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Setting out of curve (By one theodolite - Rankein’s method)

Setting out of curve (By one theodolite - Rankein’s method). Fundamentals of circular curve. Curve facilitates change in direction of alignment in between two straights (Two tangents). For a simple circular curve following terminology is used:- Tangent length = T 1 I = R tan Δ /2

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Setting out of curve (By one theodolite - Rankein’s method)

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  1. Setting out of curve(By one theodolite - Rankein’s method)

  2. Fundamentals of circular curve Curve facilitates change in direction of alignment in between two straights (Two tangents). For a simple circular curve following terminology is used:- Tangent length = T1 I = R tan Δ/2 Length of curve = T1AT2 = π R Δ /180 Length of long chord = T1T2(L) = 2 R Sin Δ/2 Apex distt. = AI = R Sec Δ/2 – R Where ‘R’ = Radius Δ = Deflection angle.

  3. A Fig -1 o ELEMENTS OF SIMPLE CIRCULAR CURVE

  4. Curve used in Railway working – Transitioned circular curve. • Terminology:- 1.Length of transition curve (LTC) ‘l’ :- Maxi of i. 0.008 Ca .Vm ii. 0.008 Cd .Vm iii 0.72 Ca ( Based on cant gradient of 1:720) 2. Shift ‘S’ = l2/24.R 3. Combined curve length ‘ CCL’ = π R Δ /180+ l Based on rate of change of cant gradient = 35 mm/ Sec.

  5. Typical calculations • Example :- Puntamba-Shirdi project - Take off curve. • Δ = 100º4’00” • LTC ‘l‘= 80.00m • Vertex Ch.= 1356.00m • Degree of curve = 5.80º (R=1750/5.8=301.724m) • ‘S’= l2/24R= 0.883m • TL (Tangent Length) =(R +S) tan Δ/2 + l /2= 401.059m Ch of T1 = 1356.00 - 401.059 = 954.941 m • ‘CCL’ = π R Δ /180 + l = 606.958 m • Length of circular curve ‘J1 to J2’ = CCL - 2 l = 446.958 m

  6. Calculations Contd… • Ch. Of T2 = Ch of T1 + CCL = 1561.899 m • Ch of Apex. ( Mid pt. of circular curve) = Ch of T1 + CCL/2 = 1258.420 m • Ch. Of J2 = Ch of Apex + Length Circular curve/2 = 1481.899 m Ch. Of T2 = Ch. Of J2 + l = 1561.899 m ( This should match with Ch of T1+CCL) Vertex distt. = (R+S) Sec Δ/2 - R = 169.375m

  7. Field Layout • Formation of Vertex :- Prolong the both straights from available references and form the vertex (Point of intersection - I). • With theodolite at one straight, mark two points app 2 m apart near probable apex. • Keep theodolite on another straight and extend the same and find ‘I’ . ( Practically a thread is held across both points marked earlier on previous straight. • This point is Point of intersection - I.

  8. Field Layout Contd… • Measure the deflection angle Δ by stationing theodolite at I . • To check accuracy of chaining etc. keep theodolite at T1 and measure angle IT1T2 and this should be equal to angle Δ /2. • Locate Apex of curve by keeping theodolite at I and measuring half of included angle (90- Δ /2 ) from one of tangent and fixing the point by chaining equal to vertex distt from point I. (This point is good check while laying curve as mid point)

  9. Field Layout Contd… • Laying of transition curve: by theodolite method:- • Set the theodolite at T1 and set o-o reading towards I. • For locating first point P1 set angle ά1 = 573(x1)²/R l (in Minutes) = 573(20) ²/(301.724x80) = 9.49 min = 0º9’29” and measure distt. of 20m from T1.

  10. Field Layout Contd… Distance x to be measured from T1 only.

  11. Field Layout Contd… • By offset method also transition curve can be laid:- • Offset from tangent y = x³/6Rl in meters • Check for J1 offset = 4S • Laying of circular curve:- • Set the theodolite at J1 . • Establish a common tangent by setting an angle equal to twice of deflection at J1 (2ά4) = 5º3’51” from line T1J1

  12. 3 α say = Ø Establishing a common tangent

  13. Field Layout Contd… • For locating point on circular curve with Rankein’s deflection angle method:- • δ = 1718.9 * C / R in minutes. • By measuring angle δ1 from common tangent (T1J1) and distance C from J1 fix first point P1. • Δ1= δ1 • Δ2=Δ1+ δ1 • Δ3=Δ2+ δ1 • Fix different points on circular curve by measuring Δ1,Δ2 &Δ3…….etc. and fixing points P1/P2 & P3 …etc at distance from each consecutive point equal to ‘C=25m” or so.

  14. Field Layout Contd… • All angles to be measured from common tangent only. • For longer curves to avoid error circular portion to be laid in four parts - from J1 and further, From Apex towards J1 , From Apex towards J2 and from J2 towards Apex. • or at convenient points common tangent be established and curve laid further wrt common tangent. • To establish common tangent at apex set theodolite at Apex see vertex and set 90º angle.

  15. Deflection angles for different points.

  16. ∆ -2Ø P2 3 P1 25m 25m 2 P3 1 25m (∆ -2Ø)/2 J2 J1

  17. Thanks

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