading
inaccurate bisectⁿ of signal → More no of observation
LOC not ⟂ HA → Mean of both face observation
Index error → By face left and face right observation
face left and face right observation → Controls error due to non parallelism of line of sight
Other errors → Minimised by dividing the cumulative angle
Error= ½ ( Face left - Face right)
Error in Theodolite work
instrumental errors → Non adjustment of plate levels, LOC not ⟂ HA, VA not ⟂ HA, LOC & axis of telescope are not parallel, Graduation being unequal, vernier being eccentric, inner & outer axis not being concentric
Observation errors → inaccurate centring & levelling, Slip, Parallax, working wrong tangent screw, non verticality of ranging rod
Natural errors → High temp causing irregular refraction, wind Storm causing vibration, unequal settlement of tripod, Sun shining on instrument
Telescope
External focusing Telescope is fitted with anallactic lens → convex lens
In external focusing telescope for focusing → Objective tube is moved
Eyepiece has high magnification power
internal focusing Telescope → Focusing is done with the help of supplementary double concave lens
Cross hair → Front of eyepiece & at optical centre of diaphragm, much closer To the eye piece then to the objective lens
Total Station Or Total station theodolite
Electronic transit theodolite + Electronic distance measurement (EDM)
To measure the sloping distance of an object to the instrument, horizontal angles, and vertical angles
Used → Remote distance and elevation measurement, Area computation, Point location
Vertical angle → as zenith → 0° vertically up 90° horizontal and 180° vertically down
EDM → light waves, infrared waves, microwaves
Measure angles → By means of Electro-optical scanning
TRAVERSING
Traverse
Series of connected lines whose length/distance & directⁿ(angle) are measured in field
Traverse Survey → Theodolite, Chain, Compass, PTS
1 min arc of longitude = 1 nautical mile
Theodolite traversing → Computation of reduce wearing of each traverse leg → Calculation of consecutive coordinate → identifying the closing error → Balancing of consecutive coordinates
Angular measurement
Accuracy → independent coordinate > included > FNM > LNM
i. Loose needle method
ii. Fast needle method
A point is taken as a reference station & MB of all points is determined & vice versa for LNM.
Most preferred
iii. Method of deflection angle: open traverse (Rd & railway)
iv. Method of include angle
Direction of progress is Counter clockwise than the included angle measure clockwise are interior angles
Direction of progress is Clockwise than the included angle measure clockwise are Exterior angles
Linear method
Taping or Chaining
Tacheometric method
EDMI
Angles in Traverse
Angle Misclosure (AM) = Actual sum of angle - Theoretical sum of angle
Permissible angle misclosure = KN → N = sides of traverse
K = 20"(generally) → Depends on least count,desire accuracy & no of repetition
Σ external angle = (2N + 4) x 90°
Σ internal angle = (2N - 4) x 90°
Error in each internal angle = Σerror of all angle / number of angles
Included angle → measured clockwise from back station
Deflection angle =180°-interior included angle
Check in Traverse
a) Closed Traverse (Loop)
Best checked
closes on the same station or whose location is known
Departure/Longitude =lsin → 0° to 180° East or West
Latitude=lcos → 0° to 180° North or South
ΣL = ΣD = 0 → No error
ex = ΣD & ey = ΣL
Direction of closing error → tanθ = ex/ey = ΣD/ΣL
Latitude and departure of station wrt preceding station is called depending co-ordinate or consecutive coordinates
Closing error or Error of closure
Actual distance by which the traverse fails to close
Closing error (e) = ex² + ey² = (L)² + (D)²
Relative error(r) = Closing error(e)/Perimeter of Traverse(P)=relative accuracy or degree of accuracy
Precision=Closing error (e)/measured length
b) Open traverse (Link)
closes on station whose location is unknown
ΣL = Latitude final - Latitude initial control point
ΣD = D final - D initial control point
Open traversing should be avoided because it is not possible to detect, adjust & balance the errors
Open traverse can be checked by Astronomical observations
Adjustment of traverse
Aim → Closing error = 0
If closing error is within permissible limit Traverser should be adjusted → Hence error is distributed among various sides of travel such that traverse geometrically closes
i) Arbitrary method
Based on Discretion of surveyor & field conditions.
ii) Bowditch or Compass rule
Adjustment of closing error in a closed traverse
∆θ = ∆L → Liner measurement and angle with same precision
Assumption → Closing error introduced in traverse are of accidental(random) nature
error in linear measurement ∝ √L
error in angular measurement ∝ 1/√L
iii) Transit rule
∆θ < ∆L → Angular precision > distance/linear precision
error in latitude of any line = ey x L / Σ L
error in departure of any line = ex x D/ Σ D
iv) Graphical method
based on Bowditch rule
used for theodolite traverse with low accuracy.
v) Axis method
Angles are measured very precisely
Correction only for length
TRIANGULATION
Principle → One side and three angles, the remaining sides can be calculated precisely → Measuring all angles and the baseline
Theodolite size for Triangulation = 14 - 25cm
System of multiplying ground control points on the earth surface
Network of triangle
in triangulation best shape of the triangle is isosceles with base angle 56°14'
Triangulation station → intervisible, easily accessible, in commanding position
Centred triangle → 3 angle conditions and one side condition
Strength of figure in a triangulation system is more → when the error is the least when computing the length of last line
Captain G.T. McCaw's solution → To check intervisibility of station
Application of Triangulation → Determining accurate locations for setting out of Civil Engineering Works, Establishing accurate control for photogrammetric survey for large areas, Establishing accurate control for plane and geodetic survey covering large area
USE → To determine the length of a bridge proposed to be built across a wide river
Triangulation system of Quadrilaterals is most suitable for railways.
Accuracy of shape is measured in terms of strength of figures & its value depends on → no. of observed directⁿ, No. of geometric conditions, magnitude of distance
Log-sine formula → To check side condition in triangulation
Well conditioned Triangle
Either isosceles or equilateral
Well conditioned triangle → 30° < θ < 120°
Equilateral triangle is most appropriate well conditioned triangle
Well conditioned Triangles are preferred → Their Apex are sharp and can be locate easily
Types of Triangulation
a) Primary triangulation (1st order)
Baseline = 5 km - 15 km
most accurate
testing defence space vehicle
b) Secondary triangulation (2nd order)
Baseline = 1.5 km - 5 km
Strengthen the network made by primary triangulation
c) Tertiary triangulation (3rd order)
Baseline = 0.5 km - 3 km
Triangulation Stations
1. Satellite / eccentric /false station
Subsidiary station est. near the True/main/Principal triangulation station as possible
Is related with control survey
2. Pivot station: no observation only for continuation
3.Main Station: control point of Triangulation network
4. Subsidiary station: additional rays to intersected points
Laplace stations → Astronomical Observation for azimuth and longitude are made
TACHEOMETRY
H & V Distance determine by taking angular observation with instrument Tachometer
Distance meas method used for rough or steep grounds
Adopted → Where obstacles, steep and broken ground deep ravines etc exist, Too many curves exists at the border
Mainly used while preparing contour plans & Traversing
Stadia diaphragm → Horizontal hair = 3
Tacheometric surveying eliminates chaining
Modern electronic tachometers → Electronic theodolite + electronic data collector + electronic distance measurement
Tacheometer
Tachometer is Transit theodolite with stadia diaphragm
Measure H & V distance
Substance bar or horizontal stave → Meas H & V distance where chaining is not possible
Stadia rod or vertical stave → 5m - 15m
Analytical Lens → Used to make staff intercept proportional to its distance from the tacheometer, Convex lens inserted between object glass and diaphragm
Methods of Tacheometry
Commonly used → Fixed stadia/hair system
Tangential method → Faster than stadia hair method
i. Stadia method
Principle → intercepts on measuring rods are proportional to the distance → Ratio of the perpendicular to the base is constant in similar isosceles triangles
Fixed hair method
ꞵ → fixed, Staff intercept → Vary
Number of horizontal crosshair in stadia diaphragm = 03
D = Ks + C → Line of sight is perpendicular to staff
Multiplying constant → K = f/i
Additive constant → C = f + d
C→ External focusing = 0, internal focusing = small
Anllactic lens (Convex) in tacheometer → K = 100, C = 0
Movable hair method
Staff intercept → Fix, ꞵ → Vary
D = Ks/m + C
K = f/p
ii. Tangential method
Stadia hair are not used
Horizontal distance → With help of two vertical angle & staff intercepts
Diff in elevation
iii. Range finding
Distance and elevation formula
When staff is Vertical
Horizontal D = Kscos²θ + C cosθ
Vertical D = Ks sinθ cosθ + C sinθ
LEVELLING
Diff of elevation or level of diff points on the earth surface
Levelling deals with → Measurements in the Vertical plane
Levelling starts with BS and end with FS
Temporary adjustment → Setting up → Centering → Levelling → Elimination of parallax
Permanent adjustment of level → Two → To make the axis of the bubble tube perpendicular to the vertical axis, To make the line of collimation parallel to the axis of the Bubble tube
Plumb line → Most fundamental line in surveying
Grade → elevation is called grade when used in reference to construction Activity
Level line → Any line lying on level surface, Constant ht. relative to MSL it must be a curved line & normal or Perpendicular to plumb line & parallel to mean spheroid of earth surface
Line of collimation is tangential to the level line
Level surface → curved surface parallel to the mean spheroidal surface of earth, Every point is perpendicular to the direction of gravity
Horizontal surface is tangential to the level surface at any point
Geoids surface → Surface of zero elevation around the earth which is slightly irregular and curved.
Mean sea level (MSL) → 19 year period & w.r.t Bombay Port
Datum → Height of any point wrt mean sea level
Reduce level → height wrt Datum surface
Level field book → book used for entering the staff reading & Reduce level of points
Back or Plus sight → 1st reading, at known elevation
Fore or minus sight → last reading, unknown elevation or elevation yet to be determined.
intermediate sight → unknown elevation b/w BS & FS
Change or Turning point → shifting of instrument or level, Both BS ,FS are taken, used to transfer elevation
Levelling on a steep slope → The instrument should preferably be set up successively along a zig-zag path
Levelling in undulating terrain → Set the level on any side of the slope
When the Bubble of the level tube of a level remains Central then line of sight is horizontal
Adjustment of horizontal cross hair is required particularly when the instrument is used for levelling
Benchmark
Fixed reference point of known elevation above Datum. & Est with help of spirit level.
Great trigonometrical survey BM (GTM) → est. By SOI wrt MSL at Bombay port with 1° latitude & 1° longitude
Permanent BM → by PWD or SOI
Temporary BM → established for a day's work or end of day work
Arbitrary BM → Survey team in beginning of project
Degree of Precision required for est of benchmark=4K
Levels
Auto Level → Has an internal compensator mechanism to automatically adjust the line of sight
Dumpy level
Telescope tube and vertical spindle are cast together
Two peg test of dumpy level → The line of collimation of the telescope is parallel to the bubble tube axis
Telescope of dumpy level is rigidly fixed to the levelling head
Dumpy level is most suitable when many reading are to be taken from single setting of instrument
Trigonometric levelling cannot be done with the dumpy level
Levelling staff
Self reading and Target staff
Self reading staff
01 m divided into 200 div
Solid → Single piece of 3m
Folded → 2 piece of 2m each, Thickness of graduation = 5 mm
Telescope → 03 piece , upper 1 piece solid & lower 02 piece hollow
METHODS OF LEVELLING
Direct & indirect methods.
Direct Levelling or Spirit levelling
Most common method
Differential or Compound L → difference b/w elevation of two points
Check L → checking of obtained elevation
Profile/Longitudinal/Sectioning → Elevation along straight line, Road, canal, terrace line, Staff, Readings & Distance b/w the point is required
Fly → Reconnaissance, Rapid but low precise & only FS & BS are taken, Differential levelling is done in order to connect a benchmark to the starting point of the alignment of any project (road, railway, canal)
Cross-section L → Levels are taken on each side of a main line at right angles to that line
Precise L → For high accuracy desired
Reciprocal levelling
Points situated quit apart & its not possible to set up the instrument mid way
Suitability → Two points at river banks, deep George
Adopted to decide the difference of Level between two points a considerable distance apart with great precision
Eliminate → Error due to curvature,refraction & Collimation And error in instrument adjustment
Not eliminate → Parallax error, non-adjustment of bubble tube
H=(1/2)[(Hb-Ha)+(Hb'-Ha')]
Collimation error =True level by formula(H)-0.0673d2
if instrument is correct → (Hb-Ha)=(Hb'-Ha')
indirect levelling
Trigonometric → Help of horizontal distance & vertical angle, Correction for axis signal is relevant
Barometric → By change in Atmospheric pressure, Quick method
Hypsometric → Determining elevations based on the boiling point of liquids
Instrument axis are at diff levels → H=(h+d.tan2)tan1 /(tan1-tan2) By trigonometric levelling
Permissible error
E = C√D, where E = error in m & D = distance in km
Precise Levelling = ± 0.006√D
Accurate levelling = ± 0.012√D
Ordinary levelling = ± 0.025√D
Rough levelling = ± 0.100√D
Optical defects of lens
Spherical Aberration → Ray incident at edge > At centre of lens
Chromatic Aberration → Dispersion of light (white light into diff colour light), In telescopes it is decreased by use of compound lenses (concave & convex)
Sensitivity of level/bubble tube (α)
Expressed in terms of → Angle in seconds subtended at the centre by the arc of one division length of the level tube or Value of 1 division of level/bubble tube
α = /n=L/R = s/nD radian
α = s/nD radian = (s/nD)x206265 seconds
error = staff intercept = s = nL/RD
1 radian =206265 seconds
L = 2mm → if not given
n = no of division, L = length of one division, R = radii of curvature of level tube, s = diff in staff reading, D = distance
Sensitivity of level tube is increased by
increasing → Radius, Length & Diameter of tube, (α ∝ dimensions), Smoothness of inner surface
Decreasing → Viscosity & Surface tension of liquid, roughness of inner wall of tube, Temperature
Level Tube → Designated by radii of level tube
Height of instrument
Elevation of the plan of sight
HI = RL of A + BS
RL of B = HI - FS = RL of A + BS - FS
inverted staff → RL of soffit = RL of floor + BS + FS(reading of inverted staff)
Σ BS > Σ FS → last point is higher than 1st point & Vice versa
Error=(Σ BS - Σ FS )-(last RL - first RL)
Reduction of level → Terms used is height of instrument
Correction
Curvature (Cc) = - d²/2R = - 0.0785d²
Refraction (Cr) = +1/7 of Cc = +0.0112d²
Combined C = Cr + Cc = - 0.0673d² =6/7 of Cc
Distance of visible horizon (d) = 3.85 √h h=0.0673d² → h in meters & d in km
Distance of observer from lighthouse(d) = 3.85 h1+3.85h2 → h1, h2 = top of lighthouse, ht of observer eye above sea respectively
Curvature → Object appears lower
Refraction → Object appears higher
Combine correction → Staff reading decreases but RL increases by 0.0673d²
Highest value of coefficient of refraction occurs during → At noon
CONTOURING
Contour → Equal elevation line
Contour bending → Generally confined to hilly area
Grade contour → Imaginary line lying on the ground and maintaining a constant slope/inclination
Area enclosed in a contour may be determined by means of planimeter
The alignment of highways are generally taken along the contour line
Contour interval
Vertical distance between two consecutive contour
it should be constant
CI depends on → Scale of map, Nature of country, Map purpose, Time, funds
CI = 25/Scale of map (cm/km)
For more precise prediction of the terrain relief the CI should be Smaller
Horizontal equivalent
Horizontal distance between two consecutive contours
Slopes b/w two point depends on horizontal equivalent
Characteristics of Contour
line passing with line of max/steep slope make angle of 90°
The directⁿ of steepest slope is along the longest distance b/w the contours
The control lines are closed curves
In a contour Map the contour lines never intersect
Zero contour line → Coastal line , flat terrain
Uniform slope → Equally spaced or parallel contour
Steep slope → Small spacing contour
Gentle slope → Same Contour interval, contour are farther apart
Watershed or ridge line contour → Crosses valley contour at 90°
Contour lines → Cross valley & ridge line at 90°
Ridge line → U shaped line, convexity towards lower ground
Valley line → V shaped contour line, convexity towards higher ground
Overhang cliff or Cave penetrating a hillside → Contour lines intersect/cross one another
Vertical cliff → Contour lines unite to form one line/contour
Hill → Close contour with higher figures inside
Lake, depression → lose contour with higher fig outside
Plane surface → Straight, parallel & equally wide spaced CL
Rough terrain → irregular Contour (uneven surface)
Vertical clear → locating & identifying points lying on contour
Water level of a still lake → Represented by Contour line
A very steep slope is scrap → A high scrap is known as Crag
Use of Contour maps
Catchment area assessment
Reservoir capacity estimate
location of route , sectⁿ determination
Method of Contouring
Direct method
Most Accurate, Slow, Tedious & Costly, For Small Areas
indirect method
Economic, fast, small scale survey of Large project, less accurate
C/S method → Route survey
Square or circle method → Plain area, small area and ground is not much undulating
Tacheometric method → Hilly terrain, Contour in rough and difficult country where ordinary levelling is tedious and chaining is slow and inaccurate
Methods of interpolation of Contour
Computation ( Arithmetic) method → Best method of contour interpolation
Estimation→ Rough method,Very crude → Small scale map
Graphical method → Rapid, Convenient & high accuracy
PLANE TABLE SURVEY
Principle → Parallelism
Most likely error → Orientation
Quick but less accurate → Used for small & medium scale survey
PT is a graphical method → field work & plotting done simultaneously.
Unaffected by local attractions
Disadvantages of PTS
It is essentially a tropical instrument
Not very accurate & Heavy → inconvenient to transport
Reproduction of maps is not possible since notes of measurement are not recorded
Accessories of PT
Board → 600 x 500, 750 x 600, 100 x 75 (all in mm)
Tripod → To support plane table
Trough Compass → To locate N-S sirectⁿ (L = 15cm) or Orientation of table
Spirit level tube → To make board horizontal
Alidade → Sighting & drawing objects
Telescope Alidade → To measure both H & V distances directly
Fiducial edge → An alidade in which one edge is bevelled
Plumbing Fork → Centering of table, with Plum bob, U-shaped metal frame
Optical plummet → Centring in windy conditions
indian Clinometer → Diff of elevation of two point
P-line intersect each other at the centre of Earth
Tachometer → Used in PT for H & V distance
Temporary Adjustments in PT
Surface board ⟂ Vertical axis of instruments
Two vanes (obj & eye) ⟂ base of the alidade
Fiducial / Working / Rolling Edge should be a straight line
Fix of PT
Failure of fix occurs when the plane table is on the great circle
Strength of fix → The accuracy with which the instrument station can be est in PT survey
Poor fix → When the station is near the great circle
The fix of a PT with 3 known point is good if the instrument station lies in the great triangle
The Fix of a plane table from three known points is good → if Middle station is nearest
Setting up the Plane Table
Setting → Levelling → Centering → Orientation
Error due to centring ≤ Scale/40
Orientation → Operation of revolving a PT about its vertical axis so that all the lines on the sheet become parallel to the corresponding lines on the ground
Orientation of PT is done by using a Trough compass by backsighting or by sighting the previous point or resection
Orientation must be carried out in plane table surveying
Method of orientation
i) By Trough Compass
N-S direction, L = 15cm, dia = 5cm
When only one point is available for orientation.
ii) By Resection
By solving 2P & 3P problem, by back ray
Back ray → It is required to go to the plotted station
2P → To orient plane table at a point with two inaccessible points
3P → Involves locating the station occupied by plane table given the position of the three known/observation points
3P is most suitable in hydrographic Survey
Three point problem is better than 2P
2P and 3P → Are methods of both resection and orientation
iii) By Back sighting (Traversing)
Best & Points are accessible
When it is not possible to set the plane table on the point
Method of Plane Table (RITR)
Radiation
Large distance, Accessible points, Clearly visible
Plotting of small areas which can be commanded from a single station
max no of ground measurement (Detail plotting)
Orientation of table not required
intersection (Graphical triangulation)
inaccessible & not intervisible point → Hilly Areas
Ground details are located
Traversing
Narrow strip survey → Road & rail
Resection
A method of orientation
est location of instrument station by drawing resectors from the known station, Require other PT
Traversing , Resection & 2P → Locating Position of inst(PT)
Radiation & intersection → Plotting Position of obj on drawing
Method of 3P problem
i) Graphical (Bessel method)
ii) Mechanical (Tracing paper)
iii. Trial & Error (Lehman's) → Most Rapid & very accurate
iv) Analytical method
v) Geometrical Construction Method
Note
Gales traverse table → Plotting points by independent coordinates
CURVES
Designation → By radius
To avoid inconvenience in horizontal curve,max centrifugal ratio → Road = ¼ & rail track = ⅛
Compound curve → Two or more simple circular curve off different radius
Vertical curve
Two straight lines at diff gradient
Generally parabolic in nature
Length=(g1-g2)/r → r = rate of change of grade(%), g = gradients
Parabolic vertical curve → Change of gradient = constant
Vertical curve ranging requires geometric surveying
Horizontal curve
Two straight line intersect in horizontal plane
Generally Circular
Reverse Curve or Serpentine curve:
Two straight lines are parallel & angle b/w them are very small
Very frequently used on hilly roads
Superelevation provided at the point of reverse curvature = 0
Deviation Curve
Combination of two reverse curves to avoid interviewing obstruction such as bend of river & building
Transition curve
A curve of varying radius introduce b/w two branches of compound curve or b/w a straight and a circular curve
Introduced to gradually change the direction
Shape of transition curve → Euler's spiral, cubic spiral, cubic parabola, Laminiscate
ideal Shape of transition curve → Clothoid
True spiral → LR = Constant
Froude’s transition curve → Cubic parabola
Polar deflection angle = Spiral angle / 3
Bernoulli's Lemniscate
Special type of transition curve
Used when deflection is very large
Objectionable in Railway but allowed on highway
Length of limiting offset
L = 0.25s / sinθ
L = meter, S = Scale (1cm:100m: s = 100), θ = Degree max allowable error in degree
Perpendicular offset from a tangent to the junction of a transition curve and circular curve = 4S
Degree of curve (D)
Angle subtended at centre by an arc or chord (Generally 30 m chord)
R = 1720/D → 30m arc or chord
R= 1146/D = 2/3 of 1720/D → 20m Arc or chord
R = 573/D = 1/2 of 1146/D → 10m Arc
Elements of a simple curve
Deflection angle (∆) = 180 - included/intersection angle
Length of curve → L = (2πR/360) x ∆=0.0174R
Tangent length → T = R tan(∆/2)
Length of long chord → L = 2R sin(∆/2)2Tangent length
External or Apex distance → E = R (sec(∆/2)-1)
Mid ordinate/versine of curve → M = R (1 - cos(∆/2)) = R versine(∆/2)
No of full chord → Curve length / Peg interval
Chainage
Chainage A = Chainage Vertex - Tangent distance
Chainage B = Chainage A + Curve length ≠ Chainage vertex + Tangent distance
Versione of Curve(V)
V = C²/8R, where V,C,R are in the same unit
V = 125C²/R, where V in mm, C & R in m
V = 1.5C²/R, where V in inches, C & R in feet
Method of setting out of Curves
i. Linear/Chain/Tape method
Perpendicular offset from tangent → Ox= R-R² - X²X2/2R
Radial offset from tangent → Ox = R² + X²- R X2/2R
Offset from chord produce → On=Cn(Cn+Cn-1) / 2R
Offset from long chord → Ox= R² - X²-R2-(L/2)2
ii. Angular/instrumental method
Rankine's/deflection method of tangential angle, Two theodolite method, Tacheometric method
Two theodolite method
Principle → Deflection to any point P from the first tangent = the angle between the long chord and the direction to P from the second tangent point
Most suitable method, Rough ground
Two angular measurement are taken → No linear measurements
FIELD ASTRONOMY
Solstice → Point at which son declination is maximum
Vernal equinox → Point at which Sun declination is zero
Sun at Autumnal equinox → 21-September
Celestial ecliptic → The great circle which the sun appears to describe on the celestial sphere with the earth as centre, in course of a year
Circumpolar stars → Which are always above the horizon and do not set, Distance from pole < Latitude of observer
Equation of time which is the difference b/w apparent solar time and mean solar time at any instant → vanishes four time during one year
Asthenosphere → Plastic layer of the mantle
S-180 → Spherical excess, S → Sum of all the angles of spherical triangle
Celestial Sphere
Poles of celestial horizon → Zenith and Nadir
Celestial horizon → The plane of great circle traced upon the celestial sphere perpendicular to zenith-nadir line and passing through the centre of earth
Zenith → Point of the celestial sphere vertically below observation point
Nadir → Point of the celestial sphere vertically below observation point, Plumb line dropped from the nodal point pierces the photograph
Co-altitude/Zenith distance → Angular distance of a heavenly body from the zenith
Co-declination → ∠ b/w star & dirctⁿ of earth axis of rotation
Hour angle → The angle b/w the observer’s meridian and declination circle of the heavenly body
Zenith Angle
Telescope of the total station will be Pointing Downwards > 90
Telescope of the total station will be Pointing Upwards < 90
PHOTOGRAMMETRY
Photogrammetry → Measurements taken using photographs as the prime non-contact medium
Terrestrial photogrammetry → Taking photographs of the terrain of the earth from cameras on ground
Aerial survey steps → Reconnaissance → est ground control → flight planning → photography → paperwork including computation and planing
Important Points
Tilt → Rotation of camera, at exposure about the line of
Substitute of map → Vertical aerial photo-mosaics
Principle point → Point where perpendicular from the optical centre of lens meets photograph
Plum point → Point where vertical through the optical centre of lens meets photograph
Perspective point → A point where rays from the object converge
Perspective view → Picture plane is assumed to be a vertical plane
Index mosaic are not true planimeter representation of the area
Index mosaic photograph are pasted on fibre board and whole assembly is photographed again
Principle distance = b/w projectⁿ centre & photograph
Tilt displacement = Radii from Nadir point
Pseudoscopic view → Overlap kept outwards & natural order is reversed
Crab → Occurred while avoiding drift, when aircraft is not oriented with Flight line photograph are not parallel to flight line
Drift → Lateral shifting of photograph
Geometrical centre of photograph is Defined by intersection of lines joining the fiducial mark
Map → Orthogonal projection, Aerial photograph → Perspective/central projection
Aerial photography → Longitudinal overlap is normally kept = 60%
Truly vertical photographic survey → Principal point, plumb point and isocenter coincide
Elevations of objects on an aerial photograph can be measured due to stereoscopic fusion
Distortions in aerial photographs → Caused by Tilt and Ground relief
Mercator projection system → Cylindrical projection
Multistage imaging (Spatial resolution)
Parallax in aerial photographs → error due to movement of camera and ground relief
Rectification → Rephotographing an aerial photograph so that the effects of tilt are eliminated
Formula
Relief Displacement → d = rh /(H - h) → Displacement radial from principal point
Scale of photograph → Sh=f/(H-h)
The product of distances of plumb point and Horizon point of a vertical photograph from its principal point → f2
Rotation angle = 360 - Swing
1:600 → Sh = 1/600
f → focal length
d = Relief displacement
r = radial distance on the image of the top of obj
h = height of object above Datum
H = flying height above the Datum
Photographs require
No of photographs required = (Length of strip/Net length of photograph) + 1
REMOTE SENSING
Active remote sensing →
Passive remote sensing → Where system has no energy source of its own but depends on external source of energy (Sun)
Spectral resolution → The wavelength to which the remote sensing system is sensitive
Error → imaging characteristics of the sensor, Stability and orbit characteristics of the platform, motion of the earth, atmospheric effects
Electromagnetic waves properties → Emitted, Reflected, Diffracted
LISS (Linear image scanning system) → 1990
Passive sensor → The instruments which sense natural electromagnetic radiations, either emitted or reflected from the earth objects
Factor affecting the microwave signature of objects
Frequency, Polarisation, incident angle, Scattering mechanism
Global positioning system (GPS)
Principle gps positioning → Analytical resection
Position of a GPS instrument → At least 3 satellite signals
Position of a point can be located → At Least 4 satellite signals
Min no of satellite for GPS receiver to draw 3D map = 4
Receivers → Quartz clocks
India → NAVIC, USA → GPS
Widely used antenna → Microstrip antenna
GNSS → Global navigation satellite system
Geostationary Satellite = 36000 km
GIS
Software used → SPANS, GENAMAP, ISRO GIS
Line in polygon method is characteristic of vector overlay
Vector data model → Topology is static and any updation/editing of vector data requires rebuilding of topology, Accurate geographic location of data can be maintain
AREA & VOLUME MEASUREMENT
Side slope 2:1 = H:V → V = x , H = 2x
Lead & lift allowed for the Earthwork → 30 m & 1.5m
Mass haul curve → Diagram prepared to work out the quantity of earth work
Voltrapezoidal formula ? Volprismoidal formula
Prismoidal formula calculates the volume of earthwork accurately
Trapezoidal formula
Also called Average end Area formula
n may be odd or even
A = ½ h( y1 + yn + 2(y2 + y3….))
Vol is over estimated hence a prismoidal correction(-ve) is applied
Assumption → The mid area of the pyramid is half the avg area of the end, End sectⁿ are in parallel plane
Simpson one third rule
A = ⅓ h(y1 + yn + 4(∑ y even) + 2(∑y odd))
No of ordinate should be odd, Area segment → even
Best if straight form Parabolic arc or line joining the top of the ordinates is parabolic
Short length formed by Parabolic arc are considered as parallel to each other
irregular & curved boundary
Coefficient of sum of the even index= 4, odd index = 2
Prismoidal formula → V = h(A1 + 4A2 + A3)/6
Avg ordinate method
A= Avg ordinate base length
A = (ordinate / no of ordinate) length of base line
Mid ordinate method
A = Avg of mid ordinate x Length of base
Mid Section method
Vol = AL = (BD + SD2) L
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