SUBJECT NAME : SURVEYING GROUP NO. 8 ENROLMENT NO. NAME ARCHIT MEWADA ANUJ PATEL HARDIK M PATEL JAIMIN PATEL

LEVELLING

Definition: “ Trigonometric levelling is the process of determining the differences of elevations of stations from observed vertical angles and known distances. ” The vertical angles are measured by means of theodolite. The horizontal distances by instrument Relative heights are calculated using trigonometric functions. Note: If the distance between instrument station and object is small. correction for earth's curvature and refraction is not required.

1) Direct Method: Where is not possible to set the instrument over the whose station whose elevation is to be determined. Combined correction is required.

AB’=AC=D L ACB = 90° Similarly, BA´=BC´=D L AC´B= 90 ° BC=D tan α AC´=D tan β

Distance between A & B is large Cc & Cr required CB´=C´A´= D 2 True difference A-B H=BB´ =BC+CB´ =D tan α D 2 Depression angle B to A AC´=D tan β [BC´=D] True Difference A-B H=AA´ =BC+CB´ = D tan β D 2

2H = D tan α +D tan β R.L of station B = R.L of station A + H=R.L. of station A +D/2[tan α + tan β ]

METHODS OF DETERMINING THE ELEVATION OF A POINT BY THEODOLITE: Case 1. Base of the object accessible Case 2. Base of the object inaccessible, Instrument stations in the vertical plane as the elevated object. Case 3. Base of the object inaccessible, Instrument stations not in the same vertical plane as the elevated object.

Case 1. Base of the object accessible B A = Instrument station B = Point to be observed h = Elevation of B from the instrument axis D = Horizontal distance between A and the base of object h1 = Height of instrument (H. I.) Bs = Reading of staff kept on B.M.  = Angle of elevation = L BAC h = D tan  R.L. of B = R.L. of B.M. + Bs + h = R.L. of B.M. + Bs + D. tan  If distance is large than add Cc & Cr R.L. of B = R.L. of B.M. + Bs + D. tan  D 2

Case 2. Base of the object inaccessible, Instrument stations in the vertical plane as the elevated object. There may be two cases. (a) Instrument axes at the same level (b) Instrument axes at different levels. 1) Height of instrument axis never to the object is lower: 2) Height of instrument axis to the object is higher:

Case 2. Base of the object inaccessible, Instrument stations in the vertical plane as the elevated object. (a) Instrument axes at the same level  PAP, h= D tan  1  PBP, h= (b+D) tan  2 D tan  1 = (b+D) tan  2 D tan  1 = b tan  2 + D tan  2 D(tan  1 - tan  2) = b tan  2 R.L of P = R.L of B.M + Bs + h

(b)Instrument axes at different levels. 1) Height of instrument axis never to the object is lower:  PAP, h1 = D tan  1  PBP, h2 = (b+D) tan  2 hd is difference between two height hd = h1 – h2 hd = D tan  1 - (b+D) tan  2 = D tan  1 - b tan  2 -D tan  2 hd = D(tan  1 - tan  2) - b tan  2 hd + b tan  2 = D(tan  1 - tan  2) h1 = D tan  1

(b)Instrument axes at different levels. 2) Height of instrument axis to the object is higher:  PAP, h1 = D tan  1  PBP, h2 = (b+D) tan  2 hd is difference between two height hd = h2 – h1 hd = (b+D) tan  2 - D tan  1 = b tan  2 + D tan  2 - D tan  1 hd = b tan  2 + D (tan  2 - tan  1 ) hd - b tan  2 = D(tan  2 - tan  1) - hd + b tan  2 = D(tan  1 - tan  2) h1 = D tan  1

In above two case the equations of D and h1 are, D h1

Case 3. Base of the object inaccessible, Instrument stations not in the same vertical plane as the elevated object. Set up instrument on A Measure  1 to P L BAC =  Set up instrument on B Measure  2 to P L ABC =  L ACB = 180 – (  +  ) Sin Rule: BC= b· sin  sin{180˚ - (  +  )} AC= b· sin  sin{180˚ - (  +  h1 = AC tan  1 h2 = BC tan  2

DIRECT LEVELLING ON STEEP GROUND: Indirect levelling can be also used where the ground is steeper.the following procedure is used to determine the difference of elevations between A and B. Select a suitable turning point C. 1.Set up the instrument at a convenient station 01 on the line AB. make the line of collection roughly parallel to the slope of the ground. Clamp the telescope.

Vertical angle θ 1 to A´. Determine the R.L of A´ as R.L of A´ = R.L of A + AA´ 3.Take a foresight CC´ on the staff held at the C change point without changing the vertical angle θ 1. Measure the slope distance AC between A and C. R.L. of C = R.L. of A´ + AC sin θ 1 – CC´ Or R.L. of C = R.L. of A + AA´ + AC sin θ 1 - CC´ 4.Shift the instrument to the station o2 midway between C and B. Make the line of collimation roughly parallel to the slope of the ground. Clamp the telescope. 5.Take a back sight CC'‘ on the staff held at the change point C. measure the vertical angle θ 2.

6.Take a foresight BB on the staff held at the point B without chaining the vertical θ 2. measure the slope distance CB. R.L. of B = R.L. of C + CB sin θ 2- BB Thus RL of B = ( R.L. of A + AA + AC sin θ 1 – CC)+ CC+ CB sin θ 2- BB If there are some intermediate points whose elevations are required, the procedure is slightly different as under. Suppose, there is an intermediate point D at slope distance AD from A. The level of the point D can be determined from the setting of the instrument at o1. R.L. of D =R.L. of A´+ AD sin θ 1- DD Where DD is the intermediate sight on the staff held at D. Same way R.L. of some intermediate points between A and C, and also between C and B can be determined as usual way.

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