12+ Surveying Formulas helpful for Civil Engineers

surveying formula

In this article, We will See some of the Important Formulas of Surveying for quick revision and overview of the Subject.

Here's What will We See:



Direct Vernier Formula



direct vernier formula


where, 

s = length of one division on the main scale

n = numbers of divisions on the Vernier scale

v = length of one division on the Vernier 



Least Count Formula


Least Count of Vernier Formula


where, 

s and n are similar as seen above.

We can calculate the Least Count of both Direct Vernier and Retrograde Vernier using the above formula.


Retrograde Vernier Formula


Retrograde Vernier Formula

where, 

n, v, and s are similar as we have seen.


Error Formula




where,

Measured Value is the value, measured by the Instrument

True Value is the actual value of quantity.


Problems

Q (1): If the measured value of the quantity is 35.5 m, and the true value of the quantity is 35m, how much error is there?

A: 0.5m

Q (2): If the true value of the distance is 300m, but the measurement taken shows 301m, what is the error?

A: 1m

Q (3): True Value of Area = 3500 square meters, Measured Value of Area = 3505.5 square meters. Find Error.

A: (-)5.5m


Correction Formula

Correction Formula

where, 

as we have seen in the error formula, True Value is the value that is actually the length of a line or the actual distance between points.

Measured Value is the value which we measure by some instruments like a chain, or tape of a given distance.


Problems

Q (1): Measured Value = 101m, True Value = 100m, find Correction?

A: (-)1m

Q (2): True Value = 500m, Measured Value = 499m, find Correction?

A: 1m

Q (3): True Value = 15m, Measured Value = 15m, find Correction.

A: 0m



Relationship between Correction and Error Formula

Relationship between Correction and Error Formula

where,

Correction and Error as we have already seen above.


Problems


Q (1): If Error = 0.1m, find Correction.

A: (-)0.1m

Q (2): If Correction = 10 m3, find Error.

A: (-)10 m3

Q (3): If Error = (-)0.01 m2, find Correction.

A: 0.01 m2



Correction for Slope Formula


Slope triangle


Correction for Slope Formula


where, 

as we see in the image, it is an approximate sketch of the slope we encounter on the surface of the earth. If we measure the Slope distance as 'L', and the approximate value of angle as 'x', then the value of correction required to get the horizontal distance such as if slope doesn't exist, is given by it.


Problems

Q (1): If the Length Measured is 10m, and Slope is 60 degrees, then what will be the Correction for the Slope?

A: (-)5m

Q (2): If the Slope is 45 degrees, and the length measured is 42.42m, then what Correction should be applied to the Slope?

A: (-)12.42m

Q (3): If the Length measured with Chain is 505m, and Slope is 2 degrees, then Correction for Slope be?

A: (-)0.31m



Correction for Temperature Formula


where,

α is the coefficient of thermal expansion

L is the length of distance being measured

Tm is the measured value of Temperature means the value of average temperature during measurement

Ts is the Standard Temperature or reference temperature.


Problems

Q (1): If Average Value of measured Temperature during Chaining is 30 degrees Celsius, Length of distance measured is 15.5 m, Standard Value of Temperature is 27 degree Celsius, and Coefficient of Thermal Expansion is 12*10(-6) ; then find Correction required for Temperature?

A: 5.58 mm

Q (2): If Correction required for Temperature is 10mm, Coefficient of Thermal Expansion is 10*10(-6) , Standard Value of Temperature is 27 degree Celsius, and measured distance is 30 m; then find what would be mean value or average value of Temperature during Measurement?

A: 60.33 degree Celsius

Hint: To find Tm, make it main part of formula. Your Formula would becomes like 

Tm = (Correction for Temperature)/(α*L) + T
  



 Correction for Pull Formula

Correction for Pull formula

where, 

Ps = Standard Pull of Tape

Pm = Measured Pull of Tape

L = Measured Length of Tape

A = Cross Section Area of Tape

E = Modulus of Elasticity of Tape


Derivation

Change in length for pull is = (-)PL/(AE).

Now, as we find a correction for pull, we have to find how much we need to add in Measured Pull to obtain Standard Pull or Correct value of Pull. Now, for Standard pull is (-)PsL/(AE) and the Measured pull is (-)PmL/(AE). 

So, correction is = (-)PsL/(AE) - (-)PmL/(AE) ; which is = (Pm - Ps)L / (AE)

And how does the Change in length formula derived? It comes from the definition of modulus of elasticity, which is 

(Modulus of elasticity) = (Stress)/(Strain)

Now, (Stress) = (Force)/(Area); and (Strain) = (Change in Length)/(Original Length)

So, (Modulus of Elasticity) = ((Force)/(Area))/((Change in Length)/(Original Length));

which is = (Force)*(Original Length)/(Area)*(Change in Length);

So, (Change in Length) = (Force)*(Original Length)/(Area)*(Modulus of Elasticity)

And we give a minus sign in a case for the pull because it is opposite to what the sign is during Stress.


Problems

Q (1): If Ps is 120 kN, Pm is 100 kN, Measured Length is 10m, and cross section area is 1000 mm2, and E is 2*105 N/mm2; then what would be Correction for Pull?

A: 100mm

Q (2): What will be formula to calculate Measured Length, if we have correction for Pull, Modulus of Elasticity, Cross Section Area, Standard Pull, and Measured Pull of Tape?

A: L = (Correction for Pull)*(AE)/(Pm - Ps)

Q (3): What will be Formula to calculate Measured Pull of Tape if we have quantities like A, E, Ps, L, and Correction for Pull?

A: Pm = (Correction for Pull)*(AE)/(L) + Ps



Relationship between Fore Bearing and Back Bearing Formula


Relationship between Fore Bearing and Back Bearing Formula

where,

Back Bearing is the clockwise angle measured of the line in the opposite direction in which the survey was carried out.

Fore Bearing is the clockwise angle measured of the line in the same direction in which the survey was carried out.

Whole Circle Bearing is the system of measurement of angle in which we measure angle clockwise from North Direction, and measure complete angle even if it exceeds 180 degrees. As it is measured from 0 degrees to 360 degrees, like a complete circle, it is called Whole Circle Bearing.

Quadrantal Bearing is the system of measurement of angle in which we measure angle either clockwise or anti-clockwise depending upon whether a measurement is in which quadrants, and it doesn't always measure from only North Direction, but North Direction and South Direction depending upon it are in which part of Quadrants. For Example, if Angle is in the first Quadrant, we measure clockwise from North Direction. 

If Angle is on the second Quadrant (Here, the Second Quadrant is below the First Quadrant instead of being left to it), then we measure it from South Direction and anti-clockwise such as the angle remains less than or equal to 90 degrees. 

Similarly, if Angle is in Third Quadrant, we measure it from South Direction but clockwise, and write the direction to it like it is between the South and West direction, So, the angle is written as "S45W". And if Angle is in the fourth Quadrant, we measure it from North Direction but Anti-Clockwise and write as an example, "N25W".


Problems

Q (1): For Whole Circle Bearing, what is the Difference between Fore Bearing and Back Bearing (in absolute number)?

A: 180

Q (2): For Quadrantal Bearing system, angle is "N45E", then what will be this angle in Whole Circle Bearing System?

A: 45 degrees

Hint: Here "N45E" means angle is in First Quadrant, therefore WCB would be equals to QB.

Q (3): If Angle is 240 degrees in WCB, then what will be its value in QB?

A: "S60W"



Latitude Formula


Latitude Formula


where, 

L is the Latitude of a line, which is a projection of a line in a North-South direction

l is the length of the line

x is an Angle of line in a Quadrantal Bearing System


Problems

Q (1): If Length of Line is 10m, and angle it makes with North direction is 60 degrees, what will be its Latitude?

A: 5m

Q (2): If Latitude of Line is 15m, and Line has made angle of 60 degrees with North Direction, what would be Length of Line?

A: 30m

Q (3): If angle some Line makes with North direction is 45 degrees, and line is in first Quadrants, what will be its Latitude to Length of Line Ratio?

A: 0.7071

Hint: As Latitude = Length * cos x; Latitude/Length = cos x, now put the value of angle as 45 degrees.


Departure Formula


Departure Formula


where, 

D is the Departure of a line, which is a projection of a given line in an East-West Direction

l is the length of the line

x is an Angle of line in a Quadrantal Bearing System

We use above Formulas in Traversing problems.


Relative Error of Closure Formula


Relative Error of Closure Formula

where,

Relative Error of Closure is the quantity that gives the value of how much error of closure is there given the perimeter of the traverse.

An error of Closure is a total error that remains between if the traverse was completely closed and where the traverse is now.

The perimeter of the Traverse is the sum of all lengths given traverse. 

This formula is used in traversing, specifically in problems of the closed traverse, where some error remains such as the first point of traverse has not met its endpoints.


Trigonometric Levelling Basic Formula


Right_angle_triangle


Trigonometric Levelling Basic Formula

where,

H is the Height of a Tower or Mountain required to find.

D is the Horizontal distance from a point to a given object.

x is the angle made by an object from a point from where horizontal distance is being measured and the top of the object, the height of which is determined.


Problems

Q (1): If horizontal distance between Object and Tower is 15m, and angle it makes with top of tower, from starting point of distance is 45 degrees, then what will be height of tower?

A: 15m

Hint: tan 45 = 1


Q (2): If height of Tower is 20m, and Horizontal Distance from tower to point where angle being made is 34.64m, then what angle will it make with top of Tower?

A: 30 degrees

Hint: As H = D tan x; tan x = (H/D); Put the values of H and D; and find x angle.


Q (3): If Height of Mountain is 100m, and angle to top of Mountain from starting point of leveling is 45 degrees, then what is the Horizontal Distance between starting point of levelling and Mountain?

A: 100m

Hint: As H = D tan x; therefore D = (H)/(tan x); put the values of H and x.



Most Probable Error Formula


Most Probable Error Formula

where,

σ is the standard deviation, which means it shows how much difference given set of quantities has in form of square and square-root and as a some number


Problems


Q(1): Standard Deviation for given set of errors has 1. What will be the Most Probable Error?

A: ±0.6745


Q (2): Most Probable Error for given set of Errors is 1. What would be Standard Deviation?

A: ±1.4826


Q (3): What is the ratio of Most Probable Error to Standard Deviation for given values of Errors?

A: ±0.6745



Tacheometry Stadia Method Formula


Tacheometry Stadia Method Formula

where,

D is the Distance between the point where Tacheometer is there and of point where staff is there.

K is the constant of Multiplication also known as Multiplying Constant.

C is Constant of addition in formula, also called additive constant..

s is the Staff Intercept or the distance between two points on staff which can be seen from Telescope.


Problems

Q (1): If K is 100, C is 0, and s is 2m, then what would be distance between staff and oint of Instrument?

A: 200m

Hint: Use D = Ks + C; Put values, so D = (100)*(2) + 0 = 200m


Q (2): If D is 150m, K is 100, and C is 0, then what would be Staff Intercept of given Staff?

A: 1.5m

Hint: As 

D = Ks +C; 

s = (D - C)/(K);

Put values, So:

s = (150 - 0)/(100)

So, s = 1.5m


Q (3): If K= 100, D = 200m, s = 1.9m; then what would be additive constant C of given Instrument?

A: 10

Hint: As, Ks + C = D;

So,

C = D - Ks;

Now, putting values;

C = (200 - (100)*(1.9))

C = (200 - 190)

C = 10



Conclusion

As we have seen various Surveying Formulas, try to practice them using various examples. You can even make an example yourself, and try to solve it. 




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