TITLE:
TO DETERMINE THE ELEVATION OF A POINT BY TRIGNOMETRIC LEVELLING.
OBJECTIVES:
THEORY:
Trigonometrical leveling is the process of determining the difference of elevation of station from observed vertical angles and known distances, which are assumed to be either horizontal or geodetic length at mean sea level. The vertical angles may be measured by means of an accurate theodolite and the horizontal distances may either be measured or computed.
Base of the object accessible:
The horizontal distance between the instrument and the object can be measured accurately.
Let
`P=` instrument station.
`Q=` Point to be observed
`A=` centre of the instrument
`Q =` projection of Q on horizontal plane through A
`D= AQ=`horizontal distance between P&Q
`h=` height of the instrument at P
`h=Q\Q`
`S=`Reading on staff kept at B.M, With line of sight horizontal.
`alpha=` angle of elevation from A to Q.
From triangle `triangle AQ\Q`,
`h=Dtan alpha`
`R.L of Q=R.L\ of\ i\nstrum\e\n\t\ a\x\is +Dtanalpha`
If the R.L. of P is known,
`R.L\ of\ Q=R.L\ of P+h+Dtanalpha`
If the reading on the staff kept at the B.M. is `S` with the line of sight horizontal,then,
`R.L\ of\ Q=R.L.\ of\ B.M+S+Dtanalpha `
Base of the object inaccessible: -
If the horizontal distance between the instrument and the object can be measured due to obstacles, two stations are used such that they are in the same vertical plane as the elevated object.
Instrument axes at the same level:-
Let,
`h=Q\Q`,
`alpha_1 = an\gl\e\ of\ el\evation\ f\r\om\ A\ t\o\ Q`
`alpha_2=an\gl\e\ of\ el\evation\ f\r\om\ B\ t\o\ Q`
`S= staf\f \ readi\ng\ on\ B.M\ tak\e\n\ f\rom\ b\oth\ A\ and\ B` (the reading being the same in both the cases.)
`b=h\o\rizontal\ dis\ta\nce\ between\ the\ i\nstrument\ stations`
`D= ho\rizontal\ dis\ta\nce\ betw\e\en\ P\ &\ Q`
From triangle `triangle AQ\Q`,
`h=Dtanalpha_1.....(i)`
From triangle `triangle BQ\Q`,
`h=(b+D) tanalpha_2....(ii)`
Equating (i) and (ii), we get,
`Dtanalpha_1=(b+D) tanalpha_2`
`D(tanalpha_1-tanalpha_2)=btanalpha_2`
`D=(btanalpha_2)/(tanalpha_1-tanalpha_2)`
Now,
`h=Dtanalpha_1`
`=(btanalpha_1tanalpha_2)/(tanalpha_1-tanalpha_2)`
`=(bsinalpha_1sinalpha_2)/(sin(alpha_1-alpha_2))`
Thus,`R.L\ of\ Q=R.L\ of\ B.M+s+h`
Instruments axes at different levels:
It is based on the assumption that the difference in level is small and the instrument station and the elevated object are in the same vertical plane.
Case(a):If instrument axis at `A` is higher
`h_1=D tan alpha_1`......(i)
`h_2=(d+D)tan alpha_2`.....(ii)
Equating (i) and (ii),
`h_2-h_1=(d+D)tan alpha_2-D tan alpha_1`
let, `S_1-S_2=h_2-h_1=S`
`D=(S-d tan alpha_2)/(tan alpha_2-tan alpha_1)`
`D=((d-S cot alpha_2)/(tan alpha_1-tan alpha_2))*tan alpha_2`
and,
`h_1=((d-S cot alpha_2)sin alpha_1 sin alpha_2)/(sin(alpha_1-alpha_2))`
R.L of G=R.L of BM`+S_1+h_1`=R.L of BM `+S_1+((d-S cot alpha_2)sin alpha_1 sin alpha_2)/(sin(alpha_1-alpha_2))`
Case(b):If instrument axis at `B` is higher
`h_1=D tan alpha_1......(i)`
`h_2=(d+D)tan alpha_2.....(ii)`
Equating (i) and (ii),
`h_1-h_2=D tan alpha_1-(d+D)tan alpha_2`
let, `S_2-S_1=h_1-h_2=S`
Thus,`S=D tan alpha_1-(d+D)tan alpha_2`
`D=(S+d tan alpha_2)/(tan alpha_1-tan alpha_2)`
`D=((d+S cot alpha_2)/(tan alpha_1-tan alpha_2))*tan alpha_2`
Thus,`h_1=D tan alpha_1`
`=((S+d tan alpha_2)/(tan alpha_1-tan alpha_2))*tan alpha_1`
`=((d+S cot alpha_2)tan alpha_1 tan alpha_2)/(tan alpha_1-tan alpha_2)`
`=((d+S cot alpha_2)sin alpha_1 sin alpha_2)/(sin(alpha_1-alpha_2))`
R.L of G=R.L of BM+S_1+h_1=R.L of BM `+S_1+((d+S cot alpha_2)sin alpha_1 sin alpha_2)/(sin(alpha_1-alpha_2))`
Case(II):When base of the objects are Inaccessible and the instruments station and the objects are not in the same vertical plane:
It is based on the assumption that instrument stations are not in the same vertical planes as the elevated object.
DERIVATION:
From triangle,`ABC'`,
`angle ACB=180^0-(theta_1+theta_2)`
Now, from sine rule,
`(AC)/(sinB)=(BC)/(sinA)=(AB)/(sinC)`
Thus,`AC=AB*(sinB)/(sinC)=(dsintheta_1)/(sin(180-(theta_1+theta_2)))=(dsintheta_1)/(sin(theta_1+theta_2))`
`BC=AB*(sinA)/(sinC)=(dsintheta_2)/(sin(180-(theta_1+theta_2)))=(dsintheta_2)/(sin(theta_1+theta_2))`
Knowing the horizontal distance AC and BC, the vertical components `h_1` and `h_2` may be calculated.
`R.L\ of\ C=R.L of B.M.+S_1+h_1`
`=R.L\ of B.M\ +S_1+ACtan alpha_1`
`=R.L\ of\ B.M + S_1+ (d sintheta_1 tanalpha_1)/(sin(theta_1+theta_2)) `
`R.L\ of\ C=R.L of B.M.+S_2+h_2`
`=R.L\ of B.M\ +S_2+BCtan alpha_2`
`=R.L\ of\ B.M + S_2+ (d sintheta_2tanalpha_2)/(sin(theta_1+theta_2))`
APPARATUS REQUIRED:
Theodolite
Staff
Measuring tape
Arrow
Peg
Ranging rod
OBSERVATIONS AND CALCULATIONS:
RESULTS:
Hence, the R.L of tower was found to be `221.364` and `222.817m` with an average of `222.090m`.
PRECAUTIONS:
The theodolite must be exactly over the pegs and centered and levelled correctly.
The theodolite must be fixed on a hard immovable surface and should be safe from erodable ground.
DISCUSSION AND CONCLUSIONS:
The top Height of the tower was determined using two cases of trigonometric levelling, both of which had inaccessible horizontal base length. This method is used to determine reduced level of points far from reach of measuring instrument and are useful in preliminary survey.Hence, the reduced level of a selected Tower was determine using trigonometric levelling.
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